Rare-Earth Borides [1 ed.] 9814877565, 9789814877565

Table of contents :
Cover
Half Title
Title Page
Copyright Page
Table of Contents
Preface
Chapter 1: Crystal Chemistry and Crystal Growth of Rare-Earth Borides
1.1:
Introduction
1.2: Principles of the Main Crystal-Growth Techniques
1.2.1:
The Czochralski Method
1.2.2:
The Flux Method
1.2.3:
Floating-Zone Melting Technique
1.2.4:
Other Techniques
1.3:
Metal-Rich Borides
1.3.1:
Rare-Earth Diborides (RB2)
1.3.2:
R2B5
1.4:
Rare-Earth Tetraborides (RB4)
1.4.1:
General Overview
1.4.2:
Yttrium Tetraboride (YB4)
1.4.3:
Cerium Tetraboride (CeB4)
1.4.4:
RB4 (R = Y, Nd, Gd–Tm, Lu)
1.5:
Rare-Earth Hexaborides (RB6)
1.5.1:
General Overview
1.5.2:
Synopsis of RB6 Crystal Growth
1.5.3:
Samarium Hexaboride (SmB6)
1.5.4:
Ytterbium Hexaboride (YbB6)
1.5.5:
Yttrium Hexaboride (YB6)
1.5.6:
Boron Isotope Effects
1.6:
Higher Borides
1.6.1:
Rare-Earth Dodecaborides (RB12)
1.6.2:
Rare-Earth Hectoborides (RB66)
1.7:
Concluding Remarks
Chapter 2: Thin Films of Rare-Earth Hexaborides
2.1:
Overview of Rare-Earth Hexaborides
2.2:
Lanthanum Hexaboride (LaB6)
2.3:
Cerium Hexaboride (CeB6)
2.4:
Gadolinium Hexaboride (GdB6)
2.5: Ytterbium Tetra- and Hexaborides (YbB4 and YbB6)
2.6:
Neodymium Hexaboride (NdB6)
2.7:
Other Hexaborides
2.8:
Samarium Hexaboride (SmB6)
2.8.1:
Fabrication of SmB6 Thin Films
2.8.2:
Proximity Effect in Nb/SmB6 Bilayers
2.9:
Superconductivity in Yttrium Hexaboride (YB6)
2.10:
SmB6/YB6 Thin-Film Bilayer Heterostructures
2.10.1:
Point-Contact Spectroscopy Measurements
2.10.2:
Dirac-BTK Theory
2.11:
Summary and Perspective
Chapter 3: Crystal Structures of Dodecaborides: Complexity in Simplicity
3.1:
Introduction
3.2: Cooperative Jahn–Teller Effect as a Driving Force behind Structural Instability in Dodecaborides
3.3: Modeling the Dynamics of the Dodecaboride Lattice Using X-Ray Diffraction Data
3.4:
Crystal Structure: Problems and Results
3.4.1: The Jahn–Teller Distortions of Structural Parameters
3.4.2: Structural Peculiarities of Dodecaborides Different in Isotopic Boron Composition
3.4.3: Formation of Charge Stripes in Voids of the Crystal Lattice
3.5:
Conclusions
Chaper 4: Magnetism, Quantum Criticality, and Metal–Insulator Transitions in RB12
4.1:
Introduction
4.2:
Electronic Band Structure of Dodecaborides
4.2.1:
Rough Estimations
4.2.2:
Metallic RB12
4.2.3:
Strongly Correlated Semiconductor YbB12
4.3:
Nonmagnetic Reference Compound LuB12
4.3.1:
Charge Transport
4.3.2:
Thermal Properties
4.3.3:
Optical Properties
4.3.4: Magnetoresistance Anisotropy and Dynamic Charge Stripes
4.3.5:
The Origin of Electron and Lattice Instability and the Energy Scales in LuB12
4.4: Magnetic Dodecaborides RB12 (R=Tb, Dy, Ho, Er, Tm) and the Solid Solutions RxLu1−xB12
4.4.1:
Magnetic Properties
4.4.2:
Electron Paramagnetic Resonance
4.4.3:
Charge Transport
4.4.4:
Thermal Conductivity
4.4.5:
Thermal Expansion and Heat Capacity
4.4.6:
Magnetic Structure
4.4.7: Magnetic H–T–φ Phase Diagrams
4.4.8: The Root of the Complexity of Magnetic Phase
Diagrams of RB12
4.4.9:
Quantum Critical Behavior in HoB12
4.5: Metal–Insulator Transition in YbB12 and Solid
Solutions YbxR1−xB12 (R = Lu, Tm)
4.5.1:
Metal–Insulator Transition in YbB12
4.5.2: Pressure-Induced Insulator-to-Metal Transition in
YbB12
4.5.3:
Field-Induced Insulator-to-Metal Transition in YbB12
4.5.4: Insulator-to-metal transition in YbxR1−xB12 (R = Y, Lu, Sc, Ca, and Zr)
4.5.5:
Metal–Insulator Transition in Tm1−xYbxB12
4.6:
Conclusions
Chapter 5: Raman Spectroscopy of Metal Borides: Lattice and Electron Dynamics
5.1:
Introduction
5.2:
Raman Scattering by Phonons
5.2.1:
Raman-Active Phonons in Hexaborides
5.2.2: Extra Phonon Features in Raman Spectra of
Hexaborides
5.2.3:
Anharmonicity vs. Electron–Phonon Interaction
5.2.4:
Phononic Raman Spectra in Dodecaborides
5.2.5:
Raman Spectroscopy of Phonons in Tetraborides
5.2.6:
Raman Spectra of Other Rare-Earth Borides
5.3:
Raman Scattering by Electronic Excitations
5.3.1:
Crystal Electric Field Transitions
5.3.2:
Electron–Hole Excitations: Collision-Limited Regime
5.3.3: Electron–Hole Excitations: Crossover from Clean to
Dirty Regimes
5.3.4:
Electron–Induced Phonon Renormalization
5.4:
Conclusions
Chapter 6: Neutron Spectroscopy on Rare-Earth Borides
6.1: Specifics of the Neutron-Scattering Technique
in Condensed Matter Spectroscopy
6.1.1: Neutron Scattering Function in Relation to the Atomic Vibrations and Dynamic Magnetic
Susceptibility
6.1.2: Characteristic Features of Inelastic Neutron Scattering with Respect to Heavy-Fermion and Mixed-Valence Phenomena
6.2: Magnetic Excitations in Hexa- and
Dodecaborides
6.2.1:
Crystal Electric Field Effects
6.2.2: Hybridization Effects: Intermediate-Valence and
Kondo Insulator Systems
6.2.3: Excitation Spectra of the Intermediate-Valence
Kondo Insulator SmB6
6.2.3.1: Intermultiplet transitions and the resonant mode in
the magnetic neutron-scattering spectra of SmB6
6.2.3.2:
The model of the exciton of an intermediate radius
6.2.3.3:
The magnetic form factor study
6.2.3.4: Resonant exciton modes at the R and X points
6.2.3.5:
Gd-impurity effect on SmB6
6.2.4:
Magnetic Excitations in the Kondo Insulator YbB12
6.2.4.1:
Resonant mode and temperature effects
6.2.4.2:
Resonant mode and impurity effects in YbB12
6.3:
Lattice Dynamics in RB6 and RB12
6.3.1: General Characterization of the Atomic
Vibrational Spectra of RB6 and RB12
6.3.2:
Electron–Phonon Interaction in RBn (n = 6, 12)
6.3.2.1: Intermediate-valence features of SmB6 in
electron–phonon interaction
6.3.2.2:
The magnetovibration interaction in YbB12
6.4:
Conclusions
Chapter 7: Competing Order Parameters in Rare-Earth Hexa- and Tetraborides
7.1:
Introduction
7.2:
Resonant and Nonresonant X-Ray Diffraction
7.2.1:
Nonresonant X-Ray Diffraction
7.2.2:
Resonant X-Ray Diffraction
7.3:
Multipolar Order in CexLa1−xB6
7.3.1:
The Parent Compound CeB6
7.3.2:
Solid Solutions CexLa1−xB6
7.4:
Rare-Earth Tetraborides (RB4)
7.4.1:
An Overview
7.4.2:
Magnetic Order in RB4
7.4.3:
Fractional Magnetization Plateaus
7.4.4:
Quadrupolar Fluctuation in DyB4
7.5:
Conclusions
Chapter 8: Multipolar Order and Excitations in Rare-Earth Boride Kondo Systems
8.1:
Introduction
8.1.1:
Conduction Bands and Fermi Surface
8.1.2: Localized 4f Shells, Their CEF States, Multipoles,
and RKKY Interactions
8.2:
Overview of RE Boride Compounds
8.3: Multipolar Hidden Order in CeB6 in the
Localized 4f Scenario
8.3.1: Pseudospin Representation of 8-Quartet
Multipoles
8.3.2: Multipole Interaction Model and Symmetry
Breakings
8.3.3: Experimental Identification of Multipolar Order Parameters
8.4:
Octupolar HO Phase IV in Diluted Ce1−xLaxB6
8.4.1: Phase Diagram and Evidence for Primary Octupolar
Order
8.5: Collective Excitations in the AFQ
Hidden-Order Phase II of CeB6
8.5.1:
Generalized Multipolar RPA Method
8.5.2: Dependence of Mode Energies on Field Strength
and Field Angular Rotation
8.6: Resonant Magnetic Excitations in the
Itinerant CeB6 Kondo Lattice
8.6.1:
Heavy Quasiparticle Band Properties in the PAM
8.6.2:
Collective Spin Exciton Modes
8.7: Dispersive Doublet Spin Exciton Mode in the
Kondo Semiconductor YbB12
8.8: Magnetic Excitations: Topological State in the
Mixed-Valent Semiconductor SmB6
8.8.1:
CEF and Collective Magnetic Excitations
8.8.2:
SmB6 as a Strongly Correlated Topological Insulator
8.9:
Conclusions and Outlook
Chapter 9: Neutron-Scattering Studies of Spin Dynamics in Pure and Doped CeB6
9.1:
Introduction
9.2: Electronic Properties and Ordering
Phenomena
9.2.1: Magnetic Structure in the Antiferromagnetic
Phases
9.2.2:
Magnetically Hidden Order in Phase II
9.2.3: Mean-Field Description of the Ordering Phenomena in CeB6
9.3: Spin Excitations in the Absence of Magnetic
Field
9.3.1: Collective Excitations in the Antiferromagnetic
Phase
9.3.2:
Quasielastic Magnetic Scattering
9.4: Magnetic-Field Dependence of the Collective
Excitations
9.4.1:
General Remarks
9.4.2:
Zone-Center Excitations
9.4.3:
Dispersion of the Field-Induced Collective Modes
9.4.4:
Anisotropy with Respect to the Field Direction
9.5:
Spin Dynamics in Ce1−xLaxB6 and Ce1−xNdxB6
9.5.1: The Influence of La and Nd Substitution on the
Electronic Structure
9.5.2: Momentum-Space Structure of the Diffuse Spin
Fluctuations
9.5.3: Temperature Dependence of the Quasielastic
Magnetic Scattering
9.5.4:
Field-Induced Collective Excitations in Ce1−xLaxB6
Chapter 10: Theory of Electron Spin Resonance in Strongly Correlated CeB6
10.1:
Introduction
10.2:
Kondo Impurity Model
10.3:
Kondo Lattice Model
10.3.1:
Paramagnetic Kondo Lattice
10.3.2:
Antiferromagnetic Kondo Lattice
10.3.3:
Kondo Lattice with Ferromagnetic Order
10.3.4:
Summary
10.3.5:
Other Theoretical Approaches and Experiments
10.4:
Antiferroquadrupolar Ordered CeB6
10.4.1: ESR in a 8 Quartet
10.4.2: ESR for Ce3+ Ions with 8 Ground State
10.4.3: g-Factor for ESR in Phase II of CeB6
10.4.4:
Ferromagnetic Correlations in Phase II of CeB6
10.4.5: Line
Width of ESR in Phase II of CeB6
10.4.6: Second Resonance at High Fields in Phase II of
CeB6
10.4.7:
Inelastic Neutron Scattering in CeB6
10.4.8:
Summary
10.5:
Longitudinal Dynamical Susceptibility
10.6:
Conclusions
Chapter 11: Bulk and Surface Properties of SmB6
11.1:
Introduction
11.2:
Crystal Growth and Structural Properties
11.3:
Theoretical Remarks
11.4:
Surface Properties
11.4.1:
dc Electrical Conductivity
11.4.2:
Tunneling Spectroscopy and Thermopower
11.4.3:
Quantum Oscillations
11.4.4:
Angle-Resolved Photoelectron Spectroscopy
11.4.5:
Thin Films and Nanowires
11.5:
Bulk Properties
11.5.1:
ac Electrical Conductivity
11.5.2:
Quantum Oscillations
11.5.3:
Thermal Conductivity
11.5.4:
Specific Heat
11.5.5:
Raman Spectroscopy
11.5.6: X-Ray, Neutron, and Mössbauer Spectroscopies
11.5.7:
Nuclear, Electron, and Muon Spin Resonance
11.6:
Concluding Remarks
Index

Citation preview

Rare-Earth Borides

Rare-Earth Borides

edited by

Dmytro S. Inosov

Published by Jenny Stanford Publishing Pte. Ltd. Level 34, Centennial Tower 3 Temasek Avenue Singapore 039190 Email: [email protected] Web: www.jennystanford.com British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. Rare-Earth Borides c 2022 Jenny Stanford Publishing Pte. Ltd. Copyright  All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN 978-981-4877-56-5 (Hardcover) ISBN 978-1-003-14648-3 (eBook) DOI: 10.1201/9781003146483

Contents

Preface 1 Crystal Chemistry and Crystal Growth of Rare-Earth Borides Natalya Shitsevalova 1.1 Introduction 1.2 Principles of the Main Crystal-Growth Techniques 1.2.1 The Czochralski Method 1.2.2 The Flux Method 1.2.3 Floating-Zone Melting Technique 1.2.4 Other Techniques 1.3 Metal-Rich Borides 1.3.1 Rare-Earth Diborides (RB2 ) 1.3.2 R2 B5 1.4 Rare-Earth Tetraborides (RB4 ) 1.4.1 General Overview 1.4.2 Yttrium Tetraboride (YB4 ) 1.4.3 Cerium Tetraboride (CeB4 ) 1.4.4 RB4 (R = Y, Nd, Gd–Tm, Lu) 1.5 Rare-Earth Hexaborides (RB6 ) 1.5.1 General Overview 1.5.2 Synopsis of RB6 Crystal Growth 1.5.3 Samarium Hexaboride (SmB6 ) 1.5.4 Ytterbium Hexaboride (YbB6 ) 1.5.5 Yttrium Hexaboride (YB6 ) 1.5.6 Boron Isotope Effects 1.6 Higher Borides 1.6.1 Rare-Earth Dodecaborides (RB12 ) 1.6.2 Rare-Earth Hectoborides (RB66 ) 1.7 Concluding Remarks

xv 1 2 9 9 10 13 18 20 20 35 43 43 56 57 58 63 63 77 97 126 136 142 146 146 177 190

vi Contents

2 Thin Films of Rare-Earth Hexaborides Seunghun Lee, Xiaohang Zhang, and Ichiro Takeuchi 2.1 Overview of Rare-Earth Hexaborides 2.2 Lanthanum Hexaboride (LaB6 ) 2.3 Cerium Hexaboride (CeB6 ) 2.4 Gadolinium Hexaboride (GdB6 ) 2.5 Ytterbium Tetra- and Hexaborides (YbB4 and YbB6 ) 2.6 Neodymium Hexaboride (NdB6 ) 2.7 Other Hexaborides 2.8 Samarium Hexaboride (SmB6 ) 2.8.1 Fabrication of SmB6 Thin Films 2.8.2 Proximity Effect in Nb/SmB6 Bilayers 2.9 Superconductivity in Yttrium Hexaboride (YB6 ) 2.10 SmB6 /YB6 Thin-Film Bilayer Heterostructures 2.10.1 Point-Contact Spectroscopy Measurements 2.10.2 Dirac-BTK Theory 2.11 Summary and Perspective

3 Crystal Structures of Dodecaborides: Complexity in Simplicity Nadezhda B. Bolotina, Alexander P. Dudka, Olga N. Khrykina, and Vladimir S. Mironov 3.1 Introduction 3.2 Cooperative Jahn–Teller Effect as a Driving Force behind Structural Instability in Dodecaborides 3.3 Modeling the Dynamics of the Dodecaboride Lattice Using X-Ray Diffraction Data 3.4 Crystal Structure: Problems and Results 3.4.1 The Jahn–Teller Distortions of Structural Parameters 3.4.2 Structural Peculiarities of Dodecaborides Different in Isotopic Boron Composition 3.4.3 Formation of Charge Stripes in Voids of the Crystal Lattice 3.5 Conclusions

245 246 246 252 254 256 257 258 260 261 267 270 272 274 278 279

293

294 298 306 312 312 314 316 323

Contents vii

4 Magnetism, Quantum Criticality, and Metal–Insulator Transitions in RB12 Nikolay E. Sluchanko 4.1 Introduction 4.2 Electronic Band Structure of Dodecaborides 4.2.1 Rough Estimations 4.2.2 Metallic RB12 4.2.3 Strongly Correlated Semiconductor YbB12 4.3 Nonmagnetic Reference Compound LuB12 4.3.1 Charge Transport 4.3.2 Thermal Properties 4.3.3 Optical Properties 4.3.4 Magnetoresistance Anisotropy and Dynamic Charge Stripes 4.3.5 The Origin of Electron and Lattice Instability and the Energy Scales in LuB12 4.4 Magnetic Dodecaborides RB12 (R = Tb, Dy, Ho, Er, Tm) and the Solid Solutions Rx Lu1−x B12 4.4.1 Magnetic Properties 4.4.2 Electron Paramagnetic Resonance 4.4.3 Charge Transport 4.4.4 Thermal Conductivity 4.4.5 Thermal Expansion and Heat Capacity 4.4.6 Magnetic Structure 4.4.7 Magnetic H –T –φ Phase Diagrams 4.4.8 The Root of the Complexity of Magnetic Phase Diagrams of RB12 4.4.9 Quantum Critical Behavior in HoB12 4.5 Metal–Insulator Transition in YbB12 and Solid Solutions Ybx R1−x B12 (R = Lu, Tm) 4.5.1 Metal–Insulator Transition in YbB12 4.5.2 Pressure-Induced Insulator-to-Metal Transition in YbB12 4.5.3 Field-Induced Insulator-to-Metal Transition in YbB12 4.5.4 Insulator-to-Metal Transition in Ybx R1−x B12 (R = Y, Lu, Sc, Ca, and Zr)

331 332 334 334 335 338 342 342 345 353 357 359 362 362 363 364 368 369 371 375 380 381 382 382 395 396 399

viii Contents

4.5.5 Metal–Insulator Transition in Tm1−x Ybx B12 4.6 Conclusions 5 Raman Spectroscopy of Metal Borides: Lattice and Electron Dynamics Yuri S. Ponosov 5.1 Introduction 5.2 Raman Scattering by Phonons 5.2.1 Raman-Active Phonons in Hexaborides 5.2.2 Extra Phonon Features in Raman Spectra of Hexaborides 5.2.3 Anharmonicity vs. Electron–Phonon Interaction 5.2.4 Phononic Raman Spectra in Dodecaborides 5.2.5 Raman Spectroscopy of Phonons in Tetraborides 5.2.6 Raman Spectra of Other Rare-Earth Borides 5.3 Raman Scattering by Electronic Excitations 5.3.1 Crystal Electric Field Transitions 5.3.2 Electron–Hole Excitations: Collision-Limited Regime 5.3.3 Electron–Hole Excitations: Crossover from Clean to Dirty Regimes 5.3.4 Electron–Induced Phonon Renormalization 5.4 Conclusions 6 Neutron Spectroscopy on Rare-Earth Borides Pavel A. Alekseev, Vladimir N. Lazukov, and Igor P. Sadikov 6.1 Specifics of the Neutron-Scattering Technique in Condensed Matter Spectroscopy 6.1.1 Neutron Scattering Function in Relation to the Atomic Vibrations and Dynamic Magnetic Susceptibility 6.1.2 Characteristic Features of Inelastic Neutron Scattering with Respect to Heavy-Fermion and Mixed-Valence Phenomena 6.2 Magnetic Excitations in Hexa- and Dodecaborides

404 419

443 443 445 445 447 453 458 463 465 465 465 466 469 476 479 489

490

493

501 505

Contents ix

6.2.1 Crystal Electric Field Effects 6.2.2 Hybridization Effects: Intermediate-Valence and Kondo Insulator Systems 6.2.3 Excitation Spectra of the Intermediate-Valence Kondo Insulator SmB6 6.2.3.1 Intermultiplet transitions and the resonant mode in the magnetic neutron-scattering spectra of SmB6 6.2.3.2 The model of the exciton of an intermediate radius 6.2.3.3 The magnetic form factor study 6.2.3.4 Resonant exciton modes at the R and X points 6.2.3.5 Gd-impurity effect on SmB6 6.2.4 Magnetic Excitations in the Kondo Insulator YbB12 6.2.4.1 Resonant mode and temperature effects 6.2.4.2 Resonant mode and impurity effects in YbB12 6.3 Lattice Dynamics in RB6 and RB12 6.3.1 General Characterization of the Atomic Vibrational Spectra of RB6 and RB12 6.3.2 Electron–Phonon Interaction in RBn (n = 6, 12) 6.3.2.1 Intermediate-valence features of SmB6 in electron–phonon interaction 6.3.2.2 The magnetovibration interaction in YbB12 6.4 Conclusions 7 Competing Order Parameters in Rare-Earth Hexa- and Tetraborides Takeshi Matsumura 7.1 Introduction 7.2 Resonant and Nonresonant X-Ray Diffraction 7.2.1 Nonresonant X-Ray Diffraction 7.2.2 Resonant X-Ray Diffraction

505 510 511

511 517 519 521 523 527 527 531 535 537 548 550 556 560

575 576 580 580 581

x Contents

7.3 Multipolar Order in Cex La1−x B6 7.3.1 The Parent Compound CeB6 7.3.2 Solid Solutions Cex La1−x B6 7.4 Rare-Earth Tetraborides (RB4 ) 7.4.1 An Overview 7.4.2 Magnetic Order in RB4 7.4.3 Fractional Magnetization Plateaus 7.4.4 Quadrupolar Fluctuation in DyB4 7.5 Conclusions

8 Multipolar Order and Excitations in Rare-Earth Boride Kondo Systems Peter Thalmeier, Alireza Akbari, and Ryousuke Shiina 8.1 Introduction 8.1.1 Conduction Bands and Fermi Surface 8.1.2 Localized 4f Shells, Their CEF States, Multipoles, and RKKY Interactions 8.2 Overview of RE Boride Compounds 8.3 Multipolar Hidden Order in CeB6 in the Localized 4f Scenario 8.3.1 Pseudospin Representation of 8 -Quartet Multipoles 8.3.2 Multipole Interaction Model and Symmetry Breakings 8.3.3 Experimental Identification of Multipolar Order Parameters 8.4 Octupolar HO Phase IV in Diluted Ce1−x Lax B6 8.4.1 Phase Diagram and Evidence for Primary Octupolar Order 8.5 Collective Excitations in the AFQ Hidden-Order Phase II of CeB6 8.5.1 Generalized Multipolar RPA Method 8.5.2 Dependence of Mode Energies on Field Strength and Field Angular Rotation 8.6 Resonant Magnetic Excitations in the Itinerant CeB6 Kondo Lattice

583 583 586 590 590 592 596 599 601

615 616 618 620 623 631 633 634 635 644 644 648 650 652 655

Contents xi

8.6.1 Heavy Quasiparticle Band Properties in the PAM 8.6.2 Collective Spin Exciton Modes 8.7 Dispersive Doublet Spin Exciton Mode in the Kondo Semiconductor YbB12 8.8 Magnetic Excitations: Topological State in the Mixed-Valent Semiconductor SmB6 8.8.1 CEF and Collective Magnetic Excitations 8.8.2 SmB6 as a Strongly Correlated Topological Insulator 8.9 Conclusions and Outlook

9 Neutron-Scattering Studies of Spin Dynamics in Pure and Doped CeB6 Pavlo Portnichenko, Alistair Cameron, and Dmytro S. Inosov 9.1 Introduction 9.2 Electronic Properties and Ordering Phenomena 9.2.1 Magnetic Structure in the Antiferromagnetic Phases 9.2.2 Magnetically Hidden Order in Phase II 9.2.3 Mean-Field Description of the Ordering Phenomena in CeB6 9.3 Spin Excitations in the Absence of Magnetic Field 9.3.1 Collective Excitations in the Antiferromagnetic Phase 9.3.2 Quasielastic Magnetic Scattering 9.4 Magnetic-Field Dependence of the Collective Excitations 9.4.1 General Remarks 9.4.2 Zone-Center Excitations 9.4.3 Dispersion of the Field-Induced Collective Modes 9.4.4 Anisotropy with Respect to the Field Direction 9.5 Spin Dynamics in Ce1−x Lax B6 and Ce1−x Ndx B6 9.5.1 The Influence of La and Nd Substitution on the Electronic Structure

656 658 661 664 664 665 670

691 692 695 695 697 702 705 705 709 713 713 715 721 726 732 732

xii Contents

9.5.2 Momentum-Space Structure of the Diffuse Spin Fluctuations 735 9.5.3 Temperature Dependence of the Quasielastic Magnetic Scattering 742 9.5.4 Field-Induced Collective Excitations in 750 Ce1−x Lax B6 10 Theory of Electron Spin Resonance in Strongly Correlated CeB6 Pedro Schlottmann 10.1 Introduction 10.2 Kondo Impurity Model 10.3 Kondo Lattice Model 10.3.1 Paramagnetic Kondo Lattice 10.3.2 Antiferromagnetic Kondo Lattice 10.3.3 Kondo Lattice with Ferromagnetic Order 10.3.4 Summary 10.3.5 Other Theoretical Approaches and Experiments 10.4 Antiferroquadrupolar Ordered CeB6 10.4.1 ESR in a 8 Quartet 10.4.2 ESR for Ce3+ Ions with 8 Ground State 10.4.3 g-Factor for ESR in Phase II of CeB6 10.4.4 Ferromagnetic Correlations in Phase II of CeB6 10.4.5 Line Width of ESR in Phase II of CeB6 10.4.6 Second Resonance at High Fields in Phase II of CeB6 10.4.7 Inelastic Neutron Scattering in CeB6 10.4.8 Summary 10.5 Longitudinal Dynamical Susceptibility 10.6 Conclusions 11 Bulk and Surface Properties of SmB6 Priscila F. S. Rosa and Zachary Fisk 11.1 Introduction 11.2 Crystal Growth and Structural Properties 11.3 Theoretical Remarks

773 774 775 778 779 781 782 784 784 786 788 790 790 794 797 800 801 803 805 807 817 817 818 821

Contents xiii

11.4 Surface Properties 11.4.1 dc Electrical Conductivity 11.4.2 Tunneling Spectroscopy and Thermopower 11.4.3 Quantum Oscillations 11.4.4 Angle-Resolved Photoelectron Spectroscopy 11.4.5 Thin Films and Nanowires 11.5 Bulk Properties 11.5.1 ac Electrical Conductivity 11.5.2 Quantum Oscillations 11.5.3 Thermal Conductivity 11.5.4 Specific Heat 11.5.5 Raman Spectroscopy ¨ 11.5.6 X-Ray, Neutron, and Mossbauer Spectroscopies 11.5.7 Nuclear, Electron, and Muon Spin Resonance 11.6 Concluding Remarks Index

822 822 828 831 833 835 836 836 839 841 844 846 848 853 858 877

Preface

This book deals with the multifarious family of binary crystalline compounds formed between the rare-earth elements and boron. Most of these compounds are not at all new, as the history of rare-earth borides extends over almost a century. Many excellent reviews about their crystal chemistry and physical properties have already been published. So why would one need another book on the subject? This review volume is motivated by a recent surge of interest in rare-earth borides across various seemingly unrelated areas of condensed-matter physics, which has intensified considerably in the past two decades. Research on the topics of magnetic frustration in the Shastry–Sutherland lattice of tetraborides, topological electronic properties in the Kondo insulators SmB6 and YbB12 , and the multipolar “hidden-order” phases in CeB6 remains at the forefront of the most actively pursued areas in the physics of correlated electron systems, for both experiment and theory. The burgeoning interest in rare-earth borides has also motivated a meticulous reexamination of their structural and electronic properties, revealing a number of delicate and exciting phenomena such as the boron isotope effect on the local lattice symmetry, structural and electronic instabilities, and an extreme sensitivity of the physical properties to crystalline defects or impurities. Our advances in understanding these subtle effects go hand in hand with the progress in crystalgrowth technology and the development of modern physical characterization methods. This book is intended for researchers and advanced undergraduate and graduate students interested in the physics of rare-earth compounds, in particular solid-state chemists, crystallographers, and both experimental and theoretical physicists. It consists of 11

xvi Preface

chapters, which are cross-referenced but can be otherwise seen as independent reviews. Every chapter begins with a short but comprehensive overview of the early research in the field before moving on to the main emphasis of the more recent developments that are discussed from the personal viewpoints of the authors. Chapters 1 and 2 are concerned with the material aspects of rare-earth borides in the form of single crystals and thin films, respectively. High-quality samples are an important prerequisite of any modern experiment, and therefore much attention is paid to the importance of sample quality characterization and the optimization of growth methods in order to achieve the desired physical properties. Chapter 3 focuses on crystallographic aspects of the lattice structure of dodecaborides and, in particular, subtle effects related to the symmetry-lowering distortions of the lattice, whereas the influence of such structural and electronic instabilities on the physical properties is discussed in detail in Chapter 4. The lattice dynamics of metal borides are discussed both in Chapter 5 from the perspective of Raman scattering and in Chapter 6 from the viewpoint of neutron spectroscopy. Chapter 6 also describes the role of neutron scattering in understanding the magnetic excitations and magnetovibration interactions in hexa- and dodecaborides. Applications of modern x-ray spectroscopy to the analysis of competing low-temperature ordered phases in hexa- and tetraborides are discussed in Chapter 7, with an emphasis on “hidden-order” phases with multipolar order parameters. The theoretical approach to treat such multipolar order in rare-earth boride Kondo systems and to calculate the associated magnetic excitations is developed in Chapter 8, followed by the experimental Chapter 9, where these predictions are verified on one of the most studied hiddenorder compounds, CeB6 , using inelastic neutron scattering (INS). In particular, neutron-spectroscopy results in magnetic fields with varying orientations to the crystal lattice are presented, which bear a close relationship to the electron spin resonance (ESR) in CeB6 . The interpretation of both INS and ESR data in strongly correlated electron systems is far from straightforward, and the corresponding theory for CeB6 is introduced in Chapter 10. Finally, Chapter 11 is devoted to the bulk and surface properties of SmB6 , which attracted

Preface xvii

much attention in recent years after it was predicted to be a strongly correlated topological Kondo insulator. In closing this preface, I would like to thank all the authors of this volume for their contributions and for very detailed, friendly, and constructive discussions. I am especially thankful to Nikolay Sluchanko and Natalya Shitsevalova for first introducing me to the exciting physics of borides during the ICM’09 conference in Karlsruhe more than a decade ago. I am grateful to Alistair Cameron, ¨ Darren Peets, Sahana Roßler, and Marein Rahn for their help in proofreading and improving some of the chapters, as well as to Anas Shahab and Mubashir Ali for the technical assistance with typesetting the book in LaTeX. Dmytro S. Inosov TU Dresden May 10, 2021

Chapter 1

Crystal Chemistry and Crystal Growth of Rare-Earth Borides Natalya Shitsevalova I. M. Frantsevich Institute for Problems of Materials Science of NAS, Krzhyzhanovsky Str. 3, 03680 Kyiv, Ukraine [email protected]

Most of the experimental results presented in this book would not have been possible to obtain without high-quality singlecrystalline samples. The history of chemical synthesis and singlecrystal growth of some families of rare-earth borides extends over 80 years, whereas the growth of some other compounds represents a challenge even at the present time. This chapter discusses the main aspects of the crystal chemistry of rare-earth borides and summarizes the existing experience in their single-crystal growth and the associated challenges. It focuses on the benefits and drawbacks of different crystal-growth methods and on resulting specifics of the quality of single crystals in terms of their size, composition, purity, homogeneity, and other parameters that may turn out to be crucial for the results of most modern physical characterization methods.

Rare-Earth Borides Edited by Dmytro S. Inosov c 2022 Jenny Stanford Publishing Pte. Ltd. Copyright  ISBN 978-981-4877-56-5 (Hardcover), 978-1-003-14648-3 (eBook) www.jennystanford.com

2 Crystal Chemistry and Crystal Growth of Rare-Earth Borides

1.1 Introduction Rare-earth (RE) borides (RBn , n ≥ 2) represent interest both for fundamental research of new physical phenomena and as potential materials in various fields of engineering. They form a large class of compounds with different types of crystal structure and a wealth of properties due to both peculiarities of their crystal and electronic structures and the nature of RE ions. The study of their properties has been going on for over half a century. However, the use of purer source substances, improvement of preparation procedures, and the development of new research methods and theories of physical phenomena constantly motivate scientists to revisit the physical properties of these compounds from the standpoint of the current state of knowledge. Samarium hexaboride (SmB6 , bcc, CaB6 structure type), discussed in more detail in Chapter 11, is a striking example. It has been studied over many years as a mixed-valence compound, heavy-fermion compound, Kondo insulator, and topological Kondo insulator, reflecting the gradual development of theoretical concepts in the physics of strongly correlated electron systems, followed by their experimental confirmation [1–12]. It should be noted that no consensus on the physics of this compound has yet been achieved despite the numerous publications devoted to it. Another example is the RE tetraborides (RB4 : R = Y, La–Nd, Sm, Gd–Lu, tetr., ThB4 structure type). By the end of the 1990s, it seemed that the magnetism in these compounds had been exhaustively studied and understood [13]. However, the development of the concepts of topological and exchange frustration and their influence on the formation of the magnetic properties of RE intermetallic compounds with the Shastry–Sutherland lattice [14] led to a complete re-examination of the mechanisms determining the magnetic properties of RB4 [15–32]. The diversity of structural types in RE borides is due to both the electronic structure of the boron atom (its acceptor capacity to extend the sp2 configuration of boron to the quasi-stable sp3 one by capturing valence electrons of a partner atom and to form covalent B–B bonds) and the high donor capacity of RE atoms, that results in a

Introduction 3

variety of stable structures. Currently there are nine unambiguously identified structural types of the RE boride phases: AlB2 , Gd2 B5 , ThB4 , CaB6 , UB12 , ScB19 , YB25 , YB50 , and YB66 (Table 1.1). This series features a progression of ever more complicated boron sublattices, from planar six-member rings that form two-dimensional (2D) boron networks in diborides (RB2 ) to the three-dimensional (3D) structure units such as octahedra (B6 ), cuboctahedra and icosahedra (B12 ), which are the basis of 3D boron sublattices in phases from hexaborides (RB6 ), up to the hectoborides (RB66 ) [33–39]. Apart from the RB2 compounds, all RE borides have a 3D boron skeleton containing voids filled with metal atoms. Due to its rigidity, there are specific restrictions on size of the RE ions that can occupy the voids in the boron sublattice. According to the conventionally accepted division, the lanthanide ions with smaller radii (mainly elements of the yttrium subgroup) form the RB2 , RB12 , RB25 , RB50 , and RB66 phases, while the larger ones (mainly cerium subgroup) form hexaborides (RB6 ). Tetraborides (RB4 ) are known for practically all lanthanides. The least number of boron phases (single) is known for the Eu – B system EuB6 , the largest (eight) is known for the Y – B system: YB2 , YB4 , YB6 , YB12 , YB25 , YB50 , YB66 (Table 1.2). According to the early works on the electronic structure of RE borides in molecular orbital (MO) models and group-theoretical analysis using a tight-binding approximation, the boron nets and boron clusters (B6 , B12 ) in RE borides are doubly electron deficient. The transfer of valence electrons from metal ions stabilizes both the covalent boron sublattice and the boride crystal structure in general [40–46]. Later calculations of the electronic structure in other approximations could be slightly different in the value of charge transfer [47–51]; however, the accepted opinion is that RE borides with trivalent metals should exhibit metallic conductivity, and those with divalent ones should be insulators or semiconductors. Generally, the experimental results agree with these expectations [37, 52, 53]. The existence of KB6 in the CaB6 -type structure with monovalent potassium is explained by electron deficiency of the B6 units [54,55]. RE borides are characterized by heterodesmic bonding with a dominant covalent component and largely decoupled high-

Formula

Basic crystallochemical information on the rare-earth boride structures

Syngony

Space group

RB2

hexagonal

1 D6h

R 2 B5

monoclinic

RB4

tetragonal

5 C 2h – P 21 /c 6 – C 2/c C 2h 5 D4h - P 4/mbm

RB6 RB12 ScB19

cubic cubic tetragonal

RB25

– P 6/mmm

The number of The limits of the Structure formula units r R /r B ratio for type per unit cell, Z the structure type AlB2

1

1.26–2.06

Gd2 B5

4

2.27–2.33

ThB4

4

1.74–2.50

Oh1 – P m3m Oh5 – F m3m P 41 21 2 or P 43 21 2

CaB6 UB12 α-AlB12

1 4 —

1.84–2.68 1.74–2.27 1.8

monoclinic

I 121, I 1m1, I 12/m1

YB25

—

1.91–1.99

RB50

orthorhombic

P bam

YB50

—

1.93–1.99

RB66

cubic

Oh6 – F m3c

YB66

24

1.74–2.32

Boron network 3D framework of planar nets formed by six-membered hexagonal rings B6 octahedra connected by two types of B2 units 3D framework of B6 octahedra connected in nets through sp2 -type borons 3D framework of B6 octahedra 3D framework of B12 cuboctahedra 3D framework based on interconnected B12 icosahedra and B22 polyhedra 3D framework consisting of B12 icosahedra and one bridging boron site 3D framework of five structurally independent B12 icosahedra and B15 polyhedra 3D framework of eight different B12 (B12 )12 supericosahedra and nonicosahedral B80 units

4 Crystal Chemistry and Crystal Growth of Rare-Earth Borides

Table 1.1

Table 1.2

Rare-earth boride phases and their properties (in the order of decreasing ionic radius)

Ionic radius ˚ [56] (A)

Eu La Yb Pm Ce Pr Nd Sm Gd Tb Dy Y Ho Er Tm Yb Lu Sc

+2 +3 +2 +3 +3 +3 +3 +3 +3 +3 +3 +3 +3 +3 +3 +3 +3 +3

1.12 1.071 1.04 1.04 1.034 1.013 0.995 0.964 0.938 0.923 0.908 0.905 0.894 0.881 0.869 0.857 0.848 0.73

Structure types YB25

YB50

UB12

AlB2

YB66

Sm2 B5

ThB4 DIA

 (+)

PM AFM PM DIA PM PM   

2D magnetism   DIA     

under P AFM AFM PM AFM AFM AFM TI, PM SC, DIA DIA†

under P FM FM FM PM FM FM FM AFM DIA PM

QAS QAS IV, QAS QAS, SG QAS, SG QAS QAS, DIA QAS, SG QAS, SG QAS QAS QAS

(+)    

(+) PM AFM, FM AFM AFM AFM AFM AFM DIA AFM AFM AFM IV, PM DIA

CaB6

other

FM DIA TI (+) AFM, AFQ AFM AFM IV, TI, PM AFM AFM AFM SC AFM

ScB19

, established phases; (+), suggested phases; † , tetragonal phase (I 4/mmm, ScB12 structure type). REE, rare-earth element; SC, superconductor; DIA, diamagnetic; PM, paramagnetic; FM, ferromagnetic; AFM, antiferromagnetic; SG, spin glass; AFQ, antiferroquadrupolar ordering; IV, intermediate valence; TI, topological insulator; QAS, quasi-amorphous semiconductor.

Introduction 5

REE

Ionic charge

6 Crystal Chemistry and Crystal Growth of Rare-Earth Borides

symmetry boron and metal sublattices [59, 60]. Each of the sublattices is responsible for certain physical properties and their temperature dependence. The rigid covalently bonded boron sublattice determines the high resistance to thermal, chemical, and mechanical stress, high melting point, and hardness. The transport, thermodynamic, and magnetic properties are governed by the nature of the metal ion: Among RE borides there are dielectrics, semiconductors, semimetals, metal-like compounds, superconductors, compounds with stable and mixed valence, and dia-, para-, ferro- and antiferromagnets showing complex magnetic structures and multiple magnetic transitions (Table 1.2). RE borides form a homologous series of compounds and are therefore a convenient subject for studying the regular changes in their properties depending on the number of 4f electrons and valence electrons of the metal ions. On the other hand, additional effects resulting from the hybridization with the boron sublattice lead to significant variations in the physical and chemical properties of borides from one structural type to another. Moreover, the majority of borides readily form mutual solid solutions under the introduction of RE atoms with different valence, which provides an ability to control interatomic distances via chemical substitution. Such a unique flexibility, combined with the relatively simple highsymmetry structures of the RE boride phases, their high purity and structural quality, turns them into model objects for studying many complex phenomena in modern solid state physics. The recent progress in understanding their physical properties, especially low-temperature phenomena, has been enabled and to a great extent determined by the continuously improved quality of single-crystal samples. This includes, in particular, well-defined crystal structure, chemical composition and homogeneity, exclusion of secondary phases, minimization of structural defects, and for certain applications also the ability to obtain materials with high isotope enrichment on the B site, RE site, or both [61, 62]. However, the preparation of perfect stoichiometric single crystals of some RE borides still appears to be impossible or very difficult and dependent on the chemical nature of the compound as well as on the specific method used for crystal growth, its growth parameters, and other technical details [63, 64]. For example,

Introduction 7

the LaB6 stoichiometric composition is outside the homogeneity region at all temperatures of its existence; its homogeneity range is also temperature-dependent, so the composition of the LaB6 single crystals obtained by the flux method (1200–1500 ◦ C) differs from that of the crystals obtained by the zone-melting method (> 2700 ◦ C) [65–67]. SmB6 single crystals that are grown by the flux method have a defective boron sublattice [68], while those obtained by zone melting have a defective metal sublattice [69, 70]. Therefore, for a reliable interpretation of the results of advanced physical investigation methods, and especially for understanding possible discrepancies between the results obtained on different samples with nominally identical composition, it is very important to evaluate the real composition and the distribution of defects in single crystals. Depending on the growth method, the composition can fluctuate within the homogeneity range, and the different concentration of structural defects can influence the intrinsic physical properties. For example, a wide range of the superconducting critical temperatures, Tc , between 1.5 and 8.4 K, has been reported for YB6 depending on the conditions of sample preparation [71–73]. In addition, for the boride solid solutions, precise control over the distribution of the “parent” RE elements in the bulk of the crystal is necessary, which may influence the structural and compositional disorder on various length scales. The availability of single-crystal samples is a prerequisite for the study of anisotropic physical properties of RE borides, for instance in an applied magnetic field, and for the investigation of electric or thermal transport properties. The first single crystals of RE hexaborides (RB6 , R = Y, La, Ce, Nd, Gd, Er, Yb) were obtained by molten salt electrolysis [74], chemical vapor deposition (CVD) [75] and zone melting [76]. A basic disadvantage of the first two techniques is the small size of the grown single crystals, so for a long time these methods were no longer used in contrast to the intensively developed method of zone melting. However, the electrolysis and CVD methods have recently received a new impetus in the development of nanoscale science, as this technology can be employed for preparation of one-dimensional (1D) single crystals such as nanowires or nanotubes [77–86]. These efforts are limited mostly to the RB6 compounds and cannot compete with the flux

8 Crystal Chemistry and Crystal Growth of Rare-Earth Borides

method and zone melting either in the diversity of compounds or in the range of applicable experimental methods, so they will not be at the focus of this chapter. A very brief summary of the activities related to the preparation of 1D single crystals of RE hexaborides will be, nevertheless, given in Section 1.2. Currently, the most widespread methods for growing RE boride single crystals are growth from molten metal fluxes and variations of the floating-zone melting technique. These techniques are routinely used to grow bulk boride single crystals of high quality for modern research. There is also a small number of publications on RE borides grown by the Czochralski method. There are a number of excellent reviews devoted to the single-crystal growth of RE intermetallic compounds [87–100]. These reviews also provide a detailed description of the growth methods. Based on these reviews, Section 1.2 gives a brief overview of the Czochralski, flux, and floating-zone methods, summarizing their advantages and disadvantages in the application to RE borides. Further in this chapter, for each structure type, information on the growth of boride single crystals is summarized, including the available methods of preparation, results of physical and structural characterization, phase diagram features, etc. The effect of the preparation method on the physical properties will be illustrated with literature data for the specific RE borides, whenever applicable. Higher borides such as RB25 (R = Y, Gd–Lu) with monoclinic YB25 -type structure (space group I 12/m1) [57] and RB50 (R = Y, Gd–Lu) with orthorhombic YB50 -type structure (space group P bam) [101] can be potentially grown in single-crystal form only with the high-temperature flux method, as these borides decompose without melting above 1850 ◦ C into phases like RB12 and RB66 in the RB25 case and into RB25 and RB66 in the RB50 case. However, I am not aware of any successful reports on such growth. On the other hand, RB25 C crystals isostructural to RB25 have been successfully grown with certain amounts of carbon atoms incorporated in the voids of the boron framework, as well as RB44 Si2 crystals isostructural to RB50 , where Si partially replaces some boron sites. Similarly, the new tetragonal phase ScB19 (space group P 41 21 2, No. 92 or P 43 21 2, No. 96) is stable up to about 1850 ◦ C, above this temperature it decomposes into ScB12 and β-boron without

Principles of the Main Crystal-Growth Techniques 9

melting. A small amount of silicon stabilizes it, and it is possible to grow ScB19+x Si y single crystals by the floating-zone method. The corresponding information on RE borides of the YB25 , YB50 , and ScB19 structure types is presented in the reviews [38, 58] and in the references therein. In this chapter, they will not be considered, because information on preparation of the original compounds in single-crystalline form is absent.

1.2 Principles of the Main Crystal-Growth Techniques 1.2.1 The Czochralski Method The Czochralski method is a liquid-solid single-crystal growth technique that consists in pulling out a single crystal from an appropriate melt placed in a crucible that is inert relatively to the melt (Fig. 1.1). The process begins with the complete melting of the source material in the crucible and the subsequent decrease in temperature almost to the point of crystallization. Then, an oriented seed rod of the same composition as the melt is introduced into the melt. When the seed is wetted by the melt due to surface tension in the liquid, a thin layer of the melt is first formed on the surface of the seed crystal, and then the seed rod fixed in its seed holder is slowly pulled from the melt. The atoms in this layer are arranged in an ordered crystalline lattice inheriting the structure of the seed crystal. The advantage of the method is the absence of direct contact with the crucible that helps to avoid critical residual stresses. However, the crucible can be a source of impurities. To equalize the temperature in the melt in order to prevent complex hydrodynamic flows, which lead to a decrease in the stability of crystallization conditions and non-uniform distribution of impurities, the growing single crystal and crucible simultaneously rotate in opposite directions. The shape and size of the crystal are determined by adjusting the temperature of the melt and the rate of pulling of the seed. The Czochralski growth process can be carried out in vacuum, in hydrogen, or in an inert gas atmosphere.

10 Crystal Chemistry and Crystal Growth of Rare-Earth Borides

Figure 1.1 Schematic sketch of a Czochralski growth setup in a vacuum chamber. Courtesy of P. Gille [97].

In relationship to RE borides, this method has only been used for single-crystal growth of yttrium hectoboride (YB66 ) [102, 103] and RE tetraborides (RB4 , R = Y, Tb, Dy, Ho, Er) [104–107].

1.2.2 The Flux Method The growth of RE-boride single crystals in high-temperature solutions (the flux method) is widely used. The essence of this method lies in the spontaneous or controlled crystallization of the RE borides in single-crystal form from the metal solvents with previously dissolved components of the desired compounds. The advantages of this method are in the enhanced diffusion of the elements dissolved in the metal solvent (the flux), wide temperature

Principles of the Main Crystal-Growth Techniques 11

range of the crystallization region, growth of crystals below their melting temperature that is very important for incongruently melting borides or having polymorphic modifications and possibility to avoid thermal and mechanical stresses due to the relatively low growth temperatures, and very slow cooling rates, which are typical for this technique [87]. One of the advantages of the flux method is that it requires relatively simple equipment. It consists of a crucible with a load of flux that is located in a vertical-tube resistance furnace or in a high-frequency installation, a supply of inert gas with a cleaning system, and a temperature control system, as described by Fisk and Remeika [88], see Fig. 1.2. Crystallization from the flux can be realized by changing the flux temperature, by changing of the fluxHe, Ar

SUPPORT WIRES (Mo, W)

FURNACE VALVES CRUCIBLE & CHARGE

EXHAUST GAS

MULLITE TUBE

Figure 1.2 Gas flow arrangement to maintain positive gas pressure while not passing gas over hot crucible and charge. Lower valve open and upper valve closed allows for initial flushing of space, the reverse arrangement being used to maintain integrity of the growth atmosphere during the heating-and-cooling cycle. The exit gas can conveniently be exhausted through a bubbler, allowing monitoring of flow rate. Reproduced from Fisk et al. [88].

12 Crystal Chemistry and Crystal Growth of Rare-Earth Borides

solute composition, and also using crystallization during a chemical reaction. Polycrystalline powders of either borides or a mixture of the individual metal with boron or metal oxide with boron in the appropriate ratio are introduced into the flux. In the latter cases, the synthesis of boride in the flux precedes crystallization. The total dissolution rate and the process of boride formation by far exceed the dissolution rate of the finished product. As a result, a high saturation is achieved and mass precipitation of fine crystals is produced. When growing hexaboride solid solutions (R1 ,R2 )B6 , it is desirable to use a finished solid solution as a starting material, because using the mixture of individual elements in powder form as the charge material may lead to undesirable gradients in the R1 : R2 ratio in the grown crystals [96]. Depending on the ratio of the main components in the charge you can get one or another boride phase (for example SmB4 or SmB6 ); depending on the temperature of the melt you can get YbAlB4 crystals from Al flux or YbB4 in the same flux but at higher temperature [88]. For the growth of RE boride single crystals, exclusively metal solutions are used as fluxes. Molten aluminum serves as the flux in most cases as it satisfies the following requirements for solvents: (1) The melting point of the flux should be significantly below the melting point of the crystallized substance under low vapor pressure during synthesis. (2) The flux and solute in the liquid state must be completely miscible and should not form a solid solution. For this, the radii of the solvent ions should differ as much as possible from the ionic radii of the substance being crystallized. (3) The flux should not react with crucible material, and must not contaminate the growing crystal with its components or interact with the RE elements. (4) The metal flux should be easy to separate from the obtained crystals by dissolving in acids or alkalis, or by distillation in vacuum. As a rule, the crystallization temperature is in the range of 700– 1500 ◦ C. For comparison, the melting point (Tm ) of RE borides

Principles of the Main Crystal-Growth Techniques 13

exceeds 2000 ◦ C, whereas the melting point of Al is 660.3 ◦ C, and its boiling temperature is 2470 ◦ C. The RE ionic radii are in the range ˚ The of 0.99–1.22 A˚ from Lu up to La, while that for Al is 0.57 A. solubility of the hexaborides in molten Al is therefore quite low, for example LaB6 has the solubility of 14 mg per gram Al near 1400 ◦ C. Aluminum interacts with boron in the melt with the formation of AlB2 , but the latter melts incongruently at 975 ◦ C, so at a higher melt temperature the desired RE boride phase can be obtained. It is also possible to use low-temperature RE – transition-metal eutectics as a solvent for growing corresponding RE borides. One example is the growth of PrB4 out of the Pr-Co eutectic with temperature 558 ◦ C [108]. The growth of RE-boride crystals from the Al flux is carried out in alumina crucibles, whereas growth from the RE-based eutectics requires Ta, Mo, or W containers. In the first case, the crystals are removed from the flux by reacting Al with a NaOH solution. For other fluxes it is necessary to find a chemical etch that will not damage the crystals. As a rule the boride crystals grown by the flux technique are small but have higher quality than crystals grown from their melts. Some of the disadvantages are that the growth rate of the crystals from the flux is approximately two orders of magnitude lower than the melt growth, and that there is a constant risk of macroscopic flux inclusions in the grown crystals.

1.2.3 Floating-Zone Melting Technique The main advantage of the crucibleless floating-zone (FZ) melting technique is the absence of the crucible-container. The melt is in contact only with its own solid phase and the gas medium created inside the installation that prevents the contamination of the growing crystal with foreign impurities. Therefore, this method is suitable for producing single crystals of compounds with a high melting point like that of the RE borides. Besides this method allows to grow single crystals of individual borides and their solid solutions. Boride solid solutions with homogeneous chemical composition can be prepared, unlike with the crucible methods. Often zone melting is used to grow single crystals of thermally unstable substances, as minimization of the molten zone height

14 Crystal Chemistry and Crystal Growth of Rare-Earth Borides

helps to significantly reduce the time that the initial substance spends in the molten state. The successful single-crystal growth of the peritectically melting YbB12 by zone melting with optical heating is a prominent example [109]. One more advantage of the zone melting is the chemical purification of the substance based on the difference in the solubility of the impurity in the solid and liquid states. Besides, impurities with a high vapor pressure evaporate from the molten zone of the base material. Minimizing the height of the molten zone results in more efficient purification. The process can be generally carried out in vacuum, reducing or inert atmosphere, or at high pressure in a closed growth chamber in the case of compounds characterized by a high vapor pressure. Specifically, in the case of refractory RE borides, the atmosphere in the growth chamber is either an inert gas or a reducing gas (gas mixture of an inert gas and H2 ), and depending on the installation design, the process is carried out either in a gas flow or in a closed chamber under an elevated gas pressure to prevent vaporization of volatile boride components from the sample and to maintain the desired composition of the single crystal. The basic idea in FZ crystal growth is the passage of a floating molten zone, which is limited by two solid-liquid interfaces, through the material. This principle is shared by all the equipment with different methods of the molten zone formation—induction, optical, arc, resistive, and indirect ones. A schematic sketch of the FZ apparatus with optical heating is presented in Fig. 1.3. A photo of the real equipment “Crystal-111A” used in our lab for zone-melting crystal growth with radio-frequency (RF) heating is depicted in Fig. 1.4(a), and its charged chamber with a seed in the upper shaft and a sintered rod in the lower shaft is shown in Fig. 1.4(b). Tips of the two rods (initial polycrystalline feed rod and singlecrystal or polycrystalline seed rod), which in alignment are fixed in the upper and lower shafts respectively (or vice versa), are located at the focal point of ellipsoidal mirrors in the optical FZ method or in the center of the induction coil in the induction FZ melting. With any method of heating, an equally small part of both the seed and the feed rods melt, connect, and form a narrow molten zone between the seed and the feed rod. This molten zone is limited by solid-liquid

Principles of the Main Crystal-Growth Techniques 15

Upper shaft

Gaseous atmosphere Quartz tube

Feed rod

Mirror Halogen lamp

Floating zone

Seed rod

Lower shaft Figure 1.3 Schematic sketch of an optical floating zone apparatus. Reproduced from Dabkowska et al. [110].

interfaces from both sides. Movement of the feed rod – seed system relative to the molten zone leads to the slow passage of the liquid zone from the seed through the feed rod, the cooling of the melt, and its crystallization on the seed, which results in the growth of a single crystal from the polycrystalline feed rod in the direction defined by the seed orientation. The seed and the feed rods also have the ability to move in the longitudinal direction independently of one another, allowing us to adjust the diameter of the grown crystal. The moving liquid zone along the rod must have a small height so that the surface tension can hold it, counteracting the hydrostatic pressure. In case of the induction heating, the melt is held by additional electromagnetic forces (levitation) due to the interaction of the external high-frequency field from the induction coil with the field induced by eddy currents in the melt. This interaction also facilitates the stirring of the molten zone, in addition to the rotation

16 Crystal Chemistry and Crystal Growth of Rare-Earth Borides

(a)

(b)

Figure 1.4 (a) The setup “Crystal-111A” for zone-melting crystal growth. (b) The charged chamber with a seed in the upper shaft and a sintered rod in the lower shaft.

of the growing crystal and the feed rod in opposite directions, that ensures a more homogeneous temperature distribution within the melt due to its active mixing and tends to flatten the crystallization front. Temperature leveling is a very important factor, because high-temperature crystallization proceeds under nonequilibrium conditions, and the process is characterized by critical temperature gradients at the liquid-solid interface that reach 100 ◦ C/mm at relatively high crystallization rates of 5–100 mm/h. A large temperature gradient along the growing crystal can cause thermal and mechanical stresses that limit the resulting crystal quality. On the other hand, due to the temperature gradient the formation of the real crystal structure does not end directly upon crystallization but continues as the crystal cools, due to annealing processes that relieve the internal stresses.

Principles of the Main Crystal-Growth Techniques 17

The absence of impurities and a high degree of chemical and physical homogeneity are common requirements for single crystals. This imposes constraints on the quality of the source feed rod: The starting material for single-crystal growth should be of the highest possible purity despite zone purification in the melting process. It is desirable that the source sintered rod has uniform density and minimal porosity to ensure the stability of the molten zone, to prevent the formation of bubbles in the melt, and to avoid the melt from flowing into the porous feed rod. The composition of the feed rod is no less important parameter because some RE borides have a homogeneity range that depends on temperature. When the phase diagram is unavailable, the source composition of the feed rod has to be determined individually in every specific case. One of the most critical parameters determining the quality of single crystals is the growth rate. It governs the structure of the crystal, formation of second phases or inclusions, composition, molten zone stability, etc. There are several general rules: The more complex the crystal structure, the lower should be the growth rate, it should also be lower for incongruently melting compounds than for congruently melting ones. But as a rule, the growth rate is determined empirically in every case. A modification of the FZ melting technique, which allows single-crystal growth of incongruently melting compounds from a molten solution, is called the traveling-solvent floating zone (TSFZ) technique. In this technique, the composition of the molten zone is different from that of the feed and seed rods and is provided by the introduction of a small solvent piece between them with their following joint melting. This composition must intersect the liquidus associated with the desired phase and secure its primary crystallization below the peritectic temperature. When growing RE boride crystals, the solvent is one of the components of the resulting boride crystal, and its introduction allows for lowering the temperature of the melt to prevent evaporation of components during growth, stabilizing the molten zone, and shifting the composition of the molten zone up to the liquidus line. If the composition of the growing crystal and the feed rod coincides, then the composition of the floating molten zone remains the same in the process of growth.

18 Crystal Chemistry and Crystal Growth of Rare-Earth Borides

Temperature in K

Temperature in °C

The Gd-B System

at % B Figure 1.5 Phase diagram of the Gd – B system. After Refs. [111–113].

For example, the peritectic phase GdB6 has been grown by using a B-rich molten zone: Composition at C L is B : Gd = 88 : 12 (Fig. 1.5). During slow growth (zoning rate ∼3 mm/h), just below the liquidus line, and the feed-rod rotation of 10 rpm to enhance mixing within the zone, the crystal of concentration C x (GdB6 ) starts to form [114].

1.2.4 Other Techniques Bridgman method This rather popular method of single-crystal growth did not work as a method for growing refractory RE borides. For instance, the attempt to grow a boride single crystal (YB66 ) by the Bridgman method was unsuccessful because of the incompatibility of the high-temperature melt with the crucible material (W) and tungsten boride formation [102]. Electrolysis The first single-crystal samples of RE hexaborides (RB6 , R = Y, La, Ce, Nd, Gd, Er, Yb) were obtained in the 1930s by molten salt electrolysis [74, 115, 116]. Details of the preparation (composition of the melts and the process parameters)

Principles of the Main Crystal-Growth Techniques 19

for works published before 1990 are summarized in the Gmelin Handbook [112]. The advantage of this method is the possibility of obtaining boride solid solutions [117]. The disadvantages are the small size of the crystals (typically less than 100 nm) [74, 112, 115, 116, 118] and the possibility of contamination by cations from the melt [119]. To obtain a LaB6 single crystal with dimensions 6 × 5 × 5 mm3 , it was necessary to carry out the electrodeposition process for 200 h using a seed of the same composition, while during deposition on a gold-wire cathode, crystallites of 4 mm on a side could be obtained within 300 h [120]. It would seem that such a method of obtaining single crystals is ineffective. However, the miniaturization of devices and the needs of nanoscale science and technology have revived the interest in obtaining nano-sized single crystals by electrolysis [121–124]. One-dimensional crystal preparation (CVD, VLS, etc.) In the last decades, the preparation and investigation of 1D single crystals such as nanowires, nanobelts, nanoawls, and nanotubes on the basis of individual RE hexaborides (RB6 , R = Y, La–Nd, Sm–Er) and their solid solutions have also received much attention. They are of interest for nanoscale materials science and technology as traditional thermionic electron sources and field emitters and appear promising for future photonic and electronic applications. Almost three decades elapsed between the first experiments in obtaining whiskers by gas transport methods [75, 125, 126] and a recent avalanche-like stream of publications devoted to the preparation of 1D single crystals by different methods. Comprehensive information on the methods of their fabrication, growth mechanisms, and investigations of physical properties can be found in four reviews [77–80] and in recently published articles not presented in the reviews [81–86]. It should be noted that nanoscaling results in new emergent properties for substances compared to their bulk state. So, ErB6 in the bulk state can be obtained only by stabilizing the CaB6 type lattice with metal ions of a larger radius than the Er ion radius, as in (Er0.8 Ca0.2 )B6 [119]. However, ErB6 nanowires are synthesized via palladium nanoparticle-assisted chemical vapor deposition method

20 Crystal Chemistry and Crystal Growth of Rare-Earth Borides

with typically a high aspect ratio with diameters between 30 and 150 nm, lengths up to several micrometers and with less than 100 ppm impurity incorporation [127]. The authors suppose that this methodology may be extended to create other novel phases of refractory materials, which cannot be obtained using conventional bulk techniques.

1.3 Metal-Rich Borides 1.3.1 Rare-Earth Diborides (RB2 ) Rare-earth diborides (RB2 , R = Sc, Y, Sm, Gd–Lu) crystallize in 1 , the hexagonal AlB2 -type structure (space group P 6/mmm – D6h No. 191) and contain one formula unit per unit cell (Z = 1) [33]. The AlB2 structure (Fig. 1.6) can be described based on trigonal prisms with metal atoms on their vertices and boron atoms in the middle forming graphite-like hexagonal boron nets. Metal atoms are located at the site (0, 0, 0); boron atoms are arranged in positions (1/3, 2/3, 1/2) and (2/3, 1/3, 1/2) and are connected by sp2 -type bonds. The center of the hexagonal boron ring lies directly above and below every metal atom. Consequently, the metal atom is surrounded by

Al

c b

a

B

Figure 1.6 The AlB2 crystal structure. The shaded prism is the unit cell. Adapted from Levchenko et al. [140].

Metal-Rich Borides 21

six other equivalent metal atoms and 12 boron atoms, whereas each boron is surrounded by six metal and three boron atoms. Due to the alternation of the 2D boron layers and hexagonal close-packed (h.c.p.) layers of metal atoms, this can be considered as a quasilayered structure that causes the anisotropic nature of the bond in diborides and defines the anisotropy of their physical properties. Crystal-chemical data for known RB2 compounds are summarized in Table 1.3. They are ordered in accordance with decreasing metallic radii, taken in coordination CN 12. All parameters decrease from Sm to Lu alongside the metallic radii. K. E. Spear [33] analyzed the crystallographic data and showed that a hard-sphere-type crystal chemical model (Fig. 1.7) is not applicable for RE diborides. One reason is rigidity of the graphitelike boron nets that hinders an increase in the a0 dimension with increasing metal size; no such hindrance occurs in the c0 direction perpendicular to the graphite-like layers [33]. Due to the limited ability of the metal ion to be deformed in the AlB2 structure, this ˚ up to Sm phase is known for metals with radii from Cr (r = 1.28 A) ˚ (r = 1.81 A). Projections of the x z planes of two AlB2 -type structures with deformed small and large metal atoms are shown in Fig. 1.8. The presented model meets the following requirements: (1) the B–B bond distances are identical in both structures, (2) the volume of each metal is identical to that of the undeformed metal, (3) the R–B bond distances are kept optimal by changes in the distance between the boron planes, and (4) the R–R bonding is enhanced over what it would be with undeformed metals [33]. The analysis of bonds within the framework of this model suggests that the stability of RB2 could be related to R-R and R–B bond reinforcement, as B–B bonds are noticeably stretched, and the role of the R–B bonds has to be the most important in this structure [33]. According to calculations of the electronic structure [144], the hybridization of the R d and B p states is sufficiently strong to lead to the formation of a pseudogap in the electron density of states. The interaction between the layers is also strong, therefore these structures can be considered only as quasi-layered. The calculations of effective charges, valence, atomic

22 Crystal Chemistry and Crystal Growth of Rare-Earth Borides

Table 1.3 Crystal-chemical data for AlB2 -type RE diboride phases: metal radii r R , experimental lattice parameters a0 and c0 , ratio c0 /a0 , shortest metal-boron distance r R–B , radius of boron assuming B–B contacts rB , and the radius of metal in the direction to the boron r R(B) Calculated values [33] rR

a0

SmB2 1.81 3.310 YB2 1.81 3.298 3.3041 3.304 GdB2 1.78 3.3153 3.3192 3.3179 TbB2 1.77 3.294 3.290 DyB2 1.77 3.287 3.287 3.2874 HoB2 1.76 3.273 3.281 3.281 3.279 3.2835 ErB2 1.75 3.263 3.271 3.28 TmB2 1.74 3.261 3.250 3.258 3.2573 3.260165 YbB2 1.74 3.2503 3.2561 3.25222 LuB2 1.73 3.246 3.2442 ScB2 1.62 3.148203 3.148 3.146 3.148 3.14599

c0

c0 /a0 r R-B

4.019 3.843 3.8465 3.843 3.9363 3.9443 3.9371 3.860 3.878 3.847 3.845 3.8393 3.814 3.813 3.811 3.811 3.8186 3.768 3.782 3.79 3.755 3.739 3.745 3.7473 3.753518 3.7315 3.7351 3.72974 3.704 3.7061 3.514835 3.516 3.518 3.517 3.51317

1.214 1.165 1.164 1.163 1.187 1.188 1.187 1.177 1.179 1.170 1.170 1.168 1.165 1.162 1.162 1.162 1.163 1.155 1.156 1.155 1.152 1.150 1.149 1.150 1.151 1.148 1.147 1.147 1.141 1.142 1.117 1.117 1.118 1.117 1.117

2.708

2.745

2.704 2.702

2.685

2.664

2.659

2.646

2.635 2.52833

rB

r R(B)

Refs.

[128]∗ 1.754 [33] [129] [130]∗∗ 0.957 1.788 [128]∗ [131] [113] 0.947 1.757 [33] [132, 133] 0.949 1.753 [33] [132] [129] 0.945 1.740 [33] [132, 133] [134] [128]∗ [129] 0.942 1.722 [33] [132, 133] [135]∗∗ 0.941 1.718 [135]∗∗ [134] [128]∗ [129] [136] 0.938 1.708 [137] [138] [139] 0.937 1.698 [33] [129] 0.9088 1.620 [140]∗∗ [141] [142] [129] [143]∗∗ 0.954

∗ Samples are obtained under pressure; ∗∗single-crystal samples. ˚ subscripts indicate measurement uncertainties. Note: All dimensions are given in A,

Metal-Rich Borides 23

a0

c0

c0

METAL

BORON

Figure 1.7 Two projections of the AlB2 -type structure. The upper panel is a view down the z axis with the metals at z = 0 and the borons at z = 1/2. The lower panel is a projection of the x z plane with the metals at y = 0 and the two boron positions at y = 1/3 and y = 2/3. Reproduced from Spear [33].

volumes, etc., in TbB2 and LuB2 , based on the Pauli metal radii and polyhedron atomic volumes, also point to the anisotropy of the metal radius in RE diborides [145]. According to the single-crystal refinement, the ScB2 interatomic distances δSc-B = 2.52833 , δB-B = 1.81762 , δSc-Sc = 3.148203 A˚ [140] are consistent with the corresponding data calculated in the deformed-metal model [33] and the conclusions that the relatively large Sc atom results in a significant stretching of the B–B bonds and in Sc–Sc bond contraction [146]. The obtained anisotropic thermal parameters [140] indicate that the metal atom is slightly

24 Crystal Chemistry and Crystal Growth of Rare-Earth Borides

c0

a0

c0

Figure 1.8 Projections of the x z plane of the diboride structure to illustrate the respective deformations of small and large metal atoms in this phase. Reproduced from Spear [33].

compressed along the c axis. The hypothesis of K. E. Spear [33] also found a confirmation in synthesis experiments on SmB2 [128]. ˚ RE diborides with a metal radius larger than that of Tb3+ (0.93 A) could not be obtained for a long time [147]. Cannon et al. [128] suggested that if the size of the metal atom is important for the formation of diborides and the compressibility of lanthanides is larger than that of boron, then synthesis under high pressure and temperature should enable the formation of diborides with lanthanides with large radii. This method was first tested on GdB2 , HoB2 , and TmB2 , which were previously obtained by other methods, and then applied to the (Sm + 2B) mixture. Under the pressure of P = 6.5 GPa and at T = 1140–1240 ◦ C, SmB2 could be synthesized. The results turned out exactly opposite to the expectations from the deformed-metal model [33], as the experimental x-ray intensities agreed well with the flattened Sm atom in the form of an oblate spheroid, i.e., slightly compressed at the poles along the c axis [128].

Metal-Rich Borides 25

Information on the physical properties of RE diborides is limited due to the difficult conditions for their preparation. Unlike RE borides of other structural groups, obtaining diborides through oxides is impossible, because the borothermal reduction of corresponding oxides is a multistage reaction with the initial formation of borate (RBO3 ) and tetraboride (RB4 ) at 600–800 K. Further exposure to elevated temperatures (up to 1900 K) does not result in the decomposition of the tetraboride into diboride and elemental boron, whose interaction with borate would lead to RB2 formation. ScB2 is an exception, because ScB4 does not exist, and therefore it is possible to obtain it by this procedure [140, 141]. The main method of RB2 preparation is the direct synthesis from elements according to the following procedures: • in an arc furnace (R = Tb–Er [132, 133, 148]) or using highfrequency induction heating (R = Ho, Tm [134]); • single-crystal growth in a molten RE flux (R = Y, Er, Tm [130, 135]); • on heating in a sealed Mo or W crucible under parent metal vapors (R = Yb [137, 138], Tm, Lu [136]); • high-pressure synthesis in a tetrahedral-anvil high-pressure device (R = Sm, Gd, Ho, Tm [128], Tb, Er, Tm, Lu [149]). Single crystals of YB2 [52, 76], ScB2 [140], TbB2 , and LuB2 [150] have been grown by the TSFZ technique. DyB2 and ErB2 zone-melting textured ingots were also prepared by the TSFZ method [150]. All source diborides except ScB2 were previously obtained by arc-melting the mixture of the corresponding elements. The source ScB2 was synthesized by borothermal reduction of Sc2 O3 [140]. In recent years, there have been a number of reports on the synthesis of RB2 (R = Sm, Gd, Tb, Dy, Ho, Tm) by the borothermal reduction of the corresponding hydrides [151, 152]. The possibility of synthesizing RB2 diborides is determined by the specifics of their phase diagrams and the thermodynamics of the corresponding reactions. In Figs. 1.9–1.11, typical phase diagrams of several R – B systems are shown, and it is necessary to emphasize that all of them are only estimated. The melting behavior

26 Crystal Chemistry and Crystal Growth of Rare-Earth Borides

Table 1.4 The melting temperatures (Tm ) and melting behavior (C – congruent, P – peritectic) of RB2 diborides (after Spear [34]) R

Sc

Y

Gd

Tb

Dy

Ho

Er

Tm

Yb

Lu

Tm (K) 2523 2373 2323 2373 2373 2473 2458 2523 2523 2523 Melting C C C C P P P P P P behavior

and melting points of RB2 have been summarized by K. E. Spear [34] and are listed here in Table 1.4. The homogeneity range for RB2 is practically absent according to studies of DyB2 [153] and TmB2 [136]. The R – B phase diagrams in the range of the atomic ratio R : B = 1 : 2 can be divided into four types by the melting behavior: (1) congruent melting of RB2 exists throughout the temperature range, as in R = Sc, Y, Tb (Fig. 1.9); (2) congruent melting exists only at high temperatures, as in R = Gd, T > 1280 ◦ C (Fig. 1.5);

Figure 1.9 Phase diagram of the Sc – B system. After Refs. [112, 154].

Metal-Rich Borides 27

Weight Percent Boron 0

10

20

2500

30 40 60 100

2360°C

L

65

2070°C

2185°C 80

2000

2083°C 92.3

2092°C

2015°C

Temperature °C

2020°C 1529°C

1500°C

1500

66.7

1000

ErB12

(Er)

ErB4

ErB2

500

(ErB65)

(βB)

0 0

10

20

30

40

Er

50

60

70

80

90

100

B

Atomic Percent Boron

Figure 1.10 Phase diagram of the Er – B system. After Ref. [155].

Weight Percent Boron

0 3000

10

20

30 40 60 100

G G + L 2500

L 2200 2150 2092°C

2000

1850°C 1500°C 1350°C

1500

1194°C

L

1000 819°C 795°C

(βYb)

10

20

30

40

50

60

Atomic Percent Boron

70

YbB12

0

Yb

80

90

YbB6-6

760°C

(αYb)

YbB4 YbB6

500

0

(βB) 800°C

YbB2

Temperature °C

2300°C

100

B

Figure 1.11 Phase diagram of the Yb – B system. After Ref. [156].

28 Crystal Chemistry and Crystal Growth of Rare-Earth Borides

(3) peritectic melting, where the melt contains tetraboride (RB4 ) and is therefore enriched in metal, as in R = Dy–Tm, Lu (Fig. 1.10); (4) this group contains only one element – ytterbium, the diagram is characterized by incongruent evaporation: at 1350 ◦ C (G + YbB2 ) ↔ G, at 1500 ◦ C (G + YbB4 ) ↔ YbB2 (Fig. 1.11). The possibility to prepare GdB2 in single-crystalline form is doubtful. Most likely YbB2 can be grown in a molten Yb flux. Diborides with melting behavior of the first and third type of the R – B phase diagrams (R = Y, Sc, Tb, Ho, Tm, Lu) have been successfully grown in single-crystalline form. Let us consider these results in detail. Single crystals of TmB2 , ErB2 [135], and YB2 [130] were grown out of a molten RE metal flux. In the case of TmB2 and ErB2 , powdered boron of 99.9995% purity and erbium and thulium metals of 99.9% purity were used as the starting materials in weight ratio R : B ≈ 4. The corresponding charges were sealed in a tantalum tube, heated to ∼1750 ◦ C for 7–20 min and then quenched to room temperature (RT). The grown single crystals were opaque platelets up to 1 mm2 × 0.01 mm thick with well-developed [001] growth facets. The as-grown crystals exhibited a high degree of crystalline perfection, and their lattice parameters were evaluated, see Table 1.3 [135]. YB2 single crystals were synthesized from YB4 and Y by the reaction equation: YB4 + Y = 2YB2 and grown in Y flux in a pyrolytic BN crucible on heating at T = 2000 ◦ C for 1 h under Ar [130]. The solution was subsequently cooled over a 3 h period to 1900 ◦ C and finally rapidly cooled down to RT. The molar ratio of YB2 : Y = 1 : 1 was chosen for the crystal growth. The as-grown YB2 crystals were black in color, with cross sections of several mm2 , primary orientation [001], and easily cleavable along the (0001) surface. One of them is shown in Fig. 1.12(a). When optimizing the growth process, lamellar-like crystals with an edge up to 5 mm were also obtained [130]. The crystals had good structural properties according to the Laue backscattering pattern and the x-ray rocking curve [130]. The typical full width at half-maximum (FWHM) of the x-ray rocking curve for

Metal-Rich Borides 29

Figure 1.12 ScB2 single crystal: (a) photograph of an as-grown crystal; (b) xray rocking curve. Reproduced from Song et al. [130].

the (0001) reflection in Cu Kα radiation is ∼ 7 , see Fig. 1.12(b). The electrical resistivity and magnetic susceptibility of YB2 have been measured [130], indicating that YB2 is a paramagnetic metal exhibiting Curie–Weiss T-dependence of the magnetic susceptibility above 20 K with the Weiss temperature of p = 39 K. YB2 single crystals have also been grown by the TSFZ method in a zone-melting furnace with RF heating [52, 76]. Source YB2 rods ∅4 mm × 25 mm were prepared by arc melting from individual metallic yttrium (99.9% purity) and pieces of boron (99.0% purity). A stable zone was established at a zone temperature near 2000 ◦ C with an yttrium-rich zone. The oblong grains obtained in the grown

30 Crystal Chemistry and Crystal Growth of Rare-Earth Borides

YB2 rods were about 1.5 cm with a small cross section. The grown crystals split off the (001) plane. These samples were used to ˚ c = 3.843 A) ˚ and measure the lattice parameters (a = 3.298 A, transport properties in the basal plane [52, 76]. The ScB2 phase diagram has an open maximum (Fig. 1.9) [112, 154], but due to the high sublimation ability of ScB2 (exposure at 2200 ◦ C in a vacuum of 10−6 Torr for 30 min results in a loss of 40% of the sample mass), it was claimed that it is impossible to obtain a molten ScB2 sample [141]. This problem can be circumvented by using the TSFZ technique. For this purpose, the source ScB2 powder was synthesized by the borothermal reduction of Sc2 O3 in vacuum at T = 1650 ◦ C [140]. Scandium oxide (99.95% purity) and amorphous boron (99.9% purity) were used as starting materials. The obtained powder was pressed in rods, sintered, and used for single-crystal growth. The ScB2 single-crystal growth by the TSFZ technique was carried out in the specialized “Crystal-111” setup with RF heating under elevated pressure of high-purity argon (1 MPa) to suppress Sc evaporation. The crystal was grown from a metal-enriched melt (a small piece of Sc metal was introduced into the initial melting zone) in order to form a stable molten zone, to decrease the melt temperature, and to prevent the appearance of the second phase – ScB12 . All subsequent ScB2 single crystals were grown with seeds from the first grown ScB2 single crystals. The optimal growth rate was 0.2 mm/min. The obtained single crystals were about 5 mm in diameter and 50 mm in length, highly pure, and with a perfect crystal structure. The primary direction of growth was [001]. A typical ScB2 crystal and its x-ray Laue pattern are shown in Figs. 1.13(a) and Fig. 1.13(b), respectively. These crystals were subsequently used for ScB2 crystal structure refinement [140], study of the de Haas–van Alphen (dHvA) effect [157] and thermoelastic properties using synchrotron x-ray radiation [143]. The latter investigations confirmed that strong covalent B–B bonds ensure stability and stiffness within the ab plane, whereas bonding along the c direction involves the weaker Sc–Sc and Sc–B interactions [143]. The TSFZ technique was also used for growing TbB2 and LuB2 single crystals, whereas DyB2 and ErB2 were prepared as textured

Metal-Rich Borides 31

(a)

(b)

Figure 1.13 ScB2 single crystal: (a) photograph of an as-grown crystal; (b) xray Laue diffraction pattern. Reproduced from Levchenko et al. [140].

zone-melted ingots [150]. These source diborides were previously obtained by arc melting of boron and the corresponding metals and contained tetraborides as a secondary impurity phase. Nevertheless, due to the enrichment of the molten zone by the metal at the beginning of the TSFZ process, tetraboride (RB4 ) was no longer present in the grown crystals. The growth was performed under He or Ar pressure (0.1–0.5 MPa), with crystallization rates about 0.2–0.5 mm/min and the rotation of crystallized and feed rods at 2–10 rpm individually for each RE diboride. The diameter of all the grown samples was about 5–6 mm and the length 25–30 mm. These crystals were characterized by x-ray diffraction (XRD), their Young’s moduli were evaluated, and Raman spectra were studied [150]. The reports on the physical properties of RB2 are rather scarce in the literature. TmB2 exhibits long-range FM order with TC = 7.2 K [136], whereas YbB2 is the only antiferromagnet in the RB2 series with the N´eel temperature TN = 5.6 K [138]. The diborides with Tb, Dy, Ho, and Er have relatively strong magnetocrystalline anisotropies and are reported to be ferromagnetic (FM) with the Curie temperatures TC = 151, 55, 15, and 16 K, respectively. With the exception of TbB2 , these ordering temperatures differ considerably from the Curie temperatures extracted from the hightemperature Curie–Weiss behavior of the susceptibility [132, 133]. In addition, magnetization of TbB2 and DyB2 vs. temperature exhibits two-step behavior, suggesting that the magnetic ordering

32 Crystal Chemistry and Crystal Growth of Rare-Earth Borides

χ

Θ

χ

Θ

Θ

Θ

Figure 1.14 Temperature dependence of the ErB2 inverse magnetic susceptibility. Courtesy of K. Flachbart [158].

phenomena in these compounds are likely more involved than a simple FM ordering. The temperature dependence of the magnetic susceptibility, the field dependence of the magnetization, the specific heat and resistivity of ErB2 vs. magnetic field and temperature [158] are shown in Figs. 1.14–1.17, respectively. These characterization results were obtained in the group of Prof. Flachbart (Institute of Experimental Physics, SAS) on single crystals grown in our laboratory [150]. Figure 1.14 shows the T-dependence of the inverse magnetic susceptibility in the fields of 10, 100 and 500 mT. The hightemperature parts of 1/χ(T ) obey the Curie–Weiss law with an effective magnetic moment μeff close to the value expected for the free Er3+ ion (9.6 μB ). The asymptotic Curie temperature p is negative for 10 mT but becomes positive for higher fields, indicating a change in the nature of spin correlations from antiferromagnetic to ferromagnetic. The presence of a narrow hysteresis loop in the magnetization (Fig. 1.15) points to FM ordering in ErB2 below TC = 13.8 K.

Metal-Rich Borides 33

Figure 1.15 Field dependence of the magnetization of ErB2 below TC . Courtesy of K. Flachbart [158].

Figure 1.16 Temperature dependence of the ErB2 heat capacity in different magnetic fields. Sharp phase transitions are observed in the range of 2–5 K and 10–18 K, respectively, as shown in the insets. Courtesy of K. Flachbart [158].

34 Crystal Chemistry and Crystal Growth of Rare-Earth Borides

Figure 1.17 Temperature dependence of the ErB2 resistivity in different magnetic fields. Courtesy of K. Flachbart [158].

The field and temperature dependences of heat capacity and resistivity are sensitive to phase transitions of both magnetic and nonmagnetic nature. Figure 1.16 shows the temperature dependence of ErB2 specific heat in different magnetic fields, revealing sharp phase transitions in two temperature regions of 2–5 K and 10–18 K that are enlarged in the insets. In zero magnetic field, there are in total three phase transitions at TC = 3.4, 13.8, and 15.6 K. The anomaly at 15.6 K, which is suppressed in larger magnetic fields, is not visible in the T-dependence of the magnetization. The anomaly at 3.4 K, in contrast, is observed in the zero-field-cooled T-dependence. In addition, temperature dependence of the resistivity in fields of 0, 1, and 3 T are presented in Fig. 1.17. In zero field, a phase transition is observed at 13.8 K, which gets suppressed in moderate magnetic fields. Thus, the first studies of the physical properties of the ErB2 samples grown by zone melting revealed complex behavior at low temperatures with several thermodynamic phases, evidenced by the presence of multiple phase transitions that are not observed in the T-dependence of the magnetization. That is, these transitions have a

Metal-Rich Borides 35

y

(a)

x

y x

(b)

x y

(c)

z

Figure 1.18 Comparison of the frameworks built by boron atoms in the (a) CaB6 , (b) ThB4 , and (c) R2 B5 structures. Reproduced from Kuz’ma et al. [159].

nonmagnetic nature and may be related to the anisotropic character of the AlB2 structure type. The exact microscopic origin of these transitions has yet to be clarified.

1.3.2 R2 B5 The binary compounds R2 B5 exist only for R = Nd, Pr, Sm, and Gd and have not received as much attention as other boride structure groups. R2 B5 compounds contain four formula units per unit cell (Z = 4) and crystallize in monoclinic symmetry but in two different space groups. Both Pr2 B5 and Nd2 B5 belong to the same structural 6 (No. 15) [159, 160], family with the space group C 2/c – C 2h whereas Sm2 B5 and Gd2 B5 share a different structure type with the 5 (No. 14) [111, 131, 161, 162] that has a space group P 21 /c – C 2h nearly twice lower unit-cell volume than the two other compounds (Table 1.5). The Pr2 B5 (Nd2 B5 ) structure is closely related to the Gd2 B5 structure and differs from it in that it has a higher symmetry, and one of the unit cell parameters is doubled (Table 1.5). The R2 B5 structure consists of a three-dimensional framework of B6 octahedra and two types of B2 units enclosed in 32 434 nets of R atoms: There are two R sites and 5 sites of B [162]. One finds R2 B5 compounds among those that have B6 octahedra as the main building block of the 3D boron sublattice. If the CaB6 structure type can be considered as a packing of B6 octahedra linked in such a way that eight-membered rings of boron atoms are formed between

Crystallographic data for R2 B5

˚ Lattice parameters (A) Phase

Space group

a

b

c

Pr2 B5 Nd2 B5 Sm2 B5

C 2/c ” P 21 /c ” ” ” ”

15.1603(4) 15.0808(7) 7.183(6) 7.179(2) 7.180(2) 7.136(1) 7.205(9)

7.2771(2) 7.2522(3) 7.191(6) 7.205(2) 7.196(2) 7.159(1) 7.202(7)

7.3137(2) 7.2841(3) 7.216(6) 7.180(2) 7.195(2) 7.183(1) 7.283(8)

Gd2 B5

β (deg.) 109.607(2) 109.104(2) 102.03(4) 102.02 102.16 102.68(1) 102.93(7)

Unit cell volume (A˚ 3 ) 760.08(7) 752.78(6) 364.5(5) 363.2(2) 363.4(2) 358.0(1) 368.3(7)

Ref. [159] [160, 163] [34] [161] [111] [162] [131]

36 Crystal Chemistry and Crystal Growth of Rare-Earth Borides

Table 1.5

Metal-Rich Borides 37

these octahedra [Fig. 1.18(a)], and in the ThB4 structure type, the B6 octahedra are linked to each other by isolated boron atoms to form seven-membered rings in the x y plane [Fig. 1.18(b)], then in R2 B5 , a similar framework of boron atoms (octahedra and seven-membered rings of boron atoms) is also observed but is projected onto the yz plane, as shown in Fig. 1.18(c) [34, 160]. However, unlike in RB4 , the R atoms in the nearly parallel metal layers in the yz plane of R 2 B5 do not lie directly above and below each other but are displaced by 1.5 A˚ along the z direction, as shown in Fig. 1.19(a) [160]. In addition, the B6 octahedra are separated by intervening B2 units, which connect and displace the octahedra in adjacent layers. The intervening B2 units (B5–B5 units) in R2 B5 are of sp2 type and similar to those in RB4 . Each boron atom is coordinated by six metal atoms arranged in a distorted trigonal prism. However, each of the B5 atoms is bonded with only two other borons, rather than three as in the case of sp2 -type boron in the RB2 and RB4 compounds [34]. This led to a suggestion that the B5–B5 bond in R2 B5 is more probably a B=B double bond [160]. A detailed x-ray single crystal analysis of Nd2 B5 was carried out by Roger et al. [160]. They showed how the Nd2 B5 structure is formed by two boron slabs of the NdB4 type [Fig. 1.19(b, c)], shifted from each other by b/2 in such a way that B6 octahedra with a different orientation from these slabs are connected by additional B2 (B5–B5) units, as shown in Fig. 1.19(d). Similar linkages are also found in the structure of Gd2 B5 [162], but in that case consecutive B6 octahedra exhibit the same orientation [Fig. 1.19(e)]. This result explains the non-doubling of the unit cell volume for Gd2 B5 , since the boron layers following one another are only very slightly shifted [160, 162]. The known crystallographic data for R2 B5 are presented in Table 1.5. The single-crystal x-ray analysis was carried out also for Pr2 B5 [159] and Gd2 B5 [162]. One can find the atomic coordinates, isotropic displacement parameters, and selected interatomic distances for Nd2 B5 [160], Pr2 B5 [159], and Gd2 B5 [162]. In addition, the anisotropic thermal parameters for two Gd atoms were presented in Ref. [162]. For Sm2 B5 , only coordinates of the Sm atoms were determined [161], and the coordinates of the boron atoms were assumed in analogy with Gd2 B5 [162], so the interatomic distances for Sm2 B5 could not be estimated.

38 Crystal Chemistry and Crystal Growth of Rare-Earth Borides

NdB4-like slab NdB4-like slab

B Nd c a

(a)

b

b

a

c

c

a

(c)

(b)

B Nd b

(d)

(e)

(f)

a

(g)

Figure 1.19 The Nd2 B5 crystal structure compared to that of NdB4 . (a) Projection on the ac plane. (b, c) Successive slabs of NdB4 type. (d) Linkage of two B6 octahedra belonging to successive slabs in Nd2 B5 , in comparison to (e) Gd2 B5 and (f) NdB4 . (g) Representation of the NdB4 structure in the ab plane, shown for comparison. Reproduced from Roger et al. [160].

Metal-Rich Borides 39

In general there are very few works on the physical properties of the R2 B5 compounds, and the information is limited only to ´ temperatures according to the unpublished observations of Neel Bucher et al. presented in Ref. [132]: TN = 15.1, 32.7, 23.5, and ∼50 K for R = Pr, Nd, Sm, and Gd, respectively (less accurate for the latter due to metallic Gd traces). Paramagnetic Curie temperatures p = 14 K were also determined for Pr2 B5 and Nd2 B5 [132]. Probably Bucher et al. obtained these compounds either by an arcmelting process or by high-temperature solid-state reaction from the elements. A direct synthesis from elements was used almost in all the studies regardless of whether the growth of crystals was done in the melt, by solid-state synthesis, or arc melting: • Polycrystalline samples of Nd2 B5 [164] and Gd2 B5 [165] were obtained by direct synthesis from the elements at 1473 ± 2 K in a calorimeter under determination of their enthalpies of formation. The presence of a minor secondary phase of approximately 5–7% of NdB4 in the resulting Nd2 B5 sample was noted. • Sm2 B5 was synthesized from elemental Sm and B by melting in an arc furnace under Ar atmosphere, the obtained ingots were then annealed. It was established that Sm2 B5 is in equilibrium with SmB4 and Sm; it is characterized by a stable composition and does not oxidize when stored in air [147, 161]. • Polycrystalline samples of Gd2 B5 were also prepared by melting of the corresponding mixtures of elemental Gd and B in an arc furnace, annealed at 1270 K for one month, and subsequently quenched in cold water [131]. • Single crystals of Gd2 B5 [162], Pr2 B5 , and Nd2 B5 [166] have also been obtained by the flux method. Gd2 B5 [162] was grown from Gd (99.99% purity) and B (99.94% purity) in the presence of GdCl3 as a flux at 1350 K. The ratio of components was 0.600 g Gd (3.8 mmol), 0.103 g B (9.5 mmol), and 0.400 g GdCl3 (1.5 mmol); the mixture was kept in a sealed Ta capsule. The reaction product was washed with H2 O and dried. After this procedure, Cl traces were removed. Gd2 B5 crystallized in the form of platelets with silver luster (typically 0.2 × 0.2 × 0.04 mm3 ) that were stable in air and soluble in dilute HNO3 .

40 Crystal Chemistry and Crystal Growth of Rare-Earth Borides

• Single crystals of Pr2 B5 and Nd2 B5 [166] were grown out of molten RE metal fluxes in sealed tantalum tubes heated to 1700 ◦ C for 10 min using a RF induction furnace. Each solution was subsequently cooled over a 10 min period to 1000 ◦ C and then quenched to RT. Large platelets of then-unknown phases, having a monoclinic crystal structure and R/B0.65 stoichiometry, were extracted from the mixtures. Later it was determined that their phase compositions were Pr2 B5 and Nd2 B5 [34]. • Nd2 B5 single crystals [160] were also grown by another synthesis method. After arc melting of suitable amounts of boron powder (>99% purity) and freshly filed chips of Nd metal (>99.9% purity), the obtained ingots were introduced in tantalum crucibles, arc-welded shut under 0.5 atm of argon, then heated up to 1770 K for 15 days in a high-frequency furnace, and finally slowly cooled step by step to RT. According to the electron backscattering image (Fig. 1.20), the Nd2 B5 formation is the result of a peritectic reaction between NdB4 and elemental Nd during annealing at 1770 K of the multiphase ingot after arc melting of the Nd and B mixture. The grown

Nd2B5 NdB4

Figure 1.20 Electron backscattering image showing the formation of Nd2 B5 (grey) from a peritectic reaction between NdB4 (black) and elemental Nd (white). The small crystallites in the middle of the image result from the imperfect preliminary reaction: NdB6 + elemental boron  NdB4 due to too rapid cooling. Reproduced from Roger et al. [160].

Metal-Rich Borides 41

Nd2 B5 single crystals showed a metallic luster and were unreactive to air. The Nd2 B5 single crystal with dimensions of 0.032 × 0.042 × 0.040 mm3 were then used for x-ray single crystal analysis. Kuz’ma et al. [159] had to wait for 2 years to obtain Pr2 B5 single crystals for x-ray structure analysis. In their experiment, samples containing 50–30 at. % of Pr and 50–70 at. % of B were melted in an arc furnace under Ar atmosphere. According to x-ray phase analysis, the ingots consisted of two phases, Pr2 B5 and Pr. However, it was impossible to remove the small shiny crystallites observed on the ingot fracture immediately. The ingots were stored in glass beakers in air for 2 years, which resulted in the oxidation of the Pr metal into a white powder. The remaining shiny Pr2 B5 crystallites could be then singled out for physical investigations. The successful growth of the R 2 B5 single crystals can be explained on the basis of the corresponding binary phase diagrams. Weight Percent Boron

Temperature °C

L

Nd

Atomic Percent Boron

B

Figure 1.21 Phase diagram of the Nd – B system. After Liao et al. [167].

42 Crystal Chemistry and Crystal Growth of Rare-Earth Borides

Weight Percent Boron

Temperature °C

L

Sm

Atomic Percent Boron

B

Figure 1.22 Phase diagram of the Sm – B system. After Liao et al. [168].

Figures 1.21 and 1.22 show the estimated phase diagrams of the Nd – B [167] and Sm – B [168] systems. The Gd – B phase diagram is shown in Fig. 1.5 [113]. The Pr – B [34] phase diagram differs from the Nd – B one only in the region with high boron content, since PrB66 does not exist, unlike NdB66 , but is otherwise very similar. These compounds melt peritectically, but if the melt is enriched with its RE metal, then R2 B5 crystals will form [162, 166]. The melting points of the Nd, Pr, Sm, and Gd metals are below those of the corresponding borides, so it can be expected that R2 B5 single crystals can be successfully grown by the TSFZ technique from starting materials previously obtained by arc melting of the elements or by a solid-state reaction. It should also represent no problem to obtain these single crystals from a flux of excess RE metal or of the corresponding RE compound with a low melting point as in the case of GdCl3 (609 ◦ C).

Rare-Earth Tetraborides (RB4 ) 43

1.4 Rare-Earth Tetraborides (RB4 ) 1.4.1 General Overview Rare-earth tetraborides (RB4 , R = Y, La–Sm, Gd–Lu) crystallize 5 , in the tetragonal ThB4 -type structure (space group P 4/mbm – D4h No. 127), Z = 4, atomic positions: R in 4g(x, 1/2 + x, 0); B(1) in 4e(0, 0, z), B(2) in 4h(x, 1/2 + x, 1/2), B(3) in 8h(x, y, 1/2) [112]. A projection of the structure along the z axis is shown in Fig. 1.23 with the unit cell indicated by the dashed lines. The structure can be described as a combination of the hexagonal RB2 and cubic RB6 units, as it is shown in Fig. 1.23 by the dotted lines [112]. Boron atoms form a continuous 3D network: Chains of the boron octahedra (B6 ) along the [001] direction, the same as in the RB6 structure, are linked in the ab plane of the RB4 structure through sp2 -type boron pairs (B2 ), creating planar seven-membered rings as can be seen in Fig. 1.24(b). The resulting 3D boron framework contains tunnels

Figure 1.23 Projection of the tetragonal RB4 structure along the z axis. The unit cell is indicated by the dashed lines. The formation of the structure from RB2 and RB6 units is illustrated by the dotted lines [112].

44 Crystal Chemistry and Crystal Growth of Rare-Earth Borides

(a)

c a

b

(b)

b c

a

Figure 1.24 Three-dimensional view of the RB4 crystal structure: (a) projection on the bc plane; (b) projection on the ab plane. Small spheres are B atoms (green – B6 units; yellow – B2 units); large purple spheres are RE atoms (R). Reproduced from Werheit et al. [170].

parallel to the [001] direction, which are filled by the metal atoms, see Fig. 1.24 [169]. The RE atoms are located on equivalent sites with the low orthorhombic point-group symmetry C 2v , above and below the centers of the seven-membered rings in the close-packed planes of the boron atoms, as in the RB2 structure. The size of the metal site depends on the structure of the boron sublattice. The 3D cage units (B6 ) are relatively rigid, but the

Rare-Earth Tetraborides (RB4 ) 45

b a Figure 1.25 Crystal structure of RB4 in the ab plane. The nearest- and second-nearest-neighbor bonds between the metal ions are represented by thick and thin lines, respectively. Reproduced from Okuyama et al. [17].

presence of the boron linear bonds (B2 ) in the ab plane significantly reduces the rigidity of the boron sublattice in general. So all RE atoms with radii between 1.73 and 1.94 A˚ are accommodated in the large voids of this structure, which is not the case for Eu (rEu = ˚ All attempts to obtain EuB4 have been unsuccessful, which 2.04 A). may be due to the dimensional factor: The size of the void in which the metal ion is located in the ThB4 structure requires that the europium ion is either trivalent or has intermediate valence between Eu2+ and Eu3+ . However, it turned out impossible to convert the Eu ion into such a state even using high-pressure synthesis [112]. The 2D layer of the R atoms in the ab plane is modified into a combination of squares and triangles as illustrated in Fig. 1.25. Here the pairs of nearest-neighbor metal ions form an orthogonal dimer lattice, which is topologically equivalent to the Shastry–Sutherland lattice (SSL) that has received much attention as a realization of magnetic models with geometrical frustration [14]. The thick lines in the [110] directions correspond to the Shastry–Sutherland dimers with the nearest-neighbor coupling J1 , the thin lines represent the

46 Crystal Chemistry and Crystal Growth of Rare-Earth Borides

Figure 1.26 Magnetization (M) of TbB4 at 4.2 K for magnetic fields B applied along [100], [110], and [001]. Inset: network of Tb atoms in the (001) plane. The solid and dashed lines denote the nearest- and nextnearest-neighbor exchange interactions, respectively. (b) Comparison of the field derivative magnetization dM/dB along the [001] axis with M. The indicated fractions denote the ratio M/Ms , where Ms is the saturation magnetization. Reproduced from Yoshii et al. [171].

Rare-Earth Tetraborides (RB4 ) 47

second-nearest-neighbor coupling J2 (Fig. 1.25). Under an external magnetic field, the antiferromagnetic (AFM) Shastry–Sutherland model is expected to show magnetization plateaus at fractional values of the saturation magnetization, which have been indeed observed in NdB4 [28, 31], TbB4 [19, 171], DyB4 [107], HoB4 [17, 20, 31, 172], ErB4 [15, 173], and TmB4 [18, 173, 174]. Such plateau states (see Fig. 1.26 for an example) are stabilized by classical and quantum mechanisms including order by disorder and various competing order effects [175–177], which are also reflected in the magnetic properties of RE tetraborides. These signatures of magnetic frustration are evidenced by their complex magnetic phase diagrams with multiple phase transitions. Apart from that, a strong structural anisotropy in these compounds determines a very strong Ising anisotropy of the magnetic moments that defines the electric and magnetic properties [171]. The magnetic behavior of frustrated magnetic systems in general can be very sensitive to small perturbations in the magnetic Hamiltonian, and in particular to the presence of impurities or defects. Therefore, a decisive factor in understanding the physical properties of RE tetraborides and the hierarchy of their microscopic magnetic interactions is the availability of high-quality single crystals, as well as their thorough characterization. The lattice parameters for RB4 are given in Table 1.6. They were obtained by powder and single-crystal diffraction studies using both x-rays and neutrons. Crystal-structure analyses are available for YB4 [178, 197], LaB4 [182], CeB4 [186], NdB4 [189], SmB4 [192], GdB4 [196], TbB4 [179], DyB4 [184], HoB4 [143] and ErB4 [179,184]. The lattice constants of RB4 are also plotted in Fig. 1.27 vs. the atomic number of the RE metal (with the exception of PmB4 and EuB4 , which do not exist). The general monotonic decrease of the lattice constants of about 5% for a and 3% for c reflects the lanthanide contraction in this system [198]. The solid lines connect the data for tetraborides from La to Lu, where the RE ions are in the trivalent state (R 3+ ). A significant deviation of the CeB4 lattice constants from this monotonic trend, as well as the magnetic properties of this compound, indicate that the oxidation state of the cerium ion corresponds to +4 rather than +3 [132, 199]. The slight deviation for YbB4 , as well as the results of its physical-property

48 Crystal Chemistry and Crystal Growth of Rare-Earth Borides

Table 1.6 Lattice parameters for RB4 (R = Y, La–Nd, Sm, Gd–Lu) R Y

˚ a (A)

˚ c (A)

7.111(3) 7.106(4)

4.017(2) 4.020(4)

c/a

Ref.

0.5649 0.5657

[178] [143]

La 7.32462(3) 4.18091(2) 0.5708 7.3268 4.1794 0.5704

Ce 7.205(4) 7.208 7.2034(8) 7.2052 Pr 7.241 7.235 7.2379 Nd 7.220 7.1775(3) 7.2264 7.21993(3) Sm 7.178 7.179 7.174(1)

4.090(4) 4.091 4.1006(5) 4.0898 4.119 4.116 4.1148 4.102 4.0996(2) 4.1048 4.10330(2) 4.071 4.067 4.0641(6)

0.5677 0.5676 0.5693 0.5676 0.5688 0.5689 0.5685 0.5681 0.5712 0.5680 0.5683 0.5672 0.5665 0.5665

Gd 7.145 7.1316(2) 7.1451 7.1435(2)

4.048 4.0505(2) 4.0479 4.0473(2)

0.5666 0.5680 0.5665 0.5666

R

˚ a (A)

Tb 7.119 7.120(3) 7.12(5) [181, 182] Dy 7.102 [170] 7.097 7.021(1) 7.0989(3) [169] Ho 7.087 [166] 7.0842(16) [186] 7.08674(3) [187] 7.0910(15) [166] Er 7.071 [183] 7.0705(1) [170] 7.0726(4) [166] Tm 7.057 [189] 7.0550(3) [170] 7.05563(8) [185] [166] Yb 7.064 [183] 7.0489(3) [192] 7.059 7.055 7.0612(5) [166] Lu 7.035(1) [196] [170] [185]

˚ c (A)

c/a

Ref.

4.029 4.042(2) 4.03(4) 4.017 4.016 3.972(1) 4.0183(2) 4.008 4.006(3) 4.00825(2) 4.0086(7) 4.000 3.99710(8) 3.9975(3) 3.987 3.9870(3) 3.98605(5)

0.5660 0.5660 0.5660 0.5656 0.5659 0.5657 0.5660 0.5651 0.5659 0.5656 0.5653 0.5657 0.5653 0.5652 0.5650 0.5651 0.5650

[166] [179] [180] [166] [183] [184] [185] [166] [143] [185] [170] [179] [185] [188] [166] [190] [185]

3.989 3.9937(2) 3.992 4.004 3.9893(3) 3.975(3)

0.5647 0.5666 0.5655 0.5675 0.5650 0.5650

[166] [191] [193] [183] [194] [195]

study, point to an intermediate valence [16,199,200]. Analysis of the χ (H c) component of the susceptibility, which is dominant at low and moderate temperatures, obtained for the YbB4 single crystal in Ref. [16], allowed an estimate of the ytterbium valence of +2.8 [199]. X-ray and neutron scattering studies of DyB4 revealed that quadrupolar order and a monoclinic structural distortion develop concomitantly below 12.3 K as second-order phase transitions [201]. The isostructural RE tetraborides TbB4 and ErB4 undergo structural phase transitions to orthorhombic symmetry around Ts ≈ 80 K and Ts ≈ 15 K, respectively [202]. In TbB4 , the structural

Rare-Earth Tetraborides (RB4 ) 49

0.2

Lattice Constants (Å)

a-7.146 Å c-4.048 Å 0.1

0

-0.1

La Ce Pr Nd Pm Sm Eu Gd Tb Dy Ho Er Tm Yb Lu

Figure 1.27 The experimental lattice constants of RB4 (R = La–Lu), showing a lanthanide contraction of about 5% for a and 3% for c. The solid lines connect the data for tetraborides, in which the R ion is in the trivalent state. Reproduced from Yin et al. [198].

transition, which occurs at a much higher temperature than the magnetic ordering (TN = 43 K), is assumed to be driven by a strong electron-lattice coupling or by an electronic quadrupole– quadrupole interaction. In ErB4 , the structural distortion is attributed to magnetostrictive effects occurring simultaneously with the magnetic ordering process (TN = 13 K) [202]. A field-induced lattice distortion is also observed in ErB4 single crystals in a magnetic field B ≥ 3 T applied along the [110] crystallographic direction in the paramagnetic range (above TN = 15.4 K). It has been suggested that the first-order transition is due to multipole interactions [203]. The lattice-parameter evolution of the HoB4 single crystal that was grown by zone melting is shown in Fig. 1.28 [143]. A structural distortion of nonmagnetic origin appears in the ab plane below a critical temperature Tc ≈ 285 K, i.e., well above ´ temperature TN1 = 7.1 K. It is supposed that symmetry the Neel lowering to the orthorhombic is caused by macroscopic strains due to strong lattice-quadrupolar coupling as in TbB4 , geometrical frustration, and crystal mosaicity. A difference in the Ho-Ho distances, corresponding to the magnetic dipolar interactions J1 and

50 Crystal Chemistry and Crystal Growth of Rare-Earth Borides

Figure 1.28 Evolution of the lattice constants a, b, and c of HoB4 with decreasing temperature. Reproduced from Wa´skowska et al. [143].

˚ respectively), testifies on the J2 in HoB4 (3.646(1) and 3.671(1) A, geometrical frustration, while the temperature dependence of the spontaneous strain points to the crystal mosaicity. The crystal is perfectly single-domain according to the Laue diffractogram, i.e., it is of high quality at the macroscopic level, but high-resolution synchrotron radiation has revealed elastic twin domains. In fact, the crystal consists of micro-clusters of orthorhombic symmetry, the generally assumed tetragonal cell being a macroscopic volume average of those clusters. This study is limited by T = 100 K, but in the author’s judgment, strain enhancement is likely to continue below 100 K, the resulting symmetry change will affect the magnetic interactions, and at a sufficiently large distortion, magnetic order will appear [143]. Once again, this example reveals the degree of accuracy required in the characterization of crystal quality for subsequent interpretation of its physical properties. There are two possible ways to synthesize RB4 phases: (i) direct synthesis from the elements using arc melting or in sealed crucibles at temperatures above 1750 K; (ii) borothermal reduction of RE oxides in vacuum above 1800 K [112, 193]. Now the last method is widespread as it provides the highest purity of the synthesized tetraborides due to the high purity of the source RE oxides. It has been applied to the tetraborides with a congruent melting point, i.e.,

Rare-Earth Tetraborides (RB4 ) 51

to the heavy RE with ions of small radii: Y, Gd–Tm, Lu. In the products of the borothermal reduction of RE metals of the cerium subgroup (R = La–Sm) and Yb, the corresponding hexaboride was present as an impurity phase. For these tetraborides, both direct synthesis from elements above 1750 K under the saturation vapor pressure of the metal in sealed molybdenum or tantalum tubes and hightemperature flux method are feasible [16, 190, 200]. SmB4 and YbB4 were also obtained by interaction of the corresponding hexaborides (SmB6 and YbB6 ) with their metals [199, 204]. The stability of the RE tetraborides on melting determines the most appropriate method of their crystal growth. According to the phase diagrams, RE tetraborides of the yttrium subgroup (R = Y, Gd– Tm, Lu) have congruent melting behavior. The typical phase diagram for this group is represented by that of the Er – B system in Fig. 1.10. So these RB4 compounds can be grown by the flux, zone melting, and Czochralski techniques. The second group consists of LaB4 , CeB4 , PrB4 , NdB4 , SmB4 , and YbB4 , which decompose on melting with preferential volatilization of the metal to give the corresponding hexaborides. Apart from YbB4 , the phase diagrams for this group are similar to the Nd – B and Sm – B phase diagrams in the range of the RB4 composition (Figs. 1.21 and 1.22). They demonstrate peritectic melting with the formation of corresponding hexaborides and a R-rich liquid. Single crystals of these tetraborides have been grown by the flux method under conditions depending on the RE element. Nevertheless, single crystals of CeB4 [187] and NdB4 [31, 185] were grown by the zonemelting technique. Let us consider these works in detail. Taking into account the features of the Yb – B phase diagram, two possibilities to grow YbB4 single crystals are the flux method and synthesis from elements at a temperature above 1300 K in a sealed crucible with long exposure. In the 1980s, the most common method for growing RB4 single crystals was a flux method with Al as solvent for R = Sm, Gd–Lu or RE self flux when R = La– Sm [166, 181–183, 205]. The typical procedure with a molten Al flux consisted of heating a mixture containing 95 at. % of Al and 5 at. % RE metal and boron powder at the tetraboride stoichiometry in an open Al2 O3 crucible at 1550 ◦ C for 2 h under argon. The melt slowly cooled at 200 ◦ C/h to 1000 ◦ C and after that was quenched to RT.

52 Crystal Chemistry and Crystal Growth of Rare-Earth Borides

Leaching of the Al flux was done in a saturated solution of NaOH. This method allowed the growth of single crystals up to 1 mm in length [166, 183]. Single crystals of the tetraborides from La through Nd were prepared from a self-flux of excess RE metal itself in a sealed Ta tube heated at 1700 ◦ C for 10 min; the solution was then cooled over 10 min to 1000 ◦ C and quenched to RT. Single-crystal platelets of LaB4 and CeB4 were grown from mixtures with the stoichiometry R : B = 1 : 0.65, while PrB4 and NdB4 were grown with R : B = 1 : 3 [166]. The individual details of the procedure (longer soak time, cooling rate, R : Bn ratio) vary when optimized to grow larger crystals. For example, single crystals of PrB4 (2 × 0.5 × 0.5 mm3 ) were prepared using an atomic ratio of boron to praseodymium of 2, the mixture was kept at 1550 ◦ C for 2 h and then was cooled at a rate of 1000 ◦ C/h [183]. PrB4 single crystals of millimeter size were grown out of the low-temperature eutectic with the composition Pr0.75 Co0.25 and an amount of 0.1 PrB4 /(Pr/Co) in the melt [108]. By the similar procedure, PrB4 single crystals were grown using a Pr/Ni flux [108]. According to previous studies, PrB4 orders ferromagnetically with TC = 24 K [183,193,200], but Wigger et al. [108] revealed two phase transitions (TN ≈ 19.5 K and TC ≈ 15.9 K) in the sample grown from the Pr/Co flux but not from the Pr/Ni flux. The authors consider that this behavior results from some as-yet-unidentified effects of trace defects but not from small concentrations of Co or Ni impurities in their samples. Attempts to grow YB4 single crystals in either an Al or Mg melt were unsuccessful. It was found that the presence of magnesium inhibits formation of the tetraboride phase so that YB6 is obtained regardless of the B : Y ratio in the mixture. The aluminum melt, on the other hand, favors formation of the tetraboride. However, the crystals grown from the melt were contaminated with aluminum. So Giese et al. [197] used synthesis from elements with a B : Y ratio of 4 : 1 at temperatures of 1600–1700 ◦ C for 15–30 min to obtain an YB4 single crystal for the structure determination. Otani et al. [206] presented results of YB4 single crystal growth from Al flux and discussed the influence of the cooling rate and the

Rare-Earth Tetraborides (RB4 ) 53

atomic ratio Y : Al to morphology of the grown crystals and their size. A correspondence of morphology with the anisotropy in hardness has been revealed. That is, the (110) plane with the highest hardness became large. But the main problem was to explain why some crystals were slightly attracted by a magnet despite being composed of nonmagnetic elements, Y and B. No unambiguous conclusion was reached, and the search for other impurities apart from iron or defects has not been performed. In addition to the contamination of grown crystals with impurities from the flux, there is also a possibility of the formation of ternary RE aluminum boride crystals. For example, TmAlB4 instead of TmB4 [190], LuAlB4 instead of LuB4 [195], and YbAlB4 instead of YbB4 [194] were grown from Al flux. Finally, optimization of the technological parameters allowed growing the individual tetraborides. In these three publications, the corresponding oxides were used as starting RE materials: Tm2 O3 , Lu2 O3 , and Yb2 O3 (99.9% purity). The optimal growth conditions of TmB4 , LuB4 and YbB4 are listed in Table 1.7. Single crystals of RB4 (R = Y, Tb, Ho, Er) with a congruent melting character (Tm > 2500 ◦ C) were grown by the Czochralski procedure in a tri-arc furnace under the flow of purified argon [104, 105]. The source tetraboride samples were prepared by borothermal reduction of the RE oxides at 1800 ◦ C in vacuum, and the polycrystalline seed consisted of melted tetraboride. Coneshaped single crystals up to ∅3 mm × 1 cm were obtained. The growth rate is sufficiently high: 1 and 2 mm/min for YB4 , 1.5 mm/min for TbB4 , and 0.7 mm/min for ErB4 . The same method has been applied also to DyB4 , with the only difference in the procedure being that single crystals of DyB4 were prepared by the Czochralski method in argon atmosphere using a tetra-arc furnace [106, 107]. Currently, RB4 single crystals are grown mainly by the FZ melting and flux methods. The first method is used when single crystals of large size, high purity, and corresponding quality are needed. The flux method is used when large samples are not required or RB4 single crystals with a peritectic melting point must be grown. Table 1.7 lists RE tetraborides that were grown using the metal-flux technique and were not mentioned earlier in the text.

Atomic ratio, R :B

Parameters of RB4 crystal growth from metallic flux

Heating Cooling Cooling Mass rate Soaking Soaking to rate ratio, (◦ C/h) temp. (◦ C) time (h) T (◦ C) (◦ C/h) RB4 : Al

Ref.

Comments

— — 1 : 50 1 : 13.8 —

[207] [24] [208–210] [211] [180, 212]

RRR = 550

— 4.8 5 — 50

1 : 14.9 1 : 50 1 : 50 1 : 60 1 : 19

[211] [213] [20] [27, 174, 214, 215] [190]

650 —

5 —

1 : 50 1 : 14.3

[16] [194]

—

50

1 : 15.7

[195]

RB4

Flux

NdB4 SmB4 GdB4 GdB4 TbB4

Nd Al Al Al Al

— — 1:4 1 : (3–3.5) 1:4

— — — — —

— — 1500 1500 —

— — — 10 —

— — 650 — —

— — 5 — —

TbB4 Ho1−x Dyx B4 HoB4 TmB4 TmB4

Al Al Al Al Al

1 : (3–4) — 1:4 1:6 1 : 5.5

— 300 — — 300

1500 1650 1500 1475 1500

10 — — — 10

— 650 650 750 —

YbB4 YbB4

Al Al

1:4 1:3

— —

1450 1550

— 10

LuB4

Al

1:3

300

1500

10

RRR = 361.9; typical size: 5 × 3 × 1 mm3 x = 0.0, 0.5, 1.0

TmB3.83 ; max. size: 0.9 × 1.2 × 1.3 mm3 YbB3.92 ; max. size: 0.3 × 0.3 × 2.4 mm3 LuB3.98 ; max. size: 0.8 × 0.8 × 0.7 mm3

54 Crystal Chemistry and Crystal Growth of Rare-Earth Borides

Table 1.7

Rare-Earth Tetraborides (RB4 ) 55

The high-temperature flux method was also used for singlecrystal growth of R1−x Yx B4 , 0 ≤ x ≤ 1 (R = Sm, Gd, Tb, Dy and Er) solid solutions in order to study the development of weak ferromagnetism in a magnetically frustrated system induced by replacing a magnetic element with a nonmagnetic one. A stoichiometric mixture of the RE metal and boron pieces with an Al flux with a mass ratio of R 1−x Yx B4 : Al = 1 : 50 was melted under a high-purity argon atmosphere at T = 1600 ◦ C and was slowly cooled to 655 ◦ C at a rate of 4.8 ◦ C/h. The synthesized crystals were extracted from the Al flux using a NaOH solution [216, 217]. Temperature and concentration dependences of magnetization for all compositions show the universality of anomalously weak ferromagnetism: The exotic weak ferromagnetism occurs only along the magnetic easy axis, which is perpendicular to the c axis in R1−x Yx B4 (R = Sm, Gd, Tb) and parallel to the c axis in R1−x Yx B4 (R = Dy, Er). The magnitude of saturated magnetization at T = 2 K has a strong dependence on Y concentration, which shows maximal saturation values near 30% Y concentration for all compounds. The origin of the exotic weak ferromagnetism is not yet clear, but the authors suggested that strong dependence on the Y concentration indicates the collective correlation effects of the doping. The method combining arc melting and self-flux was used for preparation of single crystals of Pr1−x R x B4 (R = La, Ce, Gd). Source ingots were obtained by arc melting with excess amounts of RE metals to boron. Crystals were grown in a melt after one cycle of the heating-cooling process where the excess RE metals acted as a flux. The crystals were then separated from the melt by dissolving the excess RE metals in dilute hydrochloric acid. The size of the single crystals obtained by this method is up to 2 × 2 × 0.5 mm3 . The stoichiometry of the single crystals was examined using the (004) peak in XRD by applying Vegard’s law; and it was very close to the starting ratio of Pr : R. The XRD peaks are as narrow as those of PrB4 , so distribution of the RE elements in the crystals is homogeneous [218]. The zone-melting method with induction and optical heating began to be actively used for RB4 single-crystal growth after the 2000s, when a new round in the study of tetraborides started. Unfortunately, details of the growth process and characterization

56 Crystal Chemistry and Crystal Growth of Rare-Earth Borides

results of the grown crystals are rarely presented in publications. As a rule, it is only stated that the crystal is grown by the method of zone melting, sometimes even the method of heating (optical or induction) is not indicated. Here are several recent examples of such works on RB4 compounds: (i) optical heating: NdB4 [30, 31], GdB4 [219, 220], TbB4 [171, 221, 222], DyB4 [201], HoB4 [31, 172], ErB4 [15, 203], TmB4 [223, 224]; (ii) RF heating: DyB4 [25, 225], HoB4 [17, 226]; (iii) method of heating not specified explicitly: TbB4 [19], ErB4 and TmB4 [173]. In the following, we consider the works, where the details of single-crystal growth by the zone-melting method are presented for individual RB4 compounds.

1.4.2 Yttrium Tetraboride (YB4 ) Tanaka et al. [227] prepared an YB4 single crystal ∅8 mm × 50 mm for dHvA-effect measurements [228]. Mandatory requirements to crystals for measurements of quantum oscillations are high purity and minimal concentration of crystal defects to maximize the electron mean free path. This problem was solved by using highpurity initial components (Y2 O3 and B) for the synthesis of YB4 by borothermal reduction at 1600 ◦ C in vacuum and a triple zonepass refining of the feed rod that was pressed from the synthesized powder with the optimal B : Y ratio of 4.0. The residual resistance ratio (RRR) was 32 after one pass of the zone and could be raised to 54 after three zone passes. The higher RRR value indicates the reduction in the number of both impurities and defects. The FZ procedure was carried out in He atmosphere with the feed rod driving downwards through a work coil that was used for RF induction heating. The zone temperature was kept at 2610 ± 10 ◦ C during each zone-refining process. The growth speed of 5 mm/h was most suitable for the growth process. The preferential growth direction of the YB4 single crystal is [001]. However, different crystal-growth parameters are reported in Ref. [229], where a YB4 single crystal was grown by the FZ method with RF heating: optimal feed rod composition B : Y = 4.76, the crystal grew within ±20◦ from the [110] axis with a growth rate of 7 mm/h. The source YB4 powder was commercial one, so it

Rare-Earth Tetraborides (RB4 ) 57

is possible that it was more contaminated by impurities than the synthesized YB4 powder in Ref. [228]. The conclusion one can draw from the comparison of the two works [228, 229] is that the crystal quality of RB4 is determined by a set of technological parameters such as the feed rod composition first of all, as well as growth rate, feed/crystal rotation rate, and inert gas pressure.

1.4.3 Cerium Tetraboride (CeB4 ) Otani et al. [187] grew a CeB4 single crystal by the RF-heated FZ method. The raw materials used for the crystal growth were commercial powders of cerium oxide (CeO2 ), boron (B), carbon (C), and cerium hexaboride (CeB6 ). These powders were mixed in a mortar based on the reaction 12 (CeO2 + 2C) + CeB6 → 3 CeB4 + CO↑, and 14% excess (CeO2 + 2C) was added to obtain 2 single-phase CeB4 . This mixture was placed in a rubber bag and isostatically pressed at 200 MPa. The pressed compact rod was then heated in a boron nitride (BN) crucible in vacuum at 1600 ◦ C to synthesize CeB4 and prepare the feed rod at the same time.

Figure 1.29 Phase diagram of the Ce – B system. After Massalski et al. [156]. Reproduced from Otani et al. [187].

58 Crystal Chemistry and Crystal Growth of Rare-Earth Borides

According to the Ce – B phase diagram (Fig. 1.29), CeB4 melts incongruently at 2380 ◦ C. In order to grow a CeB4 single crystal, it is necessary to use the TSFZ technique. The feed rod contained 10% excess cerium that provided the B : Ce ratio in the molten zone of less than 2.85, and the growth temperature was just below the decomposition temperature (2380 ◦ C). CeB4 single crystals were grown in 0.4 MPa of argon ambient gas, the growth rate was 0.3 cm/h, and the lower shaft with the growing crystal was rotated at a rate of 6 rpm. The initial 1–1.5 cm of the crystal rod consisted of the CeB6 phase. However, the latter part consisted of the CeB4 phase with the stoichiometric composition. The resulting crystal was silver-grey in color. The peripheral part included grain boundaries, but the central portion consisted of high-quality CeB4 crystals without any splitting of the Laue spots.

1.4.4 RB4 (R = Y, Nd, Gd–Tm, Lu) Very recently, Brunt et al. [185] published the results of RB4 singlecrystal growth at the Department of Physics of the University of Warwick (UK). They used source polycrystalline substances in powder form that were prepared by arc melting. Ingots of RE metal (R = Nd, Gd–Er, and Y, all 99.9% purity, from Alfa Aesar or Aldrich), and powder boron, either natural B (95–97% purity, from Aldrich) or isotopically enriched 11B (99.52 at. % enrichment, from Eagle Picher), were used. The starting materials were melted together according to the reaction equation: R + 4B → RB4 . A 5% excess of boron was added to account for losses during the melting. The obtained arc-melted ingots (total of 6–8 g) were then partially crushed and arc-melted together in a cylindrical mold to form feed rods (typically ∅5–7 mm × 35–45 mm) for the crystal growth experiments. Polycrystalline TmB4 was prepared by the solid-state borothermal reduction of Tm2 O3 oxide: Tm2 O3 +11B → 2TmB4 + 3BO↑ in a flow of argon gas in the temperature range 1400–1500 ◦ C. The powders of the starting oxide material, Tm2 O3 (99.99% purity, from Alfa Aesar), and boron (natural or 11B) were used. The prepared powder was then isostatically pressed into rods (typically

Rare-Earth Tetraborides (RB4 ) 59

Table 1.8 Summary of the conditions used for the growth of RB4 (with R = Nd, Gd–Tm, and Y) crystal boules [185] RB4 NdB4 GdB4 TbB4 DyB4 HoB4 ErB4 TmB4 YB4

Growth rate Pressure Feed/seed rod (mm/h) Atmosphere (bar) rotation rate (rpm) 15–18 18 12–15 18 5–18 10–15 15–18 20

Ar Ar Ar Ar Ar Ar Ar Ar

2–6 4 4 2 3–5 2–3 5–6 6

15–25 15–20 15–25 20–25 15–20 15–25 15–25 20–25

∅5–7 mm × 40–50 mm) and sintered at 1500 ◦ C in a flow of argon gas for several hours. The sintered rods were then used for crystal growth. Crystals of all RE tetraborides were grown by the FZ method using a four-mirror xenon arc lamp (3 kW) optical image furnace CSI FZ-T-12000-X VI-VP, Crystal Systems Inc., Yamanashi, Japan. The technological parameters of the FZ process (growth rate, rate of feed and seed rotation, Ar pressure in the chamber) for all the grown crystals are presented in Table 1.8. Higher gas pressures were used to minimize the loss of elements with high vapor pressure. Initially, polycrystalline rods were used as seeds, and once good quality crystals were obtained, a crystal seed was used for subsequent growths. When RB4 (with R = Nd, Ho, Er, and Tm) crystals were required for neutron scattering experiments, isotopically enriched boron, 11B, was used for the preparation of the polycrystalline material in order to reduce the neutron absorption by the 10B isotope. The prepared crystal boules were typically ∅2–6 mm × 30– 50 mm in size. Phase analysis was carried out by powder XRD measurements after grinding small pieces of the crystals. Extensive x-ray Laue analysis of the as-grown boules of RB4 usually revealed the presence of 2–3 grains, extending along the length of each of the boules. The grain boundaries were not visible by eye; however, these grains can be isolated to give single crystals typically 5 mm in length and diameter. According to Laue patterns, the authors did not

60 Crystal Chemistry and Crystal Growth of Rare-Earth Borides

observe a consistent growth direction for any of the RE tetraborides crystal boules prepared. NdB4 is the only compound from all the grown RE tetraborides that melts peritectically into NdB6 and a Nd-rich liquid. However, authors grew boules where the core represented a NdB4 single crystal, and the surface layer was NdB6 . The RRR, ρ300 K /ρ2 K , exceeded 100, confirming the low density of defects in the crystal. There is no coating of a foreign phase on the exterior of the GdB4 boule, but x-ray phase analysis indicates that besides the main phase of GdB4 , there is GdB6 present as an impurity, as well as traces of Gd2 B5 . According to the Laue pattern, the crystal is oriented along [100]. DyB4 , HoB4 , TmB4 and YB4 also demonstrate good Laue patterns with the [001] orientation, whereas ErB4 and TbB4 crystals have [110] orientation [185]. The RRR of HoB4 is larger than 20. It is surprising to find impurity phases in the XRD patterns of the DyB4 , HoB4 , GdB4 and YB4 boules, which melt congruently and have demonstrated good Laue patterns [185]. Probably, Laue patterns and diffraction analysis were obtained from different parts of the boules. Precipitation of the second phase during the growth process can be induced by an instability of the technological parameters or a nonconstant composition of the melting zone. Detection of Tm2 O3 may be connected with the oxidation of the TmB4 crystal surface that is covered by a thin layer of thulium metal due to preferential vaporization of boron [185]. In this case, the surface layer can be removed by etching in dilute hydrochloric acid. Let us now consider the growth of RB4 (R = Y, Gd–Tm, Lu) high-quality single crystals, which were used for measurements of the dHvA effect, magnetic, thermal and transport properties, nuclear magnetic resonance (NMR), and Raman scattering [18, 23, 29, 143, 170, 230–232]. They were grown in the laboratory of RE refractory compounds, Institute for Problems of Materials Science of NAS (Kyiv, Ukraine) by crucible-free inductive FZ melting under Ar or He pressure in the specially designed “Crystal-111A” apparatus (Fig. 1.4). Preparation of RB4 (R = Y, Gd–Tm, Lu) in single-crystalline form has the following common features: (i) the synthesis of tetraborides by borothermal reduction of a metal oxide in vacuum

Rare-Earth Tetraborides (RB4 ) 61

(a)

(b)

Figure 1.30 Single crystal of TmB4 : (a) photograph of an as-grown crystal; (b) x-ray Laue pattern from the cross section, deviation from [001] is about 2.5◦ .

at 1900 K; (ii) compaction of the obtained powders in the form of ∅8 mm × 60 mm rods, their sintering at 2000 K in vacuum and the use for subsequent growth of the corresponding single crystals by zone melting. Volatile impurities in the starting boron powder are removed during the synthesis of tetraborides and in the process of zone melting in contrast to RE impurities, as the zone refining process is not effective in reducing RE impurities [114]. Therefore, the purity of RE oxides by RE impurities is a very important factor, which can influence the low-temperature properties of the resulting RB4 crystals. The RB4 single crystals were obtained with amorphous natural boron (nat B: 20% 10B, 80% 11B, AVIABOR, Dzerzhinsk, Russia, 99.9 wt. % purity) or polycrystalline 11B (typical enrichment 99.5%, Ceradyne, USA) and metal oxides from the Federal State Research and Development Institute of Rare Metal Industry (Moscow, Russia): Y2 O3 (99.994%), Gd2 O3 (99.9995%), Tb4 O7 (99.996%), Dy2 O3 (99.979%), Ho2 O3 (99.9995%), Er2 O3 (99.988%), Tm2 O3 (99.986%), and Lu2 O3 (99.9985%). The chemical purity given in brackets corresponds to the proportion of the main substance (no less than) in wt. %, according to the product certification. In fact, the actual content of RE impurities in the corresponding oxides was in some cases lower than suggested by the given purity level, according to optical emission spectral analysis. Additional purification during zone melting determined the final high purity of the grown crystals

62 Crystal Chemistry and Crystal Growth of Rare-Earth Borides

such that the impurity content was less than 10−3 wt. % (besides RE), the RE impurities were determined by the purity of source oxides. Optimization of the growth parameters allowed us to prepare single crystals of all the above-mentioned compounds with the size of ∅5–6 mm × 40 mm. The technological parameters were: (i) growth rate 9–14 mm/h; (ii) feed rod rotation rate of 10–15 rpm; (iii) inert gas pressure of 1–5 bar; (iv) the seed was fixed in the lower shaft. RB4 are highly anisotropic substances (c/a ≈ 0.57), which results in a strained state of the crystals due to crystallization conditions far from the equilibrium. So sometimes the crystals burst on the (001) cleavage plane already in the growth chamber under cooling. In order to release the stress, at the end of melting the crystallization rate was kept constant, while the feed rate of the sintered rod to the melt zone was gradually decreased. Thus a ∅5– 6 mm crystal was thinned to a fraction of a millimeter on the length of ∼ 10 mm. A typical as-grown TmB4 single crystal is presented in Fig. 1.30 with its Laue backscattering pattern. The absence of splitting of the point reflections confirmed the lack of domains with a misorientation more than several tenths of a degree (procedure accuracy). An xray diffraction pattern from the crushed crystal revealed reflections of the ThB4 structure type only. The technological parameters were as follows: The source composition was stoichiometric, growth rate was 14 mm/h, Ar pressure about 1 bar, feed rod rotation rate 15 rpm. The [001] direction was the primary spontaneous growth direction as for all RB4 single crystals. The RRR of this TmB4 crystal was more than 100. When large samples with another orientation were needed, oriented seeds were specially grown. As a rule, the orientation was checked by the matching Laue backscattering patterns on both ends of the large crystal to make sure that the whole rod is a single crystal.

Rare-Earth Hexaborides (RB6 ) 63

1.5 Rare-Earth Hexaborides (RB6 ) 1.5.1 General Overview Rare-earth hexaborides (RB6 , R = Y, La–Ho, Yb) are the most studied compounds among the RE boride phases due to the wealth of their properties and the simple crystal structure. The RB6 compounds crystallize in the simple cubic CaB6 -type structure (space group P m3m − Oh1 , No. 221), Z = 1, atomic positions: R in a(0, 0, 0) and 6B in f (x, 1/2, 1/2) with x ≈ 0.2 [112]. The central metal atom is surrounded by eight B6 octahedra situated at the vertices of the cubic lattice. As a result, the RE atoms are embedded in oversized cages of 24 boron atoms in the form of B24 truncated cubes (Fig. 1.31). Each boron atom in the CaB6 -type structure is bonded with four metal atoms and five boron atoms, four of which belong to the given octahedron and the fifth one to its neighbor. So there are two ˚ and types of B–B bonds: the intraoctahedral (B–B)intra (∼1.76 A) ˚ interoctahedral (B–B)inter (∼1.66 A), the former about 6% longer than the latter [34]. This means that the rigid boron sublattice is formed by somewhat “softer” B6 octahedra, which are responsible

Figure 1.31 (a) Crystal structure of RB6 . (b) The B24 polyhedron centered at the R metal. Reproduced from Zhukova et al. [264, 265].

64 Crystal Chemistry and Crystal Growth of Rare-Earth Borides

for a slight (∼4%) change in the lattice parameters through a series of RE hexaborides whose cationic radii change by 25% (Table 1.9). The boundary values of the radii of the metal sites in the CaB6 type structure, which are calculated from the average B–R distances, ˚ are equal to assuming the atomic radius of boron is rB = 0.88 A, ˚ the R metal radii are 2.23–1.75 A, ˚ and the radii of R 3+ 2.26–2.14 A, metal ions are 1.38–0.88 A˚ [266]. So in the large cuboctahedral cavities formed between the octahedra in the boron sublattice, the weakly bonded RE ions oscillate nearly independently, leading to the quasilocal vibrational “rattling” modes seen in experiments (see Chapters 4–6 for details). The existence of RB6 with RE ions with the radius smaller than 0.88 A˚ (R = Er, Tm, Lu) is possible only under stabilization of this crystal structure by alkaline earth (AE) or RE atoms of large size, for instance in Ca0.2 Er0.8 B6 [119], Y0.25 Tm0.75 B6 , Yb0.8 Tm0.2 B6 [267], and (Tm, Yb)B6 , (Tm, La)B6 [268]. According to the crystal-chemistry analysis of the hypothetical ScB6 lattice, it was suggested that the ScB6 phase is absent in the Sc – 3+ = B system (Fig. 1.9) not because of the small size of the Sc ion (rSc ˚ but because of the strong interaction of B and Sc atoms due 0.75 A) to the large polarizing ability of scandium that leads to an excessive expansion of the B6 octahedron and the loss of its stability [269]. This is consistent with the idea of a high polarizing ability of the RE ions relative to ligands [270] and elucidates the experimentally observed fact of the existence of two types of B–B distances in the lattices of CaB6 and UB12 structure types [34, 35]. Mackinnon et al. [271] calculated the band structure and density of states in hypothetical ScB6 , which turned out to be similar in terms of the electron dispersions and gap characteristics to those of YB6 and LaB6 . Based on these calculations, the authors hypothesized that superconductivity may be realized in ScB6 if this compound can be synthesized, which is apparently impossible according to the current state of the art. A significant inaccuracy in Ref. [271] may originate from the choice of Sc3+ ionic radius as 0.87 A˚ from Shannon [272]. In reality, Shannon [272] presented two values of ˚ as he used two versions of ionic radii for Sc3+ : 0.87 and 0.745 A, 2− rion (O ) for his calculations: 140 pm (“effective” ionic radius) and 126 pm (“crystal” ionic radius). Regarding the latter, Shannon noted that “it is felt that crystal radii correspond more closely to the

Table 1.9

RB6

YB6

LaB6

CeB6

0.905

1.071

1.034

1.013

+3

+3

+3

+3

4.1021 4.1000 4.100 4.1001(5) 4.15553 4.1570 4.1569 4.1561(1) 4.149410(2) 4.1563(3) 4.1555(3) 4.13899 4.1407(1) 4.1408(1) 4.1396(4) 4.132(4) 4.1397(2) 4.13220 4.133 4.135 4.1327(1)

˚ Interatomic distances (A) R–B

(B–B)intra (B–B)inter 1.746

1.630

3.051(1)

1.766(2) 1.766

1.656(2) 1.659

3.049

1.763

1.657

Refined formula Sample specification, comments

LaB5.63(10)

LaB6 La0.8 B6 3.0439(4) 1.7511(7) 1.6644(21) 3.036(1) 1.762(4) 1.642(5) CeB5.77(7) 3.041 1.761 1.650 CeB5.67(3) CeB5.59(4)

single crystal (flux) single crystal (FZ) single crystal (FZ) single crystal (FZ) synthesized powder single crystal (FZ) single crystal (FZ) La11B6 , single crystal (Al flux) synthesized powder, synchrotron XRD synthesized powder synthesized powder synthesized powder single crystal (FZ) single crystal (FZ) single crystal (Al flux) single crystal (Al flux) single crystal (FZ) synthesized powder arc melting single crystal (FZ) single crystal (flux)

Ref. [71] [233] [234] [73] [235] [69] [233] [236] [237] [238] [238] [239] [240] [69] [241] [242] [243] [239] [244] [245] [246] (Contd.)

Rare-Earth Hexaborides (RB6 ) 65

PrB6

Cation Lattice radius Cation parameter, ˚ [56] charge ˚ (A) a (A)

Structural parameters for RB6 (R = Y, La–Nd, Sm–Ho, Yb)

RB6

NdB6

Cation radius ˚ [56] (A) 0.995

Lattice Cation parameter, ˚ charge a (A) +3

4.13299 4.1269(1) 4.1274 4.1259

(Continued)

˚ Interatomic distances (A) R–B

(B–B)intra

(B–B)inter

Refined formula

Sample specification, comments

synthesized powder single crystal (flux) arc melting single crystal (flux), neutron diffraction 4.126 1.753 1.646 synthesized powder 4.1266(1) single crystal (flux) SmB6 1.13 / 0.964 +2.59 4.13284 synthesized powder 4.1346(2) 1.743 1.668 single crystal (FZ) (Sm2+ / Sm3+ ) 4.1347(1) 3.032(1) 1.760(2) 1.639(3) Sm0.77(2) B6 single crystal (FZ) 152 4.13361(7) SmB5.76 Sm0.615 154 Sm0.385 11B6 , neutron diffraction 154 Sm11B6 , neutron diffraction 4.13319(3) SmB5.49 154 4.1332(2) SmB5.82 Sm11B6 single crystal (flux), XRD 4.1342(5) synthesized powder 154 Sm11B6 , single crystal (FZ) 4.1339(1) 4.1344(3) SmB6 , synthesized powder 4.1278(3) Sm0.8 B6 , synthesized powder 4.1334(2) SmB6 , synthesized powder EuB6 1.12 +2 4.18456 synthesized powder 4.1850 single crystal (Al flux) 4.1838 single crystal (Eu flux) 4.1849(1) 3.0783(3) 1.7596(12) 1.6965(17) EuB5.88(4) single crystal (Al flux) 3.0314(3) 1.7574(12) 1.6415(17) NdB5.88(6)

Ref. [239] [247] [248] [249] [250] [246] [235] [251] [69] [252] [68] [68] [253] [254] [238] [238] [255] [239] [256] [256] [247] (Contd.)

66 Crystal Chemistry and Crystal Growth of Rare-Earth Borides

Table 1.9

Table 1.9 GdB6 0.938

TbB6 0.923

+3

+3

+3

HoB6 0.894

+3

YbB6 1.04

+2

∗

4.09938 4.097 4.098 4.095 4.097

1.746

1.639

1.743

1.637

1.742

1.635

4.1474(5) YbB6.07 4.1479(1) 3.0495(3) 1.7525(12) 1.6695(17) YbB5.86(2) 4.1486 4.1469 YbB5.94(5.994) 4.1477 YbB5.88(5.96)

synthesized powder single crystal (FZ) single crystal (FZ) synthesized powder single crystal (flux) synthesized powder Tb11B6 powder, neutron diffraction single crystal (FZ) synthesized powder single crystal (flux) Tb11B6 single crystal (FZ), neutron diffraction synthesized powder single crystal (FZ) single crystal (FZ) single crystal (FZ) Ho11B6 single crystal (FZ), neutron diffraction single crystal (Al flux) single crystal (Al flux) single crystal (Al flux) single crystal (FZ) single crystal (FZ)

[239] [245] [257] [250] [258] [239] [259] [234] [250] [258] [260] [239] [234] [250] [234] [261] [194] [247] [262] [263] [263]

Because of the fluctuating valence, the ionic radius of Sm in SmB6 is not well defined. The reported values of 1.13 and 0.964 A˚ for Sm2+ and Sm3+ , respectively [56], are given instead. † The lattice parameters marked by a dagger were measured at low temperature.

Rare-Earth Hexaborides (RB6 ) 67

DyB6 0.908

4.10724 4.109 4.111 4.108 4.1056(1) 4.10274 4.090(2)† 4.105 4.102 4.0998(3) 4.103†

(Continued)

68 Crystal Chemistry and Crystal Growth of Rare-Earth Borides

physical size of ions in a solid”. So from his point of view, the correct ˚ A similar value for rion (Sc3+ ) of 0.73 A˚ Sc3+ ionic radius is 0.745 A. was also presented in Ref. [56]. Selected structural parameters for RB6 are given in Table 1.9. There, information on lattice parameters and interatomic distances is presented from all known single-crystal studies as well as from the data obtained from powder-diffraction studies using both x-rays and neutrons in the past two decades. Crystal-structure analyses are available for RB6 (R = Y, La–Nd, Sm–Gd) [273]; for LaB6 , CeB6 , and SmB6 [69]; LaB6 [69, 274]; MB6 (M = Ca, Ba, Sr, Y, La) [233]; CeB6 [69, 240, 241]; Nd11B6 [249]; Nd11B6 , EuB6 , and YbB6 [247]; SmB6 [68, 69, 251, 252]; and GdB6 [275]. Temperature dependences of RB6 lattice constants below 300 K were published for RB6 (R = La–Nd, Eu–Dy) [235, 239]; EuB6 and GdB6 [276]; LaB6 , CeB6 , and SmB6 [277]; SmB6 [68, 253, 278]; Nd11B6 [249]; CeB6 [240]; DyB6 [279]; GdB6 [275], and at temperatures above 300 K in Refs. [280, 281]. The Gmelin Handbook [112] contains information on the hexaboride lattice parameters for single crystals or their powders and for polycrystalline samples prepared by borothermal reduction including values for the boundary compositions from their homogeneity ranges, which have been obtained before 1990. The presence of the homogeneity region may be one of the reasons for variations in the structural parameters for the same compound. For example, the ˚ but lattice parameter of EuB6 grown in Al flux is a0 = 4.1850(3) A, in Eu flux a0 = 4.1838(3) A˚ [256]. The authors cite the results of the detailed study of the homogeneity region of thorium hexaboride— it has been shown that this compound can accommodate a very high concentration of thorium vacancies, which can even exceed that of the occupied thorium sites [282]. Therefore, they conclude that the Al-grown material is Eu-deficient (approximately Eu0.9 B6 ) as opposed to the stoichiometric EuB6 single crystals grown in Eu self-flux [256]. The problem of homogeneity ranges, the kind and density of defects in the lattice, their influence on the lattice stability, and the nature of the crystal-structure distortion at low temperatures are key questions in the analysis of hexaborides’ physical properties. Spear [34] discussed the possible relationship between the stability

Rare-Earth Hexaborides (RB6 ) 69

of a compound and the nature of its decomposition into neighboring binary compounds depending on the size of the metal atom. He showed that the end members of the series are the most unstable, therefore there is no homogeneity region for them, while the compounds with metals of average size for this series have the greatest homogeneity region for each structure type. The largest number of works devoted to these problems was performed in the 1960s –’70s. The review of Binder [283] summarizes the information on the hexaborides of AE, RE, and actinides that was obtained up to 1977. The main conclusions are that only the metal sublattice has vacancies, and vacancy concentrations up to 30% for the trivalent metals and up to 10% for divalent metals do not lead to the destruction of the lattice. This idea of large homogeneity ranges in the RE hexaborides was realized in their phase diagrams, constructed mainly on the basis of the data obtained during these decades. According to them, the RE hexaborides are stoichiometric at their metal-rich phase boundary and remain stable up to approximately 30% of metal vacancies, and these homogeneity ranges are temperature dependent [34, 65, 113, 156, 167, 168, 284–287]. Only for the La – B and Nd – B systems, the homogeneity ranges of the hexaboride phase have been studied in detail [65,248,288–290]. The homogeneity region for LaB6 is shown in Fig. 1.32 [65–67]. The works of recent decades have called into question the unambiguity of some conclusions and, first of all, the presence of large homogeneity ranges in the hexaborides and the defects of only one metal sublattice. The Smx B6 range of homogeneity has been narrowed down from Sm0.68 B6 – SmB6 [291] to Sm0.8 B6 – SmB6 [292]. The latter results were received with powder samples, synthesized by borothermal reduction of high-purity Sm2 O3 oxide for 0.5 ≤ x ≤ 1.5 and subjected to prolonged annealing to obtain homogeneous compositions. According to the Rietveld analysis, all compositions with x < 0.8 included β-boron, were deficient in samarium, and had excess “effective” thermal parameters for boron atoms relatively to the stoichiometric phase. On the other hand, Smx B6 samples with x > 1.0 were deficient in boron, had shortened “effective” interoctahedral and enlarged intraoctahedral B–B bond lengths, and included Sm2 B5 as the main impurity. In the range of

70 Crystal Chemistry and Crystal Growth of Rare-Earth Borides

Figure 1.32 Part of the La – B phase diagram [65]: (a) the homogeneity region of LaB6 ; (b) enlarged presentation of the high-temperature part of the LaB6 homogeneity region, showing the supercooling Tn . After Korsukova et al. [66].

0.8 ≤ x ≤ 1.0, only individual phases of Smx B6 are observed with a lattice parameter that changes monotonically between 4.128 and 4.134 A˚ [292], in agreement with Refs. [238, 255]. It should be emphasized that most of the pioneering works were performed with synthesized samples from starting materials of insufficient purity that could lead to stabilization of a phase that does not exist under normal conditions. For example, ErB6 stabilized by calcium ions [119]. The previously declared significant homogeneity region of europium hexaboride (at least Eu0.9 B6 – EuB6 ) with an increase in the lattice parameter upon increasing boron content [293] was later reduced to less than 1% with a constant lattice parameter [262]. Contamination of the samples with carbon from the starting materials (boron, boron carbide) explains the discrepancy with previous studies as it led to the formation of EuB6−x Cx borocarbides (0 < x ≤ 0.20) with the lattice parameters

Rare-Earth Hexaborides (RB6 ) 71

that decrease with increasing x. Similar conclusions were reached regarding SrB6 and YbB6 [262, 294]. The technological procedures of the boride sample preparation have an important effect on their composition and structural quality. The most common methods for producing RE hexaborides are: (i) the solid-state borothermal reduction of the corresponding RE oxide in vacuum, (ii) single-crystal growth from molten metal fluxes, and (iii) single-crystal growth from its own melt. Solid-state synthesis and the growth of single crystals from molten metal fluxes can be considered as equilibrium processes, while growth from the melt is a dynamic process due to the high growth rate, which can lead to a redistribution of components, internal stresses, and the formation of structural defects. Besides, these three processes occur in different temperature ranges. Let us consider the example of lanthanum hexaboride, which is the standard reference material for other RB6 when investigating their properties and, moreover, is one of the most studied. For LaB6 , the solid-state synthesis, as well as single-crystal growth from the flux and from the own melt take place at 1600–1700 ◦ C, 1200–1500 ◦ C, and ≥ 2700 ◦ C, respectively. The fragment of the phase diagram related to the LaB6 range of stability is based on the results of earlier studies [65, 288, 289] and is considered in detail in Refs. [66, 67] (Fig. 1.32). The LaB6 homogeneity range is temperature dependent, and the stoichiometric composition is outside the homogeneity range shown in Fig. 1.32(a) [65–67]. From about 2073 K, the range broadens significantly until 2325 K (the temperature of the LaB6 –B eutectic) and then gradually narrows to the melting point of LaB6 (singular point Tm ), which is shifted by about 0.2 boron atoms per La from the stoichiometric composition LaB6 , i.e., at Tm the real composition is LaB6.2 . Depending on the supercooling temperature ( Tn ) and the deviation of the initial melt composition at the beginning of crystallization in one or another direction from the point a, the grown crystal composition will correspond to points at the boundaries of the homogeneity range along the lines a–b or a–c [Fig. 1.32(b)]. Since Tn can vary from tenths of a degree to hundreds of degrees and is difficult to control (especially at high temperatures), the composition of the crystallizing substance will also change in an uncontrolled way. In fact, the composition of LaB6

72 Crystal Chemistry and Crystal Growth of Rare-Earth Borides

single crystals grown from the melt may be stoichiometric [295,296] or contain defects of boron [69, 297–299] and lanthanum [289, 300], depending on the initial composition, presence of impurities, number of zone passes, and other technological parameters. According to the phase diagram, LaB6 single crystals with an approximately stoichiometric composition can grow only below 1673– 1773 K where the left-hand boundary of the homogeneity range almost coincides with the stoichiometric composition [Fig. 1.32(a)]. Molten metal flux is the only method that allows growing LaB6 single crystals at these temperatures. LaB6 single crystals have been grown in Al solvent both from La and B and from a corresponding mixture of La2 O3 and B in the temperature range of 1473–1773 K. Comprehensive information on their preparation by the flux method is presented in reviews [66, 87] and in the Gmelin Handbook [112]. However, a detailed x-ray diffraction study of three LaB6 single crystals, which were grown in an aluminum melt containing excess boron (B : La = 9 : 1) or excess lanthanum (B : La = 5 : 1), located vacancies only at the boron positions [241, 274]. The crystals from the source charges with the B : La = 9 : 1 ratio were heated in Al flux at 1400 ◦ C (crystal 1) and 1800 ◦ C (crystal 3), and from the charge with B : La = 5 : 1 (crystal 2) at 1400 ◦ C in Ar atmosphere for 4 h and then slowly cooled down to RT. The starting materials were lanthanum rod, amorphous boron powder, and granular aluminum, all of 99 wt. % purity. The boron occupancy was refined as 96.4(7), 97.2(5), and 98.2(5)%, respectively, for the 1st, 2nd, and 3rd crystals, and their compositions were LaB5.78±0.13 , LaB5.83±0.09 , and LaB5.89±0.09 , respectively, as obtained from the crystal structure refinement [274]. It should be emphasized that the standard chemical analysis yielded results opposite to those of the x-ray analysis: These crystals contained metal vacancies. Olsen et al. [301] attempted to prepare the nonstoichiometric compositions from LaB4 to LaB8 by the Al flux method by varying the B : La2 O3 ratio in the starting mixtures, but only LaB6 single crystals were grown. Electron microprobe compositional measurements showed the average measured boron content of these crystals to be the same (85.3% with a standard deviation of 1%—no absolute standards were available to compare this value to the theoretical

Rare-Earth Hexaborides (RB6 ) 73

stoichiometric value of 85.7%). No measurable deviations from stoichiometry could be detected in samples grown from either the boron-richer or lanthanum-richer powder [301]. ¨ [290] has analyzed publications on the LaB6 hoLundstrom mogeneity range and concluded that it is very narrow, although it widens significantly above 2000 K. The data on the nature of vacancies depend on the analysis method (x-ray, NMR, microprobe, etc.), as the detection by XRD of small amounts of elemental boron in a mixture with boride is difficult because boron is a much poorer scatterer for x-rays and may be amorphous or only partially crystallized, extremely fine-grained, or even strained or heavily twinned. For example, in a synthesized sample with the composition La0.81 B6 according to the chemical analysis, XRD confirmed the presence of only a single phase, whereas a study by the transmission electron microscopy (TEM) also revealed the presence of defective crystals of β-rhombohedral boron [302]. According to Storms [248], the homogeneity range of NdB6 extends from NdB6.01 to NdB8.2 due to vacancies in the metal sublattice with a variation of the lattice parameter from 4.1269 to ˚ respectively. The experimental samples were obtained by 4.1280 A, joint arc melting of the corresponding elements. An x-ray diffraction study of a Nd11B6 single crystal grown by the flux method indicated vacancies in either the boron or metal sublattice depending on the refinement model, as the refined site occupancies of Nd11B6 correlated strongly with the extinction ˚ i.e., as that of model [247]. Its lattice parameter was 4.1269(1) A, a practically stoichiometric sample obtained by arc melting [248] ˚ [245]. or the NdB6.1 single crystal obtained by FZ melting (4.126 A) 11 It should be also noted that the Nd B6 crystal was contaminated with aluminum (0.22 wt. %). Unfortunately, Al is sometimes found as inclusions in some hexaborides [70]. Korsukova summarized the information on structural vacancies in RE hexaborides (R = La–Eu, Yb) [303, and references therein]. Except for a single publication on SmB6 grown by the FZ method [69] where a vacancy rate of up to 20% was revealed in the samarium sublattice, x-ray analysis of RB6 estimated boron vacancies in the amount of 1–9%. Similar results were obtained in single-crystal XRD studies on Ce1−x Lax B6 solid solutions

74 Crystal Chemistry and Crystal Growth of Rare-Earth Borides

(x = 0.0, 0.25, 0.50, 0.75) [241,304] grown from flux. The previous results with neutron refinement of the Ce0.75 La0.25 11 B6 crystal structure initially suggested 12% impurities on the metal sites [305] but were later identified as erroneous. To understand the reasons for disagreement in the metal- and boron-site occupation from various structure studies, x-ray and neutron diffraction investigations were carried out on the same isotopic samples of 152 Sm0.615 154 Sm0.385 11B6 , 154 Sm11B6 [68, 252], and Ce1−x Lax 11B6 [306]. X-ray and neutron diffraction methods are averaging; however, the scattering cross section of boron in comparison to RE atoms is much higher for neutrons than for x-rays. Therefore, neutron-diffraction methods are expected to be more informative than XRD when studying the crystal structure of RE hexaborides, which contain a combination of very light and very heavy atoms. The single crystals of 154 Sm11B6 , 152 Sm0.615 154 Sm0.385 11B6 , and Ce1−x Lax 11B6 (x = 1.0, 0.75, 0.5, 0.25, 1.0) were grown by the flux method using Al as a solvent. 152 Sm2 O3 and 154 Sm2 O3 of 99 wt. % purity and amorphous 11B of 95 wt. % purity served as the starting components for the growth of 154 Sm11B6 and 152 Sm0.615 154 Sm0.385 11 B6 [252]. The purities of the starting materials in the Ce1−x Lax 11B6 case were 99.7, 99.9 and 99.9 wt. % for La, Ce, and Al, respectively [306]. Amorphous boron was enriched with the 11 B isotope to 99.3 wt. %, while its purity was 95 wt. %. For powder neutron diffraction, the grown single crystals were ground. The profile refinement of high-resolution powder neutron diffraction of double-isotope samples 154 Sm11B6 in the temperature range 23 < T < 300 K gave vacancies on the boron site with an occupancy of 91.5(5)%, unlike 97(1)% for 154 Sm11B6 in the singlecrystal x-ray study in Ref. [68]. In the case of “zero-matrix”1 152 Sm0.615 154 Sm0.385 11B6 , the boron site occupancy was 96(1)% according to the x-ray and neutron study [252]. 1 Due

to opposite signs of the coherent neutron-scattering lengths for 152 Sm and the sample with the strict isotopic composition 152 Sm:154 Sm = 1.6 has zero average amplitude of coherent neutron scattering by the Sm sublattice (socalled isotopic “zero matrix”), which enables a more accurate investigation of subtle structural details in the 11B sublattice such as the concentration and distribution of boron vacancies [252].

154 Sm,

Rare-Earth Hexaborides (RB6 ) 75

The results of the Ce1−x Lax 11B6 study qualitatively demonstrated a boron-deficient structure (occupancy 93–96%) and the complete occupancy of the metal position. Unfortunately, the accurate quantitative determination of the boron occupancy is hindered by the strong correlations between this parameter and the temperature factors of boron, the temperature factors of the metal atoms, and the occupancy of the metal site as well as the mosaic block size; besides, even in neutron diffraction there is a considerable contribution from the metal atoms to the intensity of the reflections, which noticeably decreases the sensitivity of this method to variations in the structural parameters of the B atoms [252, 306]. The question of vacancies in RE hexaborides, especially their distribution and influence on physical properties, remains to a large extent open. Conflicting results may appear for the same compound depending on the preparation method, technological parameters, composition and purity of the initial components, etc. For example, GdB6 demonstrates two successive first-order AFM transitions with a simultaneous structural distortion to the AF I state at TN1 ≈ 15.5 K and AF II state below TN2 . The latter takes a value in the range TN2 ≈ 5–10 K depending on the sample [307, and references therein]. Attempts were made to demonstrate the impact of the singlecrystal growth method (flux growth, the immiscibility gap method,2 or the FZ melting) on the chemical composition, uniformity of the components’ distribution in the crystal bulk, and thermoelectric properties of the crystals by the example of individual hexaborides LaB6 , CeB6 , and their solid solutions (La1−x Cex )B6 (x = 0.01, 0.05, 0.1) in Refs. [67, 309, 310]. However, these results are difficult to compare even among themselves. All single crystals of RE hexaborides are significantly different in stoichiometry even for the same method of preparation. Cerium concentration occurs in the solid solutions when the solvent methods are used. For zone-melted crystals, the nominal and real compositions practically coincide. The analysis of presented results led to the following conclusions: (i) When using the flux method, it is possible to obtain single crystals from the same bath with a large spread in the ratio of components [309]; (ii) it is impossible to evaluate 2 A binary immiscible metal Al/Pb system is used as a solvent [308].

76 Crystal Chemistry and Crystal Growth of Rare-Earth Borides

the effect of technological parameters on the composition of the zone-melted crystals without detailed knowledge of the conditions of their preparation [310]; (iii) the large scatter in the values of La : B, Ce : B, and (La+Ce) : B ratios can be related to poor surface preparation of the crystal samples, as both energy-dispersive (EDX) and wavelength-dispersive (WDX) microprobe procedures rely on high-quality polished surfaces [67, 309]. Sato et al. [311] showed that the real compositions of the zone-melted single crystals of the Cex La1−x B6 (x = 0.03, 0.10, 0.25, 0.50, 0.75) solid solutions within the accuracy of the emission spectrochemical analysis coincided with nominal ones, and Vegard’s law is fulfilled. However, according to the electron-probe microanalysis, the gradient of Ce concentration was 1.4%/cm along the length for the Ce0.5 La0.5 B6 single-crystal rod [311]. Anisimov et al. [312] studied the resistivity and transverse magnetoresistance of LaB6 and dilute R0.01 La0.99 B6 systems. The R x La1−x B6 single crystals were grown by RF zone melting. The nominal compositions corresponded to x = 0.01 in all samples, but the real compositions estimated by microprobe analysis essentially differed from the source ones and over the RE series: 1.1, 1.0, 0.7, 0.23, 0.72, and 0.12 at. % for R = Ce, Pr, Nd, Eu, Gd, and Ho, respectively. A gradual decrease in the fraction of the substituting RE metal over the series is apparently caused by a decrease in its ionic radius and, respectively, a gradual decrease in the stability of the CaB6 -type lattice. This is most clearly seen in the example of Ho0.01 La0.99 B6 , because individual HoB6 is at the boundary of the structure-type stability, so the Ho content in the solid solution is considerably lower than in the nominal composition. A significant loss of Eu is associated with its high vapor pressure. The grown solid solutions were high-quality single crystals, and the RRR of LaB6 was equal to ρ300 K /ρ0 = 420. By far the most sensitive to any imperfections (defects, impurities, vacancies, isotopic composition, etc.) are the physical properties of SmB6 , EuB6 , and YbB6 , i.e., compounds with metals whose valence deviates from +3, and YB6 —the compound located at the boundary of the structure-type stability. A similar situation is likely in incongruently melting TbB6 , DyB6 , and HoB6 , but most of the experiments with single crystals of these compounds

Rare-Earth Hexaborides (RB6 ) 77

were carried out on the same samples [234] with just a few exceptions [260, 313]. Nevertheless, as an example, a significant ´ temperatures among single crystals spread of the reported Neel and polycrystalline samples of TbB6 (from 17.4 to 21.5 K) has been noted [314]. These problems will be discussed later in this chapter. First of all, let us make a retrospective journey into the history of the development of methodology for single-crystal growth of high-quality RE hexaborides. This work continues, as the creation of modern equipment for crystal growth, the development of new characterization methods for crystal analysis, and the study of crystals under extreme conditions such as low temperatures, high magnetic fields and pressures, stimulate the production of more perfect crystals and extend the understanding of the mechanisms responsible for their growth.

1.5.2 Synopsis of RB6 Crystal Growth The pioneering work of Lafferty [315] excited interest in RE borides and methods of their single-crystal growth. This work, devoted to the thermionic emission properties of the borides of AE and RE metals and thorium, initiated much research into the growth of high-purity single crystals of RB6 (R = La–Gd), studying the crystallographic dependence of their emission properties, surface composition, electronic structure responsible for the low work function, and creating models that explain these dependences, as well as determining the effect of the vacuum and gas medium on their emission properties. This effort was stimulated by the widespread use of RE hexaborides as thermionic and field cathodes in electronic devices (scanning electron microscopes, electron beam accelerators, plasma generators, etc.) because of the high brightness and longevity of such electron sources due to the low work function and high stability of hexaborides at high temperatures [316–320]. Several years ago, Trenary [321] published a comprehensive review summarizing the results of over 30 years of research in the surface science of RE hexaborides and their solid solutions, in particular their emission properties depending on the crystallographic surface orientation and the ambient medium (vacuum,

78 Crystal Chemistry and Crystal Growth of Rare-Earth Borides

gas) [321, and references therein]. These studies became possible due to the development and optimization of crystal-growth methods for preparing high-quality single crystals of LaB6 and other RE hexaborides, pursued along the two main directions: growth from a metal flux and using the zone melting technique. Below, the growth of RB6 single crystals and their solid solutions, the role of composition, crystal perfection, and purity in application to their emission properties will be discussed in detail. Olsen et al. [301] presented the results of growing single crystal of LaB6 , EuB6 , CeB6 , BaB6 , (La, Eu)B6 , (La, Y)B6 , (La, Ba)B6 , (La, Cs)B6 , (Eu, Y)B6 , and (Eu, Ba)B6 from Al flux in an alumina boat, using La2 O3 , Eu2 O3 , Y2 O3 , Ce2 O3 , CsNO3 , or BaCO3 and boron powders as starting materials. The crystal growth was carried out in Ar atmosphere at a typical temperature of 1500 ◦ C for 5 days, then the furnace was cooled to 600 ◦ C at a linear rate of 5–20 ◦ C/h, and then arbitrarily to RT. Rods, cubes, plates, and polyhedra about 1 mm or more in size were obtained, but no obvious correlations could be made between their morphology and temperature gradient or the linear cooling rate. However, each individual crystal usually had a single composition that was within 15 mol % of the initial powder composition. The attempt to grow nonstoichiometric compositions from LaB4 to LaB8 by varying the boron and La2 O3 ratio in the starting mixtures failed, only the LaB6 single crystals were grown. There was also no correlation between the composition of the starting charge and the composition of the grown hexaboride solid solutions, as the product crystals would tend to be richer in one metal than the starting powder. Some of the grown crystals were contaminated with both pure aluminum and aluminum oxide, some other crystals were free of these defects. Futamoto et al. [322, 323] grew single crystals of the individual hexaborides RB6 (R = La–Eu) and solid solutions Lax M1−x B6 (M = Sr, Ba, Ce, Pr, Sm, Dy) by the Al-flux method. The starting material for RB6 was a mixture of the rare-earth metal (R), boron powder (B), and aluminum (Al), with the R : B ratio chosen according to the stoichiometric composition of RB6 , the ratio of B to Al being about 1 wt. %. In the solid-solution case, the compositions of grown Lax M1−x B6 were determined by electron microprobe analysis using standard samples of the binary hexaborides of LaB6 and MB6 .

Rare-Earth Hexaborides (RB6 ) 79

The mixture was kept at a temperature between 1250 and 1300 ◦ C in Ar atmosphere for several hours to homogenize the solution, then it was cooled down at a rate of 70 ◦ C/h. After that, single crystals were obtained by dissolving the Al flux with dilute hydrochloric acid. The products consisted of bar-shaped crystals ∼0.2 mm in diameter and up to 5 mm in length, elongated along [001], needle-shaped crystals with a [110] orientation, 0.1 × 0.5 × 7 mm3 in size, and thin platelike {100} crystals 0.2 × 0.2 × 5 mm3 in size. These crystals have been used for measurements of their thermionic emission and hightemperature surface composition [324–326]. However, the preparation of cathodes for industrial applications from the crystals grown from the Al flux is unrealistic due to their small size, the impossibility to cut samples with the necessary orientation and dimension, and the presence of foreign phases from the flux, which directly impairs the crystal quality and emission properties. For these purposes, larger hexaboride single crystals are needed. At first, Lafferty [315] carried out a thermionic emission study of hexaborides with polycrystalline sintered samples. It was only after Johnson [76] demonstrated in 1963 that the method of induction heating can be used for FZ melting above 2000 ◦ C that the first works on growing single crystals of refractory hexaborides by the crucible-free FZ melting started [75, 327, 328]. Among the first experiments, LaB6 polycrystalline rods with large single-crystalline grains (about 10 mm in length) were grown by the vertical version of RF-heated zone melting in Ar flow with a rate of 3–4 L/min [327]. A preferred growth direction was not found, dislocation etch pits on the [100] cleavage plane contained 106 –107 dislocations per cm2 or more, but a very important result was obtained in these studies: Even using commercial LaB6 powder as a starting material, an essential refinement from impurities took place already after a single passage of the molten zone. Niemyski et al. [75, 328] then used both vertical and horizontal versions of the RF-heated FZ melting for the preparation of MB6 (M = Ca, Sr, Ba, La–Sm, Gd) polycrystalline rods with large areas of single crystals. For the ∅5–6 mm × 10 cm rods, the areas of single crystals amounted to ca. 1 cm of the rod length. The molten zone was passed smoothly several times along the rod in order to obtain

80 Crystal Chemistry and Crystal Growth of Rare-Earth Borides

a long, uniform, and sufficiently pure sample. The products were of high purity. Tanaka et al. [295] have grown high-purity LaB6 single crystals (about ∅7 mm × 60 mm) by RF-heated FZ melting under highpressure Ar (1.5 MPa) on the assumption that this will prevent the evaporation and dissociation of LaB6 . Two kinds of crystals were synthesized that received either one or three passes by the molten zone. The latter had higher purity than the former by at least one order of magnitude, and no impurities were detected by emission spectrography in the triply passed crystals despite the use of commercial LaB6 powder with the nominal purity of 99.9% as a starting material. The La : B ratio was held in the stoichiometric 1 : 6 ratio in every crystal. The grown crystals consisted of subgrains about 1 mm3 in size, and their crystal axes were slightly misoriented with respect to each other. The subgrains were not eliminated by slowing down the rate of crystal growth from 20 to 10 mm/h. The RRR, ρ300 K /ρ4.2 K , was about 20 for singly passed crystals and between 200 and 450 for triply passed crystals. The latter crystals were successfully used in dHvA measurements of LaB6 , which require high-purity and particularly perfect crystals [329]. The use of multiple FZ passes was applied to the growth of high-purity RE hexaborides (R = Y, La–Gd) taking into account the nature of each individual compound. In particular, the peritecticphase GdB6 single crystal has been grown at an ambient Ar gas pressure of 1.5 MPa by using a B-rich molten zone (TSFZ technique) with B : Gd = 88 : 12 and a crystallization rate vcr = 3.5 mm/h that is less than vcr (LaB6 ) = 20 mm/h [114, 330], in accordance with the principle that compounds that melt peritectically grow at lower rates than those with congruent melting. Single crystals of LaB6 , CeB6 , PrB6 , and NdB6 (about ∅8 mm × 30–40 mm) were grown by the FZ technique with three passes of the molten zone in an atmosphere of Ar at 3 MPa to prevent excessive evaporation. They had a single-crystalline core with subgrains at the periphery and were nearly stoichiometric with very low impurity concentrations; the RRR ρRT /ρ1.4 K was 740 in LaB6 , 220 in PrB6 , and 120 in NdB6 [114, 329, 331]. The grown single crystals were used for research of the structural, electronic, transport, dHvA, and emission properties [114, 273, 329, 331–335].

Rare-Earth Hexaborides (RB6 ) 81

Besides the RF heating, FZ melting with heating by an electric arc, CO2 laser, or electron beam was also used for the growth of hexaborides. The methods of single-crystal growth were reviewed by Davis et al. [336, and references therein]. He noted that of all the techniques, FZ melting was developed the most extensively. It is ideally suited to the growth of long, high-purity crystals, which may either be oriented as desired with a seed crystal during the zone pass or cut into samples of the desired orientation. By 1986 when the Davis’s review was published, RB6 crystals with the (100), (110), (111), (210), (211), (310), (346), and (321) orientations3 had been grown by the FZ seeding technique [336]. Using laser heating, filamentary LaB6 crystals with dimensions up to ∅1 mm × 150 mm have been grown [338, 339], while in the case of arc FZ melting, the grown single-crystal rods had the dimensions of ∅3.0 mm × 30 mm [296, 297, 336]. As the absorption of 10.6 μm CO2 laser radiation was low for LaB6 , the molten zone had to be surrounded with a spherical heat shield. Besides, deposition of evaporation products on to the heat shield and components of the focusing optics required the input power to be varied continually [338, 339]. So, the laser zone melting units were soon replaced by zone melting units with a more powerful heating source—xenon lamps [340]. Nowadays, zone melting units with the optical heating are the most widely used. Usually, as starting materials for zone melting, rods obtained from cold-pressed, vacuum-sintered source powders or hot-pressed rods were used. In the latter case, contamination by carbon takes place that greatly degrades the emission properties, but the FZ melting provides a high degree of purification from impurities including carbon as the major impurity. For example, at a starting carbon value of 7000 ppm by weight, after three passes of an arc FZ the carbon value was reduced to < 30 ppm [296, 297], while LaB6 work function e depends on the crystal plane and at 1600 K has the smallest values of 2.2 and 2.5 eV for the 210 and 310 planes, respectively. However, more practical tip axes are either 100 or 110, with e of 2.60 and 2.65 eV, respectively. The 100 tip features the highest brightness and a highly uniform beam but exhibits a rather large drift in current. The 110 tip shows lower brightness but a smaller drift in current than those of the 100 tip [337]. These tips are used as cathodes in various devices [321].

3 The

82 Crystal Chemistry and Crystal Growth of Rare-Earth Borides

for a double pass the corresponding values were 2160 and 40 ppm, respectively [336]. Even when the total concentration of the source impurities exceeded 3.2×104 ppm by weight, after two passes of the zone it was reduced to 100 ppm [336]. The zone refinement is also effective for oxygen removal; one pass of the zone is sufficient in this case [296, 297, 336]. High purity of the LaB6 single crystal ensures thermal emission stability [340]. Paderno et al. [298, 299, 341] studied the amount, character, and distribution of impurities in zone-melted LaB6 single crystals with RF heating, focusing on the conditions for obtaining high-purity, perfect single crystals and the effect of purity and composition on their thermoemission characteristics. Tanaka et al. [295] performed RF zone melting under high Ar pressure (1.5 MPa) to avoid dissociation and vaporization of LaB6 , but experiments with zone melting using laser heating showed that lower gas pressures cause no dissociation or intense vaporization of either boron or lanthanum [338]. Therefore, in Refs. [298, 299, 341], high-purity LaB6 single crystals were grown under normal inert gas pressure that promoted better refinement from volatile impurities (purity of the source commercial powder did not exceed 98.4 wt. %) in gaseous form from the molten zone, while the actual loss of material through evaporation was small. Coefficients of distribution for refractory transition metals were determined, which help to predict the degree of purification during zone melting [298]. After two zone passes, the concentration of the main impurities fell by 2–3 orders of magnitude and was the highest at the end of the crystallized rod; sometimes in this part inclusions of the secondary phase LaB4 were detected, and the B : La ratio reached 5.83 [299]. Inclusions of the secondary phase RB4 (R = La, Ce, Pr) may appear either due to an unstable process of zone melting or with a significant deviation of the B : R ratio from 6 : 1 in the zone-melted samples due to the preferential evaporation of boron from the overheated molten zone at high crystallization rates (>50 mm/h) [299, 338, 339, 342]. Davis et al. [336] showed that this problem can be overcome by using starting material with a B : La ratio of 6.2—after no more than two zone passes, the final material had the optimum composition of B : La = 6.09 that produced the maximal ratio of the emitted electron current to the material

Rare-Earth Hexaborides (RB6 ) 83

vaporization in LaB6 cathodes. The same behavior was found in PrB6 , whereas zone-refined CeB6 samples had the final composition CeB6.2 after starting with CeB6.0 , which may indicate less boron evaporation (or more cerium evaporation) in comparison to LaB6 or that the congruent melting point corresponds to B : Ce < 6.0 [336]. It should be noted that in the studies considered above, the zone-melted boules were not truly single crystals, they consisted of several large grains with an angular misorientation ranging from several arcseconds to a few degrees. Additionally, bulk defects were revealed on cleaved and HNO3 -etched (100) surfaces of laser and arc zone-refined LaB6 crystals with a density from 5 × 105 to 107 cm−2 , seen as square pits with sharp corners [336, 338, 339]. In comparison, Al-flux-grown single crystals, apart from some Al and Al2 O3 inclusions, had lower etch pit densities of 103 –105 cm−2 in LaB6 [343, 344] and 104 cm−2 in EuB6 [344]. Special studies of the bulk defects in LaB6 single crystals grown with a xenon arc image furnace [340,345] and RF zone melting [346] were carried out. A high-power xenon arc image furnace was first used in the crystal growth of LaB6 by Aida et al. [340, 345]. Highpurity ∅2–5 mm single crystals of LaB6 were successfully prepared, but the problem was in determining the reasons for the formation of subgrains and the appearance of dislocations in the LaB6 single crystals, and in understanding the influence of the crystal diameter and the rate of its rotation on the generation of subgrains and the dislocation density during FZ growth. Identification of the subgrains and dislocations was mainly conducted by optical microscopy, rocking curves, and Auger spectroscopy. It was shown that subgrains were generated along the [100] growth axis; in the absence of crystal rotation the number of subboundaries was only a few, and their density increased with the rotation rate during crystal growth. They appeared both parallel and perpendicular to the [100] growth axis in the form of the matrix, in which the cross bars are parallel to the square bases of the pyramid-like etch pits. The subgrains themselves are cubes of 0.2–0.5 mm in length and result in a large divergence of the [100] crystal growth axis: The rocking curve for the sample with no subgrains had only one sharp peak; in contrast, the samples with subgrains had broader rocking curves with two or more peaks. The mosaicity reached 1◦ in the sample with the high subgrain density

84 Crystal Chemistry and Crystal Growth of Rare-Earth Borides

of 1000 cm−2 , and the subboundary contained a LaB4 -like second phase with a higher impurity level representative of carbon and oxygen. The latter could be introduced by the Ar gas. Dislocation density was closely related to the crystal diameter and varied from 106 cm−2 for the larger to 104 cm−2 for the smaller diameter. The generation of subgrains was interpreted as a cellular structure produced by the compositional supercooling mechanism and the steep convexity of the melt-solid interface. The steep convexity is a consequence of the temperature gradient that results in a plastic deformation, which is the main source of dislocations in the grown crystals. To suppress the generation of dislocations, it is desirable to grow single crystals with a small diameter and low convexity in the melt-solid interface. To prevent subgrain generation, it is desirable to grow a single crystal without crystal rotation, i.e., with a stationary molten zone [340, 345]. In Ref. [346], the real structure of LaB6 single crystals grown by the RF zone melting was investigated using TEM and diffraction methods. Two structural defects were found: flat segregations and dislocations. Most of the dislocations were arranged in {100} planes, which may be slide planes for a simple cubic structure, but the dislocation loop configuration showed that during single-crystal growth their movement is not characterized by slide but by diffusion creep. It was shown that the main source of dislocations arising during growth of single crystals is the generation by flat segregations, which may be the source of the whole dislocation structure. According to the diffraction investigations, these flat segregations have a different structure from both LaB6 and LaB4 [346], which contradicts the conclusion reached in Ref. [345] on possible LaB4 phase formation in subgrains boundaries. Unfortunately, Ref. [346] gives no details of the growth parameters for their RF zone melting. Therefore, it is impossible to relate the reported dislocation structures with technological parameters such as the rates of crystallization and rotation, gas pressure, etc. Much work has been done by Otani and coworkers from the National Institute for Research in Inorganic Materials (NIRIM, Tsukuba, Japan) on improving the technology of growing highquality RB6 single crystals by FZ melting and understanding the

Rare-Earth Hexaborides (RB6 ) 85

mechanisms responsible for their real crystal structure. First of all, a method for automated crystal preparation was developed for the RF-heated FZ technique. In this method, the heating power was computer-controlled based on the changes in impedance due to small changes in the molten zone shape to keep the molten zone stable and maintain a uniform diameter of the grown crystal rod, which is the main precondition for the growth of single-grain crystals. LaB6 single crystals of 6 cm in length and 0.8 cm in diameter were prepared with two zone passes, at a crystallization rate of 10 mm/h, with a rotation rate of the growing crystal (lower shaft) of 4 rpm and under 0.7 MPa of helium [347]. Besides the control of the zone shape, they used a much lower rate of crystallization (10 mm/h) in comparison to previous studies (≥50 mm/h) [296, 297, 336, 338, 339]. The grain-boundary length density was estimated at 340 cm/cm2 [348]. As the subgrain boundaries are caused by plastic deformation due to thermal stresses as a consequence of a steep temperature gradient at the molten-zone – solid-state interface [340, 345], in order to improve the LaB6 crystal, the TSFZ method was used with La [348, 349] and B [300] as solvents. High-quality LaB6 was grown by the TSFZ method with RF heating when the optimal composition of the molten zone was B : La ≤ 3.6 with the use of a La lump as the solvent; the optimal growth rate was 5 mm/h, the rate of crystal seed rotation was 8 rpm, and the growth temperature was 200 ◦ C below the melting point of the stoichiometric LaB6 (2715 ◦ C) [348, 349]. Similarly, high-quality LaB6 crystals were obtained using the TSFZ method with a molten zone containing an excess of boron as flux [300]. First, feed rods of several compositions were prepared from mixtures of commercial LaB6 and boron powder, and an additional disk of boron, 0.05–0.2 g, was placed on the seed crystal to form an initial molten zone with high boron content. The growth speed was 10 or 20 mm/h. Only the growing crystal (lower shaft) was rotated, at a speed of 6 rpm. The grown crystal was ∅0.9 cm × 4–6 cm in size. Figure 1.33 shows the growth temperature and grain-boundary length density as a function of the feed composition. Subgrain boundaries were not observed in the crystal when the growth temperature was decreased by 170 ◦ C from the initial 2715 ◦ C. Feed rods with atomic ratios of B : La higher

2700 400

2500 200

Growth temp. (°C)

Boundary density (cm/cm2)

86 Crystal Chemistry and Crystal Growth of Rare-Earth Borides

0 6

7

8

Comp. of feed rod (B/La) Figure 1.33 Dependence of the growth temperature and grain-boundary density on feed composition. Reproduced from Otani et al. [300].

than 7.2 were used, and the atomic ratio of the zone composition of B : La was 11. The composition of the prepared crystals was in a narrow nonstoichiometric La deficiency range of less than 1.1% of La vacancy and the atomic ratio of B : La was between 6.0 and 6.06 [300]. A comparative study of the emission properties of cathodes from LaB6 and CeB6 single crystals of stoichiometric composition and from nonstoichiometric boron-rich compositions LaB6.06 and CeB6.07 showed that increasing the boron content decreased the evaporation speed, which determines the lifetime of the cathodes, without influencing their emission properties. According to this study, the CeB6.07 cathodes had a two- to three-times longer expected lifetime than the commercial LaB6 and CeB6 cathodes [350]. The next step was growing of pseudobinary hexaborides La1−x R x B6 (R = Ce, Pr, Nd, Sm, Gd) in order to avoid subgrain boundaries, which damage the emission properties [351–353]. For each RE addition, the composition ranges of grain-boundary-free crystals were established: 0.3 ≤ x ≤ 0.6 for R = Ce (Fig. 1.34) [351]; 0.15 ≤ x ≤ 0.3 for R = Pr [352] and Nd [353]. The latter two studies were carried out only up to x = 0.3. The compositions of the feed rod and the resulting crystal coincide for La1−x Cex B6 , i.e., the distribution coefficient k = 1 [351], while in La1−x Prx B6 and La1−x Ndx B6 the real R concentration in the crystals is less than in

Boundary density (cm/cm2)

Rare-Earth Hexaborides (RB6 ) 87

1000 800 600 400 200 0

0.2

0.4

0.6

0.8

1

x in (La1–xCex)B6 Figure 1.34 Dependence of the grain-boundary density in La1−x Cex B6 on the Ce content x. Reproduced from Otani et al. [351].

the initial feed rods. For example, a crystal with the composition La0.82 Pr0.18 B6 was grown from the La0.79 Pr0.21 B6 feed rod [352], and a La0.85 Nd0.15 B6 crystal was grown from the La0.82 Nd0.18 B6 feed rod [353]. Such a redistribution of metal components can be associated both with a deviation of the distribution coefficient k from unity and with a different evaporation rate of the metal components from the molten zone. For La1−x Smx B6 and La1−x Gdx B6 , the grain boundary densities in the crystals decreased but did not disappear, because the molten zone lost stability whenever more than 15 at. % of SmB6 or more than 6 at. % of GdB6 was added, causing the melt to run off from the molten zone into the sintered feed rod, which then swelled up as a result [353]. All boundary-free crystals were obtained in the range where the ˚ i.e., 0.1% smaller than that lattice constant is smaller than 4.153 A, ˚ see Fig. 1.35 [353, 354]. If this rule applies to of LaB6 (a = 4.156 A), the substitution of all RE-elements for La in LaB6 , then La1−x Smx B6 and La1−x Gdx B6 boundary-free crystals should be obtained in the composition regions x > 0.19 and x > 0.09, respectively [353].

Lattice constant (nm)

88 Crystal Chemistry and Crystal Growth of Rare-Earth Borides

0.4155

CeB6 0.415

boundary-free

PrB6

NdB6 0.4145 0

0.1

0.2

0.3

0.4

0.5

x in (La1–xRex)B6 where Re = Ce, Pr and Nd

Figure 1.35 Change in the lattice constant upon substituting Ce, Pr, and Nd ) indicate the cases where crystals for La in LaB6 . The open symbols ( ) indicate the still contained grain boundaries. The solid symbols ( cases where the crystals were boundary free. Reproduced from Otani et al. [354].

The emission properties of La0.7 Ce0.3 B6 , La0.82 Pr0.18 B6 , and La0.85 Nd0.15 B6 boundary-free single crystals were shown to be no worse than those of the LaB6 cathode. So these cathodes can be used in a scanning electron microscope as well as the LaB6 ones. This is of commercial interest, because addition of the RE borides produces boundary-free crystals [355]. The idea of decreasing the grain-boundary density by means of solid-solution hardening with the introduction of the second metal component and/or reduction of the molten-zone temperature by using the TSFZ method was verified by comparing the crystal quality and high-temperature hardness of the LaB6 and CeB6 crystals with those of the La0.7 Ce0.3 B6 , La0.82 Pr0.18 B6 , and La0.85 Nd0.15 B6 solid solutions [356, 357]. Micro-Vickers hardness measurements were carried out between RT and 1300 ◦ C. The RE-substituted LaB6 crystals were confirmed to be boundary-free as a result of solidsolution hardening. In addition, the hardness measurement revealed

Rare-Earth Hexaborides (RB6 ) 89

that boundary-free LaB6 and CeB6 crystals can be prepared when the growth temperature is decreased to about 200 ◦ C below their melting points, which agreed well with the experimental results by the TSFZ method. Unfortunately, the attempt to reduce the etch pit density failed; it was estimated at 5 ×106 to 107 cm−2 [356]. The same group also modeled LaB6 FZ melting and LaB6 TSFZ melting in an excess of La as a solvent, taking into account two kinds of convection which determine the heat and mass transfer inside the molten zone and its morphological instability [358–361]. Based on these simulations, inferences were made about the set of operational parameters that would allow growing LaB6 single crystals without the appearance of defect structures associated with the morphological instability and increasing the supersaturation gradient. It has been shown that the excess of La leads to a strong decrease of (i) the melting temperature, (ii) the heating power necessary to form the zone of a given height, and (iii) the maximal temperature inside the melt [361]. More recent works in this direction focused on the single-crystal growth of individual RE hexaborides and their solid solutions in dependence on the operational parameters, thermal-emission studies according to the ratio of RE elements in RB6 solid solutions, and were accompanied by calculations of the electronic structure and work function from first principles. Most of these works, performed in the last decade, were pursued in Chinese laboratories using optical FZ furnaces equipped with xenon lamps [362–378]. A comprehensive characterization of the grown crystals by scanning electron microscopy (SEM), XRD, x-ray Laue diffraction, and inductively coupled plasma atomic emission spectroscopy (ICP-AES) allowed optimizing technological parameters such as the number of zone passes, crystal growth rate, rotation rates of feed rod and growing crystal; and preparing high-purity, high-quality single crystals of individual hexaborides without subgrains or cracks: LaB6 [362–365, 367], CeB6 [365–367], PrB6 [368, 369], SmB6 [370], GdB6 [367], and their solid solutions Ce1−x Lax B6 [371–373], Ce1−x Gdx B6 [374], Pr1−x−y Lax Ce y B6 [375], Ce0.4 La0.2 Pr0.2 Nd0.2 B6 [377], Cex La y Pr y Nd0.05 Gd0.05 B6 [378], and Ce1−x−y Lax Pr y B6 [376]. In the following, we consider a selection of

90 Crystal Chemistry and Crystal Growth of Rare-Earth Borides

Figure 1.36 Photographic image of as-grown LaB6 samples with different diameters prepared by the optical FZ melting technique at the crystal growth rate V = 8 mm/h. Reproduced from Xu et al. [363].

these works that reflects the key points responsible for the quality of single crystals. Most recently, Xu et al. [363] prepared LaB6 single crystals by FZ melting using two zone passes in order to obtain a highpurity crystal, and the corresponding crystal growth rates, V , were 20 mm/h and 8–12 mm/h during the first and second pass, respectively. Figure 1.36 shows the as-grown LaB6 crystals with diameters between 2 and 10.24 mm prepared after the second pass of the FZ melting growth with V = 8 mm/h. After improving the quality of the LaB6 seed and optimizing the crystal-growth process, LaB6 single crystals with essentially no angular deviation from the [100] growth direction could be achieved, which resulted in a significant improvement of the thermal-emission properties [363]. Since no obvious defects such as pores, cracks, or subgrain boundaries were found using microstructure analysis [Fig. 1.37(a)], the influence of the growth rate and the diameter of the crystals on its quality was checked by measuring the FWHM of the x-ray rocking curves, shown in Fig. 1.37(b–f), which is closely related to the dislocation density of as-grown crystals [363]. The smallest value of the FWHM in Fig. 1.37(d) belongs to the LaB6 single crystal obtained at V = 10 mm/h. The best values of the emission characteristics also correspond to this growth rate (see Table 1.10). The FWHM values for the LaB6 crystals with diameters of 2 and 10 mm are 0.063◦ and 0.115◦ , respectively. The corresponding emission characteristics, presented in Table 1.11, show an

Rare-Earth Hexaborides (RB6 ) 91

θ (deg.)

θ (deg.)

θ (deg.)

θ (deg.)

θ (deg.)

Figure 1.37 (a) Microstructure morphology of the LaB6 single crystal. (b–f) X-ray rocking curves for crystals grown at different rates: (b) V = 8 mm/h, (c) V = 9 mm/h, (d) V = 10 mm/h, (e) V = 11 mm/h, (f) V = 12 mm/h. Reproduced from Xu et al. [363].

improvement for the structurally more perfect crystals, i.e., the smaller is the diameter, the more perfect is the crystal and the better are the emission properties, which confirmed the results obtained more than 30 years ago in Refs. [340, 345]. It should be noted that the emission properties deteriorate with an increase in the number of FZ melting passes, which is associated with the predominant evaporation of boron and an increasing deviation of the B : La ratio from 6 in the direction of boron deficiency. Therefore, the authors concluded that two passes are generally enough to exclude the impurities, gas, and defects in as-grown single crystals but are still

T (K)

1673 1773 1873

V = 8 mm/h J0 e (A/cm2 ) (eV) 8.36 2.52 17.98 2.58 22.38 2.70

V = 9 mm/h J0 e (A/cm2 ) (eV) 9.09 2.51 19.61 2.56 24.20 2.69

V = 10 mm/h J0 e (A/cm2 ) (eV) 9.62 2.50 22.38 2.54 26.36 2.67

V = 11 mm/h J0 e (A/cm2 ) (eV) 8.92 2.52 18.94 2.57 22.86 2.70

V = 12 mm/h J0 e (A/cm2 ) (eV) 8.15 2.53 12.94 2.63 17.14 2.75

Table 1.11 The thermionic current density J1 kV , measured at U = 1 kV and T = 1873 K; the zero-field current density J0 ; and the effective work function e for different temperatures and diameters of LaB6 [100] single crystals grown with the optimal rate V = 10 mm/h (after Xu et al. [363]) T (K) 1673 1773 1873

D = 10 mm, V = 10 mm/h J 1 kV (A/cm2 ) J 0 (A/cm2 ) e (eV) 10.15 7.10 2.55 26.57 19.05 2.56 36.10 25.23 2.68

D = 2 mm, V = 10 mm/h J 1 kV (A/cm2 ) J 0 (A/cm2 ) e (eV) 14.32 9.77 2.50 37.8 24.55 2.52 46.80 28.18 2.66

92 Crystal Chemistry and Crystal Growth of Rare-Earth Borides

Table 1.10 The thermionic current density J1 kV , measured at U = 1 kV and T = 1873 K; zero-field current density J0 ; and the effective work function e for a LaB6 [100] single crystal with the diameter of 5 mm obtained with different growth rates V (after Xu et al. [363])

Rare-Earth Hexaborides (RB6 ) 93

favorable for keeping the B : La ratio close to stoichiometric, and a LaB6 single crystal with 2 mm diameter prepared at V = 10 mm/h had the highest current density of J1 kV = 46.8 A/cm2 at T = 1873 K [363]. The role of the feed-rod density has been carefully analyzed as a parameter influencing the quality and thermionic emission properties of PrB6 single crystals grown using an optical FZ furnace by Wang et al. [369]. The PrB6 feed rods were fabricated from powder using spark-plasma sintering at P = 50 MPa, at temperatures ranging from 1700 to 1900 ◦ C, for a time t = 5 min, and the relative density of the feed rods was increased from 80.46% to 97.17%. The crystal quality of as-grown single crystals was evaluated from the FWHM of a Bragg reflection that varied from 0.234◦ for the feed rod with 80.46% density up to 0.066◦ for the feed rod with 97.17% density; the corresponding maximal current density varied from 22.90 to 45.84 A/cm2 at 1873 K under the applied voltage of 4 kV. Moreover, the calculated work function of 2.68 eV is lower than the previously published values. Therefore, the authors suggest that the density of the feed rod plays an important role as a parameter in the single-crystal growth of RE hexaborides and their thermionic emission performance [369]. The RE-element substitution in individual hexaborides is another effective method to improve their emission and mechanical properties [355–357, 371]. This motivated much of the research in the growth of multicomponent hexaboride single crystals and the optimization of their properties. For instance, high-quality Lax Ce y Pr1−x−y B6 (x = 0.6–0.8, y = 0.1–0.3) and Ce1−x−y Lax Pr y B6 (x = 0.1–0.4, y = 0.1–0.4) single crystals have been grown by the optical FZ melting method [375, 376]. CeB6 , LaB6 , and PrB6 powders with a purity of 99.9% were used as starting materials for the preparation of feed rods with the corresponding ratio of Ce, La, and Pr elements. For growing La- and Ce-based quaternary RB6 single crystals, LaB6 [100] and CeB6 [100] crystals, respectively, were used as seeds. These were tied to the lower shaft, and the feed rods were in the upper shaft; the seed and the feed rod were counter-rotated at 15 rpm with respect to each other. A two-step FZ growth with growth rates of 20 and 7 mm/h for the first and second pass, respectively, was carried out. Subgrain-boundary-free

94 Crystal Chemistry and Crystal Growth of Rare-Earth Borides

Table 1.12 The FWHM of the x-ray Bragg peak recorded from the middle part of the as-grown Lax Ce y Pr1−x−y B6 and Ce1−x−y Lax Pr y B6 single crystals [375, 376] Lax Ce y Pr1−x−y B6

FWHM

Ce1−x−y Lax Pr y B6

FWHM

La0.8 Ce0.1 Pr0.1 B6 La0.6 Ce0.1 Pr0.3 B6 La0.6 Ce0.2 Pr0.2 B6 La0.6 Ce0.3 Pr0.1 B6

0.148◦ 0.118◦ 0.226◦ 0.045◦

Ce0.2 La0.4 Pr0.4 B6 Ce0.4 La0.4 Pr0.2 B6 Ce0.6 La0.2 Pr0.2 B6 Ce0.8 La0.1 Pr0.1 B6

0.054◦ 0.036◦ 0.107◦ 0.091◦

[100] Lax Ce y Pr1−x−y B6 and Ce1−x−y Lax Pr y B6 single crystals with dimensions ∅4.5–5 mm × 30–35 mm, good crystalline quality, and homogeneous element distribution were obtained. Results of the x-ray rocking curve analysis (FWHM) from the middle part of the as-grown single crystals are presented in Table 1.12. In the Labased series, La0.6 Ce0.3 Pr0.1 B6 demonstrates the best single-crystal quality, while Ce0.4 La0.4 Pr0.2 B6 is the best in the Ce-based series. The corresponding maximal emission-current densities at 1873 K under the applied voltage of 4 kV are 105.10 and 114.21 A/cm2 ; they are the highest in their series, consistent with the crystal quality [375, 376]. The obtained values of the emission-current densities significantly exceed similar characteristics for individual hexaborides, suggesting that multicomponent hexaborides are very promising as hot-cathode materials. Wang et al. [376] also presented information on the atomic compositions of the as-grown Ce1−x−y Lax Pr y B6 single crystals, obtained by the ICP-AES analysis, as shown in Table 1.13. Deviations from the given element ratio are insignificant, and the B : (Ce+La+Pr) ratio stays above 6.0. Most likely, the excess of boron in the stoichiometry is responsible for the high quality of these crystals. In contrast to the foregoing successful works on growing the multicomponent hexaboride single crystals by zone melting, attempts to grow the polyelemental RE hexaboride with the composition La0.5 Ce0.1 Pr0.1 Nd0.1 Sm0.1 Eu0.1 B6 with the miscibilitygap method failed [379]. The resulting crystals looked like dark-blue needles (up to 8 mm in length), elongated plates (up to 5 mm), or isometric crystals (up to 2 mm in size). Microprobe analysis with a WDX analyzer showed that the ratio of metal constituents

Rare-Earth Hexaborides (RB6 ) 95

Table 1.13 Atomic compositions of the as-grown Ce1−x−y Lax Pr y B6 single crystals obtained by the ICP-AES analysis [376] Atomic ratio Sample Ce0.2 La0.4 Pr0.4 B6 Ce0.4 La0.4 Pr0.2 B6 Ce0.6 La0.2 Pr0.2 B6 Ce0.8 La0.1 Pr0.1 B6

Ce : (Ce+La+Pr) La : (Ce+La+Pr) Pr : (Ce+La+Pr) B : (Ce+La+Pr) 0.205 0.397 0.602 0.802

0.404 0.410 0.208 0.106

0.391 0.193 0.190 0.092

6.062 6.056 6.051 6.058

in the grown crystals differs both from crystal to crystal and from the initial ratio of the metal constituents. The amount of Eu in the crystals was almost an order of magnitude smaller; that of La and Sm can change in either direction in relation to the original composition. In addition, the ratio of components varied along the length of the crystals. This result was predictable because RE elements were loaded to the charge in the form of individual components but not in the form of an ingot containing all metals. This is characteristic of the flux method—selective phase formation, which has been established in earlier works on the growth of single crystals of hexaboride solid solutions by the flux method [301, 322, 323]. At present, FZ melting with optical heating is the most common among zone-melting methods, although growth of single crystals of borides with RF heating is also ongoing. For the industrial production of cathodes, high-quality LaB6 and CeB6 single crystals are grown using a process called “Inert Gas Arc Float Zone Refining” [380, 381]. This process is also used to grow crystals for experimental purposes [382]. In this section, the works on the growth of single crystals of individual RE hexaborides and their solid solutions by the FZ melting method have been reviewed with an emphasis on industrial applications. The challenge was to develop a methodology for growing perfect hexaboride single crystals in order to use them as highly efficient emission materials. Apart from their applied aspect, these developments turned out to be highly beneficial in the preparation of high-quality samples for solving a number of fundamental problems in solid state physics, which imposed similarly strict requirements on the quality of single crystals. In

96 Crystal Chemistry and Crystal Growth of Rare-Earth Borides

particular, R1−x R x B6 solid solutions (where R and R  are two different RE or AE elements) were shown to exhibit exciting low-temperature electronic, structural, and magnetic properties including novel ordering phenomena (such as noncollinear magnetism and magnetically hidden-order phases [383]) that attracted considerable attention in the low-temperature physics community. Naturally, the basic results obtained in the aforementioned studies were used to grow hexaboride single crystals for these purposes. But in addition to the zone-melting method, the flux method has also been widely used and continues to be applied in those cases when there is no need for large crystals. For example, a LaB6 single crystal used in a dHvA study had the mass of only about 200 μg and a resistivity ratio of about 200; it was grown in Al flux from lanthanum and boron taken in the approximately 1 : 6 atomic ratio [384]. Single crystals of the individual hexaborides RB6 (R = Y, La– Nd, Sm–Ho, Yb) and quasibinary hexaborides R1−x R x B6 have been grown for the investigation of their low-temperature properties with state-of-the-art experimental methods such as x-ray, electron, and neutron spectroscopies, dHvA, and a variety of transport, thermal, and magnetic measurements, each with their own specific requirements on the size, structural quality, purity, and isotopic composition of single crystals. Unfortunately, in established groups specializing on a specific method of crystal growth for a long time, it is commonly accepted to refer to one of their earlier works, where the methodology was described, without giving important technical details concerning the growth of the specific boride. For example, the group of Prof. F. Iga would commonly refer to the same work on the YbB12 growth [109], regardless of whether they grow tetra-, hexa-, or dodecaborides, or which RE element is forming the boride. Undoubtedly, successful growth of the YbB12 single crystal is a victory, but the incongruently melting YbB12 with high vapor pressure of Yb has little in common with most other borides of RE elements except for the equipment used for zone melting with optical heating. Therefore, there can be no direct transfer of methodology optimized for growing YbB12 to other borides. In the following sections, we focus on the specifics of singlecrystal growth and physical characterization of SmB6 , YbB6 , and

Rare-Earth Hexaborides (RB6 ) 97

YB6 —compounds that attracted much recent attention and therefore deserve a more detailed discussion in this chapter.

1.5.3 Samarium Hexaboride (SmB6 ) Samarium hexaboride (SmB6 ) has been under close scientific scrutiny since the discovery of the mixed-valence effect [385]. About a decade ago, on the basis of model calculations of topological indices and the spectrum of surface states, it was predicted that SmB6 should be a strongly correlated topological insulator (TI) [386–389]. This concept, discussed in more detail in Chapters 6, 8, and 11, was drawn up to explain the nature of the SmB6 ground state, and the first experimental works were published soon thereafter [390–393]. As one of them puts it, “The best current working hypothesis is that SmB6 is indeed a topological insulator” [393]. In spite of the intense research on SmB6 after adopting the concept of a topological Kondo insulator (over 300 publications in the last decade), no consensus has yet been reached regarding the question of to what extent the topology of its band structure determines its anomalous surface conductance and other physical properties. Some authors keep prudently referring to SmB6 as a “proposed topological Kondo insulator SmB6 ” [394]. One of the difficulties in reaching a consensus on the interpretation of the experimental results is that even the same experimental techniques may sometimes lead to conflicting conclusions [395–400]. It appears likely that the method of single-crystal growth and the technological parameters used for the preparation of corresponding samples may have a subtle influence on the sample properties and be responsible for at least some of these inconsistencies. The established differences in the physical properties among SmB6 single crystals obtained by different methods are due to their unusually high sensitivity to lattice imperfections and the presence of impurities, which depend in turn on the preparation procedure and the purity of starting materials. SmB6 single crystals can be grown by the flux and FZ melting methods. Typical photographs of flux-grown (FG) and FZ-grown SmB6 single crystals are shown in Figs. 1.38(a, b), respectively [70].

98 Crystal Chemistry and Crystal Growth of Rare-Earth Borides

(a)

(b)

Growth Direction (GD)

Figure 1.38 (a) A flux-grown single crystal of SmB6 (left) and the same crystal mounted onto a kapton-covered vanadium post used in a neutrondiffraction measurement (right). (b) A single crystal prepared by the FZ method. The white arrow points to a scaled image of the flux-grown single crystal. Reproduced from Phelan et al. [70].

On the one hand, the FG SmB6 single crystals have a more perfect real structure because the conditions of their growth are close to equilibrium, but they may contain inclusions of the Al solvent or impurities; these crystals are also smaller in size, which is perhaps sufficient for diffraction experiments or physical-property measurements but not for some other experimental probes. On the other hand, the FZ-grown crystals are much larger, highly pure, but their real crystal structure is less perfect due to the nonequilibrium growth character, which can result in various kinds of defects— vacancies, dislocations, cracks, etc. Besides which, samarium has a high vapor pressure that facilitates its preferential evaporation from the molten zone and may lead to a deviation from the nominal stoichiometric crystal composition. To investigate differences originating from the growth technique, Phelan et al. [70] grew several SmB6 crystals by the FG and FZ methods, comprehensively characterized them, studied their properties under identical conditions, and analyzed the results. The theme of this work was continued in a series of the subsequent studies of the individual SmB6 and its solid solutions [401–407]. Let us consider the key results illustrating the problem of choosing the right growth technique. To optimize the SmB6 single-crystal growth procedure by the flux method, Kebede et al. [408] prepared 26 batches of SmB6 crystals by precipitation in molten aluminum. Sm (99.99%), B (99.5%), and Al (99.999%) were used as starting materials. The Sm : B atomic ratio was varied from 1 : 4 to 1 : 8, and the atomic ratio of Sm to Al was

Rare-Earth Hexaborides (RB6 ) 99

varied from 1 : 17 to 1 : 393. The charge consisting of Sm, B, and Al in the appropriate weighed ratios was placed in an alumina crucible and heated slowly to 1500 ◦ C in He atmosphere. The time taken for cooling the solution to the melting point of aluminum (670 ◦ C) also varied (up to 200 h). The crystals were extracted from the charge by dissolving the aluminum in a warm, concentrated NaOH solution. The quality criterion for the crystals was the insulating gap value obtained from the temperature dependence of the resistance. Its maximum value of 80 K was at the nominal ideal ratio Sm : B = 1 : 6, but it was clear that a huge variation in the Sm : B ratio was possible in the crystals. The best samples came from the batch with the highest aluminum ratio, i.e., very dilute SmB6 . Electron microscopy studies showed regions of SmB4 , twin domains, and occasional edge dislocations within the SmB6 crystals. Scanning electron microscopy also showed many types of defects at the surfaces. The best samples were relatively free of these but contained inclusions of aluminum, comblike edge notches, and tetrahedral pyramids with their bases parallel to the crystal edges [408]. After the TI concept had been adopted for the description of the insulating state of SmB6 , the inverse resistance ratio (IRR) started to be the main quality criterion. The methodology of the SmB6 singlecrystal growth from Al flux remained unchanged. That is, the crystal growth is performed in a vertical-gradient high-temperature tube furnace under continuous high-purity Ar or He flow (Fig. 1.2), the charge is soaked at the corresponding temperature and then slowly cooled down; the remaining Al flux is then dissolved out by a NaOH solution or hydrochloric acid. The soaking temperature and time are varied in different studies within 1250–1600 ◦ C and 2–350 h, respectively, the cooling rate is usually chosen between 2 and 5 ◦ C/h. The charge consists of Sm and Al pieces and a boron powder [6,7,70, 398, 409–413] or of SmB6 powder and Al pieces [410]. The purity of all source components (Sm, B, Al) is usually not lower than 99.99%. An infamous problem with the FG hexaboride single crystals is the presence of Al inclusions in the crystal bulk. These inclusions cocrystallize with the SmB6 host crystal, with the [100] Al axis nearly aligned with the [100] axis of SmB6 . Aluminum does not substitute into the hexaboride lattice, but larger crystals often enclose Al pockets, which can be mechanically removed by polishing

100 Crystal Chemistry and Crystal Growth of Rare-Earth Borides

or chemically etched from an exposed surface with hydrochloric acid. Aluminum also crystallizes in a cubic space group, F m3m (No. 225), ˚ which is only 2% smaller than with a lattice parameter a = 4.05 A, that of SmB6 [70, 412]. Phelan et al. [70] recently grew nat SmB6 and 154 Sm11B6 single crystals by the Al-flux method in a vertical tube furnace under flowing ultrahigh-purity (UHP) argon gas. The molar ratio of Sm : B : Al was approximately 0.005 : 0.03 : 3, the rate of heating up to 1450 ◦ C was 200 ◦ C/h, the soaking time at this temperature was 10 h, and the rate of cooling down to 1000 ◦ C was 5 ◦ C/h. At RT the Al flux was removed by a NaOH etch. The penetrating power of neutrons allows probing the entire sample bulk by neutron diffraction, but in the case of SmB6 this method requires doubly isotope-enriched 154 Sm11B6 single crystals. It has confirmed that metallic aluminum is present not only on the surface of the FG crystal, but also in its bulk. Figure 1.39(a, b) shows a neutron diffraction image in the (h0l) scattering plane and the corresponding powder-averaged intensity histogram. The asterisks in Fig. 1.39(b) denote the reflections from the secondary aluminum-metal phase. Its amount of ∼4 wt. % has been estimated from the XRD measured on a powder obtained by grinding five SmB6 crystals with natural boron [70]. These five crystals were selected from the same batch and separately ground to check by XRD for variabilities in the amount of cocrystallized aluminum. According to the co-Rietveld refinement of the powder-averaged data, the amount of cocrystallized aluminum in four samples varied from ∼2 wt. % (∼13 mol %) to ∼4 wt. % (∼26 mol %), whereas only one crystal contained no aluminum above the detection limit [70]. Numerical simulations [413, see Supplementary Information] were used to demonstrate that Al inclusions should not affect the qualitative behavior of the low-temperature electrical transport properties (plateau region) as long as the inclusions are trapped inside the bulk and do not short-circuit the current path between the top and bottom surfaces. It is, nevertheless, preferable to polish both surfaces before measurements to eliminate the threat of surface inclusions [393, 413, 414]. In spite of the insulating bulk, two-dimensional Fermi surfaces were revealed in SmB6 by quantum-oscillation measurements,

Rare-Earth Hexaborides (RB6 ) 101

00l (1.521 Å-1) TOPAZ

(b)

90 K

2

-2

Intensity (arbs)

h00 (1.522 Å-1)

0

*~4 wt% Al

(a) -6

-4

-2

2

3

4

Q (Å-1)

Figure 1.39 (a) An x-ray diffraction pattern of the 154 Sm11B6 flux-grown crystal in the (h0l) plane, collected at T = 90 K using the TOPAZ singlecrystal diffractometer located at the Spallation Neutron Source and an xray CT image showing the presence of Al inclusions within the 154 Sm11B6 crystal. The companion reflections (see white arrow) correspond to epitaxial aluminum present in this flux-grown crystal. (b) A neutron diffraction histogram obtained from the radial integration of the singlecrystal neutron diffraction data. The asterisks denote reflections from the epitaxial aluminum present in the flux-grown crystal of SmB6 . Reproduced from Phelan et al. [70].

arising from the crystalline 100 and 110 surfaces according to the results obtained on FG SmB6 single crystals by one group [415, 416]. As-grown SmB6 crystals with rectangular shapes, demonstrating clean and large 110 facets, were chosen for the experiment. Etching of the crystals by a mixture of diluted acids (HCl : HNO3 : H2 O in a ratio of about 1 : 1 : 50) for 1–2 h to clean the crystal surface from possible residual Al flux was their only treatment. The samples showing dHvA oscillations were neither cut nor polished [415, 416]. Another group reported high-frequency quantum oscillations consistent with a 3D Fermi surface similar to that of metallic LaB6 [417, 418]. These results were obtained with single crystals grown by the FZ melting technique in an image furnace [419]. Powder of SmB6 with >99.9% purity from Alfa Aesar was used as a starting material, the highest impurity concentrations were

102 Crystal Chemistry and Crystal Growth of Rare-Earth Borides

Flux

10

5

5

(a)

0

B =0 T 1

T (K)

0 10

1,200 1,000 800 600 FZ 400 Flux 200 0 0.1 1 T (K)

(b)

k (B)/k (0)

15

Quantum oscillation amplitude (a.u.)

20 FZ

10

CFlux/T (mJ mol–1 K–2 )

CFZ/T (mJ mol–1 K–2 )

from Fe (0.031%) and C (0.024%), with the impurity concentration of all RE oxides being less than 0.01%. The crystal growth was carried out in about 0.3 MPa of Ar gas pressure and a flow of Ar gas of up to 10 L/min, using a maximum growth rate of 18 mm/h. Both the feed and the seed rods were counter-rotated at 30 rpm to ensure efficient mixing and homogeneity. The resulting composition of the SmB6 crystal corresponded to the stoichiometric ratio Sm : B = 1 : 6 according to an EDAX test with a standard deviation of 1%, and no RE impurities were present beyond the 0.1% ˚ detection limit. The lattice parameter of the crystal was 4.1353(1) A. The high quality of the crystal was evidenced by its sharp Laue backscattering patterns [419] and a large inverse resistance ratio, IRR = R 1.8 K /R300 K ≈ 105 [418]. In contrast, a heat-transport study [420] of the same FG SmB6 single crystals as those used by Li et al. [415] revealed no residual linear term (∼T ) in the low-temperature thermal conductivity, as one would expect in the presence of gapless fermionic excitations in the bulk. In addition, there was also no significant dependence of the thermal conductivity on magnetic field up to 14.5 T, calling into doubt the existence of a bulk Fermi surface suggested by the quantum-oscillation studies. Hartstein et al. [417, see Methods] explicitly relate this discrepancy among the results obtained on the FG and FZ-grown single crystals to their different physical properties, evidenced by the latter’s low-T upturns in the specificheat coefficient and quantum-oscillation amplitude, and by the much more field-dependent behavior of the thermal conductivity, FZ 1.6 243 mK 1.4 1.2

(c)

Flux 340 mK

1.0 10

0 2 4 6 8 10 12 B (T)

Figure 1.40 The comparison of low-temperature properties of FG and FZ-grown SmB6 single crystals: (a) specific heat divided by temperature; (b) amplitude of quantum oscillations; (c) magnetic-field dependence of the thermal conductivity, normalized to the zero-field value. Adapted from Hartstein et al. [417].

Rare-Earth Hexaborides (RB6 ) 103

as illustrated in Fig. 1.40. They point to a shorter mean free path, limited by the small thickness of the FG single crystals, as a possible cause of such a strong sample dependence. To shed light on this controversy, Thomas et al. [412] used torque magnetometry to measure quantum oscillations in the magnetization of a FG SmB6 crystal as a function of its thickness. They used a typical FG single crystal of SmB6 that is shown in Fig. 1.41(a). The measurement was repeated on the same crystal (a)

2

× 10

-8

As grown - 21.6mg Polish 1 - 11.3mg Polish 6 - 1.5mg

m /m (N m)

1

-1 -2 -3

11.3 mg

0.1

(b)

0.14 1/H (T -1)

0.16

1.5 mg

0.18

0.2

3 As grown - 21.6mg Polish 1 - 11.3mg Polish 6 - 1.5mg

2.5 Scaled FFT (A.U.)

0.12

5.5 mg

2 1.5 1

Scaled FFT (A.U.)

τ

osc

0

0

3

1 0

0.5 0

0

100

200

Polycrystalline Al

2

300 F (T)

0

200 400 F (T)

400

500

600

600

Figure 1.41 (a) Oscillatory torque versus inverse field for a FG SmB6 single crystal at different polishing steps. The torque is normalized to the sample mass (m/m0 ), and the magnetic field is applied a few degrees away from [010]. After the last aluminum deposit is removed, the oscillations vanish. The inset pictures show progressive polishing of the sample, depicting a series of aluminum deposits exposed at the surface. The initial (100) surface was left undisturbed. (b) Frequency spectra of the oscillatory torque shown above. The spectra are normalized to the sample mass. The inset shows a frequency spectrum of polycrystalline Al metal for comparison. Reproduced from Thomas et al. [412].

104 Crystal Chemistry and Crystal Growth of Rare-Earth Borides

after several polishing cycles. After polishing, the sample surface was etched to remove the exposed aluminum inclusions. The measurements were repeated until the crystal was so thin that it contained no Al inclusions, and the quantum oscillations vanished, as one can see in Fig. 1.41(a, b). This experiment confirmed that the dHvA oscillations originated from the embedded metallic aluminum in the FG crystals. For comparison, the inset in Fig. 1.41(b) shows the frequency spectrum of dHvA oscillations from a small piece of 99.999% aluminum used as flux during the growth process. There are more than five peaks in the 300–500 T range due to the fact that the pellet is composed of many randomly oriented single crystals unlike the well-defined FFT peaks exhibiting clear angular dependence in the frequency spectrum in the main panel of Fig. 1.41(b), despite the presence of multiple aluminum inclusions in the SmB6 crystal. This difference indicates that Al inclusions are preferentially aligned along the same crystallographic axis as the SmB6 crystal, as pointed out earlier [70], see Fig. 1.39(a). Nevertheless, quantum oscillations of magnetic torque have been observed in magnetic fields up to 40 T down to T = 0.4 K, and in magnetic fields up to 35 T down to T = 0.03 K also in high-purity FZgrown SmB6 single crystals [418]. These measurements, presented in Fig. 1.42, reveal multiple quantum oscillatory frequencies in a broad range from 50 to 15 000 T, consistent with a 3D Fermi surface. The oscillation amplitude is considerably enhanced below ∼0.5 K [see Fig. 1.40(b)]—a feature that could not be reproduced in FG single crystals [417]. Various theoretical interpretations of these effects have been proposed, ranging from the surface states or quenched disorder to novel types of charge-neutral in-gap excitations [417, and references therein], but they are hard to reconcile with other experimental observations, and none of the proposed theories is so far generally accepted by the community. Thomas et al. [412] point out that all the FZ-grown samples exhibiting quantum oscillations have similarly low values of specific heat at low temperatures, while the heat capacity of SmB6 samples prepared by different methods varies over more than an order of magnitude, as shown in Fig. 1.43(a). They suggest that crystallographic imperfections such as linear and planar defects

Rare-Earth Hexaborides (RB6 ) 105

28

8

24

0 0

400

800

1200

1600

20

2000

8

Landau index

Fourier transform amplitude (arb. units)

4

4

0 2000

4000

6000

16

12

8000 8

8

4

4

0 10000

12000

14000

Frequency (T)

16000

0 0.00 0.02

0.04 0.06 0.08

1/B (T-1)

Figure 1.42 Landau quantization in SmB6 . (A) Fourier transforms of the magnetic torque as a function of inverse magnetic field, from which a polynomial background has been subtracted, revealing multiple quantum oscillatory frequencies ranging from 50 T to 15 000 T. Field ranges for analysis have been chosen that best capture the observed oscillations, with the highest frequencies only appearing in the higher field ranges. (B) The maxima and minima in the derivative of magnetic torque with respect to the magnetic field, corresponding to the dominant low frequency oscillation, are plotted as a function of inverse magnetic field; the linear dependence signals Landau quantization. Reproduced from Tan et al. [418].

(dislocations, grain boundaries, and stacking faults) may cause the dHvA oscillations in the FZ SmB6 crystals, playing a similar role to Al impurities in the FG crystals. Even if this plausible scenario still awaits an unequivocal experimental verification, no consensus in the debate about quantum oscillations in SmB6 seems possible without understanding the extreme sample dependence of its physical properties.

106 Crystal Chemistry and Crystal Growth of Rare-Earth Borides

40

(a)

C/T (mJ/mol K 2)

35 30 25 20 15 10 5 0 0

2

4

6

8

10

Temperature (K)

Figure 1.43 (a) Low-temperature specific heat divided by temperature, C (T )/T , for different single crystals of SmB6 grown by the flux (triangles) and FZ (circles) methods. Adapted from Thomas et al. [412]. (b) Temperature dependence of the resistivity for different FG samples from the same batch. After Shahrokhvand et al. [395].

Kebede et al. [408] noted one more disadvantage of the flux method—the unpredictability of the single-crystal composition. As an example, let us consider the preparation and characterization of FG SmB6 single crystals used in a recent magnetoresistance study [395]. In this work, SmB6 single crystals were grown by the Al-flux method using Sm pieces (99.9%), boron powder (99.99%), and Al pieces as starting materials with a ratio of 0.5 g of Sm : B (1 : 6) to 50 g of Al (99.99%). The growth was performed in a verticalgradient furnace under a continuous flow of argon at temperatures up to 1500 ◦ C with a cooling rate of 5 K/h. The samples were Table 1.14 Dimensions [thickness (T) × width (W) × length (L)] and inverse resistance ratios (IRR) for the FG SmB6 samples (after Shahrokhvand et al. [395]) Sample S1 S2 S3 S4 S5

Dimensions, T × W × L (mm)

IRR

0.7 × 0.73 × 1.10 0.7 × 0.73 × 1.37 0.7 × 0.73 × 1.84 0.7 × 0.73 × 1.48 0.7 × 0.73 × 1.08

1600 1250 1550 8300 1000

Rare-Earth Hexaborides (RB6 ) 107

subsequently cleaned with a NaOH solution to remove the Al flux. The bar-shaped mm-sized crystals were black in color and had flat, mirrorlike surfaces. The characterization results of all the samples used in this study are given in Fig. 1.43(b) and in Table 1.14 [395]. Despite the identical conditions of the crystal growth, identical preparation procedures, and similar sizes of the experimental samples, their IRR values vary from 1000 to 8300, which most likely reflects different levels of disorder or off-stoichiometry (which has not been directly measured in this study) [395]. Phelan et al. [70] faced a similar situation: Figure 1.44(b) shows the resistance ratio R(T )/R 300 K of two randomly selected FG SmB6 single crystals from the same batch, arbitrarily designated as “1” and “2”. While the low-temperature values differ considerably, a resistance plateau is observed in both crystals, as well as in all other FG SmB6 crystals from this growth [70]. According to the phase diagram of the Sm – B system (Fig. 1.22), samarium hexaboride is limited by the compositions of Sm0.73 B6 and SmB6 . Results of the Rietveld refinement on synthesized and annealed for homogenization Smx B6 powders (0.50 ≤ x ≤ 1.50) coincide with this homogeneity range. The lattice constant mono1000

(a)

10000

Flux Grown SmB6

1000

R/R300 K

100

R/R300 K

(b)

1000

100

Floating Zone SmB6

10 0

2

4

6

8

10

T (K) 0

100

T (K)

200

100

10 300

0

2

4

T (K)

6

8

10

Figure 1.44 (a) The resistance ratio R(T )/R 300 K for the FZ cuts 1–4. The inset highlights the low-temperature region, showing that the resistance ratio and the degree of plateauing decrease from 1 to 4. (b) The resistance ratio R(T )/R 300 K for two arbitrary FG SmB6 crystals marked as “1” and “2”. While the values of R/R300 K differ at low temperatures, a resistance plateau is observed for all FG SmB6 crystals from this growth. These low-temperature data resemble closely other FG samples reported in the literature. Reproduced from Phelan et al. [70].

108 Crystal Chemistry and Crystal Growth of Rare-Earth Borides

Figure 1.45 The lattice constant of Smx B6 as a function of composition. The line serves as a guide to the eye. Reproduced from Baumer et al. [292].

tonically decreases with Sm deficiency from 4.13367(2) A˚ (SmB6 ) to 4.12822(2) A˚ (Sm0.75 B6 ), as shown in Fig. 1.45 [292]. According to the Rietveld refinement, Smx B6 samples with x < 0.75 include β-B, and samples with x > 1 include Sm2 B5 as the main impurity [292]. Based on the foregoing considerations, Smx B6 single crystals can be grown by the flux method in the range of 0.75 ≤ x ≤ 1.00. However, according to the fitting of single-crystal XRD data [70], a FG 154 Sm11B6 crystal has vacancies on the boron site with an occupancy of 0.98, i.e., its actual composition is 154 Sm11B5.88 . This is consistent with the structure refinement of another FG 154 Sm11B6 crystal from Ref. [68]. In contrast, the growth of SmB6 single crystals by the FZ method is only possible in a very narrow range near the stoichiometric composition, taking into account that this compound melts congruently, and the open maximum in the phase diagram (Fig. 1.22) corresponds to the SmB6 stoichiometry. As a rule, deviations from stoichiometry result from an uncontrolled evaporation of Sm from the molten zone. Phelan et al. [70] grew SmB6 single crystals using a fourmirror optical FZ furnace. Crystals with approximate dimensions ∅6 mm × 80 mm were prepared from polycrystalline feed rods of

Rare-Earth Hexaborides (RB6 ) 109

Growth Direction (GD)

Cut Number

Figure 1.46 The refined lattice constant a for the cuts 1–4 of the FZ SmB6 crystal, showing a decrease with the cut number. The line serves as a guide to the eye, and the error bars are contained within the data points. The dashed line shows the lattice constant of the polycrystalline feed rod used for the growth of the FZ SmB6 crystal. Reproduced from Phelan et al. [70].

SmB6 (Testbourne Ltd., 99.9%) with one zone pass, a crystallization rate of 10 mm/h, a rotation rate of the growing crystal (lower shaft) of 10 rpm, and under flowing UHP argon at a pressure of 0.2 MPa with a flow rate of 2 L/min. Four slices were cut from the FZ-grown SmB6 single crystal, which divided the crystal into 3 equal parts, as shown in Fig. 1.46 [70]. A portion of each individual cut was used for synchrotron powder XRD, while another portion was used for physical property measurements followed by trace-element analysis. The lattice parameter systematically decreases from cut 1 (zone beginning) to cut 3, while for cut 3 and cut 4 (zone end) they are similar. The R(T )/R 300 K curves in Fig. 1.44(a) show that the decrease in the lattice parameter from cut 1 to cut 3 correlates with a decrease in the resistivity ratio R/R 300 K , while the curves for cuts 3 and cuts 4 nearly coincide [70]. Both the lattice parameter and the R/R300 K values depend on the concentration of Sm vacancies,

110 Crystal Chemistry and Crystal Growth of Rare-Earth Borides

resulting from its preferential evaporation from the molten zone during crystal growth. This effect was confirmed by XRD of the evaporated products, which represented a multiphase mixture of SmB6 and SmB4 that was samarium-rich compared to the feed rod. Composition of the crystal towards the zone end was assessed as Sm1−x B6 (x = 0.01) [70]. Results of the trace-element analysis [70, Supplementary Information, Table S1]) are surprising due to the low degree of purification from impurities. On the one hand, this result is expected for RE elements, because RE hexaborides are soluble in each other and have nearly the same melting points of about 2500 ◦ C, so the partition coefficient of any RE impurities in a hexaboride matrix must be almost unity. Consequently, zone refining is ineffective in reducing the amount of RE impurities [114]. On the other hand, it is unclear, why the concentration of AE impurities with high vapor pressure decreased only slightly. As for SmB6 , the degree of contamination in the reported sample is very high, about 0.25 wt. % [70]. It is unavoidable that such a high concentration of paramagnetic impurities may alter the low-temperature magnetic properties of SmB6 . Figure 1.47 shows the differences in low-temperature electrical resistivity of three FZ-grown SmB6 single crystals [421]. Sample “a” was prepared from isotopically enriched 154 Sm and 11B elements; samples “b” and “c” were grown from natural elements with one and three zone-melting passes, respectively. All samples demonstrated high-quality Laue backscattering patterns, yet their lowtemperature resistivity differs by almost two orders of magnitude. The crystal of 154 Sm11B6 was prepared from the non-neutronabsorbing isotopes 154 Sm and 11B with enrichment of 98.6% for 154 Sm and 99.4% for 11B. The starting powder 154 Sm11B6 was synthesized by borothermal reduction of 154 Sm2 O3 , pressed into a rod and sintered in vacuum at 1900 K. The 154 Sm11B6 single crystal was grown by the RF-heated FZ melting technique from the polycrystalline rod with one zone passage, under 0.5 MPa pressure of high-purity argon gas; the growth rate was 1.25 mm/min with the feed rod fixed in the lower shaft and rotated at 20 rpm. The RT lattice parameter of a = 4.1339(1) A˚ is in agreement with the value expected for the stoichiometric composition (see Fig. 1.45).

Rare-Earth Hexaborides (RB6 ) 111

100

SmB6

ρ (Ω cm)

10

·

sample a sample b sample c

1 0.1

0.01 1E-3

2

10

100

300

Temperature (K) Figure 1.47 Temperature dependence of the electrical resistivity of various SmB6 single crystals grown by the FZ method: (a) isotopically pure 154 Sm11B6 ; (b, c) crystals from natural elements grown with one and three zone passes, respectively. Reproduced from Orend´a˘c et al. [421].

This unique doubly isotope-enriched crystal, prepared by Paderno and Konovalova, was initially used for the phonon [254] and magnetic [61] inelastic-neutron-scattering studies on triple-axis spectrometers (see also Chapter 6). In recent years, samples from this crystal were also used to study low-temperature magnetism by μSR [407] and in neutron-spectroscopy measurements of the 14 meV coherent resonant mode [62, 405] (for details, see section 6.2.3). The higher low-temperature resistivity of the 154 Sm11B6 sample (IRR ≈ 4 × 105 ) is due to the absence of isotope disorder in the lattice. Natural samarium consists of seven stable isotopes n Sm with mass numbers n = 144, 147, 148, 149, 150, 152, 154, and the natural abundance of 154 Sm is only 22.75%. As a result, the SmB6 lattice has a considerable degree of isotopic mass disorder, which may influence phonon lifetimes. An even stronger contribution to the isotopic-disorder scattering is expected from the statistical distribution of boron isotopes in the natural abundance ratio 10 B : 11B ≈ 1 : 4 because of the high mass variance of the two

112 Crystal Chemistry and Crystal Growth of Rare-Earth Borides

isotopes. For example, in a recent study of the isotopically controlled hexagonal boron nitride (α-BN), scattering due to boron-isotope disorder was claimed responsible for a threefold increase in the phonon linewidth [422]. The difference between the samples “b” (once-refined) and “c” (triple-refined) lies in the higher concentration of Sm vacancies in sample c due the uncontrolled Sm evaporation during every zone passage. These FZ SmB6 single crystals were grown from high-purity SmB6 powder that was synthesized from Sm2 O3 (99.999%) and B (99.9%) as starting materials. The Sm1−x Lax B6 (0 ≤ x ≤ 0.5) and Sm1−x Ybx B6 (0.005 ≤ x ≤ 0.2) solid solutions were also grown in single-crystalline form, and their resistivity was investigated in the same work [421]. Similarly, their starting powders were synthesized from highly pure raw materials: Sm2 O3 (99.999%), La2 O3 (99.999%), Yb3 O3 (99.99%), and B (99.9%). Boron, La, Sm, and Yb have different vapor pressures at the melting temperature of their corresponding hexaborides (B < La < Sm < Yb). To reduce the undesirable evaporation of Sm in Sm1−x Lax B6 and Yb in Sm1−x Ybx B6 , and to minimize the formation of vacancies on the metal sublattice, the zone melting was realized under high Ar pressure of about 1 MPa for SmB6 and Sm1−x Lax B6 and about 1.5 MPa for Sm1−x Ybx B6 . The growth rates were 0.45– 0.55 mm/min. To keep the Sm : B, Sm : La, and Sm : Yb ratios constant along the crystals, we actively mixed the melting zone by supplementing the inductive magnetic field with additional rotation of the feed rod fixed in the upper shaft (10 rpm) and of the growing crystal in lower shaft (5 rpm). All grown crystals (SmB6 , Sm1−x Lax B6 , and Sm1−x Ybx B6 ) were characterized by Laue backscattering patterns, XRD, SEM, microprobe and optical spectral analysis to determine their crystal quality, real composition and purity. The real composition of all samples was determined with microprobe analysis using a scanning electron microscope (SEM JSM-6490-LV, JEOL, Japan) with an integrated system of electronprobe analysis (EDS + WDS, INCA Energy), and an HKL-channel detector (Oxford Instruments, UK). Polished single-crystalline plates of the individual SmB6 , LaB6 , and YbB6 hexaborides were used as standards for determining the Sm : La and Sm : Yb ratios in the Sm1−x Lax B6 and Sm1−x Ybx B6 solid solutions.

Rare-Earth Hexaborides (RB6 ) 113

The real compositions of single crystals differ from the nominal ones due to the preferential evaporation of Sm from SmB6 and Sm1−x Lax B6 or of Yb from Sm1−x Ybx B6 . For example, the real composition of the nominally Sm0.97 La0.03 B6 sample was Sm0.964±0.002 La0.036±0.002 B6 , and in the case of Sm0.95 Yb0.05 B6 the actual composition was Sm0.976±0.002 Yb0.024±0.002 B6 . Microprobe analyses of SmB6 crystals with one (“b”) and three (“c”) passes of inductive zone melting confirmed that under the same gas pressure, the Sm concentration is reduced after every additional zone refinement. This result agrees with the conclusions of Phelan et al. [70] and with the monotonic decrease in the lattice parameter in accordance with Sm depletion [292] relative to the source SmB6 . As a result, the composition for crystal “b” with one pass was ˚ and for crystal “c” with estimated as Sm0.99 B6 with a = 4.1347(3) A, three passes as Sm0.95 B6 with a = 4.1334(2) [421]. There are several possible ways to reduce Sm evaporation: (i) to increase the pressure of inert gas in the chamber of the FZ furnace; (ii) to introduce a piece of boron to the initial zone to decrease the starting zone-melting temperature; (iii) to prepare the source feed rod with a small excess of samarium. Nevertheless, here are three examples in which FZ SmB6 single crystals were grown with a pressure in the range from vacuum up to 5 MPa, and where the authors insist on the high quality of the resulting crystals: • Liu et al. [370] grew SmB6 single crystals from commercial SmB6 powder in an optical FZ furnace under a vacuum of 10−4 Pa, with a growth rate in the range of (5–10) mm/h, and the feed and seed rods counter-rotating at 20 rpm. • Frantzeskakis et al. [6, 423] grew SmB6 single crystals in an optical FZ furnace under 5 MPa pressure of high-purity Ar gas, with a growth rate of 20 mm/h, and the feed and seed rods counter-rotating at 30 rpm; IRR = R1 K /R300 K =105 . • Tanaka et al. [114] grew SmB6 single crystals by FZ melting with RF heating. Stoichiometric SmB6 was synthesized from Sm2 O3 (99.9%) and B (97%), and the corresponding FZ single crystals (about ∅8 mm × 30–40 mm) were grown under 1.5 MPa pressure of argon with a growth rate of 10 mm/h and the seed in the lower shaft. The resistivity saturates at about 4 K.

114 Crystal Chemistry and Crystal Growth of Rare-Earth Borides

The RT resistivity and the saturated resistivity after four zone passes were 188 μ · cm and 70.7 · cm, respectively, the IRR was 4 ·105 . The residual resistivity was 8 · cm after a single FZ pass, 30 · cm after three passes, and 70 · cm after four passes. The authors wrote that vaporization from the molten zone remained low, and the chemical compositions had a tendency towards boron deficiency (SmB5.92 ) [114]. From the consideration of these three examples, we can conclude that the composition and quality of FZ-grown crystals are actually determined by the combination of all technological parameters, just as in the case of FG crystals. Let us now consider the physical properties of the samples prepared from the FG single crystals with compositions from throughout the samarium-hexaboride homogeneity range, based on the results of Stankiewicz et al. [424], in order to establish how offstoichiometry and disorder affect these properties. For comparison, we also consider the results of a similar study of the physical properties of compact sintered Smx B6 samples with stoichiometric and nonstoichiometric compositions [425]. The Smx B6 single crystals were flux-grown with different starting compositions of Sm and B within the range 0.6 ≤ x ≤ 1 [424]. The Sm content in the samples was estimated with EDX spectroscopy, and authors affirm that the nominal concentration of vacancies in the single crystals of Smx B6 is close to the actual composition, except for the sample with the highest nominal amount of vacancies (x = 0.6), see Table 1.15 [424]. The absence of crystals with x = 0.6 follows from the Sm – B phase diagram (Fig. 1.22) and the dependence of the Smx B6 lattice constant on composition x (Fig. 1.45): The lower limit of the homogeneity range is about 0.75 [255,292]. Correct determination of the Sm : B ratio with EDX is challenging, because it requires a stoichiometric SmB6 single crystal as a reference, while using reference compounds of a different structure type usually results in errors. The large uncertainties in the average lattice constants a(x) due to the scatter in compositions of individual crystals from the same batch may lead to their nonmonotonic dependence on the Sm : B ratio in Table 1.15, in contrast to the monotonic dependence in Fig. 1.45.

Rare-Earth Hexaborides (RB6 ) 115

Table 1.15 Composition and structural parameters obtained for Smx B6 single crystals from EDS and XRD measurements (after Stankiewicz et al. [424])

Nominal x 1.00 0.95 0.90 0.75 0.60

Sm : B ratio from EDS at. % Al Estimated x 0.167 ± 0.004 0.149 ± 0.002 0.147 ± 0.002 0.139 ± 0.005 0.127 ± 0.012

5±2 ≤ 0.5 3±1 ≤ 0.5 ≤ 0.5

1.00 ± 0.025 0.94 ± 0.012 0.90 ± 0.010 0.80 ± 0.03 0.75 ± 0.07

Lattice ˚ constant (A) 4.132 ± 0.0031 4.127 ± 0.0172 4.129 ± 0.0163 4.134 ± 0.0074

1

Average value from 34 published reports. Average value for 11 pieces from the same growth batch. 3 Average value for 5 pieces from the same growth batch. 4 Average value for 12 pieces from four growth batches. 2

The T-dependence of the electrical resistivity of as-grown Smx B6 single crystals after Stankiewicz et al. are shown in Fig. 1.48. The resistivity increases with a rise in vacancy concentration. This result (which is typical for conventional metals) is rather unexpected for SmB6 , where the residual resistivity at T → 0 usually decreases with an increasing number of vacancies (cf. Figs. 1.44 and 1.47). The authors explain such an unusual dependence of ρ(T ) on x by an interplay of two competing mechanisms: On the one hand, samarium vacancies introduce additional holes, increasing the conductivity; but on the other hand, they also generate scattering centers, which decrease the conductivity due to the disruption of the coherence in the hybridization [424]. A more typical behavior of ρ(T ) vs. x was observed in the Sm1−x B6 sintered polycrystalline samples (0 ≤ x ≤ 0.2), see Fig. 1.49 [425], and for FZ-grown SmB6 single crystals (Fig. 1.44(a) [70] and Fig. 1.47 [421]), where an increase in Sm vacancy concentration is accompanied by a reduction in the lowtemperature resistivity. The temperature dependence of the static magnetic susceptibility χ(T ) for Sm-deficient Smx B6 single crystals is shown in Fig. 1.50 for FG crystals [424] and in Fig. 1.51 for the sintered samples [425]. The resistivity data for the same samples were presented above in Figs. 1.48 and 1.49, respectively. The magnetic susceptibilities χ(T ) of Sm1−x B6 show the characteristic features of a mixed-valence compound, following a dome-shaped curve centered at about 50 K

116 Crystal Chemistry and Crystal Growth of Rare-Earth Borides

Resistivity (Ω cm)

101

·

100

Sm0.94B6

10-1

SmB6

Sm0.75B6

10-2

Sm0.8B6 10-3

1

10 Temperature (K)

100

Figure 1.48 Resistivity of Smx B6 as-grown single crystals vs. temperature. The inset is an image of the actual transport geometry used in the measurements. Reproduced from Stankiewicz et al. [424].

that indicates a gap structure in the density of states, whose maximum is connected with opening of the correlation gap E g . With the Sm deficiency x increasing towards 0.2, the maximum is reduced and shifted towards higher temperatures, indicating that the gap structure in the density of electronic states is retained in all the studied samples [425]. A quantitative assessment of the correlation gap E g from the resistivity and susceptibility data has shown that it remains practically constant (within the measurement accuracy) at ∼10 meV for x < 0.1, followed by a gradual reduction for higher concentrations of Sm vacancies or La impurities [425]. To resolve the magnetic nature of defects in the SmB6 samples, Pirie et al. [398] measured the magnetic susceptibility of nominally

Rare-Earth Hexaborides (RB6 ) 117

1

x=0

( .cm)

0.1

Sm1-xB6

x=0.08

0.01 x=0.1 x=0.2

1E-3

1E-4

x=0.03 x=0.05

Sm0.84La0.16B6

Sm0.72La0.28B6

2

100

300

Temperature (K) Figure 1.49 Temperature dependence of the electrical resistivity, ρ(T ), of Sm1−x B6 (x = 0, 0.03, 0.05, 0.08, 0.1, and 0.2) and Sm1−x Lax B6 solid ´ s et al. [425]. solutions (x = 0.16, 0.28). Reproduced from Prista˘

pure, stoichiometric SmB6 and Sm-deficient Sm0.95 B6 crystals, grown with a controlled number of Sm vacancies (Kondo holes). They employed the standard Al-flux technique for the single-crystal growth: a vertical tube furnace with UHP Ar flow; the mixture of Sm pieces, B powder (99.99%), and Al shots (99.999%) with the starting composition Sm : B : Al = (1 − x) : 6 : 700 (x = 0.0 and 0.05, respectively); soaking at a temperature of 1723 K for 12 h, followed by a slow cooling to 1323 K at 2 ◦ C/h. The results presented in Fig. 1.52 show that Sm vacancies lead to an enhancement of the susceptibility below the Kondo-lattice coherence temperature of 60 K [398]. The authors propose that defects in the Sm matrix manifest themselves as magnetic objects at low temperatures, referring to Ref. [426], where spin-polarized scanning tunneling spectroscopy has provided further evidence that nominally nonmagnetic defects in SmB6 may act as magnetic centers at low temperature. However, the results of Pirie et al. [398] are apparently in conflict with several other works (cf. Figs. 1.50

118 Crystal Chemistry and Crystal Growth of Rare-Earth Borides

SmxB6

χ (10 emu/mol Oe)

3.5

3.0

2.5 1.0

Fraction (a.u.)

-3

·

2.0

0

0.8

0.75

0.4

Sm2+

0.8

1.5

0.94

3+ 0.6 T = 300 K Sm

x

100

1.0

200

300

Temperature (K)

Figure 1.50 Temperature dependence of the magnetic susceptibility of Smx B6 single crystals, measured using a SQUID magnetometer. The inset shows fractions of Sm2+ and Sm3+ ions for crystals with different vacancy concentration. Reproduced from Stankiewicz et al. [424].

and 1.51) [424, 425] that demonstrated a reduction in magnetic susceptibility with increasing vacancy concentration. Eo et al. [413] also considered the influence of disorder on the energy gap in SmB6 and addressed the issue of separating the bulk and surface contributions using the inverted resistance method. They used SmB6 single crystals grown by the Al-flux technique with the starting composition Sm : B : Al = (1 − r) : 6 : 700, where r is the nominal vacancy concentration, ranging from 0 to 0.40. The actual vacancy concentration was evaluated using a combination of the Vickers microhardness, Auger spectroscopy, XRD, and high-resolution TEM measurements. It turned out impossible

Rare-Earth Hexaborides (RB6 ) 119

Susceptibility (emu/mol)

0.006

x=0 x = 0.03 x = 0.05 x = 0.08 x = 0.1 x = 0.2

0.005

Sm1-xB6

0.004

0.003

0.002 1

10

100

500

Temperature (K) Figure 1.51 Temperature dependence of the magnetic susceptibility for Sm1−x B6 (x = 0, 0.03, 0.05, 0.08, 0.1, and 0.2) and the corresponding fits, χ (T ) = χ0 + χa (T ), where χa (T ) = C /T exp(−E g /kB T ) is the activation´s type contribution with the correlation gap E g . Reproduced from Prista˘ et al. [425].

to determine the point-defect levels in the crystals unambiguously, but a qualitative tendency in the change of the physical properties was revealed from the hardness measurements of the crystals that appears to correlate well with the nominal percentage of Sm vacancies, r, and the signatures of (extended) defects in XRD patterns. The analysis of the inverted-resistance measurements showed that stoichiometrically and nonstoichiometrically grown SmB6 crystals have almost identical thermally activated behavior with a gap of E g ≈ 4 meV until the bulk resistivity plateau develops. So the bulk of SmB6 can be considered immune to disorder originating from Sm vacancies or boron interstitials [413]. Unambiguous determination of the composition of RE-boride samples remains a major problem that was encountered, in particular, in Refs. [398,413,424]. Recently, a qualitative method was proposed for the relative assessment of Sm vacancies in hexaborides using Raman spectroscopy [403, 404]. With a reference to wellcharacterized standards, it could also yield quantitative estimates.

120 Crystal Chemistry and Crystal Growth of Rare-Earth Borides

1.5

1.4

1.3 1.55

1.2

1.5 1.45

1.1

1.4 1.35 1.3

1

1.25 0

0

20 40

50

60

100

80 100

150

200

250

300

350

Figure 1.52 Temperature-dependent magnetic susceptibility of pure SmB6 and Sm-deficient Sm0.95 B6 . The data are normalized to their respective values at 350 K. Below 60 K, the relative susceptibility is higher for the Sm-deficient sample (see inset). Reproduced from Pirie et al. [398, Supplementary Information].

The aim of these works was to investigate the stability of the hybridization gap in SmB6 , measured with Raman spectroscopy, against the presence of a small number of Sm vacancies. Raman scattering can simultaneously probe Sm vacancies through phonon effects and the electronic structure through the electronic Raman response. A value of the linewidth of boron phonons was used as a parameter to characterize the quality of SmB6 samples and the degree of their structural homogeneity, while the intensity of the nominally forbidden low-energy Raman peak was used as a relative estimate for the percentage of Sm vacancies [403, 404]. For this experiment, three SmB6 single crystals were selected. One of them was grown by the Al-flux method, and the other two by the optical FZ method. The latter two crystals were described earlier in Ref. [70] and represent the cuts 1 and 4 of the single crystal shown in Figs. 1.46 and 1.44(a). Cut 1 corresponds to the

Rare-Earth Hexaborides (RB6 ) 121

Figure 1.53 Room-temperature Raman spectra of the three studied SmB6 samples with increasing number of Sm vacancies (Al-flux, FZ-pure, and FZdefc, respectively) in the (x, x) and (x, y) polarizations. The three first-order Raman-active phonons appearing at 89.6 meV (T2g ), 141.7 meV (Eg ), and 158.3 meV (A1g ) are superimposed on a broad continuum of electronic scattering. The inset shows a low-frequency fragment of the (x, x) spectra. Two symmetry-forbidden peaks appear at 10 and 21 meV due to the defect-induced and two-phonon scattering, respectively. Reproduced from Valentine et al. [403].

more stoichiometric composition (FZ-pure SmB6 ), and cut 4 to the most Sm-deficient (FZ-defc SmB6 ) with about 1% defects [70]. Figure 1.53 shows the RT Raman spectra of the three studied SmB6 samples with an increasing number of Sm vacancies (Al-flux SmB6 , FZ-pure SmB6 , FZ-defc SmB6 , respectively) in the (x, x) and (x, y) polarizations. The distortions of B6 octahedra are represented by the three first-order Raman-active phonons T2g (89.6 meV), Eg (141.7 meV), and A1g (158.3 meV). Two additional symmetry-

122 Crystal Chemistry and Crystal Growth of Rare-Earth Borides

forbidden peaks appear at 10 and 21 meV (inset), corresponding to defect-induced and two-phonon scattering, respectively [403]. The peak at 10 meV is an acoustic Raman-inactive T1u mode, which is threefold degenerate at the  point. In the cubic unit cell of SmB6 (space group P m3m), the Sm ion is located at a center of inversion symmetry, so the appearance of this peak must be connected with the local symmetry breaking induced by the presence of Sm defects and, correspondingly, with Sm movement. The authors [403, 404] correlate the intensity of the 10 meV phonon with the concentration of Sm vacancies in the studied FZgrown samples, as estimated from the variation in lattice constants by Phelan et al. [70]. This Sm-defect-induced phonon at 10 meV shows an increase in spectral weight of about 50% (Fig. 1.53) with an increase in Sm deficiency between the two FZ-grown samples. Therefore, Raman scattering can be effectively used to characterize the relative presence of Sm vacancies in SmB6 samples. The 10 meV phonon has nearly zero spectral weight for the studied Al-flux SmB6 sample, demonstrating that the sample has the lowest concentration of Sm vacancies [403, 404]. A hypothesis about the influence of subsurface cracks, arising both from surface preparation and during crystal growth, on the physical property measurements of SmB6 , was recently put forward by Eo et al. [427]. Their magnetotransport study of SmB6 samples, obtained by flux and zone-melting methods in different scientific institutions and prepared using various surface treatment procedures, confirmed the suggestion that the large, order-ofmagnitude disagreements in carrier density and mobility come from the surface preparation and the transport geometry, rather than from the intrinsic sample quality. SmB6 single crystals are hard and brittle, hence any mechanical treatment may lead to the formation of a subsurface damage layer with the depth of a few μm. The best way to eliminate it is to perform electrolytic polishing, which should be less damaging to the SmB6 surface. In order to reveal the role of magnetic and nonmagnetic impurities in the formation of the low-temperature properties of SmB6 , a number of works have been undertaken to obtain doped SmB6 single crystals. The majority of the doped SmB6 single crystals were grown with the Al-flux technique: (Sm, Fe)B6 [398, 411, 428],

Rare-Earth Hexaborides (RB6 ) 123

Temperature (K) Figure 1.54 Temperature dependence of the sheet resistance for pure and Fe-doped SmB6 samples at T ≥ 0.35 K. The inset shows the same data with a linear vertical scale. Reproduced from Akintola et al. [411].

(Sm, Y)B6 [5, 429], (Sm, Eu)B6 [430], (Sm, Gd)B6 [5, 398, 405, 429], (Sm, Yb)B6 [5], and (Sm, Er)B6 [431]. Y, Eu, Gd, Yb, and Er form solid solutions with SmB6 by replacing Sm on the metal sublattice. Where Fe is located in the SmB6 crystal remains unknown, because Fe cannot replace Sm in the lattice due its very small ionic radius, but wavelengthdispersive spectroscopy confirmed the elemental and uniform Fe concentrations in SmB6 [411]. Figure 1.54 shows the temperature dependence of the sheet resistance for SmB6 and Fe-SmB6 samples at T ≥ 0.35 K. The effect of Fe impurities on the low-T resistance plateau suggests that Fe suppresses the surface conductance below T ≈ 6 K but does not affect the bulk conductance at higher T [411]. It should be mentioned that growing a Fe-doped SmB6 single crystal by the FZ method is impossible, because Fe will partially evaporate from the molten zone and will be displaced towards the end of the crystal. Other solid solutions based on SmB6 have been successfully grown by FZ melting: (Sm,R)B6 with R = La, Eu, Yb, Sr [421, 425, 432, 433]. In all cases, the source solid solutions (Sm,R)B6 in powder form were prepared from RE oxides with purity no worse than 99.99% and amorphous boron, 99.9%.

124 Crystal Chemistry and Crystal Growth of Rare-Earth Borides

Phelan et al. [434] performed a comprehensive study of both surface and bulk physical properties of FZ-grown SmB6 single crystals, both pure and containing small levels of aluminum and carbon to check for their possible effects on these properties. Single crystals of SmB6 with approximate dimensions of ∅6 mm × 30 mm were grown from rods of polycrystalline SmB6 (Testbourne Ltd., 99.9%) using an optical FZ furnace. Some of the pure SmB6 crystals were prepared with SmB6 from Testbourne Ltd. (99.5%) and some others from Alfa Aesar (AA, 99.9%), but no difference in the lowtemperature resistances between these two types of crystals were observed. In order to simulate an “aluminum flux” growth of SmB6 , a pressed pellet with a targeted mass of ∼0.20 g was prepared with a SmB6 (AA 99.9%) : Al (AA 99.97%) mixture in a ratio of 50 : 50 wt. %. This SmB6 /Al pellet was introduced before zone melting into the gap between the seed and feed rods and passed through the molten zone during the FZ growth. A similar procedure was realized for the carbon-incorporated growth, where the initially pressed and then arc-melted pellet from a 0.5 g prehomogenized SmB6 (AA 99.9%) : C (AA 99.9995%) mixture in a ratio of 15 : 85 wt. % was used as a dopant. All crystals were grown with a single zone pass under flowing UHP Ar at a pressure of 2 bar and a flow rate of 2 L/min, with the growth rate of 10 mm/h, and rotation rates for the growing crystal and the feed rod of 10 and 3 rpm, respectively [434]. Figure 1.55 shows the normalized resistance measurements for (a) SmB6 /Al and (b) SmB6 /C. The introduction of Al results in a substantial reduction in the normalized resistance that is consistent with filamentary inclusions of Al metal that provide a low-resistance pathway and “short out” the insulating bulk at low temperatures. The data for the carbon-containing crystals demonstrate a systematic appearance of the resistance plateau below T = 6 K, which becomes less prominent as a function of the passed molten-zone position. This observation suggests that the carbon content varies systematically along the SmB6 /C crystal rod [434]. Some physical properties of SmB6 are less sensitive to crystal imperfections (vacancies, impurities, etc.), for example its lowenergy optical properties. A detailed high-resolution study of the

Rare-Earth Hexaborides (RB6 ) 125

Figure 1.55 Normalized resistance measurements for (a) SmB6 /Al and (b) SmB6 /C. Both insets highlight the R/R 300 K data below T = 10 K. The numbers in the insets are arbitrary designations representing the position of the crystal slice along the SmB6 /Al and SmB6 /C crystals (MZ, molten zone). All samples are of comparable geometry to minimize the effects of sample shape that might arise due to a mixture of surface and bulk conductivity. The slope of the low-T resistance changes systematically with the position of the slice cut along the SmB6 /C crystal but remains approximately the same along the SmB6 /Al crystal, indicating that variations in the bulk carbon content can be used to systematically tune the low-T resistance plateau. The gray line in the inset shows the normalized resistance of the pure FZ SmB6 in (a), which is off the scale in (b). Reproduced from Phelan et al. [434].

optical properties in the THz frequency range showed consistent results for SmB6 single crystals grown by both optical FZ and Alflux methods [401, 402]. The only difference between the FG SmB6 crystals as compared to the FZ-melted crystals is the poorer signalto-noise ratio of the former due to a smaller sample size. Finally, it is worth making a remark on the influence of the purity of source substances on the physical properties of the resulting crystals. Nguyen et al. [435] recently synthesized high-quality SmB6 single crystals using the high-temperature Al-flux method. Sm metal (≥99.9%, Alfa Aesar), Al (99.999%, RND Korea), and B with 99.9% or 99.9999% purity were used as starting substances. The major impurities identified in the SmB6 crystals grown with the less pure (99.9%) boron were Fe (0.0042%) and small amounts of Al (0.00092%), Cr (0.0009%), and Mn (0.0009%). The high-purity sample from 99.9999%-purity boron served as a reference. A Raman spectroscopy study of these two SmB6 crystals revealed significant

126 Crystal Chemistry and Crystal Growth of Rare-Earth Borides

differences in their spectra without noticeable differences in the XRD. Based on the high-pressure Raman measurements and phonon calculations in the uniaxial compression model, authors argue that even a small concentration of impurities (∼0.1%) in SmB6 (99.9%) is enough to induce a slight anisotropic distortion in the B6 octahedra, such as spontaneous tilting away from the [001] direction, leading to significant upshifts of the phonon-peak wavenumbers and a detectable broadening of the phonon modes of SmB6 (99.9%) in comparison with those of the purest sample SmB6 (99.9999%).

1.5.4 Ytterbium Hexaboride (YbB6 ) The first single crystals of YbB6 were grown by the Al-flux method [262]. Two types of YbB6 powders, synthesized by the borothermal reduction of Yb2 O3 having 99.9% and 99.999% purity, were used as a source material. The YbB6 : Al weight ratio was about 1 : 20. The charge was heated to 1723 K for 2 h under purified Ar flow; the cooling rate was 150 K/h down to 873 K. This technique was also used to prepare carbon-substituted YbB6−x Cx single crystals with C content less than 0.03 [262], because a considerable amount of carbon is dissolved in molten aluminum. Measurements of the magnetic susceptibility from 4.2 to 300 K on single crystals and polycrystalline samples of YbB6 with various impurity contents showed diamagnetic behavior (divalent Yb) over a broad range of temperatures, followed by a rise in χ(T ) with a transition to paramagnetic response that is connected with the impurity content. The single crystals, which were prepared with powder made from Yb2 O3 (99.999%), had the smallest susceptibility, and the sharp rise in the χ(T ) curve below 20 K was associated with the presence of magnetic impurities with a concentration of about 100 ppm [262]. A tiny diamagnetic contribution approaching zero at the base temperature was detected in YbB6 single crystal grown in Al flux, and the positive quadratic magnetoresistance without any oscillations supports bulk conduction [5]. The temperature dependence of the resistivity and its value are also governed by the sample quality (composition, purity, defects,

Rare-Earth Hexaborides (RB6 ) 127

Figure 1.56 Variation of the electrical conductivity above 200 K for an YbB6 single crystal, reproduced from Tarascon et al. [262]. Inset: the resistivity ρ(T ) of an YbB6 single crystal, reproduced from Kim et al. [5, Supplementary Information].

etc.). The variation in the electrical conductivity in the range 4.2– 800 K as a function of the reciprocal temperature for the sintered polycrystalline sample with almost stoichiometric composition is characteristic of doped semiconductors, see Fig. 1.56 [262,277,436]. For single crystals grown from Al flux, it changes to a semiconducting behavior above 300 K with a gap E = 0.11 eV, whereas below ∼200–300 K the resistivity decreases monotonically, like in a poor metal, as the temperature is lowered from RT to 2 K (Fig. 1.56, inset) [5, 262, 437]. Moreover, the temperature corresponding to the transition from metallic to semiconducting behavior is unique to every specific sample, and the RT resistivity takes values from 0.5 to 25 m · cm according to the existing publications [5, 194, 262, 277, 436, 437]. The unusual temperature dependence of the YbB6 resistivity for single-crystal samples was explained by the presence of metal vacancies [437]. It is also necessary to keep in mind the possible effect of Al contamination on the properties of

128 Crystal Chemistry and Crystal Growth of Rare-Earth Borides

this hexaboride. For example, the nonstoichiometric YbB6.07 crystal grown in Al flux was reportedly contaminated by 0.51 wt. % of Al inclusions [194]. ¨ On the one hand, Mossbauer spectroscopy [438] and hard x-ray photoelectron spectroscopy (XPS) [267] data prove that ytterbium is divalent in YbB6 . On the other hand, direct and inverse photoemission spectroscopy of the YbB6 electronic structure, measured on FZgrown single crystals, indicate that part of the Yb atoms adopts the Yb3+ configuration [439–441], with the mean Yb valence estimated as 2.2–2.3 [440]. However, susceptibility measurements on the same samples [440] showed that the contribution from the trivalent Yb3+ component is negligibly small. This contradiction was explained by the surface sensitivity of the photoemission studies in contrast to the magnetic susceptibility measurements, suggesting that the trivalent Yb3+ component is an extrinsic property originating from a surface layer which contains more Yb3+ than the bulk [440]. However, Gavilano et al. [442] revealed weak ferromagnetic order with TC > 300 K and a magnetic moment of only 0.002 μB /f.u. when studying dc magnetization of randomly oriented mm-sized single crystals of YbB6 grown in Al flux. This result implies the presence of 2% of all the Yb atoms in the Yb3+ configuration and can be reconciled with the poor-metal behavior of the resistivity that is shown in the inset to Fig. 1.56 [5]. The joint analysis of the resistivity (ρ), Hall (R H ), and Seebeck (S) coefficients in the range 1.7–300 K has shown that YbB6 is a narrow-gap semiconductor with n-type conductivity; both RH and S were negative in this temperature range [5, 262]. In subsequent years, some efforts were undertaken to study the effect of deviations from stoichiometry in the single crystals of YbB6 on their properties. Iga et al. [443] investigated the effect of Yb vacancies in FZ-grown single crystals on their magnetic properties as part of their search for a ferromagnetic transition in Yb1−x Lax B6 (0 ≤ x ≤ 0.006). The source YbB6 powders with different Yb : B ratios were synthesized by borothermal reduction of Yb2 O3 . The sintered rods from these powders were used for growing single crystals using FZ melting with optical heating. The preferential Yb evaporation from the molten zone due to its high vapor pressure above the melting point of 2300 ◦ C resulted in the formation of

Rare-Earth Hexaborides (RB6 ) 129

YbB50 as a foreign phase in the zone-melted bar. The formation of this higher boride is surprising, because it decomposes above 1700 ◦ C [444]; we are left to assume that its crystal structure was stabilized with some impurities. To avoid the appearance of the YbB50 phase, a nominal composition of Yb1.08 B6 was chosen for the sintered polycrystalline source rods, which is at the upper boundary of the Yb1±δ B6 homogeneity range that is limited by δ < 0.08 according to the binary Yb – B phase diagram in Fig. 1.11. It is difficult to assess the concentration of vacancies in single crystals grown from the Yb1+δ B6 sintered rods with the initial compositions δ = 0 and 0.08 from the corresponding lattice param˚ because the difference between eters, a = 4.1479 and 4.1481 A, these values is comparable to the experimental uncertainty of ±0.0001 A˚ [443]. However, following the analogy with CaB6 [445], it has been suggested that vacancies in the metal sublattice may be responsible for the magnetic behavior of YbB6 . To verify this hypothesis, a single crystal grown from the sintered rod with the nominal composition Yb1.08 B6 was exposed to a step-by-step heat treatment in vacuum at 1700 ◦ C for a few hours to increase the Yb deficiency in Yb1−δ B6 . Treatments for 1.5 and 3 h resulted in the weight loss corresponding to the Yb deficiencies δ = 0.005 and 0.009, respectively, under the assumption of the predominant loss of Yb only. In fact, δ could be larger taking into account the incongruent evaporation of Yb during zone melting, so the true composition is unknown. In addition, under heat treatment of the single crystal, Yb evaporates predominantly from the surface but not so much from the bulk. The corresponding results of magnetic susceptibility measurements in a 10 mT field are shown in Fig. 1.57 [443]. The diamagnetic behavior detected in the as-grown crystal (δ ≈ 0.0) changes to Curie–Weiss behavior with an increase in Yb deficiency. The absolute value of the paramagnetic Curie temperature is about 150 K for δ = 0.005 and 0.009. The magnetization curves of the Yb-deficient single crystals did not show any ferromagnetic hysteresis [443]. Unfortunately, the authors of Ref. [443] were restricted only to magnetic measurements, which are insensitive to the surface contribution, and therefore no conclusion can be drawn about the nature of conductivity and the mean valence of Yb ions in these crystals.

c (10-4 emu/mol)

130 Crystal Chemistry and Crystal Growth of Rare-Earth Borides

8

Yb1-δB6 single crystal B = 10 mT

6 4

δ = 0.009

2 0.005 0 0

100

200 T (K)

300

Figure 1.57 Temperature dependence of the magnetic susceptibility of an Yb1−δ B6 single crystal that was heat treated in vacuum at 1700 ◦ C for 1.5 h (δ = 0.005) and 3 h (δ = 0.009), compared to the as-grown crystal (bottom curve). Reproduced from Iga et al. [443].

Kim et al. [446] studied the effect of defects in the Yb and B sublattices on the transport and magnetic properties of YbB6±δ single crystals grown by the high-temperature flux method. The crystals were synthesized from the high-purity elements as starting materials: Yb (99.99%), B (99.9999%), and Al (99.999%). It was assumed that components included in the charge in the molar ratios of Yb : B = 1 : 5.7, 1 : 6.0, and 1 : 6.3 would be reproduced in the grown single crystals, resulting in YbB6−δ (B-deficient), stoichiometric YbB6 , and YbB6+δ (B-rich) samples, respectively, with a nominal δ = 0.3. Magnetic susceptibilities in the temperature range 2 K ≤ T ≤ 300 K for the three samples of YbB6 and YbB6±δ are very similar to those in Fig. 1.57 [443]. While the YbB6 crystal shows the diamagnetism expected for the divalent state of Yb, the YbB6±δ crystals show Curie–Weiss temperature dependence. Moreover, the magnetization of YbB6−δ is much larger than that of YbB6+δ . The field dependence of the isothermal magnetization curves of YbB6−δ revealed weak ferromagnetism, manifested by a hysteresis loop in M(H ) at RT with a saturation magnetic moment of 1.5 × 10−4 μB /f.u. and a ferromagnetic transition temperature higher than T = 300 K, whereas YbB6 and YbB6+δ crystals revealed

Rare-Earth Hexaborides (RB6 ) 131

6

Resistivity (mΩ∙cm)

5 4 3 Field (T)

2 1 0 0

50

100 150 200 Temperature (K)

250

300

Figure 1.58 Temperature-dependent electrical resistivity ρ(T ) for YbB6 and YbB6±δ crystals between 2 and 300 K. Inset: Hall resistivity ρx y as a function of magnetic field for the YbB6−δ crystal at various temperatures. Reproduced from Kim et al. [446].

diamagnetism and paramagnetism, respectively. Chemical etching of the YbB6±δ crystals removed both ferromagnetic and paramagnetic signals. So the bulk of the YbB6±δ crystals is intrinsically diamagnetic, and the ferromagnetism or paramagnetism are surface effects, which must be also true for the heat-treated YbB6 samples from Ref. [443]. The authors [446] concluded that the state of the surface in any form (thin film, polycrystalline or single-crystal samples) is responsible for ferromagnetism, which is not related to any extrinsic magnetic impurities on the surface of the sample according to the results of microprobe analysis and x-ray photoelectron spectroscopy. Measurements of the transport properties reveal metallic behavior in all three samples with the lowest absolute value of the resistivity in the B-deficient YbB6−δ crystal (Fig. 1.58) [446]. Even though no evidence of a transition to the semiconducting state was detected, the quite small values of RH for the YbB6 and YbB6±δ crystals and the variation of the Hall coefficient (i.e., charge carrier density) with temperature for YbB6−δ in Fig. 1.58 are consistent with

132 Crystal Chemistry and Crystal Growth of Rare-Earth Borides

YbB6 being a semiconductor (inset), and the off-stoichiometry in YbB6±δ causing a carrier doping effect [446]. An essential effect of the YbBx (x = 5.7–6.3) composition variation on the thermoelectric and transport properties was demonstrated in a study conducted on both polycrystalline YbBx samples and FZ-grown single crystals [447]. At T = 300 K the Seebeck coefficient, S, of the polycrystalline samples changes from −120 μV/K (YbB5.7 ) to +90 μV/K (YbB6.3 ) with a sharp rise in the thermoelectric power near YbB6 , while in the single crystals S ≈ +170 μV/K. So YbB6 is the only p-type semiconductor among hexaborides with divalent metals [448, 449]. Upon increasing the temperature above 300 K, the Seebeck coefficient of the YbB6.1 and YbB6.3 samples changes sign from positive to negative, while that of the n-type ones (x = 5.7, 5.9, and 6.0) remains unchanged. Above 300 K the T-dependence of electrical conductivity of the n-type YbB6 samples is characteristic of a semiconductor, whereas in p-type YbB6 samples it behaves like in a degenerate semiconductor with a high impurity concentration [447]. To shed light on the effect of intrinsic defects on the evolution of the band structure and charge-transport parameters in YbB6−δ , Glushkov et al. [263] studied electronic transport and magnetic properties of YbB6 single crystals with different levels of boron deficiency. YbB6−δ crystals were grown by inductive FZ melting in an Ar atmosphere. The initial stoichiometric YbB6 powder was synthesized from high-purity ytterbium oxide (99.999%) and amorphous boron (99.9%). Taking into account the high vapor pressure of Yb and its preferential evaporation, the Yb : B ratio in the grown single crystals was controlled by the variation of Ar pressure and by introducing some extra boron or ytterbium into the initial molten zone. Electron-probe microanalysis corroborated some deficit of B rather than Yb, which allowed designating the crystals as YbB6−δ ; the real compositions were evaluated from magnetic measurements under the assumption that the magnetism arose from the Yb3+ contribution (Table 1.16). The lattice constant was ∼0.02% larger in the more B-deficient sample. Volatile impurities from 99.9% boron were removed during synthesis and zone melting, so the total content of impurities estimated from optical spectral analysis was less than 0.001 wt. %. To remove surface defects

Rare-Earth Hexaborides (RB6 ) 133

Table 1.16 Some growth parameters and the lattice constants of the YbB6 crystals (from Glushkov et al. [263]) Crystal

Composition

No. 1 No. 2

YbB5.994 YbB5.96

Solvent

Pressure

Lattice constant

boron ytterbium

0.3 MPa 1.0 MPa

4.1469(2) A˚ 4.1477(2) A˚

induced by the spark cutting and polishing of the samples, the YbB6−δ single crystals were etched in dilute nitric acid. The parameters of charge transport in the YbB6−δ single crystals are presented in Fig. 1.59 [263]. The variation in the Yb : B ratio qualitatively modifies all the transport properties: resistivity ρ(T ), Seebeck coefficient S(T ), low-field Hall coefficient RH , and Hall mobility μH . Crystal 1 shows metal-like resistivity, whereas in crystal 2 it displays semiconducting behavior [Fig. 1.59(a)]. While the

200

a

10 8

300

c

2

0,2

2

-1

12

100

3

ρ (mΩ⋅cm)

T (K)

100 200 300 400 0

RH (cm ⋅ C )

T (K) 0

6 4

0,1

1 1

2 0

0,0

80 2

40

2

100

200

T (K)

300 0

100

40 20

0 0

60

-1 -1

1

120

2

-1

S (μV⋅K )

d

1

200

μH (cm ⋅V s )

80

b

160

0 300

T (K)

Figure 1.59 Temperature dependence of (a) resistivity ρ, (b) Seebeck coefficient S, (c) low-field Hall coefficient RH , and (d) Hall mobility μH for the YbB5.994 (1, open circles) and YbB5.96 (2, closed circles) single crystals. Reproduced from Glushkov et al. [263].

134 Crystal Chemistry and Crystal Growth of Rare-Earth Borides

Seebeck coefficient in crystal 1 behaves like in undoped hexaborides R 2+ B6 , in crystal 2 it depends on temperature in a complicated nonmonotonic manner, changing sign twice at ∼20 and 270 K [Fig. 1.59(b)]. Crystal 1 evidences p-type conductivity with the T-independent Hall constant RH = +0.066 cm3 /C. In contrast, crystal 2 has a strongly T-dependent Hall coefficient that changes from negative to positive values as the sample is cooled down from room temperature [Fig. 1.59(c)]. The hole concentration calculated in the one-valley model for YbB6−δ showed that the increase of boron deficiency from 0.09% to 0.6% decreases the concentration of bulk holes from 1 hole per 147 f.u. in crystal 1 to 1 hole per 500 f.u. in crystal 2. Analysis of RH for crystal 2 in the range of negative S at T < 20 K in the model of two groups of charge carriers suggests a noticeable T-independent contribution from some small electron pocket, and due to the high Hall mobility, the corresponding partial contribution to the total conductivity below 20 K rises and may be a possible origin of the negative Seebeck effect in YbB5.96 . The authors also conjectured that electron states originate at the surface, where an n-type inversion layer is induced due to band bending. Assuming its characteristic width to be ∼10 nm, the surface charge-carrier concentration can be estimated as n2D ≈ 1010 cm−2 , the corresponding Fermi wave vector and Fermi energy being equal to ∼2.5 × 105 cm−1 and 0.12 meV, respectively. The authors concluded that the control of stoichiometry may serve as an effective tool for tuning the parameters of electron transport in YbB6 [263]. The theoretical prediction of topological properties in YbB6 [450] attracted considerable attention to this compound in recent years as another possible candidate of a moderately correlated topological insulator similar to SmB6 . Experimental studies have been performed both on samples grown by the flux [451–456] and by the FZ melting method [423, 457]. Opinions still vary, and the experimental verification of the topological nature of electronic states in YbB6 remains the subject of an intense ongoing debate [453, 456]. Angle-resolved photoelectron spectroscopy (ARPES) measurements performed by Frantzeskakis et al. [457] did not confirm the topological nature of YbB6 . As the authors reservedly concluded,

Rare-Earth Hexaborides (RB6 ) 135

“The simplest explanation is that this material is not a topological insulator, but it is conceivable that topological character may still be found for different—as yet unobserved states or perhaps at different cleavage surfaces of YbB6 ”. A few years later, evolution of YbB6 from a topologically trivial semiconductor to a possible topologically nontrivial gapped phase at high pressures was demonstrated in a study of high-pressure transport, XRD, x-ray absorption, and Raman scattering measurements of the samples grown by the flux method [452, 454]. The YbB6 crystal structure was constant in the studied pressure range up to 50 GPa; above 18 GPa, a gradual valence change from the nonmagnetic Yb2+ to magnetic Yb3+ was accompanied by an increase in resistivity, and a transition to the topological Kondo insulator state is possible above 35 GPa [454]. According to the authors, “it remains to be seen if the chemical pressure, uniaxial strain, or their combination could stabilize the topological Kondo insulator state in this compound closer to the ambient pressure and temperature conditions” [454]. YbB6 nanowires According to the two-channel conductance model of a topological Kondo insulator, the total conductance may be tuned by changing the ratio of the surface and bulk contributions [392, 394, 454]. Keeping this in mind, Han et al. [458] synthesized highquality single-crystalline YbB6 nanowires to maximize the surface conduction. These crystals were produced at very low trigger temperature of 200–240 ◦ C by a high-pressure solid state method with a new chemical reaction using Yb, H3 BO3 , Mg, and I2 as starting materials. A comprehensive characterization has shown that the diameter and length of typical single YbB6 nanowires are about 120 nm and more than 7 μm, respectively, the growth direction is preferentially along [001]. The nanowires crystallize in the CaB6 -type structure with the lattice parameter a = 4.148(8) A˚ and a boron-deficient composition close to YbB5.8 . The temperature dependence of the resistivity shows that the YbB6 nanowires undergo a semiconductor-insulator transition below 20 K with an activation energy E ≈ 1 meV and the inverse resistivity ratio ρ2 K /ρ300 K = 49 (see Fig. 1.60). Appearance of the mixed valence of Yb ions due to the boron deficiency results in the anomalous electronic transport in the YbB6 nanowires, which the authors

136 Crystal Chemistry and Crystal Growth of Rare-Earth Borides

Figure 1.60 (a) SEM image of the YbB6 nanowire device with four Pt electrodes; (b) Temperature-dependent resistivity of the YbB6 nanowire from 2 to 300 K; (c) The normalized resistivity, ρ(T )/ρ300 K , vs. temperature; (d) Logarithm of the resistivity, ln ρ(T ), as a function of inverse temperature, 1/T . Reproduced from Han et al. [458].

describe as “striking materials with potentially novel quantum phenomena” [458].

1.5.5 Yttrium Hexaboride (YB6 ) Yttrium hexaboride (YB6 ) is a superconductor with the secondhighest transition temperature among boride compounds after MgB2 , but the reported values of the critical temperatures vary within a broad range, 1.5 K ≤ Tc ≤ 8.4 K, depending on the conditions of sample preparation. Isostructural LaB6 with nearly the same lattice parameter and a very similar electronic structure is not superconducting down to 5 mK [459]. So a lot of effort has been applied to understand the mechanisms governing superconductivity or the absence thereof in RE hexaborides. For polycrystalline YB6 samples prepared by borothermal reduction, Tc values varied depending on the B : Y2 O3 ratio of the starting charge. Samples with B : Y2 O3 = 14, 20, and 26 had

Rare-Earth Hexaborides (RB6 ) 137

Tc < 1.5, 6.8 ± 0.1, and 6.3 ± 0.3 K, respectively. In the first case, a mixture of YB4 and YB6 was obtained, while in the last two cases YB4 , YB6 , and YB12 were present in the resulting samples [460]. For synthesized YB6 powders, the following Tc values are known: ∼6.0 K [461], 5.5 K [462], 6.5–7.1 K [463], and Tc = 6.8 K for arcmelted YB6 [460]. Fisk et al. [71] grew the first YB6 single crystals, presumably nonstoichiometric (B-rich), from arc-melted buttons in a flux of 5 at. % Al and 95 at. % Ga in an Al2 O3 crucible under Ar atmosphere at 1550 ◦ C. For incongruently melting compounds such as YB6 , this is a good method that allows varying the crystal composition by changing the charge composition. Typically, nicely faceted 0.5–2 mm crystals in the form of 100 cubes or axis prisms were grown in a 24 h run. On varying the nominal composition in the starting charge from YB6 to YB9 , the lattice parameter of the product changed from ˚ and the RRR from 5.4 to 2.4, respectively. Such an 4.1021 to 4.1027 A, increase in the lattice parameter was explained by the appearance of vacancies, which are larger than yttrium ions. The superconducting temperature was 5.8 ± 0.1 K for all grown YB6 crystals, independent of the starting charge composition. This value is lower than the onset Tc of 7 K found in the arc-melted buttons and of 8.4 K in the splat-cooled sample of nominal composition YB6 . The latter value of the resistive Tc onset is the maximal Tc value for YB6 ever reported [71]. YB6 melts peritectically at 2600 ◦ C according to the Y–B phase diagram in Fig. 1.61. Besides, YB6 is located at the border of the structure-type stability due to the small yttrium ionic radius, r(Y3+ ) = 0.905 A˚ [56]; therefore, its existence is also possible in the presence of defects in both sublattices. All YB6 crystals grown later were obtained by the TSFZ technique with RF or optical heating. Details of the growing procedure including the B : Y ratio, technological parameters (the growth rate, pressure of inert gas, etc.) vary, but the essence remained the same: The feed rod needs to have an excess of boron. Kunii et al. [234, 465, 466] grew YB6 single crystals with Tc = 7.1–7.5 K. The TSFZ technique was used for the incongruently melting TbB6 , DyB6 , HoB6 , and YB6 in a furnace with RF heating

138 Crystal Chemistry and Crystal Growth of Rare-Earth Borides

Figure 1.61 Phase diagram of the Y – B system. After Liao et al. [464].

[234]. The source RE-boride powders (R = Y, Tb–Ho) were synthesized by borothermal reduction of the corresponding oxides based on the formation of hexaboride with some excess of boron to obtain a B-rich molten zone. The purity of the RE oxide powders was 99.99%, and that of the amorphous submicron boron powder was 99.999%. The synthesized powders represented a mixture of tetra-, hexa- and dodecaborides of the corresponding metals. From these powders, rods were prepared, which were used for growing crystals with a crystallization rate of 4 mm/h under 1 MPa of pressurized high-purity Ar (99.9995%) to prevent RE deficiency. The obtained crystals were about 100 mm in length and had a single-crystal core of stoichiometric RB6 composition, surrounded by a polycrystalline surface skin (∼1 mm) of a mixed composition [234]. Otani et al. [229, 467] prepared the source powders for the growth of YB6 by mixing commercial YB4 powder with amorphous boron to control the composition. The pressed rods were sintered in vacuum at 1700 ◦ C for 30 min, reaching 60% of the theoretical density. The crystal, ∅1 cm × 6 cm, was grown under a pressure of

Rare-Earth Hexaborides (RB6 ) 139

0.5 MPa of Ar gas by the TSFZ method with RF heating. No seed crystal was used. The feed rate in the upper shaft was 30% higher than the growth rate in the lower shaft in order to compensate for the low density of the feed rod. Only the growing crystal was rotated, at a rate of 6 rpm, and the crystallization rate was less than 1.3 cm/h. An YB6 crystal was prepared from a feed rod having a composition of B : Y ≈ 6.68, while the B : Y ratios in the molten zone and in the resulting crystal were 11.9 and 6.10, respectively. The starting part of the zone-melted ingot contains the YB4 phase inferred by the high power required to melt the sintered rods and in agreement with the maximum in the Y–B phase diagram corresponding to the YB4 phase (Fig. 1.61). But after 5 mm of growth, the crystal consists only of the YB6 phase without YB4 inclusions or voids. The [001] growth direction for these YB6 crystals was determined by the orientation of the initial part of the grown bar, since the [001] orientation is predominant for YB4 . The grown YB6 single crystal was characterized by a Tc = 7.5 K and RRR = 4.3. YB6 single crystals with superconducting Tc ’s in the range of 4.2–7.6 K, grown either from natural boron (nat B) or with enriched boron isotopes (10B, 11B), were studied by various experimental methods [72, 73, 468–470]. These crystals were grown by the TSFZ technique with RF heating in the specially designed “Crystal111A” apparatus (Fig. 1.4). The following starting compounds were used: nat B (20% 10B, 80% 11B); 10B (typical enrichment 98.4%), or 11 B (typical enrichment 99.5%), with a purity of at least 99.5%; and Y2 O3 with a purity of 99.999%. The source hexaboride rods were sintered from synthesized powders obtained by borothermal reduction of Y2 O3 by boron of the corresponding type (nat B, 10B, 11 B), and with an excess of boron. The optimal boron composition of the initial sintered rods was consistent with YB6.65 – YB6.85 which made it possible to (i) decrease the melting temperature below the peritectic melting point (2600 ◦ C, see Fig. 1.61), (ii) obtain singlephase ingots, and (iii) improve the real structure of the crystals. Other optimization parameters were the pressure of high-purity Ar gas in the growth chamber (0.7–1.3 MPa) and the growth rate 0.13–0.22 mm/min [72, 73]. Because of the zone-refinement effect during the process of crystal growth, the impurity concentration did

140 Crystal Chemistry and Crystal Growth of Rare-Earth Borides

not exceed 0.001 wt. %—this is very important because Tc is rapidly suppressed by impurities [461, 462]. Over the first few mm, the growth process was unstable, and the zone-melted bar was double-phase: First the tetragonal YB4 single crystal with preferential orientation [001] started to grow, but after enrichment of the molten zone by boron, the relationship between YB4 and YB6 gradually changed towards the hexaboride. The growth process was stabilized, the zone temperature was decreased, and at a certain Y : B ratio in the boron-enriched melt, the growth of the single-phase yttrium hexaboride began. Due to competitive growth, one of the grains displaced the other grains, and a perfect ∅5–6 mm × 30–40 mm single crystal grew. X-ray powder analysis of the crushed single crystal revealed reflections of the CaB6 structure type only, while Laue backscattering patterns demonstrated the absence of splitting of the point reflections and confirmed the lack of domains with a misorientation more than several tenths of a degree (procedure accuracy). The lattice constant was found to be almost insensitive to the Y : B ratio within the experimental accuracy, equal to 4.1002(5) A˚ for YB5.7 with Tc = 7.6 K and 4.1000(5) A˚ for YB5.9 with Tc = 6.6 K [72]. For all four investigated single crystals listed in Table 1.17, the lattice constant was 4.1001(5) A˚ [73]. Lortz et al. [72] also studied samples with Tc = 6.5, 7.2, and 7.4 K. Unlike LaB6 , which shows no superconductivity down to 5 mK with RRR = 160 [459], the Tc of the YB6 samples showed a positive correlation with RRR values between 3.05 to 4.58, i.e., samples with a larger RRR tend to have higher superconducting transition temperatures. This tendency was also noted in Refs. [73, 471]. In order to reveal a relationship between the composition and Tc , one of the crystals was cut into plates perpendicular to the growth axis from the range that contains YB4 up to the end of the YB6 crystal. For each of these plates, the Y : B ratio was determined by EDX, and the Tc by ac susceptibility. A clear relationship between Tc and the B : Y ratio emerged, described by an empirical formula Tc (YB6+x ) ≈ 6.25 − 4.3x ± 0.25 K. This confirmed that Tc can be enhanced by lowering the boron concentration, and the highest Tc is obtained in B-deficient samples with a B : Y ratio below 6 [72].

Rare-Earth Hexaborides (RB6 ) 141

Figure 1.62 Temperature dependence of the resistivity ρ(T ) for different YB6 samples. The ρ(T ) curve of the LaB6 crystal is also shown for comparison. Inset shows the region of the superconducting transition. Reproduced from Sluchanko et al. [73]. Table 1.17 Some characteristics of the yttrium hexaboride samples studied in Ref. [73] Sample No. 1 No. 2 No. 3 No. 4

Composition nat

Y0.95 B5.94 Y0.945 nat B5.93 Y0.96 10B5.95 Y0.97 11B5.985

Tc (K)

ρ0 (μ · cm)

RRR

7.4 7.3 6.2 4.2

8.28 9.68 13.35 25.6

4.47 4.13 2.3 2.25

142 Crystal Chemistry and Crystal Growth of Rare-Earth Borides

Sluchanko et al. [73] comprehensively studied YB6 single crystals with different chemical composition and different character of lattice defects. Figure 1.62 shows the temperature dependence of the resistivity ρ(T ) for these samples, in comparison with the corresponding curve for LaB6 . Some characteristics of the studied YB6 samples are also listed in Table 1.17, including their chemical composition, superconducting transition temperatures (Tc ), residual resistivity (ρ0 ), and the RRR, ρ300 K /ρ4.2 K . The proximity of YB6 to the border of the structure-type stability range implies that the presence of B and Y defects in the YB6 lattice is required for its stabilization. The large values of the residual resistivity, ρ0 ≈ 8– 25 μ · cm, are due to the strong scattering of charge carriers by crystal defects and inhomogeneities. And the more defects, the higher is the critical temperature, Tc . This result implies that the number of vacancies on the boron (0.25–1.1%) and yttrium (3– 5.5%) sites may be considered as a measure of this nonequilibrium state in YB6 . The increase in the vacancy concentration serves as a stabilizing factor of the YB6 lattice, leading to an enhancement of the electron–phonon scattering and, consequently, to an increase in Tc [73]. A symmetry lowering in the lattice structure was suggested from the Raman scattering data [472]. The temperature and polarization dependence of the YB6 Raman spectra in the temperature range from RT to 2.4 K point to the space group P 4/mmm for YB6 from group-theoretical considerations, rather than P m3m. Within the present experimental accuracy, no anomalies in the temperature dependence of the phonon spectra have been observed at the superconducting transition temperature [472].

1.5.6 Boron Isotope Effects Finally, let us briefly consider possible effects of the boron isotopic composition on the physical properties of hexaborides. As a rule, natural boron (nat B) is used for crystal growth. It consists of two stable isotopes: 10B (∼20 at. %) and 11B (∼80 at. %). The statistical distribution of these isotopes in the boride lattice (the mass difference is about 10%) results in the splitting of the nominally degenerate atomic vibrations in the boron clusters, and their

Rare-Earth Hexaborides (RB6 ) 143

coupling with electrons (electron–phonon interaction) leads to the local Jahn–Teller distortions and a possible symmetry reduction [473]. When the degeneracy of vibrations is lifted, symmetryforbidden vibrations are activated, and the mass difference of the isotopes may influence their frequencies depending on the positions and the number of atoms of a given isotopic species (Table 1.18) [250]. The first direct evidence of structural distortions in the cubic CaB6 -type lattice and its symmetry lowering due to the mixture of boron isotopes and the presence of vacancies in the hexaboride lattices was obtained from Raman scattering, infrared (IR), transmittance, and photoacoustic spectra of AE and RE hexaborides [250, 474, 475]. The spectra have a richer structure than expected for a simple cubic lattice of m3m symmetry. They display additional lines that are symmetry-forbidden for this lattice but are consistent with the calculations of the Raman and IR linearity spectra based on force-field calculations and the assumption of symmetry reduction due to isotopic substitution. Depending on the number and relative position of 10B isotopes in a B6 cluster, deviations from Oh symmetry are expected, and their probabilities for the natural boron with a statistical distribution of isotopes are listed in Table 1.18 [250]. Similarly, a detailed analysis of the crystal distortions in RE dodecaborides (RB12 ) due to the symmetry reduction in B12 cuboctahedral clusters and the dependence of lattice parameters on the isotopic composition of boron is given in Chapter 3 of this book. Isotope effects are also covered in Chapter 5, where Raman-spectroscopy studies of metal borides are reviewed. Here I Table 1.18 Effects of the natural isotopic composition of boron on the symmetry of B6 octahedra in RB6 Composition 11

R B6 R 10B11B5 R 10B2 11B4

Positions of 10B atoms

Symmetry

Percent probability

— Arbitrary Two axes Same axis

Oh C 4v C 2v D4h

67.70 24.40 4.75 1.19

Note: The probabilities and symmetries corresponding to 0, 1, or 2 atoms of 10B and their relative position are given. Three or more 10B isotopes occur with a negligibly small probability of < 1% and are therefore omitted [250].

144 Crystal Chemistry and Crystal Growth of Rare-Earth Borides

would like to mention a few of the most striking examples related to the influence of the isotopic composition on the properties of hexaborides. In Section 1.5.3, we have already seen that the isotopic composition of SmB6 has a profound influence on the residual resistivity [421], and that isotopically pure 154 Sm11B6 crystal displays the largest IRR value (Fig. 1.47). Paschen et al. [476] reported a similar observation on flux-grown single crystals of EuB6 . The temperature dependence of the resistivity for three samples of europium hexaboride with different isotopic compositions of boron, Eu10B6 , Eu11B6 , and Eunat B6 , are shown in Fig. 1.63. The residual resistivity for Eunat B6 with natural boron has the highest value, 40.3 μ ·cm, whereas the corresponding values for isotope-enriched Eu10B6 and Eu11B6 are 26.8 and 15.1 μ · cm, respectively. The single crystals in this study were obtained by solution growth from Al flux using the appropriate amounts of pure elements [476], but the details of preparation, purity of the raw components, and the enrichment of individual boron isotopes are not mentioned.

Figure 1.63 Electrical resistivity ρ(T ) for EuB6 samples with different boron isotopes: Eu10B6 , Eu11B6 , and Eunat B6 . Reproduced from Paschen et al. [476].

Rare-Earth Hexaborides (RB6 ) 145

Significant isotope effects were also observed in the RRR and the superconducting transition temperature of YB6 , as was discussed in Section 1.5.5 (see Table 1.17 and Fig. 1.62). The two samples with natural boron are characterized by a higher RRR and a higher value of Tc as compared to the isotopically pure samples of Y10B6 and Y11B6 [73]. The changes in residual resistivity depending on the isotopic composition in all the mentioned compounds may serve as indirect evidence of the local lattice distortions, which lead to a higher level of structural disorder and stronger electron–phonon scattering in samples with a mixture of boron isotopes. Werheit et al. [477] used Raman spectroscopy to investigate the isotopic phonon effects in nonmagnetic LaB6 single crystals with a systematically varied ratio of the 10B and 11B isotopes (see Chapter 5). The high-quality LaB6 crystals for this study were grown from amorphous natural boron (nat B, AVIABOR, Dzerzhinsk, Russia) and polycrystalline isotopically enriched boron: 10B (Institute of Stable Isotopes, Tbilisi, Georgia, typical enrichment 97.1%), and 11B (Ceradyne, USA, typical enrichment 99.5%). The chemical purity of nat B was 99.9%, and that of the 10B and 11B isotopes was 94.6% and 99.9%, respectively. First, La(10B1−x 11Bx )6 powders were synthesized with x ≈ 0, 0.25, 0.5, 0.75, 0.81 (nat B), and ∼1 by borothermal reduction of La2 O3 (99.997%) from the Federal State Research and Development Institute of Rare Metal Industry (Moscow, Russia). With the exception of the isotopically almost pure 11B and 10B, the intended isotopic composition was achieved by the respective mixtures of 10B and nat B; accordingly, samples with the lowest concentrations of isotopic impurities were La(10B0.971 11B0.029 )6 and La(10B0.005 11B0.995 )6 , which were designated as isotopically pure materials for convenience. The prepared powders in the form of pressed and sintered rods were then used for growing high-quality single crystals by the induction zone melting method under 0.2 MPa pressure of high-purity Ar gas; the growth rate was 30 mm/h with the feed rod counter-rotating at 5 rpm. Apart from “isotopic impurities”, the total chemical impurity content of these samples was less than 10−3 wt. %. A detailed group-theoretical analysis elucidated the correlation between the distortions of B6 octahedra and the splitting or broadening of phonon modes. It has been found that LaB6 does

146 Crystal Chemistry and Crystal Growth of Rare-Earth Borides

not have cubic symmetry as assumed so far. A further symmetry reduction of the B6 octahedra occurred in isotopically mixed crystals. Moreover, it has been shown that the distribution of isotopes is not random, as commonly assumed. The correlations in their spatial distribution due to the clustering of identical isotopes tends to increase for lighter atoms [477]. Finally, samples from the same LaB6 single crystals with varying isotopic content were used to study the origins of the extraordinarily low work function of LaB6 , linking it to the properties of conduction band electrons. Lanthanum hexaboride is among the most efficient electron-beam sources with a very high brightness of thermionic emission. In a study that combined IR spectroscopy, dc resistivity, and Hall-effect measurements, Zhukova et al. [264] suggested that the low work function of LaB6 is determined by nonequilibrium conduction electrons coupled with vibrations of both the Jahn–Teller unstable rigid boron cage and the rattling modes of La ions, loosely bound to the lattice [264].

1.6 Higher Borides 1.6.1 Rare-Earth Dodecaborides (RB12 ) Rare-earth dodecaborides (RB12 , R = Sc, Y, Gd–Lu) crystallize in the UB12 -type cubic structure, which was first determined for UB12 back in 1949 [478]. The first successful syntheses of RB12 (R = Y, Dy–Tm, Lu) in powder form and their crystal-structure determination were carried out in 1960 [479], followed shortly thereafter by R = Tb and Yb [480]. The first RB12 single crystals were obtained in the 1990s [109, 481–483], which enabled numerous studies of their physical properties over the last 30 years. The obtained results have been summarized in several reviews [58,484,485] and are discussed in Chapters 3–6 of this book. The UB12 -type crystal structure with the space group F m3m – 5 Oh (No. 225) is shown in Fig. 1.64(a). It can be described in terms of a modified fcc unit cell with the metal atoms located in the Wyckoff position 4a (0 0 0) in the center of the B24 truncated

Higher Borides 147

(a)

(c)

(b)

(d)

Figure 1.64 Crystal structures of RB12 : (a, b) in the cubic F m3m phase; (c, d) in the tetragonal I 4/mmm phase. In (a) and (c), the large and small spheres represent the metal and boron atoms, respectively. In (b) and (d), the red crosses mark the center positions of the B12 cuboctahedral cages. Reproduced from Liang et al. [488].

octahedron formed by B atoms in the 48i (± 12 x x) position, where x is approximately 1/6. The cubic unit cell therefore contains 4 formula units. In this structure, a boron atom is bonded with two metal atoms and five other boron atoms, four of which belong to a given cuboctahedron and the fifth to a neighboring one, ˚ within resulting in two types of B–B bonds: (B–B)intra (∼1.79 A) ˚ the cuboctahedron and the somewhat shorter (B–B)inter (∼1.71 A) between the neighboring cuboctahedra [34]. Alternatively, the same UB12 -type structure can be described as a modified rock-salt structure that consists of two interpenetrating fcc sublattices: one formed by metal atoms and the other by the B12 cuboctahedral cage units shifted by the vector ( 21 0 0), as shown in Fig. 1.64(b) with spheres and crosses, respectively [35, 486, 487].

148 Crystal Chemistry and Crystal Growth of Rare-Earth Borides

Only the smaller members of the lanthanide series from Tb to Lu and Y with R 3+ radii in the range of 0.85–0.923 A˚ form stable dodecaboride phases [35]. GdB12 with the Gd3+ ion radius of 0.93 A˚ has been synthesized under high pressure above 60 kbar and at the temperature of 2100 ◦ C due to the larger compressibility of the lanthanides than that of boron, a(GdB12 ) = 7.524 A˚ [113, 128]. The RT structural parameters of RB12 (R = Y, Tb–Lu) in the F m3m presentation are summarized in Table 1.19 [484]. Their values are mostly consistent with all the previously obtained results of crystal-structure refinements collected by Werheit et al. [489] and ˚ [490] the recent single-crystal studies of TmB12 (a = 7.47159(3) A) ˚ and LuB12 (a = 7.4645(1) A) [491], which are discussed in detail in Chapter 3. The negative thermal expansion of LuB12 in the range of 50– 120 K (see Chapter 4, Fig. 4.10) stimulated a thorough x-ray crystal structure refinement of this compound over a wide temperature range, which revealed a weak tetragonal distortion (space group I 4/mmm) at temperatures below 150–160 K, although the improvement in the R-factor was insignificant as compared to refinements assuming the cubic symmetry F m3m [492]. ScB12 was then the only known dodecaboride with a tetragonal I 4/mmm structure, as shown in Fig. 1.64(c, d) [141, 486, 493–495]. Its crystal structure with lattice parameters a = 5.236(1) A˚ and c = 7.358(1) A˚ was refined by Aksel’rud et al. [495]. It is related to the cubic structure with the same c√axis and a 45◦ rotation of the a and b axes, so ˚ the tetragonal distortion therefore that a = b ≈ c/ 2 = 5.203 A; amounts to only 0.6%. Flachbart et al. [484] suggested that the tetragonal distortions in ScB12 and LuB12 are of similar origin, and at higher temperatures ScB12 should crystallize in the cubic structure. Indeed, high-temperature x-ray diffraction on ScB12 revealed that the structure at 773 K is cubic (F m3m) with the lattice parameter a = 7.44146(6) A˚ [488]. A symmetry lowering to the tetragonal (I 4/mmm) crystal structure, associated with a distortion of the regular B12 cuboctahedra, must therefore take place at an intermediate temperature.

Table 1.19 Main structural parameters of R nat B12 (R = Y, Tb–Lu) in F m3m cubic symmetry (after Flachbart et al. [484]) YB∗12

TbB12

DyB12

HoB∗12

ErB∗12

TmB∗12

YbB12

LuB∗12

˚ a (A) V (A˚ 3 ) dcal (g/cm3 ) ˚ R–B (A) ˚ (B–B)inter (A) ˚ (B–B)intra (A)

7.5001 421.875 3.43 2.78312 1.68424 1.80924

7.500603 421.9763 4.543 2.78775 1.7184 1.792818

7.500412 421.9442 4.600 2.78745 1.7183 1.792817

7.492394 420.5924 4.653 2.78455 1.7163 1.790916

7.484034 419.1864 4.706 2.782011 1.7197 1.7874

7.475872 417.8162 4.748 2.77733 1.7062 1.79031

7.469793 416.7983 4.825 2.77614 1.7113 1.785514

7.464834 415.9684 4.865 2.77334 1.7043 1.787414

0.1690216 0.0050042 0.0056427 0.0010038 0.0054320

0.1690215 0.0046241 0.0052426 0.0010036 0.0050319

0.16883 0.0052596 0.0050663 0.0006192 0.0051249

0.1693310 0.0033328 0.0039718 0.0011523 0.0037613

0.1690213 0.0032634 0.0038922 0.0009730 0.0036816

0.1693113 0.0012333 0.0023422 0.0010632 0.0019715

Boron

Parameters

x(B) 0.1706 0.1690117 U 11 (A˚ 2 ) 0.0054645 U 22 (A˚ 2 ) 0.0060929 U 23 (A˚ 2 ) 0.0010140 U iso (A˚ 2 ) 0.0058821

R

U iso (A˚ 2 )

R1 w R2

0.061

0.0177 0.0399

0.0170 0.0378

Single-crystal x-ray diffraction study. Note: Subscripts indicate measurement uncertainties.

0.0163 0.0357

0.0341 0.0743

0.0103 0.0217

0.0146 0.0311

0.0100 0.0246 Higher Borides 149

∗

0.002684 0.002593 0.002533 0.003077 0.002582 0.002263 0.000014

150 Crystal Chemistry and Crystal Growth of Rare-Earth Borides

Earlier attempts to determine the crystal structure of ScB12 led to conflicting conclusions: Matkovich et al. [486] refined the structure as tetragonal (I 4/mmm), while Bruskov et al. [496] suggested that it was cubic (F m3m). In the first case, crystals were prepared by heating the metal oxide and boron with the B : Sc ratio of 7 : 1 at a temperature of 2000–2500 ◦ C and then quenched by dropping directly into a beaker of water. After washing with concentrated HCl, crystals of ScB12 could be easily handpicked under a microscope [486]. In the second case, single crystals were extracted from an ingot obtained by the fusion in an electric arc furnace under an Ar atmosphere of metallic scandium (purity 99.75 wt. %) and fine crystalline boron (99.4 wt. %) [496]. In both cases the ScB12 single crystals were very small, suitable only for x-ray diffraction studies. More recent high-precision x-ray diffraction studies on single crystals of RB12 (R = Ho–Lu) revealed that a weak latticesymmetry lowering below room temperature is common to all the dodecaborides [490, 497–502]. These updated results on the crystal structure of RB12 were obtained due to the development of a new formalism and modern algorithms for the analysis of temperature-dependent changes in the atomic displacement parameters [503]. Moreover, quantum-chemical calculations have shown that the microscopic mechanism of the lattice-symmetry lowering in RB12 is determined by Jahn–Teller distortions of the B12 clusters [498]. The obtained results on the RB12 crystal structure, including the boron isotope effects, are presented in Chapter 3 of this book. The dodecaborides of Y, Dy, and Er melt peritectically [34, 155, 156, 464]. Spear [34] predicted that the then-unstudied dodecaborides with R = Tb, Ho, Tm–Lu should behave like YB12 , DyB12 , and ErB12 , and their phase diagrams in the range of concentrations around R : B = 1 : 12 are similar (see Figs. 1.10, 1.11, 1.61) [34, 284]. ScB12 is the only dodecaboride that melts congruently (Fig. 1.9) [34, 112, 154]. Therefore, all single crystals of RE dodecaborides with R = Y, Sc, Ho–Lu have been grown by the TSFZ method with either optical heating (for YbB12 and its solid solutions) or RF heating (all other RE dodecaborides).

Higher Borides 151

Before discussing these works in more detail, let us consider the general principles underlying the methodology of the crystal growth, which is done in a multistage process: (1) synthesis of the individual RE dodecaborides or their solid solutions by a borothermal reduction of the corresponding individual metal oxides or their mixtures in vacuum at approximately 1900 K by routine solid-state reactions: R 2 O3 + 27 B → 2 RB12 + 3 BO ↑ (R = Sc, Y, Tb–Lu), x R2I O3

+

(1−x)R2II O3

+ 27 B → 2

II R xI R1−x B12

(1.1)

+ 3 BO ↑; (1.2)

(2) compacting the obtained powders into rods and their subsequent sintering in vacuum or inert gas medium; (3) growth of single crystals by the TSFZ technique; (4) characterization of the obtained crystals and their preparation for subsequent physical-property measurements. Boride synthesis from RE oxides and boron is chosen because it allows getting the purest materials in terms of RE impurities, as the initial oxides are usually available in higher purity than the corresponding metals. Due to the peritectic melting of RB12 (except ScB12 ), all source powders are synthesized with an excess of boron that depends on the RE element, but the technological parameters of the crystal growth are determined individually for each RE dodecaboride, also taking into account the specific features of a particular crystal-growth setup. First, let us consider the work on the growth of RB12 single crystals (R = Sc, Y, Tb–Tm, Lu), performed in our laboratory of RE refractory compounds of the Institute for Problems of Materials Science of NAS (Kyiv, Ukraine) by the TSFZ technique with RF heating. It covers all RE dodecaborides except pure YbB12 , but including R1−x Ybx B12 solid solutions with R = Tm, Zr in single-crystalline form. The following RE oxides from the Federal State Research and Development Institute of Rare Metal Industry (Moscow, Russia) were used as starting materials: Sc2 O3 (99.95%), Y2 O3 (99.994%), Tb2 O3 (99.996%), Dy2 O3 (99.979%), Ho2 O3 (99.9995%), Er2 O3 (99.988%), Tm2 O3 (99.986%), Yb2 O3 (99.998%), and Lu2 O3 (99.9985%). RB12 single crystals were

152 Crystal Chemistry and Crystal Growth of Rare-Earth Borides

prepared using pure natural amorphous boron (nat B, AVIABOR, Dzerzhinsk, Russia) or isotope-enriched boron: 10B (Institute of Stable Isotopes, Tbilisi, Georgia, typical enrichment 98.4%) and 11 B (Ceradyne, USA, typical enrichment 99.5%). The purity of the amorphous boron (nat B) was 99.3–99.9 wt. %, depending on the batch; the main impurities—oxygen, hydrogen, magnesium, chlorine—are effectively removed during the reduction of oxides in the borothermal process in vacuum and then also during zone melting. The total content of non-RE impurities in the grown crystals does not exceed 10−2 wt. %; the concentration of RE impurities is determined by the purity of the corresponding source oxides (from ≤ 5 ppm in Ho to ≤ 500 ppm in Sc). For example, the total impurity concentration in the Lunat B12 single crystal, which was used for the single-crystal structure study, was about 100 ppm, among which two magnetic impurities were Fe (10 ppm) and Yb (10 ppm), according to the optical emission spectral analysis [491]. ScB12 A single crystal of ScB12 [001] with approximate dimensions ∅5 mm × 60 mm has been grown by TSFZ melting with RF heating [494]. The synthesized polycrystalline ScB12 contained traces of ScB2 , despite the fact that the initial mixture had an excess of boron with respect to the reaction calculated for stoichiometry. Homogenizing annealing led to the narrowing of diffraction lines, but additional lines appeared that could not be identified within the known phases based on scandium and boron. Acid treatment did not lead to any change in the diffractogram. A sintered rod from the obtained powder was used for the growth of ScB12 single crystals. When the ScB12 sintered feed rod was combined with a crystal of TmB12 [001] used as a seed, a mixed Scx Tm1−x B12 single crystal with continuously varying composition (0 ≤ x ≤ 1) was obtained. According to the microprobe analysis (JEOL Superprobe), at a distance of 7–10 mm from the starting melting zone, traces of metallic Tm were already less than 100 ppm, and a perfect ScB12 [001] single crystal grew afterwards. The crystal structure of this Scx Tm1−x B12 crystal evolved from cubic (F m3m) at x = 0 to tetragonal (I 4/mmm) at x = 1. The part of the crystal with continuously varying composition was used for the Raman scattering study [489], and the homogeneous part with pure ScB12

Higher Borides 153

was used for crystal-structure refinement [495] and dilatometry measurements [504]. YB12 The first attempt to grow an YB12 single crystal by the TSFZ technique failed [76]. The source boride was synthesized from yttrium metal (>99.9%) and boron pieces (>99.0%) by arc melting under argon to form a button, which was remelted into a trough in a water-cooled copper plate to form a rod. The runs on YB12 were not very satisfactory due to the impossibility in manual mode to quickly react with power adjustments to the behavior of the molten zone that either grew too large or solidified [76]. The obtained melted bar was polycrystalline with the composition of Y0.92 B12 according to the density measurement. This result was attributed to the fact that the ionic size of yttrium is about the maximum for metals that form the dodecaborides phase, and the boron framework must “stretch” to accommodate the yttrium atoms [52]. Nevertheless, yttrium dodecaboride in single-crystalline form was subsequently grown [505]. The YB12 source powder was synthesized from Y2 O3 (grain size from microns to tens of microns) ˚ The solid-state and amorphous boron (grain size about 50 A). synthesis is limited by the diffusion of boron into oxide particles. Since synthesis time is limited by technological conditions, so in a system such as Y – B, where there are many phases, it is always possible to obtain impurity phases in addition to the main phase. Homogenizing annealing is not so much an attempt to bring the system to equilibrium as it is to obtain a single-phase composition. According to the x-ray phase analysis, the obtained YB12 powder contained YB6 and YB66 as impurity phases. So the synthesized powder was considered as a starting point for growth of a single crystal by the zone melting. The process of YB12 zone melting was unstable for the initial rods of the composition in the range Y0.90 B12 –YB12 . The growth was carried out with a crystallization rate of 1.2 mm/min, rotation of the feed rod at 20 rpm, and Ar pressure of 1 MPa. Yttrium was distilled off and accumulated in the molten zone, which at the corresponding Y : B ratio led to the crystallization of yttrium hexaboride as inclusions in the matrix from the dodecaboride phase. After this, the growth process was temporarily stabilized, and

154 Crystal Chemistry and Crystal Growth of Rare-Earth Borides

spontaneous growth of a two- or three-grain dodecaboride crystal began, until an excess of metal again accumulated in the zone. Thus, this process was periodic in nature, and regions of the single-phase growth reaching 15–20 mm alternated with two-phase regions of 2– 3 mm in length. Samples from the zone-melted YB6 tricrystal were used in dilatometry and heat-capacity measurements [504, 506]. A perfect YB12 single crystal was grown only with the initial composition of Y0.85 B12 and at the crystallization rate of 0.2 mm/min. Its real chemical composition was found to be Y0.90 B12 using an xray microanalyzer. This result is consistent with the conclusion [52] that yttrium dodecaboride exists in the form of an yttrium-deficient compound of the composition Yx B12 with x ≤ 0.92. The existence of defects in the lattice (possibly both for metal and boron) makes it possible to preserve the UB12 structure type for this compound, despite YB12 being located at the stability boundary of this structure type. This crystal was used for the 11B NMR study by Fojud et al. [505]. TbB12 Phase-pure terbium dodecaboride could not be obtained by zone melting even after exploring broad ranges of crystallization rates, 0.5–7.5 mm/min, and pressures, 0.5–2.2 MPa [481]. The obtained zone-melted bars consisted of terbium hexa- and dodecaborides. In fact, TbB12 melts peritectically, but La Placa et al. [480] obtained this phase by arc melting of the appropriate amounts of Tb metal and crystalline boron (99.9% and 99.0% purity, respectively) under 1 atm of argon. This phase can be obtained as a single crystal with the TSFZ method but only with an excess of boron, similar to yttrium dodecaborides (Y0.90 B12 ). Until now, all available studies of TbB12 have been conducted only on powder or compact sintered samples [481, 504, 507–509]. DyB12 Zone-melted dysprosium dodecaboride has been obtained only in polycrystalline form [504]. The source composition of the sintered rod was Dy0.95 B12 ; growth rate 1.5 mm/min; feed- and seedrod rotation rates 20 and 10 rpm, respectively; Ar pressure 2.2 MPa. This compound melts peritectically, but by optimizing the Dy/B ratio and choosing a lower growth rate, a DyB12 single crystal can possibly be grown.

Higher Borides 155

Table 1.20 Optimal single-crystal growth conditions for RB12 (R = Y, Ho, Er, Tm, Lu) and the experimental compositions of the resulting samples according to the hydrostatic density (D) and chemical (C) analyses [504]

RB12

Source compos.

YB12 HoB12 ErB12 TmB12 LuB12

Y0.85 B12 Ho0.96 B12 Er0.98 B12 Tm0.98 B12 Lu0.97 B12

Growth Ar Rotation rate (rpm) rate pressure (mm/min) (MPa) Feed rod Seed rod 0.20 0.40 0.55 0.45 0.45

1.0 0.9 0.6 0.5 0.5

20 5 15 10 10

0 2 0 2 4

Single-crystal compos. D

C

— HoB11.89±0.02 ErB11.81±0.02 TmB11.72±0.02 LuB11.83±0.06

Y0.90 B12 HoB11.91 ErB11.86 TmB11.99 LuB11.86

RB12 (R = Ho, Er, Tm, Lu) Single crystals of HoB12 , ErB12 , TmB12 , and LuB12 with dimensions of ∅5–6 mm × 40–100 mm were grown by the TSFZ technique in the sealed chamber of the “Crystal-111A” setup (Fig. 1.4) under high-purity (99.995% by volume) argon pressure. First, source dodecaboride powders were synthesized with a small excess of boron. In addition, a small piece of boron was introduced in the initial melting zone. Perfect single crystals have been grown due to the optimization of technological parameters (R : B ratio, crystallization rate, pressure of inert gas, etc.) individually for each dodecaboride. These optimized parameters are listed in Table 1.20. These four dodecaborides were grown in all three main orientations [001], [011], [111] using oriented single-crystal seeds. It should be noted that using a seed crystal with the [011] orientation resulted in mosaic crystals with a domain misorientation of approximately 0.5◦ –3◦ . Only the use of a seed with a deviation from the [011] growth axis of at least 5◦ –10◦ allowed us to grow highquality single crystals, which were used to prepare samples with the necessary orientation for physical measurements. Laue backscattering patterns from the central part of polished plates, cut perpendicular to the crystal growth axis, were used as an express assessment of the crystalline quality when optimizing the growth procedure. In addition, Laue patterns from both ends of the large crystal were compared to check if the orientation was preserved throughout its length. The absence of split reflections indicates that the misorientation of single-crystalline domains does

156 Crystal Chemistry and Crystal Growth of Rare-Earth Borides

not exceed a few tenths of a degree (resolution of the method), so our goal was to optimize the procedure until no splitting of Laue reflections was observed. The most complete picture of the single-crystal substructure is given by an x-ray topographic study, which allows obtaining a visual picture of the entire investigated surface of the crystal. We used this method to study the quality of three selected singlecrystal samples of LuB12 grown in the [001], [011], and [111] directions. The topograms were obtained for samples whose Laue patterns indicated the absence of a mosaic domain structure in the crystals with the [001] and [111] orientations and its presence in the [011] orientation. They were measured with Fe Kα radiation on polished plates, which were additionally etched in HNO3 + HF (10 : 1) to remove the surface layer that was potentially damaged during abrasive processing. It can be therefore assumed that the obtained reflection from certain planes corresponds to the actual structure of the crystal. According to the x-ray topograms shown in Fig. 1.65(a, c), the major part of the LuB12 [001] and [111] crystal rods (∅5–6 mm) consists of a single-crystalline core that is free of grain boundaries and slip bands, in accordance with the Laue backscattering patterns, which is surrounded by a thin ring (≤ 0.2 mm) with a small number of subgrains, misoriented relative to the core by no more than 5◦ . In contrast, the central part of the LuB12 [011] sample in Fig. 1.65(b) shows a subboundary of two grains with a mutual misorientation

(a)

(b)

(c)

Figure 1.65 X-ray topography from the lateral cross sections of as-grown LuB12 single crystals with the following growth directions: (a) [001], (b) [011], (c) [111] (after Shitsevalova [504]).

Higher Borides 157

(a)

(b)

(c)

Figure 1.66 Bright-field images of dislocation walls in RB12 [011] single crystals: (a) a subboundary is parallel to the growth axis; (b) a subboundary is inclined to the growth axis, (c) a network of dislocations (after Shitsevalova [504]).

of about 10 –15 . An analysis of grain boundaries in the RB12 [011] crystals shows that they are formed by a wall of dislocations, and their orientation relative to the main growth axis can be different, as illustrated in Fig. 1.66: (a) a subboundary is parallel to the growth axis, (b) a boundary is inclined to the growth axis, (c) a subboundary consists of a networks of dislocations, i.e., not only the inclination of the boundary takes place, but also torsion elements are present. The linear dislocation density varies in the range of 2 × 105 to 5 × 105 cm−1 [504]. The central part of crystals grown with a slight deviation of 5◦ –10◦ from the [011] growth axis was defect-free according to the electron diffraction pattern. One possible reason for the crystal growth with mosaic domains in the [011] direction is the selective deposition of impurities associated with the specifics of the crystal chemistry of dodecaborides. This direction corresponds to the R–R bond. The same situation is observed in RE hexaborides for crystals grown in the [110] direction which also corresponds to the R–R bond. According to the electron Kikuchi patterns in Fig. 1.67(a, b) and point electron diffraction patterns (not shown), which correspond only to the UB12 type structure, defects in the RB12 single crystals grown in the [001] and [111] directions are practically absent. Measured compositions of typical RB12 (R = Ho–Tm, Lu) single crystals are summarized in Table 1.20. According to chemical analysis, these RB12 crystals are metal rich. Their hydrostatic

158 Crystal Chemistry and Crystal Growth of Rare-Earth Borides

(a)

(b)

Figure 1.67 Symmetric electron Kikuchi patterns along the (a) [001] and (b) [111] directions in LuB12 single crystals (after Dudka et al. [491]).

density (ρhydro ) is lower than the corresponding x-ray density; therefore, the subtraction formula was adopted when determining the composition based on ρhydro . Errors in the density-derived compositions reflect a slight scatter in density values from sample to sample for the same dodecaboride. Despite the low accuracy of quantifying the metal and boron content by chemical analysis (±0.6 wt. % for metal, ±0.3 wt. % for boron) and a significantly better accuracy in determining ρhydro (±1 mg/cm3 or ±0.02%), the compositions obtained by both procedures are consistent with each other. The local crystal structure of RB12 (R = Ho–Lu) was studied in the temperature range of 5–300 K using EXAFS spectroscopy [510]. The displacement of 1–6% of R ions by about 0.2–0.3 A˚ from the centers of B24 clusters was explained by the presence of boron vacancies. The reproducible, sample-independent lattice parameters among different dodecaboride samples of the same metal (see Table 1.21) in combination with the composition data (Table 1.20) let us extend the conclusion about the narrow homogeneity range that was experimentally confirmed for DyB12 [153] and LuB12 [511] to all the other studied dodecaborides. Further, RB12 (R = Ho–Tm, Lu) single crystals have been grown with natural boron and with both isotopes, 10B and 11B. There were

Higher Borides 159

˚ of different R nat B12 (R = Tb–Tm, Lu) Table 1.21 Lattice constants a (A) samples [481] TbB12 DyB12

HoB12

ErB12

TmB12

LuB12

Source powder 7.505 7.50026 7.491610 7.48392 7.47502 7.46442 Beginning of — 7.50004 7.49233 7.48412 7.47485 7.46452 zone-melted rod End of zone-melted rod — 7.49993 7.49232 7.48373 7.47522 7.46442

no special differences in the technological parameters when growing single crystals with the three types of boron [489, 501, 512–517]. Samples prepared from perfect R N B12 (R = Y, Ho, Er, Tm, Lu; N = 10, nat, 11) single crystals were used in studies of the crystal structure, dHvA effect, Raman scattering, optical, magnetic, thermal, and transport properties [482, 483, 490, 491, 497–502, 505, 506, 509, 512–531]. Over the last decade, the research focus has gradually shifted from individual dodecaborides to their solid solutions, which still continue to be actively explored. For example, • Tm1−x Ybx B12 solid solutions are a convenient model system with a metal-insulator transition and a quantum critical point at xc ≈ 0.25. Long-range AFM order, observed in metallic TmB12 , is suppressed in favor of the strongly correlated paramagnetic insulator state known for YbB12 . This quantum-critical behavior is evidenced by strong charge and spin fluctuations near the critical composition [499, 502, 532–534], as discussed in detail in Chapter 4, Section 4.5.5. • Hox Lu1−x B12 solid solutions with positional disorder in the arrangement of Ho3+ ions and with statistical Ho–Lu substitution proved to be convenient model objects in studies of large negative magnetoresistance arising in the paramagnetic phase ´ temperature, magnetotransport anisotropy, and near the Neel charge-carrier scattering on dynamic charge stripes [535–539]. For details, see Chapter 4. • Substitution of Zr for Lu in LuB12 leads to a crossover from weak- to strong-coupling superconductivity in Lux Zr1−x B12 solid solutions [540, 541].

160 Crystal Chemistry and Crystal Growth of Rare-Earth Borides

This list could be continued. Let us consider the specifics of growing single crystals of the R1−x R x B12 solid solutions, based on the aforementioned examples. The initial sintered rods for single-crystal growth of Tm1−x Ybx B12 (0 ≤ x ≤ 0.81) and Hox Lu1−x B12 (0 ≤ x ≤ 1.0) solid solutions were prepared according to the same procedure as individual RE dodecaborides, with the initial synthesis described by Eq. (1.2). In the following, we consider these two systems separately, after Refs. [534] and [535, 536], respectively. Tm1−x Ybx B12 The main problems that had to be solved in growing Tm1−x Ybx B12 single crystals were to avoid the precipitation of the foreign phases RB6 and RB66 , and to maintain the constant Tm : Yb ratio both along the growth axis and in the lateral cross section of the crystals. Taking into account the difference in both vapor pressures of Tm and Yb metals and their distribution coefficients, the gas pressure (highly pure Ar, P ≤ 1.5 MPa), crystallization rate (0.2–1 mm/min), the rate of rotation of feed and growing rods (0–10 rpm), and the starting displacement of the zone composition from stoichiometry by the introduction of a boron piece had to be optimized for each Tm1−x Ybx B12 composition. The optimization criterion for crystallinity was the absence of reflection splitting in the Laue pattern, such as the one shown in Fig. 1.68(b). As a result, single crystals of Tm1−x Ybx B12 with nominal compositions in the range 0 ≤ x ≤ 0.81 and approximate

(a)

(b)

Figure 1.68 (a) General view of the lateral cross section of an as-grown Tm0.75 Yb0.25 B12 single crystal; (b) x-ray Laue backscattering pattern from the same sample, deviation from the [001] axis is about 10◦ . Reproduced from Sluchanko et al. [534].

Higher Borides 161

dimensions ∅4–6 mm × 50 mm have been grown. These crystal rods consisted of a single-crystalline core that was free of domain boundaries and was surrounded by a thin polycrystalline sheath, see Fig. 1.68(a). Occasionally, inclusions of RB6 or RB66 impurity phases were observed at the periphery of the ingot in the lateral cross section using an electron microscope. In these cases, samples were cut from the central part of the oriented polished plates so that they were free from impurity phases. The Tm : Yb ratio in the grown single crystals was estimated for all the Tm1−x Ybx B12 single crystals using a scanning electron microscope equipped with a system for energy-dispersive microprobe analysis (REM-106, with the electron probe size of about 2 μm2 ). The measurements were carried out at several points of the lateral cross section (periphery, r = 1; intermediate region, r = 1/2; and center, r = 0) on both sides of every single-crystalline rod to ensure that the Tm : Yb ratio was constant. In these experiments, either individual pure metals (Tm, Yb) or individual binary borides (TmB12 , YbB12 ) were used as reference samples; it was shown that the results of microprobe analysis for the same sample did not depend on the choice of the type of reference samples. The accuracy of the microanalysis was about 1% whenever the element concentration exceeded 10% in the compound under investigation. The real composition of the single crystal differs from the nominal source one, as shown in Fig. 1.69(a), due to the preferential evaporation

(a)

(b)

Figure 1.69 Tm1−x Ybx B12 single crystals: (a) real vs. nominal Tm : Yb ratios; (b) lattice parameters versus the real composition. Reproduced from Sluchanko et al. [534].

162 Crystal Chemistry and Crystal Growth of Rare-Earth Borides

Figure 1.70 SEM image (in back-scattered electrons) of as-grown Ho0.24 Lu0.76 B12 single crystal near the zone end. Light-grey inclusions are the Ho0.36 Lu0.64 B3.91 impurity phase.

of Yb from the molten zone. For example, the real composition of the sample with the nominal composition Tm0.75 Yb0.25 B12 is Tm0.77 Yb0.23 B12 , and it is consistent for all three measurement points at various distances from the center of the rod (r = 0, 1/2, 1). In addition, spectral emission optical analysis was used to estimate the concentration of non-RE impurities, which did not exceed 10−3 wt. %. The RE impurities were determined by the purity of source oxides. Finally, lattice parameters of dodecaborides were estimated from x-ray powder diffraction measurements on crushed single crystals in an HZG-4 diffractometer with Cu Kα radiation and a Ni filter. The lattice parameters of Tm1−x Ybx B12 do not obey Vegard’s law according to the data plotted in Fig. 1.69(b). Charge-transport measurements have been conducted for a selection of Tm1−x Ybx B12 compositions by Sluchanko et al. [534]. Hox Lu1−x B12 The Hox Lu1−x B12 (0 ≤ x ≤ 1.0) solid solutions were obtained as single crystals both for dilute magnetic (x = 0.01, 0.04, 0.1, 0.15, 0.19) and more concentrated antiferromagnetic (x = 0.23, 0.3, 0.5, 0.7, 0.8, 1.0) compositions [535, 536].

Higher Borides 163

(a)

(a)

(c)

(b)

(b)

(c)

Figure 1.71 Laue back-reflection patterns from the lateral cross sections of as-grown crystals with the nominal compositions: (a) Ho0.15 Lu0.85 B12 , (b) Ho0.3 Lu0.7 B12 , and (c) Ho0.5 Lu0.5 B12 ; after Sluchanko et al. [536, Supplementary Information].

Optimization of the growth parameters and characterization of the grown crystals were carried out following the same methodology as for the previous Tm1−x Ybx B12 system: phase analysis of the crushed single crystals, x-ray Laue patterns, and scanning electron microscopy (SEM JSM-6490-LV, JEOL, Japan) together with electronprobe microanalysis (WDS INCA Energy, OXFORD, Great Britain). Unlike the previous system, more stringent requirements were imposed on the composition of the initial sintered rods, due to possible peritectic decomposition with the formation of impurity phases based on RB4 or RB66 in the region rich in lutetium, and RB6 or RB66 in the region rich in holmium. These impurity phases would also form solid solutions. Typically, such inclusions were present at the end of the grown crystals. An example of such inclusions is shown in Fig. 1.70 for a sample with the initial composition Ho0.24 Lu0.76 B12 . Step by step, both starting compositions and growth parameters were optimized. Depending on the composition, a UHP Ar or He pressure of 0.1–0.9 MPa, crystallization rate of 0.3–0.8 mm/min, and rotation rates of the feed and seed rods of 0–10 rpm were used. As a result, domain-boundary-free crystals of all compositions have been grown. Representative Laue backscattering patterns for as-grown crystals of three compositions are presented in Fig. 1.71(a–c). Nominal and real compositions differ only slightly; the corresponding values for the zone beginning and zone end for several crystals are given in Table 1.22.

164 Crystal Chemistry and Crystal Growth of Rare-Earth Borides

Table 1.22 Nominal and real compositions obtained from the beginning and end of some Hox Lu1−x B12 single crystals by WDS measurements Real composition

Nominal composition Ho0.04 Lu0.96 B12 Ho0.15 Lu0.85 B12 Ho0.2 Lu0.8 B12 Ho0.24 Lu0.76 B12

Zone beginning

Zone end

Ho0.042±0.001 Lu0.958±0.001 B12 Ho0.150±0.001 Lu0.850±0.001 B12 Ho0.194±0.005 Lu0.806±0.005 B12 Ho0.229±0.003 Lu0.771±0.003 B12

Ho0.043±0.002 Lu0.957±0.002 B12 Ho0.143±0.005 Lu0.857±0.005 B12 Ho0.189±0.004 Lu0.811±0.004 B12 Ho0.232±0.005 Lu0.768±0.005 B12

It has been shown that the lattice parameters of Hox Lu1−x B12 also do not obey Vegard’s law [536]. The temperature dependence of the resistivity and specific heat for a series of Hox Lu1−x B12 single crystals are shown in Figs. 1.72(a, b), respectively [536, 538]. Lux Zr1−x B12 These solid solutions are superconductors throughout the whole range of concentrations from x = 0 (ZrB12 , Tc ≈ 6 K) to x = 1 (LuB12 , Tc ≈ 0.4 K) [520]. Kirschner et al. [540] investigated

N

N

ρ

μΩ

ρ μΩ

Figure 1.72 (a) Temperature dependence of the electrical resistivity, ρ(T ), for Hox Lu1−x B12 solid solutions with x = 0, 0.1, 0.15, 0.19, 0.23, 0.3, 0.5. The inset shows residual resistivity ρ0 vs. Ho concentration x. Adapted from Ref. [536]. (b) Temperature dependence of the specific heat of Hox Lu1−x B12 ´ temperature, TN , as a function of the Ho concentration, (0 ≤ x ≤ 1). The Neel x, is plotted in the inset. Adapted from Ref. [538].

Higher Borides 165

four samples of Lux Zr1−x B12 (x = 0.04, 0.07, 0.17, and 0.8) using muon spin rotation (μSR) and magnetometry measurements and found evidence for the formation of nodes in the superconducting gap for x ≤ 0.17, providing a potential new example of an sto s + d-wave crossover in the superconducting order parameter. Baˇckai et al. [541] investigated the evolution of the superconducting transition temperature, Tc , and the critical field, H c , in high-quality single-crystalline superconducting solid solutions of Lux Zr1−x B12 with Lu concentrations x = 0.55, 0.8, 0.9, 0.96, and 1.0 by measuring ac susceptibility down to 50 mK. In the preparation of the corresponding samples, the starting sintered rods of Lux Zr1−x Bn for zone melting were prepared by the identical methodology as for all other dodecaborides; the only difference is in the reaction equation of the borothermal reduction of the corresponding mixtures of oxides, given by Eq. (1.3) below, and a variable value of the amount of boron introduced into the initial charge individually for each composition.   1 x Lu2 O3 + (1−x) ZrO2 + n+2− 12 x B → Lux Zr1−x Bn 2   + 2− 12 x BO ↑ (0 ≤ x ≤ 1.0; 12.5 ≤ n ≤ 13.8). (1.3) This difference is due to the specific nature of ZrB12 melting. It is stable only in a narrow temperature range, 1696–2082 ◦ C, and melts peritectically with the formation of ZrB2 and a boron-rich melt [542]. Zirconium-rich Lux Zr1−x B12 solid solutions melt in a similar way. It is necessary to reduce the temperature of the melt in order to avoid precipitation of the Lux Zr1−x B2 phase. The melt temperature is reduced by introducing an excess of boron into the initial composition. As a result, the crystal grows from a boron-rich melt (by the TSFZ method). The amount of boron has to be selected empirically for each composition of the solid solution, in order to avoid crystallization of Lux Zr1−x B51 as an impurity phase in the Zrrich range. In fact, x-ray phase analysis of the synthesized source powders has shown that all powders are two-phase products, containing diboride and dodecaboride phases. Besides, the presence of Zr-doped β-boron (ZrB51 ) in the synthesized powders is likely, taking into account the excess of boron in the source charge, the large concentration of diboride in the synthesized products, and

166 Crystal Chemistry and Crystal Growth of Rare-Earth Borides

small intensity of x-ray reflections from this substance in addition to the RB2 and RB12 reflections. The synthesized powders were used for growing Lux Zr1−x B12 single crystals with the TSFZ technique. The main challenge was to avoid the precipitation of the diboride phase during growth and to maintain constant Lu : Zr ratio along the growth axis and in the lateral cross sections of the crystal. To resolve these problems, the main technological parameters such as gas pressure (highly pure Ar, P ≤ 1.0 MPa), crystallization rate (0.15– 0.30 mm/min), the rate of rotation of feed and seed rods (0–10 rpm) were optimized separately for each Lux Zr1−x B12 composition. A peculiarity of the zone melting of the Lux Zr1−x Bn compositions is the crystallization of the diboride phase at the initial moment of alloying of the seed and the sintered rod. Stable growth of the dodecaboride phase begins only after several millimeters of a crystallized multiphase system. As a result, single crystals of all nominal compositions Lux Zr1−x B12 (0 ≤ x ≤ 1) were grown with diameters of 5–6 mm and a length of about 50 mm. The main part of these crystal rods consisted of a single-crystal core that was free of domain boundaries, surrounded by a thin polycrystalline shell. Electron microscopy showed that the initial part of the crystal (about 3 mm along the growth axis) adjacent to the alloying zone contained inclusions of both RB2 and RB51 phases [Fig. 1.73(a)]. The periphery of the main ingot (lateral cross section) included a secondary phase of ZrB51 type; in particular, for the crystal with the nominal composition Lu0.1 Zr0.9 B13.5 , its actual composition was Zr0.984 Lu0.016 B51.32 [Fig. 1.73(b)]. In these cases, the experimental samples were cut from the central part of the oriented polished plates, free from impurity phases. The Lu : Zr ratio was estimated for all Lux Zr1−x B12 single crystals using a scanning electron microscope equipped with a system for energy-dispersive microprobe analysis (JEOL JXA-8200 EPMA; electron probe size 1 μm2 ). The measurements were carried out at several points of the lateral cross section (r = 1, 1/2, 0) on both sides of the single-crystalline rods. The individual binary borides (ZrB2 , ZrB12 , ZrB51 , and LuB12 ) were used as reference samples. The accuracy of the microprobe analysis according to the registration

Higher Borides 167

(a)

(b)

Figure 1.73 Lateral cross sections of the (a) initial and (b) main parts of a single crystal with the nominal composition Lu0.1 Zr0.9 B13.5 . The light and dark inclusions in (a) are RB2 and RB51 phases, respectively; dark inclusions in (b) at the periphery of the crystal is the RB51 phase with the Zr0.984 Lu0.016 B51.32 composition contain the RB51 phase.

certificate is several hundred ppm. The actual compositions of single crystals differ from the nominal ones and are slightly different along the crystal but remain constant within every lateral cross section. For example, the results of microprobe analysis for a single crystal grown from a source sintered rod with nominal composition Lu0.1 Zr0.9 B13.5 indicated that the actual Lu content changes from x = 0.065(1) at the zone beginning to x = 0.074(1) at the zone end. The Laue back-reflection pattern of this crystal is shown in Fig. 1.74. YbB12 The most famous representative of the RE dodecaboride family, YbB12 , was first obtained in single-crystalline form in 1998 by Iga et al. [109]. Since the neighboring YbB12 and YbB66 phases melt peritectically at temperatures of 2200 and 2150 ◦ C, respectively (Fig. 1.11), it was necessary to carry out the zone melting process in a narrow temperature range of 50 ◦ C and with the TSFZ technique, to avoid getting a mixture of YbB6 and YbB12 when the temperature exceeds 2200 ◦ C or a mixture of YbB12 and YbB66 at temperatures below 2150 ◦ C. The high-temperature stability in a newly developed image furnace equipped with four Xe lamps for zone melting allowed solving this problem: The temperature heterogeneity in radial and circumferential directions remains below 50 ◦ C at 2200 ◦ C.

168 Crystal Chemistry and Crystal Growth of Rare-Earth Borides

Figure 1.74 X-ray Laue back-reflection pattern from the lateral cross section of the as-grown single crystal with the nominal composition Lu0.1 Zr0.9 B13.5 , grown with a [100] oriented seed. Deviation of the growth direction from [100] is about 3◦ .

The initial composition of the molten zone was rich in boron, implementing the TSFZ method [109]. The source material was synthesized by Eq. (1.1); the obtained powder was pressed into ∅8 mm rods, which after sintering were used for the crystal growth in gas flow (80% Ar + 20% H2 ). A considerably higher Yb vapor pressure in comparison with that of boron results in the depletion of Yb in the molten zone and the accumulation of excess boron. This way, stable growth of YbB12 continues until the composition of the molten zone exceeds that of YbB66 . X-ray Laue photographs and powder diffraction analysis confirmed that the grown bars were large single crystals with the maximum size of ∅6 mm × 50 mm [109]. The amount of impurity phases (YbB66 ) in the central part of the crystals was reduced to 0.015 at. %, according to EPMA and the analysis of magnetic susceptibility [543, 544]. The total amount of RE impurities did not exceed 0.05% [545]. The grown crystals were used in measurements of the electrical resistivity, Hall coefficient, and magnetic susceptibility (see Fig. 1.75). The resistivity increases by five orders of magnitude as T decreases from 300 to 1.3 K. The high-temperature susceptibility shows little sample dependence; the essential difference between polycrystalline samples and a single crystal is in the upturn in χ(T ) below 20 K, which is due to paramagnetic impurities in the first case

Higher Borides 169

(a)

(b)

Figure 1.75 (a) Temperature dependence of electrical resistivity, ρ(T ), and Hall coefficient, RH , for a single crystal of YbB12 . (b) Temperature dependence of magnetic susceptibility χ (T ) for single-crystal and polycrystalline samples of YbB12 . Reproduced from Iga et al. [109].

and with a small amount of Yb3+ impurities in all samples including the YbB12 single crystal [109]. More recently, Okamura et al. demonstrated that it is possible to obtain higher-quality crystals with an RRR exceeding 105 and with no impurity-related Curie tail in the low-T magnetic susceptibility [546]. The same technique has been applied to grow other individual dodecaborides, RB12 (R = Er–Lu) [547–549], and Yb1−x R x B12 solid solutions (R = Lu [543, 544, 550–554]; Y, Sc [553, 554]; Zr [553–555]) in single-crystalline form. Samples from the perfect YbB12 single crystals were used in studies of the dHvA effect [548, 556], optical spectroscopy [546, 547, 551, 555, 557–560], μSR [549, 550], electron spin resonance [545, 561, 562], NMR [563], magnetic, thermal, and transport-property measurements [543, 544, 552–554, 564–567]. Hagiwara et al. [568] developed a cleaning method to obtain a well-defined (001) surface of YbB12 by annealing it at 1650 K in ultra-high vacuum, a preparation which is necessary for surfacesensitive photoelectron measurements. The topmost surface layer contains only B atoms, and Yb atoms exist at the subsurface. This approach allowed determining the Yb valence at the sub-surface as 2.92 ± 0.01 and 2.90 ± 0.01 at 300 and 20 K, respectively, determined from Yb 3d core-level and valence photoelectron spectra, which are consistent with those in the bulk observed in an

170 Crystal Chemistry and Crystal Growth of Rare-Earth Borides

XPS measurement [551]. Such a clean YbB12 surface was also suitable for a detailed ARPES study, where temperature-dependent reconstruction of surface states had been investigated [569]. A number of modern research methods (especially studies at extreme conditions) require miniature samples. For instance, dimensions of the sample for measurements of the electrical resistance of YbB12 at ultrahigh pressures (up to 195 GPa) were just 20 × 12 × 3 μm3 [565]. Magnetization measurements in ultrahigh magnetic fields of up to 120 T [566] and the specific heat in high magnetic fields of up to 60 T [567] were also performed with small YbB12 single crystals. In the first case, the sample had a needle shape with the cross section of about 2×10−2 mm2 to suppress the heating due to eddy currents, and the magnetic field was applied parallel to the longer axis [566]. In the second case, the experimental sample was 0.9 × 0.9 mm2 in area and 0.1 mm in thickness, its mass was only 408 μg. The sample plane corresponded to the (100) surface of the crystal, and the magnetic field was applied perpendicular to that plane [567]. Nevertheless, most other methods require larger samples. So far, there is no report of dodecaboride single crystal growth by the flux method, therefore FZ melting remains the only available method. From the same single crystal with a diameter of 5–6 mm and a random orientation, it is possible to obtain samples with the size of 10 × 1.5 × 0.1 mm3 and any given orientation, as it was done in Ref. [552], where such experimental samples were prepared to study the anisotropic magnetoresistance and the energy gap collapse in Yb1−x Lux B12 solid solutions (x = 0, 0.01, 0.05) by measuring the magnetization and electrical resistance up to 68 T at 4.2 K or 1.3 K. The long sides of the crystals were parallel to the [100], [110], and [111] high-symmetry axes. Neutron-scattering experiments were carried out on Yb11B12 single crystals with isotope-enriched 11B (99.5% enrichment level). A cylindrical sample ∅5 mm × 15 mm was used to study magnetic excitations [570]. Then, an assembly of two Yb11B12 single crystals with a total volume of approximately 0.4 cm3 , aligned to better than 1◦ with respect to each other, was used for investigating lattice and spin dynamics, finding evidence of short-range AFM fluctuations,

Higher Borides 171

and demonstrating the interplay of low-energy phonons and magnetic excitations [571–574]. Studies of Yb1−x Lux B12 solid solutions (0 ≤ x ≤ 1) were widely used for identification of the underlying mechanisms behind the physical properties of YbB12 [543,544,550–554]. Recently, the range of nonmagnetic elements used in the substitution was extended to Sc3+ , Y3+ , and Zr4+ [553–555]. Sc3+ , Y3+ , and Lu3+ ions are isoelectronic with Yb3 , whereas Zr4+ substitution dopes YbB12 with ˚ and Lu3+ (0.848 A) ˚ are electrons. The ionic radii of Yb3+ (0.857 A) ˚ Y3+ similar; they differ significantly from those of Sc3+ (0.73 A), ˚ and Zr4+ (0.80 A). ˚ Further, as already mentioned, ZrB12 , (0.905 A), YB12 , and LuB12 crystallize in a cubic F m3m structure, and ScB12 in the tetragonal structure I 4/mmm. These characteristics inherent to the substituting ions are reflected both in the lattice parameters of the obtained single crystals and in their properties. The source powders of Yb1−x R x B12 solid solutions (0 ≤ x ≤ 1) have been synthesized by Eq. (1.2) for R = Lu, Sc, Y and by Eq. (1.3) for R = Zr. Polycrystalline sintered rods produced from these powders were used for single-crystal growth with the TSFZ method using an image furnace with four xenon lamps [109]. XRD confirmed that the grown crystals were single-phase, and their lattice constants were evaluated. The values of x have been determined by x-ray fluorescence analysis and electronprobe microanalysis; these values agreed well with the nominal composition within 1 at. % [553–555]. Figure 1.76 shows the variation of lattice parameters at room temperature in Yb1−x R x B12 (R = Y, Lu, Sc, Zr) single crystals as a function of x [554]. They mostly follow Vegard’s law with the sole exception of Sc alloys due to the deviation of ScB12 crystal structure from cubic; moreover, its cubic lattice parameter a = 7.3912 A˚ raises doubt. At room temperature, the ScB12 cubic structure is possible only when stabilized by foreign ions [575]. Substitution of Y for Yb leads to an increase in the lattice constant, while the substitutions of Sc and Zr decrease the lattice constants, as stipulated by the difference in the ionic radii of these metals. Yb1−x Lux B12 shows mainly magnetic ion dilution effects and only a weak reduction in the lattice constant. The valence of Yb in Yb1−x R x B12 is almost trivalent throughout the entire range of x in the R = Y and Lu

172 Crystal Chemistry and Crystal Growth of Rare-Earth Borides

Figure 1.76 The lattice constant a of Yb1−x R x B12 solid solutions (R = Y, Lu, Zr, Sc). Reproduced from Iga et al. [554].

solid solutions [553]. In Yb1−x Zrx B12 , the effect of electron doping results in a continuous monotonic decrease of Yb valence with Zr concentration [555]. A high-quality LuB12 single crystal showed a residual resistivity ratio of about 80 between 1.4 K and room temperature. It was used for measuring quantum oscillations of elastic moduli by an ultrasonic technique at 0.4 K with a 16 T superconducting magnet. All the detected Fermi surfaces were reasonably identified in bandstructure calculations [548]. Similar to the quantum-oscillation measurements in SmB6 [417, 418], discussed in Section 1.5.3, quantum oscillations in the magnetic torque, electrical, and thermal conductivity have been observed in YbB12 [519, 556, 576]—a finding that is particularly challenging to understand because both compounds are known to be Kondo insulators. Liu et al. [519] were the first to demonstrate magnetic quantum oscillations in the Kondo insulator YbB12 , measured using torque magnetization, and discussed the potential origin of its underlying Fermi surface. A selection of their experimental results from a single crystal of YbB12 , measured in dc magnetic fields up to 45 T in the

Higher Borides 173

(a)

1.1 K 1.4 K 1.9 K 2.9 K 3.9 K

0.8

0.6

0.4

YbB12

0.2

36 T < 0.0

0

1000

2000

F (T)

(b)

0.8

1 mm

0.6 0.4 0.2

H < 45 T 0 3000

YbB12

1.0

FFT amplitude (arb. units)

FFT amplitude (arb. units)

1.0

F = 270 T m*/me = 2.6(6)

36 T < 0.0 0

1

0

H < 45 T

2

3 T (K)

4

5

6

Figure 1.77 Landau quantization in YbB12 . (a) Fourier transform of the magnetic field sweeps measured at five different temperatures. The main Fourier peak corresponds to a frequency of 270(50) T. (b) Quantum oscillation amplitude obtained from the peak height of the Fourier transforms shown in the left panel, plotted vs. temperature. Performing the Lifshitz–Kosevich fit (dashed line) yields an effective mass of m∗ = 2.6(6)me . The inset shows the YbB12 single crystal used in the measurements. Reproduced from Liu et al. [519].

temperature range 1.1 K ≤ T ≤ 3.9 K, are shown in Fig. 1.77 [519]. Crystals for these experiments were grown at the University of Warwick. Source YbB12 powder was prepared in polycrystalline form by borothermal reduction of a mixture of Yb2 O3 (99.998 wt. % purity) and amorphous boron (99.9 wt. % purity) at 1700 ◦ C under vacuum [489]. The obtained powder was pressed into a cylindrical rod and sintered at 1600 ◦ C in Ar gas flow for several hours. The YbB12 single crystal was grown by the TSFZ technique under conditions similar to those previously described in Ref. [109] using a four-mirror xenon-arc-lamp (3 kW) optical image furnace (Crystal Systems Inc., Japan). The growths were performed in a reducing atmosphere (Ar + 3% H2 ) at a growth rate of 18 mm/h with the feed and seed rods counter-rotating at 20–30 rpm. An x-ray Laue backscattering pattern from the sample on which the quantum oscillations were measured showed sharp reflection spots, evidencing that it is a good single crystal. Elemental

174 Crystal Chemistry and Crystal Growth of Rare-Earth Borides

(a)

(b)

100

3.0 2.5

Intensity (a.u.)

Intensity (a.u.)

80

60

40

2.0 1.5

Al2O3 YbB6

1.0

20 0.5 YbB2 YbB4 YbB12

0 0

10

20 2 /

30 (deg.)

40

50

0.0

10

15

20 2 /

25

30

35

(deg.)

Figure 1.78 Synchrotron x-ray powder diffraction pattern of YbB12 ˚ at room temperature. measured with 17 keV x-rays (λ = 0.774328 A) (a) The tick marks show the reflection positions of YbB12 . (b) Expanded view of (a). Circles and triangles indicate peaks of YbB6 and Al2 O3 , respectively. The tick marks show the reflection positions of YbB2 (top), YbB4 (middle), and YbB12 (bottom). Reproduced from Xiang et al. [556].

composition analysis was performed on selected single crystals using an FEI Philips XL30 sFEG scanning electron microscope (SEM), showing that the Yb : B atomic ratio is very close to the expected stoichiometric ratio of 1 : 12 (coinciding within the detection limit of SEM) and clearly distinct from the ratios for YbB6 and for YbB66 . EDX microanalysis on multiple samples provided comparable results to SEM. Rietveld refinement performed using the Bruker TOPAS software on powder XRD data yielded a lattice ˚ in agreement with the published data for constant of 7.4686(1) A, YbB12 [489]. Unlike SmB6 , which displays quantum oscillations only in magnetization (the dHvA effect), YbB12 also exhibits the Shubnikov– de Haas (SdH) effect, i.e., quantum oscillations in the resistivity. Both effects have been investigated in Refs. [556, 576]. Here we consider the preparation and characterization of the YbB12 single crystals on which these measurements have been carried out. The crystals were grown by the TSFZ method [109] and characterized by highresolution synchrotron x-ray powder diffraction measurements performed at the BL02B2 beamline at the SPring-8 facility (Japan) ˚ at RT. YbB12 single crystals with 17 keV x-rays (λ = 0.774328 A)

Higher Borides 175

taken from the same batch as those used in the electrical resistivity and magnetic torque measurements were ground into fine powder in an alumina mortar. Figure 1.78 shows the corresponding powder diffraction pattern. Almost all peaks can be indexed with the crystal structure of YbB12 . Tiny peaks of the impurity phases YbB6 and Al2 O3 can also be identified. The volume fraction of the YbB2 and YbB4 phases is estimated to be less than 10 ppm, below the experimental detection limit, so the only confirmed impurity is YbB6 . Three crystals were cut from the as-grown ingot and polished into a cuboid shape with six (100) surfaces. Crystals 1 and 2 were taken from the same growth batch, whereas crystal 3 was from a different growth batch. The dimensions of the samples were 4.0 × 0.52 × 0.51 mm3 (crystal 1), 1.0 × 1.0 × 0.2 mm3 (crystal 2), and 1.8 × 0.81 × 0.13 mm3 (crystal 3). The temperature dependence of the magnetic susceptibility χ of the YbB12 single crystals was measured at 1 T for crystals 2 and 3, as shown in Fig. 1.79(a) [576, Supplementary Information]. The Curie contribution at low temperature is very small compared to that in the previous report [109], indicating high quality of the samples. A slight distinction in their susceptibilities in the range of intermediate and high temperatures [Fig. 1.79(a)] is consistent with the difference in resistivity of these samples obtained from different batches [Fig. 1.79(b)]. The possible reason is a difference in the Yb : B ratio in these two samples that results in an increase of susceptibility and a decrease of resistivity, for example, with a slight increase in the number of Yb3+ ions due to vacancies in the lattice in one of the samples. Figure 1.79(b) shows a significant difference in the resistivity among the three samples. The RRR, measured between RT and 0.1 K, is about 104 –105 ; moreover, there is a difference of nearly one order of magnitude between samples grown in different batches, whose susceptibilities are only slightly different. The presence of a plateau for all samples below T ≈ 2 K, typical for SmB6 , is attributed to 2D metallic surface states. This conclusion is confirmed by a 2.2 times larger value of the residual resistivity in crystal 1 than in the thinner crystal 2, as the surface-to-volume ratio increases with reduced sample thickness. Above 5 K, where surface conduction is negligible, the resistivity of crystals 1 and 2 overlaps, while that of crystal 3

176 Crystal Chemistry and Crystal Growth of Rare-Earth Borides

(a)

12 0H

=1T

No. 2 (H || [100])

8 6

6

5

-3

-1

(10 emu mol )

YbB12

No. 3 (H || [110])

10

4

4 3

2 0 0

2 0

10

50

100

20

150

30

200

250

300

Temperature (K)

(b)

101

10–1 –2

Δ = 54.2 K = 4.7 meV

10–2 ln[ρ]

ρxx (Ωcm)

100

–4

10–3

Δ = 46.3 K = 4.0 meV

–6

0.05

10–4 0.1

No. 1 No. 2 No. 3

0.10 0.15 –1 1/T (K )

0.20

1

10

100

T (K)

Figure 1.79 Temperature dependence of the magnetic susceptibility and electrical resistivity for three YbB12 crystals (crystals 1 and 2 from the same batch and crystal 3 from another batch). (a) Magnetic susceptibility, measured at 1 T using a Quantum Design VSM; inset shows the expanded view at T < 30 K. (b) The comparison of resistivity curves; the inset is an Arrhenius plot above 5 K. Reproduced from Sato et al. [576].

Higher Borides 177

remains somewhat lower until approximately 20 K [Fig. 1.79(b), inset]. The authors conclude that the resistivity value depends on the quality of the sample [556].

1.6.2 Rare-Earth Hectoborides (RB66 ) Rare-earth hectoborides, described by the general formula RB66 (R = Y, Nd, Sm, Gd, Tb–Lu), are refractory semiconductor compounds having one of the most complex crystalline structures among binary higher borides. They are characterized by a wide homogeneity range, so a change in the R : B ratio makes it possible to obtain hectoborides with compositions ranging from RB48 to RB100 . Only metals with atomic radii between 1.73 and 1.82 A˚ are likely to form stable hectoborides [36, 577–585]. The high melting points and hardness of RB66 are determined by the predominantly covalent bonds in the boron sublattice. The most studied in this class of compounds is YB66 , which has found wide applications as a soft x-ray monochromator for synchrotron radiation [586–589]. Hectoborides possess promising hightemperature thermoelectric properties that can be controlled either by a change in composition within the homogeneity range or by doping with d-block transition metals [590–592]. Comprehensive, detailed information on the properties of RB66 is presented by Mori et al. in the latest reviews [593, 594]. The structural type of RB66 hectoborides was first determined and described in detail by Richards and Kaspar [577] by the example of YB66 single crystal accidently grown in one of a series of yttrium and boron mixtures after melting in vacuum in a highfrequency induction furnace [595]. The unit cell of YB66 is cubic with a lattice parameter a = 23.440(6) A˚ (space group F m3c – Oh6 ). The main structural unit is the B12 icosahedron, whose symmetry elements include a fifth-order rotation axis. Therefore, a periodic lattice based on regular icosahedra alone would be forbidden, so they must be bonded through isolated atoms or other groups of atoms. The YB66 unit cell, depicted in Fig. 1.80(a), contains 1584 boron atoms and 24 yttrium atoms. The majority of boron atoms (1248) are grouped into eight supericosahedra (B156 ). Each supericosahedron consists of 13 icosahedral clusters of B12 , one

178 Crystal Chemistry and Crystal Growth of Rare-Earth Borides

Figure 1.80 (a) Crystal structure of YB66 viewed along the z axis; the B12 icosahedra forming the B12 (B12 )12 supericosahedra are shaded in grey (central icosahedron) and red (outer icosahedra); all other boron atoms (the B80 cluster) are shown as small red spheres; metal atoms are depicted as dark blue spheres. Reproduced from Akopov et al. [39]. (b) The B12 (B12 )12 supericosahedron consisting of 13 smaller icosahedral units (shaded in green). (c) The B80 cluster unit. The excessive bonding in panel (c) is because it assumes that all sites are occupied, whereas the total number of boron atoms is only 42. Reproduced from Tanaka [596].

of which is located at the center, surrounded by 12 others. The individual B12 icosahedra are directed with their vertices along the 12 fifth-order rotation axes of the supericosahedron. The resulting B12 (B12 )12 supericosahedral cluster, which also inherits the fifth-order rotation symmetry, is shown in Fig. 1.80(b). Direct evidence of the existence of stable supericosahedral clusters in YB66 was obtained using scanning tunneling microscopy [597]. The supericosahedra are rotated relative to each other by 90◦ in the unit cell. The remaining boron atoms that do not belong to the icosahedral clusters are arranged in a B80 structural element shown in Fig. 1.80(c). However, only some of the positions (between 28% and 71%) in the B80 cluster are occupied, so that the average number of boron atoms in these clusters is only 42. The metal atoms are located in channels formed by the supericosahedra packed in the cubic lattice. The length of the B–B bonds inside the icosahedron ˚ the intericosahedral bonds from varies from 1.719 to 1.855 A,

Higher Borides 179

˚ and the length of the Y–B bonds from 2.691 to 1.624 to 1.823 A, ˚ 2.768 A [577]. Two high-quality single crystals of YB66 were grown by an indirect-heating FZ method [583]. One crystal was grown with a congruent composition of B : Y = 62, another one by self-flux under an incongruent melting condition with the composition B : Y = 56. Their structures were studied by single-crystal x-ray diffractometry. The boron framework of these crystals is almost identical to that of YB66 [577]. However, there is a significant increase in the yttrium site occupancy; 0.532(4) for YB62 and 0.575(5) for YB56 , whereas in YB66 it equals 0.5. A marked anisotropy of thermal parameters was observed for the Y site, suggesting a form of static disorder with some simultaneous occupation of neighboring Y sites, resulting in a longer Y–Y distance due to their repulsion. The anisotropic thermal motion results from the thermal displacement along the Y–Y direction [583]. Higher concentration of Y atoms leads to a significant increase in the Y–Y distance: 2.554(2), 2.638(2), and 2.761(3) A˚ for YB66 , YB62 , and YB56 , respectively. At the same time, the change in the lattice constant within the homogeneity range was insignificant: a = 23.4600(9) A˚ for YB56 ; 23.4364(6) A˚ for YB62 ; and 23.440(9) A˚ for YB66 [577, 583]. The aforementioned structural features of the YB66 crystal— (i) the presence of structural units with the fifth-order rotation axis, connected through groups of isolated atoms; (ii) variations of the short-range crystalline order parameters within the unit cell and preservation of the long-range order; and (iii) more than a thousand atoms in the unit cell, resembling the model of a disordered continuous network—led to a suggestion that hectoborides may serve as “natural structural models of an amorphous semiconductor” [36, 598, 599]. The Gmelin Handbook [112] summarized information on the hectoboride lattice parameters for polycrystalline samples prepared by borothermal reduction or arc melting, including values for the boundary compositions from their homogeneity ranges, which were available up to 1990. All known lattice-parameter data for RB66 from single-crystal studies as well as from powder-diffraction data obtained in the last 20 years are listed in Table 1.23. An exception is made only for NdB66 and TmB66 for the sake of completeness,

Lattice parameters and compositions of RB66 (R = Y, Nd, Sm, Gd, Tb–Lu)

RB66

Cation radius ˚ [56] (A)

Cation charge

Lattice ˚ constant (A)

Refined formula

YB66

0.905

+3

+3

GdB66 TbB66

0.995 1.13/0.964 (Sm+2 / Sm+3 ) 0.938 0.923

DyB66 HoB66 ErB66

0.908 0.894 0.881

+3 +3 +3

YB47.3 YB48.3 YB56 YB62 YB61.75±0.2 YB66 Nd-rich∗ SmB60 SmB62 GdB61.5 TbB62.7 TbB48.1 DyB66.2 HoB62.4 Er1.00(1) B65(1)

TmB66 YbB66

0.869 0.857

+3 +3

LuB66

0.848

+3

23.4292 23.4307 23.4600(9) 23.4364(6) 23.445(5) 23.440(9) 23.508 23.466(1) 23.493(8) 23.3939 23.3916 23.596(5) 23.3900 23.3886 23.440(3) 23.408(3) 23.387 23.433(3) 23.3587(6) 23.4142(6) 23.3850

NdB66 SmB66

∗

+3 +3

ErB63.4 B-rich∗ YbB56.7 Yb1.162 B67.617 LuB65.9

Sample specification, comments

Ref.

single crystal (FZ) [590] ” [590] ” [583] ” [583] ” [103] ” [577] arc melting (mixture of B : Nd = 80:1) [579] single crystal (FZ) [600] ” ” synthesized powder [601] synthesized powder [601] ” [602] synthesized powder [601] synthesized powder [601, 603] arc melting (mixture of B : Er = 70 : 1) [604] single crystal (FZ) [605] synthesized powder [601] arc melting (mixture of B : Tm = 66 : 1) [580] single crystal [584] single crystal (FZ) [585] synthesized powder [601, 603, 606]

The ingots obtained by arc melting contained two phases: either RB66 with RB12 or RB66 with β-boron, indicated here as R-rich or B-rich, respectively.

180 Crystal Chemistry and Crystal Growth of Rare-Earth Borides

Table 1.23

Higher Borides 181

as they were only investigated in the 1970s [579, 580]. Solely Spear et al. [579] succeeded in obtaining the NdB66 phase of the YB66 structure type; this phase is stable only at high temperatures. Structural data are available for YB66 across its homogeneity range [577, 583, 590], as well as for SmB66 [600] and YbB66 [584, 585]. Temperature dependence of the RB66 lattice parameters for R = Gd, Tb–Ho, and Lu [601]; HoB66 [603]; and LuB66 [603,606] has been measured from 5 to 300 K. The first sizable YB66 single crystals (up to ∅5 mm × 20 mm) were grown by Oliver and Brower [102]. They tested several techniques for YB66 crystal growth using preliminarily synthesized polycrystalline YB66 . The Bridgman method using a sealed tungsten crucible and pulling of YB66 crystals using pyrolytic BN crucibles (Czochralski technique) were unsuccessful because of the incompatibility of the high-temperature melt with the crucible material. Only by pulling the YB66 rod from the YB66 melt located in a water-cooled boat (a pedestal-pulling technique), was a single crystal with the composition YB64±2 successfully grown [102]. Slack et al. [103] grew yttrium hectoboride single crystals with the same pedestal-pulling technique. The source material was synthesized from yttrium, distilled in ultra-high vacuum, and boron. The congruent composition was determined as YB61.75±0.2 and the stoichiometric value as YB68 . The measured density of a large single crystal of the congruently melting composition was 2.569(5) g/cm3 , and the lattice parameter determined from x-ray ˚ Measurements of elastic constants, diffraction, a0 = 23.445(5) A. acoustic attenuation, electrical resistivity, and optical absorption were also performed on the same samples. A comparison of the properties of YBn (n = 61 ± 3) with those of β-boron showed many similarities. In subsequent years, single crystals of RB66 hectoborides (R = Y, Sm, Gd, Dy, Er, Yb) were grown by zone melting with optical heating, taking into account their semiconducting nature of conductivity. Golikova with colleagues made a significant contribution to the preparation of the samples and studies of their transport, optic, and thermal properties, and the development of ideas about the physical mechanisms that determine these properties. Single crystals of RB66 (R = Y, Sm, Gd, Dy, Er, Yb) were prepared in three steps:

182 Crystal Chemistry and Crystal Growth of Rare-Earth Borides

(i) powder preparation; (ii) cold pressing; (iii) zone melting in He atmosphere or in vacuum [607, 608]. The crystals were grown across the homogeneity range, and it was shown that the physical properties of all RB66 compounds are similar. For instance, their electrical properties are practically insensitive to the choice of the RE metal [582, 608, 609]. Based on the obtained results, Golikova [36, 598, 599] concluded that the only difference between the MB66 crystals and amorphous structures is the existence of long-range atomic order in the former and its absence in the latter, yet this difference is apparently irrelevant for the semiconducting properties. YB66 single crystals were used for the development of a soft-xray monochromator for synchrotron radiation in the energy range 1–2 keV [586–589]. The benefits of this material are in the large d-spacing (2d = 11.72 A˚ for the (400) reflection), offering high resulting monochromator resolution of E /E ≈ 5 × 10−4 , as well as sufficiently high reflectance, low internal absorption, vacuum compatibility, and stability against radiation damage [587, 588]. Such monochromators opened a new spectroscopic window to investigate low-Z materials containing Si, Al, and Mg with K -edge x-ray absorption fine structure (XAFS) spectroscopy and enabled L- and M-edge spectroscopy of many of the 4p elements and rareearths whose respective edge energies fall in the 1–2 keV region. The problem of growing monochromator-grade YB66 single crystals was solved after numerous iterations of improving the growth process, supported by feedback of characterization results, by the team of Japanese scientists led by Tanaka [587, 610– 612], in collaboration with scientists from other institutions, who provided systematic hard-x-ray characterization with rocking-curve measurements and x-ray topography to map out subgrain structure and growth-induced defects [586–588, 613]. YB66 is a p-type semiconductor with RT resistivity of several hundred · cm [103]. Therefore YB66 crystals were grown by the FZ method with an indirect heating of the molten zone by radiation from a RF-heated tungsten ring placed between the work coil and the molten zone in He atmosphere under pressures of P = 0.2–0.5 MPa [610]. To ensure high purity of the grown crystals, the source sintered rods were prepared from powder

Higher Borides 183

synthesized by borothermal reduction of yttrium oxide as the starting material [611]. Polycrystalline feed rods have a relatively low sintered density of about 60%. To obtain high-quality single crystals, two passes of the molten zone were used, with the crystallization rate optimized individually for each pass. The first pass ensured the compaction of the rod; therefore, the driving rates of the polycrystalline feed rod and grown crystals were 45– 50 and 25 mm/h, respectively; in the second pass occurring in the opposite direction, both rates remained identical, 10–12 mm/h. The crystal and the feed rod rotation rates were 6 and 0–6 rpm, respectively [587]. The congruent melting composition is B : Y = 62 [611], but the highest-quality crystals have been grown by self-flux under incongruent melting conditions [587]. The chemical composition of the feed rod and the molten zone were controlled to be B : Y = 56 and 40, respectively. This condition lowered the growth temperature compared to the melting point of the growing crystals with a congruent composition and allowed improving the crystal quality. The highest-quality crystals were grown in the [110] direction. Crystals grown this way are more uniform and exhibit no subgrain boundaries or have only a few small subgrain boundaries at the periphery of the crystal. FWHM values are about 20% narrower than for the incongruent growth in the [001] direction, and the lattice misorientations of the crystals are 60 –100 . The existence of defects and mosaicity in these crystals had no significant effect on monochromator resolution. The grown crystals had dimensions of about ∅12 mm × 60–70 mm [587]. Despite the widespread use of YB66 soft x-ray monochromators at many synchrotron radiation facilities, the search for new materials with improved capabilities was continued, because the output beam intensity is less than 10−3 from the incident one. Moreover, low thermal conductivity of YB66 , more typical for amorphous solids, restricts its applications in high-brilliance beamlines at the thirdgeneration synchrotrons [614]. By analogy with β-boron, it was proposed to modify the YB66 structure by doping d-transition metals to improve the monochromator characteristics. YB66 doping was supposed to reduce randomness in the crystal structure, which is inherently characterized by about 50% site occupancy of yttrium

184 Crystal Chemistry and Crystal Growth of Rare-Earth Borides

ions and statistical distribution of only 42 boron atoms in a B80 cluster [614]. Therefore, a number of works on doping YB66 have been performed [591, 592, 614, 615]. Tanaka et al. [614] doped YB66 with 3d (from Ti to Cu) and 4d (from Zr to Mo) transition metals by arc-melting of the corresponding mixtures and measured the resulting structures by x-ray diffraction. Powder XRD patterns of transition-metal-doped YB66 , except for the samples with Co and Ni doping, showed considerable differences from that of undoped YB66 ; for example, the (200) reflection, which is very weak in the parent YB66 , became so strong as to be comparable with the (400) reflection. Other reflections also underwent changes. But for analysis of XRD of Cu-, Ni-, and Mo-doped YB66 single crystals, the intensity ratio of the (200) and (400) reflection can be used to quantify the structural changes. Single crystals of Cu-, Ni-, and Mo-doped YB66 were grown by the FZ method using a xenon-lamp image furnace. The source YB62 Cu1.0 feed rod was obtained by arc melting of an YB62 sintered rod and Cu granules; YB62 was previously synthesized from commercial YB4 and amorphous boron. During the first floatingzone pass, Cu evaporated partially, and on the second pass it could have evaporated completely, but without at least two zone passes achieving the high crystal quality required for a monochromator was impossible. The situation with Ni was also disappointing, as for the YB62 Ni1.0 rod the molten zone was unstable because of the accumulation of Ni in the melt, and the final composition of the ingot had a considerably reduced Ni content. For the composition YB62 Ni0.2 , a single crystal was finally grown with much effort. A crystal from the YB56 Mo0.6 feed rod grew stably initially, but apparently the melting temperature of the molten zone gradually decreased. Finally, the molten zone became unstable and the zone pass could not be continued. Compositions of the crystal parts at the seed end, in the middle, and at the zone end were YB56.5 Mo0.26 , YB56.0 Mo0.29 , and YB47.5 Mo0.6 , respectively. The decrease in the melting temperature during growth is related to the accumulation of Mo in the molten zone. Powder XRD analysis of these three samples showed that the intensity of the (200) reflection was less than that of the (400) reflection [614]. Therefore, the search for a more suitable doping element was continued.

Higher Borides 185

An XRD study of YB66 doped with Nb has shown that this metal enters a particular ( 14 41 41 ) site in the unit cell, and the occupation of this site by Nb increases the intensity of the (400) reflection, which is used for dispersing soft x-ray synchrotron radiation, by a factor of about 2. Moreover, the boron site at (0.235, 0.235, 0.235), acting as a phonon scattering center, was suggested as a cause for amorphouslike low thermal conductivity. This scattering can be reduced if this special site is occupied by a transition metal, leading to an increase in the thermal conductivity [614]. These results stimulated followup work to grow Nb-doped YB66 single crystals using FZ melting in an ellipsoidal mirror-type image furnace equipped with four xenon lamps [615]. Raw powders with a desired composition for feed rods were synthesized by reacting YB4 , NbB2 (Japan New Metals Co. Ltd., Japan) and amorphous boron (SB-Boron Corp., USA) powders at about 1700 ◦ C for 10 h in vacuum using an RF heating furnace. The compacted polycrystalline feed rods were sintered at about 1800 ◦ C, but they had a relatively low sintered density of about 60% because of poor sintering behavior of strongly covalent YB66 . So the first passage of the heating zone just below the melting temperature was used for additional sintering of the feed rods. The zone pass rate was set at 400–500 mm/h and power density was about 90% of that used for the molten-zone pass. At the second pass, both the feed rod and the growing crystal were driven downwards at 7–10 mm/h and were counter-rotated at 6–20 rpm. Growth orientation was adjusted to either [001] or [011] direction. The growth was carried out in flowing Ar gas [615]. The initial desired composition of the molten zone was determined by placing a pellet with the given composition on the seed crystal, but the growing crystal composition always differed from that of the molten zone. As an example, the chemical composition of the feed rod, molten zone, and crystal in one of the experiments were YNb0.24 B56.3 , YNb0.63 B45.5 , and YNb0.22 B58.4 , respectively [615]. The maximal achieved site occupancy of Nb was 95%. The FWHM of rocking curves measured for the Cu Kα (10 0 0) reflection was 40 –50 [615]. Thermal conductivity of the crystal with the maximal site occupancy was about twice higher than in undoped YB66 crystals [591]. The obtained crystals partly showed high quality

186 Crystal Chemistry and Crystal Growth of Rare-Earth Borides

comparable with undoped YB66 , but the uniformity was insufficient for practical applications [615]. Results of the work on YB66 doping with Nb and carbon to improve high-temperature thermoelectric properties of yttrium hectoboride are presented in Ref. [592]. Synthesized powders (YNbx )B66 and YB66 Cx were compacted and sintered to prepare feed rods for FZ melting in a procedure similar to the one described above. The difference was in the details: The feed and seed crystals were counter-rotated at 40 rpm; YB66 was used as a seed crystal; the growth rate was kept at 4 mm/h; all the grown YB66 crystals were prepared at the boron-rich end with an actual chemical composition close to B : Y = 66 [592]. An undoped YB66 crystal, three Nb-doped YB66 crystals ranging from YNb0.30 B66 to YNb0.33 B66 , and a C-doped compound with the formula YB66 C0.6 were grown. These were studied by measuring resistivity, Seebeck coefficient, and thermal conductivity in the temperature range of 300–1100 K. The final goal was to obtain the figure of merit zT = α 2 T /ρκ, with α, T, ρ, and κ being the Seebeck coefficient, temperature, electrical resistivity, and thermal conductivity, respectively, to evaluate the possibility of using this modified hectoboride as a high-temperature thermoelectric material. The results looked promising for the carbon-doped YB66 , as for this material an increase in the figure of merit by a factor of 2 in comparison with the undoped YB66 was achieved at 1000 K [592]. In recent years, as part of the search for novel high-temperature thermoelectric materials, a number of hectoboride single crystals including YB48 , SmB62 , and YbB62 of the YB66 structure type were grown, and their high-temperature thermoelectric properties were investigated [584, 585, 590, 600]. The boundary of the yttrium hectoboride homogeneity range from the metal-rich side was previously limited by the YB56 composition, according to an YB66 crystal-structure study [583]. Therefore, when under FZ melting a strikingly metal-rich phase of YB47 with the YB66 structure type grew instead of yttrium borosilicide, YB44 Si2 , it came as a surprise [590]. The existence of YB50 phase with the space group Pbam is known. Above 2100 K, this boride decomposes without melting into YB12 and YB66 , but it can be stabilized by Si to form the YB44 Si2 phase, which is isostructural to YB50 [58]. Initially, Hossain

Higher Borides 187

et al. [590] planned to grow YB44 Si2 single crystals by FZ melting and to investigate their thermoelectric properties. A polycrystalline feed rod was obtained from the mixture of YB4 (New Metals Co., 99%), B (SB Boron, 99.9%), and Si (Wako, 99.9%) powders in the corresponding ratio, this charge was pressed to form a rod and reaction-sintered in an inductive RF furnace at 1700 ◦ C for 8 h in vacuum. To obtain a high-density feed rod, the synthesized rod was ground, formed again into a rod and sintered under the same conditions. The final sintered feed rods had a density of about 75– 80%. The FZ crystal growth was carried out using a four-mirror infrared image furnace, the seed was in the lower shaft, drive rates of the feed and seed rods were 10 and 8 mm/h, respectively, with the two rods counter-rotating at 16 rpm under Ar atmosphere. The grown crystal was crack-free, silvery in color, but its composition was YB47.3 with only a negligible amount of Si, the rest of which was lost during the repeated feed-rod sintering in vacuum. After such an unexpected result, the experiment with the growth of YB47 single crystal was performed on purpose under similar conditions. According to microprobe analysis (EPMA), the composition of the grown crystal was YB48.3 . For both the unintentionally and deliberately grown crystals, single-crystal structure refinement was performed, and their thermoelectric properties were studied. The lattice constants of the ˚ respectively. YB47.3 and YB48.3 crystals were 23.4292 and 23.4307 A, The total number of yttrium atoms per unit cell was 33.1, which is much larger than in YB62 (26.2) or YB56 (29) [591]. The structure refinement of the YB48 single crystal evidenced the presence of another yttrium site Y3 at the center of the octant of the unit cell—the same site as that occupied by dopant transition-metal atoms [614]. Hossain et al. [590] concluded that the creation of a new Y site is further proof of the unprecedented metal-rich nature of these single crystals. SmB62 , SmB60 , and YbB62 single crystals were grown by the FZ method with focused light from four xenon lamps. Corresponding powders for the feed and seed rods were synthesized by the borothermal reduction of the RE oxides powders (R = Sm, Yb). The purity of Sm2 O3 , Yb2 O3 , and boron was 99.95%, 99.9%, and 99.9%, respectively. The optimal growth rate for SmB62 and SmB60 was

188 Crystal Chemistry and Crystal Growth of Rare-Earth Borides

found to be 8 mm/h. The seed rod was rotated clockwise at 10 rpm, while the feed rod was rotated counterclockwise at 15 rpm. The crystals were grown approximately 10 mm in diameter, and both compositions contained a small amount of SmB6 as an impurity phase; probably the homogeneity range of SmB66 is more narrow than for YB66 . Rietveld refinement yielded lattice parameters of 23.493(8) A˚ and 23.466(1) A˚ for the SmB62 and SmB60 crystals, respectively [600]. An unexpected result was obtained when studying the magnetic susceptibility of SmB62 : According to the Curie–Weiss fit, an effective magnetic moment of 0.42 μB per Sm atom in SmB62 is approximately half the expected effective magnetic moment, μeff , of a free Sm3+ ion. This has been suggested to be evidence for a mixed valence of samarium in this compound, a suggestion which requires verification. It was also shown that the thermoelectric properties of both crystals practically coincide [600]. The YbB62 feed rod was lowered at a rate of 8 mm/h, and the feed rod and crystal were counter-rotated at 10 rpm during the crystal growth. The grown crystal is shown in Fig. 1.81, cleaved along the growth direction. Its crystal structure has been refined with the

Figure 1.81 Cleaved surface of the YbB59 single crystal. Reproduced from Sauerschnig et al. [585].

Higher Borides 189

Figure 1.82 Temperature dependence of the dimensionless figure of merit of YbB66 with SmB66 [600], YB66 [592], and YB48 [590] plotted for comparison. Reproduced from Sauerschnig et al. [585].

˚ The space group F m3c and the lattice parameter of 23.4142(6) A. XRD pattern of the cleaved surface shows only reflections with Miller indices (2h 0 0), which indicates that it was grown along the a-axis. The theoretical density from the single-crystal structure solution gives ρXRD = 2.89 g/cm3 , with the number of B and Yb atoms per unit cell NB = 1623.6 and NYb = 27.79. The chemical composition was determined as YbB58.7 by ICP-AES, which is slightly more Ybrich than the initial nominal composition of YbB62 [585]. This result agrees with the composition of YbB56.7 obtained by Sologub et al. for a single crystal isolated from a sample of nominal composition YbB70 , annealed at 1825 ◦ C [584]. Thermoelectric properties of YB48 [590], SmB62 [600], YB66 [592], and YbB59 were investigated, and their dimensionless figures of merit calculated. The temperature dependence of zT is plotted in Fig. 1.82 [585]. At 1000 K, the dimensionless figures of merit of YB48 , SmB62 , and YbB62 single crystals are close to 0.1, which is approximately 30 times higher than previously reported for YB66 samples. This value is not very high in comparison to other

190 Crystal Chemistry and Crystal Growth of Rare-Earth Borides

thermoelectric materials, but considering the steep increase in the figure of merit with increasing temperature, which extrapolates to ∼0.4 at 1500 K (the expected working temperature for topping cycles in thermal power plants), these hectoborides may be of interest for very-high-temperature applications. It is assumed that the origin of the good performance of these hectoborides is owed to their metal-rich composition [585, 594].

1.7 Concluding Remarks To conclude, in this chapter I summarized the progress in singlecrystal growth of six families of rare-earth borides. Special attention was paid to the works published in recent decades, reflecting the technical progress in crystal-growth setups and modern developments in the physical characterization of the samples. The two most widely used growth methods—flux and zone melting—have been compared in terms of their advantages, disadvantages, and specific features, which should be considered when planning an experiment. The ideology of these two methods and the corresponding factors determining the conditions of crystal growth are very different, which may influence the stoichiometry, level of impurities, distribution of crystal defects, and physical properties of the resulting samples. Therefore, to grow RE boride single crystals with appropriate properties, crystal quality and homogeneity must be optimized empirically through accurate understanding and precise control of growth parameters. The choice of such experimental conditions for the growth of a specific boride requires individual approach, taking into account both the nature of the compound (type of melting, vapor pressure of the components, homogeneity range, etc.) and the design features of the specific crystal-growth setup. After all, single-crystal growth still remains, first and foremost, art, and only then science. The information on various details of the growth process and comprehensive characterization of the resulting single crystals are extremely important for a complete insight into the quality and properties of the grown crystal and an informed interpretation of their possible spread among seemingly equivalent samples prepared with different methods or in different groups. In

References 191

the present chapter, these general principles have been illustrated with numerous examples of sample-property relationships across all known families of binary metal borides.

Acknowledgments I am deeply grateful to D. Inosov for the invaluable help in preparing this publication. I also want to thank my external collaborators P. Alekseev, N. Bolotina, A. Dudka, K. Flachbart, S. Gabani, V. Glushkov, K. Siemensmeyer, and N. Sluchanko for fruitful cooperation over the years and the implementation of their scientific ideas using various boride single crystals grown in our Laboratory of the RE refractory compounds, which was established by Yuriy Paderno; studies of these crystals formed a major part of this chapter. I also thank colleagues V. Filipov, A. Levchenko, S. Polovetz, and S. Sichkar from our laboratory for many years of successful joint work that has enabled these results.

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Chapter 2

Thin Films of Rare-Earth Hexaborides Seunghun Lee,a,b Xiaohang Zhang,a and Ichiro Takeuchia a Department of Materials Science and Engineering and Maryland Quantum Materials

Center, University of Maryland, College Park, Maryland 20742, USA b Department of Physics, Pukyong National University, Busan 48513, South Korea

[email protected]

In the field of condensed matter physics and materials science, thin-film technologies are often employed to study the physical properties of materials and explore device applications. Over several decades, thin films of metal borides have been extensively studied for the applications to wear- and corrosion-resistant or thermionic coating on tools, engineering components, etc. [1], due to their high melting points, exceptional hardness, extraordinary chemical stability, and high thermal and electric conductivity. Most studies on the thin films of rare-earth borides are focused on the hexaboride system because rare-earth hexaborides exhibit a variety of complex ordering phenomena and unique physical properties. In this chapter, we review the literature on rare-earth hexaboride thin films by giving a brief introduction to their synthesis and summarizing the main experimental results. We particularly focus the discussion on the studies of SmB6 thin films, which recently gained great interest as a promising platform to realize exotic quantum states based on the topologically nontrivial nature of SmB6 .

Rare-Earth Borides Edited by Dmytro S. Inosov c 2022 Jenny Stanford Publishing Pte. Ltd. Copyright  ISBN 978-981-4877-56-5 (Hardcover), 978-1-003-14648-3 (eBook) www.jennystanford.com

246 Thin Films of Rare-Earth Hexaborides

2.1 Overview of Rare-Earth Hexaborides The family of rare-earth (i.e., Ce, Dy, Er, Eu, Gd, Ho, La, Lu, Nd, Pr, Sc, Tb, Tm, Yb, and Y) hexaborides is characterized by the simplecubic crystal structure with rare-earth ions sitting at the centers and boron octahedra located at the corners. Despite the simple crystal structure, rare-earth hexaborides demonstrate important physical properties including complex orbital and magnetic ordering phenomena, strong correlation effects, topological electronic states, etc. For instance, lanthanum hexaboride (LaB6 ) has a high melting point and a low work function, and thus it has been widely used as a cathode material [2]; europium hexaboride (EuB6 ) is a ferromagnetic semimetal with a Curie temperature of 15.3 K [3] and a spin polarization of about 50% [4]; yttrium hexaboride (YB6 ) is an s-wave superconductor with a superconducting transition temperature of 8.4 K [5]; cerium hexaboride (CeB6 ) is a heavyfermion compound showing an interesting antiferroquadrupolar (AFQ) order between a high-temperature paramagnetic phase and a low-temperature antiferromagnetic (AFM) phase [6]; and samarium hexaboride (SmB6 ) is a topological Kondo insulator with a truly insulating bulk [7–10]. Because of the simple crystal structure and rich physical properties, the rare-earth hexaboride family provides an ideal platform for theoretical and experimental studies of novel physics and emerging quantum phenomena.

2.2 Lanthanum Hexaboride (LaB6 ) Lanthanum hexaboride (LaB6 ) has been widely used in many applications, for instance as a thermionic and field emitter due to its low work function (2.66 eV), as well as high thermal, mechanical and chemical stabilities [11]. Theoretical calculations have suggested that the metallic behavior (∼10−5 · cm) of LaB6 is due to the [La3+ (B6 )2− + e− ] electronic configuration [12]. For emitter applications, high-quality thin film fabrication is crucial, because some secondary phases, such as La2 O3 or B2 O3 , and crystalline defects have been suggested to significantly affect the conductivity

Texture coefficient

Lanthanum Hexaboride (LaB6 ) 247

Work Function (eV)

Work Function (eV)

Fracture load (N)

annealing temperature [K]

Nitrogen Concentration (%)

Oxygen Concentration (%)

Power (W)

Figure 2.1 (a) SEM photograph of the surface of LaB5.85 thin film after scratching by diamond stylus. The thin film was fabricated by radiofrequency sputtering with the deposition pressure of 0.532 Pa. Reproduced from Kajiwara et al. [11]. (b) Texture coefficient of LaB6 thin films deposited onto W substrates as a function of the annealing temperature, reproduced from Waldhauser et al. [15]. (c) Influence of dc power on the bonding strength of LaB6 thin films, reproduced from Xu et al. [17]. (d) Work function of the LaB6 film as a function of nitrogen (left) and oxygen (right) concentrations in the target, reproduced from Ishii et al. [35].

of thin films. The reduced conductivity for the thin-film form as compared to the bulk of the material is detrimental for the emitter application, but this adverse effect can be overcome by fabricating the film of LaB6 on a suitable conducting substrate [13]. LaB6 thin films have been fabricated using a variety of methods, including sputtering [11, 14–17], pulsed laser deposition (PLD) [18–21], ebeam evaporation [22–28], molecular beam epitaxy (MBE) [29], and others [30]. Metallic substrates such as Mo, W, etc., have been adopted in general to improve the emission properties [13]. The fabrication of LaB6 thin films enables the wearing and decorative function of the material [31], and also allows the material to be used as an emission filament and oxygen absorber in vacuum chambers [16]. As a highly desirable film deposition method for practical application, sputtering process has been widely employed for the

248 Thin Films of Rare-Earth Hexaborides

fabrication of LaB6 thin films. Mroczkowski [14] deposited LaB6 thin films on W and Re filaments using magnetron dc sputtering and then examined the thermal emission density and durability of the filaments. The obtained work function of the thin film in their work is in the range of 2.4–2.6 eV. In addition, Kajiwara et al. [11] suggested that the working pressure has an influence on the stoichiometry (i.e., the atomic ratio of B to La) of sputtered LaB6 thin films [Fig. 2.1(a)], and Waldhauser et al. [15] and Xu et al. [17] also systematically investigated the influence of the deposition parameters used in dc sputtering on the properties of LaB6 thin films, see Fig. 2.1(b, c). Waldhauser et al. have also investigated thermionic properties of other sputtered rare-earth hexaborides thin films (CeB6 , SmB6 and YB6 including LaB6 ) for comparison [32]. They suggested that LaB6 and CeB6 thin films are promising as the emissive material for coated cathodes. Moreover, Kirley et al. [33] investigated the thickness dependence of the field emission properties of sputtered LaB6 thin films fabricated on knife-edge shaped Cu; and Nakano et al. [34] systematically studied the mechanical properties of sputtered LaB6 thin films as a function of deposition (i.e., working) pressure. In particular, as reported in Ref. [34], the deposition pressure affects the preferred crystalline orientation in the thin films, which further leads to the observed variation in the mechanical properties. Further, the effects of nitrogen and oxygen impurities in sputtered LaB6 thin films on the work function were studied in [35], as shown in Fig. 2.1(d), and the photoemission properties of sputtered LaB6 thin films were investigated in Ref. [36] because of the potential use of the material as a low-work-function photocathode. Recently, it was demonstrated that the sputtering process is applicable to the fabrication of nanostructured LaB6 thin films on a flexible substrate [37]. PLD is a method that has been widely used for the fabrication of oxide thin films; however, this technique has also been successfully adopted to fabricate LaB6 thin films. A number of references [18–21] have provided detailed description of the relation between the general properties of LaB6 thin films fabricated by PLD [Fig. 2.2(a)] and the deposition temperature (in a range of 300–850 ◦ C). Liu et al. [20] further investigated the electronic structures of nanocrystalline LaB6 thin films and single crystals by studying

Lanthanum Hexaboride (LaB6 ) 249

(b)

(a)

(c)

(d)

Figure 2.2 (a) XRD spectra of a LaB6 thin film deposited with a laser fluence of 10 J/cm2 at a substrate temperature of 850 ◦ C under a residual vacuum of the order of 9×10−7 Torr, reproduced from Craciun et al. [18]. (b) X-ray absorption near edge structures (XANES) for B2 O3 and LaBO3 , the XANES of LaB6 PLD film recorded in total electron yield (TEY), reproduced from Liu et al. [20]. (c) SEM image of the LaB6 thin film deposited on a tungsten tip. (d) Field emission current–voltage (I –V ) characteristics of LaB6 deposited on the tungsten tip (left) and the Fowler–Nordheim (F–N) plot (right). Reproduced from Late et al. [38].

the x-ray absorption near edge structure (XANES) [Fig. 2.2(b)]. Late et al. [38, 39] also examined the emission properties of LaB6 thin films fabricated by PLD on various substrates and tips (W, Rh, Mo, Ta, and Si) [Fig. 2.2(c, d)]. Several groups have reported the fabrication of LaB6 thin films on Ta [26] and Si with an adhesive Mo layer [28] using the e-beam evaporation method. In particular, Bessaraba et al. [26] deposited LaB6 thin films at 600–800 ◦ C and investigated the thermionic emission properties of the films [Fig. 2.3(a, b)], whereas Wang et al. fabricated LaB6 thin films at 300 ◦ C and investigated their field emission properties as shown in Fig. 2.3(c, d) [28]. In a series of works, Uchida et al. [23–25, 27] have thoroughly discussed the formation of a Schottky barrier between LaB6 thin film and GaAs

250 Thin Films of Rare-Earth Hexaborides

(a)

(b)

j, mA/cm2 1.2

1

1.0 0.8

2

0.6 0.4

3

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200 Ua, V

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Forward Current (A)

(f)

(a)

Forward Bias (V)

(b)

Reverse Bias (V)

Figure 2.3 (a) Reflection electron diffraction photography of 4.57-μm thick LaB6 thin film. (b) Volt–ampere characteristics of aged cathode at a temperature of: (1) 1300; (2) 1220; (3) 1150 ◦ C. Reproduced from Bessaraba et al. [26]. (c) SEM micrograph for the fabricated LaB6 coated Sitip field emitter array (FEA). (d) I –V curves taken from a LaB6 -coated Si-tip FEA, a pure Si-tip FEA and a Mo-coated Si-tip FEA. Reproduced from Wang et al. [28]. (e) Forward and reverse I –V characteristics before and after annealing at various temperatures for LaB6 chemically etched GaAs(001) Schottky diodes. Reproduced from Uchida et al. [25]. (f) Schematic energy band diagram at the interface between LaB6 and GaAs. It shows the relation between the Schottky barrier height φB and the Ga 2p binding energy E b . E VAC : vacuum level, E C : conduction band, E F : Fermi level, E V : valence band. Reproduced from Uchida et al. [27].

Lanthanum Hexaboride (LaB6 ) 251

˚ Figure 2.4 (a) Electron diffractions of LaB6 thin film grown at 0.2 A/s on air cleaved MgO substrate at 750 ◦ C: from left to right, without preheating, preheating at 900 and 1200 ◦ C before deposition. Reproduced from Muranaka and Kawai [22]. (b) XRD pattern of LaB6 thin film on sapphire substrate without SrB6 buffer layer. Inset: RHEED image viewed along the sapphire [1010] axis. (c) XRD pattern of LaB6 thin film on sapphire substrate with SrB6 buffer layer. Insets: magnified (×35) view of the XRD pattern in the 2θ range of 25–40◦ and a RHEED image viewed along the sapphire [1010] axis. Reproduced from Kato et al. [29].

prepared by e-beam evaporation and have suggested the possible use of LaB6 thin film as a self-aligned gating material for GaAs-based integrated circuits [Fig. 2.3(e, f)]. Muranaka and Kawai [22] have reported the epitaxial growth of LaB6 thin films on MgO substrates using e-beam evaporation. They systematically investigated the effects of the substrate temperature (200–850 ◦ C) and the deposition ˚ rate (0.2 or 1 A/s), as well as different substrate preparation (i.e., substrate cleaving in air or vacuum) on the epitaxy of LaB6 thin film [Fig. 2.4(a)]. The growth of an epitaxial LaB6 thin film on Al2 O3 substrate with a SrB6 buffer layer using the MBE technique was reported [29]. The lattice mismatch between LaB6 and Al2 O3 is 12.7%, but SrB6 has ˚ as LaB6 , and thus, with almost the same lattice parameter (∼4.2 A) a SrB6 buffer, the lattice mismatch at the bottom surface of LaB6 can be effectively reduced [Fig. 2.4(b, c)].

Build-up time (ns)

252 Thin Films of Rare-Earth Hexaborides

Pressure (Torr)

Figure 2.5 (a) Boron distribution in the thermoemissive layer from LaB6 to the dielectric substrate discussed in Ref. [40]: At the initial moment of activation (left) and after saturation of the barrier layers (BL) with boron atoms (right). (b) Schematic diagram of the macro-cell configuration used for investigation of discharge properties of LaB6 thin films and (c) The discharging characteristics of ac PDP test cell with LaB6 thin film and MgO thin film. Reproduced from Deng et al. [41].

Moreover, the fabrication of LaB6 thin film using pulse-plasma spray deposition has been demonstrated by Shaginyan et al. [30]. Vasil’ev et al. [40] discussed boron diffusion at the interface between the LaB6 thin film and the substrate, schematically depicted in Fig. 2.5(a). In addition, because of the low work function of LaB6 , the use of LaB6 thin films as protective layers in alternatingcurrent plasma display panels (ac-PDPs) is expected to improve the discharging properties [see Fig. 2.5(b, c)] [41, 42].

2.3 Cerium Hexaboride (CeB6 ) Similar to LaB6 , cerium hexaboride (CeB6 ) has been widely used as a thermionic cathode material. CeB6 was expected to have lower work function and lower volatility as compared to LaB6 [43]. In this

Cerium Hexaboride (CeB6 ) 253

regard, Xu et al. [43] investigated the field-emission properties of CeB6 thin films fabricated through direct thermal evaporation of raw micron-sized CeB6 powders using an induction furnace [Fig. 2.6(d)]. From the J–E and F–N plots (i.e., ln J/E 2 vs. 1/E ), they evaluated the turn-on field and the field enhancement factor of CeB6 thin films, respectively. Although the field-emission performance of the CeB6 thin films was not comparable with that of LaB6 micro/nanostructure, the deposition technique of using induction heat was regarded as a simple approach that can facilitate the synthesis of hexaboride thin films, and thus through further optimization, the performance can be improved. CeB6 has also been suggested as a promising thermoelectric material at low temperatures. Kuzanyan et al. [44] investigated the temperature-dependent resistivity and Seebeck coefficients of CeB6 thin films fabricated on different substrates (Al2 O3 , AlN, Si, Mo, and W) through e-beam evaporation. The low-temperature resistivity of the thin films is found to be higher than that of CeB6 single crystal, and the inconsistency in the resistivity is attributed to the presence of oxygen impurities in the films. However, the Seebeck coefficients of the thin films are all found to be close to that of the single crystals. There are a couple of reports on the fabrication of CeB6 thin films using the MBE method [45, 46]. Shishido et al. fabricated CeB6 thin films on MgO substrates at 750 ◦ C (lattice mismatch of 1.8%) [45]. θ-2θ x-ray diffraction (XRD) measurements reveal the (100)-oriented structure of the CeB6 thin film, but reflection high-energy electron diffraction (RHEED) and in-plane XRD patterns indicate random in-planar orientation. The temperature-dependent resistivity of the CeB6 thin film is found to be roughly consistent with that of CeB6 single crystal, which reflects the antiferroquadrupolar (AFQ) ordering below TQ = 3.2 K. However, the kink structure observed in CeB6 single crystals due to the AFM ordering below TN = 2.3 K is not present in the data measured on the thin films, as can be seen in Fig. 2.6(b). Hatanaka et al. also reported the successful synthesis of CeB6 thin films on MgO substrates through MBE at a growth temperature of about 550–650 ◦ C [46]. Similar to the thin films reported by Shishido et al., the CeB6 thin films have a (001)-oriented structure

Intensity (cps)

Current Density (μA∙cm-2)

r (μΩ∙cm)

r, μΩ∙cm

254 Thin Films of Rare-Earth Hexaborides

Binding Energy (eV)

Applied Field (V∙μm-1)

Figure 2.6 (a) Temperature dependence of the resistivity of CeB6 films on different substrates, reproduced from Kuzanyan et al. [44]. (b) Temperature dependence of the electrical resistivity of CeB6 film under various magnetic fields (inset: RHEED pattern of CeB6 thin film), reproduced from Shishido et al. [45]. (c) XPS narrow spectrum of the surface of CeB6 thin film, reproduced from Hatanaka et al. [46]. (d) J–E plot from CeB6 thin films fabricated by direct evaporation of CeB6 powders (inset: high-resolution transmission electron (HRTEM) image and selected electron diffraction pattern (SAED) of CeB6 thin film). Reproduced from Xu et al. [43].

but are randomly oriented in the plane. The temperature dependent resistivity of the thin films is also found to be similar to that of the film fabricated by Shishido et al. Based on x-ray photoemission spectroscopy (XPS) studies, Hatanaka et al. [46] further discussed the possible effects of secondary phases and impurities on the electrical properties of CeB6 thin films [Fig. 2.6(c)].

2.4 Gadolinium Hexaboride (GdB6 ) Gadolinium hexaboride (GdB6 ) thin films have also been studied for possible electron emitter applications. Similar to other rare-earth

Gadolinium Hexaboride (GdB6 ) 255

(a)

(b)

(c)

Figure 2.7 (a) SEM images of pulsed laser deposited GdB6 thin films on Re flat substrate (top) and Re tip (bottom). (b) F–N plots of GdB6 thin films on Re flat substrate and Re tip. (c) Field-emission micrograph recorded during emission from GdB6 /Re flat substrate. Reproduced from Suryawanshi et al. [47].

hexaboride thin films, due to the low work function (1.5 eV), the high melting point (2500 ◦ C), and the low sputtering yield, GdB6 thin film is also expected to be an ideal field-emission material. Suryawanshi et al. [47] fabricated GdB6 thin films on tungsten (W) and rhenium (Re) tip and foil substrates at 700 ◦ C using the PLD method, and then systematically studied the field emission characteristics of these composite structures (Fig. 2.7). The GdB6 /W and GdB6 /Re tip emitters are both found to deliver a large emission current density: 1.4 mA/cm2 at an applied field of 6.0 V/μm and 0.811 mA/cm2 at an applied field of 7.0 V/μm, respectively. The linear behavior in F– N plots indicates the metallic nature of these GdB6 -based emitters. The work function of the GdB6 thin film is found to be 1.6 eV, which is consistent with the reported value for a GdB6 single-crystalline nanowire [48].

256 Thin Films of Rare-Earth Hexaborides

2.5 Ytterbium Tetra- and Hexaborides (YbB4 and YbB6 ) ´ Guelou et al. investigated the thermoelectric properties of YbB4 and YbB6 thin films on Al2 O3 substrates [49]. The thin films were fabricated by hybrid physical chemical vapor deposition (HPCVD) using elemental ytterbium and decaborane (B10 H14 ), which is a rapid thin film fabrication method (deposition rate: ∼0.1 μm/min) compared to other PVD techniques. The substrate temperatures were set to 750–950 ◦ C, and all the films have a thickness in a range of several μm. The crystallinity of the fabricated films, including the preferred orientation, impurity phases, etc., was found to depend strongly on the growth conditions, including the partial pressures of Yb and B10 H14 , the substrate temperature, etc. [Fig. 2.8(a)]. Electrical transport properties, Seebeck coefficient, and corresponding power factor of YbB4 and YbB6 thin films in the hightemperature range (300–563 K) were also investigated [Fig. 2.8(b)]. (b)

S2ρ-1 / μW m-1 K-2

Intensity / a.u.

S / μV K-1

Intensity / a.u.

ρ / mΩ m

(a)

2θ / °

T/K

Figure 2.8 (a) X-ray diffraction patterns of YbB6 and YbB4 thin films. (b) Temperature dependence of electrical resistivity, Seebeck coefficient and power factor of ytterbium-boride thin films. Reproduced from ´ Guelou et al. [49].

Neodymium Hexaboride (NdB6 ) 257

The YbB4 films show metallic behavior with low resistivity, and the Seebeck coefficient is small, suggesting that such thin films may not be a promising candidate for thermoelectric applications. On the other hand, the YbB6 films show very encouraging thermoelectric properties with relatively low electrical resistivity and reasonable Seebeck coefficients measured on the highly crystalline films. The electrical resistivity remains independent of the temperature, while the magnitude of the Seebeck coefficient increases from 30 μV/K at 325 K to 60 μV/K at 563 K with the power factor of about 200 μW m−1 K−2 . The authors attributed the inconsistent thermoelectric properties of the YbB6 thin films to the possible variation in the stoichiometry and the formation of defects in the films. This result was discussed in comparison to the results reported for alkaline-earth hexaborides such as SrB6 and CaB6 , which have been widely considered as ideal thermoelectric materials. Note that YbB6 and YbB12 were recently suggested as possible topological materials [50], but their natures are still not clear and a controversy remains [51–53].

2.6 Neodymium Hexaboride (NdB6 ) Kher et al. reported the growth of NdB6 thin films using a chemical vapor deposition (CVD) method in which nido-pentaborane (B5 H9 ) or nido-decaborane (B10 H14 ) with neodymium(III) chloride (NdCl3 ) were used as precursors [54]. Among all the samples fabricated (b) Intensity

(a)

2Θ (Degrees)

Figure 2.9 (a) SEM image of NdB6 films deposited from neodymium(III) chloride (NdCl3 ) and nido-pentaborane(9) (B5 H9 ) at 850 ◦ C on a quartz substrate. (b) XRD pattern of NdB6 films. Reproduced from Kher et al. [54].

258 Thin Films of Rare-Earth Hexaborides

using the CVD method, thin films grown at 835 ◦ C with a growth rate of 1 μm/h are found to be polycrystalline NdB6 without any secondary phase (Fig. 2.9).

2.7 Other Hexaborides In Ref. [55], LaB6 , PrB6 , and SmB6 thin films are fabricated by electron-beam evaporation for optical studies. During the fabrication of these hexaboride thin films, the substrates were kept at a temperature in the range of 500–900 K. A post-annealing process was carried out at a temperature equal to or higher than the deposition temperature. The optical reflectivity of the thin films was measured within the spectral wave-number range of 100–50,000 cm−1 , which is used to determine optical constants (e.g., reflective index, absorption coefficients, and dielectric constants) based on the Kramers–Kronig relationships (Fig. 2.10). A comparison between the optical and the electrical characteristic

Figure 2.10 (a) The dispersion dependences of 1 and 2 for the LaB6 film (top) and the LaB6 melted sample (bottom). (b) The reflection coefficient R and the energy loss function P (top) and the dispersion dependences (bottom) of 1 and 2 for the SmB6 film. Adapted from Bessaraba et al. [55].

Other Hexaborides 259

values of the thin films and those obtained on melted bulk samples is also provided. The effective masses of the thin films are found to be larger than the melted samples, which is attributed to the influence of the film’s integral boundaries on the kinetic parameters of the carriers. The mobility, the relaxation time, and the conductivity determined by the optical methods are somewhat lower than the corresponding values obtained through electrical measurements, and the results are attributed to the effect of additional light absorption in an inter-band transition. Vasil’ev et al. investigated thermoelectric emission properties of rare-earth (Ce, Pr, Nd, Sm, Eu) hexaboride thin films fabricated by e-beam deposition [56]. The substrate temperatures were in the range of 800–1000 K, and a post-annealing process was employed. The details on structural properties of the thin films are not provided, but the information on the electron diffraction measurement for the PrB6 thin film is presented. The thermoelectric emission properties were examined with a thin-film emission structure consisting of a metal contact layer (Mo)—metal carbide (ZrC or TiC)—metal diboride barrier layer (TiB2 )—rare-earth hexaboride layer. The measurements of temperature dependent emission current density in a temperature range of 1400–1600 K were carried out to plot Richardson–Dushman dependences, which are further used to estimate the work functions of the rare-earth hexaboride thin films. The work functions of the thin films are found to be lower than the reported values of single crystals, and Vasil’ev et al. noted that a metal carbide layer affects the estimated work function in the measurements performed in the study. After the measurements, no significant degradation can be identified in the surface morphologies of most thin-film emission structures. However, cracks can be found in the layered structures of SmB6 and NdB6 , which are attributed to the difference in the thermal expansion coefficients between the layers of these two different materials (Fig. 2.11). It is suggested that the thickness of emission layers and the layer combination in the thin-film emission structures are both crucial for the durability (i.e., service life) of the cathode device containing the rare-earth hexaboride thin film.

260 Thin Films of Rare-Earth Hexaborides

Figure 2.11 Surface of thin-film emission structures (TES) based on rareearth hexaborides which had operated for 20 h under the thermoemission condition: (a) Mo-TiC-TiB2 -CeB6 ; (b) Mo-ZrC-TiB2 -CeB6 ; (c) Mo-TiC-TiB2 SmB6 ; (d) Mo-TiC-TiB2 -NdB6 . Reproduced from Vasil’ev et al. [56].

2.8 Samarium Hexaboride (SmB6 ) SmB6 is an exemplary Kondo lattice system, and the basic ingredients of local-itinerant hybridization in the material have long been recognized: The electronic configuration of the Sm ions fluctuates between 4f 6 (Sm2+ ) and 4f 5 (Sm3+ ), giving an effective valence of about 2.5 at low temperatures [57, 58]; there is a

Samarium Hexaboride (SmB6 ) 261

20 meV gap in the optical conductivity [59]; the low-temperature magnetic susceptibility is indicative of a nonmagnetic ground state [60]; the temperature dependence of the electrical resistivity is insulating, increasing by several orders of magnitude upon cooling [61], while the carrier density decreases to 1017 cm−3 [62]. However, the observed finite electrical resistivity at very low temperatures has been a puzzle for decades, which essentially raised a question: Whether the material is a metal or an insulator? Recent electronic transport studies [7, 63] and point-contact spectroscopy (PCS) measurements [64] on SmB6 single crystals suggest that a topological surface channel and an insulating bulk channel coexist in SmB6 , indicating that the Kondo lattice system is a topological insulator (TI) [65]. Topological insulators have attracted tremendous interest in the condensed matter physics community due to the topologically protected metallic surface state that has been regarded as a promising platform for exploring exotic quantum phenomena [66, 67] in solid materials. In addition, the spin-momentum locking of topological surface states lends itself naturally to a promising candidate for quantum and spintronic device applications [68, 69]. Towards this end, preparation of SmB6 in the form of thin films is particularly important to facilitate device patterning, for the fabrication of heterostructures, and for surface-sensitive measurements that will give direct insight into the topological nature of this material and lay a foundation for building quantum or spintronic devices based on this material.

2.8.1 Fabrication of SmB6 Thin Films According to Yong et al., growth of SmB6 thin films using a stoichiometric target results in grossly boron-deficient films [70]. Therefore, a combinatorial composition-spread approach by co-sputtering of SmB6 and boron targets was adopted in order to achieve the correct stoichiometry. During deposition, a physical shadow mask was placed over a Si (001) wafer with a 300 nm amorphous SiO2 to naturally separate the film into 2 mm × 2 mm [Fig. 2.12(a)], and the deposition temperature was kept at 800 ◦ C. After deposition, the composition variation of the Smx B1−x spreads was mapped by wavelength dispersive spectroscopy (WDS) measurements. The

262 Thin Films of Rare-Earth Hexaborides

Figure 2.12 (a) Sm concentration x, measured by WDS, as function of lateral distance. (b) Temperature dependences of sheet resistances (left axis) and resistivities (right axis) for three Sm1−x Bx films with x = 0.10, 0.16, and 0.22, respectively. The films are about 100 nm thick and resistances are measured using the Van der Pauw method. (c) Crosssectional TEM image on the interface between SmB6 film and MgO. (d) PCS on the SmB6 film with a PtIt tip (green filled squares) and a superconducting Nb tip (black open circles). The blue curves are the best fits to the Fano line shape. The red curve is the fitting to both Fano line shape and the modified BTK model. Reproduced from Yong et al. [70].

spots with the right 1:6 stoichiometric ratio show resistivity saturation at low temperatures [Fig. 2.12(b)], consistent with the behavior observed in the SmB6 single crystals. The films grown in this manner are polycrystalline in nature [Fig. 2.12(c)]. A parallel conductance model consisting of a surface channel and a bulk channel is able to describe the temperature dependent resistance behavior of these SmB6 thin films [71]. A PCS study performed on these stoichiometric SmB6 films [70] further indicates that the conductance spectrum shows a typical asymmetric Fano line shape, revealing the Kondo nature of the material [Fig. 2.12(d)]. Through static THz time domain spectroscopy and dynamic THz conductivity

Samarium Hexaboride (SmB6 ) 263

(b)

MgO CoFeB W

15

Mz

1

(c)

CFB/W

10

R H (ohm)

Hx

ISmB /Itot 6

(a)

5 0 –5

0.9

0.8

–10 z

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CFB/W/SmB6(50) × 150

–15

y x

–100

–50

0

50

0.7

100

20

H (Oe)

(e) R H (ohm)

R H (ohm)

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I (mA)

50

50

0

–0.05

–0.05

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F

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(d)

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–0.1 –40 –30 –20 –10 0 10 20 30 40

I (mA)

–100 –2 –1.5 –1 –0.5 0 0.5 1 1.5 2

H (kOe)

Figure 2.13 (a) Schematic drawing of the device for spin orbit torque (SOT)-induced switching experiment. (b) Hall resistance for the trilayer MgO(1.5)/CFB(1)/W(0.8) (blue) and MgO(1.5)/CFB(1)/W(0.8)/SmB6 (50) (red). The latter is magnified 150 times. (c) Current distribution in SmB6 of CFB/W/SmB6 multilayers for various SmB6 film thicknesses. The dashed line shows the calculation using longitudinal resistivity. SOT switching for CFB(1)/W(0.8)/SmB6 (50) measured at 20 K (d) and 300 K (e), with external magnetic fields of 500 Oe (blue) and −500 Oe (orange) applied along the x direction. (f) Switching phase diagram of CFB(1)/W(0.8)/SmB6 (50) at 20 K. Reproduced from Li et al. [73].

measurements (using optical-pump THz probe), Zhang et al. [72] suggested the emergence of the surface states based on abnormal behaviors observed in the SmB6 thin film: (i) a sudden transition of 2D Drude weight and scattering rate of carriers at 20 K and (ii) a deviation from the expected thermally excited quasiparticle density in the Rothwarf–Taylor model below 20 K. Li et al. also reported the fabrication of SmB6 thin film using the sputtering technique [73]. Specifically, a stoichiometric SmB6 target was used for sputtering, and an additional dc bias (−90 V) with respect to the electrical ground was applied to the sample to prevent developing boron-deficient films during the deposition process. In layered structures consisting of MgO/FM/β-W/SmB6 thin films, strong magnetic switching of an adjacent ferromagnetic layer (CoFeB or Co) induced by spin-orbit torque (SOT) is observed (Fig. 2.13), which could be attributed to the spin-momentum locking of the topological surface states. However, the observed SOT at the

264 Thin Films of Rare-Earth Hexaborides

Figure 2.14 (a) A schematic of spin pumping experiments on SmB6 thin film on YIG. (b) Spin-pumping signals measured using magnetic fields (H) and an input microwave power Pin = 100 mW. (c,d) σ (V1, 0 /a)d data at 297 and 10 K, respectively, as a function of SmB6 thickness, d. Here σ is the conductivity of SmB6 and a is the sample length. σ (V1, 0 /a)d thus represents the signal strength independent of the sample dimension and resistivity. The red curves are fits to an equation, which has been widely used to evaluate the spin-pumping signals in the heavy-metal/ferromagnet. Reproduced from Liu et al. [74].

room temperature cannot be ascribed to the presence of topological surface states because the emergence of the topological surface states as well as the Kondo hybridized gap is expected to be under the Kondo temperature (∼100 K) of SmB6 [Fig. 2.13(e)]. Therefore, Li et al. attributed the observed SOT to an alternative origin, that is, the bulk spin Hall effect or the interfacial Rashba effect, which is expected to have no marked temperature dependence. The monotonic enhancement of the current-SOT conversion efficiency as the thickness of the SmB6 film increases also supports the idea that SOT comes from the bulk.

Samarium Hexaboride (SmB6 ) 265

(c)

(a)

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MgO

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400

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SmB4

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300 0

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8 6

B/Sm = 5.7

4

2

40

200 T (K)

U (m:·cm)

310

10 U (m:·cm)

U (P:·cm)

U (P:·cm)

200

TN

SmB6/MgO (003)

SmB6 (5.7) SmB4 (4.0)

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0 1

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6 8

0.10 2

10 T (K)

4

6 8

100

2

10

20

30

40 50 60 2 TT(deg)

70

80

90

100

Figure 2.15 (a) Reflection high-energy electron diffraction (RHEED) patterns for SmB4 with the deposition rate ratio of B:Sm = 4.0 and SmB6 with B:Sm = 4.9, respectively. (b) Temperature dependence of the electrical resistivity for SmB4 (4.0) (left) and SmB6 (5.7) (right). The inset of the right panel shows the Arrhenius plot of the resistivity. (c) Interplane x-ray diffraction pattern for SmB4 (4.0), SmB6 (4.9) and SmB6 (5.7), respectively. Reproduced from Shishido et al. [75].

Liu et al. [74] studied the penetration depth of the topological surface states of SmB6 , sputtered on Y3 Fe5 O12 (YIG) in the same manner, with a spin-pumping measurement [Fig. 2.14(a)]. Spin current and lateral electrical voltage generated by spin pumping in this system can be attributed to the presence of topological surface states with the spin-momentum locking [Fig. 2.14(b)]. In the thickness dependence measurements [Fig. 2.14(c)], the spinpumping signal strength at low temperature (10 K) shows different behavior to that at room temperature and starts to rise sharply at d = 50–80 nm. They attributed this behavior to the decoupling of the topological surface states at the top and bottom surfaces, resulting in the enhanced spin-pumping signal. The penetration depth of the topological surface states was estimated to be ∼32 nm. Shishido et al. [75] fabricated SmB6 as well as SmB4 thin films on MgO substrates through an MBE method by evaporating separated Sm and B sources in Knudsen cells (K cell) (Fig. 2.15). The B:Sm deposition ratio was controlled to be 4.0–5.7. The SmB4 thin films fabricated with a B:Sm deposition ratio of 4 have a structure that is highly c-axis oriented but has an essentially random

266 Thin Films of Rare-Earth Hexaborides

in-plane orientation. The temperature-dependent resistivity of SmB4 thin films exhibits metallic behavior, which is qualitatively consistent with the results of SmB4 bulk. However, the resistivity at room temperature is about 60 times larger than that of bulk samples, and the difference was attributed to in-planar disorders in the thin film. The resistivity of thin film increases below 24 K [Fig. 2.15(b)] and they attributed it to the emergence of AFM ´ temperature, TN = 25 K). However, such electrical ordering (Neel behavior is not observed in the single crystal [76], which shows only a kink structure around TN . The thin films fabricated with B:Sm deposition ratios of 4.9 and 5.7 are both found to have the SmB6 -like crystal structure with partial epitaxial orientation of SmB6 [100] MgO[100]. The overall temperature dependence of the resistivity of SmB6 thin films shows insulating behavior and saturates at low temperatures, which is in reasonable agreement with that of the bulk crystals. Shaviv Petrushevsky et al. [77] fabricated a SmB6 thin film on MgO substrate through PLD and investigated in-plane anisotropic magnetoresistance (AMR) of SmB6 thin films depending on the film thickness, see Fig. 2.16. To compensate boron deficiency in the film fabricated from the stoichiometric SmB6 target, a SmB6 target and a boron target were alternately ablated with a pulse rate ratio of 3:1, respectively. The sign of the AMR changes from negative (ρ < ρ⊥ ) at high temperatures to positive (ρ > ρ⊥ ) at low temperatures [Fig. 2.16(b)]. The temperature, Ts , at which this sign change occurs, decreases with increasing the thickness, t of the film, and eventually vanishes as t becomes greater than 30 nm. The results are ascribed to the competition between two components: a negative bulk contribution and a positive surface AMR. By removing the natively-formed amorphous SiO2 layer on the Si (001) wafer through a pre-cleaning process and adjusting the plasma density, Lee et al. [78] were able to fabricate highly crystalline epitaxial SmB6 thin films (Fig. 2.17). High-resolution transmission electron microscopy measurement confirms the absence of interfacial gradation or extra phases. Selected area electron diffraction (SAED) patterns of the interface region reveal the epitaxial relation, SmB6 [100] Si[110], as shown in Fig. 2.17(d, e), which is consistent with a small lattice mismatch (∼7%) between

Samarium Hexaboride (SmB6 ) 267

(a)

(b)

Figure 2.16 (a) High-resolution (HR, phase contrast) TEM (HRTEM) image reveals a polycrystalline cubic structure with random orientations. On the left of the HRTEM image, typical power spectra calculated from different regions of the sample are shown. Region 1 is the MgO substrate aligned to a [100] zone axis. Region 2 shows an example of an SmB6 crystal in zone axis (also [100] zone axis). Due to the random crystallographic orientation of the film and the small crystallographic domain size compared to the film thickness, reflections from additional crystals not in zone axis can also be observed. At the SmB6 /MgO interface, the layer is of reduced crystallinity. The top part of the film (region 3) is oxidized and amorphous. (b) Normalized planar Hall effect (PHE) amplitude thickness dependence. The amplitude is determined by fitting the PHE data at 5 and 70 K (blue diamonds and red rectangles, respectively (left). Thickness dependence of the transition temperature, Ts , at which the PHE amplitude changes sign. For the 32-nm-thick sample the PHE does not change sign, and the red triangle represents an estimation by extrapolation to low temperatures (right). Reproduced from Shaviv Petrushevsky et al. [77].

Si (110) and SmB6 (100). In addition, aberration-corrected scanning transmission electron microscopy was utilized, and the atomicresolution image taken from the SmB6 film displays its cubic structure [Fig. 2.17(f)]. The θ-2θ XRD pattern in Fig. 2.17(g) shows a c-axis-oriented structure of SmB6 . The XRD diffraction pattern exhibits sharp SmB6 peaks, which are associated with the {001} ˚ which is planes only. The lattice parameter is found to be 4.13 A, close to the bulk value [10].

2.8.2 Proximity Effect in Nb/SmB6 Bilayers Superconducting proximity effect occurring at an interface between a TI and a superconductor has been suggested as a way to induce chiral topological superconductivity [79–81] and Majorana fermion excitations [82–87]. To this end, concerted experimental

268 Thin Films of Rare-Earth Hexaborides

(a)

(b)

SmB6 ZA [100]

b

(c)

Si ZA [110]

(d)

Interface SmB6

d c 20 nm

f

4. 13 3 (a 6

Sm

B

(g) Intensity (a.u.)

=

[001]

g

(f)

Å)

(e)

Si (a = 5.430 Å)

Si (004)

SmB6/Si (001)

SmB6 (001) SmB6 (002)

2 nm

lattice mismatch = 7 %

20

40

60

80

2θ (deg.)

Figure 2.17 (a) High-resolution cross-sectional transmission electron microscopy image of a SmB6 thin film. The yellow squares correspond to the regions of the selected-area electron diffraction (SAED) measurements shown in (b)–(d). (b–d) SAED measurements of SmB6 (b), Si substrate (c) and SmB6 /Si interface regions (d). (ZA: zone axis) (e) Epitaxial relationship between the SmB6 and the Si substrate. (f) Aberration-corrected scanning transmission electron microscopy cross-sectional image of a SmB6 thin film. (g) θ-2θ x-ray diffraction pattern of a SmB6 thin film on a Si (001) substrate. Reproduced from Lee et al. [78].

efforts have been made to study the superconducting proximity effect in superconductor/TI bilayer structures. However, in most Bi- and Te-based TI such as Bi2 Se3 , Bi2 Te3 , Sb2 Te, etc., the overwhelming conducting bulk electronic states hinder interfacial quantum phenomena and preclude the investigation of the interplay of the topological surface states and Cooper pairs [88–91]. In addition, low transparency at the superconductor/TI interface due to a non-pristine surface of the TI can significantly reduce the extent of the proximity effect [86, 92, 93]. Therefore, suppressing the bulk conductivity and securing high interfacial transparency in superconductor/TI bilayers has been an outstanding issue in this field. The superconducting proximity effect describes a phenomenon at a superconductor-normal metal interface where Cooper pairs diffuse into the normal metal, which results in suppression of the critical temperature (Tc ) of the superconductor and also induces surface or local superconductivity into the normal metal [94].

Samarium Hexaboride (SmB6 ) 269

(a)

(b)

Figure 2.18 (a) Model for the proximity effect in a superconductor-TI bilayer system. The plot in the schematic represents the evolution of the superconducting pair potential (z). In this model, we consider the case where the normal coherence length of SmB6 (ξSmB6 ) is larger than the thickness of the surface state of SmB6 (tsurface ). (b) Tcb /Tcs (i.e., the ratio of Tc of the Nb/SmB6 bilayer to the Tc of a single Nb layer) as a function of the Nb-layer thickness, dNb . The data points from bilayers with different SmB6 layer thicknesses (10 and 25 nm) are also included. The solid line is a fit using the Usadel equation. The star indicates Tcb /Tcs of the Nb/SmB6 bilayer where a 20 nm Nb layer is deposited on SmB6 (50 nm) after its surface is first exposed to air, forming an ex situ interface. Reproduced from Lee et al. [71].

To study the superconducting proximity effect on SmB6 , Lee et al. fabricated superconducting Nb/SmB6 bilayer structures and investigated the change in Tc with the varying thickness of the Nb layer [71]. Due to the superconducting proximity effect at the interface, the Tc of the bilayer decreases compared to that of the single superconductor (Fig. 2.18). The change in Tc can be quantitatively characterized by the Usadel equation, which is a function of the superconducting (normal) coherence length of the superconductor (normal metal), the thickness of superconductor, and the proximity coupled thickness in the normal metal layer [95–97]. Based on the picture of the surface states coupled with the adjacent superconducting Nb layer, the best fit to the change in the measured Tc as a function of the thickness of the Nb layer results in a normal coherence length of ∼10 nm, and the corresponding Fermi velocity is ∼105 m/s. This value is consistent with the values of the surface states reported from

270 Thin Films of Rare-Earth Hexaborides

angle-resolved photoelectron spectroscopy (ARPES) and quantum oscillation measurements on the SmB6 single crystals, implying that the observed superconducting proximity effect is attributed to the surface state of the SmB6 thin film. The importance of securing a high-quality interface for the superconducting proximity-effect study is confirmed by directly comparing the Tc suppression in the Nb/SmB6 bilayers fabricated without breaking the vacuum (i.e., by an in-situ method) to the Tc suppression in samples where the SmB6 layer was exposed to ambient air before the deposition of the Nb layer (i.e., by an ex situ method) [71]. The comparison clearly indicates that the Tc suppression observed in the ex situ samples was significantly limited [Fig. 2.18(b)]. Since the proximity effect is expected to be sensitive to the nature of the interface, the degradation of the topmost surface layer of SmB6 (e.g., due to oxidation, etc.) may be able to reduce the boundary transparency, resulting in a reduction of the proximity effect. This result demonstrates that a pristine interface ensured by the in-situ fabrication method is crucial for the proximity effect at the interface of SmB6 and Nb, and thus is a key to further exploring the novel quantum phenomena proposed in this system.

2.9 Superconductivity in Yttrium Hexaboride (YB6 ) Among all known borides, yttrium hexaboride (YB6 ) shows the second highest superconducting transition temperature (Tc ) after MgB2 [99, 100]. Experimental studies of superconductivity indicate that the onset Tc of YB6 is about 8.4 K, and the material is an s-wave superconductor [5]. It has also been suggested that the inevitable chemical variation, i.e., the variation of the B:Y stoichiometric ratio in YB6±δ single crystals, leads to a variation in the Tc [100]. However, no further systematic study of the superconducting properties in a broad range of compositional variation has been reported. Schneider et al. fabricated YB6 thin films on Al2 O3 substrates for electron tunneling experiments using dc sputtering [98]. The thin films were fabricated at a temperature of 1000 ◦ C, and after the

Superconductivity in Yttrium Hexaboride (YB6 ) 271

Figure 2.19 (a) Derivative measurement dI /dV vs. voltage for a YB6 /YB6 oxide/In tunnel junction with the YB6 in both the superconducting (straight line) and normal (dashed line) state taken at 1.2 K and slightly above the critical temperature (T = 8 K), respectively. The superconductivity of the In-counterelectrode is suppressed by a magnetic field (H = 0.08 T). (b) Measured reduced density of states (RDOS) (solid line) and a fit calculated by the McMillan and Rowell (MMR) program (dashed line) for YB6 ; 0 is the zero-temperature gap. The inset shows the RDOS of a diode with a slightly smaller gap, measured up to 120 meV using a broad modulation signal of ∼1 meV. (c) The Eliashberg function resulting from the MMR analysis of the measured RDOS shown in panel (b). Reproduced from Schneider et al. [98].

deposition, each film was exposed to the atmosphere at 170 ◦ C for forming a native oxide tunnel barrier. The thin films show a Tc of 7.1 K in resistivity measurements, and the superconducting gap size is found to be 1.24 meV [Fig. 2.19(a)]. From the second-derivative analysis of the current–voltage characteristic [Fig. 2.19(b, c)], Schneider et al. concluded that in the superconducting cluster compound YB6 , the superconductivity is solely due to the coupling between the electrons and the acoustic phonons. Lee et al. investigated the stoichiometry effect on the Tc of sputtered YB6±δ thin films [78]. Due to the substantial difference in the atomic mass between Y and B, and the variation in the distance from the target to different locations of a 3 inch wafer, “natural composition spread” films of YBx are fabricated on Si (001) substrates by sputtering a stoichiometric YB6 target. The deposition process was performed under a deposition pressure of 10 mTorr and at a growth temperature of 860 ◦ C. The B:Y stoichiometric ratio for films deposited at different positions was examined by WDS

272 Thin Films of Rare-Earth Hexaborides

(a)

(b)

YBx thin film B/Y ratio (x) 5.3 5.6 5.9 6.1 6.4 6.9 7.0 7.3 7.5

0.5

7 YBx thin film 6 5

Tc (K)

R/RN

1.0

4 3 2

0.0 2

3

4

5

Tc (K)

6

7

8

5

6

7

8

x

Figure 2.20 (a) Temperature-dependent resistance curves of YBx thin films with different B:Y stoichiometric ratios. (b) Change in Tc as a function of the B:Y stoichiometric ratio (x). Reproduced from Lee et al. [78].

measurements. The temperature dependence of the normalized resistance (R/R N , where R N is the normal-state resistance) of the YBx thin films indicates that the superconducting transition temperature, Tc , varies with the B:Y stoichiometric ratio (Fig. 2.20). The highest Tc is observed in a slightly boron-deficient region (with a B:Y ratio of 5.6).

2.10 SmB6 /YB6 Thin-Film Bilayer Heterostructures In the earlier section, we have discussed that the superconducting proximity effect occurring at an interface between a TI and a superconductor has been suggested as a way to induce chiral topological superconductivity and Majorana-fermion excitations [82, 101]. In this regard, SmB6 and YB6 could be the best material ingredients for building up such a system. Especially, the use of the isostructural rare-earth-hexaboride superconductor as the layer underneath SmB6 enables the fabrication, by sequential hightemperature growth, of a pristine SmB6 /YB6 interface, which is necessary for achieving a robust proximity effect [71]. SmB6 is a topological Kondo insulator, in which the bulk gap at low temperatures ensures the existence of an insulating bulk sandwiched by topologically protected conducting surface layers. This is a critical

SmB6 /YB6 Thin-Film Bilayer Heterostructures 273

prerequisite for the observation of effects that originate solely from the topologically protected states. When a TI layer is placed on top of the heterostructure, various measurements can be used to directly access proximity-induced superconductivity in the TI layer. Earlier, angle-resolved photoelectron spectroscopy (ARPES) [101, 102], scanning tunneling microscopy (STM) [103, 104], and PCS measurements [101, 105] have been used to investigate the proximity-induced superconductivity in Bibased topological insulators on NbSe2 . Point-contact spectroscopy is usually one of the tools that is immediately adopted to probe the superconducting properties [106, 107]. Specifically, when a pointcontact junction is formed between a superconductor and a normal metal, within the superconducting energy gap, an electron from the normal metal needs to be paired up to enter the superconductor, and to maintain conservation in charge, spin, and momentum, a hole with an opposite spin is generated at the interface and bounced back into the normal metal. The above process is known as the Andreev reflection [108]. Because of the Andreev reflection process, the junction conductance is expected to be doubled within the superconducting energy gap (± ) as compared to the outside of the gap. Hence, the PCS measurements performed on a superconductor are also referred as to point contact Andreev reflection (PCAR) measurements. In most junctions, however, normal reflection that leads to single electron tunneling and suppressed in-gap conductance is inevitable. To represent the interfacial barrier strength that leads to normal reflection due to the reduced transparency at the interface, a dimensionless parameter Z was introduced by a theory developed by Blonder, Tinkham and Klapwijk (BTK theory) [109]. According to the BTK theory, perfect conductance doubling in the superconducting gap requires Z = 0. However, it is known that a finite Z can arise not only from the presence of a physical barrier but also from the mismatch in the Fermi velocities of the superconductor and the normal metal [106, 107]. Therefore, perfect Andreev reflection (i.e., the exact conductance doubling in ) has been rarely observed in PCAR measurements [78].

274 Thin Films of Rare-Earth Hexaborides

2.10.1 Point-Contact Spectroscopy Measurements To investigate how the presence of Dirac states at a TI surface affects the processes of particle transport governed by Andreev reflection, a PtIr tip was used to form a point-contact interface at the top surface of the SmB6 /YB6 bilayer heterostructure, see Fig. 2.21(a). As theoretically expected [82] and experimentally confirmed [101], the superconducting proximity effect occurring z

(a)

(b) PtIr 0.7 meV SmB6

20–30 nm PtIr tip

YB6

100 nm

z

SmB6 Incident electron

ky

ky Electron re ection Cooper pair

Δ

1.2 meV

y

kx Hole generated

x

kx

SmB6 pz = 0 YB6

1.5

SmB6 (20 nm) /YB 6

1.0 –4

–2

0 Bias (mV)

2

4

(e) Dirac–BTK

2.0 1.5

(meV)

Normalized dI/dV

Dirac–BTK

2.0

Normalized dI/dV

(d)

(c)

SmB6 (30 nm) /YB 6

1.0

Dirac–BTK BCS t

0.5

SmB6 (20 nm)/YB6 1.0 –4

–2

0 2 Bias (mV)

4

0.0

0

1

2

3 4 T (K)

5

6

Figure 2.21 (a) Schematic of PCAR measurement on SmB6 /YB6 heterostructures. Owing to the lack of bulk states in SmB6 , only electrons with momentum parallel to the plane of the surface states of SmB6 (that is, pz = 0) contribute to transport. The inset shows variation of in a SmB6 (20–30 nm)/YB6 heterostructure. (b) Andreev reflection process at the interface between PtIr and superconducting SmB6 . The surface of SmB6 has topologically protected helical states exhibiting spin-momentum locking. Irrespective of barrier height, normal electron reflection is not allowed because it requires a spin flip. (c, d) Perfect Andreev reflection due to Klein tunneling, indicated by exact doubling of the normalized differential conductance (dI /dV ), is observed in the PCS of (c) PtIrSmB6 (20 nm)/YB6 (100 nm) and (d) PtIr-SmB6 (30 nm)/YB6 (100 nm) heterostructures measured at 2 K, respectively. The red lines are fits to the experimental data using a BTK model modified with a Dirac Hamiltonian (Dirac-BTK) with (c) = 0.75 ± 0.06 meV and (d) = 0.73 ± 0.05. (e) The temperature-dependent (extracted using the Dirac-BTK model) from an Au-SmB6 (20 nm)/YB6 structure in which a gold thin film was used to form the junction, displaying Bardeen-Cooper-Schrieffer behavior (cyan line). Reproduced from Lee et al. [78].

SmB6 /YB6 Thin-Film Bilayer Heterostructures 275

in such TI/superconductor heterostructures creates helical Cooper pairing on the surface of a TI. Due to the constraints imposed by the two-dimensional surface states and the insulating bulk, incoming electrons with a finite pz (momentum perpendicular to the surface) do not participate in the transport at the interface between a normal metal and TI/superconductor heterostructure [110]. Thus, the PtIr-SmB6 /YB6 contact creates an interface where only in-plane transport (i.e., the momentum parallel to the plane of the surface states, pz = 0) is allowed. In addition, induced spin-momentum locking in a normal metal in contact with a TI has previously been observed as a result of the topological proximity effect [111, 112]. Due to the spin-momentum locking on both sides, incident electrons are forbidden to reflect back, as shown schematically in Fig. 2.21(b). The perfect electron transmission to superconducting SmB6 and the concomitant hole generation result in the observed conductance doubling for energies within the proximity-induced energy gap. For SmB6 /YB6 heterostructures with a SmB6 layer in a thickness range between approximately 20 and 30 nm, the PCS measured at 2 K displays normalized differential-conductance (dI /dV ) curves with doubling of the conductance within the bias voltage corresponding to the induced gap , as shown in Fig. 2.21(c, d). The observed doubling of the conductance is exact within the uncertainty due to the fitting procedure. In this thickness range, the SmB6 layer is thick enough to have fully-developed topologicallyprotected surface states, while the superconducting proximity effect from the YB6 can still be observed at the top surface. The best model fit to the data, based on the BTK theory [109], results in a proximity-induced of ∼0.7 meV, see Fig. 2.21(e). The obtained value is, as expected, smaller than the bulk of YB6 (∼1.3 meV) [113]. To further confirm the correlation of the observed perfect Andreev reflection and the topological superconductivity achieved in the SmB6 /YB6 bilayer heterostructures, Lee et al. carried out a series of experiments: SmB6 thickness dependency (0–50 nm), Y substitution effect, and magnetic-field dependency, summarized in Fig. 2.22 [78]. When the thickness of the SmB6 film is less than ∼20 nm, the effect of hybridization of the top and bottom surface states becomes pronounced, opening a gap in the surface-states dispersion and

276 Thin Films of Rare-Earth Hexaborides

Surface

Bulk

Bulk

Thick SmB 6 (t ≥ 20 nm)

Thin SmB6 (t < 20 nm)

Normalized dI/dV

(d)

BTK 2.0 Sm0.5Y0.5B6 (20 nm)/YB6 1.5 1.0 –4

–2

0 2 Bias (mV)

4

SmB6 (10 nm)/YB6 1.5 1.0 –4

–2

0 2 Bias (mV)

(e)

Normalized dI/dV

Surface

EF

BTK 2.0

BTK 2.0 1.5

BTK

–4

4

SmB6 (50 nm)/YB6

0.05

0.8 –4

0 2 Bias (mV)

Fano line 0.06

YB6

–2

(f)

1.2 1.0

Sm0.8Y0.2B6 (20 nm)/YB6

1.0

4

dI/dV

EF

(c)

(b)

Bulk

Normalized dI/dV

Bulk

Normalized dI/dV

(a)

–2

0 2 Bias (mV)

4

–60

–30

0 30 Bias (mV)

60

Figure 2.22 (a) Band structures of different thicknesses of SmB6 . (b) Pointcontact spectrum of a SmB6 (10 nm)/YB6 heterostructure. Reduced conductance at zero bias (normalized dI /dV ≈ 1.4) is observed. (c–e) Point-contact spectra of Y-substituted SmB6 (Sm1−x Yx B6 (20 nm))/YB6 heterostructures with x = 0.2 (c) and x = 0.5 (d) and of the YB6 layer only (e). (f) The point-contact spectrum of a SmB6 (50 nm)/YB6 exhibits an asymmetric Fano-type spectrum due to the inherent Kondolattice electronic structure of SmB6 . The orange line is the best fit to the Fano line shape. All point-contact spectra were obtained at 2 K. The blue lines are best fits to the standard BTK theory. Reproduced from Lee et al. [78].

weakening the topological protection, as shown schematically in Fig. 2.22(a) [101, 102]. Experimentally, the conductance enhancement obtained from a contact made on a SmB6 (10 nm)/YB6 heterostructure is found to be significantly reduced at 2 K [Fig. 2.22(b)]. When the surface of a YB6 film is directly probed (with no SmB6 layer on top), the point contact spectrum at 2 K shows an entirely different characteristic [Fig. 2.22(e)]: The junction is in the regime where tunneling has significant contribution, exhibiting reduced conductance in the gap region of YB6 with a substantial barrier strength at the interface (Z ≈ 1, extracted using the standard BTK model). The gap value ( ≈ 1.3 meV) determined from the fit is consistent with the full of bulk YB6 [113]. Moreover, measurements were also performed on Sm1−x Yx B6 (20 nm)/YB6 heterostructures, where Sm in the top layer is partially substituted by Y to modify the electronic structure. The chemical doping effect is expected to induce transport channels that are not subjected to spin-momentum locking. The point-contact spectra taken on Sm0.8 Y0.2 B6 /YB6 and Sm0.5 Y0.5 B6 /YB6 heterostructures at

SmB6 /YB6 Thin-Film Bilayer Heterostructures 277

Figure 2.23 (a) A schematic of the bilayer consisting of an SmB6 film and a YB6 film. Parallel microwave magnetic field (H 0 ) is applied to the top surface of the SmB6 layer (red arrows). (b) Temperature dependence of the effective penetration depth λeff (T ) of the SmB6 /YB6 bilayers for various SmB6 layer thickness (tSmB6 ). (c) Schematic spatial profile of the order parameter N,S (blue) and the local penetration depth λN,S (red) through the normal-layer (N) / superconductor (S) bilayer sample for the case of the absence (top) and presence (bottom) of an insulating bulk. Here z is the thickness direction coordinate and tN (tS ) is the thickness of the normal layer (superconductor). In the presence of an insulating bulk, the proximitized thickness dN < tN since the insulating bulk blocks propagation of the order parameter to the top surface. Reproduced from Bae et al. [114].

2 K indeed show a substantially reduced zero-bias conductance enhancement (∼1.5), indicating that increasing bulk conduction electrons diminishes the topological protection [Fig. 2.22(c, d)]. Further, when the thickness of SmB6 is larger than 40 nm, the dI /dV spectrum at 2 K does not show any feature corresponding to proximity-induced superconductivity. Instead, the entire dI /dV spectrum shows Fano resonance [Fig. 2.22(f)]—a familiar signature of the Kondo lattice physics of bulk SmB6 [64]. Bae et al. [114] have studied microwave Meissner screening properties of this system as shown in Fig. 2.23(a). The temperaturedependent effective penetration depths of SmB6 /YB6 bilayers

278 Thin Films of Rare-Earth Hexaborides

for different SmB6 thicknesses depicted the existence of a bulk insulating region in the SmB6 layer, see Fig. 2.23(b). From spatially dependent electrodynamic screening model analysis illustrated in Fig. 2.23(c), the normal penetration depth, normal coherence length, and the thickness of the SmB6 surface states are given.

2.10.2 Dirac-BTK Theory Standard BTK theory captures spectra from a variety of pointcontact junctions well, including the point contact spectrum of the YB6 thin film. However, the perfect conductance doubling observed in PtIr-SmB6 /YB6 junction cannot be elucidated by the standard BTK theory; according to the standard BTK theory, the observed perfect conductance doubling implies Z ≈ 0 for contacts to the SmB6 /YB6 heterostructures with the thicknesses of the SmB6 layer in the 20–30 nm range. In the case of a contact between PtIr and SmB6 , a substantial barrier is expected based just on the significant Fermi velocity mismatch between them (the Fermi velocity of the surface states of SmB6 is 12. All the dodecaborides, except YbB12 , are good metals similar to RB6 and RB4 , whereas RBn with n > 12 are insulators. Both B12 cuboctahedra and icosahedra are electrondeficient by two electrons. The trivalent rare-earth atoms can supply three electrons, so there is one excess conduction electron per unit cell. The only exception is the narrow-gap semiconductor YbB12 , known also as a Kondo insulator, where Yb takes an intermediate valence. The summary table of structural, electronic and magnetic characteristics of the dodecaborides of the UB12 type different in isotope boron composition is presented in Ref. [9]. In addition to the rare-earth dodecaborides TbB12 – LuB12 mentioned above, this table contains ZrB12 , HfB12 , pseudo-cubic ScB12 , GdB12 synthesized under high pressure [10] as well as dodecaborides of heavy metals ThB12 , UB12 , NpB12 , PuB12 , and several solid solutions R1x R21−x B12 of the UB12 type. In the same article [9], the lattice constants of RB12 as well as the B–B distances in and between the B12 cuboctahedra versus ionic radii of the metal atoms are plotted and

296 Crystal Structures of Dodecaborides

discussed. A large amount of the reference data on the structures and properties of higher borides, including dodecaborides RB12 , is contained in the overview chapter of a recent PhD thesis [11]. Metal dodecaborides MB12 attract particular interest as multifunctional materials. For instance, in contrast to conventional superhard materials like diamond, which are insulators or semiconductors, many dodecaborides are superhard compounds with high electrical conductivities that can be used as conductors at extreme conditions [12]. Owing to the simple cubic structure, dodecaborides are convenient objects for studying physical properties of the metal atoms possessing relative freedom in the large B24 cavities of the boron framework. There are a large number of publications on the physical properties of dodecaborides. Their crystal structures, however, have not been studied in such detail and are almost not studied at low temperatures (the ZrB12 structure at 140 K [13] is a rare exception), although low-temperature physical properties often reveal features that require explanations based on the crystal structure. Moreover, clear explanations are not always easy to obtain in the framework of a simple cubic model. For example, in Refs. [14, 15], linear thermal expansion coefficients α of RB12 single crystals, R = Y, Ho, Er, Tm, Lu, were measured in the temperature range of 5–300 K. The values of the coefficients at low temperatures varied nonlinearly for all the compounds studied. The nonlinear α(T ) dependences had two minima, sometimes negative in magnitude. Two temperature intervals with negative thermal expansion (NTE) were found for LuB12 crystals: the first one was 60–130 K with a minimal negative at 90 K and the second one was 10–20 K with a minimum at 12 K. One NTE interval 50–70 K was revealed for YB12 with a negative minimum near 60 K, whereas the second minimum at 15 K was close to zero but positive. The relationship between anomalies of the thermal expansion and crystal structure of the dodecaborides is poorly understood so far. Until recently, published data on the observed anomalies of the RB12 structure was actually limited to discussions about observed disordering of the metal atoms near the 4a position of the F m3m group [16, 17] and to a short report on a small

Introduction 297

tetragonal distortion of the LuB12 lattice at temperatures below 150 K [18], which was discovered in the analysis of thermal expansion using x-ray data. Reports of a tetragonal distortion of the dodecaboride lattice even at room temperature appeared before, but they only concerned ScB12 and Sc-containing solid solutions Sc1−x R x B12 (R = Y, Zr) [5, 12, 19, 20]. As noted in [20], the origin of the transformation from cubic to tetragonal structure in ScB12 is unclear. The influence of the size of scandium is not obvious, as the radius of scandium is located within the limits of atomic radii of metals that form cubic RB12 phases. The cell parameters of known dodecaborides are graphically presented in [20] as functions of their d, 4f, or 5f metal radii. All data fit on three straight lines with an individual slope for each group of elements. Yttrium, gadolinium and thorium dodecaborides that finish the corresponding series have almost identical radii but considerably different lattice constants. The d-elements form the Hf – Zr – Sc – Y line with intermediate Sc which enters into the composition of the tetragonal ScB12 structure. Structural stability and physical properties of MB12 containing transition-metal elements (M = Sc, Y, Zr, Hf) were studied [12] using first-principles calculations supplemented with the x-ray diffraction experiments for ScB12 and YB12 . The tetragonal I 4/mmm structure was predicted to be the thermodynamic ground state of ScB12 and a metastable state of YB12 , ZrB12 , and HfB12 . Tetragonal ScB12 was shown to transform reversibly into the cubic F m3m symmetry group at 510 K, which corresponded to the thermodynamic ground state of YB12 , ZrB12 , and HfB12 at room temperature. The temperatures of the phase transition between tetragonal and cubic phases of the yttrium and zirconium dodecaborides could be lower than 100 K as predicted in [12]. Neutron diffraction measurements have revealed that HoB12 , TmB12 and ErB12 have incommensurate magnetic structures [21– 23]. The complex magnetic structure of these materials seems to result from the interplay between the RKKY and dipole-dipole interactions. Strong frustration of an antiferromagnetic order in the fcc symmetry of the dodecaborides could also play an important role.

298 Crystal Structures of Dodecaborides

3.2 Cooperative Jahn–Teller Effect as a Driving Force behind Structural Instability in Dodecaborides The origin of small structural distortions in RB12 dodecaborides, which distinctly manifest in the background of an almost unchangeable basic cubic structure of the robust boron network, is a challenging and intriguing problem. In most of the works on rareearth dodecaborides RB12 , discussion of structural instability is usually based on the two main points: (a) The boron network is rigid and undistorted; (b) the dimension of the B24 cavity is oversized with respect to the central metal ion R; this results in a rattling character of thermal vibrations of the metal atoms R in the cavities [9,17,24–26]. According to this approach, at low temperatures some fraction of the metal ions shifts from the central position in the B24 cages to form a cage-glass state [24–27]. These displacements may be caused by lattice defects such as impurities and boron vacancies. These models were extensively used in the analysis of low-temperature structural and magnetic characteristics of RB12 compounds [24–27]. However, as will be shown below, the basic assumptions of these models need some revision, since they do not take into account some important features of the electronic structure of the RB12 dodecaborides and their structural units. Namely, apart from displacements of R atoms in the oversized B24 cages, low-temperature lattice distortions in RB12 may originate from intrinsic structural instability of B12 cuboctahedra related to the JT effect. This point can be best illustrated for isolated B12 units. Similarly to other high-symmetry molecules, such as B12 icosahedra in boron and higher metal borides [28, 29], the cuboctahedral boron clusters B12 may have an orbitally degenerate ground state resulting in JT distortions of the regular cubic structure. In terms of molecular orbitals (MOs), the orbital degeneracy of the ground state is associated with partial electron occupation of the highest occupied molecular orbital (HOMO) of B12 , which is represented by triply degenerate MOs of a t-type symmetry (Fig. 3.2).

Cooperative Jahn–Teller Effect as a Driving Force behind Structural Instability 299

Figure 3.2 On the origin of the Jahn–Teller effect in isolated cuboctahedral B12 clusters in RB12 compounds. The highest occupied molecular orbital (HOMO) is triply degenerate. Partial population of the HOMO orbitals with electrons produces a triply degenerate many-electron ground T-state, which leads to Jahn–Teller distortions.

Essentially, the character of the JT distortion depends on the electric charge of the B12 clusters. In an electrically neutral [B12 ]0 cluster, the HOMO accommodates two electrons (t2 configuration); in negatively charged clusters [B12 ]n− (n = 1–4), the number m of t-electrons in HOMO increases from three to six (Fig. 3.3). Four electronic configurations tm with m = 2–5 produce a triply degenerate many-electron ground T-state [Fig. 3.3(a–d)] This implies that neutral [B12 ]0 cluster and charged [B12 ]n− clusters (n = 1, 2, 3) are JT-active systems, which would tend to distort the cubic structure, except the [B12 ]4− cluster with fully occupied HOMO (t6 configuration), which has a nondegenerate ground state and thus is not JT-active [Fig. 3.3(e)]. For JT systems with the T-type ground state, the character of distortions is determined by two active vibrational modes of e and t2 symmetry; such situation is

300 Crystal Structures of Dodecaborides

Figure 3.3 Molecular structure of neutral and negatively charged isolated [B12 ]n− clusters (n = 0−4) obtained from DFT geometry-optimization calculations [27]. The optimized structures correspond to the deeper local JT minima of the corresponding charged cluster. Principal atomic distances ˚ and bond angles are indicated. (A)

referred to as the T × (e + t2 ) Jahn–Teller problem [30]. In this case, depending on the ratio between the strength of the e and t2 electronvibronic couplings, three types of minimum points on the groundstate potential energy surface can occur: trigonal (D3d ), tetragonal (D4h ) (Fig. 3.2), and approximately orthorhombic (D2h ) points [30]. More quantitative information on the amplitude and type of the JT distortions has been obtained from density functional theory (DFT) calculations for the neutral cluster [B12 ]0 and negatively charged [B12 ]n− clusters (n = 1–4) [27]. Calculated structures of [B12 ]n− (n = 0–4) clusters resulting from the DFT geometry optimization are shown in Fig. 3.3. These results indicate that the JTactive clusters [B12 ]n− (n = 0–3) are slightly distorted cuboctahedra. The character of JT distortions depends strongly on the charge of the cluster: The neutral cluster and charged clusters with n = 1, 2 exhibit trigonal type of JT distortions, while the [B12 ]3− cluster shows tetragonal JT distortion. It is important to note that the overall magnitude of the JT distortions in isolated B12 clusters is rather small, as the bond lengths and bonding angles vary within ∼0.1 A˚

Cooperative Jahn–Teller Effect as a Driving Force behind Structural Instability 301

and ∼5◦ , respectively (Fig. 3.3); DFT calculations show that the energy gain resulting from JT distortions is within 0.2–0.3 eV per B12 cluster. These results suggest that the JT structural lability of the B12 units in the crystal lattice of metal dodecaborides RB12 may play an important role in the microscopic mechanism of lattice distortions of RB12 at low temperature. It should be borne in mind, however, that in the actual crystal structure of RB12 , the B12 clusters are connected by B–B covalent bonds to form an extended 3D covalent boron network, in which HOMOs of individual B12 clusters may have non-integer electron occupation. Nevertheless, one can expect that in RB12 crystals some fraction of the JT activity of B12 clusters may retain in the three-dimensional boron network because the local triply degenerate HOMOs of B12 cuboctahedra remain partially filled, as can be seen from the overall electron balance between the metal and boron sublattices. Due to interactions between the nearest B12 clusters in a RB12 crystal, local JT distortions of B12 cuboctahedra become mutually consistent resulting in a symmetrylowering distortion of the lattice; this phenomenon is known as the cooperative JT effect, which is well documented in the literature [30–32]. In a concentrated JT system, the full JT Hamiltonian of the crystal is given by the equation [31, 32]:  1 + ˆ − m)Q(m), Hˆ = Hˆ JT (n) + Q (n) K(n (3.1) 2 n, m n (n=m)

where the vector indices n and m enumerate unit cells of the crystal, Hˆ JT (n) is the one-center JT Hamiltonian for unit cell n, and the last term represents pairwise interactions between the local JT centers n and m. Here Q(n) is a vector whose components are the local JTactive vibrational modes and Kˆ (n − m) is the operator describing interactions between the local JT vibrational modes on sites n and m. The electronic and geometric structure of a cooperative JT system is determined from the minimization of the total energy of the crystal resulting from the competition of the local distortions, the first term in Eq. (3.1), and the interaction between the different sites (the second term). In the general case, cooperative JT interactions can lead to a variety of situations, depending on the specific character of

302 Crystal Structures of Dodecaborides

orbitally-degenerate moieties and interplay between the on-site and inter-site JT interactions. In particular, lower symmetry structures in RB12 resulting from these cooperative interactions can lead to a parallel alignment of all the local distortions of B12 cubooctahedra (which is termed as ferrodistortive case) or to a more complicated geometrical arrangement of the local B12 distortions (so-called antiferrodistortive case). In the ferrodistortive phase, the local JT centers are coupled to a strain of the lattice, which changes the shape of the crystal and its unit cell parameters; this strain mode coupling provides an effective long-range interaction between the JT centers. It is important to note that in most cooperative JT systems the coupling is predominantly to a strain of the lattice [31, 32]. This fact gives an idea of the origin of observed anomalous behavior of RB12 dodecaborides: In fact, formation of a ferrodistortive JT phase with a long-range ordering of JT distortions mediated by the strain of the lattice is the most likely scenario in metal dodecaborides. More specifically, the ferrodistortive JT phase of RB12 with the elastic strain axis parallel to the [111] direction seems to be the actual (a)

Axial elongation parallel to the [111] direction

(b)

Compression in the (111) plane

Figure 3.4 Structure of the ferrodistortive JT phase in RB12 dodecaborides and the character of the strain of the crystal lattice. (a) The B12 cuboctahedra are all elongated along the trigonal axis [111] and compressed in the orthogonal plane; this geometry corresponds to one of the local trigonal JT minima of B12 shown in Fig. 3.3; (b) The crystal lattice of RB12 is compressed in the (111) plane.

Cooperative Jahn–Teller Effect as a Driving Force behind Structural Instability 303

situation that may account for the unusual behavior of RB12 . In this case, when the strain mode corresponds to elongation, the B12 cuboctahedra are all elongated along the trigonal axis [111], that refers to one of the trigonal JT minima shown in Fig. 3.3; the local distortion JT axes are all parallel to each other [Fig. 3.4(a)]. Since the elastic strain does not change the volume of the crystal, elongation in the [111] directions is followed by compression in the (111) plane, as depicted in Fig. 3.4(b). This leads to important changes in the electronic band structure caused by the trigonal strain mode, as the compression in the (111) plane gives rise to some shortening in the R–R distance between the neighboring metal atoms [Fig. 3.5(a,b)]. This would change orbital interactions between the R and B atoms responsible for the formation of the electronic conduction band of RB12 , which is mainly represented by 2 p(B) and 5d(R) atomic orbitals. The largest changes are expected for the 5dz2 metal orbitals that have the strongest σ -type overlap with the 2 p valence orbitals of boron [Fig. 3.5(c)]. Accordingly, enhanced 5dz2 (R)-2 p(B) orbital overlap increases the energy dispersion of the electronic conduction band, thereby increasing the overall number of the filled conduction band state below the Fermi level of RB12 . This results in larger electron population of the 5dz2 (R) orbitals oriented along the local R–R lines connecting neighboring R atoms in the (111) plane [Fig. 3.5(c)]. Considering the elongated shape of 5dz2 (R) orbitals, this leads to increased electron density along the R–R lines being parallel to the side diagonals [110], [011], [101]), which are shown with red solid lines in Fig. 3.6. These findings are in excellent agreement with the recent experimental results on LuB12 , which reveal lowersymmetry electron density distribution (charge stripes) correlating with the filamentary structure of conduction channels observed in the magnetoresistance measurements [33]. Remarkably, the general character of the residual density distribution near the metal atom in the (100) and (010) planes at low temperatures (50 K) shown in Fig. 3.15 strongly resembles the shape of 5dz2 (R) orbitals oriented along the R–R lines, as depicted in Fig. 3.5 and Fig. 3.6. Thus, a theoretical model based on the cooperative JT effect provides a new insight into the microscopic origin of the mysterious structural behavior of RB12 cubic dodecaborides. The main reason

304 Crystal Structures of Dodecaborides

Axial elongation parallel to

(a)

Compression In the R plane

(b)

(c)

Figure 3.5 On the origin of charge stripes in the ferrodistortive JT phase in RB12 dodecaborides, (a) The general character of the local distortions in the ferrodistortive JT phase, (b) reduction of the R–R distance between the neighboring metal atoms due to compression in the (111) plane, (c) orientation of 5dz2 metal orbitals having the strongest σ -type overlap with the 2 p valence orbitals of boron; shortening of the R–R distance leads to a maximal change in the 5dz2 (R)-2 p(B) orbital overlap, which ultimately causes transfer of excess electron density to the 5dz2 orbitals and formation of the charge stripes (see Figs. 3.15 and 3.16 later in the text).

behind the manifestation of low-symmetry effects of RB12 lies in the inherent structural lability of the B12 cuboctahedral units resulting from their orbital degeneracy and the JT effect. Assuming a ferrodistortive JT ordering in RB12 with deeper trigonal JT minimum of B12 clusters enables one to rationalize in a natural way the main

Cooperative Jahn–Teller Effect as a Driving Force behind Structural Instability 305

Figure 3.6 Orientation of the charge stripes (the solid red lines) in the (111) plane of RB12 resulting from the electron density transfer to the 5dz2 (R) orbitals caused by the elastic strain of the crystal lattice in the ferrodistortive JT phase (see Fig. 3.5).

experimental results on dodecaborides (see the following sections of this chapter for more details): (1) The cooperative JT model explains the presence of small distortions of the cubic lattice of RB12 at all temperatures, including room temperature. Indeed, in the ferrodistortive JT state distortions persist at all temperatures, without a structural phase transition. Theoretical treatment of the ferrodistortive JT state is similar to that for a magnetic spin lattice in an external magnetic field, which always retains some magnetization [31, 32]; the latter serves as an order parameter, whose analog in RB12 dodecaborides corresponds to the relative magnitude of the deviation from the regular cubic structure. (2) The JT model provides a physically transparent insight into the origin of lower symmetry electron density distribution in RB12 , including appearance of charge stripes in LuB12 and the filamentary structure of conduction channels resulting in anisotropic magnetoresistance [33]. These effects are mainly

306 Crystal Structures of Dodecaborides

due to enhanced electron occupation of the 5dz2 metal orbitals resulting from the larger 5d(R)-2 p(B) orbital overlap caused by elastic shortening in the (111) plane in the ferrodistortive JT state (Figs. 3.4 and 3.5). (3) The above consideration indicates that low-symmetry distortions are a unique property of all rare-earth dodecaborides, as they result from the JT structural lability of B12 units, not from the ground-state characteristics of the metal ions R. (4) This theoretical approach evidently demonstrates that subtle structural departures from the regular cubic structure of RB12 dodecaborides are by no means an artifact of x-ray diffraction analysis, but they are an inherent property of all rare-earth RB12 compounds resulting from the JT activity of the B12 units. Indeed, in the ferrodistortive JT state, arbitrarily weak JT interactions always lead to a static lattice distortion, by analogy with the nonzero magnetization of the spin system in an external magnetic field. More specifically, this property originates from the fact that the energy gain resulting from the JT distortions grows linearly with the strain value e, while the competing elastic energy is proportional to e2 ; therefore, at small strains the low-symmetry JT state lies lower in energy [30–32]. (5) In fact, the presence of significant 10 B/11 B isotope effects in itself suggests the JT origin of the structural instability, since it documents a breakdown of the Born-Oppenheimer approximation for the orbitally degenerate systems, in which electronic and vibrational motions are no longer independent. Generally, isotope effects are more pronounced for the dynamic JT effect, when the JT stabilization energy competes with the vibrational energy; this situation is likely to occur in RB12 .

3.3 Modeling the Dynamics of the Dodecaboride Lattice Using X-Ray Diffraction Data Experimental conditions, such as the sample temperature, can vary to better identify the barely perceptible symmetry violations,

Modeling the Dynamics of the Dodecaboride Lattice Using X-Ray Diffraction Data 307

which may appear due to the cooperative JT effect. The most known phenomenon is the temperature dependence of the unitcell values, which should be monotonous in the absence of a lattice transformation (see below). Additional information can be obtained by analyzing atomic displacement parameters (ADPs) at different temperatures using both experimental and theoretical temperature curves. The key role in the structure analysis of crystals is assigned to structure factors F (H), which provide a transition from measured intensities of diffraction peaks to the distribution of electron density in the crystal. The expression for F (H) is as follows: F (H) =

N 

fν (|H|) exp(2πi rν · H)Tν (H).

(3.2)

ν=1

fν (|H|) is the atomic Here H = ha∗ + kb∗ + lc∗ is a scattering vector;  scattering factor of the atom at rν ; Tν (H) = p(uν ) exp(2πi uν·H)d3 u is the temperature factor, which is known also as the Debye-Waller factor that accounts for the atomic displacements uν from the lattice points. Summation is carried out over all atoms in the unit cell of the crystal. As one can see, the temperature factor Tν (H) is the Fourier transform of the probability density function p(uν ) whose coefficients are atomic displacement parameters (ADPs) discussed below. The p(uν ) function can be approximated respectively by univariate or trivariate Gaussian in case of isotropic or anisotropic harmonic vibrations of an atom, and it can be more complicated in case of anharmonic vibrations as a result of heating, for instance. In any case, however, the temperature does not participate directly in the calculations either as a fixed parameter or as a refined variable. Moreover, the conventional approach does not require any assumption of the atomic displacement nature. The ADP values may correspond to thermal vibrations supplemented with static shifts [34]. Along with the atomic coordinates, ADPs are the refined parameters of the structural model. The least-squares refinement procedure consists in approximation of |F calc (H)|2 calculated by the formula (3.2) with |F obs (H)|2 , whose values are proportional to the measured intensities of the diffraction reflections. The displacements of each atom are represented in the structural model by one or more parameters, depending on the chosen formalism

308 Crystal Structures of Dodecaborides

(isotropic, anisotropic harmonic or anharmonic displacements). Harmonic ADPs form a second-rank matrix {ui j }, 1 ≤ i , j ≤ 3, the trace of which gives an estimate of the equivalent atomic displacements u2 eq or ueq in short notation, ueq = (u11 + u22 + u33 )/3. This parameter often appears in studies of the thermal properties of solids. An alternative method of quantifying atomic displacement parameters is not tied directly to a structural model. Thermal vibration amplitudes ucalc (R) of the metal atoms in the large cavities of the dodecaboride structure well correspond to the Einstein model [35] for independent harmonic oscillators supplemented with a temperature independent static component u2 shift or ushift in short notation:   1 2 1 (3.3) + + ushift (R) ucalc (R) = kB ma TE 2 exp(TE /T ) − 1 The expanded Debye model [36] is suitable for atoms of the boron framework whose displacements strongly correlate with each other: 

 2 TD T 1 y dy 32 T + + ushift (B) (3.4) ucalc (B) = kB ma TD 4 TD 0 exp(y) − 1 The agreed notations are as follows:  = h/2π is the Planck constant; kB the Boltzmann constant; ma atomic mass; TE (TD ) the characteristic Einstein (Debye) temperature; T the temperature of the experiment. The problem is that the characteristic Einstein (Debye) temperature and the value of ushift must be known in advance to calculate the values of ucalc from Eq. (3.3) or (3.4). Still, it is possible to solve the inverse problem of calculating the characteristic Einstein (Debye) temperature and ushift using the values of ueq determined from diffraction data. For this purpose, one should collect the multi-temperature data sets {h, k, l, |F obs |, σ F } and refine the crystal structure at each temperature using conventional techniques. As a result, each atom is supplied at each temperature with a value of uobs = ueq . The multi-temperature set of these parameters then  serves as an input for a least-squares procedure |u2calc − u2obs | → min to fit the model curve to the set of uobs . The characteristic Einstein (Debye) temperature TD (TE ) and the value of ushift are adjustable parameters of this procedure [37].

Modeling the Dynamics of the Dodecaboride Lattice Using X-Ray Diffraction Data 309

Thus, the proposed approach [37] allows us to solve several problems at once: (1) to obtain the Einstein (Debye) characteristic temperatures estimated otherwise from the heat capacity or somehow else; (2) to describe the temperature dependence of the thermal atomic vibrations using an appropriate analytical function; (3) to separate the contributions of the static and dynamic components into the equivalent parameter ueq of atomic displacements. The Debye and Einstein models were previously used to fit the multitemperature ADPs and to estimate the Debye (Einstein) temperatures in crystals of various compositions including hexaborides RB6

(a)

(b) Figure 3.7 Temperature dependences of ueq in the crystals of Lu N B12 (N = 10, 11, nat). The Einstein (a) and Debye (b) models are used respectively for Lu and B atoms. The fit is based on the uobs values marked with squares [38].

310 Crystal Structures of Dodecaborides

(R = Y, La – Gd) [39–42]. This approach, being first applied to the dodecaborides Lu N B12 (N = 10, 11, nat), revealed a difference in the static components ushift depending on the isotope composition of boron [38]. The abbreviation ‘nat’ is hereinafter used to refer to natural boron with the ratio 10 B :11 B ≈ 19.8 : 80.2. Refined values of TE (TD ) and ushift were substituted in the Eqs. (3.3) and (3.4) to draw the curves for Lu and B presented in Figs. 3.7(a) and (b), respectively. The ADPs sum up mean-square zero vibrations u2 zero , temperature-dependent thermal vibrations u2 (T ), and static shifts

u2 shift . The curves in Fig. 3.7 are plotted for three crystals with very close Debye (Einstein) temperatures, so that ueq (T ) mostly differ in their temperature independent components u2 c = u2 zero +

u2 shift . As shown in [38], these components are maximal in Lunat B12 both for Lu and B atoms. Static distortions of boron polyhedra are combined with static shifts of Lu atoms from the lattice points, which can be explained by disorder in 10 B -11 B substitution in the crystal with natural boron.

Figure 3.8 Experimental ADPs (ueq ) in HoB12 are fitted using the Einstein (for Ho) and Debye (for B) models in temperature ranges 86–180 and 210–  500 K. R = |u2obs − u2calc |/ u2obs .

Modeling the Dynamics of the Dodecaboride Lattice Using X-Ray Diffraction Data 311

, ,

The structure of single-crystal HoB12 was studied by x-ray diffraction analysis in the F m3m group at 29 temperatures in the range of 86–500 K [43]. Temperature variations of ueq (B) and ueq (Ho) lose stability near 200 K. To improve the fit, one has to divide each of the experimental sets of ueq into two parts obtained in the 86–180 K and 210–500 K temperature ranges, and to build two curves for each atom (Fig. 3.8) for better modeling of the experimental curves. The instability of the unit-cell values could be clearly determined from the x-ray data not only in HoB12 in the temperature range 150–200 K, but also in RB12 (R = Ho, Tm, Yb, Lu) below 200 K (see Figs. 3.9–3.13 in the next section).

, ,

Figure 3.9 Linear (a, b, c) and angular (α, β, γ ) unit-cell parameters of HoB12 in the temperature range 85–500 K [43].

Figure 3.10 Linear (a, b, c) and angular (α, β, γ ) unit-cell parameters of TmB12 in the temperature range 85–300 K [46].

, ,

312 Crystal Structures of Dodecaborides

Figure 3.11 Linear (a, b, c) and angular (α, β, γ ) unit-cell parameters of YbB12 in the temperature range 85–300 K [47].

The development of similar lattice instability with decreasing temperature was also reported earlier in Lu N B12 crystals with different isotopic boron composition (N = 10, 11, nat) that were studied using low-temperature heat capacity and Raman scattering data [24]. The maximum density of vibrational states was observed at the temperature near 150 K [24]. It was noted that the mean free path of phonons reaches the Ioffe–Regel limit in the vicinity of this temperature, being compared with their wavelength. Remarkable spectral changes in the zero-field spectra and a sharp maximum in the relaxation rate were recorded near 150 K in μSR experiments for dodecaborides RB12 (R = Yb, Lu) and solid solutions Lu1−x Ybx B12 . It has been suggested that the large-amplitude dynamic features arise from atomic motions within the B12 clusters [44, 45]. Most likely, the instability of ueq is caused by changes in the phonon structure of the rare-earth dodecaborides and is not a unique feature of only HoB12 .

3.4 Crystal Structure: Problems and Results 3.4.1 The Jahn–Teller Distortions of Structural Parameters Active studies of the RB12 structure at various temperatures, which were started in the early 2000s but not continued at that time, were resumed later after the tetragonal distortion of the LuB12 structure

Crystal Structure 313

had been confirmed in the temperature range 50–75 K [48]. The structure of a LuB12 single crystal was then thoroughly studied at room temperature [49]. The single crystals of LuB12 were grown by modified crucibleless inductive floating zone melting using highpurity source materials: lutetium oxide Lu2 O3 and boron [17]. One of the purposes of the re-examination of the known structure was to assess the suitability of the grown single crystals for accurate structure analysis. The refinement of the structural model in the F m3m symmetry group with a uniquely low residual factor R = 0.2% was made possible due to the high diffraction quality of the single crystals combined with a set of original experimental techniques [50–52] that ensured the accuracy and reliability of the x-ray data measured. Besides that, accurate measurements of the periods of the LuB12 crystal lattice were carried out in the temperature range 20–295 K [53]. Two periods a ≈ b did not differ within the limits of the standard uncertainty (σ ), but the third period c steadily deviated downward by 2σ or more over practically the entire temperature range. In absolute values, the difference in the lattice constants is ˚ which is an order of magnitude less than very small (about 0.002 A), in the lattice of ScB12 . Such a small difference in the lattice constants does not give grounds for a revision of the structural model, especially in view of what was said above about the excellent results of the refinement of the cubic structure of LuB12 . However, even very small differences in the lattice constants can have a significant effect on the physical properties of crystals. The lattice parameters must be determined for many dodecaborides of different composition in a wide temperature range without symmetry restrictions in order to collect experimental information on the Jahn–Teller distortions. This work is still far from complete, both in the number of crystals studied and in the number of temperature points measured. After the first experiments with LuB12 , linear and angular unit-cell parameters have been determined at various temperatures for HoB12 , TmB12 , and YbB12 as shown in Figs. 3.9–3.11. The linear parameters always manifest small tetragonal-type distortions. Note that one lattice constant of LuB12 and TmB12 is smaller than the other two: a ≈ b > c (the unit cell is slightly compressed along an edge), whereas one lattice constant of HoB12

314 Crystal Structures of Dodecaborides

Figure 3.12 The lattice parameter vs. temperature in LuB12 containing natural boron and 10 B isotope [54].

is slightly elongated: a > b ≈ c. Both linear and angular parameters of each unit cell undergo the most noticeable nonlinear changes in the same low-temperature region between 100 and 150 K. It is noteworthy that lattices of TmB12 and YbB12 undergo opposite changes despite the proximity of Tm and Yb in the series of rareearth elements. Closer to the middle of the mentioned temperature range, the lattice constants of YbB12 abruptly decrease and return to the former, even slightly larger values with a further decrease in temperature. The obliquity of the YbB12 lattice slightly increases, but then the angles return to their previous values. At the same temperatures, the periods of the TmB12 lattice slightly increase, and the angles become slightly closer to 90◦ .

3.4.2 Structural Peculiarities of Dodecaborides Different in Isotopic Boron Composition Since the cooperative JT effect is determined by the dynamics of light boron atoms, one can suppose that isotope substitutions 10 B -11 B may affect both properties and crystal structure of dodecaborides. Until recently, the research was mainly limited to physical properties [9, 17, 24–26]. Thermal expansion of Lu10 B12 and Lunat B12 was studied based on the x-ray powder diffraction

Crystal Structure 315

Figure 3.13 Temperature dependences of the lattice parameters for Lu N B12 over the temperature range 88–293 K: (a) lattice constants; (b) unit-cell angles. Experimental values are connected by dashed lines; solid lines in panel (a) are linear fits. Standard uncertainties do not exceed 0.0002 A˚ and 0.001◦ , respectively [38].

data in the temperature range 10–290 K [54]. Both samples showed negative thermal expansion between 50 and 100 K (Fig. 3.12). This is consistent with the temperature region in which the negative thermal expansion was previously observed for Lunat B12 by the three-terminal capacitive method [15]. The lattice constant of Lu10 B12 is increased relative to that of Lunat B12 by 0.001–0.002 A˚ over the measured temperature range. The β-rhombohedral boron lattices have the same property, but the difference between the lattice parameters in the crystals with 10% and 97% content of 10 B ˚ as established in Ref. [55] whose is more noticeable (about 0.03 A) authors presented a theoretical justification for such an expansion of the 10 B lattice. An influence of the isotopic composition on the structure and properties of Lu N B12 , N = 10, 11, nat, was studied in [38]. Taking into account both linear and angular distortions of the

316 Crystal Structures of Dodecaborides

unit cells of the three crystals, one can conclude that the Lu10 B12 lattice is distorted rather by tetragonal type a ≈ b > c, whereas the distortions of the Lu11 B12 lattice are more similar to pseudo-trigonal ones a ≈ b ≈c, α ≈ β ≈ γ > 90◦ . Lattice distortions of Lunat B12 have an intermediate character (see Fig. 3.13). At temperatures below 140 K, the distortions are nonlinear, as can be assumed despite the small number of the points measured. Nonlinear distortions of the parameters, which occur at close temperatures in three different crystals, are hardly explained by a sheer accident. At a temperature of about 120 K, the trigonal-type distortions of the lattices of Lu11 B12 and Lunat B12 are amplified, as well as the pseudo-tetragonal lattice distortions of Lu10 B12 crystal, but the situation changes again with a further decrease in temperature. Thus, we observe the same jump in the parameters of the unit cell approximately in the middle of the temperature range 100–150 K as in other three dodecaborides mentioned above. It is worth noting that 120 K is close to the upper boundary of the temperature interval with negative thermal expansion of Lunat B12 according to Refs. [14, 15].

3.4.3 Formation of Charge Stripes in Voids of the Crystal Lattice The numerical differences between the lattice parameters are very small and do not require a transition to the low-symmetry structure model. The crystal structures of LuB12 , TmB12 , HoB12 and many other dodecaborides can be successfully refined in the cubic group F m3m with low values of R-factors. It should be noted, however, that the completeness of the structural analysis is judged not only by the R-factor value but also by the distribution of the residual electron density (ED) on the difference Fourier maps. Fourier synthesis of the electron density is a computational procedure, which starts with a set of both experimental and previously calculated parameters. The computational formula can be written in general terms as follows: G(r) =

1  A(H) exp[i ϕ(H)] exp(−2πi H·r). V H

(3.5)

Crystal Structure 317

Here G(r) is either full (g) or residual ( g) electron density, resulting respectively either from a “regular” or difference Fourier  ∗ synthesis; V is the unit-cell volume; H = i hi ai is a scattering vector; and ϕ(H ) is a scattering phase. A(H) are coefficients dependent on the type of the Fourier synthesis. In case of difference Fourier synthesis, A(H) = |F obs (H)| − |F calc (H)| is a difference between observed and calculated absolute values of the structure factor. The first value is the square root of the reflection intensity, whereas the second one is calculated from atomic coordinates and ADPs, whose values are refined using a least-square technique. As follows from Eq. (3.5), the Fourier synthesis of the electron density does not require any data on the crystal symmetry. It can be performed independently in each point of the crystal lattice. Nevertheless, the symmetry of the crystal is usually taken into consideration in the algorithms that implement Fourier synthesis of the electron density. It means that the measured intensities of x-ray reflections are averaged in the corresponding Laue class and the Fourier synthesis is performed in a symmetrically independent region of the unit cell. As a result, the symmetry of the Fourier map exactly corresponds to the space group, information about which is fed to the input of the computational procedure. Certainly, any measurement is not free from the influence of instrumental errors and the data processing methods. On the one hand, the abovementioned techniques of calculations are designed to improve the accuracy of the results and to ensure visual consistency of the Fourier maps with the stated symmetry of the crystals. On the other hand, averaging can harm since the symmetry of the electrondensity distribution over the cell can be overestimated. In the case when accuracy and reliability of measured x-ray data are ensured by reliable measurement of literally each reflection, with subsequent consideration of experimental corrections using special techniques [50–52], one may feed a less symmetrical group to the input of the Fourier procedure. In [27], this approach was applied to LuB12 whose structure was first refined in the highsymmetry F m3m group at temperatures 295 and 90 K. After that, the measured values of |F obs | were averaged in the mmm Laue class instead of m3m, and the orthorhombic F mmm group was fed to the input of the difference Fourier procedure, skipping the tetragonal

318 Crystal Structures of Dodecaborides

(a)

(b)

Figure 3.14 Difference Fourier maps (residual electron density g in e/A˚ 3 ) in the x = 0 face of the LuB12 unit cell at (a) 90 K and (b) 295 K. Red circle is the Lu site; green circles are B sites. The panels (c) – (g) and (d) – (e) are the surface plots of difference Fourier maps in the vicinity of the Lu ion, in the x = 0, y = 0, and z = 0 faces of the unit cell, respectively. The first and second rows of the figure correspond to temperatures 90 K and 295 K, respectively [27].

I 4/mmm group, which would require a transition to another unit cell. The difference Fourier maps built from low-temperature (90 K) x-ray data clearly showed residual electron-density peaks oriented along [001] at distances of about 0.5 A˚ from the central position of Lu. As can be seen from Fig. 3.14, similar peaks are absent along [010], which is symmetrically equivalent to [001] in the cubic group. This result agrees with the result obtained in the same work [27] concerning unequal magnetoresistance in the LuB12 sample in two directions of the 100 family. In the next work [33], the crystal structure of LuB12 was studied at the four temperatures 293, 135, 95, and 50 K. To eliminate possible dependence of the results on systematic instrumental errors and on the features of the crystalline sample, the x-ray experiments were performed on three different-type diffractometers and on two LuB12 crystals. To analyze the electron-density distribution in the crystal at room temperature, the same data were used that were previously collected on a CAD4 diffractometer (Enraf Nonius) for a precise analysis of the cubic structure of LuB12 [49]. The x-ray data

Crystal Structure 319

Figure 3.15 Residual electron-density distribution in the x = 0, y = 0, and z = 0 planes of LuB12 . Difference Fourier synthesis is done in F 1 using data collected at four temperatures. Contour intervals are 0.2 e/A˚ 3 (295, 135, 95 K) and 1 e/A˚ 3 (50 K). Positive (pink) and negative (light-green) residual electron density is highlighted. The central Lu(0, 0, 0) site (lime green circle) is surrounded by eight boron sites (dark green circles); [−0.5, 0.5] intervals are periods of the crystal lattice [33].

320 Crystal Structures of Dodecaborides

Figure 3.16 Maximum-entropy-method maps are calculated from the LuB12 data sets collected at temperatures 293, 135, 95 and 50 K. Three columns from left to right present thin slices of the electron-density distribution in three planes of the crystal lattice. The central Lu is surrounded by eight boron atoms; [−0.5, 0.5] intervals are periods of the crystal lattice [33].

Crystal Structure 321

Figure 3.17 (a) Magnetoresistance anisotropy of LuB12 in polar coordinates: ρ/ρ0 = [ρ(ϕ, B) − ρ(ϕ0 , B)]/ρ(ϕ0 , B), ϕ0 = 270◦ corresponding ¯ to B [1¯ 10]; (b) anisotropic electron-density distribution in a thin layer of the electron density reconstructed by the maximum entropy method [33].

at 135 and 95 K were collected on an Xcalibur EOS S2 diffractometer with a two-dimensional CCD detector. The experiment at 50 K was obtained on a four-circle Huber-5042 diffractometer equipped with a point detector and a closed-cycle helium cryostat Displex DE-202. The structure was first refined in F m3m as before, but information on triclinic F 1 symmetry was fed to the input of the Fourier procedure. Non-standard abbreviation F 1 instead of P 1 is due to the reluctance to move to another (non-cubic) cell, which would correspond to the standard setting. The difference Fourier maps were built for each temperature in three sections of a crystal with the (100), (010), (001) planes. As seen from Fig. 3.15, the symmetry of the residual electron-density distribution is clearly lower than orthorhombic. The selected directions remain but lose their exact orientation along the canceled axis 2 of the orthorhombic group, turning in the direction closer to the face diagonal of the unit cell. The residual electron density increases almost by an order of magnitude at the temperature of 50 K forming a continuous diagonal strip in the (010) section. The formation of the electron-density strip at 50 K is confirmed by the maximum entropy method (MEM) as shown in Fig. 3.16. We associate this observation with the formation of a filamentary structure of conductive channels—charge stripes along selected

322 Crystal Structures of Dodecaborides

directions in the crystal [33]. In the same paper, two results were compared, which were obtained on LuB12 samples cut from one block. The same sample could not be used in all experiments due to different requirements for its size and shape for x-ray experiments and measurements of transport and magnetic properties. Moreover, the x-ray measurements were carried out at significantly higher temperatures and in the absence of an external magnetic field. The more surprising is the exact orientational coincidence of two pictures in the left and right parts of Fig. 3.17, one of which (left) illustrates the anisotropy of the transverse magnetoresistance in LuB12 , whereas the second picture demonstrates the anisotropy of the residual electron-density distribution in LuB12 at 50 K. Another structure of a single-crystal Tm0.19 Yb0.81 B12 was analyzed according to the same scheme at room temperature [56]. Extreme members TmB12 and YbB12 in a series of solid solutions Tm1−x Ybx B12 vary greatly in their properties, despite the proximity

Figure 3.18 (a) Difference Fourier and (b) maximum-entropy-method maps of Tm0.19 Yb0.81 B12 are created in (100), (010), (001) faces of the unit cell. Electron density (g) in the layer of any given thickness is automatically divided into several levels from gmin to gmax , each of them is assigned to a definite color from dark-blue over green to red. The values of gMEM are cut at the level gmax = 0.075% of the maximal gMEM value to show fine electrondensity gradations in the thin layer. Difference electron-density values are cut at ±0.5 e/A˚ 3 [56].

Conclusions 323

of Tm and Yb in the series of rare-earth elements. Unlike metallic TmB12 with antiferromagnetic properties, YbB12 is a narrow-gap semiconductor known as a Kondo insulator (see Section 4.5.5 in Chapter 4 of this book). In order to analyze the loss of metallic properties when thulium is replaced by ytterbium, the information is needed on the corresponding changes in the crystal structure. It has been determined that the crystal lattice of Tm0.19 Yb0.81 B12 has the same type of distortion as that of LuB12 , with a ≈ b > c and a small difference of about 0.002 A˚ between the smaller lattice constant and the other two. The residual electron density is oriented predominantly along the three face diagonals of the unit cell. They are connected by a spatial diagonal, which is one of the threefold axes of the undistorted cubic structure. The residual electron density forms a strip along one of the face diagonals even at room temperature, as can be seen in Fig. 3.18.

3.5 Conclusions The results presented in this chapter demonstrate the complexity of the atomic structure of the dodecaborides, a complete description of which does not fit into the framework of a simple cubic model. Both atomic coordinates being expressed in fractions of the lattice constants and ADPs of almost all dodecaborides correspond well to cubic symmetry and do not require revision of the structural model despite the Jahn–Teller distortion of lattice parameters. Symmetry violations manifest themselves in difference Fourier syntheses as an asymmetric distribution of the residual electron density in the interstices of the crystal lattice along symmetrically equivalent directions. These results are in good agreement with the observed asymmetry of physical properties (conductive, magnetic). Another prospective direction of the structural analysis of dodecaborides is the quantitative analysis of the temperature behavior of the atomic displacement parameters using multi-temperature xray data. The dynamics of the crystal lattice can be traced without going beyond the cubic structural model, by matching the equivalent

324 Crystal Structures of Dodecaborides

atomic displacement parameters to the extended Einstein or Debye models. The analysis of the dodecaboride structure is thus not limited to the refinement of the structural model in the high-symmetric group at one or several temperatures. The multi-temperature data on ADPs must be supplemented with the temperature dependent unitcell parameters, which are not bound by the symmetry constraints, and with the difference Fourier maps built without reliance on the symmetry of the structure model. The transition from single experiments to systematic research of the structure-property relationship in dodecaborides requires the creation of a database of diffraction data. For reliable characterization of a single dodecaboride of a certain composition, it is necessary to carry out a series of diffraction experiments in a wide temperature range with the maximum possible coverage of the low-temperature region. The temperature step should be selected individually for each composition in order to monitor the structural parameters.

Acknowledgments The authors are grateful to N. E. Sluchanko and N. Yu. Shitsevalova for useful discussions. This work was supported by the Ministry of Science and Higher Education within the state assignment of the Federal Scientific Research Center (FSRC) “Crystallography and Photonics” of the Russian Academy of Sciences in the part related to the development of structural analysis methods. Crystal structures and properties of HoB12 and ErB12 crystals were studied with the support of the Russian Foundation for Basic Research, Grant No. 18-29-12005; similar studies on TmB12 , YbB12 , and LuB12 were supported by the Russian Science Foundation, grant No. 17-1201426. The diffraction data were collected using the equipment of the Shared Research Center of the FSRC “Crystallography and Photonics” of the Russian Academy of Sciences and was supported by the Russian Ministry of Education and Science (project RFMEFI62119X0035).

References 325

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Chapter 4

Magnetism, Quantum Criticality, and Metal–Insulator Transitions in RB12 Nikolay E. Sluchanko Prokhorov General Physics Institute, Russian Academy of Sciences, Vavilova str. 38, 119991 Moscow, Russia [email protected]

This chapter presents a review of the physical properties of rareearth (RE) dodecaborides RB12 that are characterized by a cageglass crystal structure with loosely bound RE ions. The analysis of available literature leads to a conclusion that the RE dodecaborides are strongly correlated electron systems with simultaneously active charge, spin, orbital, and lattice degrees of freedom. This explains the complexity of all RB12 compounds including antiferromagnetic (TbB12 – TmB12 ) and nonmagnetic (LuB12 ) metals, on one hand, and the so-called Kondo insulator compound YbB12 and Yb-based Ybx R1−x B12 solid solutions, on the other. It is shown that the reason for the complexity is based on the development of the cooperative dynamic Jahn–Teller instability of the covalent boron network, which produces trigonal and tetragonal distortions of the rigid cage and results in the symmetry lowering of the fcc lattice in the dodecaborides. The ferrodistortive dynamics in the boron sublattice generates both the collective modes and quasilocal vibrations

Rare-Earth Borides Edited by Dmytro S. Inosov c 2022 Jenny Stanford Publishing Pte. Ltd. Copyright  ISBN 978-981-4877-56-5 (Hardcover), 978-1-003-14648-3 (eBook) www.jennystanford.com

332 Magnetism, Quantum Criticality, and Metal–Insulator Transitions in RB12

(rattling modes) of the heavy RE ions, causing a modulation in the density of conduction electrons and the emergence of dynamic charge stripes in these strongly correlated compounds. We consider the manifestation of the charge stripes both in the properties of the nonmagnetic reference compound LuB12 and in the phase diagrams of the RB12 antiferromagnets that exhibit multiple magnetic phases with anisotropic field-angular phase diagrams in the form of the Maltese cross. We will also discuss the metal–insulator transitions in YbB12 and in Yb-based dodecaborides in terms of the instability of the Yb 4f -electron configuration, which appears in addition to the Jahn–Teller instability of the boron cage, providing one more mechanism of the charge and spin fluctuations. The experimental results lead to conclusions that challenge the established Kondoinsulator scenario in YbB12 , providing arguments in favor of the appearance of Yb–Yb vibrationally coupled pairs which should be considered as the main factor responsible for the charge- and spingap formation.

4.1 Introduction Rare-earth (RE) dodecaborides RB12 with a “cage-glass” structure [1] attract considerable attention of researchers due to a unique combination of their physical properties, including high melting temperature, microhardness, high chemical stability, etc. In these ´ temperature decreases antiferromagnetic (AFM) metals, the Neel monotonically from TbB12 (TN ≈ 22 K) to TmB12 (TN ≈ 3.2 K) in the RB12 series, while the conduction band remains similar, consisting of 5d (R) and 2 p (B) atomic orbitals and changing only the filling of the 4f shell of the RE ion (8 ≤ n4f ≤ 14) [2, 3]. In these RE antiferromagnets, the principal interaction which couples the magnetic moments of 4f orbitals is the indirect RKKY-type exchange. The long-range and oscillatory character of this coupling in the presence of other interactions (for example, the crystal-electricfield (CEF) anisotropy, magnetoelastic coupling, dipole–dipole, or two-ion quadrupolar interactions and many-body effects) may lead to a competition (frustration) among the interionic interactions, resulting in complicated magnetic structures. So, these AFM metals

Introduction 333

with an unfilled 4f shell and incommensurate helical or amplitudemodulated magnetic structures, having the simple-cubic crystal structure with a single type of magnetic ions located in high˚ between symmetry positions at large enough distances (∼5.3 A) them, look, at the first glance, like very suitable systems for testing theories and models developed for the magnetic properties of RE intermetallics. The 4f filling between n4f = 12 and n4f ≈ 13 leads to dramatic changes both in the magnetic and charge transport characteristics [4–6], demonstrating the transition from an AFM metal (TmB12 ) to a paramagnetic semiconductor with the intermediate valence of Yb ions (YbB12 ). The metal–insulator transition (MIT) brings a considerable growth at helium temperatures of the dc resistivity, from 4 μ ·cm in TmB12 to 10 ·cm in YbB12 , changing its behavior from metal-like to semiconductor-like [2]. In spite of intensive investigations, the nature of this nonmagnetic semiconducting state in YbB12 (so-called Kondo insulator) remains a subject of debate [7–10]. There are several models proposed to explain the insulating nature of YbB12 . One of them is based on the coherent band picture where the energy gap is formed due to the strong hybridization of the f electrons with the conduction band [11, 12]. The inversion between 4f and 5d bands that accompanies this process is considered as an essential characteristic of the topological Kondo insulator [13–15]. Other models are based on a local picture, where the electrons contributing to the Kondo screening are captured by the local magnetic moments of Yb ions, resulting in an excitonic local Kondo bound state [16,17]. The singlesite scenarios were proposed also by Liu [18] and by Barabanov and Maksimov [10]. To shed more light on the nature of both the charge and spin gaps in the quasiparticle spectra of YbB12 , a number of studies have been undertaken during the last decade since the previous reviews devoted to the physics and chemistry of RE dodecaborides1 were published [19, 20]. Some of these results are presented in this book in the chapters devoted to the details of fine crystal structure 1 During

the preparation of this chapter, one more review article devoted to the studies of RE dodecaborides has been published [21].

334 Magnetism, Quantum Criticality, and Metal–Insulator Transitions in RB12

(Chapter 3), Raman scattering spectra of RB12 (Chapter 5), and magnetic excitations (Chapter 6). In studies of the charge transport, magnetic and thermal properties of the nonmagnetic LuB12 , antiferromagnetic HoB12 – TmB12 compounds, and solid solutions Tm1−x Ybx B12 , it was recently established that the cooperative Jahn– Teller (JT) dynamics of the B12 clusters should be considered as the main factor responsible for a strong renormalization of the quasiparticle spectra, electron phase separation, and the symmetry breaking in the RE dodecaborides with the face-centered cubic (fcc) crystal structure. In this chapter, I present the analysis of both wellknown and recent results in this area, arguing in favor of a spatial inhomogeneity and a non-Kondo physics which should be used to interpret the unusual magnetic ground states and the MIT in these exemplary strongly correlated electron systems. The chapter is organized as follows. In Section 4.2, we discuss the results on the electronic band structure of the RE dodecaborides, Sections 4.3 and 4.4 are devoted to the nonmagnetic reference compound LuB12 , magnetic dodecaborides RB12 (R = Tb, Dy, Ho, Er, Tm), and the solid solutions R x Lu1−x B12 . Section 4.5 collects the numerous results accumulated in the studies of the MIT in YbB12 and Ybx R1−x B12 (R = Lu, Tm) solid solutions, and Section 4.6 concludes the chapter.

4.2 Electronic Band Structure of Dodecaborides 4.2.1 Rough Estimations A very rough description of the electronic structure of the RE dodecaborides is produced if all valence electrons of the elements are taken into account. Some of these electrons fill the bonding orbitals, the other are delocalized in the conduction band. Thus, according, for instance, to the model of Lipscomb and Britton [22], the stabilization of the boron network in RB12 requires the addition of two electrons from each RE atom. Indeed, the cuboctahedral arrangement of the boron atoms leads to six different bonding states (molecular orbitals) which can accommodate 26 electrons when we take their degeneracy into account. Since three valence

Electronic Band Structure of Dodecaborides 335

electrons are provided by each boron atom, there are altogether 36 valence electrons coming from one B12 unit, of which 12 are required for the external bonds with a neighboring B12 unit. Therefore, the remaining 24 electrons plus 2 additional valence electrons of the RE metal are required to fill up all bonding orbitals of the cuboctahedral cluster, leaving one remaining electron per metal ion to occupy the conduction band. Consequently, the conduction band is half-filled, and the dodecaborides are metals [23]. Thus, even if this model does not account for the boron–metal and metal–metal bonding [24], it can qualitatively predict the metallic properties of RE dodecaborides with a trivalent state of R ions and one electron per unit cell in the conduction band. This electron transfer generates ionic bonding between RE atoms and B12 clusters, whereas covalent bonding prevails in the rigid boron cage [25]. Hence, all the RE dodecaborides discussed here are good metals with an exception of the intermediate-valence system YbB12 (so-called Kondo insulator with the Yb valence between ∼2.9 and 3) [4, 5, 26, 27], which undergoes a MIT and becomes insulating at low temperatures.

4.2.2 Metallic RB12 During the last 40 years, stable B12 nanoclusters have been considered as basic structural elements of the fcc lattice of dodecaborides in band-structure calculations. The corresponding UB12 -type crystal structure is similar to the simple rock-salt lattice, where U atoms and B12 cuboctahedra occupy the Na- and Cl-sites, respectively (see Chapters 1 and 3 for details). Calculations of the energy band structure of LuB12 by Harima et al. [28, 29] suggest that there are two conduction bands intersecting the Fermi level. The upper band corresponds to a simply connected Fermi surface located around the X point. The other band shows certain similarities to the well-known noble metal “monster” Fermi surface [30], with the difference that in LuB12 the occupied electronic states lie near the boundary of the Brillouin zone, and the necks are wider [31]. The calculations of Heinecke et al. [31] confirmed the conduction band structure for LuB12 and corrected the position of 4f levels from 4.9 eV [28, 29] to 6.4 eV below the Fermi level (E F ). The difference E F − E 4f obtained in Ref. [31] is closer to the results

336 Magnetism, Quantum Criticality, and Metal–Insulator Transitions in RB12

200

total

LuB12

100

0

DOS

40

B

20 0

Lu 5

0

-15

-10

-5

0

E [eV]

Figure 4.1 (a) Band structure for the nonmagnetic reference compound LuB12 . Courtesy of G. Grechnev [32]. (b) Total density of states (DOS) (top) and local partial DOS components for LuB12 in units of states per Rydberg and per formula unit (center: B s states, solid line; B p states, dashed line; bottom: Lu eg states, dotted line; Lu t2g states, dashed line; Lu f states, solid ¨ line). Reproduced from Jager et al. [23].

of x-ray photoelectron spectroscopy (XPS) measurements of TmB12 and LuB12 [5], which show that the binding energy of the 4f level in these two dodecaborides is about 5 and 7.7 eV, respectively. Later on, ab initio electronic structure calculations were carried out for the paramagnetic (PM), ferromagnetic (FM), and collinear AFM phases of RB12 (R = Ho, Er, Tm) in Refs. [25, 33] for a number of lattice parameters close to the experimental ones. These calculations provided total ground-state energies and the corresponding equations of states E (V ) with sufficient accuracy. In this way, the magnetic stability of AFM ordering in RB12 (R = Ho, Er, and Tm, see below) was confirmed by comprehensive totalenergy calculations for PM, FM, and AFM phases. Also, more detailed calculations of the band structure, Fermi surface, total and partial densities of electronic states (DOS) were carried out for the reference compound LuB12 [23, 25, 33, 34]. They reveal principal features of the electronic spectra, which are common for the whole RB12 series. The corresponding electronic structure and the total electron DOS of LuB12 in the close vicinity of the Fermi level are presented in Fig. 4.1. It was found that in addition to the lower and upper conduction bands and the corresponding two Fermi surface sheets — the first multiply connected in the (111) directions [L direction in the Brillouin zone, Fig. 4.1(a)] and the second simply

Electronic Band Structure of Dodecaborides 337

Figure 4.2 (a) Calculated band structure of LuB12 . Several characters are projected out of the eigenvectors at each k point, and the resulting weight is indicated by a circle of proportional size. Green circles are Lu f states, orange circles are Lu d-states, and violet circles are boron states. (b–d) The Fermi surfaces of LuB12 shown together in (b) and separately for clearer viewing in (c) and (d). (e) Fast Fourier transform (FFT) of the magnetic field sweeps taken at different temperatures. The inset shows the FFT fit (dashed line) to the frequency F = 4.17 kT. Reproduced from Liu et al. [34].

connected which forms “pancake”-like electron surfaces centered at X symmetry points [see Figs. 4.1 and 4.2(a–b)], there is a third Fermi surface sheet that consists of small electronlike lenses centered at K points of the Brillouin zone [33]. As it follows from the results of calculations, the main features in the conduction band structure of LuB12 are governed by hybridization of the RE 5d states (predominantly the Lu eg orbitals pointing toward the centers of the square faces of the B12 cuboctahedron [23]) with the 2 p states of boron. At energies above about −7 eV and up to the Fermi level, mainly π bonds between adjacent B12 clusters are found. These hybridized states are forming a conduction band of about 1.6 eV in width and exhibit a strong dispersion at the Fermi level (see Fig. 4.1). The calculated effective masses of conduction electrons in LuB12 appeared to be comparatively small with respect to the bare electron mass, m∗ ≤ m0 [33]. The m∗ estimations are confirmed in quantumoscillation experiments [31,34,35], where effective masses of 0.44−

338 Magnetism, Quantum Criticality, and Metal–Insulator Transitions in RB12

1.88 m0 were deduced from the Fourier transform data analysis. It is worth noting that in addition to the fundamental peaks in the frequency spectrum of the de Haas–van Alphen (dHvA) oscillations (i.e., quantum oscillations in the magnetic susceptibility), distinctive higher harmonics appear in these spectra with an unusually large amplitude. As deduced by Pluzhnikov et al. [36], the cyclotron masses for HoB12 , ErB12 , and TmB12 are in the range of (0.4– 1.45) m0 , in agreement with the dHvA results on LuB12 [31, 34, 35, 37]. Moreover, for ErB12 and TmB12 , the same authors [36] found branches in the dHvA oscillation spectrum that originated from Fermi-surface fragments with spins along and opposite to the applied magnetic field. Thus, the authors have discussed the exchange splitting of the conduction band on spin-up and spindown sub-bands. The splitting is proportional to the magnetic moment and the exchange integral, and the energy of the exchange interaction was estimated to be 8.2 meV in ErB12 and 14.5, 28.5 and 18.3 meV for different branches in TmB12 . However, the splitting is not observed in HoB12 with the largest value of the RE magnetic moment, but this fact was not explained in Ref. [36].

4.2.3 Strongly Correlated Semiconductor YbB12 The energy band structure of YbB12 has been first calculated by Yanase and Harima [38] using the self-consistent linearized augmented plane wave (LAPW) method in the local spin-density approximation (LSDA) including the spin-orbit interaction. YbB12 was found to be a semimetal with a holelike Fermi surface around the L points and an electronlike Fermi surface centered at the W points. Later Antonov et al. [39] mentioned that LSDA calculations generally provide an inadequate description of the 4f electrons due to improper treatment of correlation effects. In particular, LSDA calculations cannot account for the splitting of filled and empty f bands determined, for example, by photoelectron spectroscopy, which is expected to be 7–8 eV in YbB12 [40]. The authors [38] noted that they cannot definitely assign an energy-gap from LDA, which is only a ground-state theory. In the detailed band-structure study of YbB12 [39], three independent fully relativistic spin-polarized calculations were performed,

Electronic Band Structure of Dodecaborides 339

considering 4f electrons as (1) itinerant electrons using the local spin-density approximation; (2) fully localized putting them in the core and (3) partly localized using the LSDA+U approximation. The sharp Yb 4f peaks in the DOS calculated within the LSDA cross the Yb 5d states just below E F and hybridize with them, resulting in a small direct gap of ∼65 meV at the Fermi level. It was found that there is a small hybridization gap at E F also in the case of LSDA+U energy bands for divalent Yb ions, indicating a nonmagnetic semiconducting ground state in YbB12 [39]. For the trivalent Yb ion, thirteen 4f electron bands were found well below the Fermi energy and hybridized with the B 2p states. The 14th unoccupied 4f hole level is about 1.5 eV above E F [39]. Authors note that such an electronic structure is appropriate for the development of the Kondo scenario. The energy band structures of YbB12 with 4f electrons in the core are characterized by partly occupied Yb 5d bands which cross the Fermi level and overlap slightly with the B 2p states. On the one hand, according to the conclusion of Alekseev et al. [27] about the trivalent state of Yb ion in YbB12 , the last two cases describing YbB12 as a metallic strongly correlated electron system with a 5d-2p conduction band of about 1 eV in width should be considered as preferable. On the other hand, the experimentally estimated energy gap in YbB12 is less than 10 meV from the activation-energy measurements [26, 41– 43], 25 meV from optical spectroscopy [44, 45], and 200–300 meV from tunneling experiments [46], which stimulated a search for the mechanisms responsible for the MIT upon cooling below 70 K. Recently, two investigations of the YbB12 electronic structure were carried out. In the first study, Weng et al. [14] applied the local density approximation with spin-orbit coupling (SOC) combined with the Gutzwiller (GW) density functional theory (DFT) to compute the ground state and the quasi-particle spectrum. The LDA-SOC-GW band structure, shown in Fig. 4.3, indicates (i) two 5d bands strongly hybridized with the 2s and 2p states of boron, (ii) the 4f occupation number nf ≈ 13.28 and (iii) the hybridization gap E g ≈ 6 meV between the 4f and itinerant 5d bands which is only slightly below the experimental values [26, 41–43]. In the second study, Liu et al. [34] used DFT with the modified Becke– Johnson potential (a semi-local approximation to the exact exchange

340 Magnetism, Quantum Criticality, and Metal–Insulator Transitions in RB12

(a) YbB12 (LDA+SOC)

0.8 0.4

0.4

Energy (eV)

0

Energy (eV)

(b) YbB12 (LDA+SOC+GW)

0.8

-0.4 -0.8

0 -0.4 -0.8

-1.2 -1.2

-1.6 -2

-1.6

Γ

KX W

L

Γ

X

Γ

KX W

L

Γ

X

Figure 4.3 The band structure of YbB12 : (a) obtained from LDA+SOC calculations and (b) quasiparticle band structure calculated from LDA + SOC + Gutzwiller with U = 6 eV [14].

plus a screening term [47]), resulting in a nonmagnetic ground state of YbB12 with an indirect band gap of 21 meV and a direct gap of 80 meV (see Fig. 4.4). Integrating over all k points, it was found that out of the 14 states of the Yb f complex, 13.2 lie below the Fermi energy and 0.8 above (Fig. 4.4), which corresponds well with the experiment and band structure calculations [14]. Small needle-shaped Fermi surfaces of YbB12 with an effective mass m∗ ≈ 6m0 were detected in the de Haas–van Alphen measurements and explained using the modified Becke–Johnson potential for a small positive energy shift [34]. I would also like to mention recent observations of the Shubnikov–de Haas (SdH) oscillations (i.e., quantum oscillations in the resistivity) in YbB12 [48, 49] that have been discussed by the authors as a big surprise. Indeed, in spite of the large charge gap inferred from the insulating behavior of the resistivity, YbB12 , as well as the SmB6 compound, apparently hosts a Fermi surface at high magnetic fields. The authors argue that in YbB12 , where the mean valence of the Yb ions is close to +3 (4f 13 state) [27, 50], the 3D nature of the SdH signal demonstrates that the quantum oscillations in the resistivity arise from the electrically insulating bulk. On the basis of the symmetry analysis and simulations, they deny the possibility of SdH oscillations arising from a minority portion of

Electronic Band Structure of Dodecaborides 341

Figure 4.4 Calculated band structure of YbB12 shown over a wide energy range in (a) with an expanded view around the Fermi energy E F in (b). Several characters are projected out of the eigenvectors at each k-point and the resulting weight is indicated by a circle of proportional size. Green circles are Yb f -states, orange circles are Yb d-states, and violet circles are boron states. (c) Small needle-shaped Fermi surfaces of YbB12 with effective mass m∗ /m0 ≈ 6 obtained using the modified Becke–Johnson potential for a small positive energy shift. Expected angular dependence of the quantum oscillation frequencies F (θ) in the [001]–[111]–[110] rotation plane are evaluated by approximating the shown Fermi surfaces as prolate ellipsoids. (d) Small peanut-shaped Fermi surfaces of YbB12 with effective mass m∗ /m0 ≈ 9 obtained using the modified Becke–Johnson potential for a small negative energy shift. Expected angular dependence of the quantum oscillation frequencies in the [001]–[111]–[110] rotation plane are evaluated by approximating the shown Fermi surfaces as prolate ellipsoids. (e) A schematic of the conventional fcc Brillouin zone used for the band structures within the cubic Brillouin zone used for the Fermi surfaces [34].

the sample or from metallic domains resulting from impurities or strain [48, 49].

342 Magnetism, Quantum Criticality, and Metal–Insulator Transitions in RB12

To summarize the results presented in this section, the main postulates of both the aforementioned band-structure calculations and the interpretation of quantum-oscillation experiments are (i) homogeneous state in all RB12 single crystals including the intermediate-valence YbB12 compound and (ii) the stable B12 nanoclusters which are considered as basic structural elements of the fcc lattice of dodecaborides. It will be shown below in this chapter that both these assertions are, strictly speaking, not valid for RB12 , and the dynamic JT instability of the boron cage is among the main factors responsible for phase-separation effects and inhomogeneity in these entangled strongly correlated electron systems.

4.3 Nonmagnetic Reference Compound LuB12 4.3.1 Charge Transport Lutetium dodecaboride is a good metal with the measured residual resistivity ratio ρ(300 K)/ρ(10 K) ≈ 70 for the natural boron content (Lunat B12 , “nat” means the natural isotopic composition with 18.8% 10 B and 81.2% 11 B), as shown in Fig. 4.5. The high value confirms the high quality of the single crystals [51–53]. For pure isotopic Lu10 B12 and Lu11 B12 crystals, the residual resistivity turns out to be considerably higher, with the maximum value of ρ0 ≈ 0.83 μ ·cm observed in Lu10 B12 [53, 54]. Bolotina et al. [54] note that for all three boron-isotope compositions, the resistivity reaches its residual value below 20 K (Fig. 4.5), and the resistivity changes with temperature ρ(T ) can be analyzed in terms of the Einstein formula [55]:

ρ = ρ − ρ0 = ρE =

T



eE /T

A ,  − 1 1 − e−E /T

(4.1)

where E is the Einstein temperature and A is a proportionality constant. Eq. (4.1) is valid in the range T < T ∗ ≈ 60 K where the strong electron–phonon scattering on the quasi-local vibrations of the Lu3+ ions dominates. Figure 4.6 shows a fit of the resistivity data by Eq. (4.1), which allows estimating the Einstein temperature

Nonmagnetic Reference Compound LuB12

Figure 4.5 Temperature dependences of the specific heat and resistivity of the isotopic Lu N B12 crystals with N = 10, 11 and nat. Reproduced from Ref. [54].

E = 162–170 K [54]. At higher temperatures, above T ∗ , Umklapp processes dominate in the charge-carrier scattering of LuB12 , and the relation (4.2)

ρ = ρU = B0 T exp(−T0 /T ) may be considered a good approximation for resistivity in the temperature range 100–300 K. The authors [54] note that the T0 parameter in Eq. (4.2) estimated from these fits is close to the Einstein temperatures E , and the crossover between these two regimes described by Eqs. (4.1) and (4.2) coincides very well with the T ∗ transition region (see Fig. 4.6). Taking into account that T0 ≈ vs q/kB , where vs is the sound velocity, q is the wave number of the Umklapp phonon, and kB is the Boltzmann constant, the authors concluded that the quasi-local modes of Lu ions are the vibrations also prevailing in the Umklapp processes in LuB12 . The Hall effect in Lu N B12 has been studied in Refs. [53, 56]. Figure 4.7 shows the dependences of the Hall coefficient R H (T , H ) obtained for the Lu11 B12 and Lu10 B12 crystals in fixed external

343

344 Magnetism, Quantum Criticality, and Metal–Insulator Transitions in RB12

Figure 4.6 (a) Temperature-dependent component of the resistivity curves

ρ(T ) (curves are shifted for convenience). Fitting results from Eqs. (4.1) and (4.2) are shown by dashed and solid lines, respectively. (b) Temperature dependences of the derivative of specific heat dC /dT and the Hall coefficient RH (T ). The vertical dashed line is located at T ∗ ≈ 60 K. Reproduced from Ref. [54].

magnetic fields. It follows from the data of Fig. 4.7 and from the results of Ref. [56] for Lunat B12 that the change in the Hall coefficient with temperature in the lutetium dodecaboride is essentially nonmonotonic with a strong anomaly near T ∗ ≈ 60 K, which is observed in the entire range of magnetic fields H ≤ 80 kOe. Moreover, in strong magnetic fields (40 and 80 kOe curves in Fig. 4.7), the high-accuracy measurements allow discerning two closely spaced features near T ∗ in the temperature dependences of RH .

Nonmagnetic Reference Compound LuB12

Figure 4.7 Temperature dependences of the Hall coefficient R H for (a) Lu11 B12 and (b) Lu10 B12 dodecaborides in magnetic fields H = 5, 40, and 80 kOe [53].

The Seebeck coefficient S(T ) in Fig. 4.8 demonstrates more or less pronounced negative minima at intermediate temperatures 30– 150 K [54]. The amplitude of the anomaly, which is observed for all the LuN B12 crystals, is the largest in Lu10 B12 and the smallest in Lunat B12 . At low temperatures, the Seebeck coefficient changes linearly, and the largest slope of this Mott (diffusive) thermopower is detected in the isotopically pure lutetium dodecaborides. When discussing the S(T ) behavior presented in Fig. 4.8, the authors conclude that the negative minimum is a typical feature for metals with electron conduction and it appears as a crossover from phonon-drag thermopower with the dependence Sg ∝ 1/T at higher temperatures to a linear diffusive low-temperature component S = B T. These two parts of thermopower are approximated by solid lines in Fig. 4.8, and the authors argue that electron–phonon scattering on quasi-local vibrations (Einstein modes) of Lu3+ ions dominates in the intermediate temperature range, hence these Einstein modes are determinants for thermopower of the phonon dragging [54].

4.3.2 Thermal Properties For Lu N B12 (N = 10, 11, nat), temperature dependences of the specific heat C (T ) at a constant pressure and intermediate

345

346 Magnetism, Quantum Criticality, and Metal–Insulator Transitions in RB12

Figure 4.8 Temperature dependence of the Seebeck coefficient of the crystals Lu N B12 with N = 10 (bottom), 11 (middle) and nat (top). Solid lines show the data approximation by the Mott dependence (T < 40 K) and phonon-drag thermopower (at intermediate temperatures). Reproduced from Bolotina et al. [54].

temperatures T > 30 K almost coincide with each other in the log–log plot (Fig. 4.5), whereas noticeable differences in C (T ) are observed at low temperatures [1, 54, 57]. The largest C (T ) values in the range of 2–20 K were detected for the Lunat B12 assuming effects in the specific heat of this compound of random isotope substitution and boron vacancies. A steplike anomaly in the C (T ) dependence in the range 20–40 K for Lunat B12 (Fig. 4.5) was observed and discussed previously [1, 3, 58] in terms of the Einstein type contribution to the specific heat from quasi-local vibrations of RE ions embedded in the large B24 cavities in the rigid boron cage. According to the data in Fig. 4.5, boron isotope substitution affects only slightly the behavior of the Einstein component in the specific heat of Lu N B12 (N = 10, 11, nat), which confirms the loosely bound state of the Lu3+ ions. The contributions from B and Lu atoms in the vibration heat capacity have been considered in terms of the Debye and Einstein

Nonmagnetic Reference Compound LuB12

models given by Eqs. (4.3) and 4.4), respectively [1]: CD 9 r R D /T x 4 ex = dx T3 (ex − 1)2 TD3 0 CE 3R = 3 3 T E



E T

5

e−E /T , (1 − e−E /T )2

(4.3)

(4.4)

where R is the gas constant and r = 12 is the number of boron atoms in the unit cell. The electronic specific heat C el = γ T with γ ≈ 3 mJ/(mol·K2 ) and the Debye contribution with D ≈ 1100 K detected from refined atomic displacement parameters (ADP) of the structure model were applied in Ref. [54] to calculate the difference  C ph = C − C el − C D = C E (T ) + i =1, 2 C Sch(i ) (T ). The two Schottky terms C Sch(i ) were used additionally to approximate lowtemperature anomalies in the specific heat by contributions from the two-level systems TLS1 and TLS2 . It has been argued [1, 57] that these two Schottky components given by   C Sch(i ) e E i /kB T R Ni E i 2 = (4.5) 3 3

E T T kB T (e i /kB T + 1)2 (Ni is the concentration of TLSi ) are necessary to describe the effect of boron vacancies (TLS2 ) and divacancies (TLS1 ) in the specific heat of RB12 . Indeed, in view of a weak coupling of the RE ions in the boron network in combination with a significant number of boron vacancies and other intrinsic defects in the UB12 type structure [59], the formation of various two-level systems arranged in double-well potentials (DWP) should be expected at the displacements of Lu3+ ions from their central positions in the B24 cuboctahedra. All the above-mentioned specific heat contributions are shown in Fig. 4.9 in the C (T )/T 3 plot together with the experimental data [54]. Based on the approach developed in Refs. [1, 57], the energy

E 2 /kB = 54–64 K in Eq. (4.5) should be attributed to the barrier height in the DWP (or, in other words, the energy difference in the two-level system). The normalized concentration of TLS2 of N2 = 0.047–0.08 corresponds to the number of Lu ions displaced from the central positions in B24 cells. This result approximately agrees with the concentration of 0.036 deduced from EXAFS measurements of LuB12 powders at low temperatures [60]. Taking into account that each boron vacancy ensures the displacement of two neighboring

347

348 Magnetism, Quantum Criticality, and Metal–Insulator Transitions in RB12

Figure 4.9 Separation of the low-temperature vibrational contribution C ph /T 3 to the specific heat of Lu N B12 into Debye (C D ), Einstein (C E ) and two Schottky (TLS1 and TLS2 ) components. Reproduced from Ref. [54].

RE ions from the center of the B24 octahedron in the structure of RB12 , the number of boron vacancies was estimated as nv = N2 /2 = 2.4–4%, being dependent on the boron isotope composition [57]. According to Junod et al. [61], (5/4)Rπ 4 C ph /T 3 vs. T gives an approximate picture of the one-dimensional phonon density of states (DOS) ω−2 F (ω) for ω = 4.928 T , where ω is expressed in kelvins. On a logarithmic scale the response of C ph /T 3 to a δ-function (Einstein peak) is a bell-shaped peak, shown as the C E component in

Nonmagnetic Reference Compound LuB12

4

0.10

-6

-1

α (10 K )

0.15

α (10 –6 K–1)

3

0.05 0.00

-0.05

2

5

10 15 20 25 Temperature (K)

30

LuB12 LuB12

1

0

0

50

100 Temperature (K)

150

200

Figure 4.10 The temperature dependence of the thermal expansion coefficient α for the LuB12 100 and 110 single crystals. Inset: α versus T in an expanded scale. Adapted from Shitsevalova [62].

Fig. 4.9. In this way, the main contribution to C ph for LuB12 at low temperatures [1, 54, 58] is related to the mode equal to 14.1 meV that is very close to the energy of the δ-like peak (14.2 meV) from the neutron phonon spectrum [63] (see also Chapter 6) and from the point-contact spectrum (14.5 meV) [64]. When discussing the nature of the anomalies at T ∗ ≈ 60 K (see Fig. 4.6), the authors [1, 54] concluded that it should be attributed to the order-disorder phase transition, and below T ∗ the Lu3+ ions are “freezing” at different positions of DWP minima induced by the random distribution of boron vacancies, 10 B and 11 B atoms, and impurities. In this scenario the barrier height E 2 /kB in the DWP is close to the cage-glass transition temperature T ∗ = 54–65 K in crystals of Lu N B12 [1, 53], and the disordered state below T ∗ is a mixture of two components: the crystal (rigid boron covalent cage) and glass (clusters of Lu ions displaced from their central positions in the B24 cuboctahedra). Thermal conductivity measurements of Lunat B12 display a large amplitude maximum of λ(T ) at ∼25 K [65]. The separation of electron and phonon contributions was calculated from the

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350 Magnetism, Quantum Criticality, and Metal–Insulator Transitions in RB12

Wiedemann–Franz law, indicating strong thermal conductivity anomalies at 25 K both for λe and λph components and detecting at higher temperatures the λph ∝ 1/T dependence which is typical for dielectrics [65]. The ratio λph /λe ≈ 3–4, that is also unusual for good metals, may be estimated from the thermal conductivity data in the range 4–300 K [65]. Keeping in mind that vibrations in the DWP are anharmonic in principle, it is important to investigate the thermal expansion originating from the anharmonic atom vibrations in LuB12 . The temperature dependences of the thermal expansion coefficient α of the Lunat B12 single crystals are shown in Fig. 4.10 [58]. Here α(T ) reveals a negative thermal expansion (NTE) in two intervals: the “low-temperature” one with a minimum around 12–15 K and the “high-temperature” one with a minimum around 90–100 K. The authors [58] discuss a slight difference in the thermal expansion of the LuB12 crystals with 100 and 110 orientations, which could be produced either by the anisotropy of the chemical bonds (the 100 axis corresponds to the Lu–B bond direction, and the 110 axis to the Lu–Lu bond), or the low-temperature distortion of the LuB12 fcc crystal structure. Based on the experimental heat capacity, thermal expansion and bulk modulus (cb ) data, Czopnik et al. [58] evaluated the ¨ temperature dependence of the generalized Gruneisen parameter Gr = 3αcb V/C v (where V is the molar volume) in LuB12 . In Fig. 4.11, Gr (T ) is presented for LuB12 in the direction 100. In the 8–17 K and 60–130 K ranges, Gr for LuB12 is negative in accordance with its two ranges of NTE, whereas C p from 2 K to 300 K is a monotonically increasing function of temperature (Fig. 4.5), and cb is a smooth monotonically decreasing function of temperature that changes from 229.5 down to 224.9 GPa [58]. As a result, the ¨ generalized Gruneisen parameter Gr , but not the heat capacity, mainly determines the α(T ) behavior for LuB12 in both 100 and

110 crystal orientations. The authors [58] suggested that a defectinduced soft mode is responsible both for the NTE and the Schottky anomalies of heat capacity at temperatures below 20 K, having the same origin — a formation of the two-level tunneling systems based on the metal ions and defects [66]. As well, according to [58], the existence of the negative Gr at intermediate temperatures 60–130 K

Nonmagnetic Reference Compound LuB12

200

8

150

LuB12

100 ΓGr

6

50

ΓGr

0

4

-50 -100

5

10

15

20

25

30

T (K)

2

0

-2

0

50

100 T (K)

150

200

¨ Figure 4.11 The LuB12 generalized Gruneisen parameter Gr . Inset: Gr versus T for LuB12 in the expanded low-temperature scale 5–30 K [58].

should be attributed to the presence in the phonon spectra of LuB12 of a flat transverse acoustic mode characterized by E ≈ 164 K ≈ 14.1 meV (see also Chapter 6), as it was predicted for the first time by Dayal [67] and Barron [68] for tetrahedral semiconductors Si and Ge. Dove and Hong Fang pointed out recently [69] that in many cases NTE is associated with a combination of transverse acoustic and optical modes in compounds where the lattice instability is developed. Then, it is worth noting here two characteristic temperatures in the α(T ) and Gr (T ) dependences of LuB12 (Figs. 4.10 and 4.11). The sign inversion at 60 K on these two curves matches precisely both the cage-glass transition T ∗ and the barrier height in DWP, E 2 /kB ≈ 60 K. Additionally, the knee-type anomaly observed on these curves near 130–150 K correlates very well with the renormalization of the low-frequency vibration spectra (see, for example, Fig. 4.12) and to the appearance of the boson peak in the Raman spectra attributed to the disordering in the positions of the RE ions in the LuB12 matrix [1]. Applying the approach proposed in Refs. [70, 71], the position of the boson peak in the

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352 Magnetism, Quantum Criticality, and Metal–Insulator Transitions in RB12

g(ω,T 0) × 10−5

1.2

g(ω,T 0) × 10−5

2.5

(a)

(b)

2.0 0.8 1.5

130 K 162 K 102 K 211 K 300 K 87 K 6K

1.0 10 K

0.4

74 K 98 K

0.5

300 K

0

100

200

300

400 ω, cm−1

0

100

200

300

400 ω, cm−1

Figure 4.12 Temperature variation of the low-frequency vibrational density of states g(ω, T0 ) for (a) Lu10 B12 and (b) Lu11 B12 dodecaborides. Reproduced from Ref. [1].

Raman spectra of disordered systems was used in Ref. [1] to evaluate quantitatively the spatial size DN of regions with lowfrequency quasi-local vibrations (vibrational clusters) in the RB12 structure. Using the relation for boson peak frequency ωmax ≈ (0.7– 0.85) vs /c DN [70–72] and the sound velocity vs ≈ 962 m/s found for LuB12 at T = 78 K [25], authors [1] obtain D10, 11 = 12–15 A˚ for Lu10 B12 and Lu11 B12 , and Dnat = 18–22 A˚ for Lunat B12 , arguing that in the presence of additional substitutional 10 B-11 B disorder, the correlation length in the system of interacting harmonic oscillators increases by a factor of about 1.5 and reaches Dnat ≈ 3a0 (a0 ≈ 7.5 A˚ is the lattice constant). It was noted in Ref. [1] that the development of the lattice instability with a decrease in temperature leads to a sharp increase in the vibrational DOS g(ω, T0 ) that is plotted in Fig. 4.12. Around the temperature T ≈ TE = 5TC max ≈ 150 K (here TC max is the temperature of the heat-capacity maximum), the mean free path of phonons reaches the Ioffe–Regel limit and becomes comparable with their

Nonmagnetic Reference Compound LuB12

wavelength [73]. Near TE , in addition to the maximum in the vibrational DOS g(ω, T ) (see Fig. 4.12), the sharp maximum of the relaxation rate is observed in muon-spin relaxation (μSR) experiments with RB12 dodecaborides (R = Er, Yb, Lu) and Lu1−x Ybx B12 solid solutions [74, 75]. The authors suggested that the large amplitude dynamic features arise from atomic motions within the B12 clusters.

4.3.3 Optical Properties The optical reflectivity R(ω) experiments on single crystals of LuB12 have been conducted for the first time by Okamura et al. [44], and the optical conductivity σ (ω) spectra were obtained from R(ω) using the Kramers–Kronig relations [76]. It was found in Refs. [44,77] that R(ω) spectra have a clear plasma cutoff (ωp ) near 1.6 eV and sharp structures above 2 eV due to interband transitions [78]. Below ωp , the optical conductivity demonstrates a sharp rise due to a metallic response of free carriers. Recently, more detailed optical studies of the high-quality Lu N B12 single crystals with different boron isotopes (N = 10, 11, nat) were performed at room temperature by Gorshunov et al. [79]. Figure 4.13 shows a broadband spectrum of the reflection coefficient of Lunat B12 (dots). At low frequencies, the spectrum looks typically metallic [76] with the plasma edge at ∼14 000 cm−1 (∼1.75 eV). It was found in Ref. [79] that it is impossible to model the measured spectrum within the Drude conductivity model [76] alone (see Fig. 4.13), keeping the measured dc conductivity value fixed to σDC = 95 000 −1 cm−1 and varying only the scattering rate γD . This result is a strong indication that the infrared optical response of LuB12 is not determined by just free charge carriers which provide its metallic conductivity, and that there are additional IR excitations and specific mechanisms governing the electronic properties of the compound. Additional evidence for such excitations is seen in the reflectivity spectrum as kink-like features at ∼1100, 4500, and 7000 cm−1 (indicated by arrows in Fig. 4.13 and enlarged in the inset), which are not completely screened by the charge carriers. The authors introduce, together with the free-carrier Drude term, the minimal set of excitations (Lorentzians) that provides a suitable

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354 Magnetism, Quantum Criticality, and Metal–Insulator Transitions in RB12

Figure 4.13 Room-temperature reflection coefficient spectrum of the Lunat B12 crystal. Dots show experimental data obtained using the Fouriertransform spectrometer and ellipsometer. The dotted line corresponds to high-frequency reflectivity data from [78]. The solid red line shows the results of fitting the spectrum using the Drude term for the free charge carrier response and Lorentzians responsible for absorption resonances. The dashed line shows best fit of the spectrum that can be obtained using the Drude conductivity term alone with σDC = 95 000 −1 cm−1 and γD = 290 cm−1 . Kink-like features observed in the spectrum are indicated by arrows and presented in more detail in the inset. Reproduced from Gorshunov et al. [79].

model of the measured reflectivity spectrum (red solid line in Fig. 4.13). Figure 4.14 presents the obtained broadband conductivity spectrum of LuB12 including five additive contributions detected in Ref. [79]. The excitations L1 and L2 (Fig. 4.14) are the dominant and rather unusual in having unexpectedly large dielectric contributions, especially the L1 peak ( ε = 8000 ± 4000), and being strongly overdamped (γ /v0 > 2.5). Gorshunov et al. [79] estimate the concentration of conduction electrons involved in the free carrier conductivity and in the formation of the collective excitations L1 and L2 using an 2 = ne2 /(πm∗ ) = f for the plasma frequency and expression vpl oscillator strength f (here n is the concentration of free electrons,

Nonmagnetic Reference Compound LuB12

Figure 4.14 Room-temperature spectrum of the real part of the optical conductivity of the Lunat B12 single crystal (solid red line). The dashed spectrum above 20 000 cm−1 corresponds to the result obtained by fitting the reflectivity data from Ref. [78] (dashed line in Fig. 4.13). Dots at 4000–27 000 cm−1 represent conductivity spectrum from ellipsometry measurements. The spectrum is obtained by least-square fitting of the reflection coefficient spectrum shown in Fig. 4.13 using Drude conductivity term and four Lorentzian terms. The Drude term is shown separately by a dash-dotted line and the Lorentzian terms by dashed lines. The L1 + L2 line corresponds to the sum of L1 and L2 contributions. To visualize separate contributions, same spectra are shown in a double logarithmic plot in the inset. Reproduced from Gorshunov et al. [79].

e – elementary charge). With the Drude unscreened plasma frequency vpl = 21 700 ± 3200 cm−1 (ε∞ ≈ 2.5) and m∗ = 0.5m0 [31, 34, 36], they obtained nD = 2.6 × 1021 cm−3 ± 30%. From the combined oscillator strength of the L1 and L2 Lorentzian terms, fL1+L2 = 1.3 × 109 cm−2 assuming m∗ = 0.5m0 , they get the concentration of the charge carriers participating in the formation of the collective excitation nL1+L2 = 7.2 × 1021 cm−3 ± 30% and the total concentration of charges in the conduction band ntot = nD + nL1+L2 = 9.8 × 1021 cm−3 ± 30%. The obtained value ntot coincides very well with the concentration of Lu-ions n(Lu) = 9.6 × 1021 cm−3 in LuB12 assuming that every Lu ion delivers one

355

20K 70K 120K 170K 220K 270K

10

1

V1 ( :-1 cm-1)

VDC ( :-1 cm-1)

356 Magnetism, Quantum Criticality, and Metal–Insulator Transitions in RB12

0.1

10

100

1000

10000 -1

Wavenumber (cm ) Figure 4.15 Temperature dependence of the real part of the widefrequency-range optical conductivity spectra of LuB12 . Direct current conductivity values obtained from resistivity measurements are shown in the left panel. Reproduced from Teyssier et al. [80].

electron in the conduction band. According to the above estimates, about 70% of charges in the conduction band are involved in the formation of the collective excitation. Similar estimates were done in Ref. [79] for Lu10 B12 and Lu11 B12 crystals with the parameters giving the percentage of about 70% and 80%, respectively. Thus, it was concluded that nonequilibrium electrons involved in the collective modes dominate in the LuB12 charge transport. The temperature dependence of the broadband dynamic conductivity spectra has been studied in Lunat B12 by Teissier et al. [80] where the narrowing of the Drude term frequency range with the temperature lowering down to 20 K (Fig. 4.15) was observed, providing better conditions for the separation of Drude and collective-mode contributions. Our crude estimations allow concluding that during cooling both the oscillator strength (intensity) of the Lorentzian describing the broad bump seen between 1000 and 10 000 cm−1 (see Fig. 4.15) and the squared plasma frequency of the Drude contribution slightly increase. So, the collective mode (overdamped oscillator), which is present even at 20 K, may be considered as a fingerprint of a quantum motion (zero-temperature vibrations) in the matrix of LuB12 .

Nonmagnetic Reference Compound LuB12

It is worth noting also that the positions of L1 (∼250 cm−1 ≈ 31 meV) and L2 (∼1000 cm−1 = 125 meV) collective modes in the IR spectra of LuB12 (Fig. 4.14) correlate very well with those reported by Bouvet et al. [63], Nemkovski et al. [81], and Rybina et al. [82] in the inelastic neutron scattering (INS) studies of the boron phonon modes in Lu11 B12 (see Chapter 6) and by Werheit et al. [83] in the Raman spectra of Lu N B12 with N = 10, 11, nat (see Chapter 5).

4.3.4 Magnetoresistance Anisotropy and Dynamic Charge Stripes The transverse magnetoresistance (MR) of LuB12 was measured for current directions I 110 and 100 by Heinecke et al. [31]. The strongest magnetic anisotropy was observed at μ0 H = 12 T and T = 0.5 K, reaching the values between 4 for H 111 and 18 for H 001. As the authors found a saturation of MR with I 110 and H 111, they concluded that LuB12 is an uncompensated metal with more than one direction of open orbits on the Fermi surface. The local MR maxima were suggested to be caused by open orbits in the current direction. The drift and Hall mobilities of the charge carriers were estimated to be ∼2000 cm2 V−1 s−1 , and it was concluded that the impurity scattering dominates in LuB12 below 20 K [84]. An alternative interpretation was proposed in recent transverse MR studies of LuB12 [85, 86]. The transverse MR anisotropy has been measured for two [010]- and [001]-elongated rectangular monodomain single crystals cut from one ingot of Lunat B12 with equally oriented [100], [010], and [001] faces [85]. A significant anisotropy (up to 20%) was observed below T ∗ ≈ 60 K at H = 80 kOe for one of these crystals, in spite of the fact that the field directions [001] and [100] are symmetry equivalent in the cubic lattice. In addition, a minimum was detected below T ∗ in the temperature dependences of the resistivity, ρ(T , H = 80 kOe), with the growth of resistivity at low temperatures. To shed more light on the nature of the unusual anisotropy, angular dependences ρ(φ) were measured at 2.0–4.2 K in the magnetic fields up to 80 kOe by rotating these two crystals around their current axes [85].

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358 Magnetism, Quantum Criticality, and Metal–Insulator Transitions in RB12

Figure 4.16 (a) Field-angular dependences of resistivity ρ1 (φ) and ρ2 (φ) obtained by rotating the crystals #1 and #2 around their current axes I1 [010] and I2 [001], respectively, in various magnetic fields up to 80 kOe at temperatures of 2.0–4.2 K. (b) The anisotropy of magnetoresistance ρ1 (φ) − ρ2 (φ) = f (φ, H ) presented in polar coordinates. Reproduced from Ref. [85].

Families of curves ρ1, 2 (φ) for the samples are shown in Fig. 4.16(a) demonstrating the great difference for H [100] (φ = 0◦ ) and H [001] (φ = 90◦ ), although these directions are equivalent in cubic crystals. It is clearly discerned from the polar plot in Fig. 4.16(b) that conduction channels appear in the LuB12 matrix [85]. The orientation of the conduction channels was detected certainly in Ref. [86], where the detailed x-ray diffraction study was carried out in combination with the angular MR measurements of Lunat B12 crystals. A new approach was developed that consists in the difference Fourier synthesis of the residual electron density (ED) as well as in the reconstruction of the ED distribution using maximal entropy method (see Chapter 3). It was found [86] that the ED peaks become stronger with the temperature lowering and form a filamentary structure of conduction channels — unbroken charge stripes oriented almost along the [110] axis (see Figs. 3.16 and 3.17 in Chapter 3). These observations are in accordance with the cubic symmetry distortions of the LuB12 crystals [86] (see

Nonmagnetic Reference Compound LuB12

Fig. 3.13 in Chapter 3). It has been shown also that the asymmetric ED distribution correlates very accurately with the anisotropy of transverse MR. Thus, the same conduction channels along the direction [110] observed from both the x-ray and charge-transport data should be considered as the reason for the strong increase of MR in the direction H [001] which is transverse to the dynamic charge stripes. Below we are discussing the nature of both the lattice instability and inhomogeneity of electron density distribution in the nonmagnetic reference compound LuB12 .

4.3.5 The Origin of Electron and Lattice Instability and the Energy Scales in LuB12 The mechanism responsible for the development of electron and lattice instability in RB12 was discussed in Refs. [54, 85] in terms of the dynamic cooperative JT effect on B12 clusters. More specifically: (1) It has been shown in Ref. [85] that because of triple orbital degeneracy of the electronic ground state, the B12 molecules are JT active and thus their structure is labile due to JT distortions (see Chapter 3 for details). The quantum-chemistry calculations and geometry optimizations for a charged [B12 ]2− cluster, whose doubly negative charge state is regarded as the most relevant in RB12 compounds, allow the conclusion in favor of strong trigonal and tetragonal distortions and a mixture via the electron-vibronic nonadiabatic coupling of electronic states on each B12 cluster [88]. (2) In the dodecaboride matrix, the collective JT effect on the lattice of these B12 complexes is at the origin of both the collective dynamics of boron clusters and large amplitude vibrations of the RE ions embedded in cavities of the boron cage as it is shown in Fig. 4.17. (3) Strong coupling of these Lu rattling modes that are located at 110 cm−1 [89] and the JT vibrations is the reason both for the lattice instability with a dramatic increase in the vibrational DOS near TE ≈ 150 K (see Fig. 4.12) and for the emergence of the collective excitation in the optical spectra (Fig. 4.14).

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360 Magnetism, Quantum Criticality, and Metal–Insulator Transitions in RB12

Figure 4.17 (a) Crystal structure of RB12 . (b) B24 truncated cuboctahedra surrounding two adjacent R 3+ ions. (c) Schematic view of R ions vibrations in the double-well potentials. The red arrow shows the direction [110] of dynamic charge stripes in RB12 . Reproduced from Ref. [87].

(4) The Lu rattling modes at 110 cm−1 [1, 89, 90] are responsible also for the damping in the Drude term (solid line in Fig. 4.13, γD = 90 ± 30 cm−1 [79]). These large-amplitude Einstein oscillators in the DWP, schematically illustrated in Fig. 4.17(c), necessarily initiate strong periodic changes in the 5d-2p hybridization between the RE and the electronic states of boron. Because the states in the conduction band are composed by the 2p orbitals of the B12 cluster and the 5d orbitals of the Lu atoms [25, 28, 29, 31, 33, 38], the variation in the 5d-2p hybridization will lead to the modulation of conduction bandwidth (see Chapter 3) and consequently generate numerous (up to ∼70% of the total number of conduction band electrons [79]) nonequilibrium (hot) charge carriers manifested in the collective mode in the optical conductivity spectra at room temperature (Fig. 4.14). (5) In the cage-glass state of RB12 at T < T ∗ ≈ 60 K [1], two additional factors appear: (i) the positional disorder in the arrangement of Lu3+ ions in the B24 truncated cuboctahedra (static displacements of R 3+ ions in the DWP), which is accompanied by the formation of vibrationally coupled nanometersize clusters in the RB12 matrix, and (ii) the emergence of dynamic charge stripes (ac current with a frequency ∼200 GHz

Nonmagnetic Reference Compound LuB12

[43]) directed along the single [110] axis in LuB12 [86] that accumulate a considerable part of nonequilibrium conduction electrons in the filamentary structure of fluctuating charges. Based on the precise investigations of the crystal structure, heat capacity, and charge transport, Bolotina et al. [54] argue that in the family of Lu N B12 crystals (N = 10, 11, nat), Lunat B12 has the strongest local atomic disorder in combination with long-range JT trigonal distortions, providing optimal conditions for the formation of the dynamic charge stripes below T ∗ ≈ 60 K. As a result, the resistivity and Seebeck coefficient of the Lunat B12 heterogeneous media decrease strongly in comparison with the characteristics detected for pure boron-isotope enriched crystals (see Figs. 4.5 and 4.8). Thus, the authors concluded that defects are supposedly used as the centers of pinning facilitating the formation of additional ac conductive channels — the dynamic charge stripes in the metallic matrix of RB12 . To summarize, three energy scales that determine the properties of RE dodecaborides at intermediate and low temperatures can be highlighted. The JT splitting of the triply degenerate electronic ground state of the B12 clusters is estimated to be ∼100–200 meV [85] (see also Chapter 3), so the cooperative JT dynamics is expected to be observed in the range of 100–1500 cm−1 . The energy of the rattling mode E ≈ 110 cm−1 ≈ 15 meV is very close to the characteristic temperature TE ≈ 150 K of the fcc lattice instability, which develops in RB12 and leads to a strong increase in the vibrational DOS in approach to the Ioffe–Regel regime of the lattice dynamics. The order-disorder cage-glass transition at T ∗ ≈ 60 K is regulated by the barrier height in the DWP for RE ions, and below this temperature, the static displacements of R 3+ ions in the fcc lattice are accompanied by an emergence of dynamic charge stripes in the dodecaboride matrix. Taking into account the loosely bound state of the R ions in the rigid boron covalent sublattice, it is natural to expect that all these energy scales along with nonequilibrium and many-body effects produced by the JT effect in the B12 clusters will be realized also in magnetic and semiconducting RE dodecaborides, as will be discussed in the following sections.

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362 Magnetism, Quantum Criticality, and Metal–Insulator Transitions in RB12

4.4 Magnetic Dodecaborides RB12 (R = Tb, Dy, Ho, Er, Tm) and the Solid Solutions Rx Lu1−x B12 4.4.1 Magnetic Properties The temperature dependence of the magnetic susceptibilities χ(T ) of RB12 (R = Tb – Yb) was investigated over the temperature range 90–1200 K by Moiseenko and Odintsov [91]. The linear Curie–Weiss type behavior of χ −1 (T ) was found for all compounds, and the authors concluded that the paramagnetic properties of these dodecaborides are caused mainly by the 4f electrons with AFM exchange interaction between the localized magnetic moments of the RE ions. The calculated effective magnetic moments μeff = 4.3–10.6 μB were close to the values of the corresponding 3+ charged free RE ions, and the RKKY indirect exchange is suggested to be responsible for the interaction between them [91]. The low-temperature magnetization measurements of RB12 (R = Tb, Dy, Ho, Er, Tm) [62, 92] indicated ´ temperatures of 22, 16.4, 7.4, the AFM phase transitions with Neel 6.7, and 3.3 K, respectively. It was mentioned by Kohout et al. [93] that in the AFM state toward low temperature, the magnetization does not approach zero; almost independent of the crystalline orientation, the magnetization extrapolates to approximately 70% of its maximum at TN . Thus, the magnetization data for single crystals resemble the response from a powder sample with equal mixing of the transverse and longitudinal components indicating a significant disorder in the orientation of the 4f magnetic moments. The high-field magnetization measurements have been carried out for HoB12 and TmB12 by Siemensmeyer et al. [94]. The saturation is not fully reached even in 14 T field due to very strong interactions. The comparison of the experimental results with calculation allows concluding in favor of a triplet 51 ground state of Ho and Tm ions which is isotropic in low magnetic field. At high fields, the induced magnetization and thus the single-ion energy in a magnetic field depend strongly on the field orientation and the

111 direction is preferred over 110 and especially the 001 direction [94].

Magnetic Dodecaborides RB12 (R = Tb, Dy, Ho, Er, Tm) and the Solid Solutions Rx Lu1−x B12

The substitution of Ho atoms by nonmagnetic Lu in Hox Lu1−x B12 ´ solid solutions has been studied by Gabani et al. [95]. They showed that the TN (x) dependence demonstrates a linear increase from a quantum critical point (QCP) with TN = 0 detected at xc ≈ 0.1. At temperatures above 3TN , the measured χ(T ) dependences obey the Curie–Weiss law and the effective magnetic moments per formula unit μeff were well fitted by the relation √ μeff (Hox Lu1−x B12 ) = μeff (HoB12 ) 1 − x, indicating the suppression of indirect exchange interaction between Ho3+ moments with an increase in the concentration of nonmagnetic Lu ions.

4.4.2 Electron Paramagnetic Resonance

N

Recently, measurements of the high-frequency (60 GHz) electron paramagnetic resonance (EPR) were carried out on Hox Lu1−x B12 solid solutions at intermediate and low temperatures in a wide range of the holmium content 0.01 ≤ x ≤ 1 [96]. For the samples with x ≥ 0.1, it was demonstrated (see Fig. 4.18) that the electron paramagnetic resonance in the form of a single broad line with a g-factor of ∼5 appears in the cage-glass phase at T < T ∗ ≈ 60 K because of the decrease in the relaxation rate for magnetic moments of Ho3+ ions. For the compounds with x > 0.3, the pronounced

T

Figure 4.18 (a) Evolution of the EPR spectra with the concentration x in Hox Lu1−x B12 at T = 15 K [96]. (b) Concentration dependence of the EPR linewidth (left vertical axis) and g-factor (right vertical axis). ´ temperature [97]. (c) Concentration dependence of the Neel

363

364 Magnetism, Quantum Criticality, and Metal–Insulator Transitions in RB12

broadening of the resonance line and a steep decrease in the gfactor related to AFM correlations (short-range order effects) are observed on cooling below 30 K. The performed simulation of the EPR spectra has indicated [96] that the exchange interaction and positional disorder in a location of magnetic ions play a decisive role in the specific features of spin dynamics in Hox Lu1−x B12 .

4.4.3 Charge Transport Resistivity In the magnetic dodecaborides HoB12 , ErB12 , and TmB12 , the temperature dependence of the resistivity ρ(T ) is metallic-like with a power-law behavior ρ(T ) ∝ T α at intermediate temperatures with the exponent varying in the range from α ≈ 1.2 for HoB12 to α ≈ 0.8 for TmB12 . In the range from 10 to 20 K, the ρ(T ) curves exhibit a minimum, and the resistivity demonstrates a moderate increase with the temperature lowering to TN . The transition to the AFM state at 7.4, 6.7, and 3.2 K for HoB12 , ErB12 , and TmB12 , respectively, is accompanied by a steplike upturn in ρ(T ), and in the case of ErB12 , a second magnetic phase transition appears at 5.8 K [84]. At liquid helium (LHe) temperatures, the crossover to the coherent regime of charge transport in the magnetically ordered phase of RB12 is accompanied by a monotonic decrease in resistivity. In an external magnetic field, three characteristic regimes of MR behavior have been revealed in the AFM metals: (i) the positive MR similar to the case of LuB12 is observed in the paramagnetic phase at T > 25 K; (ii) in the range TN ≤ T ≤ 15 K, the MR becomes negative and depends quadratically on the external magnetic field; and, finally, (iii) upon the transition to the AFM phase (T < TN ), the positive MR is again observed and its amplitude reaches 150% for HoB12 [84]. The low-temperature resistivity increase in the range TN ≤ T ≤ 15 K in combination with a strong negative MR has been explained in terms of charge carrier scattering on nanosize magnetic domains (short-range AFM order of the RE-ion moments) and spinpolarization effects [84, 98], which are suppressed dramatically in external magnetic field. It was found [84, 98, 99] that the negative MR component in magnetic RB12 is proportional to the square of 2 , and in the vicinity of the the local magnetization − ρ/ρ ∝ Mloc ´ temperature it demonstrates the critical behavior ρ/ρ ∝ Neel

Magnetic Dodecaborides RB12 (R = Tb, Dy, Ho, Er, Tm) and the Solid Solutions Rx Lu1−x B12

Figure 4.19 Temperature dependences of Hall coefficient RH (T ) for RB12 compounds (R = Ho, Er, Tm, Lu). Reproduced from Ref. [56].

(1 − T /TN )2β with the exponent β = 0.36–0.44 which is about equal to that one predicted for magnetization and observed in model antiferromagnets [100]. Hall effect The temperature-independent behavior of the Hall coefficient, which was observed in the nonmagnetic metal LuB12 at low temperatures T < 20 K and small magnetic fields (Fig. 4.19), is not typical for the magnetic dodecaborides HoB12 , ErB12 and TmB12 [56, 101]. Indeed, distinct maxima of RH (T ) are observed at Tmax ≈ 10–12 K in HoB12 and TmB12 (Fig. 4.19). Additionally, the pronounced anomalies of the Hall coefficient associated with AFM transitions can be clearly seen in the temperature dependences of RH (T ) for all the magnetic compounds RB12 in the vicinity of their ´ temperatures, TN (Fig. 4.19). The estimated values of TN agree Neel very well with the results of magnetic and thermal measurements of RB12 [3, 93, 102]. However, the observed discrepancy between the increase of R H in ErB12 and the decrease of RH in HoB12 and

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366 Magnetism, Quantum Criticality, and Metal–Insulator Transitions in RB12

TmB12 below TN (Fig. 4.19) was not explained in Ref. [56]. Additional anomalies were observed in the range of 130–300 K where several extrema were detected in the temperature dependence of the Hall coefficient [56, 101]. High-precision measurements of the Hall effect have been carried out on HoB12 at LHe and intermediate temperatures for different field directions in magnetic fields up to 80 kOe [101]. It has been shown that, along with the normal component of RH (H , T ), the strong (up to 30%) anomalous magnetic contribution to the Hall effect, which is suppressed in high magnetic field, is observed in both the paramagnetic and ´ phases of HoB12 . In Ref. [101] the ferromagnetic component Neel of the Hall signal was revealed above 20 kOe in the AF1 phase of HoB12 [94]. The singularities in the Hall coefficient RH (T , H ) were compared with the H –T magnetic phase diagram of HoB12 and explained by the considerable spin-polaron effects that are typical for strongly correlated electron systems [101]. A similar suppression of the temperature-dependent contribution to the Hall effect was found in the strong-magnetic-field studies of Hox Lu1−x B12 solid solutions [95] where it was shown that below T ∗ ≈ 60 K, the RH (T ) becomes temperature-independent at high magnetic fields ∼80 kOe. These changes in the charge transport in the cage-glass state of Hox Lu1−x B12 were explained in terms of spin-polaron effects and reconstruction of the filamentary structure of the conduction channels (dynamic charge stripes) in strong magnetic field. Seebeck coefficient The temperature dependences of the Seebeck coefficient S(T ) for magnetic compounds RB12 (R = Ho, Er, Tm) demonstrate a singularity at TE ≈ 130–150 K (see Fig. 4.20), which is connected both with the Hall-effect minimum (Fig. 4.19) and the dramatic changes in the μSR scattering rate [75]. The amplitude of the negative minimum in S(T ) near TE is the largest for the nonmagnetic LuB12 and decreases along the HoB12 – TmB12 sequence, as can be seen from Fig. 4.20. When lowering the temperature, the thermopower of all magnetic dodecaborides changes sign. The authors [56, 103] noted that the different signs of thermopower and Hall effect observed at low temperatures for magnetic dodecaborides can be ascribed to the peculiarities of strong quasiparticle interactions similar to the canonical heavy-

Magnetic Dodecaborides RB12 (R = Tb, Dy, Ho, Er, Tm) and the Solid Solutions Rx Lu1−x B12

10

100 2

LuB12

HoB12

ErB12

TmB12

1

-1 2

S (PV/K)

-2

S (PV/K)

0

-3

1 S

TN 0

-4 TN

TN

TN

-5

-1

-6 2

3

4

5

6

7

8

Figure 4.20 Temperature dependences of Seebeck coefficient S(T ) for RB12 compounds (R = Ho, Er, Tm, Lu). The vicinity of magnetic transitions is presented in the inset on larger scale. Reproduced from Ref. [56].

fermion system CeB6 [104]. The AFM transition in RB12 is accompanied by the appearance of a large positive contribution to the Seebeck coefficient, preceded by the positive spin-polaron component (Fig. 4.20) resulting from short-range order effects. It was noted in Refs. [56, 103] that the change in the 4f -shell population in the sequence HoB12 – LuB12 starting from HoB12 (4f n , n = 10) induces a significant decrease in the charge-carrier mobility and in the amplitude of the Seebeck coefficient at T > T ∗ ≈ 60 K, as both parameters depend monotonically on the filling number n in the range 10 < n < 13 [105]. This behavior of the transport characteristics evidently contradicts the drastic reduction of the de Gennes factor (g − 1)2 J (J + 1), where J is the total angular momentum of the 4f shell, which is generally used to characterize the magnetic scattering of charge carriers. This unusual tendency

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368 Magnetism, Quantum Criticality, and Metal–Insulator Transitions in RB12

seems to be connected with the enhancement of spin fluctuations in the magnetic RB12 compounds that will be discussed in detail in the next section.

4.4.4 Thermal Conductivity The thermal conductivity values for magnetic dodecaborides RB12 (R = Dy, Ho, Er, Tm) are quite close to λ(LuB12 ) in the intermediate temperature range (150–300 K) [52, 65], in accordance with the development of the fcc lattice instability that leads to a strong increase of the vibrational DOS in approach to the Ioffe–Regel regime of the lattice dynamics near TE ≈ 150 K [1]. On the contrary, a very strong (∼10 times) suppression of λ(T ) is observed in these magnetic compounds at low temperatures [52]. Besides, the authors note that the main feature of all the magnetic dodecaborides is a sharp decrease in the thermal conductivity within the AFM state in the vicinity of the antiferromagnet-paramagnet (AF-P) phase boundary as the magnon heat conductivity near the magnetic phase transition falls to a negligibly small value [52, 65]. They concluded

Figure 4.21 Temperature dependences of the thermal expansion coefficient for RB12 . Inset: low-temperature part of α(T ) for ErB12 [3, 62].

Magnetic Dodecaborides RB12 (R = Tb, Dy, Ho, Er, Tm) and the Solid Solutions Rx Lu1−x B12

also that the electronic thermal conductivity λe dominates over the phonon component λph with the ratio λe /λph ≈ 2–3 within the temperature range of 4–300 K [65].

4.4.5 Thermal Expansion and Heat Capacity Temperature dependences of the linear thermal expansion coefficient α of HoB12 , ErB12 , and TmB12 are shown in Fig. 4.21 [3, 62]. The behavior of α(T ) with two minima at low and intermediate temperatures is quite similar to that one observed for LuB12 . It is worth noting also that one of the α(T ) inflection points in RB12 is detected near the cage-glass transition temperature T ∗ ∼ 60 K. In the ordered state of HoB12 and ErB12 , an additional anomaly in ´ temperature (see Fig. 4.21, the thermal expansion occurs at the Neel inset). Temperature dependences of the heat capacity of magnetic dodecaborides have been studied in Refs. [3, 62] (see Fig. 4.22). The

Figure 4.22 Temperature dependence of the heat capacity of the magnetic RE dodecaborides [3, 62].

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370 Magnetism, Quantum Criticality, and Metal–Insulator Transitions in RB12

most important features on the C (T ) curves are a discontinuity C at the AFM transition and the heat-capacity maximum below TN . In the ordered state, TbB12 also reveals two first-order metamagnetic transitions [106], whereas ErB12 shows a small anomaly of unclear origin below the N´eel temperature [3, 62]. In the paramagnetic region of HoB12 –TmB12 , a significant contribution from short-range correlations is observed in the wide vicinity of TN . A Schottky contribution to the specific heat above 40 K [3,62] has been analyzed in terms of the CEF splitting of the magnetic ground state. The observed behavior of the heat capacity in the critical region of all AFM dodecaborides is intrinsic for amplitude-modulated (AM) magnetic systems. Indeed, in the framework of the periodic field model (PFM) [107], C AM of a system ordered with an amplitudemodulated magnetic structure is equal to 2/3 of the corresponding

C EM expected for the structure with equal magnetic moments. In the absence of the crystal-field contribution, C AM is described by [108] 10 J (J + 1)

C AM = , R 3 2J 2 + 2J + 1

(4.6)

where J is the total angular momentum. The first excited CEF level in HoB12 , ErB12 and TmB12 is located too high in energy, so that practically only the ground state (51 triplet for HoB12 and TmB12 [3, 62, 109, 110], and 83 quartet for ErB12 [3, 62, 111]) is thermally populated at temperatures near TN . In this case, HoB12 and TmB12 may be characterized by an effective spin Seff = 1 and C AM = 11.09 J/(mol·K), ErB12 by Seff = 3/2 and C AM = 12.23 J/(mol·K), while the experimental C AM values are equal to 10.75, 11.36, and 12.14 J/(mol·K), respectively, in good agreement with the C (T ) measurements and the PFM estimation. The low-temperature specific heat of Hox Lu1−x B12 solid solutions with 0 < x ≤ 1 has been studied in the cage-glass state [97]. For the Ho0.01 Lu0.99 B12 composition in the regime of isolated magnetic impurities, the authors have found the specific heat anisotropy becoming as high as 15% in an applied magnetic field up to 9 T. The increase in specific heat for the magnetic field orientation H [001] has been attributed to the existence of dynamic charge stripes (i.e., high-frequency, ∼200 GHz, [43] oscillations of the inhomogeneous

Magnetic Dodecaborides RB12 (R = Tb, Dy, Ho, Er, Tm) and the Solid Solutions Rx Lu1−x B12

charge carrier density) along the direction [110] in the fcc lattice of dodecaborides.

4.4.6 Magnetic Structure The amplitude-modulated magnetic structures in TbB12 [106], HoB12 [93, 94, 112, 113], ErB12 [112], and TmB12 [109, 112] have been confirmed by elastic neutron scattering. The structure can be described as a modulation of magnetic moments that have parallel orientation within the 111 sheets and antiparallel orientation between the neighboring sheets, which propagates along the three crystallographic directions. In the case of the terbium compound, the modulation persists well below the lower first-order transition. In this temperature range, the two magnetic phases, described by the same type of the propagation vector q = ( 12 ±δ, 12 ±δ, 12 ±δ) with different incommensurabilities δ1 = 0.022 and δ2 = 0.059, coexist. The phase transition at 14.6 K is due to the disappearance of an additional magnetic phase and the simultaneous rearrangement of the δcomponents of the q vector describing the sine-wave modulation of magnetic moments. Even the most careful examination of the neutron data revealed no difference in the temperature range where the second (T2 = 18.2 K) first-order transition was observed in specific heat. The authors [3, 62] concluded that, most likely, TbB12 preserves the sine-wave ordering down to 0 K. Therefore, the CEF ground state has to be a nonmagnetic singlet. In fact, the 2 singlet ground state has been proposed also from an analysis of the Schottky heat capacity [3, 62]. On the contrary, in HoB12 , ErB12 , and TmB12 , the CEF ground state is the magnetic one [3, 62, 109–111]. In this case, due to entropy effects associated with the modulation of the magneticmoment amplitude, the magnetic system should either (i) suddenly jump through a first-order transition from the amplitude-modulated to an equal-magnetic-moment structure, or (ii) evolve to an antiphase structure at T = 0 through a progressive squaring up of the modulation. This latter process should be accompanied by the growth of higher-order harmonics of the propagation vector.

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372 Magnetism, Quantum Criticality, and Metal–Insulator Transitions in RB12

Powder neutron diffraction on HoB12 in zero magnetic field revealed an incommensurate amplitude-modulated magnetic structure below TN with the basic reflections that can be indexed by q = ( 21 ±δ, 12 ±δ, 12 ±δ) with δ = 0.035 [93]. The data presented in Ref. [93] show three phases in an applied field. The authors [93] showed that the two AF1 and AF2 phases can be characterized by the same incommensurate ordering vector, and the difference between them is in the ferromagnetic moment that is present along the field direction in the AF2 phase, whereas in the zero-field AF1 phase a significant amount of magnetic moments remain disordered. The latter result supports an incommensurate amplitude-modulated structure and seems to be in qualitative agreement with the observation of Kalvius et al. [75] using μSR. Indeed, according to the μSR experiments, the mean local magnetic field and its distribution width are approximately of the same magnitude, indicating a significant spin disorder on the short-range scale (≤ 5 lattice constants) in ErB12 . Assuming that the spin disorder is predominantly directional, the μSR-data point toward a complex amplitude-modulated spin structure with a disordered component [75]. The magnetic structure of Ho11 B12 , Er11 B12 and Tm11 B12 has been investigated by neutron diffraction on isotopically pure single crystals [112]. Results in zero field as well as in magnetic fields up to 5 T reveal modulated incommensurate magnetic structures in these compounds with the basic reflections indexed with q = ( 21 ±δ, 12 ±δ, 1 ± δ), where δ = 0.035 both for HoB12 and TmB12 , and with q = 2 ( 32 ±δ, 12 ±δ, 12 ±δ), where δ = 0.035 for ErB12 . The authors suggested that the formation of these structures can be understood taking into account the interplay between the direct RKKY interaction, which favors AFM ordering with qAFM = ( 12 21 21 ), and the dipole– dipole interaction that splits this basic propagation vector [112]. As long as the amplitude-modulated magnetic structure is not stable, a tendency towards squaring upon decreasing temperature is expected in these RB12 compounds with a magnetic CEF ground state, so that the different moment values (magnitudes) of the magnetic structure get compensated. In this case, harmonics of basic reflections should appear, and the 3rd harmonic satellites were observed in HoB12 indicating that there is no “simple” amplitude

Magnetic Dodecaborides RB12 (R = Tb, Dy, Ho, Er, Tm) and the Solid Solutions Rx Lu1−x B12

Figure 4.23 Diffuse neutron-scattering patterns measured on HoB12 at 7.8 and 24 K above TN = 7.4 K [94, 113, 114].

modulation, and that likely a multi-q structure of the type ±( 12 −δ, 1 −δ, 12 −δ) has to be considered [112]. With increasing magnetic 2 field up to 5 T, the basic propagation vector ( 12−δ, 12−δ, 12−δ) remains in the higher-magnetic-field phase AF2 with the ferromagnetic component [112]. In ErB12 only AFM reflections of the type q = ( 32 ±δ, 12 ±δ, 12 ±δ) have been observed and explained in terms of the possible changes in the crystal structure (symmetry breaking from fcc to tetragonal) well below TN [112]. A more detailed investigation of the magnetic structure was undertaken for HoB12 [94] below and above the N´eel temperature. By studying the harmonics to the basic qAFM vector along the (111) direction, the authors concluded against the multi-domain single-q structure and argued in favor of the quadruple-q magnetic ordering at least in zero field. It was shown in Ref. [94] that in the fieldinduced AFM phase the quadruple-q structure changes to a double-q structure with a ferromagnetic component, which coexists with the quadruple-q order. Figure 4.23 shows the diffuse neutron-scattering patterns observed in HoB12 above TN [94, 112]. As follows from these patterns, a strong modulation of diffuse scattering appears at former magnetic reflections, e.g., at ( 32 23 23 ) but not at the typical crystal reflections. This fact points to strong correlations between the magnetic moments of Ho3+ ions. The scattering patterns observed in Refs. [94,112] can be explained by the appearance of correlated one-

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374 Magnetism, Quantum Criticality, and Metal–Insulator Transitions in RB12

Figure 4.24 Correlation length versus temperature above TN = 7.4 K along ¯ and [311]. The with two mutually perpendicular crystal directions: [23¯ 3] data are derived by numerical integration of the diffuse scattering along the directions indicated. The straight line indicates a T −1 behavior. For the [311] direction, a fit to (T −TN )−1/4 was used to construct the curved line. Adapted from Siemensmeyer et al. [94].

dimensional spin chains (short chains of Ho3+ -ion moments placed on space diagonals of the elementary unit), similar to those in lowdimensional magnets [115]. They can be resolved both well above ´ temperature, where the 1D TN (up to 70 K) and below the Neel chains seem to condense into the ordered AFM structure [94, 112, 113]. Thus, the direct interpretation of the results of diffuse scattering in HoB12 is based on the fact that the AFM correlation length along the 011 and 100 directions is much shorter than along 111, and it demonstrates a quite different temperature dependence along 111 and transverse to this direction (see Fig. 4.24 and Ref. [94] for more details). There is no diffuse signal at reflections of the crystal structure above TN , thus there are no ferromagnetic correlations. The authors [94] discussed the following scenario for the occurrence of long-range order in HoB12 : Far away from TN , strong interactions lead to correlations along 111, they are

Magnetic Dodecaborides RB12 (R = Tb, Dy, Ho, Er, Tm) and the Solid Solutions Rx Lu1−x B12

essentially one-dimensional and would not lead to long-range order at finite temperature. As TN is approached, the 1D-correlated regions grow in the perpendicular directions (Fig. 4.24), possibly due to other interactions. One may imagine cigar-shaped AFM-correlated regions that become more spherical when TN is approached. Within this picture, the ordering temperature is determined by the point where spherical symmetry is reached. Only then 3D behavior sets in, and the system exhibits long-range AFM order.

4.4.7 Magnetic H –T –φ Phase Diagrams Results of the thermodynamic and charge-transport experiments were used to construct the H –T phase diagrams of RB12 (R = Ho, Er, and Tm) [3,36,62,84,93,94,98,116–118] with three AFM phases in an external magnetic field applied along the three principal directions in the fcc lattice, H 001, H 110, and H 111 (see, for example, Fig. 4.25 for ErB12 [36, 62]). Scans at various temperatures in magnetic fields between 0 and 5 T were carried out in a neutron diffraction study of HoB12 [94] in order to test whether there are indications for phase boundaries between the AFM phases. It was concluded [94] that the principal reflections remain; i.e., no change in the AFM propagation vectors in a low magnetic field was observed. Also, it was deduced [94] that the changes in the neutronscattering data of HoB12 , both as a function of temperature and field, correlate well with the location of the phase boundaries found in the specific-heat and magnetization studies, thus confirming the phase boundaries inferred from these experiments. Much more detailed and complex magnetic phase diagrams resulted from the higher-precision studies of specific heat, magnetization, and MR of the mono-domain single crystals of HoB12 [98], Ho0.8 Lu0.2 B12 [87] and Ho0.5 Lu0.5 B12 [99, 119, 120]. Among the number of magnetic phases in the AFM state of Hox Lu1−x B12 , only the low-field one (phase I in Fig. 4.26) is common for all three principal directions of magnetic field: H 001, H 110, and H 111, but each of these directions has its own individual set of magnetically ordered states in higher magnetic fields [87, 119, 120]. In these measurements, summarized for Ho0.8 Lu0.2 B12 in Fig. 4.27,

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376 Magnetism, Quantum Criticality, and Metal–Insulator Transitions in RB12

Figure 4.25 Magnetic H –T phase diagrams of ErB12 for the field applied along three principal directions in the fcc lattice. Reproduced from Refs. [36, 62].

Magnetic Dodecaborides RB12 (R = Tb, Dy, Ho, Er, Tm) and the Solid Solutions Rx Lu1−x B12

Figure 4.26 Magnetic H –T phase diagrams of Ho0.5 Lu0.5 B12 (left), Ho0.8 Lu0.2 B12 (middle), and HoB12 (right) for three principal field directions in the fcc lattice [87, 119, 120].

the authors detected sharp radial phase boundaries separating three main sectors φ001 , φ110 , and φ111 in the angular phase diagrams (see Fig. 4.28). It was shown [87, 119, 120] that the boundaries form a Maltese cross in the (110) plane (see Fig. 4.28), implying that the magnetic phases corresponding to these sectors differ significantly from each other in their magnetic structure. Such a complicated and low-symmetry magnetic H –T –φ diagrams are unusual and unexpected for a RE dodecaboride with the fcc crystal structure. When discussing the nature of the observed dramatic symmetry lowering, the authors [87,119,120] noted that significant MR changes are observed already in the paramagnetic phase, as one can see, for example, in Fig. 4.27(d) for Ho0.8 Lu0.2 B12 , T0 = 6.5 K. They argue that numerous magnetic phases observed below TN and a lot of phase transitions in the H –φ–T phase diagram in the AFM state are the result of two factors: (i) formation of

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378 Magnetism, Quantum Criticality, and Metal–Insulator Transitions in RB12

Figure 4.27 3D view of magnetoresistance dependences of Ho0.8 Lu0.2 B12 on magnetic field (a) T = 2.1 K, (b) T = 4.2 K and (d) T = 6.5 K, and on temperature (c) at H = 25 kOe. The rotation axis is I [110], the three principal cubic directions in the [110] plane are shown by arrows. Reproduced from Ref. [87].

the dynamic charge stripes along the 110 axis in the matrix of dodecaborides which leads to uniaxial scattering anisotropy in the paramagnetic phase and (ii) the magnetic ordering due to indirect AFM exchange interaction mediated by the conduction electrons according to the RKKY-mechanism, see Fig. 4.29. The conclusion is also confirmed by the results of more detailed studies [121], where it was shown that the MR in the paramagnetic phase of Ho0.8 Lu0.2 B12 can be represented as a sum of (i) isotropic negative component associated with charge-carrier scattering on magnetic clusters of Ho3+ ions (nanometer-size domains with AFM shortrange order) and (ii) anisotropic positive contribution which is due to the cooperative dynamic JT effect in the boron sublattice. As one can see from Fig. 4.28, the Maltese-cross anisotropy was detected for charge-carrier scattering, and two more MR mechanisms dominant in the AFM state were separated and analyzed quantitatively [87]. It was suggested [87, 98, 121] that the main positive linear in magnetic

Magnetic Dodecaborides RB12 (R = Tb, Dy, Ho, Er, Tm) and the Solid Solutions Rx Lu1−x B12

Figure 4.28 Color plots of magnetoresistance vs. magnitude (H ) and direction (φ) of magnetic field as it is rotated in the [110] plane for (a) HoB12 , (b) Ho0.8 Lu0.2 B12 , and (c) Ho0.5 Lu0.5 B12 at a temperature of 2.1 K. For comparison, panel (d) demonstrates the anisotropy of magnetoresistance in the paramagnetic state of Ho0.8 Lu0.2 B12 at T = 6.5 K. The phase boundaries (solid lines) are derived from the magnetoresistance data (see symbols in Fig. 4.26). Roman numerals are the same as in Fig. 4.26 and denote various AFM phases [87, 119, 120].

field contribution may be attributed to charge-carrier scattering on a spin density wave (SDW), i.e., the periodic modulation in the 5d-component of the magnetization density that contributes to the magnetic structure in addition to the more localized 4f moments, whereas the negative quadratic term ρ/ρ ∝ H 2 is due to the scattering on local 4f -5d spin fluctuations of the Ho3+ ions. These two (SDW and localized moments of magnetic ions) components in the AFM ordering in combination with dynamic charge stripes are the main factors responsible for the complexity of the magnetic phase diagrams with numerous phases and phase transitions [87, 120].

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380 Magnetism, Quantum Criticality, and Metal–Insulator Transitions in RB12

4.4.8 The Root of the Complexity of Magnetic Phase Diagrams of RB12 To summarize in more detail, in the cage-glass state of the RE magnetic dodecaborides at T < T ∗ ≈ 60 K [1], two additional factors appear: (i) the positional disorder in the arrangement of magnetic R 3+ ions in the B24 truncated cuboctahedra (static displacements of R 3+ ions in the DWP, see Fig. 4.17), which is accompanied by the formation of vibrationally coupled magnetic nanometer-size clusters in the RB12 matrix and (ii) the emergence of dynamic charge stripes (ac current with a frequency ∼200 GHz [43]) directed along the single [110] axis in RB12 , which accumulate a considerable part of nonequilibrium conduction electrons in the filamentary structure of fluctuating charges [79, 86]. Thus, due to the cooperative dynamic JT effect, when the B12 polyhedra are consistently distorted (ferrodistortive effect, see Fig. 3.5 in Chapter 3 for details), both static displacements of magnetic RE ions and 10 B−11 B substitutional disorder provide centers of pinning that facilitate the formation of additional ac conductive channels — the dynamic charge stripes. The positional disorder in the arrangement of magnetic R 3+ ions in B24 truncated cuboctahedra in the cageglass state (T < T ∗ ≈ 60 K) leads to a significant dispersion of exchange constants (through indirect exchange, RKKY mechanism) and formation of both nanometer-size domains of magnetic ions in the RB12 matrix (short-range AFM order effect above TN ) and creation of strong local 4f -5d spin fluctuations responsible for the polarization of 5d conduction band states (the spin-polaron effect). The last one produces spin polarized sub-nanometer-size ferromagnetic domains (“ferrons,” according to the terminology used in Refs. [122, 123]) resulting in the stabilization of SDW antinodes in the RB12 matrix. The spin-polarized 5d-component of the magnetic structure (ferrons) is on the one hand very sensitive to the external magnetic field, and, on the other hand, the applied field suppresses 4f -5d spin fluctuations by destroying the spin-flip scattering process. Moreover, along the direction of the dynamic charge stripes [110], the huge charge-carrier scattering destroys the indirect exchange between the nearest-neighbor localized magnetic moments of R 3+ ions (see Fig. 4.29) and renormalizes the RKKY

Magnetic Dodecaborides RB12 (R = Tb, Dy, Ho, Er, Tm) and the Solid Solutions Rx Lu1−x B12

Figure 4.29 (a) Crystal structure of RB12 . The color shows spin-density oscillations of conduction electrons that mediate the RKKY interactions, as shown in panel (b). (c) Dynamic charge stripes along the 110 direction destroy the indirect RKKY exchange between the nearest-neighbor (first NN) localized magnetic moments of the R 3+ ions. Courtesy of K. Krasikov.

interaction, accumulating a noticeable part of charge carriers in the filamentary electronic structure. Thus, the complex H –T –φ phase diagrams of RB12 antiferromagnets may be explained in terms of the formation of a composite magnetically ordered state of localized 4f moments of R 3+ -ions in combination with spin-polarized local areas of the 5d states — ferrons involved in the formation of SDW in the presence of a filamentary structure of dynamic charge stripes.

4.4.9 Quantum Critical Behavior in HoB12 When long-range order (in the case of HoB12 , magnetic order below TN ) is suppressed to zero temperature by tuning an external variable, such as pressure or magnetic field, the system is said to cross a QCP [124, 125]. The MR measurements were carried out at 0.5 K and in the magnetic field up to 12 T, revealing the QCP in HoB12 at Bc ≈ 8.2 T [113]. The analysis of obtained electrical resistivity dependences (using the ρ(T ) ∝ T x relation) at temperatures below 4.2 K shows that in magnetic fields close to Bc the electrical resistivity exhibits an unusual T 3 -dependence, where theoretical models expect a power law with x = 3/2 [124, 125]. For B ≥ 10 T a ρ(T ) ∝ T 2 dependence was observed [113], which points to the fact that in higher fields, the Fermi-liquid behavior is recovered.

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382 Magnetism, Quantum Criticality, and Metal–Insulator Transitions in RB12

4.5 Metal–Insulator Transition in YbB12 and Solid Solutions Ybx R1−x B12 (R = Lu, Tm) 4.5.1 Metal–Insulator Transition in YbB12 Yb valence YbB12 is an archetypal strongly correlated electron system that has been actively investigated since the 1980s as a fluctuating-valence semiconductor [4, 5, 26]. It was mentioned that the studies of fluctuating-valence Yb compounds are especially attractive because the single 4f hole of the Yb3+ ion is in the J = 7/2 ground state, separated from the J = 5/2 excited state by an energy splitting of 15 000 K, while the Yb2+ ions are in the 1 S0 nonmagnetic state, leading to a simple energy-level scheme similar to that of the Ce-based compounds [126]. It was found that at low temperatures YbB12 shows semiconducting-like and nonmagnetic (paramagnetic) properties similar to the fluctuating-valence compound SmB6 [5, ¨ 126]. At the same time, existing data on Mossbauer effect of YbB12 suggest that the Yb ion is nearly trivalent at 4.2 K [127]. Moreover, as a bulk-sensitive technique, the Yb LIII (2 p3/2 ) edge x-ray absorption spectroscopy (XAS) has been performed, and the Yb valence is estimated to be very close to 3.0 (larger than 2.95) at 20 K [27]. The Yb valence v(Yb) = 2.92–2.93 has been estimated in YbB12 at room temperature both by a bulk-sensitive photoelectron spectroscopy (PES) [128] and hard x-ray PES (HARPES) [50, 129]. The detected v(Yb) values show a moderate decrease to 2.9 on cooling down to 20 K [50, 128, 129]. Thus, YbB12 in which the Yb ion is nearly in a trivalent state and in the so-called Kondo regime, can be considered to be a fluctuating-valence material characterized by the formation of a gap at the Fermi energy [126]. Charge transport As the temperature decreases from 300 K down to 15 K, resistivity ρ increases exponentially with the activation energy E g /2kB ≈ 62 K. With further cooling, ρ increases with a smaller activation energy E p /kB ≈ 28 K and becomes nearly temperature independent upon subsequent temperature lowering [126]. Similar values E g /2kB ≈ 68 K and E p /kB ≈ 25 K were estimated in studies of high-quality YbB12 single crystals, where ρ(T ) increased by five orders of magnitude with the temperature lowering from 300 to

Metal–Insulator Transition in YbB12 and Solid Solutions Ybx R1−x B12 (R = Lu, Tm) 383

1.3 K [4]. The authors also found two intervals 15 K < T < 40 K and 7 K < T < 15 K of the Arrhenius-type behavior in the Hall coefficient measurements with the activation energies E g /2kB ≈ 90 K and E p /kB ≈ 28 K, respectively, which have been attributed to the opening of indirect charge gap (E g ) and to the charge transport through the intra-gap states (E p ). The mobility μH of the single-band conduction electrons was estimated to be as small as 3 and 10 cm2 /(V·s) at 300 and 1.7 K, respectively, and the μH (T ) dependence demonstrates the maximum values of ∼50 cm2 /(V·s) near the temperature T ≈ 15 K, separating these two activation intervals [126]. The temperature and field dependences of longitudinal MR of YbB12 have been studied in Ref. [130] for three principal directions in the cubic lattice. At 1.5 K, MR in all these directions shows a large negative value up to about 70% with an increase of magnetic field up to 15 T. With increasing temperature, the negative MR decreases gradually and becomes zero at 70 K for 110 and 111. Only for

100, the positive MR appears most remarkably in the range 5– 15 K, where the smaller energy gap appears in the resistivity and the strong MR anisotropy persists at least up to 70 K [130]. A similar anisotropic contribution in positive MR of YbB12 at 88 mK was reported by Kawasaki et al. [131]. Taking into account that the 4f -state of Yb ion is assumed to be the 8 quartet, and the wave function of these 4f -electrons is elongated in the 100 direction, Iga et al. [132] have attributed the MR anisotropy to the anisotropic distribution of the 4f electron density. At the same time, it is worth noting the similarity with the largest positive MR in the direction H 100 discussed previously in this chapter both for LuB12 and the magnetic RB12 RE compounds. The analogy allows us to seek a general explanation of the MR anisotropy in terms of the formation of dynamic charge stripes in the RE dodecaborides. The Seebeck coefficient S(T ) exhibits a giant negative peak of ∼140 μV/K at 10 K in addition to the second peak at 35 K and a shoulder at the temperature slightly above 100 K. Between 40 and 80 K, S(T ) demonstrates semiconducting behavior, being proportional to 1/T [133]. According to an estimation in the framework of the single-impurity Anderson model [134], it was concluded [133] that the maximum temperature T Smax ≈ 35 K can

384 Magnetism, Quantum Criticality, and Metal–Insulator Transitions in RB12

be related to the Kondo (or spin fluctuation) temperature Tsf = 3T Smax ≈ 100 K. Magnetic properties The high-temperature susceptibility χ(T ) obeys the Curie–Weiss law in the range 150–1200 K [4, 91, 126], and the effective Bohr magneton number μeff was evaluated to be 4.3–4.4 μB , which corresponds to 2.9–2.94 for the Yb valence (the calculated value is 4.54 μB for Yb3+ ). The large negative values of the paramagnetic Curie temperature p between −112 and −135 K [4, 91] indicate the AFM exchange interaction between Yb ions, and it is ascribed to the spin-fluctuation effects (Tsf ≈ p ) because the RKKY interaction in YbB12 is only of the order of 1 K [91]. As the temperature decreases, χ(T ) demonstrates a pronounced peak at 75 K and then decreases rather rapidly corresponding to the magnetic gap opening in YbB12 [126]. Below 20 K a slight upturn is observed in the χ(T ) curves of single crystals, which is usually discussed in terms of a small amount of isolated Yb3+ impurity ions in the matrix of YbB12 . Nuclear magnetic resonance To separate the impurity component in the susceptibility, it is important to investigate the intrinsic χ(T ) at low temperatures. For this purpose, nuclear magnetic resonance (NMR) Knight-shift measurements on 11 B [126] and 171 Yb [37] have been performed. The local symmetry of boron sites in RB12 is orthorhombic mm2. However, the line shape observed in the 11 B NMR was found to have an essentially axial symmetry [126]. The authors analyzed the Knight shift in terms of the isotropic Kiso and axial (anisotropic) Kax parts. It was found that the proportionality between χ(T ) and Kax (T ) exists in the temperature range 30–300 K; the 11 B hyperfine coupling constant A ≈ 650 Oe/μB has been estimated and attributed to the dipolar field at 11 B sites which comes from the induced dipole moment on Yb ions [126]. The authors noted that the constant value of Kax below 20 K is the characteristic of the nonmagnetic ground state of the fluctuatingvalence compound. The large isotropic Knight shift of about 66% was observed below 10 K in studies of the 171 Yb NMR on an YbB12 single crystal [37]. The detected hyperfine coupling constant of 1150 kOe/μB agrees very well with the calculated value for the

Metal–Insulator Transition in YbB12 and Solid Solutions Ybx R1−x B12 (R = Lu, Tm) 385

J = 7/2 state of free Yb3+ ions, indicating that the magnetic susceptibility in the low-temperature limit is dominated by the Van Vleck contribution within the J = 7/2 multiplet. The presence of an activation spin gap in YbB12 was clearly established from the NMR measurements of the 10 B and 11 B nuclear spin-lattice relaxation rate, 10,11 (1/T1 ) [126, 135]. It was shown in [126, 135] that 11 (1/T1 ) drastically decreases below 80 K, which was explained by an excitation gap in the DOS at the Fermi level of about 80–100 K, depending on the external magnetic field. The 11 (1/T1 ), however, has a minimum value near 15 K which becomes deeper and moves to lower temperature when the field increases up to 16 T [37, 135]. The ratio of (1/T1 ) for 11 B and 10 B nuclei indicates that the anomaly below 15 K is caused by dilute paramagnetic impurities assisted by nuclear spin diffusion. It was argued that the minimum in (1/T1 ) suggests that the impurity moments are created only after most of the bulk magnetic excitations die away to make a stable singlet spin-gap state. This suggests that the origin of the moments is likely to be associated with defects of magnetic states or nonstoichiometry of conduction electrons as opposed to the extrinsic magnetic ions such as Gd [135]. No anomaly below 15 K was observed at the Yb sites. The absence of such process for Yb nuclei can be explained by the strong hyperfine coupling of Yb nuclei to the 4f electrons [37], since the resonance frequency of Yb nuclei near impurities should be largely shifted, and the mutual spin-flip process will be prohibited [135]. Moreover, the nuclear spin-lattice relaxation rate at the Yb sites shows an activated temperature dependence below 15 K with the activation energy of 87 K, which is completely different from the behavior at the B sites [37, 135]. The authors noted that the recovery of the nuclear magnetization is not strictly exponential but shows a fast relaxing component with small amplitude, indicating inhomogeneous distribution of 171 (1/T1 ). The appearance of the full gap E g activation in the spin-lattice relaxation on Yb sites in the temperature interval T < 15 K, where the charge transport through the intra-gap states is observed, cannot be explained in terms of the hybridization gap scenario. It will be shown below that, on the contrary, the formation of vibrationally coupled Yb–Yb pairs

386 Magnetism, Quantum Criticality, and Metal–Insulator Transitions in RB12

should be considered as a natural interpretation for this kind of behavior. Electron paramagnetic resonance The EPR has been studied on Gd ions as spin markers inserted in YbB12 . In the framework of the exciton dielectric model, Altshuler et al. [136] determined the temperature dependence of the spin gap E g ≈ 12 meV, which is nearly temperature independent below 40 K and then decreases and disappears at ∼115 K. Additional singularity in the temperature behavior of the EPR linewidth was found at 13–14 K, pointing to the existence of intra-gap states with a finite electron density [136]. It was suggested that the intra-gap states may be due to manybody excitations [136]. In the follow-up work [137], an EPR signal from Yb ions with an integer valence of 3+ was detected in the single crystals of YbB12 intermediate-valence compound at LHe temperatures. The authors [137] observed two main lines symmetrically displaced from a g value of 2.55 and exhibiting a modulation with 5% anisotropy when the magnetic field was rotated in the [110] plane from the cubic axis to the [110] direction. The EPR spectra have been explained by the existence of Yb–Yb pairs which are coupled by the isotropic exchange but interact also with the other pairs by dipole and exchange coupling. The concentration of Yb3+ ions in these pairs was estimated to be 0.2–0.5%, and the origin of the EPR signal has been attributed to the effect of defects and/or vacancies stabilizing the valence of Yb ions [137]. Taking into account dipole–dipole splitting of the resonance lines, the authors have estimated the distance between the interacting ˚ It was suggested that ap is the average Yb pairs, ap ≈ 9.1 A. interpair distance, and pairs are distributed randomly so that their interaction energy fluctuates in space, contributing to fluctuations of the local CEF and resulting in an inhomogeneous broadening of the EPR line [137]. It was noted especially that the formal description of the EPR results points to the alignment of all Yb pairs, i.e., the appearance of a spontaneously chosen direction in the otherwise cubic crystal, and this fact is equivalent to the existence of a sort of phase transition induced by the interpair coupling. The occurrence of a slight anisotropy in a cubic semiconductor should be attributed to a spontaneous symmetry breaking which, according to Altshuler

Metal–Insulator Transition in YbB12 and Solid Solutions Ybx R1−x B12 (R = Lu, Tm) 387

et al. [137], is specific to the ground state of a Kondo insulator. A strong temperature dependence of the EPR amplitude was found on cooling in the temperature range of 1.6–4.2 K [137]. The finding was interpreted in terms of the capture of electrons by Yb3+ ions from electron traps with a binding energy of 18 K, which transforms the Yb centers into EPR-inactive ions with a fluctuating valence. μSR spectroscopy Positive-muon spectroscopy (μSR) and 170 Yb ¨ Mossbauer absorption measurements have been performed on the cubic Kondo insulator YbB12 down to 50 mK in temperature. Yaouanc et al. [138] observed a paramagnetic fluctuation mode at low temperature, with a weak moment amplitude (∼10−2 μB ) and a slow fluctuation frequency (∼60 MHz) which remains constant between 0.04 and 4.2 K. No indication of a magnetic phase transition was found down to 0.04 K. It was argued [138] that the smallness of the observed moments points to an itinerant spin picture within the strongly hybridized 4f band. The μSR data therefore imply that at a temperature of a few tens of mK, excitations with almost zero energy (100 neV ≈ 1 mK) are possible within the narrow band at the Fermi level, i.e., they suggest that the gap in YbB12 (almost) closes along some direction qc in q-space, for which χ(qc ) remains paramagnetic [138]. μSR spectra of the single crystals of Yb1−x Lux B12 (x = 0, 0.125, 0.5, 1) and ErB12 were measured between 1.8 and 300 K [74, 75]. The authors found similar spectral shapes in all magnetic and nonmagnetic compounds studied over the whole temperature range, which excludes any contribution from Yb magnetic moments. In zero field the spectra alter their appearance around 20, 100 and 150 K. In a longitudinal field of 100 Oe, which largely suppresses the contribution from 11 B nuclear moments, the μSR relaxation rate remained constant up to ∼150 K, where it suddenly peaks. It was concluded that the spectral shape is determined by the field of the 11 B nuclear moments. These fields show dynamic behavior, but nuclear spin relaxation can be excluded, because its rate is below the μSR time window. It is suggested that the dynamical features arise from atomic motions within the B12 clusters [74, 75].

388 Magnetism, Quantum Criticality, and Metal–Insulator Transitions in RB12

Neutron scattering Magnetic excitations in YbB12 have been studied by neutron scattering [111, 139–142]. According to the INS experiments [142], three peaks M1, M2, and M3 appear at low temperatures, T < 60 K, showing a significant dispersion. Both M1 (14.1 meV, L point) and M2 (17.9 meV, L point) correspond to an excitation from the Kondo singlet state to the magnetic state with 8 symmetry [142]. M1 is affected by the AFM interaction between Yb ions and is regarded as an intra-gap exciton-like mode [142]. A significant 2D character of AFM short-range correlations has been found in YbB12 [141, 142]. In particular, the correlation lengths that correspond to the couplings within (ξ ) and between (ξ⊥ ) the (001) planes were estimated in Ref. [142], and the values ξ = 5.4 ± 1.4 A˚ and ξ = 3.4±1.1 A˚ have been obtained. The spin gap was evaluated by the lower edge of the M2 peak, and it is close to the M1 peak position. It is worth noting that the spin gap in YbB12 was found to be around 15 meV (for more details, see Chapter 6), which is nearly the same as the charge gap. Thermal conductivity and specific heat Thermal conductivity κ(T ) shows a pronounced increase below 60 K and exhibits a sharp peak at 15 K [133]. This behavior below 60 K is a result of a strong increase in the phonon mean free path due to the gap formation in the electronic DOS. The electronic contribution to the thermal conductivity, κel , was estimated from the κ(T ) data by using the Wiedemann–Franz law. It was shown that κel is less than 10% of the total value even at 100 K [133]. There is no second anomaly below the κ(T ) peak, which indicates that there is no thermal activation corresponding to E p observed in ρ(T ) below 15 K. The heat capacity C (T ) of YbB12 single crystals has been studied for the first time by Iga et al. [41]. To estimate the magnetic contribution C m , the authors used as a phonon heat-capacity reference the C (T ) dependence of the isostructural nonmagnetic compound LuB12 , taking into account a small difference in the Debye temperatures D between the two dodecaborides. The Schottky anomaly in C m (T ) was detected with an activation energy E g /kB ≈ 170 K [41]. It was shown that the magnetic entropy in YbB12 is close to the value of R ln 4 (per Yb ion) near room temperature. It was mentioned that this result indicates the existence of four nearly

Metal–Insulator Transition in YbB12 and Solid Solutions Ybx R1−x B12 (R = Lu, Tm) 389

generate states up to 300 K irrespective of the presence of the energy gap. If the fourfold degenerate state originates from the CEF splitting of Yb3+ in a cubic symmetry, this state should be either the 8 quartet or two closely located doublets 6 and 7 [41]. It is worth noting that the finding contradicts the CEF splitting of the J = 7/2 multiplet of the Yb3+ ion. Indeed, the overall splitting of 11.2 meV (∼130 K) estimated previously from the INS experiments [110, 111] suggests that both the 6 and 7 excited doublets should contribute also to the entropy at room temperature. Optical properties Measurements of the optical reflectivity, R(ω), were conducted on single crystals of YbB12 at various temperatures [44, 143, 144] in order to obtain their optical conductivity, σ (ω). Similar to LuB12 , the reflectivity spectra of YbB12 have a clear plasma cutoff (ωp ) near 1.6 eV and a sharp structure above 4 eV due to interband transitions [78]. The optical conductivity spectrum of YbB12 clearly showed an energy gap formation below 80 K. The gap development involved a progressive depletion of σ (ω) below a shoulder at ∼40 meV. In addition, the authors observed a strong midinfrared (mIR) absorption in σ (ω) peaked at ∼0.25 eV, which was also strongly temperature dependent [44]. The temperature- and photon-energy ranges of the experiment have been extended from T = 20–290 K and ≥ 7 meV in Ref. [44] to T = 8–690 K and ≥ 1.3 meV in Ref. [143]. The obtained σ (ω) reveals the entire evolution of the electronic structure from the metallic to semiconducting behavior with the temperature lowering, as illustrated in Fig. 4.30. Below 20 K, σ (ω) has revealed a clear onset at 15 meV, which was identified as the energy gap width. The energy of 15 meV agrees well with the gap widths obtained by other experimental techniques [4, 145, 146]. The authors concluded that the observed energy gap of 15 meV in σ (ω) arises from an indirect gap, predicted by the band model of the Kondo semiconductor; the feature at 0.2–0.25 eV was ascribed to a direct gap [143]. By employing the terahertz quasioptical technique [147], Gorshunov et al. performed reliable reflectivity measurements of YbB12 down to the frequency of 8 cm−1 [45], as shown in Fig. 4.31. In addition to the reflection experiments, direct measurements of the GHz and THz spectra of the dynamical conductivity σ (ω)

390 Magnetism, Quantum Criticality, and Metal–Insulator Transitions in RB12

1.00

1.00 690 K 500 K 295 K 200 K 80 K 8K

YbB12 R(ω)

0.90

σ(ω) [104 Ω-1cm-1)

(a) 8K 80 K 200 K 295 K 400 K 500 K 600 K 690 K

1.5 690 K

1.0

(b)

0.5 0

8K

0

0.2

0.4

Photon Energy (eV)

0.6

8K 20 K

6 4

YbB12 80 K 100 K 120 K

0.90 0.85

0.85

50 K 60 K 70 K

0.95

σ(ω) [103 Ω-1cm-1]

R(ω)

0.95

100 K 80 K

40 K 30 K

(c) (d)

120 K

70 K 60 K 50 K

2

40 K 30 K 20 K

0

0

8K

20

40

60

Photon Energy (meV)

Figure 4.30 (a) Optical reflectivity R(ω) and (b) conductivity σ (ω) of YbB12 between 8 and 690 K. (c, d) R(ω) and σ (ω), respectively, below 120 K in the low-energy region. The blue arrow in (c) indicates the hump in R(ω), and the black and red arrows in (d) indicate, respectively, the shoulder (at 40 meV) and the onset (at 15 meV) in σ (ω). The broken curves are the extrapolations. Reproduced from Ref. [143].

and dielectric constant (ω) have been performed at the lowest temperature of 5 K by measuring the complex transmissivity of a thin (∼20 μm) crystal [45]. The progress in the lowestfrequency measurements at low temperatures allowed the authors to perform a quantitative analysis of the charge-carrier parameters in dependence on temperature in order to explain the electronic properties of YbB12 . Assuming a single band with one type of carriers, they used the Hall data obtained on crystals from the same batch in order to evaluate the carrier concentration n = 1/(ec RH ). As a result, the plasma frequency ωp = 2π(ne2 /πm∗ )1/2 , the effective mass m∗ = ne2 /(ωp /2π)2 and the mobility of the charge carriers μ = e/(2πm∗ γ ) (where γ is the relaxation frequency) have been estimated [45], see Fig. 4.32. It is seen from Fig. 4.32 that upon cooling from 300 to 100 K, the effective mass m∗ (T ) increases fivefold, while the scattering rate γ simultaneously decreases by a similar factor. This behavior is characteristic of a phonon-

Metal–Insulator Transition in YbB12 and Solid Solutions Ybx R1−x B12 (R = Lu, Tm) 391

Figure 4.31 Spectra of reflectivity, conductivity and dielectric constant of YbB12 at different temperatures, reproduced from Gorshunov et al. [45]. The absorption peak at 22 cm−1 seen at T = 5 K in the conductivity spectrum was recorded by measuring complex transmissivity of a thin (∼20 μm) sample; the solid line shows a Lorentzian fit with the eigenfrequency v0 = 22 cm−1 , dielectric contribution  = 75 and damping γ = 15 cm−1 . Inset: broadband reflectivity of YbB12 showing two plasma edges at low (heavy quasiparticles) and high (unscreened electrons) frequencies. The dashed line corresponds to the data of Okamura et al. [44].

assisted scattering and contradicts the simple Kondo mechanism for which one would expect an increase in magnetic scattering upon cooling [148], which should suppress the mobility, while the latter is almost constant at these temperatures (Fig. 4.32). Also the temperature-dependent resistivity of YbB12 , which shows an

392 Magnetism, Quantum Criticality, and Metal–Insulator Transitions in RB12

nHall, 1021 cm–3 16 K

100

20 m*/mb

10–2

0

0.08

γ, cm–1

10

0

0.16 T –1, K–1

νp /103 T*

4

γ

τ, 10–13 s

μ

100 2 YbB12 10

20

τ 50 100 T, K

300

0

Figure 4.32 Temperature dependences of charge-carrier parameters in YbB12 . Upper panel: mobility μ in cm2 V−1 s−1 , effective mass m∗ /mb (mb = 2.85m0 ), plasma frequency vpl in cm−1 . The inset shows an Arrhenius plot of the Hall concentration of charge carriers. Lower panel: scattering rate γ and relaxation time τ . T ∗ ≈ 70 K indicates the crossover temperature. Reproduced from Gorshunov et al. [45, 150].

activated behavior rather than a Kondo-type dependence ρ ∝ − log T, is in contrast to the Kondo scenario. A possible reason for these observations is the presence of a gap in the DOS of YbB12 at temperatures above 100 K. Indications of a gap at T > 100 K are also seen in photoemission experiments [149]. At temperatures below T ∗ ≈ 70 K, the gap opens in the DOS at E F , which produces a shoulder in the σ (v) spectrum at 40 meV (Figs. 4.30, 4.31) and leads to the above-mentioned anomalies in

Metal–Insulator Transition in YbB12 and Solid Solutions Ybx R1−x B12 (R = Lu, Tm) 393

the transport, magnetic, and thermodynamic properties. At T ∗ , an abrupt change in the charge-carrier characteristics is observed: The scattering rate starts to follow the Fermi-liquid-type dependence γ (T ) ∝ T 2 [151], and the effective mass starts to increase (see Fig. 4.32). These findings indicate that below T ∗ a transition occurs into a state with full coherence among the f sites [148,152], leading to the emergence of the heavy fermions. In other words, the lowfrequency (v < 100 cm−1 ) dispersion which the authors [45, 150] observe in the spectra of YbB12 at low temperatures (T < T ∗ ≈ 70 K) is determined by the response of a Fermi liquid composed of heavy fermions, m∗ (20 K) ≈ 34m0 , with a strongly enhanced relaxation time τ (20 K) = 4 × 10−13 s and mobility μ(20 K) = 24 cm2 /(V·s), as it is also shown in Fig. 4.32. Gorshunov et al. consider T ∗ ≈ 70 K as the coherence temperature in YbB12 . It was noted in Refs. [45, 150] that the findings depicted in Figs. 4.30 –4.32 qualitatively resemble the optical response of heavy-fermion compounds where a Drude-like behavior of heavy quasiparticles is expected [151] and observed [153, 154] in the coherent state. Another feature of the heavy-fermion state in YbB12 is the presence of two plasma edges in the reflectivity spectra that originate from plasma oscillations of heavy (renormalized) and light (unrenormalized) quasiparticles (Figs. 4.30, 4.31). One of the most important findings of the optical study [45, 150] is a pronounced absorption peak which was observed at 22 cm−1 (∼2.7 meV) at the lowest temperature of 5 K. The origin of the low-frequency feature in the σ (ω) spectrum (see Fig. 4.31) has been connected with the formation of exciton-polaronic many-body intra-gap states close to the bottom of the conduction band. The authors [45] noted that the energy position of 2.7 meV of the low-temperature conductivity peak, which can be associated with the photon-assisted breaking of the intra-gap many-body states, is in accord with the activation energy of the charge transport characteristics. Indeed, at the temperature of ∼15 K, a crossover from E g ≈ 20 meV (excitations across the hybridization gap) to E p ≈ 2.2 meV (thermal activation of carriers from the “impurity band” into the conduction band) is observed [4, 126]. It was suggested that the 22 cm−1 peak in the conductivity spectrum of YbB12 can correspond to the many-body states arising from the

394 Magnetism, Quantum Criticality, and Metal–Insulator Transitions in RB12

coupling of free electrons to soft valence fluctuations on Yb sites. Within this picture, the localization radius of these heavy fermions has been estimated to be ∼5 A˚ [45, 150], close to the Yb–Yb distance ˚ in the YbB12 lattice. (∼5.3 A) Tunneling spectroscopy Measurements of the tunneling spectra have been carried out on YbB12 single crystals using the breakjunction technique [46]. It was found that the differential conductance shows very sharp gap-edge peaks with quite a low leakage at zero bias. The representative gap magnitude of 2 ≈ 220–260 meV at 4.2 K was found to be unexpectedly larger than that from the transport measurements [46]. At the same time, the 2 value agrees very well with the energy of the mid-infrared absorption peak in the σ (ω) spectra of YbB12 , i.e., ∼250 meV [45, 143]. Photoemission spectroscopy The temperature-dependent energy gap formation in both the valence bands and the Yb 4f states of YbB12 has been examined by means of high-resolution PES [155]. It was found that an energy gap (< 15 meV) is gradually formed in the valence band upon cooling. Two characteristic temperatures have been detected at T1 ≈ 150 K and at T ∗ ≈ 60 K. It was shown that the 55 meV peak at 250 K is shifted toward lower binding energy (∼35 meV) upon cooling below T1 . The appearance of a 15 meV peak in the Yb 4f and Yb 5d states was clearly observed below T ∗ [155] and enhanced near the L point. The authors concluded that the results give direct evidence that the coherent nature of the Yb 4f state plays an important role in the energy gap formation via the d- f hybridization below T ∗ ≈ 60 K [155]. Then, the low-energy electronic structure of YbB12 and its transient properties were investigated in more detail [156] using ultra-high-resolution PES and time-resolved PES (TrPES). In the T -dependent laser-PES spectra, the authors found two different (pseudo)gaps with sizes of 25 and 15 meV, which were attributed to the single-site effect and the insulating hybridization-gap opening, respectively. The characteristic temperature T1 was determined to be ≤150 K, where the hybridization gap begins to open, although the Fermi edge remains as the in-gap state down to the lowest temperatures. In TrPES measurements, Okawa et al. [156] found that the long-lived (≥100 ps) component of the photoexcited

Metal–Insulator Transition in YbB12 and Solid Solutions Ybx R1−x B12 (R = Lu, Tm) 395

electrons gradually develops upon cooling below 150 K, which was interpreted as a feature of the hybridization-gap evolution. Thus it was experimentally determined that the characteristic temperature T1 ≈ 150 K corresponds to a metal-to-insulator crossover in YbB12 .

4.5.2 Pressure-Induced Insulator-to-Metal Transition in YbB12 In the case of YbB12 , the electrical resistivity has been first measured under pressure up to 8 and 20 GPa in Refs. [157] and [130], respectively. However, it has shown semiconducting behavior over such a narrow pressure range. The pressure dependence of the activation energies E g and E a estimated from the electrical resistivity suggested that the semiconductor-to-metal transition may be expected at a much higher pressure of about 100 GPa [130, 157]. More recent measurements of the electrical resistance R(T ) on YbB12 single crystals were carried out under pressures up to 195 GPa [158]. It was then found that at pressures under

Figure 4.33 Pressure dependences of two activation energies of YbB12 estimated from the resistivity measurements R(T ) up to 150 GPa. E 1 = E g and E 2 = E p were detected in the intervals 18 K < T < 33 K and 100 K < T < 280 K, correspondingly. Reproduced from Kayama et al. [158].

396 Magnetism, Quantum Criticality, and Metal–Insulator Transitions in RB12

150 GPa the semiconducting behavior persists, but after that the low-temperature resistance is dramatically suppressed with increasing pressure. The pressure dependences of the two activation energies E g and E p estimated from the R(T ) dependences in the two temperature regions, 18 K < T < 33 K (for E g ) and 100 K < T < 280 K (for E p ) decrease monotonically with increasing pressure and are expected to vanish at ∼160 GPa by extrapolation (see Fig. 4.33). This marks a pressure-driven insulator-to-metal (I–M) transition. Moreover, metallic R(T ) behavior was observed at 164 GPa [158], and at 195 GPa a drop in the resistance was observed below 0.8 K, which was attributed to the onset of superconductivity in YbB12 .

4.5.3 Field-Induced Insulator-to-Metal Transition in YbB12 Strong negative MR is naturally expected for systems with fieldinduced metallization. The results of high-field magnetization and MR measurements indicate that the energy gap in YbB12 collapses in a strong magnetic field by a metamagnetic transition [130, 132]. An extreme drop in the resistivity ρ(B) by two orders of magnitude was observed and considered as the manifestation of the first-order transition. At the critical field Bc1 , the resistivity drops and then becomes constant, and the authors suggested that the electronic state above Bc may be a metallic one. The critical field Bc1 shows an anisotropy with respect to the field direction: 47 T for H 100 and 53 T for H 110 and H 111. The hysteresis on the magnetization and MR dependences also indicates this transition to be of the first-order type. The authors supposed that the anisotropy in the energy gap may originate from the k-dependence of the mixing integral between the conduction and 4f -electrons [130]. Recently the magnetization measurements were extended to ultrahigh magnetic fields up to 120 T with B 111 to explore the possibility of additional magnetic transitions [159]. Indeed, two clear metamagnetic transitions at Bc1 ≈ 55 T and Bc2 ≈ 102 T, and the tendency of the magnetization to saturate above 112 T, have been observed (see Fig. 4.34). A two-gap structure in the

Metal–Insulator Transition in YbB12 and Solid Solutions Ybx R1−x B12 (R = Lu, Tm) 397

2.5

(a) YbB12 4.2 K

B // [111]

M (μB/Yb)

2.0

powder

1.5

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[111] Single crystal [001]

1.0

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[110] 0.0 0

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dM/dB (μB/T Yb)

40

60

0.10

(b) Powder

0.06

20

[111] Single crystal

0.05

0.04

0.00 0

20 40 60 80 100

0.02 BC1

0.00

0

20

40

BC2

60

80

100

120

B (T) Figure 4.34 (a) Magnetic-field dependence of the magnetization of YbB12 . The red dashed and black solid curves denote the magnetization in the powder sample measured by a nondestructive pulsed magnet and by the horizontal single-turn coil (HSTC), respectively. The green dashed and gray solid curves denote the magnetization in a single-crystal sample measured by a nondestructive pulsed magnet and by the HSTC, respectively. The inset shows the magnetization process of single crystals with different magnetic field directions. (b) The field derivative of the magnetization (dM/dB) curves for the results of the nondestructive pulsed magnet (red dashed curve) and the single-turn coil method (black solid curve). The inset shows dM/dB for the single crystal in the B 111 direction. The green solid curve was obtained with a nondestructive magnet and the gray solid curve was obtained with the HSTC. Reproduced from Terashima et al. [159].

398 Magnetism, Quantum Criticality, and Metal–Insulator Transitions in RB12

(a) 0 T

(b) 55 T

E

(c) 102 T

= 23.5 meV

meV

EF

meV

DOS

DOS

DOS

Figure 4.35 (a) Schematic of the assumed density of states (DOS) at zero magnetic field. (b) DOS expected in magnetic fields where the insulatorto-metal transition at Bc1 takes place. (c) DOS expected in magnetic fields where the second metamagnetic transition takes place at Bc2 . Reproduced from Ref. [159].

DOS, probably due to an anisotropic hybridization of the 4f states with the conduction electrons (Fig. 4.35), has been suggested [159] to explain the characteristic magnetization process. The authors have argued that the second metamagnetic transition at 102 T may correspond to the transition from the heavy-fermion metallic state to another metallic phase that either possesses a weaker Kondo effect than the phase below 102 T or conserves the strong AFM interaction between Yb magnetic moments and reduces the magnetization. One more very recent study of YbB12 in the pulsed magnetic field may be also mentioned here to complete the section. Specificheat measurements of a YbB12 single crystal have been conducted in high magnetic fields of up to 60 T [160], and the obtained data were used to deduce the magnetic field dependence of the Sommerfeld coefficient. It has been found that the linear coefficient of the electron heat capacity increases considerably at the I–M

Metal–Insulator Transition in YbB12 and Solid Solutions Ybx R1−x B12 (R = Lu, Tm) 399

transition, being as large as 67 mJ/(mol·K2 ) at high fields, which is an indication of heavy quasiparticles [160]. Unfortunately, the simplest data analysis presented in Ref. [160] did not consider the low-temperature magnetic components C m (T ) that are typical for inhomogeneous systems with magnetic ions. But, at temperatures below T ∗ ≈ 60 K in the cage-glass state of RB12 , the formation of AFM nanosize domains of Yb3+ ions (short-range AFM order) should be expected, having a strong influence on the low-temperature heat capacity [161, 162]. Such an indication of “false heavy fermions” has been discussed in detail previously, for instance, by Gschneidner et al. [161] and by Coles [162]. On the other hand, the dramatic increase of the heat capacity in between 39 and 49 T may be certainly considered in terms of the metamagnetic transition in YbB12 [160].

4.5.4 Insulator-to-metal transition in Ybx R1−x B12 (R = Y, Lu, Sc, Ca, and Zr) Another approach to investigate the I–M transition in YbB12 is to study the substitutional solid solutions Yb1−x Rx B12 , where R is a nonmagnetic ion. In particular, the substitution of Lu3+ for Yb3+ is expected to remove the 4f magnetic moment at the corresponding RE site. At the same time, it does not change the number of conduction electrons, one per formula unit in an atomic picture [163]. In addition, the substitution does not change significantly the electronic structure beside the 4f -related states. Hence, the replacement is expected to “dilute” the magnetic moments in this system, without changing the filling of the conduction band or influencing other electronic states. This dilution is expected to lower the overlap among the Yb orbitals at different Yb sites, which is believed to be essential in the hybridization gap scenario of the Kondo semiconductor [163]. On this way the combined results of resistivity [41, 126, 164], magnetic susceptibility and specific heat [41, 164], optical conductivity [77, 163], tunneling spectroscopy [46] of Yb1−x Lux B12 have suggested that the gap structure remains up to x = 0.5, and a mid-gap state develops near the Fermi level with increasing Lu concentration. From the analysis of specific heat, the authors have

400 Magnetism, Quantum Criticality, and Metal–Insulator Transitions in RB12

proposed that the ground state of Yb1−x Lux B12 (x < 0.5) is the 8 quartet, but the magnetic entropy values (∼R ln 4/mol Yb at 300 K) contradict the CEF scheme with small enough total splitting (∼11 meV) suggested in Ref. [111]. The transport experiments [41, 126] on Yb1−x Lux B12 have shown that the electrical resistivity of YbB12 at low temperatures is strongly reduced by substituting a small amount of Lu for Yb. These results indicate that the DOS at E F increases strongly with a small amount of Lu substitution. In addition, ρ(T ) shows thermally activated T dependence only for x ≤ 1/2, which indicates the absence of a transport gap in dilute Yb regime (x > 1/2) [41, 126]. The magnetic field induced a collapse of the energy gap in the semiconducting solid solutions Yb1−x Lux B12 , revealed in the measurements of magnetization and electrical resistivity in the pulsed magnetic fields up to 68 T [132]. It was found that for x = 0.01, positive MR appears in the range up to 20 T for the three principal directions and then the MR drops at almost the same field Bcs (47 T for B 100 and 53 T for B 110 and 111) as in YbB12 [132]. It should be pointed out that the largest positive MR is observed for B 100 both in the case of x = 0.01 and especially for x = 0.05 [132], and the finding is very similar to the MR anisotropy detected both for LuB12 and magnetic dodecaborides RB12 , which is shown, for example, in Fig. 4.28(d). When discussing the evolution of the optical conductivity spectra of Yb1−x Lux B12 , studied in detail by Okamura et al. [77, 163], it is worth noting that similar to YbB12 , in solid solutions with x < 0.5 a shoulder at ∼40 meV remains nearly unshifted in a wide range of temperatures (up to 70 K), but the energy gap in σ (ω) below 20 K is rapidly filled in from the bottom rather than by narrowing (Fig. 4.36). This result is consistent with the observation of the rapid filling of the gap in resistivity measurements at small x [41, 126]. In contrast, the σ (ω) data [77, 163] show that the DOS at E F is slightly reduced even at x = 3/4, as evidenced by the small depletion of spectral weight below ∼50 meV (Fig. 4.36). According to the conclusions of Okamura et al., the rapid filling of the gap in σ (ω) of Yb1−x Lux B12 at small x clearly shows the importance of lattice effects in producing a well-developed gap. On the other hand, the shoulder position (E ∗ ≈ 40 meV) is almost unchanged over a wide range of x, which strongly

Metal–Insulator Transition in YbB12 and Solid Solutions Ybx R1−x B12 (R = Lu, Tm) 401

Figure 4.36 Optical reflectivity (R) and conductivity (σ ) spectra of Yb1−x Lux B12 . (a) R of x = 1 (LuB12 ) at 295 K and those of x = 0 (YbB12 ) at 295, 80, and 9 K. (b) σ for various values of x at 295, 80, and 9 K. Reproduced from Okamura et al. [77].

implies that E ∗ is closely related to some single-site energy scale of Yb3+ in YbB12 [77, 163]. Another important feature of the σ (ω) spectra is a broad mIR peak which was observed in YbB12 at ∼250 meV [44, 77, 163]

402 Magnetism, Quantum Criticality, and Metal–Insulator Transitions in RB12

(Fig. 4.36). It was clearly demonstrated in Refs. [77, 163] that the peak position depends dramatically on the Lu concentration, changing monotonically to 120 meV for x = 7/8 (Fig. 4.36). In addition to the gradual shift of the mIR peak toward lower energy with increasing x, the displacement of the anomaly toward higher energy with decreasing temperature has also been observed [77, 163]. To perform quantitative analyses of the σ (ω) spectra, the authors have fitted the spectral shape of the mIR peak for Yb1−x Lux B12 using the classical Lorentz oscillator model, and the broad continuum toward ω = 0 using the Drude model. Figure 4.37(a) shows an example of the fitting. Using the optical sum rule, an effective carrier density Neff contributing to σ (ω) below ω = ωp was obtained as n 2m0 ωp Neff = ∗ = σ (ω)dω, (4.7) m πe2 0 where n is the carrier density, and m∗ is the effective mass in units of the free electron mass m0 . In Figs. 4.37(b,c), Neff for the total σ (ω) (the sum of fitted Drude and Lorentz contributions) and that for the mIR peak (fitted Lorentz) are plotted as functions of x and T, together with the position and the width of the mIR peak. Figure 4.37(b) shows that Neff contributing to the mIR peak is strongly nonlinear in 1 – x, and hence it does not scale with the number of Yb 4f electrons. Then, the peak energy and the width of the mIR feature show large, linear decreases with increasing x. Taking into account that a similar mIR anomaly has been found for the nonmagnetic reference compound LuB12 at room temperature (see Fig. 4.14), and that the extrapolation of the peak width and position in Yb1−x Lux B12 [Fig. 4.37(b)] to x = 1 provides the same values ∼800 cm−1 (∼100 meV) that were measured in LuB12 [79], it is natural to conclude about the common nature of this collective mode in YbB12 and LuB12 . Thus, the origin of the mIR peak in Yb1−x Lux B12 can be attributed to the collective JT dynamics of the boron clusters (B12 , ferrodistortive effect, see Chapter 3 for details) in the RE dodecaborides which is renormalized additionally by the fast charge and spin fluctuations on the Yb ions. Okamura et al. [77,163] noted that not only the mIR peak but also the Drude component shows considerable x- and T -dependences, while the combined intensity (Drude + mIR peak) is kept almost

Metal–Insulator Transition in YbB12 and Solid Solutions Ybx R1−x B12 (R = Lu, Tm) 403

Figure 4.37 Fitting results of the mIR peak in the optical conductivity σ (ω) spectra of Yb1−x Lux B12 . (a) An example of the fitting for x = 1/4. The solid line shows the measured spectrum, and dotted and dashed lines show the fitting. (b) The effective carrier density per formula unit (Neff ) for the total (Drude + Lorentz) intensity and that for the mIR (Lorentz) peak, and the position and the width of mIR peak are plotted as a function of x at four temperatures. (c) The same data as in (b) but plotted as a function of temperature for four values of x. In (b) and (c), the solid lines are guided to the eye, and the error bars are derived from the uncertainty in the measured R. Reproduced from Okamura et al. [77].

404 Magnetism, Quantum Criticality, and Metal–Insulator Transitions in RB12

constant, Neff ≈ 1.8 upon varying either x and T. Thus, a spectral weight transfer exists between the Drude component and the mIR peak, which demonstrates that the dynamics of both the Drude free carriers and the conduction electrons involved in the collective mode are strongly connected to each other. In other words, despite of the appearance of the energy gap in YbB12 only below 70 K, while the mIR peak is already observed at room temperature [163], one can conclude that (i) the depleted spectral weight in the gap region is transferred to the wide energy range that contains the mIR peak and (ii) both the gap and the mIR peak show similar responses to T and x. It will be shown below for Tm1−x Ybx B12 that the cageglass transition at T ∗ = 60–70 K to the disordered state of RE ions in the rigid boron sublattice may be responsible for the emergence of vibrationally coupled Yb–Yb pairs randomly distributed in the dodecaboride matrix. Let us note at the end of the section that in Ref. [165], the nonmagnetic-element substitution effects have been reported for YbB12 . It has been found that YbB12 in a virtual gapless state would have a peak at about 25 K from magnetic susceptibility of Yb1−x R x B12 (R = Y, Lu, Sc, and Zr). The peak temperature T χmax can be extrapolated to 25 K from the high-x (Yb-dilute) region. The linear variations in the x-dependence of T χmax show kinks at around x = 0.5 for all substituted element R due to the development of an energy gap E g . Furthermore, upturns shown in the magnetic susceptibility below 20 K in Yb1−x R x B12 may be also related to the formation of the second gap E p [165]. It is worth noting also the results of Matsuhra et al. [166] who succeeded in the synthesis of Yb1−x Cax B12 solid solutions up to x = 0.15 by using high-pressure synthesis. In these substituted alloys, a low-temperature increase in both χ and C /T according to − log T are remarkably larger than those in powdered YbB12 , which, as concluded in Ref. [166], may originate from a giant increase in the DOS at E F due to an in-gap state.

4.5.5 Metal–Insulator Transition in Tm1−x Ybx B12 As the concentration x varies, the properties of the substitutional solid solutions Tm1−x Ybx B12 continuously transform from the AFM

Metal–Insulator Transition in YbB12 and Solid Solutions Ybx R1−x B12 (R = Lu, Tm) 405

metal TmB12 (x = 0, TN ≈ 3.2 K) [2] to the paramagnetic insulator YbB12 (x = 1) with strong charge and spin fluctuations [4, 45, 167]. Thus, the study of the charge transport, magnetic characteristics, heat capacity, and fine details of the crystal structure were undertaken in Refs. [2, 42, 168–170] to reveal the singularities of both the MIT and the AF-P phase transition in these compounds with an unstable electron configuration of the Yb ion. In strongly correlated electron systems, the most significant changes in the physical characteristics take place near QCPs [124, 125, 171–173] and at the MITs [145, 167, 173–175]. Near QCPs corresponding to zero-temperature phase transitions, such changes are due to both thermal and quantum fluctuations. This leads to an enhancement of correlation effects and to the formation of complicated manybody states [124, 125, 171–173]. Similarly, insulator-type features arising at the MITs are usually closely related to the formation of bound many-body states in the systems of itinerant charge carriers, resulting to a drastic drop in the density/mobility of conduction electrons [145, 167, 173–175]. Resistivity and Hall effect The resistivity of Tm1−x Ybx B12 crystals on cooling at small x values exhibits a decrease typical for ´ metals in combination with low-temperature features near the Neel temperature TN (see Fig. 4.38). With an increasing Yb content, the AFM phase transition becomes suppressed at the QCP, xc ≈ 0.25, the ρ(T ) dependences demonstrate a low-temperature increase, and for x > 0.5 the plots correspond to the semiconducting behavior on cooling below 300 K. As a result, the resistivity in the Tm1−x Ybx B12 series with MIT increases by a factor of about 2 × 106 at low temperatures and by a factor of about 7 even at 270 K (inset in Fig. 4.38). The results of room-temperature Hall effect measurements allowed us to detect a variation in the charge carrier density n(x) = (R H e)−1 versus the ytterbium content x in the Tm1−x Ybx B12 series [170]. It was shown that the single-band carrier concentration n decreases by more than a factor of 2, and the Hall mobility μH (x) = RH /ρ also exhibits a significant decrease (by a factor of about 3) [170]. The room-temperature behavior of n(x) and μH (x) near the QCP at xc ∼ 0.25 was found to be monotonic within the experimental error. The band gap in the Tm1−x Ybx B12 for x ≥ 0.5 was estimated to

406 Magnetism, Quantum Criticality, and Metal–Insulator Transitions in RB12

N

Figure 4.38 Temperature dependence of the electrical resistivity for Tm1−x Ybx B12 . The inset shows the dependence of resistivity on the ytterbium concentration at T = 270 K. Reproduced from Ref. [170].

be E g /kB ≈ 200 K [42]. Thus, a metallic behavior can be expected for all x(Yb) compositions at T > E g /kB . It was concluded in Ref. [170] that strong electron correlations and many-body effects are apparently responsible for the renormalization of the chargecarrier density and mobility observed at room temperature. Then, to determine the mechanism responsible for the increase in ρ(x) at room temperature, the authors have analyzed the equivalent atomic displacement parameters in the boron U B and cation U R subsystems, as determined from the x-ray diffraction data. Figure 4.39 shows the U B (x) and U R (x) dependences, demonstrating a considerable growth of the amplitude of atomic displacements with the increase in x both in the boron sublattice (more than by a factor of 2) and for the RE ions (by a factor of about 6) [170]. It was pointed out in Ref. [170] that the situation in Tm1−x Ybx B12 at x ≥ 0.7 is very

B–B inter

B–B intra

Metal–Insulator Transition in YbB12 and Solid Solutions Ybx R1−x B12 (R = Lu, Tm) 407

(Å2)

c

c

Figure 4.39 (a) Boron-boron distances in the B12 clusters, r(B–B) intra , and between the clusters, r(B–B) inter , and (b) the equivalent atomic displacement parameters U B and U R of the boron and cation atoms, respectively, in Tm1−x Ybx B12 versus the ytterbium content x. Reproduced from Ref. [170].

unusual, because the displacement amplitude of heavy RE ions with mA ≈ 170 exceeds by far that for the light boron atoms (mA ≈ 11), as shown in Fig. 4.39. It was suggested in Ref. [170] that the increase in the amplitude of atomic displacements by a factor of 2–6 must be responsible for the enhancement of charge-carrier scattering in the crystal lattice of dodecaborides and for the corresponding (by a factor of about 7) decrease in the conductivity (inset in Fig. 4.38). The detailed measurements of the resistivity in the Hall geometry in Tm1−x Ybx B12 have been carried out in Ref. [42]. The comprehensive analysis of the angular, temperature, and field dependences of the resistivity from the Hall probes’ allowed separating and classifying two components: the Hall and transverse even (TE) effects in the charge transport of these strongly correlated electron systems. It was found that the second-harmonic contribution ρTE

408 Magnetism, Quantum Criticality, and Metal–Insulator Transitions in RB12

Figure 4.40 Temperature dependences of (a) the Hall coefficient RH (T , H 0 ) at low (H 0 = 15 kOe, solid symbols) and high (H 0 = 70 kOe, open symbols) magnetic fields and (b) mobility μH (T, H 0 = 15 kOe) of charge carriers for Tm1−x Ybx B12 with x = 0.31, 0.54, 0.72, and 0.81. For comparison the R H (T ) and μH (T ) dependences for TmB12 at H 0 = 3.7 kOe are shown in panels (a) and (b), respectively. Reproduced from Ref. [42].

(TE effect) appears in the angular dependences of the Hall probe’s resistivity of the Tm1−x Ybx B12 dodecaborides near the QCP (xc ≈ 0.25), and it enhances drastically as the concentration x increases in the interval x ≥ xc . A pronounced negative maximum in the temperature dependence of the Hall coefficient for x ≥ xc and the sign reversal of RH (T ) at LHe temperatures for Tm1−x Ybx B12 with x ≥ 0.5 have been revealed [42] (see Fig. 4.40), and a decrease in RH (T ) at T < 30 K has been identified as a signature of the coherent regime of charge transport. It was shown that at temperatures below the negative maximum of the Hall coefficient (interval III in Fig. 4.40), the temperature dependence RH (T ) ∝ exp(−T0 /T ) is observed with the estimated values T0 ≈ 3.5–7 K. A similar behavior of RH (T ) was previously observed for heavy-fermion compounds CeAl2 [176] and CeAl3 [177], and it was interpreted in terms of the dependence predicted in Ref. [178] for the Hall coefficient in a system with Berry-phase effects, where the carrier moves by

Metal–Insulator Transition in YbB12 and Solid Solutions Ybx R1−x B12 (R = Lu, Tm) 409

Figure 4.41 (a–b) Arrhenius-type dependences of the reduced concentration v = (RH eN4f )−1 of charge carriers for Tm1−x Ybx B12 compounds with x = 0.31, 0.54, 0.72, and 0.81 in the different temperature ranges I–III. Reproduced from Ref. [42].

hopping in a topologically nontrivial spin background. According to Refs. [178, 179], the Hall effect is modified in such a situation because of the appearance of the internal magnetic field H int ∼ (1/kB T ) exp(−T0 /T ), which is added to the external field H . It has been shown that the external magnetic field drastically enforces the TE effect [42], also suppressing the coherent regime of the spin-flip scattering of charge carriers on the magnetic moments of the RE ions [see Fig. 4.40(a)]. For the Tm1−x Ybx B12 compounds with x ≥ 0.5 in the intervals 120–300 K (I) and 50–120 K (II), the authors [42] found an Arrhenius-type behavior of the Hall coefficient with the activation energies E g /kB ≈ 200 K and E p /kB ≈ 58–75 K (Fig. 4.41) and the following microscopic parameters: effective masses m∗ ≈ 20m0 and localization radii of the heavy-fermion many-body states ∼5 A˚ (120– 300 K) and ∼9 A˚ (50–120 K). We have argued in Ref. [42] that the

410 Magnetism, Quantum Criticality, and Metal–Insulator Transitions in RB12

MIT, which develops both in YbB12 as the temperature decreases and in Tm1−x Ybx B12 solid solutions as the Yb concentration increases in the range 0 < x ≤ 1, is induced by a formation of Yb–Yb pairs randomly distributed in the RB12 matrix. In the framework of the spin-polaron approach, the appearance of the odd RH (T , H ) and even ρTE (T , H ) anomalous components in the resistivity detected in the Hall-effect geometry in the Tm1−x Ybx B12 series has been discussed in terms of the interference effects between local 4f 5d and long-range spin fluctuations, leading to the formation of a filamentary structure (network) of the interconnected many-body complexes in the dielectric matrix of these compounds. Charge stripes and magnetotransport anisotropy To testify fine details of the crystal and electron structure, accurate x-ray diffraction experiments and structural data analysis were developed for single crystals of the semiconducting Tm0.19 Yb0.81 B12 solid solutions [43]. Combined with the reconstruction of difference Fourier maps of residual electron density and applying the maximum entropy method to deduce the normal electron density, the approach allowed us to visualize the dynamic charge stripes oriented predominantly along a face diagonal of the unit cell in Tm1−x Ybx B12 (for more details, see Chapter 3). For single-domain crystals of Tm0.19 Yb0.81 B12 , measurements of the charge transport anisotropy at temperatures in a coherent regime (interval III in Fig. 4.40) were performed in a scheme corresponding to the transverse MR and Hall resistance, recorded as the crystal was rotated around the current axis I parallel to one of the 110 ⊥ H. Both the transverse even effect ρTE (φ, H 0 ) and the transverse MR have been separated (see Fig. 4.42), demonstrating uniaxial anisotropy with the anisotropy axis predominantly oriented along the H [110] direction [43]. Besides, on the metal side of the MIT, single-domain crystals of Tm0.96 Yb0.04 B12 with an AFM ground state (TN ≈ 2.6 K [42, 169]) have been studied in detail by low-temperature MR, magnetization, and heat capacity measurements. The angular H − φ − T0 AFM phase diagram in the form of a Maltese cross has been deduced for this AFM metal from precise MR measurements, reproduced here in Fig. 4.43 [180]. The authors argued that the observed dramatic symmetry lowering is a consequence of strong renormalization of

Metal–Insulator Transition in YbB12 and Solid Solutions Ybx R1−x B12 (R = Lu, Tm) 411

Figure 4.42 Transverse magnetoresistance ρanis /ρ0 (φ, H 0 ) (left) and the transverse even effect ρTE (φ, H 0 ) (right) for Tm0.19 Yb0.81 B12 as a function of magnetic-field magnitude and direction in polar coordinates. Reproduced from Ref. [43].

Figure 4.43 (a) The magnetoresistance of Tm0.96 Yb0.04 B12 as a function of magnetic field and its direction at T = 2 K. The rotation axis is I [110], three principal cubic directions in the [110] plane are shown with arrows. (b) The H –φ magnetic phase diagram at T = 2 K. The phase boundaries are derived from the magnetoresistance measurements. Roman numerals denote various AFM phases. Reproduced from Ref. [180].

the indirect exchange interaction (RKKY mechanism) due to the presence of dynamic charge stripes in the matrix of this AFM metal. They proposed that additionally to the development of the fcc lattice instability (cooperative dynamic JT effect in the boron sublattice) and appearance of the nanometer-size electronic phase separation (dynamic charge stripes), the electron density fluctuations on Yb sites also play an essential role both in the suppression of the AFM state and in driving the changes in orientation of the stripes and the

412 Magnetism, Quantum Criticality, and Metal–Insulator Transitions in RB12

Figure 4.44 (a) Temperature dependences of the Seebeck coefficient S(T ) of the Tm1−x Ybx B12 solid solutions for the orientation of the temperature gradient ∇T [110]. Solid lines show the approximations of the S(T ) curves by the activation (T > 50 K, S ∝ 1/T ) and linear (T < 30 K, S(T ) ∝ A(x)T ) dependences. (b) The S(T ) curves in the Arrhenius plot. The temperature ranges of the activation behavior of the thermopower are denoted as I and II. Adapted from Ref. [2].

phase-diagram boundaries in Tm1−x Ybx B12 antiferromagnets [180, 181]. Seebeck coefficient According to Ref. [2], strong variations in the charge-transport characteristics of Tm1−x Ybx B12 have been observed in the temperature dependences of the Seebeck coefficient S(T ). Indeed, the low magnitudes of the thermopower for TmB12 (|S(T )| < 3 μV/K, see Figs. 4.44 and 4.20), typical of a compound with metallic conductivity, change to |S(T )| > 15 μV/K already for the Tm0.96 Yb0.04 B12 solid solution. Further substitution of Yb for Tm increases the Seebeck coefficient magnitude up to 580 μV/K (Fig. 4.44), and the negative minimum on the S(T ) curves shifts down in temperature with increasing x. The authors [2] noted that the S(T ) dependences for all solid solutions with x > 0.01 in the intermediate temperature range T ≥ 50 K are characterized by a complex activation behavior (Fig. 4.44) and S(1/T ) representation allows estimating both the gap E g /kB = 100–160 K (interval I)

Metal–Insulator Transition in YbB12 and Solid Solutions Ybx R1−x B12 (R = Lu, Tm) 413

and the binding energy of intra-gap many-body states E p (interval II) [2]. It is worth noting that the estimated values E g /kB = 100– 160 K are close to the estimate of the gap value found for YbB12 in the measurements of the Hall effect and resistivity (180 and 134 K, respectively [4]) as well as NMR on the Yb ions and specific heat (∼170 K) [37, 41]. They are also comparable to the spin-gap value ∼14 meV determined by neutron spectroscopy [141, 142] and EPR [136]. With a decrease below 30 K, the S(T ) curves approach about linear asymptotic behavior (Fig. 4.44), which points to a metallic conductivity via the band of many-body states in Tm1−x Ybx B12 . It has been argued [2] that a considerable increase (about 5 times) in the A(x) slope in the S(T ) ∝ A(x)T curves when Yb is substituted for Tm in the Tm1−x Ybx B12 series may be attributed to the Mott term (diffusion thermopower) that points to the DOS renormalization near E F . Similar effects were detected in the investigation of the electronic specific heat C /T = γ (T ) of the Ybx Lu1−x B12 compounds in Ref. [164], where an increase by about an order of magnitude in the Sommerfeld coefficient has been observed upon varying the Yb content between 10 and 80 at. %. Magnetic properties In the temperature range 40–300 K, Tm1−x Ybx B12 compounds with x < 0.9 are paramagnets whose magnetic susceptibility χ(T ) is well described by the Curie– Weiss law with a negative paramagnetic Curie temperature p corresponding to the AFM exchange interaction [2]. The estimation of the effective magnetic moment μeff per unit cell was done by considering the additive contributions from the localized magnetic moments of the Tm3+ (μ(Tm) = 7.5μB ) and Yb3+ (μ(Yb) = 4.5μB ) ions present in the Tm1−x Ybx B12 solid solutions. Analysis of χ(T ) dependences [2, 168] indicates that p (x) increases monotonically, and in addition to the Curie–Weiss contribution, a magneticfluctuation component appears in the low-temperature magnetic response of Tm1−x Ybx B12 . Both to reconstruct the H –T –x magnetic phase diagrams and to clarify the nature of the additional paramagnetic response, the authors [168] have investigated the permanent (up to 11 T) and pulsed (up to 50 T) field magnetization of the Tm1−x Ybx B12 compounds in the composition range 0 < x ≤ 0.81. It was found that at low temperatures there are two paramagnetic

414 Magnetism, Quantum Criticality, and Metal–Insulator Transitions in RB12

contributions including a Pauli-type component, which corresponds to the response of the heavy-fermion many-body states that appear in the energy gap in the vicinity of the Fermi level (DOS of (3–4)·1021 cm−3 meV−1 ), and a contribution which saturates in high magnetic fields and may be attributed to the localized magnetic moments (0.8– 3.7 μB per unit cell) of the nanosize clusters of Tm(Yb) ions with AFM exchange (short-range order nanodomains) [168]. The last finding may be interpreted in terms of the Griffiths-phase formation [182] and the spin-polaron effect, which are the two dominant factors responsible for the low-temperature changes in the magnetic properties of Tm1−x Ybx B12 . Heat capacity For the refinement of the magnetic T − x phase diagram of Tm1−x Ybx B12 , the temperature dependences of the specific heat have been investigated in detail, as shown in Fig. 4.45 [170]. Surprisingly, the log–log plots nearly coincide for different Yb concentrations. These plots exhibit a steep decrease below 40 K, which is fitted with a good accuracy by the Einstein model for the specific heat related to the nearly universal quasilocal vibrations of both Yb and Tm ions in the rigid boron cage [183]. Near TN , the C (T ) curves exhibit a sharp increase related to the formation of the long-range AFM order [183], and with the growth of the Yb content in Tm1−x Ybx B12 the step-like C (T ) anomaly changes to a peak for x > xc ≈ 0.25. Taking into account that at low temperatures the Yb ion becomes nonmagnetic, we have concluded [170] that the substitution of ytterbium for magnetic Tm3+ results in disordering and the formation of a mictomagnetic state (spin glass). Thus, it was argued [170] that the spin glass phase exists above the AFP transition, and a hidden QCP appears in the T − x plane of Tm1−x Ybx B12 compounds [see Fig. 4.46(a)]. Then, an isosbestic point was observed at T ∗ ≈ 6 K on the C (T, H = 0) curves [170, 184] (Fig. 4.45), and it was found that the temperature dependence of the specific heat for the entire composition range 0.004 ≤ x ≤ 0.81 linearly scales with respect to the Yb concentration. It was shown that at low temperatures a linear in x decrease in C (T0 , x) should be attributed to the formation of a charge gap, which leads to a decrease in the electron DOS at E F . Within the proposed approach, the magnetic contribution

Metal–Insulator Transition in YbB12 and Solid Solutions Ybx R1−x B12 (R = Lu, Tm) 415

Figure 4.45 Temperature dependence of the specific heat for Tm1−x Ybx B12 , nonmagnetic reference compound LuB12 (solid line), and the sum of the magnetic contribution from Tm3+ ions (C CEF ) and specific heat of LuB12 (dashed line) [184]. The inset depicts the low-temperature range corresponding to the magnetic phase transitions (indicated by arrows). The antiferromagnetic, paramagnetic, and spin glass phases are denoted as AF, P, and SG, respectively [170].

to the specific heat of Tm1−x Ybx B12 , consisting of several additive components, can be distinguished [184]. It was shown that for all Tm1−x Ybx B12 compositions, the magneto-vibronic contribution to the heat capacity with a maximum near T ∗ ≈ 60 K, which

416 Magnetism, Quantum Criticality, and Metal–Insulator Transitions in RB12

–

c

c

Figure 4.46 (a) The magnetic phase diagram in the T –x plane with the QCP at xc ≈ 0.25. (b) Average lattice parameter for the cubic unit cell and boron-cation distance versus the ytterbium content for Tm1−x Ybx B12 solid solutions. Reproduced from Ref. [170].

is practically independent of the Yb concentration, is dominant (Fig. 4.45) [184]. An analysis of the low-temperature contributions to the specific heat allows one to distinguish the Zeeman component from Tm3+ ions with the triplet ground state 5 of the 3 H 6 multiplet, as well as to find the g-factors (g1 ≈ 2.5, g2 ≈ 5) and their changes in the series Tm1−x Ybx B12 . It was found that one more contribution to C (T ) with a maximum near T ≈ 10 K is independent of the magnetic field and apparently should be attributed to the effects of a nanosize clustering of Yb ions [184]. Optical spectra To shed more light on the characteristics of the dynamic charge stripes detected in the x-ray diffraction experiment on the semiconducting Tm0.19 Yb0.81 B12 solid solution (see Chapter 3), broadband reflectivity measurements have been carried out at room temperature, see Fig. 4.47(a) [43]. Then,

Metal–Insulator Transition in YbB12 and Solid Solutions Ybx R1−x B12 (R = Lu, Tm) 417

using the Kramers–Kronig analysis, the spectrum of dynamical conductivity was calculated, which is shown here in Fig. 4.47(b). Among the most important results, two issues have been mentioned in Ref. [43]: (i) The conductivity at terahertz frequencies, σ (30– 40 cm−1 ) ≈ 1500 −1 cm−1 , is about an order of magnitude below the measured dc conductivity σDC ≈ 13 000 −1 cm−1 and (ii) modeling the mismatch with the Drude conductivity term (dashed line in Fig. 4.47(b)) provides the scattering rate of carriers γ ≈ 8 cm−1 , which coincides very well with the damping of two quasilocal vibrations of the heavy R ions (Tm and Yb) located at 107 cm−1 (∼154 K) and 132 cm−1 (∼190 K), marked by two asterisks in Fig. 4.47(b). For comparison, similar values of the Einstein temperature E = 160–206 K have been detected previously in EXAFS [60], heat capacity [1] and INS [89, 110] studies of the dodecaborides. Taking into account that these rattling modes induce a variation in the 5d-2p hybridization in RB12 [85], the authors [43] have suggested that this kind of “modulation” of the conduction band is responsible for the dynamic charge stripes formation in the dodecaboride matrix. Besides, the frequency ∼2.4 × 1011 Hz (8 cm−1 ) can be deduced from the conductivity spectra as the characteristic of the quantum motion of charges in the dynamic stripes. As a result, the room-temperature dc conductivity in Tm0.19 Yb0.81 B12 should be attributed to the charge transport in dynamic stripes that are more conductive than the surrounding semiconducting matrix. Consequently, the value of the dc conductivity is determined by the stripes that percolate through the crystal, while the relatively smaller ac conductivity is provided by the THz-FIR reflectivity of the “whole” sample (conducting stripes + semiconducting matrix). Quantum critical behavior It has been predicted [185, 186] that the AFM instability arising near the QCP at T = 0 should substantially modify the characteristics of strongly correlated electron systems in a wide temperature range, including room temperature. To examine these predictions, we have studied fine details of the crystal structure of the system Tm1−x Ybx B12 in a wide vicinity of the QCP at xc ≈ 0.25 [170]. It was found that in addition to the decrease in the lattice constant a upon substitution of ytterbium for thulium

418 Magnetism, Quantum Criticality, and Metal–Insulator Transitions in RB12

Figure 4.47 (a) Room-temperature spectra of the reflection coefficient for a Tm0.19 Yb0.81 B12 single crystal. The data points at 20 and 38 cm−1 show measurements with the backward-wave oscillator √ spectrometer. The dotted line is the Hagen–Rubens reflectivity R = 1 − 4ν/σDC calculated with the measured σDC = 13 000 −1 cm−1 . The solid line is the result of fitting the spectrum with a set of Lorentzians. Corresponding excitations are shown separately in panel (b) together with the optical conductivity spectrum (black solid line) obtained by Kramers–Kronig analysis of the reflectivity. Red dots above 4000 cm−1 correspond to a direct measurement on an ellipsometer. The dashed line models the mismatch between the THz-FIR and dc conductivity with the Drude expression σ (v) = σDC (1 − i ν/γ )−1 . Asterisks denote quasi-local vibrations of heavy R ions at 107 and 132 cm−1 with the damping γ ≈ 8 cm−1 . Reproduced from Ref. [43].

(lanthanide compression), the a(x) plot near the QCP at xc ≈ 0.25 exhibits an anomaly, which evidently results from the abrupt decrease in the boron-cation distance near xc , see Fig. 4.46(b). The high accuracy in determining the structural parameters allowed the authors to determine also the variation in the distance between

Conclusions 419

boron atoms within the B12 clusters, r(B–B) intra , and between the neighboring B12 clusters, r(B–B) inter . In Fig. 4.39(a), it is seen that the distance between the B12 clusters in Tm1−x Ybx B12 with x near xc decreases stepwise, whereas the size of the B12 clusters remains the same. In contrast, above the QCP, the distance between the boron clusters tends to a constant value, whereas their size decreases gradually in the composition range of x = 0.25–0.7. It has been emphasized in Ref. [170] that the amplitude of changes in the structural parameters a, dR-B , r(B–B) intra , and r(B–B) inter is too small to explain the nature of the MIT in the composition range x > 0.5, so, a pronounced (by a factor of about 7) increase in resistivity in the Tm1−x Ybx B12 series at T = 270 K (Fig. 4.38) should be attributed to the development of the fcc lattice instability in combination with the unstable electron configuration of the Yb ion. To test the quantum-critical behavior in the Tm1−x Ybx B12 series near xc ≈ 0.25 at low temperatures, we have also studied the temperature dependence of the magnetic contribution to the specific heat C m of the Tm0.74 Yb0.26 B12 single crystal [183, 187]. A logarithmic divergence of the renormalized Sommerfeld coefficient C m /T ∝ − ln T was found at the temperature below 4 K which is typical for a system at QCP [188], and it is attributed usually to the dramatic renormalization of the quasiparticles’ effective mass and to the issue of a non-Fermi-liquid behavior of heat capacity. It was noted in Refs. [183, 187] that the magnetic field H > 30 kOe suppresses completely the quantum critical regime, and at low temperatures the specific-heat curves C m (T ) demonstrate the Schottky anomaly.

4.6 Conclusions Until recently, it was commonly believed that the filling of the 4f shell in the RE dodecaborides RB12 (R = Tb, Dy, Ho, Er, Tm, Yb and Lu) in the range from TbB12 (n4f = 8) to LuB12 (n4f = 14) has only a single singularity at YbB12 due to the Yb-ion valence instability (n4f ≈ 13.05). The results presented in this chapter clearly demonstrate the complexity of all the RB12 compounds including the AFM (TbB12 –TmB12 ) and nonmagnetic (LuB12 ) metals.

420 Magnetism, Quantum Criticality, and Metal–Insulator Transitions in RB12

The reason for this complexity is the cooperative dynamic JT instability of the rigid boron covalent network, which produces trigonal distortions and results in the symmetry lowering of the fcc lattice. The ferrodistortive dynamics in the boron sublattice (see also Chapter 3) generates both the collective modes (overdamped oscillators in the frequency range 250–1000 cm−1 ) that can be seen in the dynamic conductivity spectra of all the metallic RB12 compounds [189] and the rattling modes — quasilocal vibrations of the heavy RE ions inside the oversized B24 cavities (the radius ∼1.2 A˚ of the B24 complex considerably exceeds the ionic radius of the RE ˚ The large amplitude displacements of the R ions ions ∼0.9–0.95 A). (Einstein oscillators, kB E = 14–18 meV) immediately cause (i) the development of vibrational instability at intermediate temperatures TE ≈ 150 K on reaching the Ioffe–Regel limit and (ii) strong changes in the hybridization of the R 5d and B 2p atomic orbitals by varying their overlap. Accordingly, these overlap oscillations along the [110] axis (the direction of the shortest R–R distance in the fcc lattice) are responsible for the modulation of the conduction band constructed from these R 5d and B 2p electron states. The modulation of the conduction-electron density with the frequency ∼2 × 1011 Hz [43] should be discussed in terms of the emergence of dynamic charge stripes which are the feature of a nanoscale electron instability and electronic phase separation. These nonequilibrium charge carriers dominate in the RE dodecaborides, taking 50–70% from the total number of the conduction electrons, and these hot electrons can no longer participate in the indirect exchange (RKKY interaction) between the RE magnetic moments which are located at the distance of ∼5.3 A˚ from each other. As a result, the suppression of the magnetic exchange interaction along the [110] direction of the nanosize filamentary structure should be considered as the main consequence of the electron instability, providing a magnetic symmetry lowering and leading to the field-angular phase diagrams of the RB12 antiferromagnets in the form of the Maltese cross. Because of the loosely bound state of RE ions in the rigid boron cage, there is also an order-disorder phase transition to the cageglass state at the temperature T ∗ ≈ 60 K resulting in the random displacements of the R ions from their positions in the fcc lattice. The disordering induces immediately a formation of the nanosize

Conclusions 421

clusters of RE ions (couples, triples, etc.) with AFM exchange, and these nanodomains should be apparently taken into account to explain the nature of the short-range magnetic order detected in the RB12 antiferromagnets at temperatures far above TN . It was certainly established that these AFM nanosize clusters, for example, in HoB12 have a cigarlike form, elongated parallel to the 111 axis [94]. The situation becomes much more complicated in the dodecaborides with Yb ions. Indeed, in this case, in addition to the JT instability of the boron cage, the instability of Yb 4f -electron configuration appears, providing one more mechanism of the charge and spin fluctuations in RB12 . When this additional charge degree of freedom is switched on, a strong and monotonic shift of the collective modes to high frequencies is observed in the Ybx Lu1−x B12 series, until the two Lorentzians appear in the dynamic conductivity spectra of YbB12 at 40 meV (∼320 cm−1 ) and 250 meV (∼2000 cm−1 ) [143]. An appearance of Yb–Yb pairs with the temperature lowering can be probably considered the main factor which is responsible for the charge- and spin-gap formation in YbB12 , and the many-body states are characterized by the localization radius ∼5 A˚ [42, 45], which is about equal to the Yb–Yb distance in the fcc lattice. Taking into account that this indirect gap opening is a phonon-assisted process [45, 143], and there is a coupling between the intensities of the anomalous acoustic phonons and the M1, M2 magnetic excitations in YbB12 [190], one has to conclude that the Yb dimers should be vibrationally coupled complexes. The process of the Yb binding in pairs develops gradually below the temperature T ∗ ≈ 60 K, in the cage-glass state where static displacements of Yb ions occur in the double-well potentials [42, 137]. It is worth emphasizing here that (i) the mechanism underlying the formation of heavy fermions in the vicinity of Yb ions is different from the Kondo one, (ii) the localization radius for these many-body states (heavy fermions) turns out to be nearly equal to the R–R ˚ being much smaller than that of the “Kondo cloud” distance (∼5 A), ˚ (> 20 A), and (iii) the strong local spin fluctuations near the RE ions lead to the formation of both heavy fermions and the spin-polarized nanodomains [123, 191] which represent the 5d component of the

422 Magnetism, Quantum Criticality, and Metal–Insulator Transitions in RB12

magnetic structure in the RB12 antiferromagnets [87, 98]. Taking into account that the anisotropic Kondo Hamiltonian is a special case of the “spin-boson” Hamiltonian describing the properties of a dissipative two-level system [192], it is natural to argue [193] that the origin of spin fluctuations in YbB12 is related to the quantum oscillations of a heavy particle (magnetic ion) between the states in the DWP, and the Kondo physics is invalid in this case. In the framework of the common scenario for the RB12 family, the symmetry lowering which develops at low temperatures, T < 30 K, should be attributed to the formation in YbB12 of a filamentary structure of conduction channels — dynamic charge stripes. The intra-gap states with a binding energy of 2.7–5 meV are characterized by a localization radius ap ≈ 9 A˚ [42, 137]. The ap ˚ and the arrangement of value exceeds the lattice constant (∼7.5 A), both the Yb-pairs and the charge stripes leads to the spontaneous selection of a special direction in the formally cubic crystal. Thus, the coherent regime of charge transport can be attributed to the percolation through the network of the many-body states and charge stripes in YbB12 and Yb-based dodecaborides [42]. To summarize, when discussing the mechanisms responsible for the strong charge-carrier scattering in the RB12 compounds, it is worth emphasizing that the local on-site spin fluctuations play the dominant role in the charge transport of all magnetic RE dodecaborides. Indeed, when n4f increases from HoB12 to LuB12 in the range 10 ≤ n4f ≤ 14, one should expect a monotonic increase in the mobility of charge carriers μH (n4f ), which follows from a decrease in the de Gennes factor G = (g J − 1)2 · J (J + 1) characterizing the magnetic scattering intensity in the RB12 series. Instead, below the QCP in the range 10 ≤ n4f ≤ 12.3, a decrease in μH (n4f ) is observed (Fig. 4.48), and the mobility minimum near x = 0.3–0.5 in Tm1−x Ybx B12 corresponds to the antiferromagneticparamagnetic transition with the QCP near xc ≈ 0.25. Furthermore, the substitution of Yb for Tm in the range x ≥ 0.5 (n4f ≥ 12.5 in Fig. 4.48) leads to an increase in μH (x), while a monotonic and sharp increase in the resistivity ρ(x) (Fig. 4.38) and the Hall coefficient R H (x) (Fig. 4.48) at low temperatures is observed in the whole range 0 < x < 1 [42]. The metal–insulator transition is accompanied by rather low mobility values μH (T ) ≈ 27 cm2 /(V·s)

Conclusions 423

2

103

4 RB12

(a)

102

TN, K

8

μH , cm2/(V s)

G

0

101

4 AF

P

0 QCP

(b)

−RH , cm3/C

Ea

10−3

10−4

10−1

EF EF − Eg/2 EF + Eg/2

M

10

11

10−2 I

12 n4f

13

SC

14

ρH2, μΩ cm

x1 < x2 < x3

10−2

10−3

Figure 4.48 (a) The de Gennes factor G = (g J − 1)2 · J (J + 1), the Hall mobility μH (n4f ) at the LHe temperature (the data for HoB12 (n4f = 10), ErB12 (n4f = 11), TmB12 (n4f = 12), and LuB12 (n4f = 14) are from Refs. [2, 42, 56], and the data for YbB12 (n4f ≈ 13.05) are from ´ temperature TN (n4f ) versus the filling of the Refs. [45, 150]) and the Neel 4f shell of the RE ions in the RB12 compounds; AF and P denote the antiferromagnetic and paramagnetic phases; M, I, and SC stand for the metal, insulator, and superconductor. (b) The Hall coefficient R H (n4f ) and the even harmonic contribution ρH2 (n4f ) for H = 15 and 80 kOe. The inset shows the renormalization effect in the intra-gap density of states in the series of Tm1−x Ybx B12 compounds. Reproduced from Ref. [42].

in YbB12 , which are obviously caused by the scattering of carriers on strong spin and charge 4f -5d fluctuations. Evidently, these fast fluctuations of electron density should be discussed among the factors responsible for the suppression of the AFM ground state in the Yb-based dodecaborides. And finally, in passing to the nonmagnetic superconducting metal LuB12 (n4f = 14), the Hall

424 Magnetism, Quantum Criticality, and Metal–Insulator Transitions in RB12

mobility increases strongly (∼100 times, see Fig. 4.48), but the nonmagnetic reference compound also demonstrates the same kind of structural and electron instabilities.

Acknowledgments This work was supported by the Russian Science Foundation, projects No. 17-12-01426 and 21-12-00186. The author is grateful to S. Demishev, V. Glushkov, K. Flachbart, G. Grechnev, N. Shitsevalova, B. Gorshunov, S. Gabani, K. Siemenmeyer, N. Bolotina, A. Dudka, V. Mironov, T. Mori, and V. Moshchalkov for helpful discussions and to K. Krasikov for his assistance in preparing the manuscript. Many of the presented results would have been impossible without highquality samples provided by N. Shitsevalova and V. Filipov.

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176. Sluchanko, N. E., Bogach, A. V., Glushkov, V. V., Demishev, S. V., Ignatov, M. I., Samarin, N. A., Burkhanov, G. S., and Chistyakov, O. D.; “Genesis of the anomalous Hall effect in CeAl2 ”; J. Exp. Theor. Phys. 98, 793–810 (2004). 177. Sluchanko, N. E., Glushkov, V. V., Demishev, S. V., Burkhanov, G. S., Chistyakov, O. D., and Sluchanko, D. N.; “Heavy fermions in CeAl3 ”; Physica B 378–380, 773–774 (2006). 178. Ye, J., Kim, Y. B., Millis, A. J., Shraiman, B. I., Majumdar, P., and Teˇsanovi´c, Z.; “Berry phase theory of the anomalous Hall effect: application to colossal magnetoresistance manganites”; Phys. Rev. Lett. 83, 3737 (1999). 179. Kim, Y. B., Majumdar, P., Millis, A. J., and Shraiman, B. I.; “Anomalous Hall effect in double exchange magnets”; arXiv:cond-mat/9803350 (1998). 180. Azarevich, A., Bogach, A., Demishev, S., Glushkov, V., Shitsevalova, N., Fil´ S., Pristas, G., Flachbart, K., Gavrilkin, S., and Sluchanko, ipov, V., Gabani, N.; “Magnetic phase diagram of Tm0.96 Yb0.04 B12 antiferromagnet with dynamic charge stripes and Yb valence instability”; Acta Phys. Pol. A 137, 788–790 (2020). 181. Sluchanko, N. E., Azarevich, A. N., Bogach, A. V., Glushkov, V. V., Demishev, S. V., Gavrilkin, S., Gabani, S., Flachbart, K., Shitsevalova, N., Filipov, V. B., Vanacken, J., Moshchalkov, V. V., and Stankiewicz, J.; “Magnetoresistance anisotropy and magnetic H-T phase diagram of Tm0.996 Yb0.004 B12 ”; Acta Phys. Pol. A 126, 332–333 (2014). 182. Bray, A. J.; “Nature of the Griffiths phase”; Phys. Rev. Lett. 59, 586 (1987). 183. Sluchanko, N. E., Bogach, A. V., Glushkov, V. V., Demishev, S. V., Gavrilkin, ´ S. Y., Shitsevalova, N. Y., Filipov, V. B., Gabani, S., and Flachbart, K.; “Anomalies of the specific heat near the quantum critical point in Tm0.74 Yb0.26 B12 ”; JETP Lett. 91, 75–78 (2010). 184. Azarevich, A. N.; “Magnetoresistance and heat capacity of Tm1−x Ybx B12 solid solutions”; PhD thesis; Moscow Institute of Physics and Technology; Moscow (2014). 185. Kopp, A., and Chakravarty, S.; “Criticality in correlated quantum matter”; Nat. Phys. 1, 53–56 (2005). 186. Lonzarich, G. G.; “Quantum criticality: magnetic quantum liquid enigma”; Nat. Phys. 1, 11–12 (2005). 187. Sluchanko, N. E., Azarevich, A. N., Bogach, A. V., Glushkov, V. V., Demishev, S. V., Gavrilkin, S. Y., Shitsevalova, N. Y., Fillipov, V. B., Gab´ani,

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Chapter 5

Raman Spectroscopy of Metal Borides: Lattice and Electron Dynamics Yuri S. Ponosov Institute of Metal Physics UB RAS, S. Kovalevskoy str. 18, 620137 Ekaterinburg, Russia [email protected]

Raman spectroscopy is usually used as a standard technique for studying the vibrational spectra in solids. In the study of boride systems, the possibilities of the method turned out to be much richer, allowing one to study not only phononic but also electronic excitations and various types of electron and phonon interactions. The main objective of this chapter is to review the experimental and theoretical work on the study of the lattice dynamics, electronic excitations, and various interactions (electron–phonon, phonon– phonon, magnetic, etc.) in rare-earth and transition-metal borides.

5.1 Introduction Raman scattering is a process of inelastic scattering of an electromagnetic wave, in which the energy and momentum conservation laws are fulfilled. Since the photon momentum in the optical region of the spectrum is small, Raman spectroscopy in the first order of Rare-Earth Borides Edited by Dmytro S. Inosov c 2022 Jenny Stanford Publishing Pte. Ltd. Copyright  ISBN 978-981-4877-56-5 (Hardcover), 978-1-003-14648-3 (eBook) www.jennystanford.com

444 Raman Spectroscopy of Metal Borides

scattering (with the participation of a single excitation) is restricted to excitations with the wave vector q ≈ 0 in the case of a perfect lattice. These can be vibrational, magnetic, or electronic excitations, allowed by the selection rules for a given structure. In fact, the Raman spectrum is a kind of imprint of a certain substance. There are several reasons for the appearance of additional lines in the Raman spectrum, exceeding the number of allowed lines in the first-order scattering for a given structure. So in Raman scattering of the second order (for example, with the participation of two phonons or magnons), excitations with a nonzero wave vector also appear in the spectrum, as soon as the sum of their wave vectors is close to zero. Even in perfect lattices, intensities of the two-phonon (or two-magnon) excitations in some cases exceed the intensities of one-phonon lines, which helps to obtain information on the density of phonon (magnon) states. The appearance of defects in the structure also leads to the appearance of excitations with a nonzero wave vector in the Raman spectra. This causes violation of the selection rules for the wave vector, which is manifested in the appearance of additional lines in the spectra beyond those allowed for a given structure. All the mentioned excitations are quite vividly manifested in the Raman spectra of boride compounds. It is obvious that the main features of the lattice dynamics of boride systems are determined by their structures, which are fundamentally similar for both metal-rich and higher borides. The most characteristic feature of the crystal structure of boron compounds is the presence of clusters in the boron sublattice. The metal ions are located in the voids between the boron clusters or inside them. Due to the strong covalent bonding, boron atoms form flat grids in MB2 , octahedra in MB6 , cuboctahedra and icosahedra in MB12 , and icosahedron chains in MB66 . Relatively weak interactions between metal and boron atoms in borides result in two weakly coupled sublattices. This circumstance, together with a large mass difference, to a large extent determines the phonon spectrum of borides characterized by decoupling of metal and boron vibrations. The main contribution to the density of states of acoustic phonons comes from the metal-atom sublattice, the high-frequency phonons stem from the boron ions. However, the bonding with the boron sublattice substantially depends on the type of metallic ion. The presence of

Raman Scattering by Phonons

transition and rare-earth (RE) metals in the boride lattice leads to a wide diversity of electrical and magnetic properties, as well as high hardness, melting points, and characteristic temperatures of these materials, due to strong covalent boron–boron bonds. Inelastic scattering spectroscopies, including Raman scattering in the optical photon range and inelastic x-ray scattering in the keV range, as well as neutron scattering, are able to provide information not only about the lattice dynamics but also about the structural features of the material, its electronic spectrum, and existing interactions. Our following presentation will be devoted mainly to the study of hexaborides and dodecaborides of rareearth and transition metals by these methods. These materials have attracted widespread attention from researchers for several decades because of the wealth of their phase states.

5.2 Raman Scattering by Phonons 5.2.1 Raman-Active Phonons in Hexaborides Numerous Raman studies of hexaborides have been performed since the mid-seventies of the last century [1–9]. The hexaborides MB6 crystallize in a simple cubic structure of the CsCl type (space group ¯ P m3m = Oh1 ), which is formed from octahedral B6 clusters and a single metallic ion. The factor-group analysis predicts a reduced representation of the form (5.1)  = A1g + Eg + T1g + T2g + 2T1u + T2u for the 18 optical vibrations of the unit cell. Vibrations of symmetries A1g , T2g , and Eg are Raman-active; all of them represent internal vibrations of the B6 octahedra. These three vibrations, located in the frequency range of 600–1400 cm−1 , dominate all published experimental Raman spectra, as shown in Fig. 5.1(a). In turn, the calculated dispersion curves agreed well with both the Raman data and the dispersion of low-frequency acoustic and optical branches of the phonon spectrum measured by neutron spectroscopy [9,11–19]. All studies found the decrease of phonon energies with increasing lattice parameter by substitution of M ions (divalent Ca, Sr, Ba, Yb or trivalent La, Ce, Pr, Nd, Sm, Eu, Gd, Dy), see Fig. 5.1(b). Such

445

446 Raman Spectroscopy of Metal Borides

Figure 5.1 (a) Raman spectra of MB6 (M = Ca, La, Ce, Pr, Sm, Gd, Dy, and Yb) at room temperature and at the excitation wavelength of 488.0 nm. The spectra are depicted in the order of the decreasing lattice constants from top to bottom. The dotted and solid arrows denote extra modes discussed in this paper. The peaks under dotted arrows appear in the spectra excited by 602 nm laser light. Triangles denote CEF excitations. (b) Observed peak energy vs. lattice parameter. The closed marks denote trivalent cations, and open ones are for divalent Ca and Yb and intermediate-valent Sm cations. The open and closed triangles denote the observed peaks in the lower-energy excitation spectra. The dotted lines are a guide for the eyes. Reproduced from Ogita et al. [10].

behavior was attributed to especially strong interoctahedral B–B bonding. For the T2g mode, the energy difference of ∼100 cm−1 was found between the trivalent and divalent crystals. The valence dependence of the T2g mode was discussed with examples of SmB6 and Gd1−x Eux B6 [2, 4, 6, 20]. These observations suggest that the contribution of the M–B interaction in the energies of the B6 modes decreases with increasing conduction-electron concentration. Ogita et al. [10] suggested the existence of an interference effect between the sharp phonon and a broad electronic

Raman Scattering by Phonons

continuum due to expected electron–phonon interaction in trivalent crystals.

5.2.2 Extra Phonon Features in Raman Spectra of Hexaborides In addition, most researchers reported the observation of the splitting of certain lines, as well as the observation of weaker additional lines in the spectra of many hexaborides [Fig. 5.1(a)]. Natural boron consists of two stable isotopes, 10 B (19.6%) and 11 B (80.4%). The random distribution of these isotopes in the MB6 lattice results in the splitting of degenerate vibrations and the activation of vibrations that are forbidden under the Oh1 lattice symmetry. Furthermore, the relative mass difference of ∼10% has an important effect on the frequencies of certain vibrations. There is no doubt that research on pure isotopic samples can resolve a number of controversial interpretations of the spectra. Two kinds of extra peaks have been commonly observed near 200 and 1400 cm−1 [6, 10, 20–24]. The authors suggested that twophonon light scattering is the most plausible cause of the appearance of wide peaks in the 1400 cm−1 region. Their energies are close to the double energy of the T2g mode, which has a flat dispersion in all directions of the Brillouin zone. The overtones and combinations of acoustic and optical modes at the X , M, and R critical points and along the  X ,  M, and  R directions of the Brillouin zone are allowed for all Raman-active polarization geometries. The calculations also show noticeable peaks in the density of phonon states in the half-energy region (∼700 cm−1 ). Another argument in favor of such an interpretation of the extra lines is illustrated in Fig. 5.2(a), where the Raman spectrum of isotopically pure La11 B6 is shown for different polarization geometries [25]. There are clearly visible peaks in the region of 2300 cm−1 . Like the lines near 1400 cm−1 , they have A1g symmetry (characteristic of overtone excitations) and, unambiguously, are overtones of the Eg mode, also having a flat dispersion along the Brillouin zone. This figure also shows another phonon of A1g symmetry at 1150 cm−1 , which was previously attributed to the Eg vibration [26]. Its frequency is close to the doubled energy of the T1g optical branch, giving a large peak

447

448 Raman Spectroscopy of Metal Borides

11

La B6

a

532 nm

300K

b

LaB6

2

Intensity (a.u.)

2 *

* *

X, X

*

1

X+Y, X+Y

CeB6

1

X+Y, X-Y SmB6

X,Y 0

0

0

1000

2000 -1

Raman shift (cm )

50

100

150

200

250

-1

Raman shift (cm )

Figure 5.2 (a) Raman spectra of an isotopically pure La11 B6 crystal measured in different polarization geometries. X and Y correspond to the crystal axes of [100] and [010], respectively. The asterisks indicate peaks uniquely identified as two-phonon scattering, and the arrows indicate weaker features. (b) Low-frequency peaks in some RE hexaborides measured with 633 nm excitation wavelength. Reproduced from Ref. [25].

in the density of states [18], which clearly indicates its two-phonon origin. The origin of anomalous low-energy peaks at around 200 cm−1 is also controversial. It was assigned as the optical T1u mode at the  point [20], or as the defect-induced Raman-inactive LO(X ) mode, which couples most strongly to the valence fluctuations [6], or as a two-phonon scattering of the longitudinal acoustic (LA) phonon at the Brillouin-zone boundary [10, 20]. The dependence of the observed peak energy of extra low- and high-frequency modes on the lattice parameter is also included in Fig. 5.1(b). All extra modes below 200 cm−1 show opposite correlation against the Raman-active phonons, which is consistent with the dependence on the lattice parameter of the force constant describing the metaloctahedron interaction [9]. It was noted [10] that the extra modes have a more complex and broad structure for the sample with the

Raman Scattering by Phonons

smaller lattice parameter, e.g., for Dy, Gd, and Pr. Much narrower lines were observed for La, Ce, and Sm hexaborides, where the second low-energy lines were found at frequencies of 100 cm−1 , as one can see in Fig. 5.2(b). All these lines do not have a clear polarization dependence. Using the found correlation of energy and intensity of low-frequency vibrations and the size of cage space for the metal atom, Ogita et al. [10] came to the conclusion that the extra modes near 200 cm−1 should be assigned to second-order excitations of the acoustic branch, and their anomalous properties

Figure 5.3 Raman spectra of YB6 , measured at various temperatures for the symmetries A1g + Eg (XX) and T2g (XY) with the 633 nm excitation wavelength. Reproduced from Ref. [27].

449

450 Raman Spectroscopy of Metal Borides

are related to the M ion movement in the cage constructed by the boron framework. This conclusion is consistent with the measured and calculated phonon-dispersion curves of the hexaborides [9, 11–19]. Acoustic branches in all of them have flat areas in a large part of the Brillouin zone, which provides fairly pronounced peaks in the density of phonon states. The doubled peak energy of acoustic phonon density agrees well with the frequencies of the lines observed near 200 cm−1 . As can be seen in Fig. 5.2(b), the shape of the lines at ∼100 and 200 cm−1 for LaB6 , SmB6 , and CeB6 is similar, and their energies differ by a factor of 2. A similar situation is observed in hexaborides where these lines are more structured (DyB6 , GdB6 , and YB6 ), see Fig. 5.1(a). Figure 5.3 shows the YB6 crystal spectra measured in two polarization geometries in a wide temperature range. The low-frequency part of the spectrum of YB6 at high temperatures features two bands at ∼80 and 160 cm−1 in both polarization geometries shown in Fig. 5.3. We attribute the former line to the density of states of acoustic phonons. Its energy is close to those calculated by Xu et al. [28] and observed in tunneling experiments [29–32]. Using “thermal spectroscopy”, Lortz et al. [33] showed that the spectral electron-phonon scattering function of YB6 is dominated by the lowfrequency feature at ∼8 meV. These vibrations are inactive in the Raman spectra of hexaborides with an ideal structure. Consequently, the emergence of this line was associated with the violation of selection rules for the wave vector owing to structural imperfections, e.g., vacancies of one of the sublattices [34]. A similar situation occurs in carbides and nitrides of transition metals, where the one-phonon spectra forbidden for the NaCl structure are observed due to the presence of carbon and nitrogen vacancies and reflect the phonon density of states [35]. The second line at the doubled frequency is caused by overtones of acoustic modes, which are allowed in the Raman spectra and appear in many hexaborides [6, 10, 20, 21, 23]. In LaB6 , the two-phonon line appears near 200 cm−1 , while a weak first-order line is seen near 100 cm−1 in Fig. 5.2(b). The frequencies of both extra lines are independent of the isotopic composition, which additionally confirms that these excitations are associated with vibrations of metallic ions. The intensity of these low-frequency bands in LaB6 increases significantly with decreasing

Raman Scattering by Phonons

100

LaB6

-1

Frequency (cm )

90 80

YB6

70 60 50

a

40

b

-1

A1g frequency (cm )

1320

YB6

1300

1240

LaB6

1220 1200 0

200

400

600

Temperature (K) Figure 5.4 (a) Temperature dependence of the frequencies of acoustic phonons in YB6 and LaB6 . In the case of YB6 , different symbols represent the frequencies of peaks found by decomposition of the first- and secondorder spectra (the frequencies of the second-order spectra are divided by 2) into components. The lines show trends. In the case of LaB6 , the average energy of the two-phonon spectrum divided by 2 is shown. (b) Temperature dependence of the frequencies of optical A1g phonons in YB6 and LaB6 . The lines show the contributions of thermal expansion to the phonon frequencies. Reproduced from Ref. [27].

excitation energy of the spectra, while the intensity of Raman-active phonons, especially T2g , on the contrary, decreases. A similar trend is observed for other RE hexaborides [10]. This indicates possible resonance effects, although the contribution of surface phonons to the observed effect is assumed [26, 36–38].

451

452 Raman Spectroscopy of Metal Borides

Obviously, the first- and second-order spectra of YB6 in the XX and XY geometries are almost structureless at high temperatures. With a decrease in temperature, a low-frequency shoulder appears in the two-phonon XX spectrum; this shoulder is shifted from 130 cm−1 at 430 K to 85 cm−1 at 8 K. All spectra become more structured. In addition, one-phonon spectra exhibit a shoulder at ∼42 cm−1 at low temperatures, the energy of which corresponds to half of the frequency of the low-frequency shoulder in the twophonon XX spectrum. It can be seen from Fig. 5.3 that in both spectra, the frequencies of the specific features are close to each other, although they are more clearly pronounced in the two-phonon spectra. This is associated with the fact that the second-order Raman scattering is an allowed process, the main contributions to which are made by phonons from the critical points of the Brillouin zone in this structure, whereas the observed first-order Raman scattering is activated by the disorder with a little known photon–phonon coupling constant. Assuming that the two-phonon spectra are primarily caused by overtone scattering, we fitted the first- and second-order spectra to three Gaussian peaks (for two transverse and one longitudinal branch). The found estimates of the frequencies of the peaks are shown in Fig. 5.4(a). Undoubtedly, the contributions of each branch to various spectra differ in intensity, which leads to a scatter of the experimental points. However, the general tendency is beyond doubt. All acoustic branches in YB6 exhibit softening with a decrease in temperature. Their energies just above the transition to the superconducting state are 42 cm−1 (∼5 meV) and 60–70 cm−1 (∼8 meV). These values agree well with the peak energies found in tunneling experiments [29–32]. A similar trend was observed in a limited temperature range in the Raman spectra of YB6 , DyB6 , and GdB6 [10, 21, 23, 24]. Figure 5.4(a) also shows the temperature dependence of the energy of the acoustic-phonon peak of LaB6 (half-energy of the twophonon band), which was fitted to a Gaussian. The width of this band is small (∼20 cm−1 ). For this reason, we did not use decomposition into components. Although this peak is more structured at low temperatures, its frequency remains almost unchanged with an increase in temperature to 200 K. Some hardening occurs at even higher temperatures. However, the magnitude of the effect is lower

Raman Scattering by Phonons

than that in YB6 by a factor of ∼5. The energies of high-frequency vibrations of the boron sublattices of both hexaborides demonstrate an increase in frequencies with a decrease in temperature, which is seen for the A1g modes in Fig. 5.4(b). Thus, the appearance of most extra lines in the Raman spectra of hexaborides can be explained without invoking assumptions about the distortion of the crystal structure, which have been put forward in a number of publications [24, 39]. The lattice-dynamical calculations that assume such distorted structures, however, cannot explain the observed spectra [39]. In addition to the discussed extra lines, several weaker features are also observed. Most likely, their presence is explained by the reasons already given.

5.2.3 Anharmonicity vs. Electron–Phonon Interaction The spectra of YB6 and LaB6 under pressure are shown in Fig. 5.5. In both materials, the frequencies of single-phonon and two-phonon spectra increase with pressure, and their pressure-induced shifts nearly coincide. This is another confirmation that the origin of the low-frequency feature is due to the first-order defect-induced scattering, while the high-frequency peak represents the secondorder scattering. A linear increase in the energies of acoustic phonons is identical for both hexaborides and is 1.5 cm−1 /GPa. The use of the elastic moduli of 166 GPa for YB6 [40] and 164 GPa for ¨ LaB6 [41] yields the following estimates of the isothermal Gruneisen coefficients: γT = ln ω/ln V = 3.87(6) and 2.15(8) for YB6 and LaB6 , respectively. For high-frequency optical phonons, γT ≈ 1– 1.5 for both compounds. The latter value for LaB6 agrees with the experiments [41, 42] and calculations [18, 19]. Actually, the experimental γT values for acoustic phonons in YB6 appeared to be quite high. However, these values are almost three times lower than the estimates from thermal-expansion data [33] and from the temperature behavior of the resistance under pressure [40, 43]. It is also worth mentioning that the volume anharmonicity of acoustic phonons in LaB6 is also quite high: γT is only a factor of 1.5–2 lower than in YB6 and exhibits a similar frequency behavior. The results obtained show that the thermal effects are much larger than the volume ones and define the negative sign of the isobaric

453

454 Raman Spectroscopy of Metal Borides

Figure 5.5 Raman spectra of YB6 and LaB6 measured at various pressures at 300 K. Reproduced from Ref. [27].

¨ Gruneisen parameter γ P for acoustic modes in both hexaborides under investigation, its absolute value being more than an order of magnitude larger than the positive coefficient γT = 3.87 measured at 300 K. The situation in the case of high-frequency vibrations of the boron sublattice is more complicated. Solid lines in Fig. 5.4(b) show contributions of the thermal expansion to the frequencies of A1g phonons calculated using thermal-expansion data [33, 44] and the measured coefficient γT. The measured energy shifts with temperature in YB6 exceed the contribution of the thermal

Raman Scattering by Phonons

expansion, whereas in LaB6 they turned out to be smaller. A similar result was also obtained for other Raman-active phonons. The observed difference in the temperature behavior of high-frequency modes in YB6 and LaB6 is surprising, because the frequencies of these phonons differ only by ∼5%, and their isothermal ¨ Gruneisen coefficients are identical. Additional contributions to the temperature shifts of phonon energies are usually attributed to phonon–phonon interactions described by higher than secondorder terms in the expansion of the potential energy in powers of ion displacements. However, such effects can also occur due to nonadiabatic effects in the electron–phonon interaction [45], which clearly manifest themselves in the asymmetric shapes of some phonon lines in hexaborides. The obtained result may indicate significantly different contributions of 3rd and 4th order anharmonicities to the self-energies of B-atom vibrations, which should be taken into account when considering the role of these phonons in the enhancement of the superconducting critical temperature, Tc , in YB6 in the framework of an electron–phonon pairing mechanism. Early experiments on Raman scattering in hexaborides [10, 21, 23, 24] showed that the vibration energies of metal atoms (the aforementioned extra low-frequency lines) depend on the relation between the lattice parameter and the radius of the metal ion inside the “cage” of boron octahedra. The peak energies decrease with an increase in this parameter. They also exhibit anomalous softening with a decrease in temperature. The above figures are one more confirmation of the following fact: The vibrations of lanthanum atoms possess higher frequencies and exhibit less pronounced softening effects. The ratios of the lattice parameter to the radius of the metal ion nearly coincide for YB6 and DyB6 . The energies of acoustic phonons are very close to each other [10, 21, 23, 24, 27]. Recent measurements of phonon dispersion by means of inelastic x-ray scattering [46] revealed general softening of acoustic modes in DyB6 , which leads to a dip in the [100] dispersion curve at the X point at the edge of the Brillouin zone and to a Kohn anomaly in the [110] direction, as illustrated in Fig. 5.6(a,b). Similar effects were observed in the dispersion curves and their temperature dependences for GdB6 and TbB6 [47, 48]. The energy of the longitudinal phonon in DyB6 at the X point decreases by

455

456 Raman Spectroscopy of Metal Borides

∼30% to 5 meV (40 cm−1 ) with a decrease in temperature from 300 to 23 K, as can be seen in Fig. 5.6(a). A similar softening is found in YB6 [Fig. 5.4(a)]. It can also be associated with the phonon anomaly in the [100] direction. It should be mentioned that the calculated energy of the longitudinal branch at the X point of the Brillouin zone of YB6 [28] is indeed anomalously low. The existing explanations of anomalies in the behavior of acoustic phonons in

10 (a) 8

GdB6

(b)

6

6 DyB6

TbB6 Q = (5 + ξ, 0, 0) RT 25 K Q = (5.5, 0.25, 0) 25 K

4 2 0 0.0

0.1

Energy [meV]

Energy [meV]

8

0.2

0.3

0.4

4 TbB6, Q = (5, ζ, ζ) RT 25 K 5K

2 0 0.0

0.5

0.1

0.2

0.3

0.4

0.5

ζ [r.l.u.] of Q = (5, ζ, ζ)

ξ [r.l.u.] of Q = (5 + ξ, 0, 0)

2.0

(c) TbB6 RT

2.0 (d) TbB6 RT

25 K FWHM [meV]

FWHM [meV]

2.2

1.8 1.6

0.1

1.8 1.6 resolution

resolution 1.4 0.0

25 K

1.4 0.2

0.3

0.4

ξ [r.l.u.] of Q = (5 + ξ, 0, 0)

0.5

0.0

0.1

0.2

0.3

0.4

0.5

ζ [r.l.u.] of Q = (5, ζ, ζ)

Figure 5.6 (a) Dispersion of the longitudinal phonon modes propagating along the [100] axis of TbB6 at room temperature (circles) and 25 K (squares). The thick dashed curve and dotted curve are the dispersion relations of GdB6 and DyB6 at room temperature, respectively. (b) Dispersion of the transverse mode in TbB6 , propagating along the [011] axis, Q = (5 ζ ζ ), at several temperatures. (c, d) ξ - and ζ -dependences of the full width at half maximum (FWHM) of the INS spectra for different wave-vector directions and temperatures. Broken lines at 1.5 meV show the instrumental energy resolution. Reproduced from Iwasa et al. [48].

Raman Scattering by Phonons

cage compounds assume an important role of anharmonicity. In particular, the presence of large space in the cage of boron octahedra in hexaborides leads to an increase in the vibration amplitude of metal atoms and to a decrease in the contribution of the direct ion-ion interaction [46–48]. This, in turn, increases the relative contribution of the electron–phonon interaction, which facilitates the softening of phonons with an increase in the ratio of the lattice parameter to the radius of the metal ion and a stronger softening with a decrease in temperature. However, the magnitude of the electron–phonon coupling in hexaborides of RE metals is low, which follows from small widths of phonon lines found in the experiments as shown in Fig. 5.4(a,b) [48]. Iwasa et al. [46–48] suggested that the observed softening of the acoustic phonons of RE hexaborides is a precursor of static structural distortions, which metastably occur at higher temperatures, and it is possibly associated with magnetic degrees of freedom. Calculations of the lattice dynamics of RE hexaborides using the superatom model have suggested that the valence and the number of f electrons in the inner atomic shell have a significant influence on the anomalous softening of longitudinal acoustic phonons [49]. The observation of similar effects in dmetal hexaboride YB6 , which experiences a transition to the superconducting state, is very interesting in this regard and requires a clearer understanding of the anomalies of the acoustic branches in hexaborides. The similarity of phonon behavior in YB6 and in hexaborides of trivalent RE metals supports the assumption that the contribution of acoustic modes to the electron–phonon interaction in YB6 is small, although the interaction is stronger than in LaB6 owing to softer phonons. Observations of significant temperature dependences in the dispersion of acoustic phonons of RE hexaborides and similar temperature changes of low-frequency extra lines in Raman spectra of YB6 indicate that these lines cannot originate from purely localized phonons. The latter represent independent vibrations of the guest M atom inside the cage without M-M interactions (rattling mode). Experiments, however, confirm the relation of extra lines with acoustic phonons, in the formation of the dispersion of which both M-M and M-B interactions are important [10, 27].

457

458 Raman Spectroscopy of Metal Borides

5.2.4 Phononic Raman Spectra in Dodecaborides Metal dodecaborides possess a crystal structure of the type UB12 5 ¯ (space group P m3m-O h ). Another concept is the NaCl structure, where the lattice of chlorine atoms is replaced by the lattice of B12 clusters. A reduced representation of the form  = A1g + A2g + A2u + 2Eg + Eu + 2T1g + 3T1u + 2T2g + 2T2u (5.2) is predicted for 36 optical zone-center phonons. As in the hexaboride structure, vibrations of symmetries A1g , T2g and Eg are Ramanactive; all of them represent internal vibrations of the B12 cuboctahedra. In comparison with hexaborides, a very small number of Raman studies have been carried out on dodecaborides [50–58]. Werheit et al. [52] performed a very detailed study of rare-earth (Gd, Dy, Ho, Er, Tm, Yb, Lu) and transition-metal (Y, Zr, Sc) dodecaborides, and both crystals with a natural isotopic composition and enriched with isotopes 10 B and 11 B were studied. Five Raman-active bands dominate all spectra (Fig. 5.7), and their energies fit satisfactorily

Figure 5.7 The Raman spectra of lanthanide dodecaborides with natural boron. Reproduced from Werheit et al. [52].

Raman Scattering by Phonons

Figure 5.8 Phonon frequency of the Raman-active phonons in dodecaborides versus lattice parameter a. Reproduced from Werheit et al. [52].

to linear dependences on the lattice parameter a in Fig. 5.8. The authors [51, 52] noted that in addition to the Raman active modes, the spectra of all compounds contain a number of noticeably weaker modes. They suggested that their appearance indicates distortions in the crystal structures, and discussed the effect of the surface layer on changes in the Raman spectra with decreasing wavelength used to excite the spectra. However, the large depth of the skin layer and the absence of its explicit correlation with the excitation energy of the spectra does not make this argument convincing. As in the case of hexaborides, detailed polarization measurements on LuB12 and ZrB12 crystals provide confirmation that twophonon excitations appear in the spectra (Fig. 5.9). The threepeak high-frequency structure in the region of ∼2000 cm−1 can be unambiguously interpreted as the overtones of a strong multipeak structure in the density of states near 1000 cm−1 [59–62]. It has A1g symmetry characteristic of overtone scattering. The same situation was found in LuB12 spectra. Noticeable peaks near 500 and 1150 cm−1 in ZrB12 also have the prevailing symmetry A1g . Their frequencies are also close to the doubled peak energies in the

459

460 Raman Spectroscopy of Metal Borides

2

4 *

300K

11

Zr B12 300K

Eg

5

*

*

*

2 1

2

T2g

T2g

Intensity (a.u.)

1

Intensity (a.u.)

10 b

a

Eg A1g

LuB12

A1g+Eg

Eg T2g 0 6

11

10K A1g+Eg

B

4 10

B 2

nat

90

c 0 0

500

1000

1500

2000 -1

Raman shift (cm )

2500

0 0

155 100

175 200

300 -1

Raman shift (cm )

Figure 5.9 (a) Raman spectra of 11 B-enriched zirconium dodecaboride, measured in different polarization geometries with 633 nm excitation wavelength. Asterisks indicate peaks uniquely identified as two-phonon scattering, arrows indicate disorder-induced features. (b, c) Low-frequency phonon spectra measured at the excitation wavelength of 633 nm. The vertical solid line separates the one-phonon and two-phonon regions of the spectra. (b) Different symmetry components of the phonon spectra of Zr11 B12 and Lu11 B12 (the intensity of the spectrum of the Lu11 B12 sample is increased by a factor of 5, T = 300 K, the vertical dotted arrows indicate peaks in the spectra). (c) Phonon spectra of ZrB12 samples of different isotopic compositions (T = 10 K, the numbers in the spectra are the frequencies of the peaks indicated by the arrows). Adapted from Ref. [58].

density of phonon states which has a very spiked structure in dodecaborides. In the low-frequency range, LuB12 Raman spectra contain rather narrow lines at ∼115 and 230 cm−1 , the frequencies of which do not change for samples of different isotopic compositions. The energy of the first line coincides with the energy of the densityof-states peak of acoustic phonons [59, 61, 63, 64], whereas the energy of the second peak is higher by a factor of two. In the Raman spectra of the ZrB12 single crystals, the low-frequency structure is observed up to frequencies of ∼160 cm−1 , and the boundary of the high-frequency structure has the doubled energy [Fig. 5.9(b,c)]. The

Raman Scattering by Phonons

well-defined peaks in the vicinity of 140 cm−1 and at the doubled frequency of 280 cm−1 correspond to the measured and calculated density-of-states peaks of acoustic phonons in ZrB12 [59, 60, 62]. In both materials, the low-frequency structures are attributed by us to the density of states of acoustic phonons, which are inactive in the Raman spectra of the ideal structure but can be activated because of the violation of the selection rules for the wave vector. For the structure of the dodecaborides, overtones and combinations of acoustic transverse and longitudinal vibrations at the critical points , X , L, and W of the Brillouin zone are allowed in the second-order spectra for all the Raman-active polarization geometries. In the case of the appearance of maxima or minima in the dispersion curves, the same is true for the directions  K and LW. Therefore, we believe that the appearance of the structures at the doubled frequencies in the two compounds is associated with the two-phonon light scattering by the overtones of acoustic phonons. The intensity of these features is comparable to the intensity of one-phonon lines. The results of detailed temperature investigations of the twophonon density of states [58] are presented in Fig. 5.10, where the intensities of the spectra were normalized to the intensity of the peak in the region of 280 cm−1 , which represents the overtone of the dominant peak of the phonon density of states in ZrB12 [59–62, 65]. An increase in the temperature obviously leads to a hardening of the frequencies of the low-frequency components of the twophonon density of states, namely, the narrow feature in the vicinity of 175 cm−1 and the broad peak near 220 cm−1 . At the same time, for the high-frequency component near 280 cm−1 , these changes are significantly less pronounced. If the two-phonon spectrum is dominated by the overtone processes, then the two-phonon density of states is proportional to the one-phonon density of states on the doubled frequency scale. The low-frequency peak of the onephonon density of states near 90 cm−1 (Fig. 5.9) is located in the region of the low-frequency peak near 175 cm−1 of the two-phonon density of states (Fig. 5.10). The dependences of the energies of one-phonon and two-phonon peaks consistently increase with increasing pressure, which confirms the proposed interpretation. Thus, our results indicate that the low-temperature spectrum of acoustic phonons in ZrB12 is more structured than the spectrum of

461

462 Raman Spectroscopy of Metal Borides

Figure 5.10 Two-phonon density of states of the ZrB12 sample at different temperatures. The vertical lines indicate the frequencies of the main features in the spectrum. Reproduced from Ref. [58].

LuB12 . The data obtained for ZrB12 suggest that, in addition to the peak in the region of 140 cm−1 , there are softer phonons localized in particular regions of the Brillouin zone. The above facts do not confirm the hypothesis of the transition of ZrB12 to the cage-glass phase at T  90 K [55], because the low-frequency peak in the spectrum of acoustic phonons in the vicinity of 100 cm−1 exists even at high temperatures, demonstrating only a slight softening upon cooling. The published results of the calculations for ZrB12 [59, 61, 63] suggested that there is a softening of the transverse acoustic branch to 13 meV (∼105 cm−1 ) at the X point of the Brillouin zone, which was not found in the neutron-scattering experiment. The observed softening of the acoustic branch confirms its important role in the mechanism of superconducting pairing of electrons [31, 66–68], which causes a rather high transition temperature Tc = 6 K in ZrB12 . The results obtained prove that, as in hexaborides, the extra low-frequency modes reflect the density of states of the acoustic

Raman Scattering by Phonons

branches and are not independent localized vibrations (rattling modes). In the low-temperature spectra of ZrB12 presented in Fig. 5.9(c), attention should be paid to the appearance of narrow lines at 155 cm−1 . The width of this line is equal to only ∼2 cm−1 . This suggests that the line under consideration is attributed to the one-phonon process. However, the energy of this line (19 meV) is lower than the energy of the optical phonon at the  point of the Brillouin zone (21 meV), which was measured in the neutronscattering experiment [59, 61, 63]. Consequently, either there are lower-frequency optical vibrations, or a decrease in the temperature leads to the appearance of precursors of another phase.

5.2.5 Raman Spectroscopy of Phonons in Tetraborides The Raman spectra of single-crystalline RE tetraborides RB4 (R = Y, La, Pr, Nd, Sm, Gd, Tb, Dy, Ho, Er, Tm, Lu) have been measured and analyzed in [69]. Tetragonal RB4 tetraborides crystallize in the ThB4 structure (space group P 4/mbm). A group-theoretical analysis yields 19 Raman-active modes in tetraborides:  = 5A1g + 5A2g + 2A1u + 4A2u + 4B1g + 3B1u + 4B2g + 1B2u + 6Eg + 10Eu

(5.3)

The lack of polarization data precluded the observation of all 19 Raman-active lines; only 10 of them were confidently measured. The phonon frequencies varying with the increasing lattice parameter are shown in Fig. 5.11. Phonons with frequencies  600 cm−1 shift down monotonically; those near 600 cm−1 remain nearly unchanged, and the phonon frequencies near 400 cm−1 increase. The frequencies of the low-lying phonon modes (100–200 cm−1 ) are largely determined by the vibrating metal atoms. Comparison of the observed frequency shifts allowed the authors to conclude that for the inter-octahedral B–B vibrations, the force parameters are nearly the same as in hexaborides. This is because both structures are essentially composed of B6 octahedra and metal atoms in between, but the tetraborides contain additional B2 units joining the octahedra.

463

464 Raman Spectroscopy of Metal Borides

Figure 5.11 Phonon frequencies of tetraborides, obtained from the peak maxima of the spectra vs. unit-cell volume. The polarization dependent phonon frequencies of CaB4 are indicated according to the inset. The RE elements are marked at the upper fringe; bold, non-lanthanides. Reproduced from Werheit et al. [69].

Raman Scattering by Electronic Excitations 465

5.2.6 Raman Spectra of Other Rare-Earth Borides Experimental data on the dynamics of the diboride lattice are absent with the exception of MB2 with M = Ti, V, Nb, and Y, for which the density of phonon states was measured by inelastic neutron scattering [70], and Raman spectra of commercially available powder and bulk TiB2 samples were presented [71]. They confirm calculations of the phonon spectrum of a series of transitionmetal [72] and rare-earth diborides [73] showing that, as in higher borides, the acoustic region of the spectrum is separated by a gap from the region of optical vibrations, where the boron-sublattice vibrations dominate. Raman spectra of other RE borides are little studied. Limited data on YB66 and GdB66 are presented in Refs. [74–76]. In this complex structure with more than 1584 boron atoms in the unit cell, the phonon modes are smeared, showing rather broad peaks. A major part of the spectra is essentially determined by vibrations of the boron framework. The spectrum is rather similar to the density of phonon states, which, as for other icosahedral boron structures, occupies an energy range up to 1200 cm−1 .

5.3 Raman Scattering by Electronic Excitations 5.3.1 Crystal Electric Field Transitions The first Raman studies of crystal electric field (CEF) excitations have been performed by Zirngiebl and Pofahl for CeB6 [7] and NdB6 [77]. Magnetic excitations have been observed in CeB6 near 372 cm−1 and in NdB6 near 95 cm−1 . This made it possible to verify and even revise the existing schemes for the f -electron levels in these borides, which sometimes cannot be done with the help of neutron spectroscopy. Ogita et al. [10] identified three peaks around 210, 250, and 300 cm−1 in the Raman spectrum of PrB6 as the CEF excitations. The observed three peaks in the XY (T2g ) spectra satisfy polarization selection rules for the proposed transition scheme, and their intensity increases with decreasing temperature. A recent Raman study of CeB6 [78] has revealed, in addition to the known excitation at 380 cm−1 , additional transitions between the f

466 Raman Spectroscopy of Metal Borides

levels at high frequencies of 2060, 2200, and 2720 cm−1 . All three ∗ have been identified as inter-multiplet excitations between  8 –  6 , ∗ ∗  8 –  8 , and  8 –  7 states, respectively.

5.3.2 Electron–Hole Excitations: Collision-Limited Regime Raman scattering is useful not only to study the vibrational spectrum but also to obtain information about the ground and low-energy excited states of correlation-gap insulators and metals. In studies by Nyhus et al. [79, 80], a broad electronic Ramanscattering background has been found in SmB6 , which rises linearly at low frequencies up to about 1200 cm−1 (Fig. 5.12). An abrupt suppression of electronic scattering below ∼290 cm−1 was observed for T ≤ 50–70 K together with a corresponding enhancement of electronic scattering intensity between 300 and 400 cm−1 . The energy scale ∼290 cm−1 , where the redistribution of electronic

Figure 5.12 Comparison of the A1g + Eg + T2g symmetry Raman-scattering response functions of SmB6 at T = 70 K and 15 K. Inset: room-temperature, high-frequency Raman-scattering response function. Reproduced from Nyhus et al. [79].

Raman Scattering by Electronic Excitations 467

response happens, is substantially larger than the characteristic temperature T ∗ for gap development, c = 6–8 kB T ∗. The authors suggested that the development of the gap in SmB6 is not primarily influenced by the f -d hybridization but rather by strong Coulomb correlations in the d band. The second high-frequency feature of predominantly A1g symmetry develops near 800 cm−1 as the temperature decreases to 130 K in all samples with Sm vacancies [81]. Both features in the electronic Raman scattering at 41 and 100 meV were assigned to the excitations between hybridized 4f -5d bands using recent band structure calculations [82]. These calculations predict the formation of a semiconductor-like gap with band inversion in the vicinity of the X point of the BZ as a result of hybridization between 5d and 4f bands. The mixed parity of the hybridized 5d-4f orbitals in SmB6 allows the 800 cm−1 peak to appear in optical conductivity [83]. In addition, at T < T ∗ , four narrow peaks were observed within the gap [80, 81]. The field dependences of their energies suggest their relationship with CEF excitations of the Sm3+ ion, split by magnetoelastic coupling. The extremely low line width of 4 cm−1 for the in-gap exciton at 130 cm−1 suggests its long lifetime, as it is protected from particle– hole decay by the hybridization gap. Valentine et al. [81] suggested that in the most stoichiometric SmB6 samples, the hybridization gap should be fully opened, and the topological Kondo insulator state is possible. The search for a possible transition to the topological Kondo insulator state in YbB6 under pressure has uncovered a transition to an insulating state at a pressure of 30 GPa, as well as anomalies in the pressure dependence of the T2g Raman mode, the boron atomic position, and the line widths of x-ray (101) and (111) reflections, suggestive of shear stress-induced broadening [84]. Authors suggested that the compressed YbB6 may become a topological Kondo insulator above 35 GPa. A Raman-scattering study of the metal–semiconductor (MS) transition has also been performed on EuB6 and Eu1−x Lax B6 [85,86]. EuB6 is a paramagnetic semiconductor at high temperatures that becomes a ferromagnetic metal below TC = 12 K. In the hightemperature phase (T0 ≥ 60 K), the Raman scattering cross-section

468 Raman Spectroscopy of Metal Borides

Figure 5.13 Raman spectra of EuB6 at various temperatures. The dashed line in the top spectrum is a fit to Eq. (5.4). Reproduced from Nyhus et al. [85].

Raman Scattering by Electronic Excitations 469

is well described by a “collision-limited” response [87] that is typical of that observed for single-particle excitations in degenerate or doped semiconductors: 2 ωγ (T ) Iαβ (ω) ∝ [n(ω) + 1] γαβ 2 , (5.4) ω + γ 2 (T ) where Iαβ is the Raman intensity (α and β denote the polarizations of the incident and scattered light), (T ) is the carrier scattering 2 rate, [n(ω) + 1] is the Bose–Einstein thermal factor, and γαβ is the square of the Raman scattering vertex. The electron relaxation rate decreases significantly with decreasing temperature, and in the ferromagnetic metal phase, a featureless continuum is observed, the shape of which does not change in a magnetic field. The spectrum in the intermediate temperature range TC ≤ T ≤ T0 in Fig. 5.13 shows two features (ωo1 and ωo2 ) which are identified as spinflip Raman scattering peaks. Their development just above the MS transition reveals the spontaneous formation of bound magnetic polarons.

5.3.3 Electron–Hole Excitations: Crossover from Clean to Dirty Regimes Inelastic light scattering by electrons can provide information on the structure of the Fermi surface, electron velocities, and electron scattering mechanisms [88]. Usually metals are associated with isotropic systems having a parabolic dispersion. In such case, charge-density fluctuations created by light are largely screened by conduction electrons, which reduces the nonresonant contribution to the scattering cross section (proportional to q 2 for the scalar component) at small wave vector q [88, 89]. However, most elemental metals have anisotropic multi-sheet Fermi surfaces. This leads to nonvanishing unscreened low-frequency scattering even for q → 0. An increase of the effective q vector because of the strong absorption at a metallic surface [90] (up to 2 × 106 cm−1 for the exciting laser energies ωi in the visible range) also enhances the scattering intensity. With electron velocities vF being as high as 108 cm/s, the electronic excitation’s energies ω = q vF in the Raman spectra can spread up to 1000 cm−1 . This indicates that the treatment of the electron Raman scattering in metals in the q → 0

470 Raman Spectroscopy of Metal Borides

limit, see Eq. (5.4), may lead to incorrect conclusions if q vF  , where  is the electron relaxation rate. Optical properties of boride systems make it possible to vary the wave vector of probed electronic excitations, having a narrow distribution, which makes them a convenient object for a detailed experimental and theoretical study of electron dynamics in light scattering. Figure 5.14 shows the electronic response obtained at various temperatures for a 11 B-enriched LaB6 single crystal [91]. At temperatures up to 300 K, the results coincide with previous measurements performed on a sample with the natural boron composition [25]. The impurity scattering in LaB6 is small: the 10K

10K 100K

300K

χ"(ω)

χ"(ω)

160K b 300K

310K

470K 640K

470K

730K 630K

a 0

500

1000

c

1500 0 -1

Raman shift (cm )

1000

2000

3000

4000

-1

Raman shift (cm )

Figure 5.14 (a) Electronic response χ  (ω) in LaB6 at various temperatures in the XX polarization geometry with the excitation by the 532 nm line. For convenience, phonon lines are subtracted from the two high-temperature spectra. The solid lines are the calculations of χ  (ω). (b,c) Dependences χ  (ω) in YB6 obtained at various temperatures in the XX polarization geometry with the (A1g + Eg ) symmetry: (b) Low-temperature χ  (ω) under 633 nm excitation. (c) High-temperature χ  (ω) under 532 nm excitation. The solid lines are the calculations with the coupling constant λe-p = 0.40 and the phonon spectrum F (Ω) with the constant α 2 (Ω). Reproduced from Ref. [91].

Raman Scattering by Electronic Excitations 471

residual resistivity ratio is ρ300 K /ρ4.2 K ≥ 400. A small damping of electronic states at low frequencies ensures the collisionless regime. This allows for the observation of a peak caused by the term qvF at low temperatures. The energy position and shape of this peak are determined by the wave-vector transfer and the distribution of electron velocities on the Fermi surface. A significant broadening and the shift of the pronounced peak near 200 cm−1 at 10 K toward high frequencies with an increase in temperature are due to a change in the electron–phonon scattering, leading to a mass renormalization and to a finite electron lifetime. For comparison with the experiment, the frequency dependences of the spectrum of light scattering by intraband electronic excitations were calculated within the polarization operator formalism with allowance for the electron–phonon scattering effects. The frequency dependence of the scattering cross-section is determined by the integral of the electronic susceptibility χα, β (q, ω) over the distribution of wave vectors (for details, see Refs. [45, 92]):

2 ∞ dSF  γα, β (k) d [ f () − f ( + ω)] χα, β (q, ω) = |vF | −∞ 1 . × Im ω − q·vF −   ( + ω) +   () − i [  ( + ω) +   ()] (5.5) Here f () is the Fermi function, vF is the Fermi velocity vector, γα, β (k) are the matrix elements of the electron–photon interaction, and the integration is performed over the Fermi surface (SF being an element of the Fermi surface in k space). The retarded and advanced quasi-particle electronic self-energies () and ( + ω) determine the electron spectrum renormalization near the Fermi level due to different interactions. In the case of electron–phonon scattering, the real and imaginary parts are [93]:   () = dΩ α 2 F (Ω)      1 +Ω 1 −Ω × Re ψ +i −ψ +i , (5.6) 2 2T 2 2T   () = π dΩ α 2 F (Ω) × [2nB (Ω) − f ( − Ω) + f ( + Ω) + 1] + ν,

(5.7)

472 Raman Spectroscopy of Metal Borides

where ν is the impurity relaxation frequency, ψ — the digamma function, Ω — the phonon energy, α 2 F (Ω) — the Eliashberg spectral function for the electron–phonon interaction. The electronic structure, Fermi surface, and velocities of electrons on the Fermi surface were calculated using the linearized muffin-tin orbital (LMTO) method in the local electron density approximation. The distribution of wave vector transfers U (q) [94] was calculated with the optical data from Ref. [95] and the electron–phonon coupling constant was calculated with the phonon density of states F (Ω) from Ref. [18]. The only varying parameter  was λ = 2 dΩ α 2 F (Ω)/Ω, which determines the electron selfenergies (ω). Its initial value was determined from the estimate of the relaxation rate  ≈ 2 ·   (ω) (  (ω) is the imaginary part of the electron self-energy) at high temperatures. In this case, the collisionless contribution can be neglected, and   (ω) can be specified by the above expression for the relaxation mechanism, given by Eq. (5.7). Although lanthanum hexaboride has a parabolic dispersion law, its Fermi surface is anisotropic. This is one of the reasons responsible for the observation of electronic light scattering. In addition, the observed scattering is determined by a small finite wave vector. The spectra calculated with λ = 0.25 for several temperatures are shown in Fig. 5.14(a) along with the measured electronic response χ  (ω). The calculated dependences χ  (ω) are in good agreement with the experimental dependences throughout the entire temperature range. Some discrepancy at high frequencies is due to the contribution of luminescence with a peak near 2.1 eV. The obtained agreement confirms the correctness of the estimate of the electron–phonon interaction constant and makes it possible to estimate the average velocity of electrons as vF = 6.5 · 107 cm/s. The position of the continuum with the wave vector direction q (001) in the experiment is shifted by 10–15% with respect to the maximum of the continuum with q (011). The continuum energies measured in different polarization geometries also differ. Our calculations were performed with the constant matrix element of the electron–photon interaction γαβ , which implies resonance conditions for the entire Fermi surface. The

Raman Scattering by Electronic Excitations 473

calculation for real resonance conditions can likely lead to a difference between spectra measured with different excitation energies. Moreover, the curvature of the Fermi surface serves as the matrix element in the nonresonant situation [88]. This can possibly lead to a change in the frequencies of the continua for different directions of the momentum. The indicated effects require additional investigations. The calculated electronic structures and Fermi surfaces, as well as the phonon spectra, of YB6 are very close to the respective characteristics of LaB6 , which implies the similarity of their spectra of electronic light scattering. However, the spectra χ  (ω) in YB6 [Fig. 5.14(b,c)] significantly differ from similar spectra in LaB6 : the maximum of the electronic continuum even at low temperatures was observed at a frequency of ∼800 cm−1 [91], which is much higher than the calculated energies of maxima using rather large magnitudes of the electron–phonon constant λe-p up to 1.1 [33]. This certainly indicated the impossibility of observing spectra of the electronic light scattering in YB6 in the regime of conservation of the wave vector. The description of the observed spectra required the introduction of an additional relaxation mechanism with an energy of ∼600 cm−1 (∼75 meV). The general damping of electronic excitations was increased by this constant, which made it possible to satisfactorily describe the experimental spectra χ  (ω) up to temperatures of ∼400 K with λe-p ≈ 0.4 (Fig. 5.3). As in LaB6 , an increase in the temperature leads to the shift of the energies of maxima of the electronic continuum toward high frequencies and their broadening, which indicates the phonon mechanism of the renormalization of the electronic spectrum. Thus, the comparison of the experiment with the calculation confirms that the observed temperature renormalization of the electronic spectrum in YB6 is really due to the interaction with phonons, but the electron–phonon coupling constant is much smaller than the previous estimates. Our estimate of λe-p is close to the value λe-p = 0.5 calculated in Refs. [28, 96] in the rigid muffin-tin potential approximation but is much smaller than the value λe-p = 1.44 obtained in Ref. [28] by the linear response method.

474 Raman Spectroscopy of Metal Borides

Polarization measurements showed the presence of T1g symmetry scattering in YB6 , which is characteristic of magnetic excitations. The scattering of such symmetry was also observed in the electron scattering spectra of GdB6 , which undergoes a transition to the magnetic state at low temperatures [97]. Electronic Raman scattering and optical phonon self-energies are studied on single crystals of LuB12 with different isotopic composition in the temperature region 10–650 K and at pressures up to 10 GPa [56, 57]. Raman response χ  (ω) for different temperatures is shown in Fig. 5.15. The spectra observed with 633 nm (1.96 eV) laser excitation [Fig. 5.15(a)] are very similar to those measured in Ref. [54]. The low temperature continuum near 185 cm−1 in these spectra shifts to higher energies and

10K

a 532 nm

100

U(q)

2

-1

Σ" (cm )

χ"(arb.units)

300K 650K 0

10K

b

0

633 nm 50 100 4

150

200

-1

q (10 cm )

50

2

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T=10K

1 650K 0

0

c

200 400 600 800 1000 1200 1400 -1

Raman shift (cm )

0

0

200

400

600

-1

ω (cm )

Figure 5.15 Raman response χ  (ω) in LuB12 , obtained from the (001) plane at various temperatures in the polarization geometry (XX) – A1g + Eg of symmetry: (a) 633 nm excitation, (b) 532 nm excitation. The solid line represents χ  (ω) calculated with λ = 0.32. The dotted line shows the response calculated with λ = 0. (c) Energy dependence of the imaginary part of the electron self-energy, given by Eq. (5.7), at 10 K. Wave-vector distributions U (q) [94] for the excitation radiation wavelengths used in experiments are shown in the inset. Reproduced from Ref. [56].

Raman Scattering by Electronic Excitations 475

A1g+4Eg 4

10

11

B

Lu B12

10K

q||[011]

LuB12 11

B

q||[001]

χ"(ω)

200K A1g+Eg +T2g 2 T2g 10K

A1g+Eg q||[001] b

a 0 0

200

400

600 -1

Raman shift (cm )

0

200

400

600 -1

Raman shift (cm )

Figure 5.16 (a) Raman response χ  (ω) in Lu11 B12 , measured from the (001) plane in various polarization geometries at T = 10 K with an excitation wavelength of 633 nm. (b) Raman response χ  (ω) in Lu11 B12 , measured from the (001) planes of samples of various isotope compositions at two temperatures. Reproduced from Ref. [56].

broadens with increasing temperature. As one can see, the intensity of the broad continua goes to zero at all temperatures when ω → 0. Significant broadening and a shift of the peaks to ∼800 cm−1 was found upon increasing the temperature to 650 K. When using 532 nm (2.33 eV) laser excitation, the maximum of low-temperature continuum is observed near 300 cm−1 (Fig. 5.15b) due to a change in the magnitude of the probed wave vector. Since at low temperatures low-frequency phonons are frozen, an electronic damping χ  (ω) is really low at T = 10 K at low energies [Fig. 5.15(b)]. In essence, this fact ensures a collisionless regime for electrons and makes it possible to observe a rather narrow peak at 185 cm−1 . The fact that a wave vector is retained in this process can be used to estimate the renormalized electron velocity on the Fermi surface at vF = 7.3 · 107 cm/s. The frequency dependent increase in the electron damping leads to the appearance

476 Raman Spectroscopy of Metal Borides

of incoherent scattering at high energies, as compared to the absence of scattering by phonons. As the temperature increases, the electron lifetime decreases at all frequencies, and the imaginary part of the electron self-energy   (ω) becomes higher than q. In this case, it can be neglected and one can pass to the limit q → 0. Then, the susceptibility is described by the relaxation expression (5.4). As in LaB6 , the shape and energy position of the spectral peaks depend on the magnitude of the probed wave vector, temperature, and symmetry of the excitations (Fig. 5.16). The continuum is strongly polarization dependent, with large Eg + T2g symmetry and negligible A1g symmetry contributions. The shape and position of the observed continuum do not change in measurements on the samples with different isotopic composition while the phonon lines show shifts in consistence to the atomic mass ratio. As it was shown in Refs. [25, 56, 91], such scattering in borides originates from intraband electronic transitions near the Fermi level, but it was previously interpreted [54] as a boson peak.

5.3.4 Electron–Induced Phonon Renormalization The shape of the Eg and T2g phonon lines at 650 and 790 cm−1 [57] is another striking evidence of the electronic origin of broad continua. They show a strong asymmetry depending on the excitation wavelength [Fig. 5.17(a)] that suggests interference between the phonon line and the continuum. In this case the phonon line shape may be described by the asymmetric Breit–Wigner–Fano (BWF) profile expression [98]: (Q + )2 , 1 + 2 ω − ω0 − V 2 R(ω) = ,  V (Tp /Te + V 2 R(ω) , Q= π V 2 ρ(ω)

I (ω) = πρ(ω)Te2

(5.8) (5.9) (5.10)

where ω0 and R(ω) are the bare (uncoupled) mode frequency and the Hilbert transform of continuum density of states ρ(ω). Te and Tp are scattering amplitudes for the continuum and phonon. Electronphonon interaction with matrix element V determines the phonon

Raman Scattering by Electronic Excitations 477

a

633 nm

b

40

-1

Width (cm )

Intensity(a.u.)

6

4

785 nm

2

30

20

532 nm 10 550

600

650

700 -1

Raman shift (cm )

750

0

200

400

600

Temperature (K)

Figure 5.17 (a) The line shapes of E1g phonon in Lu11 B12 measured with different excitation wavelengths at 300 K. The solid lines show fit curves using Eq. (5.6). (b) Temperature dependence of the width for the E1g phonon at 650 cm−1 . The solid line is the calculated temperature dependence using both anharmonicity and electron–phonon contributions. Width dependences are shown in the cases when the low-temperature phonon damping is fully determined anharmonicity (dashed line) and electron–phonon interaction (dotted line). Reproduced from Ref. [57].

width  = 2[0 + π V 2 ρ(ω)] (0 is the phonon width in the absence of interaction). This interaction also shifts the phonon energy to ω = ω0 + V 2 R(ω). A distinguishing feature of the Fano resonance in the Raman spectra is the dependence of the asymmetry parameter Q and, consequently, of the profile of the phonon line on the wavelength of the exciting light. It is seen very clearly in Fig. 5.17(a) where all E1g line profiles were fitted by Eq. (5.6). In spite of a rather large visual difference in line shapes, the extracted phonon frequencies and widths are very similar for different excitation wavelength used: ω = 649.6 cm−1 ,  = 24 cm−1 , and Q = 4.7 for the 532 nm excitation; ω = 649.2 cm−1 ,  = 23 cm−1 , and Q = 2 for the 633 nm excitation; and ω = 649.8 cm−1 ,  = 25 cm−1 , and Q = −1.4 for the 785 nm excitation. The frequencies of all 5 phonons in LuB12 soften, and their widths broaden with increasing temperature. At first glance, such a behavior may come from anharmonic effects. Really, the temperature behavior of the A1g phonon self-energy may be well explained by the three-phonon coupling approximation. Inability to describe

478 Raman Spectroscopy of Metal Borides

the damping dependences for the E and T phonons indicates the existence of additional decay mechanisms. These may be an electron–phonon interaction or higher-order anharmonic processes. The latter provide a quadratic dependence of the phonon line width vs. temperature. The former really manifests itself through the above-mentioned Fano interference, which leads to the noticeable asymmetry of the E1g and T12g phonon profiles. Line shape asymmetry is smaller for the E2g and T22g modes and absent for the A1g phonon. Obviously, this is due to the presence of the electronic continua with the Eg and T2g symmetry and the absence of electronic excitations with A1g symmetry. Asymmetric profiles of phonon lines are also observed in other dodecaborides, which indicates the existence of electron continua of various nature interacting with phonons. The phonon spectral function was calculated on the basis of an ab initio electron structure by taking into account the frequency and temperature dependences of the phonon self-energies due to both the anharmonicity and electron–phonon interaction [45, 57]. These calculations showed that neither the anharmonic mechanism nor electron–phonon mechanism separately is able to explain the observed temperature behavior of the phonon damping. Therefore, both the mechanisms of anharmonicity and electron– phonon interaction give contributions to line widths and, possibly, to frequency shifts. The calculated temperature dependence of the line width for the E1g phonon at 654 cm−1 is shown in Fig. 5.17(b). There is a nearly perfect agreement with experimental data. The electron–phonon coupling constant in this calculation was set λ = 0.32. As one can see, the electron–phonon mechanism provides the same temperature trend for the phonon self-energies as the anharmonic one. The reason for this behavior is clear from the following evaluation of the phase velocity for the E1g phonon at 654 cm−1 , which upon excitation at 633 nm yields 2.5 × 108 cm/s. This value is several times larger than the average velocity of the electrons vF  7.3 × 107 cm/s. This means that E1g and other higher-frequency phonons are in the nonadiabatic regime. No Landau damping, which has a threshold at ∼1.2 × 106 cm−1, is possible for these phonons. The phonon damping, calculated for the

Conclusions 479

probed wave vectors, is small at low temperatures but increases with increasing temperature until a maximum and then decreases. This behavior superimposed on anharmonic growth explains the appearance of humps on the temperature dependences of phonon damping [Fig. 5.17(b)] for E and T phonons [57]. We note that a similar situation was observed in the superconducting boride MgB2 [99] with the highest for the phonon mechanism Tc .

5.4 Conclusions Most of the early Raman work was devoted to the study of the dynamics of the hexaboride lattice, although metal–insulator and metal–semiconductor transitions were also affected. Recent studies of electronic excitations in the metallic state have shown the ability of the method to obtain information about electron dynamics. Such excitations were observed both in hexaborides and in dodecaborides, therefore further studies are necessary in other borides. In mixed crystals, they can give a trace of the evolution of the system with the introduction of magnetic impurities and the transition to the Kondo state for such systems as Ce1−x Lax B6 or Yb1−x R x B12 . Investigations of the unusual electron scattering symmetry T1g , found in borides with different ground states, YB6 and GdB6 [97], also need to be continued and understood.

Acknowledgements The author warmly thanks N. Yu. Shitsevalova, V. B. Filipov, and A. A. Makhnev for their cooperation in obtaining results for this work. Calculations of electronic structures of S. V. Streltsov significantly advanced the understanding of the experiment. A part of the presented results was obtained with the support of the Russian Foundation for Basic Research (Grants No. 14-02-00952 and No. 1952-18008).

480 Raman Spectroscopy of Metal Borides

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98. Klein, M. V.; “Electronic Raman scattering” in M. Cardona (ed.), “Light scattering in solids”; Topics in Applied Physics, vol. 8, pp. 147–204 (Springer, Berlin, Heidelberg, 1975). 99. Ponosov, Y. S., and Streltsov, S. V.; “Raman-active E2g phonon in MgB2 : electron-phonon interaction and anharmonicity”; Phys. Rev. B 96, 214503 (2017).

Chapter 6

Neutron Spectroscopy on Rare-Earth Borides Pavel A. Alekseev, Vladimir N. Lazukov, and Igor P. Sadikov National Research Centre “Kurchatov Institute,” pl. Akademika Kurchatova 1, 123182 Moscow, Russia pavel [email protected]

Applications of the thermal neutron spectroscopy to the study of lattice and magnetic dynamics in rare-earth borides is discussed. A brief presentation of the main specifics of the neutron spectroscopy method is given. The results of the phonon excitation study as well as 4f-electron excitation spectra, based on the data obtained with single-crystal and polycrystalline samples, are discussed in detail. The attention is focused on the hexa- and dodecaborides which demonstrate a wide range of the physical phenomena related to strong electron correlations. The features of the intermediatevalence state of samarium in SmB6 are discussed in connection with the lattice dynamics and magnetic excitation spectra. The latter are analyzed over a wide energy scale, from intermultiplet transitions to the resonant mode in the spin gap and quasielastic excitations. Then, we present and discuss the main results of the neutron spectroscopy study on the typical Kondo insulator YbB12 . Special attention is focused on the resonant mode (spin exciton) and

Rare-Earth Borides Edited by Dmytro S. Inosov c 2022 Jenny Stanford Publishing Pte. Ltd. Copyright  ISBN 978-981-4877-56-5 (Hardcover), 978-1-003-14648-3 (eBook) www.jennystanford.com

490 Neutron Spectroscopy on Rare-Earth Borides

the spin-gap transformation due to the temperature and impurity effects.

6.1 Specifics of the Neutron-Scattering Technique in Condensed Matter Spectroscopy Neutron spectroscopy or inelastic thermal-neutron scattering (energy transfer range 10−1 –10−3 eV) is currently considered one of the most universal methods for studying lattice vibrations and magnetic dynamics of materials. In other words, this versatile method provides access to phonons and magnetic excitations in solids. The neutron scattering technique was invented in the period of 1950– 1955 and continues to be improved in our days. Bertram Brockhouse and Clifford Shull were awarded the Nobel Prize in Physics in 1994 “for pioneering contributions to the development of neutron scattering techniques for studies of condensed matter,” in particular “for the development of neutron spectroscopy.” The interesting point is that neutron spectroscopy was developed simultaneously in two directions: the time-of-flight and the triple-axis techniques, based on the inherent quantum-mechanical duality of neutrons, i.e., their particle- or wavelike behavior. The competitiveness of the method follows from a number of specific properties of thermal neutrons important for the applications in condensed matter research. The most important of them are: (1) Charge neutrality secures the high penetration depth for neutrons in comparison to x-rays or any charged particles. (2) The energy and wavelength of thermal neutrons match the typical excitation energies and the interatomic distances in solids, respectively. (3) The presence of a magnetic moment due to the spin S = 1/2 of the neutron enables magnetic scattering that has comparable amplitude with that of the nuclear scattering, which makes it possible to study lattice and magnetic excitations of solids using the same instruments. Nuclear scattering amplitudes are not systematically dependent on the nuclear charge and mass, which makes neutron scattering complementary to the x-ray-

Specifics of the Neutron-Scattering Technique in Condensed Matter Spectroscopy 491

based synchrotron radiation spectroscopy, especially when it is necessary to distinguish the contributions of light atoms or atoms with similar atomic number in multi-component substances. (4) Finally, the characteristic interaction time between neutrons and the solid (∼10−11 –10−13 s) appears to be well tuned to the frequency range of spin fluctuations for strongly correlated electron systems with the so-called valence instability. More details on this subject can be found in existing textbooks [1, 2]. The above-mentioned features ensure that the neutron scattering technique will remain in demand for a long time in the future. The two main types of neutron spectrometers—the triple-axis spectrometer (TAS) and the time-of-flight (TOF) spectrometer—can typically provide the information about the dispersion of particular excitation modes and general characteristics of the whole excitation spectra, respectively. For phonons, the latter characteristics can be, for instance, the phonon density of states (PhDOS) as a function of the energy, g(E ), or frequency, g(ω). The wave-particle duality of neutrons allows the determination of their energy either from the Bragg law by diffraction on a monochromator or analyzer crystal, or by measuring the time necessary for a neutron to travel the known distance from the source to the counter. These are the underlying principles of the TAS and TOF instruments that are schematically depicted in Fig. 6.1. Figure 6.1(a) shows the working principle of a TAS instrument. The polychromatic continuous neutron beam is incident on the crystal monochromator, which selects the reflected monochromatic beam with the initial energy E i (wave vector ki ) according to the Bragg condition. This monochromatic beam arrives at the sample position, where neutrons change their energy and wave vector due to the scattering processes in the sample. A particular value of the final energy E f and wave vector kf is then selected by the crystal analyzer, and these neutrons are then counted by the detector unit. The scattering angle 2θ at the sample position corresponds to the change in angle between the directions of the initial and final neutron beams in the scattering process, (ki , kf ).

492 Neutron Spectroscopy on Rare-Earth Borides

a)

Sample

Neutrons

Detector

Ei, ki

Monochromator

b)

2d sin4 = nO

Ef, kf Analyzer

Choppers

Sample

Ei L1

V=L / t

L2

Detectors

E=mV2 / 2 Figure 6.1 The principal schemes of the TAS (a) and TOF (b) instruments.

Figure 6.1(b) schematically shows a TOF spectrometer that uses a pulsed neutron beam from a spallation neutron source or from a continuous source, which is then “chopped” into pulses by a rotating beam chopper. A pair of such choppers, set at a given distance L1 from each other, can be synchronized to select a sequence of monochromatic neutron pulses from a polychromatic beam with the initial energy E i that is determined by the ratio of the phase shift between choppers to the flight pass L1 . After the neutrons scatter at the sample, their final energy E f is determined by measuring the arrival time of each particular neutron to one of the counters located at the fixed distance L2 from the sample. The use of large areasensitive detectors in TOF spectrometers enables the simultaneous counting of neutrons with different finite wave vectors, which makes this technique particularly efficient when it is necessary to map out large volumes of the energy-momentum space at the constant sample conditions.

Specifics of the Neutron-Scattering Technique in Condensed Matter Spectroscopy 493

The results of neutron spectroscopy measurements are typically presented in the form of the double differential cross-section d2 σ/dΩ dE , which corresponds to the relative number of neutrons scattered in the element of solid angle dΩ and in the interval of energy transfer from E to E + dE . It is related to the scattering function S(Q, E ) (also known as the dynamic structure factor), which represents the spectral characteristics of a substance under study, with the corresponding excitation energy1 E = ω and momentum transfer Q according to the following general equation: N kf d2 σ S(Q, E ), = dΩ dE  ki where N is the number of nuclei in the sample, ki and kf are the initial and final neutron wave vectors in the scattering process, Q = ki − kf and E are the momentum and energy transfer from the neutron to the sample, respectively.

6.1.1 Neutron Scattering Function in Relation to the Atomic Vibrations and Dynamic Magnetic Susceptibility Lattice vibrations To agree on the consistent notation for the rest of this chapter, we will start by introducing several specific features of the neutron scattering technique which will be important for an adequate and qualitative understanding of the results presented below. Further details of the corresponding formalism can be found in the specialized literature [1, 3, 4]. At first, we would like to remark on the specifics of the instrumentation. TAS instruments were initially designed for investigating the details of particular phonon (or magnon) dispersion curves in single-crystalline samples that consist of chemical elements or their isotopes with predominantly coherent nuclear (or magnetic) scattering, described by the scattering length bc . This mission is energy is defined as E = ω, where ω is the frequency of the measured excitation; in the following, both notations will be used in the sense of “energy transfer,” i.e., the exchange by energy between neutrons and excitations (quasiparticles) in the sample. An approximate relation between different units used in spectroscopy of solids is 1 meV ≈ 0.242 THz ≈ 8.066 cm−1 .

1 The

494 Neutron Spectroscopy on Rare-Earth Borides

realized by TAS due to its ability to assume a configuration that corresponds to an arbitrary energy transfer and the momentum transfer vector within a large enough volume of the energymomentum space. The momentum transfer is usually restricted to the particular horizontal scattering plane. First TOF spectrometers, on the other hand, were focused on studies of the PhDOS spectra, initially operating with predominantly incoherent scattering materials. Inelastic incoherent scattering of neutrons, characterized by the scattering length binc , lacks momentum dependence, but its energy dependence is under certain assumptions proportional to the PhDOS function g(ω), making it possible to perform the simplest phonon spectroscopy measurements even without using position-sensitive detectors. Namely, for a sample consisting of a single chemical element in the one-phonon scattering approximation (neglecting multi-phonon processes), and under the assumption of an isotropic Debye-Waller factor (which is certainly valid for a cubic crystal), the incoherent scattering crosssection is given by  ω     2 ± kf 2 −2W Q2 g(ω) 1 d σ e kT , (6.1) = N binc e ω dΩ dω inc ki 6M ω e kT − 1 1 where ± corresponds to the scattering process with neutron energy loss (+) and neutron energy gain (−), M is the atomic mass of the scatterer, T —sample temperature, binc —incoherent neutron scattering length, e−2W —Debye-Waller factor, kB —Boltzmann constant. Later on, it was established that with position-sensitive detectors that cover a large range of scattering angles in modern TOF spectrometers, the so-called “incoherent approximation limit” can be reached even for a predominantly coherent scatterer after powder averaging. This becomes true especially for powder samples measured at high enough momentum transfer |Q| (above 6– 7 A˚ −1 for the most of inorganic materials). Using modern TOF spectrometers, one can get not only the PhDOS spectrum for any material in polycrystalline form (in this case the scattering intensity given by Eq. (6.1) is proportional to the sum of coherent and incoherent scattering) but also the dispersion relationships for coherent scatterers (i.e., phonons or magnons) using singlecrystalline samples. The PhDOS function g(ω) is related to the

Specifics of the Neutron-Scattering Technique in Condensed Matter Spectroscopy 495

phonon dispersion modes by 3r   dC (ω)|gradq ωs (q)|−1 dq, g(ω)dω = (2π)3 s=1

(6.2)

where  is the volume of the crystal, and dC (ω) is an element of the surface of constant energy between ω and dω. The dispersion relation ωs (q) corresponds to a particular s-phonon branch among the 3r dispersion curves, r being the total number of atoms in the structural primitive cell. The second set of remarks concerns the specific relation between the dynamics of the atomic crystal lattice or array of magnetic moments (generally related to the electrons in solids) and particular spectra obtained from the neutron spectrometers. The effective interaction of the neutron with a nucleus is described by the scattering length b, which generally has two components, the coherent (bc ) and incoherent (binc ). Their ratio is determined by the properties of the nucleus and depends strongly not only on the element but also on its particular isotope. The scattering length has no simple systematic dependence on the atomic number and can differ considerably even for neighboring isotopes of the same element. Therefore, the scattering function S(Q, ω) measured with powder in a TOF experiment is directly proportional to the PhDOS function g(ω) only for single-element materials, according to Eq. (6.1). The function g(ω) is what is considered in calculations of such physical properties as heat capacity or the average vibration energy. In the case of a multi-component material, the relation between g(ω) and S(Q, ω) is no longer so straightforward. For complex materials consisting of more than one element, the scattering function S(Q, ω) measured in an experiment is related to the so-called neutron-weighted generalized PhDOS G(ω) as given by  ω  1 e kT ± 2 G(ω) , (6.3) S(Q, ω)inc = Q ω ω e kT − 1 1 where G(ω) =

P  ν=1

cν

2 binc, ν gν (ω). Mν

(6.4)

Here P is the number of different sorts of atoms with the scattering length bν , concentration cν , and mass Mν , and the function gν (ω) is the partial PhDOS (see below).

496 Neutron Spectroscopy on Rare-Earth Borides

The relation presented by the Eqs. (6.3) and (6.4) is true for most of the presently studied materials with multi-component structure. The function G(ω) provides the most general insight to the structure and other characteristics of the lattice-vibration spectrum and is determined by the partial (related only to the particular sort ν of atoms in the substance) DOS gν (ω), which represent the contribution of the particular ν-sort of atoms to g(ω), as it is defined by Eq. (6.5) below. Thus, the contribution of each gν (ω) to the scattering function S(Q, ω) is scaled by the particular ratios of neutron scattering length to atomic mass for atoms of the sort ν. Correspondence between the partial gν (ω) and total g(ω) is determined by the square of polarization vectors |ξν (ω)|2 ∝  2 s |ξν, s (ω)| for the ν-sort of atoms at particular energy ω for all the phonon branches (numbered by s) contributing to the physical density of states g(ω): gν (ω) = g(ω)e−2Wν |ξν (ω)|2 .

(6.5)

It is obvious that in cases when ν > 1, the experiment does not directly provide g(ω), which is only physically meaningful from the thermodynamic point of view. To get it, one would need to make a model assumption for polarization vectors or to make more experiments with isostructural samples that differ only by the scattering length bν . This possibility may be realized for elements which have stable isotopes with markedly different scattering lengths. The corresponding method, known as the “isotopic contrast method,” was developed in Refs. [5, 6]. It allows the extraction of the partial density of states gν (ω) from experimentally obtained spectra after analyzing P sets of measurements that differ from each other only by the values of bν . These approaches were successfully applied, in particular, to high-temperature superconductors [7, 8] and quasicrystals [9, 10]. Magnetic excitations The use of inelastic neutron scattering (INS) for the study of magnetic excitations is based on the magnetic dipole interaction between the neutron spin and the localized magnetic moment. Under magnetic excitations, most generally, one understands both single-ion excitations between crystal-electric-field (CEF) levels and collective excitations of the magnetic subsystem, such as spin waves in ordered magnets or various sorts of diffuse

Specifics of the Neutron-Scattering Technique in Condensed Matter Spectroscopy 497

inelastic or quasielastic scattering. In rare-earth (RE) compounds, mainly discussed below, there is no quenching of the orbital angular momentum as in the case of many transition-metal ions. Therefore, the magnetic moment originates from the total angular momentum J of the f-electron shell, which has both spin and orbital contributions, J = S + L. Because of the strong spin-orbit coupling, spin and orbital angular momenta are not conserved individually, and only the total angular momentum remains a good quantum number [11]. The spinorbit interaction produces an energy difference between the states with different J that is typically much larger than room temperature, so as a matter of fact only the spin-orbit multiplet corresponding to the ground-state value of J, defined by the Hund’s rules, is relevant for the material’s properties at low temperatures. In addition, under the influence of the CEF, the degeneracy of the spin-orbit multiplet of the RE ion is reduced as it splits into states distinguished by the different projections Jz . This splitting is typically much smaller than the spin-orbit coupling and is relevant for the low-temperature physical properties of RE compounds. Transitions between the CEF levels can be probed by INS, which becomes especially valuable in the case of metallic samples that cannot be effectively studied by optical spectroscopy in the relevant range of energies. The scattering of unpolarized thermal neutrons on the local magnetic moments is described by the double differential crosssection for neutron scattering [12, 13]: 1 N N kf d2 σ S(Q, E ) = = dE dQ  ki 2π 



γ re μB

2

kf χ  (Q, E , T ) , ki 1 − e−E /kB T

(6.6)

where γ ≈ −1.913 is the gyromagnetic ratio for a neutron, re = e2 /(me c 2 )—classical electron radius, μB = e/(mc)—Bohr magneton, χ  (Q, E , T ) ≡ Im χ(Q, E , T )—imaginary part of the dynamic magnetic susceptibility. According to the Kramers-Kronig relations, χ  (Q, E , T ) can be presented using the real part of the dynamic magnetic susceptibility χ  (Q, E , T ) ≡ Re χ(Q, E , T ):   χnm (Q, 0, T )Pnm (E , T ), (6.7) χ  (Q, E , T ) = π E n, m

498 Neutron Spectroscopy on Rare-Earth Borides

 where χnm (Q, 0, T ) is the Van Vleck susceptibility related to the transition between two states of local moments |n and |m (in the case n = m it corresponds to the Curie susceptibility for the   (Q, 0, T ) = χ  (Q, 0, T ). Each function state |n), so that n, m χnm Pnm (E , T ) represents a singularity (peak) with the maximum at the energy E = E m − E n , where E n and E m are the energies of RE ions in the states |n and |m, respectively, with the normalization +∞ Pmn (E , T )dE = 1. (6.8) −∞

In the absence of inter-ion interactions or notable hybridization with phonons due to magnetoelastic interactions (CEF-phonon bound states [14, 15]), the Q-dependence of χ  (Q, 0, T ) is described only by the single-ion magnetic dipole form factor F (Q). Therefore,  2  2 χ  (Q, 0, T ) = F (Q) χ  (0, 0, T ) ≡ F (Q) χst (T ), (6.9) where χst (T ) is the static magnetic susceptibility which is measured in classical magnetometers. Thus, from Eq. (6.6) we obtain   2 1 γ re 2 N kf  d2 σ F (Q) = dE dQ 2 μB  ki 

(6.10)  E  × χnm (0, 0, T )Pnm (E , T ) 1 − e−E /kB T n, m If we assume for simplicity that Pnm (Q, E , T ) can be presented by the succession of δ-functions, Eq. (6.5) will take the form [12, 13, 16]  2   N kf  d2 σ ˆ ⊥ |n 2 δ E − (E m − E n ) , F (Q) ρn m| = (γ re )2 dE dQ  ki n, m (6.11)   −E i /kB T −E n /kB T is the probability to find the where ρn = e i e ˆ ⊥ (Q) ≡ RE ion at a given temperature T in the state |n, and    −2 ˆ × Q] is the operator for magnetic interaction of a Q × [ Q neutron with the RE ion, 1 ˆ ˆ = Lˆ + S.  (6.12) 2 In the case when states |n and |m belong to the same spin-orbit Jˆ may be presented using the operator of multiplet,2 the operator  2 In the case when the initial and final states belong to different multiplets, the matrix

ˆ as well as the magnetic form factor have a special form [16]. element for 

Specifics of the Neutron-Scattering Technique in Condensed Matter Spectroscopy 499

the total angular momentum of the f-shell: ˆ = 

1 g J Jˆ, 2

(6.13) 

where g J = 1 + [J (J +1)+ S(S +1)− L(L+1)] [2J (J +1)] is the Lande´ g-factor. After all the above transformations, the expression for the double differential scattering cross-section (6.6) for the RE ion in CEF takes the form 2  2   d2 σ N kf  F (Q) ρn m|Jˆ⊥ |n δ E − (E m − E n ) . = (γ re )2 dE dQ  ki n, m (6.14) As it was mentioned above, for the presentation of neutron spectra, the so-called scattering function (or spectral scattering function) S(Q, E ) is used. It is more convenient in comparison to the double differential cross-section, because it represents the spectral properties of the studied material as a function of the momentum and energy transfer and the thermodynamic temperature. According to Eqs. (6.7), (6.11), and (6.14), in the case of magnetic scattering S(Q, E ) represents a combination of peaks at energies defined by the difference between energies of the states coupled by magnetic dipole transitions. The peak intensity is determined by the probability of such a transition (along with the population of the initial states of the system), which is proportional to the ˆ ⊥ (Q)|, thus being directly related square of the matrix element | to the structure of the wave functions corresponding to the abovementioned states. As a result, the magnetic INS spectrum contains the information on the energy and wave functions of f-electron states. Usually, experimental spectra for f-electron systems with CEF excitations are interpreted using Eqs. (6.14) or (6.10), depending on its relaxation characteristics and the extent of cooperative interaction between magnetic ions. It is necessary to mention two important conservation laws that hold for the spectral function S(Q, E , T ). (i) The principle of detailed balance implies that the energy integral over the spectrum has a constant value. One particular consequence is that at T → ∞, the spectral parts for neutron energy loss (E > 0) and neutron energy gain (E < 0) have equal intensity, but at T → 0 the whole intensity

500 Neutron Spectroscopy on Rare-Earth Borides

is collected on the energy-loss side. (ii) According to the sum rule, the mentioned integral (taking into account the elastic scattering, resulting from the Curie-type contribution to the susceptibility, and quasielastic magnetic intensity, when it is present) after the integration over the whole solid angle is a constant defined by the total magnetic scattering cross-section of the RE ion given by the square of its total magnetic moment M2 , according to the equation 2π S(Q, E , T )dE dQ = (γ re )2 M2 σmag = 3 (6.15) 2 barn ≈ · 0.913 2 · M2 ≈ (0.61 barn) · (g J )2 J (J + 1). 3 μB Quantitatively this relationship implies that the magnetic moment M = 1.3μB provides the total magnetic scattering cross-section of σmag ≈ 1 barn. It is necessary to note that typical values of the magnetic moments for the ground-state multiplets of the RE ions are in the range of 1–10 μB . Therefore, the magnetic crosssections have approximately the same order of magnitude as the nuclear ones. This fact provides the opportunity to perform neutron spectroscopy studies of f-electron excitations using the same experimental techniques that have been designed for spectroscopy of atomic vibrations (phonon spectroscopy). On the other hand, the close values of the scattering amplitudes result in the necessity of quantitative separation between the magnetic and nuclear scattering contributions in the measured spectra. This is realized by several algorithms, for instance by using an isostructural La- or Lu-based compound as a nonmagnetic reference. The most direct way to separate the magnetic scattering from the nuclear one is the polarization analysis of the neutron spin orientation in the scattering process. Unfortunately, this method entails a dramatic loss of the signal intensity (typically by more than an order of magnitude). Practical aspects of all these procedures are discussed in the literature (see, for instance, Ref. [13]). Finally, it is necessary to mention the effects of neutron absorption. Compounds based on a number of chemical elements in their natural isotopic composition (Sm, Gd, Cd, B and several others) cannot be studied by INS due to the very high absorption cross-section of certain isotopes of these elements. For the abovementioned elements, a layer of ∼0.1 mm or even less would result

Specifics of the Neutron-Scattering Technique in Condensed Matter Spectroscopy 501

in the attenuation of the thermal neutron beam by at least one order of magnitude. Therefore, an enrichment by low-absorbing isotopes (such as 154 Sm or 11 B) to a high enough level (typically not less than 99%) provides the only possibility to make spectroscopic measurements with reasonable statistical accuracy due to the resulting increase in the sample volume participating in the neutron scattering. This remark is of special importance with respect to the study of borides, where only the isotopically enriched 11 B can be used, and in the most unfavorable case of SmB6 , discussed in Section 6.2.3, isotopic enrichment of both elements is essential. At the same time, isotope enrichment offers the opportunity to study the influence of the boron atomic mass on the physical properties (boron isotope effect), because the isotopic masses of 10 B and 11 B differ by approximately 10%, which is in principle sufficient to cause measurable changes in some structural and thermodynamics properties.

6.1.2 Characteristic Features of Inelastic Neutron Scattering with Respect to Heavy-Fermion and Mixed-Valence Phenomena Spectroscopy is the tool for studying dynamical properties, which provides the information about the origin and character of interactions that determine the physical properties of a material. Let us recall the particular problems at this level in strongly correlated electron systems which have been actively investigated using neutron spectroscopy in the last decades. Below we provide a short overview that will precede the discussion of the results on RE borides. The starting point was the problem of directly observing the splitting of the ground-state spin-orbit J-multiplet of the RE ion’s f shell under the influence of the CEF in the metallic matrix. Here, neutron spectroscopy is the only direct method for studying the CEF splitting of f-electron states in metallic substances. Afterwards, the interest was focused on cooperative effects, resulting from the RKKYinteraction among the magnetic moments in the periodic lattice of RE intermetallic compounds (ordered alloys).

502 Neutron Spectroscopy on Rare-Earth Borides

From the beginning of the 1980s, the focus has shifted more to understanding the formation of the “heavy fermion” state in intermetallic compounds with RE ions from the beginning (Ce) and the end (Yb) of the lanthanide series. A little later it was expanded to the “intermediate (mixed, fluctuating, unstable) valence” systems with Sm, above-mentioned Ce, Yb, and then Eu. The study of the essence and driving forces of this phenomenon, its relation to the Kondo-effect approach, which is basically the dilute impurity-limit consideration, and the principal conditions for transferring this concept to the periodic lattice of RE ions (Anderson model) takes on special significance. All these problems found a manifestation in relation to the physics of borides. The qualitative analysis of many subsequent INS experiments leads to several important conclusions: (i) about the relationship of the CEF potential with the local crystal symmetry in the vicinity of the RE ion serving as a “sensor” of the CEF; (ii) about the existence of a considerable contribution of conduction electrons to the CEF potential; (iii) about the influence of the exchange interaction on the experimentally observed parameters of the excitations (energy, dispersion). Moreover, the rapid development of the neutron-scattering method in application to strongly correlated electron systems at the initial stage in the 1980s and ’90s (a number of detailed reviews exist from that period [17– 21]) resulted in the development of a rather simple and vivid classification scheme for the multitude of synthesized binary and more complicated systems based on f elements. It is based on the combination and competition of the CEF effects (characteristic splitting scale CEF ), exchange-driven magnetic ordering (interaction energy parameter Jex ), and hybridization interaction (energy scale Vcf ). Depending on the extent of hybridization with respect to other parameters, first of all to the CEF splitting, the ground state has a local magnetic moment when CEF , Jex > Vcf ), a Kondotype ground state (in the ordered lattice of Kondo ions—“heavy fermion”) is formed when CEF ≈ Vcf , and when the hybridization dominates ( CEF < Vcf ), the “intermediate-valence” state is realized, corresponding to the partly delocalized f electron. In the latter case, strong spin fluctuations are observed in the form of a

Specifics of the Neutron-Scattering Technique in Condensed Matter Spectroscopy 503

70 60

units) S(Q,,E) ( arb. u

50 40

Ce0.9 0 9Y0.1 0 1Al3

* ( meV)

quasielastic (centered at E = 0) contribution to the magnetic INS spectra. In the heavy-fermion regime, the existence of broad (the line width is comparable with the energy of excitation) inelastic peaks related to the CEF transitions are observed, along with the quasielastic signal like the one shown in Fig. 6.2(a) with a quite specific temperature dependence of its width. The width of the quasielastic signal serves as an important characteristic of the felectron state—it probes the energy scale of spin fluctuations, kB Tsf , which appears to be closely related with the concept of the Kondo temperature (kB TK ≡ 12 limT →0 qe = kB Tsf ). A sketch of the relationship between temperature and quasielastic line width for different physical regimes of strongly correlated electron systems is presented in Fig. 6.2(b). In this way, based on the characteristic features of neutron spectra, one can distinguish what type of regime is realized for a particular sample under investigation. This serves as a good support

T=4K

15

a)

b) * (T=0)/2 ~ W -1 sf

~const const IV

~A+T1/2

10

30

Г=kBT

HF

20

5

10 0 -3 3

-2 2

-1 1

0

Energy transfer (meV)

1

2

0

~DT LM

50

T (K)

100

Figure 6.2 (a) A typical spectral function in the range of the quasielastic signal is shown for the heavy-fermion system Ce0.9 Y0.1 Al3 at low temperature (T = 4 K) when some characteristic asymmetry of the scattering function is observed due to the temperature factor (  T ). (b) The width of the quasielastic line in the magnetic excitation spectrum vs. temperature is schematically presented as the “classification” characteristic of strongly correlated electron systems in different physical regimes: local magnetic moment (LM), heavy fermions (HF), intermediate valence (IV). Here τsf is the period of spin fluctuations, —the quasielastic line width at halfmaximum.

504 Neutron Spectroscopy on Rare-Earth Borides

in addition to the thermodynamic methods of classification and, in some cases, as the unique objective criterion. An important premise for the significant role of neutron spectroscopy is the characteristic time scale of 10−11 –10−13 s for the interaction of neutrons with excitations in solids. This has a principal meaning in the study of the “intermediate-valence” systems. This name originates from the experimental fact that for some compounds (typically based on Ce, Sm, Eu, Yb), partial population of the f shell is observed under certain external conditions (chemical composition, pressure, temperature). The intriguing fact is that the observed particular “image” for the state of the f shell appears to be dependent on the method of measurements. On the one hand, all the “fast” methods (such as x-ray absorption spectroscopy or photoelectron spectroscopy with a characteristic time of 10−15 –10−17 s) demonstrate the coexistence of two f-shell configurations with the integer number of electrons differing by 1. On the other hand, the relatively “slow” method—isomer shift in ¨ the γ -resonance (Mossbauer) spectroscopy (characteristic time of the order of 10−9 s)—shows some average intermediate position of the energy of the γ -quanta corresponding to the mixture of the two “competing” configurations for the f electrons. Based on these observations, a simplified concept about the dynamic origin of the “intermediate-valence” phenomenon arose [22], characterized by the time of inter-configuration fluctuations between 10−9 and 10−15 s. As we can see from above, characteristic time scales of neutron spectroscopy are in the middle of this interval. This fortunate fact allows neutron spectroscopy to be used for the study of corresponding effects in the electron (magnetic) excitation spectra as well as in the lattice dynamics where the electron–phonon interactions are also modified due to the mixed valence. Below, in Sections 6.2 and 6.3, we review a number of results concerning magnetic excitations (Section 6.2) and lattice dynamics (Section 6.3) of hexa- and dodecaborides. These results are related to the manifestation of certain phenomena in strongly correlated electron systems with non-negligible hybridization. Many of the presented works represent comprehensive studies of the magnetic and lattice dynamics, as well as their mutual influence and interconnection.

Magnetic Excitations in Hexa- and Dodecaborides 505

6.2 Magnetic Excitations in Hexa- and Dodecaborides The main contribution of neutron scattering to condensed matter physics is the experimental study of magnetism on the microscopic level and its different manifestations on the local and cooperative levels. In particular, measurements of CEF excitations in metals is a valuable contribution to the study of Kondo and unstablevalence systems, provided by the advances in magnetic neutron scattering in application to the physics of strongly correlated electron systems [18, 20, 23]. In this section, achievements of inelastic magnetic neutron scattering in the field of RE borides are discussed.

6.2.1 Crystal Electric Field Effects CEF effects in hexa- and dodecaborides are expected to be described within the cubic local symmetry for the RE site with two independent parameters of the model Hamiltonian, according to point-charge model [24], with its famous parametrization suggested by Lea et al. [25] and calculations of the dipole-magnetic transition probabilities presented by Birgeneau [26]. The latter two classical works show how neutron spectroscopy data, described by the crosssection in Eq. (6.14), can be treated to extract the CEF splitting scheme and describe it with the phenomenological CEF parameters A n r n  for the particular J-multiplet of any RE ion in any cubic material [25]. The simplest splitting scheme is realized for the Ce3+ ion with the spin-orbit ground state J = 5/2, which splits in the cubic CEF potential into the 8 quartet and the 7 doublet [25]. In CeB6 , the splitting energy appears to be 46 meV, with 8 suggested as the ground state according to the results of neutron spectroscopy on isotopically enriched3 samples [27] and Raman spectroscopy [28]. The symmetry of the ground state was confirmed recently [29] by means of core-level nonresonant inelastic x-ray scattering (NIXS), 3 In

the following we will always suppose by default the use of 11 B-based samples of borides in a neutron study.

506 Neutron Spectroscopy on Rare-Earth Borides

for details see Chapter 7. This splitting value is relatively high for metallic compounds in general [23], and 8 appears as a quite isolated ground state. CeB6 became a model compound exhibiting antiferroquadrupolar (AFQ) order, its magnetic properties being typically interpreted within localized models including Kondo effect. More recently, however, the observation of strong and sharp magnetic exciton modes forming in its antiferromagnetic (AFM) state at both ferromagnetic and AFQ wave vectors [30, 31] suggested a significant contribution of itinerant electrons to the spin dynamics [32, 33]. These excitations develop in the energy range of the order of ∼1 meV or less and serve as the subject of an extended ongoing study [34–36] that will be discussed in Chapters 7–10 of this book. The two RB6 compounds based on RE elements nearest to Ce, namely PrB6 and NdB6 , have been studied by inelastic magnetic

Г7

NdB6

PrB6

CeB6 540K Г1

464K 6.7

Г4 4.4

Г3

377K 9.3

314K 278K

Г6 9.3 93 4.0 3.5

Г8

(1)

135K 9.2 0.4

Г8

0 Г5

0 Г8

(2)

0

Figure 6.3 Crystal-field level schemes for CeB6 , PrB6 , and NdB6 deduced from INS experiments. Arrows indicate magnetic dipole transitions observed in neutron spectra, and numbers are the squares of matrix elements in Eq. (6.14). Reproduced from Loewenhaupt et al. [37].

Magnetic Excitations in Hexa- and Dodecaborides 507

¨ scattering in Julich [37]. From the measured spectra, the crystalfield parameters A 4 r 4  and A 6 r 6  of the order of −200 and +6 K, respectively, were determined. The data are consistent with the value A 4 r 4  ≈ 230 K corresponding to the splitting of 46 meV observed for CeB6 [27]. Negative value of the A 4 parameter seems to be specific for hexaborides in contrast to other metallic systems like RAl2 and RAl3 , for instance. The CEF splitting schemes for the three compounds are presented in Fig. 6.3. Note that the ground state of the Pr3+ ion is the 5 triplet, but not a singlet as it could in principle be for the integer J, contrary to the case of Kramers-type ions such as Ce3+ and Nd3+ with half-integer J. This essential difference underlies the specific low-temperature physics of PrB6 , related to the magnetic phase transitions based on the 5 ground-state configuration, that was studied by complementary methods including neutron spectroscopy [38–40] on singlecrystal samples. It was demonstrated that some inhomogeneous correlated magnetic states appear at temperatures below 20 K, preceding the long-range order that sets in at ∼7 K. It is necessary to mention the most recent works performed using resonant inelastic x-ray scattering (RIXS) with synchrotron radiation, which allows the determination of the CEF-type transitions even for the intermediate-valence state. In particular for SmB6 , Amorese et al. [41] interpret their data as the splitting of the J = 5/2 multiplet of the Sm3+ partial state with an energy of about 20 meV. Another synchrotron-radiation-based technique, the corelevel nonresonant inelastic x-ray scattering (NIXS) (this method is discussed in Chapter 7), has been employed by Sundermann et al. [42] to investigate the CEF ground state of the 4f manifold of SmB6 . The observed anisotropy of the scattering function was interpreted as the unambiguous evidence of the 8 type of symmetry for the ground state of the Sm ion. The mentioned x-ray methods introduce a lot of energy into the system in comparison to the characteristic energy scales of basic interactions that are relevant for the particular physical properties. As it will be shown in the following, neutron spectroscopy did not register any signal related to the CEF splitting of the groundstate J-multiplet in the intermediate-valence systems, including SmB6 .

508 Neutron Spectroscopy on Rare-Earth Borides

150

100

50

0 0

10

20

30

Figure 6.4 Magnetic part of the Lu0.92 Tm0.08 B12 spectra at 2, 50, and 100 K, associated with the Tm3+ CEF splitting. The solid line is a fit to the spectrum at T = 2 K by Gaussian line shapes. Reproduced from Ref. [43].

In RE dodecaborides, the CEF effects have been studied in the YbB12 matrix by neutron spectroscopy of the Er3+ [44] and Tm3+ [43] impurity ions, whereas no clear CEF excitations could be observed for pure YbB12 . Experimental spectra for the Lu0.92 Tm0.08 B12 powder, measured at the TOF spectrometer IN4 with the incoming neutron energy E i = 36 meV, integrated within low scattering angles, 2θ ≈ 15◦ , and corrected for the weak phonon contribution, are shown in Fig. 6.4. The width of the CEF excitation peaks is limited by the experimental resolution. Taking into account the extreme sensitivity of the crystal-field splitting to the local symmetry and displacement distortions in the immediate vicinity of the RE ion, one would expect that local distortions resulting from the structural instability due to the cooperative Jahn-Teller effect proposed for LuB12 [45, 46] (see also Chapters 3 and 4) should result in the broadening of the CEF lines. One can, therefore, conclude from the INS data that such distortions do not exceed a few percent of the interatomic distance, on average, or influence only a minority (< 10%) of the RE ions. The splitting schemes for 4 I15/2 of Er3+ and 3 H 6 of Tm3+ are shown schematically in Fig. 6.5. The low concentration of magnetic

Magnetic Excitations in Hexa- and Dodecaborides 509

meV

Tm3+

Er3+ Г7

30 Г2 Г8(1) 20

Г5(2) Г6 Г1 Г4

Г8(2)

10 Г3

0

Г5(1)

Г8(3)

Figure 6.5 CEF splitting for Er3+ in Yb0.9 Er0.1 B12 and Tm3+ in Lu0.92 Tm0.08 B12 obtained from INS data [43, 44]. In both cases the ground state is degenerate (5 triplet for Tm3+ and 8 quartet for Er3+ ) and carries some magnetic moment. Arrows indicate magnetic dipole transitions directly observed in INS spectra at low temperatures.

ions was used to exclude magnetic interactions between them at low temperatures. The parameters of the cubic-symmetry CEF potential A 4 and A 6 are in good correspondence for both ions. The extrapolation of these values for the Yb3+ ion’s F 7/2 multiplet, split into three levels, provides the estimation of the total expected CEF splitting of 11 meV [44]. This is about twice less than the energy of the single broad inelastic peak at about 20–25 meV observed in the “high-temperature” (T > 70 K) spectra of YbB12 , which is thought

510 Neutron Spectroscopy on Rare-Earth Borides

to represent the CEF splitting coexisting with the incoherent spinfluctuation regime (Tsf ≈ 100 K) realized for the Yb ion (see below). It appears that the hybridization of the Yb f states with conduction electrons results in a strong increase in the energy splitting of the multiplet of Yb atoms, forming the host lattice for the impurity ion serving as a simple “CEF sensor.”

6.2.2 Hybridization Effects: Intermediate-Valence and Kondo Insulator Systems Among the hexa- and dodecaborides there are a few systems (YbB6 , SmB6 , YbB12 ) which demonstrate hybridization-driven physical phenomena and related “anomalous” ground states such as Kondo insulator or intermediate valence behavior. As discussed in Section 6.1, neutron spectroscopy is an effective tool for analyzing these features, but the high neutron absorption cross-section of the 10 B isotope with 20% natural abundance makes any neutron experiment on borides not a routine task. Furthermore, natural samarium cannot be used for neutron scattering for the same reason as natural boron. Therefore, neutron-scattering measurements of SmB6 are most challenging, as they require samples with double isotope enrichment. An additional advantage of using isotopically purified elements for the preparation of samples like 154 Sm11 B6 , especially in the form of single crystals grown by the floating zone technique (see Chapter 1), is that it offers a unique opportunity to work with a highly pure and homogeneous material in terms of the extremely low concentration of other RE elements as impurities. This is important in some cases when delicate and experimentally subtle effects in electronic subsystem are discussed. The specific and physically rich properties of the CeB6 compound and its solid solutions are discussed in Chapter 9 of this book. Below we consider neutron spectroscopy results for the intermediatevalence Kondo insulator SmB6 and another “classical” Kondo insulator YbB12 . Both compounds demonstrate the formation of a small (of the order of 100 K) gap at Fermi energy at low temperatures [47, 48], which has been interpreted as a metalsemiconductor transition. It is necessary to mention that SmB6 was the first compound classified as an intermediate-valence system

Magnetic Excitations in Hexa- and Dodecaborides 511

more than 50 years ago in Yu. Paderno’s laboratory [49], and up to now the scientific output of this laboratory at the I. M. Frantsevich Institute for Problems of Material Sciences in Kyiv, Ukraine, remains very high. The unusual properties of SmB6 are particularly exciting due to the phenomenon of a “topological insulator” that was proposed for this compound [50, 51] among others (see Chapters 8 and 11 for details). The data obtained by neutron spectroscopy as a bulk-sensitive method provided some background for the analysis of SmB6 from this point of view.

6.2.3 Excitation Spectra of the Intermediate-Valence Kondo Insulator SmB6 As mentioned in the introduction, neutron spectroscopy is quite “compatible” with valence fluctuations in terms of the range of characteristic interaction times of neutrons with phonons and magnetic moments [13]. The use of neutrons allows establishing a number of previously unknown features of the intermediatevalence state which become apparent in the atomic and magnetic dynamics [52], and against this backdrop some new generalizing ideas dealing with the quantum-mechanical origin of this state [53, 54] have been developed. Comprehensive studies of the lattice and magnetic excitation spectra in a wide range of temperatures and intermediate-valence values for poly- and single-crystalline samples of SmB6 and SmS (produced from the pure isotopes 154 Sm and 11 B) have been carried out. The influence of the intermediate-valence state on the lattice dynamics was studied at the level of phonon dispersions [55–58] (see Section 6.3). The magnetic excitation spectra including intermultiplet transitions of both competing configurations f 6 (Sm2+ ) and f 5 (Sm3+ ) were studied in Refs. [59–65]. The induced magnetic form factor was measured with polarized-neutron diffraction [66].

6.2.3.1 Intermultiplet transitions and the resonant mode in the magnetic neutron-scattering spectra of SmB6 The first successful measurements of the magnetic INS spectra of the intermediate-valence Sm ion in the wide range of energy

512 Neutron Spectroscopy on Rare-Earth Borides

transfer (up to 180 meV) have been carried out on polycrystalline samples of SmB6 and LaB6 by the method of TOF spectroscopy at the pulsed neutron source ISIS [59]. The single-crystal samples of these materials have been studied at the TAS positioned at the thermal neutron beams of the high-flux research reactor at ILL ´ at LLB (Saclay) [61, 62, (Grenoble) [55, 56] and the reactor Orphee 65]. The important fact which was obtained by the TOF measurements at T = 20 K [59, 63–65] is the existence of characteristic peaks (see Fig. 6.6, inset) in the f-shell excitation spectra, just as

Figure 6.6 Intensity ratios of the intermultiplet transitions for Sm2+ (I2 ) and Sm3+ (I3 ) in the neutron-scattering spectra of intermediate-valence Sm-based systems (broad peaks in the scheme from the inset) are shown as a function of the average valence of Sm, the points are experimental data for (Sm,M)B6 , the solid line is the calculation, average valence is determined from the L3 -edge (XANES) spectroscopy experiments. The triangles show data for Sm3 Te4 [67] and (Sm0.3 Y0.7 )S [52]. The inset shows a schematic presentation of the experimentally observed spectra of the magnetic neutron scattering for SmB6 . The correspondence between dipole intermultiplet transitions (the magnetic scattering cross-sections σm shown in red) and the observed spectral peaks is indicated by arrows. The narrow peak (14 meV) appears in the spectrum only at temperatures below 50 K.

Magnetic Excitations in Hexa- and Dodecaborides 513

expected from the traditional concept of intermediate valence (i.e., the coexistence of two “competing” f-electron configurations with inter-configurational fluctuations between them as the basis for the intermediate-valence phenomenon). The observed peaks are quite broad (half-width at half-maximum /2 ≈ 10–15 meV) and originate from the intermultiplet transitions specific to the two “competing” Sm-ion-configurations f 6 and f 5 , with energies of 36 and 125 meV (J = 0 → J = 1 and J = 5/2 → J = 7/2 for f 6 and f 5 ), respectively.4 The width of the neutron peaks unambiguously indicates the relatively fast relaxation of spin-orbital states (∼10−12 – 10−13 s). A very important but unexpected in the above-mentioned sense result obtained by INS is that at low temperatures (below 80 K) the very narrow peak at the energy of ∼14 meV was unambiguously observed in the magnetic excitation spectra. The first evidence of its existence was demonstrated in the pioneering Holland-Moritz and Kasaya’s [68] neutron experiments with SmB6 . The experimental width for this peak has been limited [59] by the resolution of the spectrometer, and in this sense was at least an order of magnitude smaller than for the above-mentioned intermultiplet transitions. There is no place for such an excitation in the intermediate-valence fluctuation concept. The magnetic form factors of the broad intermultiplet excitation are described well by the single-ion calculations for L-S multiplets [69]. But for the 14 meV excitation, the Q-dependent form factor appears to be much steeper with respect to the “normal” f electron. The form factor of the 14 meV excitation was experimentally studied in detail on the doubly isotope-enriched single crystal of 154 Sm11 B6 in Refs. [61, 62, 65] and in the more recent work by Fuhrman et al. [70] who investigated the energy and Q-dependence of the excitation and its localization in reciprocal space. The form factor of the quasielastic spectral component that gradually replaces the 14 meV peak when the temperature increases above 50 K was obtained experimentally in Ref. [71]. Of course, diffraction measurements of the induced magnetic form factor of 4 It

is necessary to note that these values are the lowest ones in the RE series for the intermultiplet transitions with | J | = 1.

514 Neutron Spectroscopy on Rare-Earth Borides

Sm in SmB6 at T = 7 K in the magnetic field of 5 T were also done [66], with similar “puzzling” results as in the previously studied intermediate-valence system SmS [72]. The Q-dependence shows no evidence of the contribution from the “anomalous” form factor of the Sm3+ (J = 5/2) ground state (with a maximum at ∼5 A˚ −1 ) to the “true” form factor of the intermediate-valence ground state, expected due to the valence value v ≈ 2.5. One of the essential results of the works [63, 64] is the experimental study of the connection between the magnetic neutronscattering spectral parameters and the valence state of samarium, the latter was determined independently by the x-ray absorption spectroscopy (XANES). The INS measurements have been carried out on SmB6 -based samples with a variation of the average Sm valence v in a wide range between 2.2 and 2.7 realized by the variation of composition (substitution of La, Ca, Ba for Sm in SmB6 provides “negative” and “positive” chemical pressure in some analogy with the substitution of Y for Sm in SmS). It appears that the experimentally observed intensity ratio of the two broad intermultiplet transitions (related to the Sm3+ and Sm2+ ions) is in good correspondence with the variation of the average valence (see Fig. 6.6), the widths are not changed too much, and the energy positions are stable in correspondence with the single-ion values. In connection to the discussion of the intermultiplet excitations in the homogeneous intermediate-valence systems like SmB6 , let us pay attention to the results of the neutron study of the Sm3 Te4 compound [67] where the Sm valence v = 2.66 results from the real chemical mixture of stable Sm2+ and Sm3+ ions in the identical positions of the crystal structure. The magnetic neutron-scattering spectra show well-defined narrow intermultiplet transitions for both configurations with clear CEF splitting resulting in the temperature-variation-driven transformation of the excitation spectra. The integrated intensity ratio of the intermultiplet transitions for Sm3 Te4 is also shown in Fig. 6.6 by a triangle, and its value corresponds to the general tendency of the direct relation between the peak intensities and average valence.

Magnetic Excitations in Hexa- and Dodecaborides 515

Figure 6.7 Dependence of the resonant-mode energy on the Sm valence: circles correspond to Sm1−x Lax B6 [64]; the triangle corresponds to Sm0.5 Ba0.5 B6 [63]; the square represents the energy of the intermultiplet transition for the f 6 configuration of Sm2+ ; solid line shows the calculation for the model of Kikoin and Mishchenko [54].

Roughly speaking, the picture of the intermultiplet spectra appears as well consistent with the fluctuation representation of the intermediate-valence state as a mixture of the two short-lived competing Sm2+ and Sm3+ f-electron configurations. Nevertheless, as it will be seen in the following, the properties of the 14 meV excitation are inconsistent with this simple idea. The first contradiction is its narrow energy width. Another one results from the analysis of the Q-dependence of the intensity and from the data on the influence of valence variations on the parameters of this excitation. In the following the 14 meV excitation is termed “resonant mode” due to its quite narrow intrinsic width and some analogy with the strongly correlated high-Tc cuprates and iron-based superconductors. The effect of the valence variation on the parameters of the resonant mode is opposite in character to the described above

516 Neutron Spectroscopy on Rare-Earth Borides

Figure 6.8 Integrated intensity of the resonant mode as a function of the momentum transfer Q along [111], measured with single-crystal samples. Small black circles correspond to SmB6 , larger circles and squares—to Sm1−x Lax B6 samples with x = 0.1 and 0.22, respectively. Experimental data from Refs. [64, 65] have been used. The dotted line shows the calculated f 6 form factor of Sm2+ . ZB and ZC denote the Brillouin-zone boundary and center, respectively.

features for the intermultiplet single-ion transitions. A selection of the experimental data is summarized in Figs. 6.7 and 6.8. Both energy and intensity of the resonant mode change systematically as a function of the average valence v. On approaching v = 2+, the energy (Fig. 6.7) and intensity (Fig. 6.8) increase, while the form factor falls off more slowly, approaching the Q-dependence expected for the Sm2+ ion. The excitation energy changes almost linearly with valence, and the extrapolation to v = 2 corresponds to the energy near 36 meV, which coincides with the value for the J = 0 → J = 1 transition in the f 6 configuration. Thus it appears natural to conclude that there is a close link of the valence state to the resonant mode.

Magnetic Excitations in Hexa- and Dodecaborides 517

6.2.3.2 The model of the exciton of an intermediate radius The above observations in comparison with the preceding detailed lattice-dynamical study of SmB6 (see Section 6.3) resulted in the formulation of an idea [53, 54] that the true ground state of the homogeneous mixed (intermediate) valence Sm ion is the singlet state of the wave function corresponding to the quantummechanical mixture of f 6 and f 5 with loosely bound f-electron states. This new wave function corresponds to the so-called “intermediate-radius exciton” state when one of the six f electrons (the loosely bound one) appears to be located in some double-well potential. This new wave function can be represented by   g (6.16) α| f 6  J=0 + 1 − α 2 f 5 B ( f ) J=0 = J ∗ =0 .

Figure 6.9 The presentation of the double-well potential for the loosely bound electron of the Sm-ion f shell shared between the f 5 and f 6 configurations according to the excitonic model [54]. As a result, two types of excitations from the ground state become possible.

518 Neutron Spectroscopy on Rare-Earth Borides

Here α represents the deviation of the valence from 3+ , and B ( f ) describes the loosely bound electron in the outer well. The J = 0 subscript denotes the symmetry type of the partial states. This way, the excitonic state of finite radius is formed near each RE ion due to the hybridization, the corresponding state of loosely bound electron may be described by a double-well potential as schematically shown in Fig. 6.9. This wave function possesses the excitation spectrum with two types of excited states. The first type corresponds to the reorientation between the spin and orbital moments and provides the magnetic type of excitations without a charge transfer, they can be observed in magnetic neutron scattering. The second type is related to the excitations between states of these two wells that correspond to different probabilities of finding the electron in the particular well, which implies some charge transfer. The latter is just the quantum-mechanical analogue of the initially supposed “charge fluctuations” and can be observed by neutrons spectroscopy only in the phonon spectra due to the electron–phonon interaction effects (see Section 6.3). This idea is consistent with the neutron spectroscopy observations of the f-electron and phonon excitation spectra of both SmB6 and Sm1−x Yx S intermediate-valence systems [23, 53]. The resonant mode can be interpreted as some kind of excitation specific for the true singlet ground state with the new wave function. Therefore, it is quite narrow in contrast to the broad intermultiplet excitations of the “partial” states of the combined wave function represented by Eq. (6.16), like f 6 or f 5 (the socalled “parent states”), which are short-lived with the decay time τ ∝ sf−1 . The resonant mode is intrinsically related to the new wave function, and therefore it is strongly dependent an the loosely bound electron state, i.e., the extent of the mixture (or representation) of the f 5 and f 6 states characterized by the average valence. Therefore, its energy and form factor are strongly related to the average valence v as observed in Figs. 6.7 and 6.8. Of course, the limitation of the described model is its purely single-site character.

Magnetic Excitations in Hexa- and Dodecaborides 519

6.2.3.3 The magnetic form factor study A dedicated study was initiated to find out the form factor of the quasielastic signal, which gradually replaces the resonant mode upon increasing the temperature above ∼50 K. This quasielastic magnetic signal corresponds to the spin-fluctuation spectra after the gap closes. The task is not trivial due to the low intensity of the quasielastic signal in combination with its relatively large width (about 10 meV) and the need to explore a broad range of Q values. Moreover, the phonon contribution may be superimposed on the magnetic signal with increasing temperature and Q. Therefore, we used a single crystal and chose the Q point close to the Brillouin-zone boundary to avoid the acoustic phonon signal in the measured spectra at energy transfer below 10 meV. The results of this experimental study [71] present the first-time clear evidence of Sm2+ J = 5/2 multiplet-specific form factor contributing to the experimentally observed Q-dependence of the spectral intensity. The experimental data are reproduced here in Fig. 6.10 with open symbols. The large error bars are the consequence of the relatively weak quasielastic signal (see inset to Fig. 6.10). Nevertheless, a clear deviation from the typical form factor for Sm2+ is established up to the maximum reachable momentum transfer Q ≈ 5 A˚ −1 . For the interpretation of the observed particular Q-dependence, the presentation of Eq. (6.16) for the Sm wave function was used. The source of the neutron signal has been identified with the combination of the “partial” non-eigenstates contributing to Eq. (6.16). The main idea of the interpretation is that both the field-induced static form factor and the excitations to higher-J multiplets are related to Van Vleck-type susceptibility terms out of the intermediate-valence singlet ground state, whereas quasielastic scattering corresponds to Curie terms. The Curie-type contribution can occur only within degenerate energy levels (ground state or thermally populated excited states). Such states can be provided by the lower multiplets of the quantum-mechanically mixed 4f 5 (Sm3+ ) and 4f 6 (Sm2+ ) configurations that contribute to the excitation spectra only when spin gap is closed at intermediate temperatures (T ≈ 100 K in the present experiments). A calculation based on a simple phenomenological model (the effective temperature of

520 Neutron Spectroscopy on Rare-Earth Borides

Figure 6.10 Experimental Q-dependences of different components of the magnetic response in the mixed-valence SmB6 . Squares: induced form factor from the polarized-neutron diffraction [66]; stars: the resonant mode (14 meV excitation) at T = 2 K [65]; circles: normalized quasielastic scattering intensity at T = 100 K [71]. The lines show calculated Qdependences of the static Sm2+ (solid) and Sm3+ (dashed) magnetic form factors. The dotted line is a guide to the eyes through the star markers. Inset: the quasielastic magnetic signal for Q = 1.4 A˚ −1 at E f = 14.7 meV (the region contaminated with elastic scattering is excluded). The line is a fit with a Lorentzian spectral function that has a half-width at half-maximum of QE /2 = 8(2) meV. Reproduced from Ref. [71].

the system is estimated form the spin-fluctuation width in INS spectra) qualitatively reproduces the experimental Q-dependence of the quasielastic scattering intensity. A gradual enhancement of the maximum in the Q-dependence of the form factor for the quasielastic neutron scattering signal is predicted as the Sm valence gets closer to 3+. Generally speaking, in a low-temperature neutron diffraction study of the induced magnetic form factor of SmS or SmB6 intermediate-valence compounds (like those in Refs. [66, 72]), there is no reason to expect any evidence for the Sm3+ J = 5/2 multiplet form factor contribution because of the spin-gap structure with

Magnetic Excitations in Hexa- and Dodecaborides 521

a singlet ground state of the spectrum of magnetic states for f electrons. Only the spin-gap suppression by increasing temperature above 50 K results in a formation of the spectrum where the J = 5/2 multiplet contribution (modified, of course, by the spin-fluctuation energy scale) can be detected. The supposed above interpretation of the essential part of experimental spectroscopy data based on the model of the exciton of an intermediate radius [53, 54] is, of course, oversimplified. The measurements of single crystals at modern TOF spectrometers, equipped with position-sensitive detectors with a large solid angle, provided a more extended picture of the intensity and energy dispersions in the reciprocal space of the momentum transfer for the resonant mode [70]. This TOF technique, based on a spallation neutron source with a pulsed proton accelerator, provides a larger range of the momentum transfer in comparison to conventional TAS measurements due to the opportunity of optimizing the ratio between the energy of the studied excitations and the incoming neutron energy. The concept of a “topological insulator” [50, 73] and the consideration of SmB6 as a representative of this class of materials (see Chapter 8) gave rise to the analysis of the bulk properties of SmB6 obtained by the INS technique, which we briefly present below.

6.2.3.4 Resonant exciton modes at the R and X points Fine details of the Q-dependent intensity distribution of the 14 meV excitation, showing strong anisotropy with the intensity concentrated along the (111) direction, was initially studied using TAS [61, 62] and later using state-of-the-art TOF spectroscopy [70]. In Fig. 6.11(a), the results of the TAS study [62] are shown for this excitation as an intensity map around the R( 12 21 21 ) point, with |Q| = 1.3 A˚ −1 , within the (100) – (011) plane in the Brillouin zone. In the more recent TOF measurements [70], a similar signal was observed at two X points at the zone boundary in the first and the second Brillouin zones, in addition to the R-point resonant mode. These experimental data are presented as a color map of INS intensity in Fig. 6.11(b). The right- and left-hand sides of the panel correspond to the (HHL) and (HK0) scattering planes, respectively,

522 Neutron Spectroscopy on Rare-Earth Borides

Figure 6.11 Intensity distribution of the 14 meV resonant mode: (a) from TAS (contour intensity map) after Refs. [62, 65]; and (b) from TOF spectroscopy (the color map of INS intensity) after Fuhrman et al. [70], measured on the 154 Sm11 B6 single crystal. (c) Illustration of the irreducible part of the simple-cubic Brillouin zone in reciprocal space with several highsymmetry points. The shaded planes indicate the (HK0) and (HHL) crosssections shown in panel (b).

as shown by the green and red shading in Fig. 6.11(c). The measured intensities are in accordance with the Q-dependence of the form factor of the resonant mode from Ref. [62], which results in the higher intensity of the X -point resonance in the first Brillouin zone, Q = ( 12 00), as compared to the equivalent reciprocal-space point shifted by the reciprocal lattice vector at Q = ( 12 10). The momentum-dependent dynamic structure factor of the INS signal is an evidence of cooperative effects and interactions between the localized moments which can result from band effects [70] leading to the formation of the resonant mode in the framework of the excitonic model [54]. The theoretical model of Fuhrman et al. [70] is based on the calculations of the band structure which take into account the

Magnetic Excitations in Hexa- and Dodecaborides 523

f-d hybridization (Anderson model). It results in the R and X points forming poles of intensity as a result of exciton formation within the gap of the f-d zone due to the Coulomb interaction of electron-hole quasiparticle pairs [54]. The electron density distribution around the magnetic Sm3+ ion favors an efficient superexchange along the body diagonal of the unit cell, defining the elongated shape of the exciton along the (111) direction. The spin exciton in this sense is considered as a collective excitation inside the gap of hybridized narrow bands, and the intensity dependence of the signal in Q-space reflects the topology of its band structure. The general point of Refs. [70] and [54] is that the resonant mode is treated as a collective electronic excitation in the insulating gap from the Kondo-singlet ground state. The experimental estimation of its intensity provides the value of the order of 0.2 barn, which corresponds to the effective magnetic moment μ2eff  ≈ 0.3μ2B /Sm. The specific steep Q-dependence of the form factor of the resonant mode is related [70] to the strong hybridization with the 5d electrons. The latter seems to be in some contradiction with the observed variation of form factor with the change in valence [64]. Also, the f-d hybridization approach to the formation of the exciton does not provide a hint to understanding its energy variation with valence (see Figs. 6.7, 6.8). Therefore, we should conclude that some serious questions still exist in the interpretation of the features in the f-electron excitation spectra of SmB6 .

6.2.3.5 Gd-impurity effect on SmB6 In a topological insulator, the electronic structure at the surface is different from that of the bulk material, yet it constitutes only a small fraction of the sample volume. In this sense, the effect of impurities on the physical properties of the bulk is of principal interest. In a recent study [74], the effect of Gd3+ impurities with the pure spin magnetic moment on low-temperature properties has been investigated. As a reference system with zero impurity concentration, the same highly pure 154 Sm11 B6 single crystal was used, in which the upper bound for the Gd content was estimated as not more than 0.04%. The Gd ion can be treated as a spinful isoelectronic Kondo hole contrary to the case of La, which is a

524 Neutron Spectroscopy on Rare-Earth Borides

nonmagnetic Kondo hole. Several samples with Gd concentrations of 0.65%, 1.85%, 3.8%, and 4.8%, as well as a 5% La-doped sample have been studied by magnetization and heat capacity measurements in the temperature range from 1 to 10 K. Also the triple-axis neutron spin-echo technique (TRISP spectrometer, FRM-II, Garching) has been used to investigate the intrinsic relaxation characteristics of the reference SmB6 single crystal with experimental resolution of the order of 10 μeV [74]. Results of these magnetization studies gave direct evidence that the increase in the concentration of paramagnetic impurities results in the suppression of the local magnetic moment from 7.74μB for Gd 0.04% to 5.84μB for Gd 4.8% at T = 10 K. The screening of magnetic moments is unexpected for isolated Gd3+ given the halffilled 4f 7 electron configuration, which carries no orbital moment and is particularly stable [77]. The temperature range for this deviation coincides with an upturn in the linear portion of the specific heat, which is greatly enhanced with doping. The enhanced linear specific heat and moment screening bear resemblance to the Kondo impurity effect seen in metals. In the s-d Kondo impurity model, magnetization and susceptibility are renormalized by Jη— a dimensionless parameter formed by the product of an exchange constant J and the electron DOS η. Estimation of this parameter from the magnetization data by using the Kondo model, μeff = μGd (1+Jη), results in the estimate Jη = −0.26 for the 4.8% sample. The values of Jη extracted from the magnetization data (inset in Fig. 6.12) allows establishing a scaling of the field dependence of the effective Gd moments and the linear specific heat coefficient (the latter is shown in Fig. 6.12) for all the samples. Although SmB6 is nonmetallic, the Kondo impurity model still provides a qualitative aid in the description of our magnetization and specific-heat data. The intermediate valence intrinsic to SmB6 may facilitate electronic and magnetic fluctuations of magnetic Sm3+ and nonmagnetic Sm2+ adjacent and exchange-coupled to Gd3+ impurities, resulting in a dynamic screening of localized Gd moments which mimics the well-known Kondo impurity effect. Whether a Kondo-impurity-like effect or another dynamic exchange mechanism, the net result is an insulator with screened impurity moments and substantial low-energy DOS as evidenced by specific

Magnetic Excitations in Hexa- and Dodecaborides 525

a

4.8% Gd

5% La

300

3.8% Gd

250 mJ

0.65% Gd

200

Isotopic (0.04% Gd)

150

T

C

(

mol K2

)

1.85% Gd

100 50 0

b

1.2 T

1.

–1

0.3

T⎮J ⎮

⎮J ⎮

J C –C lat

(

mol K 2

)

0.03% Gd

0.8

ESR

0.2 0.1

0.6

∝c 0

0.4

0

1

2

3 c (%)

4

5

6

0.2 0

2

4

6

8

10

T (K)

Figure 6.12 Heat capacity of Gd- and La-doped SmB6 . (a) Raw heat capacity over temperature. The upturn is most dramatic in the heavily Gd-doped samples, whereas 5% La introduces a predominantly linear heat capacity with much lower magnitude per impurity than Gd doping. (b) Heat capacity of Gd-doped samples after subtracting the lattice contribution (βT 3 ), scaled by J η as determined from magnetization data. The same value of β = 0.2 mJ K−4 mol−1 is used for all samples. Previously published results for the heat capacity of isotopic SmB6 at low temperatures are scaled and included in red [75]. The scaled C /T data are fit by T −1+ with  = 0.02. The inset shows J η as a function of Gd concentration, which is proportional to c α with α = 0.7(1). The previously published ESR result (from g-factor shift) is included as an open circle [76]. Reproduced from Fuhrman et al. [74].

526 Neutron Spectroscopy on Rare-Earth Borides

1 h (meV)

14

*(meV)

0.8

13

0.6

12 0

5

10 15 20 25 30 T (K)

0.4 0.2 0

*J 0

5

10

15

20

25

30

35

T (K)

Figure 6.13 Resonance mode width  and energy ω (inset) vs. temperature. The solid line shows 0 + A exp(− /kB T ), with nonthermal lifetime 0 = 0.11 meV and temperature dependence from the exponential activation to the mode energy = 12.8 meV. Dashed line is the line width estimated from coupling the collective mode to the DOS at E F ( J η = 0.080 meV). Reproduced from Fuhrman et al. [74].

heat and magnetization. The same consideration is applicable to the explanation of the temperature dependence of the relaxation width related to the resonant-mode lifetime for the SmB6 single crystal presented in Fig. 6.13. The temperature dependence of the relaxation rate was modeled as the sum of an intrinsic width and exponential activation across a gap:  = 0 + A exp(− /kB T ). Fixing the gap energy to the exciton mode energy ∼13 meV gives an intrinsic width 0 = 0.11(1) meV and bandwidth A = 0.7(2) eV. In an analogy to spin-resonance modes in cuprate superconductors, coupling of the exciton to a Fermionic DOS at E F leads to a relaxation rate 0 = Jη = 4π(Jη)2 Ω, where J is an exchange interaction (J ∝ t2 /U for the Hubbard model), η is the DOS, and Ω is the mode energy [74]. Finally the estimation of 0 ≈ 0.1 meV for Gd: 0.04% (“isotopic pure sample” SmB6 ) seems to be realistic as seen from Fig. 6.13. Intrinsic width 0 dramatically depends on Jη, which characterizes the effect of impurity on the electronic system: just a few percent of Gd will increase the relaxation by two orders of magnitude. This explains the fact observed in studies of another Kondo insulator, YbB12 , where

Magnetic Excitations in Hexa- and Dodecaborides 527

the resonant mode was totally suppressed by a few percent of magnetic Kondo holes like Tm3+ (see Section 6.2.4). In addition to the systematic variation of physical properties with doping, a neutron spectroscopy study at very low temperatures down to 200 mK [74] revealed no apparent magnetic excitations below the long-lived 13 meV spin exciton in high-purity SmB6 . Magnetic neutron scattering is well suited for probing magnetic excitations such as those proposed for SmB6 (spinons, excitons, etc.), and the present experiments place a tight upper limit on any excitations with a 4f electron form factor of μ2eff  ≤ 0.05μ2B on any fluctuating moment in the sub-exciton energy range. For comparison, the total fluctuating moment associated with the exciton is μ2exciton  = 0.29(6)μ2B .

6.2.4 Magnetic Excitations in the Kondo Insulator YbB12 6.2.4.1 Resonant mode and temperature effects Among the dodecaboride-based strongly correlated electron systems, YbB12 presents a well-known example of the Kondo insulator behavior in macroscopic and spin-dynamical studies. A detailed description of the results of the experimental study of its properties is presented in the original papers and review articles. Here we should mention a few of them, dealing with powders and single crystals to study the magnetic dynamics, including the pioneering publication of Bouvet et al. [78] and the follow-up works of Nefedova et al. [79], Mignot et al. [80], and the review article [23]. Many of the features of its magnetic excitation spectra are well documented, for instance the temperature effects [81–83]. In Fig. 6.14, the low (10 K) and high (125 K) temperature neutron magnetic scattering spectra obtained by polarized-neutron spectroscopy on TAS instruments with a single crystal [80, 82] are shown. The use of polarization analysis results in the most straightforward suppression of the phonon contribution to the measured spectra and turned out to be quite effective in the YbB12 single-crystal study. A clear evolution from the spin-gapped spectra with three peaks of cooperative origin to a single-site spectrum with a gapless spin-relaxation quasielastic signal is observed.

528 Neutron Spectroscopy on Rare-Earth Borides

Figure 6.14 Magnetic excitation spectra of the single-crystal YbB12 measured at the temperatures (a) T = 5 K and (b) 125 K in experiments with neutron polarization analysis. The spectrum at T = 5 K was measured with the momentum transfer Q = ( 23 32 32 ) or q = ( 21 12 12 ). The spectrum at T = 125 K was obtained by combining similar (in the limits of error bars) data observed at the points L( 32 32 23 ) or q = ( 21 12 12 ); X (112) or q = (001); and (113) or q = (000) for the bcc Brillouin zone. Symbols are the experimental data points, lines represent the results of Lorentzian fits. Reproduced from Nemkovski et al. [84].

The low-temperature limit (T < 20 K) for the magnetic excitation spectrum is characterized by the spin gap of about 20 meV with a resonant mode at 15 meV and two other broad peaks, one above 20 meV and another one around 40 meV [79]. The Q-dependence of the energy and intensity studied on a single crystal with polarized neutrons [80, 82] obviously demonstrates the cooperative character of all these excitations, in particular the quasi-2D character of the magnetic correlations associated with the resonant mode, which results in the formation of a system of “rods” in momentum space

Magnetic Excitations in Hexa- and Dodecaborides 529

Figure 6.15 Color maps of the integrated resonant-mode intensity for YbB12 in the (HHL) plane (top left, from Ref. [83]) and in the (HK0) plane (top right, from Ref. [84]). The intensity maxima correspond to the line (HH 12 ) in the (HHL) plane and to the point ( 12 12 0) in the (HK0) plane. A 3D schematic image of the “intensity rods” for the resonant mode in the Brillouin zone of YbB12 (fcc crystal structure) is shown at the bottom right.

corresponding to the maximal intensity and minimal energy of this excitation [85]. These rods are parallel to the {100} directions and cross the Brillouin zone at the L symmetry points (see Fig. 6.15). Their structure may be considered as evidence of short-range 2D correlations in the ground state of Yb within the crystallographic planes of (100) type. Another puzzling feature is the character of the transition to the high-temperature type of the spectrum, which appears as an uncorrelated single-ion response with two CEF-like broad peaks (similar to the heavy-fermion regime of Ce-based systems) and the quasielastic spin-fluctuation signal with a width of the order of qe /2 = 10(1) meV. The latter value is quite large in comparison with the qe for Ce-based HF systems and is not influenced by temperature. The crossover from coherence to the single-site regime takes place [82, 83] in quite a narrow temperature range between

530 Neutron Spectroscopy on Rare-Earth Borides

Figure 6.16 Temperature evolution of the low (LT) and high (HT) temperature spectral features (LT—intensity of three inelastic peaks and HT—quasielastic + single-site inelastic peak, see Fig. 6.14): a step-like transition for YbB12 (solid lines) around 50 K and a smooth crossover for Yb0.8 Zr0.2 B12 (dashed lines), discussed later in the text. In the latter case, LT is represented by the intensity of the inelastic peak M3, and HT is represented by the quasielastic intensity. Reproduced from Ref. [86].

50 and 80 K and looks like a change of decoration in a theater: that means, the “low-temperature signal” simply diminishes in intensity and the “high-temperature signal” rises simultaneously instead (see Fig. 6.16). In principle, one could introduce some effective coherence temperature Tcoh specific for this heavy-fermion system, which is at least two orders of magnitude higher than for systems like CeAl3 , CeCu2 Si2 , or CeCu6 . To understand these observations, several theoretical models have been suggested: one considers the balance between the CEF effects and the exchange under the condition of hybridization interaction [87], another one is based on the gap structure of the spectrum [88], and one more is based on the effects of dimerization

Magnetic Excitations in Hexa- and Dodecaborides 531

in the Yb-ion magnetic sublattice [89]. But for the explanation of the observed change in regimes in a very narrow temperature interval and at relatively low temperatures (compared with other characteristic physical parameters, such as the excitation energy and the width of the quasielastic peak), between 50 and 70 K, no significant models have been suggested so far, except for the assumption of the existence of a “hidden parameter” which is responsible for the “changeover” of the system from one regime to another. So, a clear ultimate physical model for a number of such features specific to YbB12 has not taken shape yet.

6.2.4.2 Resonant mode and impurity effects in YbB12 It is interesting to analyze the influence of impurities on this transformation of the spectrum, as well as on the low-temperature regime representing the cooperative f-electron state. Three types of impurities were used in our study: iso-electronic Lu3+ (nonmagnetic Kondo hole), Tm3+ (magnetic Kondo defect, CEF ground state is a triplet with the magnetic moment of about 4.7μB comparable with the full magnetic moment of Yb3+ ), and Zr ion which has another structure of the electronic atomic shells with 4d electrons and without f electrons. The influence of these impurities substituted for the Yb ions in the crystal lattice is quite specific and also different. Lu substitution has been studied in a wide range of concentrations on polycrystalline Yb1−x Lux B12 (x = 0.25, 0.75, 0.90) samples, starting from Ref. [44] and later generalized in Ref. [86]. A selection of results for two characteristic temperatures is presented in Fig. 6.17. It was shown that the single-site spectra at high temperatures are quite similar for all values of x. The spin gap at low temperature persists up to x = 0.75, and only the resonant mode is broadened with increasing x, but it is nevertheless observable in the spectra up to the highest measured Lu concentration of 75%. It should be noted that the character of the temperature crossover studied in details for x = 0.25 [44] is absolutely identical to that observed for pure YbB12 (see Fig. 6.16). This is apparently no longer the case either for Zr or Tm substitution. In the work of Nemkovski et al. [90], it has been shown that 20% of Zr

532 Neutron Spectroscopy on Rare-Earth Borides

Figure 6.17 Magnetic excitation spectra of Yb1−x Lux B12 (x = 0, 0.25, 0.75) at T = 10 and 120 K, measured by TOF neutron spectroscopy at the HET instrument (ISIS) with the incoming neutron energy E i = 80 meV. Reproduced from Ref. [86].

substitution results in the essential suppression of the resonant mode at the lowest temperature of 5 K and its substitution by the quasielastic signal (qe /2 ≈ 3 meV). The temperature evolution of the magnetic excitation spectra is also changed and becomes more consistent with a thermodynamically driven transition. The transformation is developing quite gradually (see Fig. 6.16) in the wide temperature range, with some features of the low-temperature spectrum persisting even up to 250 K [86]. The influence of magnetic impurities, in some analogy with (Sm,Gd)B6 (see Section 6.2.3.5), has been studied in Ref. [43] on polycrystal samples with 8% and 15% substitution of Tm for Yb. The

Magnetic Excitations in Hexa- and Dodecaborides 533

(a)

(b)

(c)

(d)

Figure 6.18 Magnetic scattering spectra for YbB12 at (a) T = 2 K and (b) T = 80 K, and for Yb1−x Tmx B12 with (c) x = 0.08 and (d) x = 0.15 at T = 2 K after subtraction of the direct Tm contribution. The dashdotted line in panels (b) and (d) corresponds to a fit of the spinfluctuation contribution by a quasielastic Lorentzian line shape (half-width qe ≈ 9 meV), taking into account the detailed-balance temperature factor. The dashed lines represent the decomposition of the inelastic magnetic signal into partial contributions according to our interpretation of the YbB12 magnetic response in Refs. [79, 80]: one narrow peak at 15 meV and two broad peaks at 20 and 40 meV at low temperature, and one single broad peak at about 25 meV at high temperature. Reproduced from Ref. [43].

most dramatic changes develop in the spin-gap energy range at low temperatures. The resonant mode was totally suppressed even for 8% Tm, a further increase of Tm concentration (to 15%) results in the appearance of quasielastic scattering intensity in the spin-gap energy range. A comparison of Yb1−x Tmx B12 and YbB12 spectra is presented in Fig. 6.18. The suppression temperature of the low-temperature regime decreases with increasing Tm concentration and approaches 30 K for the 15% Tm sample. In this sense, the effect of Tm is opposite

534 Neutron Spectroscopy on Rare-Earth Borides

to that of Zr ions. Addressing to the previous section (Fig. 6.13 in Section 6.2.3), where we discussed the relaxation of the resonant mode in SmB6 , it is natural to relate the effect of the exciton suppression with an increase in the concentration of the magnetic impurity. Due to the relationship 0 ∝ (Jη)2 Ω, the increase in the DOS (related to η) which is roughly proportional to the increase in the concentration x, may result in the quadratic increase of the intrinsic relaxation width 0 . This means that a few percent of the magnetic impurities with respect to the residual level (estimated roughly as 0.1%) may give rise to a 103 –104 times higher relaxation width with respect to the nominally pure compound. This corresponds to a change of  from 0.1 meV to more than 10 meV, sufficient to wipe off the excitation in the neutron spectra. The influence of Yb ions on the magnetic properties of Tm has also been observed in the same work [43]. We did not study the possible screening of the Tm moment but found a transformation of the CEF effects which can also be considered as evidence of some modification of the effective magnetic moment. There was a small but clear reduction of the CEF splitting which could be related to the modification of the hybridization effects in the electron subsystem of dodecaborides. The latter provides an essential contribution to the CEF potential as was suggested in Ref. [44]. It is natural to expect a difference in the influence of the substitution of different character on the static magnetic susceptibility as a function of temperature, χ(T ), related by the integration equation to the dynamical susceptibility χ  (E , T , Q) measured in neutron spectroscopy. The effects of Yb substitution were studied experimentally and analyzed from the point of view of correspondence with the magnetic excitation spectral features [91]. The essential modification of χ(T ) in the full temperature range from 5 to 300 K was observed for Zr substitution. It is in correspondence with the strong influence of this ion on the temperature evolution of the magnetic spectral function. The formation of the maximum in the χ(T ) of YbB12 is in agreement with the spin-gap opening at T < 80 K. A sharp decrease in the static susceptibility below 50 K is a consequence of the development of the cooperative Q-dependent effects like the resonant mode, first of all, as well as the spin gap in

Lattice Dynamics in RB6 and RB12

the excitation spectrum. It is necessary to mention that the energy of spin fluctuations (i.e., the quasielastic peak width or the “Kondo temperature”) is almost unchanged in YbB12 for all the studied types of impurities. A quite prominent effect observed while studying the influence of the substitution on the f-electron excitation spectra in YbB12 is the strong difference between the effect of nonmagnetic Kondo hole (Lu) and the magnetic (Tm) impurities as clearly illustrated by the comparison of Figs. 6.17 and 6.18. It is necessary to mention that the introduction of Lu has almost no effect on the spin-gap spectrum up to very high concentrations (above x ≈ 0.5) in spite of the clear cooperative origin of the low-temperature spectral function. The magnetic Tm ions destroy the low-temperature regime in the concentrations even below 10%. This may be the evidence of the specific mechanism for the formation of the cooperative low-temperature (T ∗ < 80 K) Kondo insulator state. It implies the importance of the equivalence among all existing Kondo scattering centers in the Kondo lattice, as the appearance of another type of magnetic scattering centers is much more destructive for the cooperative Kondo state than the introduction of nonmagnetic defects, which was termed Kondo undercompensation effect [43].

6.3 Lattice Dynamics in RB6 and RB12 The lattices of RB6 hexaborides and RB12 dodecaborides can be characterized as structures based on the framework of boron clusters: B6 octahedra and B12 cuboctahedra, respectively. RB6 can be related to the cubic system with the CsCl structure type, and RB12 corresponds to an fcc system with the NaCl-type structure. Hexaborides MB6 exist for M = Y, La–Dy, and Yb. The “small” metal ions, M = Sc, Ho, Er, Tm, and Lu, do not form stable hexaborides in bulk form5 [95], whereas dodecaborides exist in the whole range of the rare-earth series. 5 Exceptions

in the form of ErB6 nanowires [92] and nonepitaxial TmB6 thin films [93] have been reported. According to Mar [94], ErB6 can also be stabilized at elevated temperatures due to thermal expansion effects, whereas below ∼1975 ◦ C it decomposes into ErB4 and ErB12 .

535

536 Neutron Spectroscopy on Rare-Earth Borides

The lattice dynamics of these compounds strongly reflects the hierarchy of the interactions between atoms and clusters. The discovery of a small electronic gap forming on decreasing the temperature in SmB6 first [47], and then in YbB12 [48], which serves as a manifestation of the metal-semiconductor transition (Kondo insulator phenomenon, see Section 6.2.2) in connection with the f-electron instability and/or hybridization, incited the increasing interest in the physics of these systems and, in particular, in the effects of electron–phonon interactions. Early studies of the lattice dynamics started in the 1970s. Several Raman-active modes have been observed initially in the range of 150 to 1400 cm−1 (approximately 20–180 meV) [96, 97]. Systematic analysis of the Raman data shows [98] that only one optical mode (T1u ) at the lowest measured phonon frequency has an opposite frequency dependence on the lattice spacing in the series of RE hexaborides with respect to the other three modes (T2g , Eg , and A1g ) observed in the spectral range 650–1300 cm−1 . The latter ones expectedly increase their frequencies with decreasing lattice spacing. Moreover, the particular softening of the T1u mode for SmB6 with respect to the interpolation line from other RB6 compounds was interpreted as the first evidence of the interplay of phonons with “valence fluctuations.” The first neutron spectroscopy measurements of the phonon density of states for rich borides were performed on La11 B6 [99] and compared to model calculations [99, 100]. All three data sets did not come to a reasonable mutual agreement. The development of the TOF technique based on the pulsed neutron sources allowed extending the accessible energy range far above 100 meV and essentially improve the energy resolution. Therefore, the study of the PhDOS of RE borides6 became accessible (of course, with the use of the 11 B isotope) from the end of the 1980s.

6 It

is necessary to mention that model calculations of Schell et al. [99] appear to be in quite better agreement with our measurements [101] than with the authors’ own data, which reflects the progress in the measurement technique.

Lattice Dynamics in RB6 and RB12

6.3.1 General Characterization of the Atomic Vibrational Spectra of RB6 and RB12 Below we are briefly discussing the main and specific features of the atomic oscillation spectra of hexa- and dodecaborides studied by INS. The details of this method and of the results are presented in the review article [101] and in Section 6.1. As it was mentioned, the main feature of the compounds under consideration is the boron-cluster-based structure. It reflects the specific interatomic interaction for these materials. The typical spectra of LaB6 and LuB12 measured with TOF spectroscopy on polycrystalline 11 B-enriched samples are shown in Figs. 6.19 and 6.20. The high incident energy and the relatively good energy resolution provide the general view on the neutron-weighted generalized PhDOS G(Q, E ), which is directly related to the scattering function S(Q, E ) in accordance to the Eq. (6.4). The phonon spectra of S(Q, E ) for hexa- and dodecaborides measured by the TOF method [63, 79] (see Fig. 6.19 for SmB6 and Fig. 6.20 for LuB12 ) show that the upper boundary of the phonon-

Figure 6.19 S(Q, E ) spectra for LaB6 measured at T = 250 K with the incident neutron energy E i = 200 meV on the TOF instrument HET (ISIS) at low (10◦ ) and high (120◦ ) scattering angles 2θ with the resolution (FWHM) of 4 and 6 meV at zero energy transfer, respectively [63].

537

538 Neutron Spectroscopy on Rare-Earth Borides

20

20

RE vibraons

Q,E) (a..u.) S(Q

15

YbB12 LuB12

10

S(Q,E)) (a.u.)

E0=200 200 meV V

Ei=80 meV

15

YbB YbB12 12 LuB12 LuB 12

10 5 0 0

10

20

30

40

50

Energy transfer (meV)

60

70

b boron vibraons ib 

5 0 0

50

100

Energy transfer (meV)

150

Figure 6.20 S(Q, E ) spectra for LuB12 (circles) and YbB12 (dots), measured at HET (ISIS) with E i = 200 meV at T = 15 K and high scattering angle (2θ ≈ 133◦ ), providing the conditions for purely phonon contribution to the scattering function. Inset: spectra measured with E i = 80 meV (twice better resolution with respect to E i = 200 meV) and at T = 160 K. Reproduced from Ref. [79].

mode energies lies in a range of 130–160 meV (higher values correspond to RB6 ), i.e., at relatively higher energies in comparison to most other inorganic compounds. Measurements with different RE elements in the RBn series allow the realization of the isotopic contrast method (see Section 6.1) due to the considerable (more than threefold) difference in the neutron-scattering cross-sections between, for example, Yb and Lu, while these nearby elements in the 4f series possess only a small mass difference of the order of 1%. From this type of experiments, partial gRE (ω) and gB (ω) [see Eqs. (6.4), (6.5)] have been determined [102]. It appears that the thermal oscillations of heavy RE atoms are mainly concentrated in the range of energies below 20 meV (acoustic branch), actually giving rise to the lowest-in-energy peak of the PhDOS. The vibrations at higher frequencies mainly originate from the rigidly bound boron

Lattice Dynamics in RB6 and RB12

network, including local cluster vibrations that correspond to their deformations of different symmetry types (“breathing,” “torsion,” etc.). A significant boron isotope effect is expected to be observable due to the 10% mass difference of 11 B and 10 B isotopes, which occur in the natural boron with the 80% and 20% abundance, respectively. However, the very high neutron absorption crosssection of the 10 B isotope (∼3800 barns for thermal neutrons) precludes neutron experiments with 10 B-based samples. In contrast, Raman spectroscopy allows studying borides based on both isotopes of boron. The corresponding results [45] clearly show energy shifts (softening) of the Raman-active modes at the zone center (Q = 0) for LuB12 in the energy range 600–1100 cm−1 (75– 140 meV). This range corresponds to the boron-specific vibrations in the neutron spectra, as seen in Fig. 6.20. Another feature in the Raman spectra is a broad peak at about 22 meV with a remarkable temperature dependence, showing a sharp increase in intensity when the temperature is lowered below 80 K. It exhibits no isotope effect and cannot be directly related to any particular feature in the PhDOS of LuB12 in Fig. 6.20. The first neutron experiments performed with single crystals of RB6 (LaB6 [103] and SmB6 [55–57]) show a quite unusual (but expected from the PhDOS spectra) picture of phonon dispersions in the range of relatively low phonon energies (below 15 meV). The dispersion curves of acoustic phonons demonstrate quite extended (from |q| ≈ 0.2qmax to the zone boundary) flat parts in all highsymmetry directions, as seen in Fig. 6.21. The optical phonon band is typically separated from the acoustic branches by the energy gap of about 5 meV. The flat part of the acoustic dispersion curve can be qualitatively explained by the presence of a hierarchy of interion interactions, such that the RE atoms are loosely coupled to one another and to the boron clusters. Nevertheless, the essential difference in energies for these phonons between LaB6 and SmB6 cannot result from the simple mass renormalization and is discussed below in Section 6.3.2. It is necessary to note that in some cases the energy band of flat acoustic phonon curves coincides with the energy range of magnetic excitations within the f-electron

539

540 Neutron Spectroscopy on Rare-Earth Borides

Figure 6.21 Phonon dispersion for SmB6 at room temperature, measured on the doubly isotope-enriched single crystal at ATOS (IR-8, KIAE) and IN8 (ILL) [55]. Open circles and squares correspond to transverse acoustic (TA) and optical (TO) modes with polarization in (1) and perpendicular to (2) the (100)–(011) plane. Dots correspond to longitudinal (LA, LO) phonon modes. Triangles denote a dispersionless unpolarized mode in the gap between the optical and acoustic bands (see Section 6.3.2). Dashed lines and star symbols show the data for LaB6 single crystal from Refs. [55, 103]. The reduced wave vector is given in reciprocal lattice units for each direction.

Lattice Dynamics in RB6 and RB12

spectrum. This may have interesting physical consequences due to the formation of hybrid magnetoelastic excitations. We should mention that the extra vibrational mode at ∼4.5 THz within the gap between the optical and acoustic phonon branches, as well as kinks in the longitudinal acoustic modes, are specific only to SmB6 and are not observed in LaB6 [56] and other hexaborides [104]. It has been attributed to the intermediate-valence state of SmB6 . The results of a 149 Sm nuclear resonant inelastic ¨ scattering experiment (that is, in fact, Mossbauer spectroscopy) with SmB6 at the synchrotron Spring-8 [105] clearly demonstrate that this extra mode belongs to the spectrum of atomic vibrations of Sm. Similar behavior of acoustic branches with a large region of nearly flat dispersion at |q|  0.2 has been observed in studies of LuB12 and YbB12 [106–108]. It should be noted that the volume of the “void” in the center of the B24 cuboctahedron is estimated to be approximately twice larger than the atomic volume of the RE ion, providing some free space for the “rattling” motion of the RE atom. Phenomenological calculations based on the force-constant model (UNISOFT software package [109], with 9 coordination spheres taken into account [106, 107]) provided quite a good description (see Fig. 6.22) of all the experimentally observed dispersion curves in the energy range7 up to 50 meV [108]. The strong hierarchy of interactions results from the above modeling: RE–RE  RE–B  B–B, with a difference on every stage of about an order of magnitude, similarly to the SmB6 treatment [55]. On the basis of this model and the corresponding phenomenological parameters, the generalized density of states was calculated and compared with the one extracted from experimental data. Quite a good description of the PhDOS for acoustic and lower optical branches (up to 40 meV) has been obtained, but the pure boron vibrations with higher energies are not described even qualitatively [102], and the only success of the mentioned calculation in this energy range is the good correspondence between the calculated and experimentally determined boundary energies of the phonon spectrum (see Fig. 6.23). 7 This

value is limited by the typical energy range of operation for TAS at the thermal neutron reactors.

541

Energy (meV)

542 Neutron Spectroscopy on Rare-Earth Borides

65 Γ 60 Δ 2 55 50 Δ 2 45 40 35 Δ 1 30 25 20 15 10 5 0 0.0

[001]

X Δ1

Δ2

Σ1

Σ2 Δ5

Δ5

K/U [110]

Δ1 Δ5

Σ3 Σ1 Σ4

0.5

Σ4

Σ1

q (r.l.u.)

Λ3

Λ1

Σ4

0.5

Λ3 Λ3

Σ1,Σ3

1.0

Λ2

Λ1

Σ1

(a) (b)

L Λ1

Λ3

Δ1 Δ5

Λ2

Σ4 Σ1

Σ4

Δ1

[111]

Γ

Λ1 Λ3

0.0

0.25

(c)

(d)

0.50 0.00 0.01 0.02 PhDOS (meV-1)

Figure 6.22 (a–c) Phonon dispersion for YbB12 (open symbols) and LuB12 (closed symbols) as a function of the reduced wave vector q (in reciprocal lattice units). Longitudinal and transverse branches are marked with circles and diamonds, respectively. Model calculations are shown with solid lines [107]. Symmetries of phonon modes are labeled according to the Bouckaert-Smoluchowski-Wigner notation. (d) Calculated PhDOS for YbB12 . Boron and ytterbium contributions to the partial PhDOS are hatched and cross-hatched, respectively. Reproduced from Ref. [108].

The phonon spectra for MB12 (M = Lu, Yb [102], and Zr [110]) have been used for the analysis of results from ab initio calculations (DFT-LDA). It appears that in a region of boron frequencies (Fig. 6.23) this type of calculations gives quantitative agreement with the experiment, contrary to the phenomenological model [102, 108, 111]. However, for phonon energies below 30 meV, the problems are aggravate. In particular, the calculation correctly predicts the position of the first peak in the DOS but practically does not describe a peak at 25 meV. These features can be captured in more details in the analysis of single-crystal data in the framework of a very simple physical model of a superatom (see below). It is necessary to mention that the phenomenological approaches based ´ an ´ approximation for the force constant, on the Born–von Karm such as the UNISOFT package, provide the possibility to elaborate a sufficiently adequate model for phonon descriptions in the range

Lattice Dynamics in RB6 and RB12

Figure 6.23 (a) Experimental (dots) and calculated (lines) PhDOS for YbB12 . The boron and ytterbium contributions to the partial PhDOS are hatched and cross-hatched, respectively. The solid line is the UNISOFT calculation (see text) of PhDOS for YbB12 convoluted with the resolution function of the TOF instrument [102]. (b) The experimental PhDOS of LuB12 in comparison with DFT-LDA ab initio calculations [110] (solid line).

of low-energy vibrations. On the other hand, the electron influence f (d) is not described accurately enough in the pseudo-potential approach of DFT-LDA. In analogy with RB6 and RB12 , the Zr ions in ZrB12 appear to be loosely bound within the rigid boron network, which results in large nearly flat regions in the dispersion of acoustic phonons, as shown in Fig. 6.24. The energy of these Zr-related modes is approximately 17.5 meV at q = qmax , which is a relatively low value with respect to the simple scaling by the ratio of atomic masses RE/Zr, since the Einstein model would predict a value of ∼21 meV. Thus an essential role of the electronic contribution to the formation of the force potential for atomic vibrations is demonstrated experimentally in rich borides. The results of both ab initio calculations and INS measurements further indicate that the Zr vibration modes have an appreciable contribution to the electron–phonon interactions and may thus take part in the formation of the superconducting state with a rather high transition temperature Tc = 6 K. Recent studies of heavy RE hexaborides such as GdB6 , DyB6 , and TbB6 , which are metals with an AFM ground state, revealed another interesting feature of these materials: the anomalous softening of phonon modes in several directions of the Brillouin zone [112–114], see Fig. 6.25. This softening manifests itself as a

543

544 Neutron Spectroscopy on Rare-Earth Borides

Figure 6.24 Low-energy phonon dispersion in ZrB12 . Symbols: TAS experiment (circles—LA, LO modes; triangles—TA, TO modes). Lines: force-constant calculation derived from the model similar to the one for LuB12 [107] by taking into account the difference in atomic masses and adjusting the Zr-Zr and Zr-B interactions. Reproduced from Rybina et al. [110].

considerable decrease in the acoustic mode energy at certain wave vectors. A similar anomaly can be observed in Heusler alloys and other compounds with a martensitic phase transformation, where a softening of the transverse acoustic TA2 phonon mode in the (110) direction is observed at the reduced wave vector q ≈ 0.33 [116]. An anomalous phonon dispersion of the longitudinal acoustic branches is observed in the (100) direction for GdB6 , TbB6 [112, 114], and for DyB6 [113] in the (100) and (110) crystallographic directions at the wave vectors q = ( 12 00) and q ≈ (0.38 0.38 0), respectively. There is a softening of the acoustic mode, not typical for the structural analogs such as LaB6 , SmB6 , and YbB6 . In the latter case, Yb2+ is in the valence state variable by external pressure [117]. Note that LaB6 and YbB6 have no uncompensated f-electrons in the electron shells. Two inflections points of the acoustic curve measured for DyB6 are observed in the (110) direction, see Fig. 6.26. Considering the boron clusters as rigid objects enables the socalled superatom model [115,118] for the analysis of the low-energy

Lattice Dynamics in RB6 and RB12

14

12

10

8

6 12 10

4

8 6 4

2

2 0 0.0

0 0.0

0.1

0.2

0.3

0.2

0.4

0.4

0.5

Figure 6.25 Longitudinal acoustic phonon dispersion measured at room temperature in various hexaborides in the (100) crystallographic direction. The symbols denote the experimental data [55,101,103,112–114]; the lines are spline fits to the experimental data taken from [115]. The inset shows the temperature effect for the intermediate-valence system SmB6 .

( ωph ) can be presented as extreme variants of the existence of an excitonpolaron in the crystal lattice (exciton is producing a polarization of the crystal lattice in its vicinity), which results in the deformation of the potential relief for atomic vibration of the “cluster” formed by atoms located close to the exciton. Formation of the extra vibrational modes (“local” and “coherent”) results from the interaction of electron excitations corresponding to a deformation of the electronic density with optical and acoustic (longitudinal) modes. An exciton of intermediate radius is itself responsible for the renormalization of the “normal” adiabatic phonon spectrum [54, 55, 124–126]. The finite size of the exciton defines the shape of its form factor in momentum space. The form factor of the exciton affects the strength of the phonon renormalization. The form factor has a maximum at some intermediate value of the reduced wave vector

Lattice Dynamics in RB6 and RB12

which defines the position of the maxima of renormalization effects approximately halfway between the center and boundary of the Brillouin zone. That coincides with the location of the “kinks” in LA phonon dispersion curves observed experimentally for SmB6 . For completeness, the temperature effects observed for SmB6 should be also mentioned. They are quite strong (up to the 15% at the zone boundary) for (100) and (111) directions (see Fig. 6.25 here and Fig. 1 in Ref. [56]) and are absent in either LaB6 or PrB6 . One of the possible reasons for these effects in SmB6 could be a strong renormalization of the q-dependence, including specific anisotropy selective for the (100) and (111) directions, and the energy structure of the f-electron excitation spectra [62] at the zone boundary for these symmetry directions at temperatures in the range of 50–100 K. The strong influence of the electron subsystem on the acoustic phonons with wave vectors in the range of the second half of the Brillouin zone is directly confirmed by the recent study of the REvibration contribution to the acoustic branch of DyB6 [113] and GdB6 [112] using inelastic x-ray scattering discussed in the previous section. It shows strong renormalization of phonon energy as a result of change in the RE element (the relative change in energy exceeds the relative difference of masses between Gd and Dy fourfold) and moreover, as a result of temperature variation in the vicinity of the long-range magnetic ordering transition. In fact, the energy of the flat part of the acoustic phonon dispersion at room temperature for DyB6 is located at about 7 meV, which should be compared with 8 meV for GdB6 , 11 meV for SmB6 and PrB6 (see Fig. 6.21 and Fig. 6.25), and 13 meV for LaB6 . Of course, interpretation of these observations requires special theoretical analysis, because unfortunately the well-known theoretical approaches to the phonon description in the vicinity of a phase transition [127] seem to be not directly suitable in the present case. In summary, we can conclude that the electron–phonon interaction in the range of acoustic phonon energies is quite strong and diverse in the manifestations for this type of compounds.

555

556 Neutron Spectroscopy on Rare-Earth Borides

6.3.2.2 The magnetovibration interaction in YbB12 YbB12 is a so-called Kondo semiconductor [128]—a system which demonstrates the transition between Kondo-metal behavior at relatively high temperatures and a low-temperature nonmagnetic state with a narrow gap in the electronic DOS at Fermi energy [129– 131]. This transition is accompanied by a dramatic transformation of the magnetic excitation spectrum. In particular, a spin gap of the order of 200 K develops during this transition in the narrow temperature range above 50 K [44, 78, 79] for YbB12 . On the other hand, almost no difference was observed in a wide temperature range between the phonon energies of YbB12 and it nearest nonmagnetic neighbor compound LuB12 . From the point of view of electron–phonon interaction in strongly correlated systems, it is interesting to analyze a possible connection of the joint metal-insulator and magnetic-nonmagnetic transitions with features in the lattice dynamics of YbB12 . In this system, as already noted [78, 82], there is an overlap of magnetic excitations with some phonon modes in the energy range 15– 20 meV (see PhDOS in Fig. 6.22) near the gap edge in the magnetic excitation spectra (Fig. 6.14). This fact gives rise to the assumption of a possible interaction between them and even to ideas about the certain role of lattice dynamics in the formation of conditions for a sharp transformation in the magnetic excitation spectrum from spinfluctuation to gap-like one, which looks similar to some kind of phase transition with a “hidden parameter.” The first evidence of a possible connection between phonons and the rearrangement of magnetic spectrum (see also Chapter 8) has been found by analyzing the temperature dependence of the intensity of the first peak (at about 15 meV) in the spectrum of PhDOS of YbB12 as a function of temperature [108], which is shown in Fig. 6.29. Experiments have been done by the TOF technique with incident neutron energy of 80 meV (the inset of Fig. 6.29). A noticeable increase (∼25%) in the intensity of the phonon peak [see Fig. 6.29(b)] was observed [108] in the temperature range of the gap formation in magnetic excitation spectra as well as in the charge density of states near the Fermi energy. It was observed at sufficiently high Q values (≥ 10 A˚ −1 ) when a direct magnetic

Lattice Dynamics in RB6 and RB12

Figure 6.29 Parameters of the low-energy phonon peak in YbB12 spectra as a function of temperature: (a) phonon peak width (circles) and energy (triangles), reproduced from Ref. [108]. (b) Integrated intensity of the 15 meV phonon peak in YbB12 (closed circles) and LuB12 (open circles) measured on powder at Q = 10 A˚ −1 , and in Yb0.8 Zr0.2 B12 (open squares, Ref. [132]), divided by the Bose factor and normalized to unity at T = 150 K. The inset shows experimental TOF spectra for YbB12 powder measured on the HET spectrometer (ISIS) with the incident energy E i = 80 meV at 2 = 114.9◦ . Spectra are shown for T = 15 K (closed circles) and 150 K reduced to 15 K using the Bose factor (open circles). Reproduced from Ref. [108].

contribution from the f electrons (in the form of the resonant mode) to the INS intensity did not exceed a few percents of the overall phonon signal. It is interesting to note that in the same temperature range of the electronic gap formation the variation of the phonon peak width in PhDOS was also fixed [see Fig. 6.21(a)], which looks like another manifestation of the electron–phonon interaction. Earlier such an effect was observed in neutron studies of superconductors. Further studies connected with this mysterious observation have been realized using TAS technique on a single crystal of YbB12 [132] and were based on the symmetry analysis given in Ref. [108]. Measurements of the phonon parameters were performed at two temperatures of 80 and 10 K, which are in correspondence with the two regimes of the electron subsystem: the spin-fluctuation (“high-T”) and the gap-like one (“low-T”) for the magnetic excitation spectrum. Energy and intensity of a number of phonon modes were analyzed in the energy range from 0 to 30 meV. No significant energy

557

558 Neutron Spectroscopy on Rare-Earth Borides

changes have been discovered; however, the intensities of some longitudinal and transverse phonons demonstrated substantial deviation from the temperature behavior normally determined by the Bose factor. There was a 15–30% increase in intensity for some acoustic branches [132] with the wave vector close to the Brillouin-zone boundary. The corresponding phonon energy is about 15 meV, which is consistent with observations for the polycrystalline sample PhDOS by the TOF technique [108]. The effect of intensity increase (see Fig. 6.30) was observed for those branches whose symmetry assumed the possibility of interaction with magnetic dipole excitations, and in that area of wave vectors, where dispersive low-temperature magnetic modes denoted as M2

Figure 6.30 (a) Relative temperature change between 80 and 10 K (Int10 – Int80 )/Int80 , of the integrated intensities of acoustic phonon excitation along the main symmetry directions in YbB12 : 1 (LA) and 5 (TA) along (001), 1 (LA) and 4 (TA) along (110), and 1 (LA) and 3 (TA) along (111). Closed and open symbols denote the longitudinal (LA) and transverse (TA) modes, respectively. (b) Integrated intensities of the low-energy magnetic excitations M1 (open squares) and M2 (closed square) in the Kondo insulator regime at T = 5 K as a function of the reduced q vectors along the main symmetry directions. Reproduced from Rybina et al. [108].

Lattice Dynamics in RB6 and RB12

and M1 (20 and 15 meV, respectively) are present in the magnetic spectrum at low temperature [82] (see Figs. 6.14 and 6.17). The observed features obviously indicate the manifestation of some kind of magnetovibrational interaction—quite an unusual effect is the presence of strong renormalization of intensity without any renormalization of phonon energies. The microscopic reasons for such behavior could be related to the strong hierarchy of forces of interatomic interactions obtained from the analysis of the phonon dispersion. It corresponds to the picture when the formation of the acoustic mode at large q is defined by the vibrations of loosely coupled heavy ions, and the remaining 36 optical branches are related to the hard boron framework with the RE-B and B-B interactions. Under the conditions of high crystalline symmetry and the large number of atoms in a primitive cell, the large number of the vibration modes appears to be symmetry-connected, which is exactly the case for RB12 . It is possible that their interplay with each other results in a specific manifestation of inter-subsystem (in the present case—magnetovibrational) interaction. Renormalization of polarization vectors10 is observed instead of the renormalization of phonon energies. It should be noted that such effects have also been observed in the phonon study of high-Tc superconductors [133,134] in a range of temperatures corresponding to the superconducting transition with the electron energy gap formation. In the discussion it was connected by the authors to the specifics of the electron– phonon interaction under special condition just applied for the case of YbB12 (high enough crystal symmetry and the large number of atoms). It is possible that the observed interaction evidences a certain role of the lattice subsystem in the sharp change of the regime of the magnetic subsystem (from gapless spin fluctuations to a gapped spectrum) in the narrow temperature range around 50 K. Additional evidence for this scenario are provided by the results of a recent INS study of the excitation spectra of Yb0.8 Zr0.2 B12 [86, 90]. It was established that a strong damping of the sharp transformation for 10 Its modulus just defines the amplitude of atomic displacement in the given phonon

mode, which is connected to the signal intensity at particular energy in the neutron scattering spectrum, see Eq. (6.3).

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560 Neutron Spectroscopy on Rare-Earth Borides

the Yb-ion magnetic spectrum and the suppression of anomalous temperature dependence of the intensity for the 15 meV peak in the phonon spectrum are simultaneously observed upon partial substitution of Yb with Zr. The Zr ion is different from the Yb one not only by the atomic mass but also by the number of valence electrons and the type of d band (4d instead of 5d) near the Fermi energy. Both factors are foremost important for the atomic dynamics, and, consequently, it is possible to suppose that the lattice dynamics plays a substantial role (some kind of a “hidden parameter”) in the formation of the Kondo-insulating ground state in YbB12 .

6.4 Conclusions In spite of the specifics of boron from the point of view of neutron-scattering studies, a large number of unique and physically important results in the field of atomic and magnetic dynamics have been obtained by neutron spectroscopy on both RB6 and RB12 systems, in particular (1) Clear specific features in the f-electron spectra of the intermediate-valence SmB6 and the Kondo insulator YbB12 , which provide the starting point for the search and analysis of any realistic model for these strongly correlated states. The INS data can serve as a backdrop for the possible generalization of the microscopic models for Sm- and Eu-based systems with intermediate-valence phenomena, one of the recent attempts is presented in the work of Savchenkov et al. [135]. (2) The strong hierarchy of interatomic interactions, which results in the specific cage-cluster structure of the boron network with RE as donors of electrons located in the high-symmetry and weakly bonding positions. (3) The peculiar manifestations in the electron–phonon interaction are observed, related to the resonant violation of the adiabatic approximation realized for SmB6 . It originates from the existence of quite “soft” electron excitations related with a charge transfer excitations specific for the intermediate-valence ground state of Sm.

References 561

(4) Some kind of “magnetovibrational” effects which need to be further analyzed from the point of view of interconnection of phonon and local magnetic-moment subsystems for the case of strong electron hybridization realized in the Kondo-insulating regime of YbB12 . In conclusion, it is necessary to mention the essential importance of interplay between lattice and electronic excitation spectra in strongly correlated electron systems like the hexa- and dodecaborides. The combination of neutron spectroscopy with spectroscopic optical and synchrotron-radiation methods provides a unique possibility to implement a comprehensive study from both sides simultaneously. The particular knowledge of these properties may be of practical importance for understanding the thermal, thermodynamic, electrical, and magnetic properties and for the developments in the field of thermoelectricity, controlled thermal expansion materials, etc.

Acknowledgments We express gratitude to the colleagues without whom this work would be impossible: N. Yu. Shitsevalova, V. B. Filipov, J.-M. Mignot, K. S. Nemkovski, A. S. Ivanov, P. P. Parshin, R. I. Bewley, A. V. Kuznetsov, P. S. Savchenkov. We acknowledge fruitful discussions with A. Yu. Rumiantsev, A. F. Barabanov, A. S. Mishchenko, A. P. Menushenkov and feel indebted for the memorable brainstorming discussions with Yu. B. Paderno, K. A. Kikoin, L. A. Maksimov, and Yu. M. Kagan that served as a source of inspiration for the future.

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Chapter 7

Competing Order Parameters in Rare-Earth Hexa- and Tetraborides Takeshi Matsumura Department of Quantum Matter, Graduate School of Advanced Sciences of Matter (ADSM), Hiroshima University, Higashi-Hiroshima 739-8530, Japan [email protected]

In rare-earth borides we encounter many cases where not only the magnetic dipole but also the electric quadrupole and magnetic octupole are involved in the ordering phenomena, as summarized in several recent review articles [1–4]. A typical case is the rich magnetic phase diagram of cubic Cex La1−x B6 with the 8 quartet crystal-field ground state. Among many experimental and theoretical efforts dedicated to observe and establish the quadrupole and octupole ordered states, in this chapter we focus on the role of resonant x-ray diffraction. Another topic in this chapter is a competition of order parameters and ordered structures in tetragonal rare-earth tetraborides, RB4 . With the so-called Shastry– Sutherland lattice, a network composed of orthogonal dimer bonds, the magnetic structures of RB4 result in a macroscopic degeneracy due to geometrically frustrated exchange interaction. In this series of compounds, interesting successive ordering phenomena are observed, which is possibly associated with a competition of Rare-Earth Borides Edited by Dmytro S. Inosov c 2022 Jenny Stanford Publishing Pte. Ltd. Copyright  ISBN 978-981-4877-56-5 (Hardcover), 978-1-003-14648-3 (eBook) www.jennystanford.com

576 Competing Order Parameters in Rare-Earth Hexa- and Tetraborides

magnetic and quadrupolar order parameters. Another competition among the geometrically frustrated short-ranged interactions, longranged RKKY interaction, and the Zeeman energy in magnetic fields, leads to an emergence of interesting and unpredictable arrangements of magnetic moments, manifested in the appearance of fractional magnetization plateaus.

7.1 Introduction Antiferroquadrupolar order The discovery of antiferroquadrupolar (AFQ) ordering in CeB6 at TQ = 3.3 K dates back to 1970s [5]. However, the anomaly found in specific heat was initially regarded as extrinsic because it was much smaller than that of the antiferromagnetic (AFM) transition at TN = 2.3 K. It was not until 1980, when the tiny anomaly at 3.3 K was found to sharpen in applied magnetic fields, that it was recognized as indicating an intrinsic phase transition [6]. The true nature of the phase, however, remained a mystery. The paramagnetic phase above 3.3 K, the intermediate phase, and the AFM phase below 2.3 K were named phase I, II, and III, respectively. The nature of the anomalous magnetic phase diagram was studied extensively by bulk property measurements, NMR, and analysis of the magnetic structure by neutron diffraction [7–14]. It was around 1983 that the idea of orbital ordering, or quadrupolar ordering, was proposed to explain the nature of phase II [15–17]. However, since the crystal-electricfield (CEF) ground state had initially been considered to be the 7 doublet, it still took a while until it was finally recognized that the AFQ order takes place in the 8 quartet ground state [18–22]. A strong piece of evidence for the AFQ order was obtained by the observation of field-induced antiferromagnetism by neutron diffraction [23–25]. It was shown that AFM order with the propagation vector q = ( 21 21 21 ) and moments pointing along the [001] direction is induced when a magnetic field is applied along the [110] axis. No magnetic order was detected in zero field. This result can be understood by considering some sort of primary orbital ordering that confines the direction of the field-induced magnetic moments. The double-q-q magnetic structure in phase III, which is described

Introduction 577

by the q1 = ( 14 41 21 ), q2 = ( 14 14 21 ), q1 = ( 14 41 0), and q2 = ( 41 14 0) ordering vectors as depicted in Ref. [23], would be consistent with the assumption of an underlying Ox y -AFQ order, which confines the direction of the magnetic moment along the [110] direction for the

Ox y  = +Q site and [110] for the Ox y  = −Q site. The induction of the AFM moment along [001] for B [110] was rationalized by considering an AFQ order of Oyz  or Ozx  [24]. It is noted that the double-q-q AFM structure proposed in Ref. [23] for CeB6 , which seems to explain the observations successfully, however, is still not conclusive [26–28].

Magnetic octupoles While detailed studies on the magnetic phase diagram proceeded based on the concept of AFQ order [29–32], a serious problem remained in CeB6 : The hyperfine field at the boron site observed by NMR in phase II could not be explained in terms of the field-induced AFM order with q = ( 12 21 21 ) determined by neutron diffraction [33]. A breakthrough on this problem was achieved with the proposal of field-induced magnetic octupoles [34– 36]. This is a reformulation of the theory of Ref. [21] by Ohkawa into a rigorous form using multipolar moments, while taking into account the higher-order interaction originating from spin and orbital degeneracies. Because of the underlying AFQ order with anisotropic charge distributions, the hyperfine fields at the boron sites are also affected. This effect can be understood by taking into account the antiferromagnetically arranged octupole (AFO) moments of Tx yz type induced on the Ce 4f orbitals. It was also shown that the transition temperature TQ increases with increasing magnetic field because of the inter-site interaction among the fieldinduced Tx yz octupoles [36]. By introducing the Ox y -AFQ interaction and the Tx yz -AFO interaction, many of the characteristics of the AFQ and AFM ordered phases can be explained with a mean-field calculation [37]. The role of octupolar moments in the AFM ordered structure was also discussed [38]. It is noted that the suppression of quadrupolar fluctuations by applying magnetic fields should also play an important role in the increase of TQ [39, 40]. Surprisingly, TQ = 3.3 K at zero field increases up to 10 K at 35 T before the AFQ phase starts to get suppressed by the applied field [41].

578 Competing Order Parameters in Rare-Earth Hexa- and Tetraborides

Figure 7.1 Magnetic phase diagrams of CeB6 and Ce0.7 La0.3 B6 for B [001] and the calculated electric and magnetic charge distributions of the antiferro-type ordered states. Redrawn from Refs. [32, 41–44]. The figure of phase II shows the Ox y (5g )-AFQ order with a magnetic field applied along [001], which results in the appearance of the field-induced magnetic dipole J z (4u ) and the Tx yz (2u ) octupole. The figure of phase III shows the AFM state within the Ox y -AFQ phase, with the ordered dipole moment along [110] on one site and along [110] on the other site. The octupole moments of T α (4u ) and T β (5u ) are simultaneously induced. The figure of phase IV shows the T β (5u )-AFO order, where the Oyz + Ozx + Ox y ferroquadrupolar order arises simultaneously. These figures were drawn by the method described in Ref. [45].

The mystery of phase IV In the middle of 1990s, a new thermodynamic phase was found in Cex La1−x B6 solid solutions. When Ce is partially replaced by nonmagnetic La, TQ decreases more rapidly than TN , and the phase lines cross at around x = 0.8. Below this concentration, an enigmatic new state labeled “phase IV” evolves [42]. In spite of many experimental studies by bulk property measurements and by μSR, the nature of phase IV was not resolved,

Introduction 579

and so it remained an unidentified nonmagnetic hidden-order phase [46–52]. An important aspect of phase IV was clarified by the discovery of trigonal lattice distortion [43]. A theory was presented, showing that the physical properties in phase IV can be explained by the T β -AFO order [53, 54]. The response of uniform magnetization to a uniaxial stress is also consistent with this theory [55]. The magnetic phase diagrams of Cex La1−x B6 for x = 1 and 0.7 and the schematics of the ordered wave functions are illustrated in Fig. 7.1. The research history of Cex La1−x B6 is reviewed in more detail in Ref. [4]. Neutron diffraction To probe the ordered structures of multipolar moments as those illustrated in Fig. 7.1, microscopic methods based on x-ray and neutron diffraction are quite effective. Firstly, neutron diffraction is a very powerful method to study crystal and magnetic structures, from which we can infer the arrangement of ordered moments. The nuclear and magnetic scattering cross section of neutrons from a periodic structure of a single crystal is written as  2  dσ  = N b j ei Q·r j ≡ N 2 |F N |2 , (7.1) d Nuc j g 2  dσ  j ˜ i Q·r j = r02 N F j (Q){Q˜ × ( J j  × Q)}e d Mag 2 j  dσ 

≡ N 2r02 |F M⊥ |2 ,   ∗ · P, = r0 N 2 F N∗ FM⊥ + F N FM⊥

(7.2)

(7.3) d NM where b j , g j , F j , J j  represent the nuclear coherent scattering length, magnetic g-factor, magnetic form factor, and magnetic moment vector of the j th atom located at r j in a unit cell [56]. r02 = 0.289 × 10−24 cm2 is a well known constant for the magnetic scattering cross section. N is the number of unit cells, and Q = k −k is the scattering vector. When we use a polarized neutron beam with the polarization vector P, a nuclear-magnetic interference term arises. Eq. (7.1) shows that we can investigate the crystal structure in terms of an arrangement of nuclei. The magnetic structure, i.e., the arrangement of magnetic dipole moments J, can be studied using Eq. (7.2). In some cases, more details of the magnetic structure may be investigated using the interference term. However, we see that

580 Competing Order Parameters in Rare-Earth Hexa- and Tetraborides

it may be very difficult to observe ordered quadrupolar moments, because they represent a nonmagnetic degree of freedom that does not break time-reversal symmetry. Therefore, there is no direct coupling with the neutron scattering cross section. Octupolar moments, on the other hand, are magnetic. However, the only route to distinguish their contribution from conventional dipolar scattering may be through the magnetic form factor F (Q), which reflects the spatial distribution of anisotropic magnetic moment density [57, 58]. To determine F (Q), one needs to measure many intensities of magnetic Bragg peaks [59]. However, since the Q-dependence is mostly determined by the isotropic lowQ contribution reflecting the magnetic dipole moment, precise quantification of the directional dependence of F (Q) needs to be performed in the high-Q region, where the magnetic signal becomes weak and the energy resolution broadens (making it impossible to distinguish Bragg scattering from low-energy inelastic contributions), which makes accurate experiments challenging. Even if one succeeds in detecting the Q-dependence originating from magnetic octupoles under a specific condition, it would be difficult to measure the temperature and magnetic field dependences of the multipolar order parameter in one experimental session. For this purpose, resonant x-ray diffraction (RXD) may often be more suitable.

7.2 Resonant and Nonresonant X-Ray Diffraction 7.2.1 Nonresonant X-Ray Diffraction In this section, we briefly review the formalism of resonant and nonresonant x-ray diffraction to highlight similarities with and differences from neutron diffraction [60–62]. The intensity of x-ray diffraction is proportional to the square of the total structure factor F , which consists of nonresonant and resonant scattering terms. The energy (ω) and momentum (Q) dependent structure factor F (ω, Q) that gives rise to a Bragg diffraction is expressed as   fc, j (Q) + fm, j (Q) + f j (ω) + i f j (ω) ei Q·r j , (7.4) F (ω, Q) = j

Resonant and Nonresonant X-Ray Diffraction 581

where fc, j and fm, j represent the nonresonant Thomson and magnetic scattering factors of the j th atom, respectively. They are expressed as  ei Q·rn , (7.5) fc, j = ( ∗ · ) n

fm, j

  i ω  i Q·rn = −i 2 e (pn × Q) · Am + sn · Bm , mc n Q2

(7.6)

where pn and sn represent the momentum and spin, respectively, of the nth electron belonging to the j th atom. The vectors Am and Bm are Am = −4 sin2 θ( ∗ × ),  Bm = ( ∗ × ) − ( · kˆ  )( ∗ × kˆ  )

 ˆ ˆ − (kˆ  ×  ∗ ) × (kˆ × ) . + ( ∗ · k)( × k)

(7.7)

(7.8)

ˆ The polarization vectors (,   ) and the unit wave vectors (k, kˆ  ) of the incident and scattered x-rays are defined as shown in Fig. 7.2. It is noted that all electrons contribute to the Thomson scattering, leading to a strong scattering amplitude. By contrast, the magnetic scattering arises only from a small number of electrons that are responsible for the magnetic moment. This makes it difficult to detect nonresonant magnetic scattering in the background of Thomson scattering and other types of incoherent scattering such as x-ray fluorescence.

7.2.2 Resonant X-Ray Diffraction The energy-dependent anomalous scattering factor f j (ω) + i f j (ω) is expressed as  ba a|  · J∗ (k )|b b| · J(k)|a , (7.9) f j (ω) + i f j (ω) = ω ω − ba + i /2 b where the transition from the initial state |a to the intermediate state |b is expressed by the momentum density operator J(k) = ei k·r (p − i k × s). It represents the electric and spin polarization induced by the electromagnetic field of the x-ray. Since the magnetic transition amplitudes through k × s are generally much weaker, only electric transition amplitudes through p are relevant. A rough

582 Competing Order Parameters in Rare-Earth Hexa- and Tetraborides

Figure 7.2 Scattering geometry of a typical resonant x-ray diffraction experiment at a synchrotron facility. σ and π represent x-ray beams with linear polarization perpendicular and parallel, respectively, to the scattering plane spanned by k and k . The rotation angle ψ of the sample around the scattering vector is called the azimuthal angle. The polarization state of the scattered beam is analyzed by rotating the analyzer crystal by angle φA . An appropriate crystal is selected so that the scattering angle 2θA is close to 90◦ . When φA = 0◦ , only the σ -polarization beam is diffracted by the analyzer, whereas only the π -polarization beam is diffracted at φA = 90◦ . It is also possible to manipulate the incident polarization state by using phase plates, which allows full polarization analysis [63, 64].

estimate of k/ p is ∼e2 /c = 1/137, i.e., the intensities of magnetic transitions are weaker than those of the electric transitions by 10−4 –10−5 . Nevertheless, magnetic ordered states can be detected through the electric transition because the transition probability is sensitive to the spin-split intermediate state [65]. ba and  represent the energy difference and the lifetime broadening, respectively. When the transition from |a to |b takes place, the quantum number of the orbital angular momentum changes by 1 and 2 for the E 1 (dipole) and E 2 (quadrupole) processes, respectively. In rareearth elements, the E 1 and E 2 processes for an L edge correspond to the 2p ↔ 5d and 2p ↔ 4f transitions, respectively. The L-edge energies of the rare-earth elements lie in the region from 5 to 10 keV, which usually allows scattering vectors to access several Brillouin zones.

Multipolar Order in Cex La1−x B6

One drawback in using the resonant process is that it is difficult to deal with the intermediate state |b exactly. By putting the 1/(ω − ba + i /2) term outside the summation and treating the resonance as a single oscillator, a convenient formalism is obtained. However, we are sometimes confronted with a limitation caused by not treating the energy dependence properly, particularly when dealing with interference effects between different origins of multipolar scattering. In a more appropriate formalism using multipole operator equivalents, the resonant atomic scattering factors can be written as (7.10) f  (ω) + i f  (ω) = f E 1 (ω) + f E 2 (ω), where the functions 2 2ν+1   (ν) (ν) α E 1 (ω) P E 1, μ (,   ) zμ(ν) , (7.11) f E 1 (ω) = f E 2 (ω) =

ν=0

μ=1

4 

2ν+1 

ν=0

(ν)

α E 2 (ω)

P E 2, μ (,   , k, k ) zμ(ν)  (ν)

(7.12)

μ=1

describe the E 1 and E 2 resonances, respectively. Here zμ(ν)  represents the expectation value of the multipole operator equivalent zμ(ν) for rank ν and the component number μ [66, 67]. For example, (1)

(2)

(3)

z1 = J x , z3 = Oyz , and z1 = Tx yz , etc., as tabulated by Nagao and Igarashi [66]. The E 1 and E 2 resonances have sensitivities up to rank-2 and rank-4 moments, respectively. The sensitivity to even-rank orbital degrees of freedom is a distinctive characteristic (ν) of RXD, which is a great advantage over neutron diffraction. P E 1, μ (ν) and P E 2, μ are the geometrical factors for zμ(ν) for the E 1 and E 2 resonances, respectively, which are also tabulated in Ref. [66]. For (0) (1) example, P E 1, 1 = (  · ) and P E 1, μ = −i (  × )μ , etc. The functions (ν) (ν) α E 1 (ω) and α E 2 (ω) are the rank-dependent spectral functions for the E 1 and E 2 resonances, respectively.

7.3 Multipolar Order in Cex La1−x B6 7.3.1 The Parent Compound CeB6 Long after the observation of field-induced AFM order in CeB6 by neutron diffraction [23, 24], a strong piece of evidence for the

583

584 Competing Order Parameters in Rare-Earth Hexa- and Tetraborides

AFQ order at zero field was given by resonant and nonresonant xray diffraction [68, 69]. The data were compared with theoretical calculations, which supported Ox y (5g )-AFQ order [70–72]. In particular, nonresonant x-ray diffraction by Thomson scattering directly revealed the staggered ordering of the anisotropic charge distributions of Ce 4f electrons in the AFQ phase [73–75]. The Q-dependence of the intensity was consistently explained by a theoretical calculation [70, 76]. Field-induced octupoles Another important aspect of the AFQ order in CeB6 is the field-induced magnetic octupolar order in phase II, which contributes to the increase of TQ through the AFO interaction [36]. The theory predicts that Tx yz -octupoles are induced for any field directions in the AFQ phase, because the order parameter of the underlying AFQ order is expressed by

α Oyz + β Ozx + γ Ox y , where (α, β, γ ) represents the unit vector of the field direction (see also Section 8.3.3 on page 639). In the case for B [110], the magnetic dipolar, electric quadrupolar, and magnetic octupolar order parameters are J z , Oyz − Ozx , and Tx yz , respectively, each with q = ( 12 21 21 ). The experimental evidence for this scenario was given by RXD, measuring the asymmetric field dependence of the intensity caused by the interference effect [77, 78]. As shown in Fig. 7.3(a), the energy spectra for opposite field directions exhibit a significant difference, especially at the E 2 resonance energy around 5.718 keV. This effect is caused by the interference between the E 1 and E 2 resonances; to be more specific, between the rank-1 and rank-2 resonances of E 1 and the rank-3 resonance of E 2. Notably, the even-rank multipoles are invariant under field reversal, whereas the odd-rank multipole changes sign because phase II is paramagnetic. Therefore, the energy dependence of the intensity at +H and −H fields is expressed as (2) (1) (3) I (ω, ±H ) = |F E 1 (ω, H ) ± i {F E 1 (ω, H ) + F E 2 (ω, H )}|2 . (7.13) By analyzing the data using this formula, we can extract the field dependence of the three order parameters of different ranks: J z ,

Oyz − Ozx , and Tx yz , which is shown in Ref. [78] and is introduced in Chapter 8. Continuous response of the AFQ order parameter It is also shown experimentally that the AFQ order parameter in phase II in magnetic

Multipolar Order in Cex La1−x B6

Figure 7.3 (a) Energy dependence of the ( 23 23 12 ) Bragg intensity of CeB6 for the π -σ  channel in magnetic fields parallel to the [110] direction, in the AFQ phase II at 2.4 K. Filled (+H ) and open (−H ) squares show the data for opposite field directions. (b) Azimuthal angle (ψ) dependence of the E 1 resonance intensity at 1 and 4 T. The solid lines are the calculations of |α F yz + β F zx + γ F x y |2 , expected for a single-domain linear-combination AFQ order parameter. The dotted line shows |α F yz |2 +|β F zx |2 +|γ F x y |2 , expected for a selected domain state that aligns with the field direction. The dashed line shows |F yz |2 + |F zx |2 + |F x y |2 )/3, expected for a three-domain state with equal populations. Adapted from Ref. [78].

fields is expressed by the linear combination α Oyz + β Ozx + γ Ox y , which had been suggested theoretically and by the analysis of NMR line splitting [36, 79]. This means that the AFQ order parameter is not fixed but varies continuously as a function of the field direction. As shown in Fig. 7.3(b), when the sample is rotated around the scattering vector, the E 1 resonance intensity varies continuously. The data are compared with the calculated curves for three models of the AFQ structure as explained in the figure caption. F yz , F zx , and F x y are the E 1 resonance structure factors for the rank-2 quadrupoles Oyz , Ozx , and Ox y , respectively. It is evident that the data can be reproduced only by assuming the singledomain linear-combination type AFQ order parameter that varies continuously with the field direction. In such an ordered state, no magnetic anisotropy arises: the uniform magnetization α J x +β J y + γ J z , AFQ moment α Oyz + β Ozx + γ Ox y , and the AFO moment

585

586 Competing Order Parameters in Rare-Earth Hexa- and Tetraborides

Figure 7.4 Variation of the anisotropic charge distributions of the 4f electron in the two Ce sites (A, B) of CeB6 , in phase II, for the field direction between [001] and [110], calculated in the mean-field approximation.

Tx yz  are all constant, irrespective of the field direction. This is an important consequence of the AFO interaction, which allows an easy and continuous reorientation of the AFQ moment. In Fig. 7.4, the calculated charge distributions in the AFQ phase are shown for the field directions between [001] and [110]. For B [001], the to Oyz + Ozx + order√ parameter is Ox y . This changes continuously √ Ox y / 3 for B [111], and to Oyz − Ozx / 2 for B [110]. The associated changes in the magnetic-excitation spectrum in phase II under field rotation [80] are discussed in Chapters 8 and 9. There is a phase boundary at low fields of approximately 0.1 T in phase II as shown by the dotted line in Fig. 7.1 [29, 32, 44]. At zero field and up to this boundary, the ordered state is presumed to be the three-domain state of Oyz , Ozx , and Ox y . In this state, anisotropic behavior is clearly observed in the magnetic susceptibility [44]. However, so far, the three-domain state itself has not been directly confirmed by any microscopic experimental method.

7.3.2 Solid Solutions Cex La1−x B6 Antiferrooctupolar order The ground state of a Ce3+ ion with J = 5/2 in the cubic CEF is the 8 quartet, which is expressed √ √ as |α = |1/2, |β = |−1/2, |γ  = 5/6 |5/2 + 1/6 |−3/2, and √ √ |δ = 1/6 |5/2 + 5/6 |−5/2. In the Ox y -AFQ ordered state, the

Multipolar Order in Cex La1−x B6

8 quartet splits into √ are ordered alternately: √ two doublets, which one is (|β − i |γ )/ 2 and√(|δ − i |α)/ 2 with√eigenvalue +1, and the other is (|β + i |γ )/ 2 and (|δ + i |α)/ 2, with eigenvalue −1. From the four eigenfunctions of the 8 quartet, we can construct other linear combinations, which diagonalize the 5u magnetic octupole (Txβ + T yβ + Tzβ ) [53, 54]. In this AFO ordered state, the 8 quartet splits into singlet-doublet-singlet levels, where the ground state and the highest singlet have opposite eigenvalues. When these two singlets are ordered alternately, the T β -AFO order is formed. Since both states are the eigenfunctions of (Ox y + Oyz + Ozx ) with the same eigenvalue, a ferro-type quadrupole (FQ) order is simultaneously realized, as observed by a thermal expansion measurement [43]. The charge and magnetic moment density of this AFO ordered state is schematically illustrated in Fig. 7.1. The direct observation of the AFO order with q = ( 21 21 21 ) was achieved by RXD [81–83]. The energy dependence around the L2 edge and the azimuthal dependence of the E 2 resonance intensity with polarization analysis are shown in Fig. 7.5. The signal is dominated by the E 2 resonance. Notably, no E 1 resonance is observed for σ -σ  , which suggets that the order parameter is not of rank 2 but must be a higher-order multipole.

Figure 7.5 (a) Energy dependence and (b) azimuthal dependence of the E 2 resonance intensity of the ( 23 32 32 ) reflection in Ce0.7 La0.3 B6 , in phase IV. Figures redrawn from Ref. [81]. The solid and dashed lines in (b) are the calculations based on the 5u -AFO order [66, 82].

587

588 Competing Order Parameters in Rare-Earth Hexa- and Tetraborides

To analyze the azimuthal dependence, we need to consider the domain distribution. There are four domains corresponding to the octupole order parameters (Txβ + T yβ + Tzβ ), (Txβ − T yβ + Tzβ ), (Txβ + T yβ − Tzβ ), and (−Txβ − T yβ + Tzβ ). The calculated lines in Fig. 7.5(b) assume a domain ratio of 3:1:1:1 [66]. The reason for this unequal population might somehow be associated with the (111) surface of the sample used in the experiment. T β -AFO order is supported by another experiment studying both the incident and final polarization dependences [84]. The possibility of a triple-q AFO order with the (00 12 ) propagation vector should not be ignored. Since the T β -type AFO interaction at q = (00 12 ) is expected to be large, the scattering from randomly distributed Ce ions may give rise to a Bragg peak at ( 12 12 21 ) [85]. The rhombohedral lattice distortion induced by the simultaneous FQ order has been confirmed by x-ray diffraction [86]. Due to the multidomain character of this state, it was difficult to unambiguously deduce the crystal system only from dilatometric measurements [43]. The direct observation of the lattice distortion by x-ray diffraction, which is consistent with the FQ order by the T β -AFO, therefore, provides an important piece of experimental evidence. Field-induced multipoles in phase IV If a magnetic field is applied in phase IV with the T β -AFO order, various kinds of multipoles are induced: the 4u -dipole J x ± J y , 3g -quadrupoles O20  and O22 , 5g -quadrupoles Oyz +Ozx  and Ox y , or the 4u -octupole Txα ±T yα , depending on the T β -AFO domain [84]. These are the hidden order parameters existing in the 8 ground state and interacting with each other behind the T β -AFO order, which should be induced in magnetic fields. As shown in Fig. 7.6, the field dependence of the resonance peak is very unusual, especially for σ -σ  . In contrast to the AFQ phase in CeB6 , the signs of even-rank multipoles are reversed under field reversal, while those of odd-rank multipoles remain the same. A detailed analysis of this field dependence shows that this asymmetric behavior is due to the interference between the 5u (T β )-AFO and the field-induced 5g - and 3g -AFQ. Crucially, the induced amplitude of 5g -AFQ is much larger than that of 3g -AFQ. However, the prediction of mean-field calculation is contrary to this

Multipolar Order in Cex La1−x B6

σ-σ' 2

(3/2, 3/2, 1/2)

π-σ'

H

Intensity (counts/s)

Ce0.7La0.3B6 T = 1.0 K, E = 6.160 keV (E 2), H || [-1 1 0] II III

II

III

IV

III

II

IV

I T

1

0

-2 0 2 Magnetic Field (T)

-2 0 2 Magnetic Field (T)

Figure 7.6 Magnetic field dependence of the E 2 resonance intensity for σ σ  and π-σ  scattering processes. Adapted from Ref. [84]. The solid lines are the calculations assuming the interference among field-induced dipole, quadrupole, and octupole moments and the underlying T β -AFO order.

experimental result. This indicates that the mean-field model is not appropriate to estimate quantitatively the field-induced multipoles. Since all kinds of interactions of AFM, AFQ, and AFO coexist in the Cex La1−x B6 system, a rich phase diagram is realized in this 8 system [87–89]. Although a mean-field model is convenient to a rough explanation of these states [90], it is not possible to explain all the properties. A typical unsolved issue is the cusp anomaly in χ(T ) on entering the phase IV, as shown in Fig. 7.7. This anomaly can also not be explained by mean-field calculations, if the unique AFQ character of the system is to be taken into account [84, 90, 91]. In the mean-field calculation of Refs. [53] and [54], which well reproduces the cusp anomaly, the AFQ interaction is not included. If we include the 5g -AFQ interaction that must coexist with the AFO interaction, the cusp anomaly is not reproduced. Hybridization with the conduction electrons, which is treated in Chapter 8, should also play an important role in the formation of the true ground state [92–99]. Modeling the emergence of the AFQ order from first-principles calculation is a fundamental theoretical challenge [100].

589

590 Competing Order Parameters in Rare-Earth Hexa- and Tetraborides

0.15

0.15 0.10 0.4 0.2 0 0

CexLa1-xB6 H || [100]

x=0.7

IV

0.10 χ (μB/T)

x=0.5

IV

III IV

I

x=0.75 II

x=1 III 1

2 T (K)

3

I 4

Figure 7.7 Temperature dependences of the magnetic susceptibility of Cex La1−x B6 . Figure redrawn from Ref. [42].

7.4 Rare-Earth Tetraborides (RB4 ) 7.4.1 An Overview The rare-earth ions of RB4 , with a tetragonal crystal structure belonging to the space group P 4/mbm, form a network of orthogonally connected dimers in the ab-plane, as shown in Fig. 7.8 [101]. This two-dimensional structure is topologically equivalent to the so-called Shastry–Sutherland lattice (SSL), where a spinsinglet ground state can form in quantum spin systems such as SrCu2 (BO3 )2 [102–104]. Although RB4 is a three-dimensional system with a strong coupling along the c-axis, and therefore cannot be strictly speaking equivalent to the SSL, the orthogonal dimer structure can still lead to a macroscopic degeneracy of the magnetic structure, giving rise to a rich variety of notable magnetic phases due to geometrical frustration. Of special interest is the appearance of fractional magnetization states in magnetic fields, which is a common characteristic in geometrically frustrated magnetic systems [104]. Notable cases are those of NdB4 [106, 107], TbB4 [108], DyB4 [109, 110], HoB4 [107, 111–113], ErB4 [114, 115], and TmB4 [115–118] (see Fig. 1.26 on page 46 for an example).

Rare-Earth Tetraborides (RB4 ) 591

Figure 7.8 Crystal structure of RB4 . Drawn with VESTA [105]. Thick solid lines show the nearest-neighbor bonds (J 1 ), the thick dotted lines the second neighbor bonds (J 2 ). The third neighbor bond (J 3 ) is along the c-axis. Macroscopic degeneracy can be lifted by introducing J 4 in the c-plane.

In addition, in heavy rare-earth tetraborides, the quadrupolar degree of freedom becomes active because of the large orbital angular momentum. This gives rise to interesting successive ordering phenomena as a result of competition between magnetic and quadrupolar interactions. The successive transitions in NdB4 , TbB4 , DyB4 , and HoB4 are very interesting in this respect. Although RB4 has a long history of research [119–126], renewed interest in this system was revived by the study on the successive phase transitions in DyB4 [109]. As shown in Fig. 7.9, DyB4 exhibits successive transitions at TN1 = 20.3 K and at TN2 = 12.7 K. The temperature dependence of magnetic susceptibility χ(T ) for B c and B ⊥ c indicates that only the c-axis component orders at TN1 , while the c-plane component orders at TN2 . Given the released magnetic entropy of ∼R ln 4 at TN1 , the ground state is considered to be a quasi-quartet consisting of two doublets separated by less than ∼20 K. Interestingly, the elastic softening of the C 44 mode is enhanced on entering phase II, which suggests the appearance of incipient Ozx -type ferroquadrupolar fluctuations that form long-range order below TN2 . These results stimulated interest in the relationship between multipolar degrees of freedom and the geometrical frustration in this system.

592 Competing Order Parameters in Rare-Earth Hexa- and Tetraborides

(b) 2 III II 0.4 H || [100], [110] 0.3

R ln 4 10 1 R ln 2

0 0

10 20 Temperature (K)

15

-2

H || [001]

DyB4 0.0

I

30

0

20 40 Temperature (K)

5

-1

0.1

II

-1

0.2 0.1 T

III

-1

Cmag/T (J mol K )

I

S (J mol K )

χ (emu/mol)

(a)

0 60

C44 (10

10

3

J/m )

(c) 2.0

1.5

1.0

0

TC = 4.61 K Θ = 2.26 K III 0

II

I

10 20 30 Temperature (K)

40

Figure 7.9 Temperature dependences of (a) magnetic susceptibility, (b) specific heat, and (c) elastic constant C 44 of DyB4 . Figures redrawn from Ref. [109]. The solid line in (c) is a theoretical fit of C 44 (T ) at higher temperatures.

7.4.2 Magnetic Order in RB4 In most magnetically ordered states of RB4 compounds, the unit cell does not change from the chemical one, i.e., the magnetic structures are mostly described by the propagation vector q = (000). To describe the magnetic structure, it is helpful to consider the irreducible representations for q = (000), which are shown in Fig. 7.10. Since there are four R ions in the unit cell, there are 12 basis structures to express the magnetic structure. The actual structure is expressed by one or a few types of representations. The transition temperatures and the magnetic structures of RB4 determined by neutron diffraction are summarized in Table 7.1.

Rare-Earth Tetraborides (RB4 ) 593

Figure 7.10 Irreducible representations of the magnetic structure of RB4 with q = (000) [127, 128].

Table 7.1 Magnetic phase transitions and magnetic structures in successive orderings of RB4 [123, 127–129, 132–135]. The name of the phase and the transition temperature are numbered in descending order of temperature. IC denotes an incommensurate structure. The structure at the bottom in each column is the ground-state structure

I TN1 II TN2 III TN3 IV

NdB4 Para 17.2 K 4 +10 7.0 K 4 +10 +IC2 4.8 K 4 +10 +IC3

GdB4 Para 42 K 2

TbB4 Para 44 K 2 22 K 2 +6

DyB4 Para 20.3 K 10 12.7 K 10 +2

HoB4 Para 7.1 K IC1 5.7 K 10 +6

ErB4 Para 15.4 K 10

TmB4 Para 11.7 K IC4 11 K IC5 10 K 5

IC1 : (δ δ δ  ), δ = 0.022, δ  = 0.43; IC2 : (δ δ δ  ), δ = 0.14, δ  = 0.4; IC3 : (δ 0 δ  ), δ = 0.2, δ  = 0.4; IC4 : (δ  +δ  δ  0)+(δ  +3δ  δ  0), δ  ∼0.13, δ   0.012; IC5 : (δ  0 0), δ  ≈ 0.13.

594 Competing Order Parameters in Rare-Earth Hexa- and Tetraborides

Figure 7.11 Magnetic susceptibilities of RB4 . Figures redrawn from Refs. [106, 108, 109, 113, 114, 117, 122].

In all RB4 , the magnetic structures are noncollinear, even in GdB4 , which has a vanishing orbital moment (S = 7/2 and L = 0) [129]. Although the magnetization process of GdB4 is almost isotropic [130], the 2 structure at zero field suggests that the S = 7/2 spins experience a local anisotropy due to the CEF, which indicates that the spin-orbit coupling does not vanish in GdB4 [131]. Figure 7.11 summarizes the temperature dependences of the magnetic susceptibilities of RB4 , which illustrates the diversity of this system. The χ(T ) curves for ErB4 and TmB4 show that these

Rare-Earth Tetraborides (RB4 ) 595

are Ising systems with strong anisotropy along the c-axis. This is consistently understood by the 10 and the 5 structures for ErB4 and TmB4 , respectively. Although the orderings of ErB4 and TmB4 at the lowest temperature seem to be simple at first sight, TmB4 features an unusual multistep magnetization process [116, 117]. As for DyB4 , the successive transitions can be understood by the appearance of the 10 structure at TN1 and by the additional appearance of the 2 structure below TN2 . In TbB4 , the 2 structure appears below TN1 = 44 K, followed by a superposition by the 6 structure below TN2 = 22 K [133]. The 2 structure, therefore, is preferable for the local magnetic anisotropy due to the CEF. Below the second transition at TN2 , the spins tilt toward [100] or [010] to form a uniaxial AFM order. Since this AFM structure is not preferable for the local anisotropy, there should be a quadrupolar interaction behind this ordering, as also suggested by an ultrasonic measurement [136]. It is quite anomalous that both χ c and χ⊥c exhibit a cusp at TN1 = 44 K. Also at TN2 = 22 K, similar anomalies are observed for both field directions. Since the magnetic moments in both the 2 and the 6 structures lie in the cplane, it is a big question why χ c corresponding to a perpendicular susceptibility exhibits a cusp. For the case of NdB4 , it is unexpected that χ⊥c , which corresponds to the hard axis susceptibility, shows a cusp at a higher temperature TN1 = 17.2 K, whereas χ c along the easy axis continues to increase and shows a cusp at a lower temperature TN2 = 7.0 K (note that the labeling of TN here is different than in the original papers). Using neutron diffraction with polarization analysis, it was concluded that the primary magnetic order below TN1 is described by the irreducible representation 4 , with moments in the c-plane, and the 10 along the c-axis is a secondary order parameter that is gradually induced below TN1 with decreasing temperature [128, 132]. Although the anomaly in χ⊥c (T ) at TN1 is rather weak, specific heat shows that the transition is very sharp [106]. These results suggest that the transition at TN1 is not the magnetic dipolar order but a quadrupolar order (accordingly, TN1 is named T0 in Ref. [106]).

596 Competing Order Parameters in Rare-Earth Hexa- and Tetraborides

7.4.3 Fractional Magnetization Plateaus Multistep magnetization processes observed for B c in RB4 are summarized in Fig. 7.12, where we can see various kinds of fractional magnetization plateaus. In the plateau state, the magnetization stays at a constant value in spite of the increase in the external field, which indicates that the whole system is in an eigenstate. This is an important characteristic of a frustrated system with macroscopic degeneracy at zero field. However, it is not easy to understand what kind of interaction exists behind this multiparticle system and why this fractional plateau state appears. Among these plateau states, the 1/2-plateau state might be rather easy to understand. One of the four spins in the 10 or 5 structures flips from ↓ to ↑ if the unit cell is preserved, for which, however, we do not have any experimental evidence. Another possibility could be that the ferromagnetic planes of 3 and the AFM planes of 10 or 5 stack alternately along the c-axis, giving rise to a (00 12 ) diffraction peak.

Figure 7.12 Magnetization processes of RB4 . Figures redrawn from Refs. [107, 108, 110, 112, 114, 117].

Rare-Earth Tetraborides (RB4 ) 597

TbB4 The fractional plateau states other than 1/2 observed in TbB4 , HoB4 , NdB4 , and in TmB4 are very interesting. In TbB4 , many steps are observed between 15 and 30 T [108]. It is surprising that these plateau states are realized for B c, starting from the magnetic structure ordered totally within the c-plane, which is unusual. To investigate the magnetic structures of these plateau states, high-field experiments were performed using both resonant x-ray diffraction (RXD) and neutron diffraction [137, 138]. The Bragg peaks corresponding to q = (000) within the basal (HK0) scattering plane were investigated and were shown to survive up to the saturation field. From the limited amount of data, which nevertheless involves quite important information, it was inferred that the spins still lying within the c-plane and those flipped to the c-axis are arranged so that the total magnetization stabilizes at a fractional value. A possible structure for the 1/2 plateau phase is shown in Fig. 7.13(a) [137]. The structure for the 1/3 plateau phase is also considered to be a mixture of the spins lying in the c-plane and standing along the c-axis [138]. To clarify these structures in the plateau phases, it is necessary to search for Bragg peaks arising from the periodicity with larger unit cells.

Figure 7.13 Models of fractional magnetization states of RB4 . Figures were drawn from the information in Refs. [113, 135, 137–139]. Drawn with VESTA [105].

598 Competing Order Parameters in Rare-Earth Hexa- and Tetraborides

HoB4 A detailed study of the magnetic structure of the plateau phase by neutron diffraction has been performed in HoB4 . This is shown in Fig. 7.14(b), which includes the Bragg peak appearing outside the (HK0) horizontal scattering plane [113]. The 10 + 6 structure at zero field, which gives rise to (100) and (210) reflections, vanishes on entering the 1/3 plateau phase. Then, the (21 13 ) intensity appears in the 1/3 plateau phase, directly indicating the appearance of a new stacking sequence along the c-axis with a periodicity three times the unit cell. The magnetic structure analysis, though not complete, suggests a ferrimagnetic stacking as shown in Fig. 7.13(b). As pointed out in the original paper, the presence of a c-plane component, which may be anticipated from analogy to TbB4 , is not ruled out. Another remarkable feature of HoB4 is the competition between commensurate (10 + 6 ) and incommensurate (δ δ δ  ) magnetic structures [127]. As shown in Fig. 7.14(a), the two peaks coexist at high temperatures above TN1 = 7.1 K as diffuse scattering, reflecting short-range orders. With decreasing temperature, the intensity of (δ δ δ  ) increases more than that of (101) and incommensurate longrange order forms below TN1 . Simultaneously, however, the (101) peak also begins to develop below TN1 , which ultimately leads to

Figure 7.14 (a) Temperature and (b) magnetic field dependences of neutron diffraction intensities of HoB4 . Reproduced from Ref. [113] and Ref. [127]. 10 gives rise to (100), (101), and (210), 6 to (100), (101), but hardly (210), and 3 (ferromagnetic) to (210). The propagation vector (0.022, 0.022, 0.43) corresponds to IC1 .

Rare-Earth Tetraborides (RB4 ) 599

the first-order transition to commensurate order at TN2 . Surprisingly, this incommensurate structure revives in magnetic fields at low temperature. It seems from Fig. 7.14(b) that the (δ δ δ  ) structure competes with the (10 + 6 ) structure, but can coexist with the 1/3 plateau and the ferromagnetic structures. TmB4 Many fractional plateau phases appear in TmB4 , depending on the history of temperature and magnetic field: 1/7, 1/8, 1/9, and even higher fractional states [116, 117, 135, 139]. Since the magnetic moments in TmB4 have a strong anisotropy along the caxis, the appearance of such a diverse structure of Ising spins is quite surprising. Illustrated in Fig. 7.13(c) is a model structure for the 1/7 plateau phase constructed from the information in Refs. [135, 139]. Starting from the base structure of 5 , a 1/7 fractional magnetization state can arise by flipping two spins in seven unit cells, which also changes the sequence of the spin arrangement. The appearance of 1/8, 1/9, and other fractional states suggests that there are several local minima in the total free energy; they are very close in energy and are almost degenerate. Therefore, an unpredictable emergent state can be realized from such a situation.

7.4.4 Quadrupolar Fluctuation in DyB4 The quadrupolar fluctuation suggested in Fig. 7.9 was studied by RXD [140, 141]. Figure 7.15(a) shows the temperature dependence of the RXD intensity at (100), where the signals from 10 and 2 structures are detected. Since the E 1 resonance structure factor for the scattering from a magnetic dipole moment m is proportional to (  ×)·m, the structure factor at ψ = 0◦ (90◦ ) is proportional to the magnetic moment along the c-axis (ab-plane). Therefore, the data at ψ = 0◦ reflects μc , whereas that at ψ = 90◦ reflects μab . With respect to the structure in the c-plane, it is difficult to distinguish between 2 and 6 by RXD; but the 2 irrep can be confirmed by powder neutron diffraction [134, 142]. Notably, the signal at ψ = 90◦ exists in the intermediate phase II, indicating that the in-plane moments are already ordered. Furthermore, the width of the peak profile in phase II for μab is broader than the resolution as shown in Fig. 7.15(c), and it suddenly

600 Competing Order Parameters in Rare-Earth Hexa- and Tetraborides

Figure 7.15 (a) Temperature dependence of the intensity of the resonant x-ray diffraction at (100) in DyB4 for two scattering geometries at ψ = 0◦ ˆ and 90◦ (k×k cˆ ). An analyzer crystal is used to isolate the σ -π  (k×k b) scattering. (b) Temperature dependence of the half-width at half-maximum (HWHM) of the peak profile. (c) Comparison of the peak profiles at 6 and 15 K for the geometry (ψ = 90◦ ) detecting μc . Adapted from Refs. [140, 141].

drops below TN2 . The existence of the c-plane component in phase II has also been confirmed in a detailed single-crystal neutron study [143]. Thus, it was concluded that the c-plane component is short-range-ordered in phase II, where the c-axis component is long-range-ordered. However, the data in Fig. 7.9(a, c) show that they are definitely paramagnetic in phase II. This contradiction can be rationalized by considering the time scale of the measurement. The quasielastic scattering due to the fluctuation of the c-plane component in phase II could be within the energy resolution of the Bragg peak. It is estimated that the time scale of the fluctuation in phase II is longer than approximately / E ∼ 10−12 s. The fact that

Conclusions 601

the critical scattering is observed at TN1 in neutron scattering but is not observed at TN2 also shows that the dynamics in phase II is very slow [143]. Although a ferro-type quadrupolar interaction is apparently important, the role of dipolar interaction cannot be discarded.

7.5 Conclusions In this chapter, we have reviewed the multipolar ordered phases in Cex La1−x B6 and the unusual ordered phases in RB4 , from experimental viewpoints. The antiferroquadrupolar (AFQ) ordered phase in CeB6 (phase II), first evidenced by the observation of the field-induced AFM order by neutron diffraction, was confirmed directly by resonant x-ray diffraction (RXD). In a magnetic field along (α, β, γ ), the AFQ order parameter of phase II is expressed by

α Oyz + β Ozx + γ Ox y , which varies continuously with the field direction. Although the AFQ order at zero field is considered to be a three-domain state of Oyz , Ozx , and Ox y , direct evidence has not yet been obtained. The field-induced antiferrooctupolar (AFO) state features an inter-site interaction, which is theoretically proposed to be playing a fundamental role in the increase of the transition temperature with increasing field. This state was also confirmed with RXD by measuring the interference between the E 1 resonance due to AFQ and the E 2 resonance due to AFO. The two contributions were separated by measuring the field reversal effect. In the La-doped system, pure T β (2u )-AFO order without magnetic dipole order takes place, which was also confirmed by resonant xray diffraction. By applying magnetic field in the AFO phase, various kinds of multipole moments are induced, with the Ox y -quadrupole being most prominent, as can naturally be anticipated from the appearance of the AFQ order in magnetic fields. However, it is difficult to explain it by a mean-field calculation. The cusp anomaly in magnetic susceptibility also remains unexplained. It may be necessary to go beyond the localized picture of the Ce 4f electron and include the hybridization with the conduction electrons. We also reviewed the geometrically frustrated family of RB4 . We compared the magnetic structures and magnetization curves

602 Competing Order Parameters in Rare-Earth Hexa- and Tetraborides

for R = Tb, Dy, Ho, Nd, Er, and Tm, where various kinds of fractional magnetization plateau states are realized for B c. In TbB4 , it is very unusual and interesting that many plateau states are realized starting from the magnetic structure where the moments lie within the c-plane. In HoB4 , the incommensurate (δ δ δ  ) phase (δ = 0.022, δ  = 0.43), which appears as the intermediatetemperature phase at zero field, revives at the lowest temperature when the commensurate 10 + 6 structure is suppressed in magnetic fields and survives up to the saturation field. In TmB4 , with strong Ising anisotropy along the c-axis, many fractional plateau phases appear, depending on the history of temperature and magnetic field. Long periodic structures are formed along the a-axis by flipping a few of the spins. These are associated with macroscopic degeneracy caused by the geometrical frustration. Competition between magnetic and quadrupolar order parameters also plays an important role in RB4 . Successive phase transitions in R = Tb, Dy, Ho, and Nd are possibly associated with the quadrupolar interaction. In DyB4 , a short-range order of the c-plane component of the magnetic moments is observed, which fluctuates on a time scale longer than 10−12 s. This is associated with the strong elastic softening and the Ozx -quadrupolar fluctuation.

Acknowledgments The author acknowledges valuable discussions with D. Okuyama, R. Watanuki, K. Suzuki, S. Michimura, F. Iga, T. Inami, S. Yoshii, H. Nakao, K. Iwasa, Y. Murakami, and many other collaborators. This work was supported by the Japan Society for the Promotion of Science (JSPS) through the Grant-in-Aid for Scientific Research (KAKENHI).

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11

B-NMR in phase II of

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Chapter 8

Multipolar Order and Excitations in Rare-Earth Boride Kondo Systems Peter Thalmeier,a Alireza Akbari,b,c,d and Ryousuke Shiinae a Max Planck Institute for Chemical Physics of Solids, 01187 Dresden, Germany b Asia Pacific Center for Theoretical Physics, Pohang, Gyeongbuk 790-784, Korea c Department of Physics, POSTECH, Pohang, Gyeongbuk 790-784, Korea d Max Planck POSTECH Center for Complex Phase Materials, POSTECH,

Pohang 790-784, Korea e Department of Physics and Earth Sciences, University of Ryukyus, Nishihara, Okinawa 903-0213, Japan [email protected]

The cubic rare-earth boride series displays diverse electronic states like localized 4f electron multiplets split by the crystal electric field (CEF), itinerant heavy-fermion quasiparticle bands of the Kondo lattice as well as gapped Kondo insulator or mixed-valent semiconductor states. Furthermore, at low temperatures fairly exotic ordered states may appear due to the “hidden” order of multipoles carried by degenerate CEF multiplets, in addition to common (dipolar) magnetic order present in many RB6 (R = rareearth) systems. Most prominent are CeB6 and its La diluted alloys which exhibit quadrupolar and octupolar ordering enabled by the cubic 8 quartet state. The associated collective excitations are

Rare-Earth Borides Edited by Dmytro S. Inosov c 2022 Jenny Stanford Publishing Pte. Ltd. Copyright  ISBN 978-981-4877-56-5 (Hardcover), 978-1-003-14648-3 (eBook) www.jennystanford.com

616 Multipolar Order and Excitations in Rare-Earth Boride Kondo Systems

multipolar waves with a dispersion characteristic for the underlying order and accessible by inelastic neutron scattering. This localized multipolar-moment picture of RB6 has to be complemented by the itinerant Kondo lattice approach. Due to the presence of hybridization and collective ordering gaps, a singular magnetic response can lead to the appearance of collective spin exciton modes inside the gap around symmetry points of the Brillouin zone (BZ). This has been observed in heavy-fermion metal CeB6 and in particular in the Kondo insulators YbB12 and SmB6 . The latter, which has no Landau-type local symmetry breaking is also the prime candidate for a strongly correlated insulator with topological order, caused by odd number of band crossings of 4f and 5d bands in the BZ. The signature of topological order is the existence of massless Dirac surface states with helical spin polarization, a topic of intense investigation in SmB6 .

8.1 Introduction The RB6 (R = rare-earth) compounds are a versatile model series for strongly correlated 4f electron materials. These compounds ¯ where the B6 have the cubic CaB6 structure (space group P m3m) octahedra play the role of anions (Fig. 8.1). The rare-earth (RE) ions are mostly in the 3+ configuration, but 2+ and mixed valence also occur (see Table 8.2). The continuous interest in the series in the last fifty years stems largely from the fact that they show a great variety of low-temperature ordered phases. They are triggered by the lifting of the degeneracies by the 4f-electron CEF ground state multiplets and the interplay with RE 4f-5d conduction electron intersite hybridization. In most cases the ordered phases show magnetic order of (non-)collinear type depending on temperature and external field. The degenerate CEF ground states, however, not only support ordering of dipolar (rank 1) magnetic moments but also of higher quadrupole (rank 2), octupole (rank 3), and generally rank p ≤ 2l (l = 3) multipoles of 4f electrons. Here even and odd rank p correspond to the preservation and breaking of timereversal symmetry by the order parameter [1, 2]. The hybridization with conduction states leads (via a generalized Schrieffer-

Introduction 617

Figure 8.1 Cubic CaB6 -type structure of RB6 borides. R 3+ ions (blue) form a simple cubic sublattice. The B6 octahedra (red) play the role of the anions at body-centered positions forming a cage for RE. The lattice constant for ˚ Boron sites 1 and 3 are inequivalent in NMR R = Ce is a ≈ 4.14 A. (Section 8.3.3).

Wolff mechanism) to effective intersite interaction between those multipoles which drive their ordering. Multipole order with rank p ≥ 2 is generally termed “hidden order” (HO) because it cannot easily be detected by the conventional (dipolar) neutron and x-ray scattering which yield no diffraction peaks except when it induces a considerable secondary lattice distortion. More involved methods like neutron diffraction (ND) in external field [3], high-momentumtransfer neutron scattering [4], resonant x-ray scattering [5–7], or ultrasonic investigations [8, 9] and NMR method [10] have to be applied. This has given new impetus to investigate the multipolar ordering in f-electron compounds. The most well-known examples of higher-rank HO are cubic hexaborides CeB6 and Ce1−x Lax B6 (rank 2 quadrupole and rank 3 octupole) [11], NpO2 (rank 3 octupole) [12], and tetragonal URu2 Si2 (proposed rank 5 dotriacontapole) [13–16]. Further examples are found in the cubic 4f skutterudites [17–19] and 1-2-20 cage compounds [20]. In fact, the hexaborides may also be considered as cage compounds where rare-earth ions on the simple cubic sublattice are surrounded by a cage of eight B6 octahedra (Fig. 8.1). In

618 Multipolar Order and Excitations in Rare-Earth Boride Kondo Systems

both series this leads to the interesting possibility of ‘rattling’ or strongly anharmonic motion of RE ions in the cages. It is also known from the RE clathrate cage compounds [21] where it strongly influences transport properties, in particular thermoelectric power. The rattling motion in the boride series leads to flat phonon branches that can be interpreted as low-energy (ωE  10 meV) Einstein modes of RE ions in oversized B6 cages. It is most prominent for some heavier RE (Gd, Dy, Tb) hexaborides due to the lanthanide contraction of RE ionic radii [22–24]. The anharmonic low-energy rattling phonons are in contrast to the extremely stiff motion of the boron cage as witnessed by the very large longitudinal elastic constants [25].

8.1.1 Conduction Bands and Fermi Surface In this review we focus exclusively on the correlated electronic properties of the series, in particular hidden 4f multipole order and its excitations. We also discuss the consequences of Kondo effect and associated 4f -5d hybridization, i.e., Kondo insulator state, spin resonance formation, band crossing, and topological properties like protected helical surface states. Therefore, it is useful to get first a schematic picture of the electronic degrees of freedom, itinerant 5d and localized 4f of the rare-earth as well as 3p valence state of the B6 cages. A sketch of the position and dispersion of these electronic states (excluding hybridization effects) is given in Fig. 8.2 for typical cases of the RE valences.1 The 5d-like bands show large dispersions and, except for the two 2+ valence cases of semimetal EuB6 and semiconductor YbB6 , lead to the large 5d-type electron pockets in the Fermi surface (FS) (see Fig. 8.2) around the X point (0 12 0) (in reciprocal lattice units, 1 r.l.u. = 2π/a) and equivalent ones. The FS ellipsoids are touching and form small necks between them. In the cases of LaB6 , CeB6 , PrB6 , and NdB6 where the dimensions of the X-point pockets have been determined by dHvA experiments (Table 8.1) [28, 29], the orbital cross sections are very close for all compounds, and their field-angular dependence identifies an almost spherical shape. 1 In all figures, panels a, b, ... are labeled from left to right and top to bottom.

Introduction 619

Figure 8.2 (a-c) Schematic bulk band structure (without hybridization, Fermi level F = 0) consisting of RE 5d (purple) and 4f (blue) as well as B 2p (green) type bands for exemplary compounds with integer as well as mixed valences; Yb2+ , Sm2.5+ and Ce3+ , respectively. Here YbB6 is a p-d band semiconductor due to the filled 4f shell (4f 7/2 binding energy | f |  1 eV. Due to (intersite) f-d hybridization and on-site f-f correlation SmB6 becomes a mixed valence semiconductor and CeB6 (| f |  2.1 eV) a Kondo heavyfermion metal (adapted from Ramankutty et al. [26]). (d) Main large Fermi surface sheet (α3 -orbit on cubic faces) of hexaborides. The dimensions of connected X-point ellipsoids is almost identical for LaB6 , CeB6 , and SmB6 . Adapted from Tan et al. [27].

Although the large electron FS (corresponding to an α3 -orbit of dHvA results shown in Fig. 8.2) are similar in LaB6 and CeB6 , their effective masses are vastly different [28, 30], while that of PrB6 is in between (Table 8.1). In LaB6 there are no f electrons and the 5d band mass is observed. In CeB6 the valence is close to 3+ with a 4f 1 8 CEF ground state, but 4f-5d hybridization and 4f-4f Coulomb repulsion lead to very narrow 4f-quasiparticle bands due the Kondo lattice formation (Section 8.6.1). They may be interpreted as latticeperiodic coherent bands formed by the sharp single-site Kondo resonance states (dotted blue line in Fig. 8.2). In PrB6 the integer 3+ valent 4f 2 CEF 5 ground state has a much smaller hybridization that leads only to a small perturbative renormalization to an effective 5d mass, while in NdB6 there is no mass enhancement as compared to LaB6 due to negligible hybridization. The boron 2p states do not cross the Fermi level in the series. Nevertheless they have an important indirect influence in SmB6 where the f band obtains an upward dispersion in ΓX direction (thin blue line in Fig. 8.24) due to the (on-site) hybridization with lower but close 2p bands at X .

620 Multipolar Order and Excitations in Rare-Earth Boride Kondo Systems

Table 8.1 Area and the effective cyclotron mass (H [001]) of the α3 orbits (intersection of X-point pockets in Fig. 8.2 with the cubic [001] faces). An average α3 area of 8000 T corresponds to about 1/3 of the cubic BZ face area (2π/a)2 in Fig. 8.2 α3 orbit F α3 [T] m∗ /me Ref. ∗

LaB6 7890 0.64 [28]

CeB6 8670 14–21 [28]

PrB6 8190 1.95 [28]

NdB6 7980 0.60 [29]

SmB6 7800 ∗ [27]

Reliable mass assignment was not possible.

This effect is essential for being able to form the topological insulator state as discussed in Section 8.8.2.

8.1.2 Localized 4f Shells, Their CEF States, Multipoles, and RKKY Interactions In the cubic structure of RB6 the RE ion point group is Oh . This leads to a cubic CEF potential for the shell of localized spherically symmetric 4f n states. For finite temperature and low energies one may restrict to the ground state characterized by shell angular, spin, and total angular momentum (L, S, J), the latter being determined by the large spin–orbit coupling [ζs.o. = 0.045 eV (Ce) − 0.364 eV (Yb)] in the R 3+ ions. In the common Stevens representation, the cubic CEF potential is written as an operator in terms of symmetrized polynomials of (J x , J y , J z ) in the (2J + 1)-dimensional Hilbert space of the total angular momentum J ground-state multiplet. This is the well-known expression     (8.1) HCEF = B40 O40 + 5O44 + B60 O60 − 21O64 where O4m and O6m represent, respectively, fourth and sixth order symmetrized polynomials of J z (m = 0) and J ± , J z (m = 4) [31] corresponding to the real-space tesseral harmonics. Furthermore B40 , B60 are CEF parameters that may be formally given within a pointcharge model [31]. The latter determine the splitting into generally degenerate CEF multiplets that belong to the Oh representations α (dα ) where α = 1, 2, 3, 4, 5 for non-Kramers ions (integer J ) with corresponding degeneracy dα = 1, 1, 2, 3, 3 and α = 6, 7, 8 for Kramers ions (non-integer J ) having corresponding degeneracy

Introduction 621

Γ7

540 Γ1 Γ4 Γ3

464 377 314 Γ 6

278

(1)

135

Γ8 Γ8

0

CeB6

(2)

Γ5

Γ8

Γ8

PrB6

230

Γ7

NdB6

SmB6

Figure 8.3 Spectroscopically determined CEF splittings of RE hexaborides in kelvins. Results from INS (Ce–Nd) [36] and from RIXS (Sm) [35].

dα = 2, 2, 4. The degenerate states of each CEF multiplet are designated by |αi  with 1 ≤ i ≤ dα . They are tabulated in Ref. [32] for all J as a function of CEF parameters B4 , B6 or alternatively related parameters W, x. In practice the splittings and wave functions, i.e., the CEF parameters, have to be determined experimentally, mainly by two methods: (i) analysis of high-temperature (singleion) susceptibility and (ii) fitting to peak positions and intensities of inelastic neutron scattering (INS) spectra that determines directly the splitting and dipolar magnetic matrix elements between the CEF states. In CEF schemes with high J (or inequivalent RE sites) this is, however, not a unique procedure to determine the CEF parameters. Recently x-ray techniques like NIXS and RIXS have also contributed to unravel the CEF states and energies [33–35]. The CEF level schemes and, in particular, CEF ground states for the RB6 series known so far are summarized in Fig. 8.3. They concern mostly the light RE, as there is surprisingly little information on the heavier RB6 in the literature. For more information on the CEF level schemes in hexa- and dodecaborides, see also Chapter 6. A CEF level scheme with (2J + 1) states can carry 1 ≤ n ≤ (2J + 1)2 − 1 multipole operators X n (the identity has been subtracted), which is simply equal to the number of standard ij j basis operators (minus the identity) defined by Lαβ = |αi  β |. The multipole operators are linear combinations of the standard basis operators that belong to specific cubic representations. They may be expressed as rank p polynomials P p (J x , J y , J z ) and their

622 Multipolar Order and Excitations in Rare-Earth Boride Kondo Systems

explicit form (for p ≤ 4) is tabulated in [7]. Their treatment is discussed in more detail in Section 8.3 for a special case. These multipoles are the physical 4f-shell degree of freedoms at every site. When the hybridization with 5d conduction electrons is taken into account, the localized 4f multipoles may be effectively coupled at adjacent sites by a generalized RKKY mechanism well known for the rank p = 1 multipoles (magnetic dipoles). For a degenerate CEF ground state then at low temperatures the multipole with the maximum effective intersite coupling at a particular wave vector q will be the primary order parameter. Most frequently this is either a multipole of rank 1 (magnetic order) or rank 2 (quadrupolar HO), but more general HO, in particular rank 3 (octupolar order), can also occur (see Section 8.4). General expressions for the effective RKKY-type multipole interactions may be derived [37–40], and quantitative first-principles results for CeB6 were presented recently Table 8.2 Compilation of essential data for RB6 compounds with hidden (e.g., quadrupolar) and/or magnetic (dipolar) order characteristics. CeB6 is a Kondo lattice heavy-fermion metal while semiconducting SmB6 is the only compound with strongly mixed valence Compound RB6 ∗

J

Valence

Oh CEF g.s. (degeneracy)

CeB6 (HFM)

5 2

3+

8 (4)

3.3 ( 12 12 21 )

PrB6 (m)

4

3+

5 (3)

( 12 12 0)

4.2 ( 12 12 0) 4.2 (C)

( 14 41 12 )

3+

—

—

8

(00 12 )

2+

(L = 0)

—

—

12.5

(000)

GdB6 (m)

9 2 7 2 7 2

(2) 8 (4)

3+

(L = 0)

—

—

16

( 14 41 12 )

TbB6 (m)

6

3+

2 (1) or 3 (1)

—

—

20

( 14 41 12 )

DyB6 (m)

15 2

3+

8 (4)

31

(000)

26

( 14 41 12 )

HoB6 (m)

8

3+

5 (3)

6.1

(000)

—

—

YbB6 (sc)

0

2+

—

—

—

—

—

2.55+

8 (4)

—

—

—

—

NdB6 (m) EuB6 (sm)

SmB6 (MV,TI)

0∼

5 2

(1)

THO [K]

qHO r.l.u.

TN [K]

qm [r.l.u.]

2.3

( 14 41 12 )

7 (IC) ( 14 −δ 14 12 )

∗ HFM = heavy-fermion metal; m = metal; sm = semimetal; sc = semiconductor; MV = mixedvalent; TI = topological insulator.

Overview of RE Boride Compounds 623

[41]. They demonstrate that quadrupolar and octupolar nearestneighbor (n.n.) interactions are maximally enhanced, supporting the parameterized form [42] used in the following sections. When the k,q-dependent multipole matrix elements between conduction band states are replaced by a constant, their generalized RKKY interaction is proportional (for every multipole) to the Lindhard  function χL (q) = q ( fk+q − fk )/(k − k+q ) where k and fk are conduction band and Fermi function. For the FS topology of Fig. 8.2 it has (sub-)maxima at the wave vectors Q = ( 41 41 0) and Q = ( 12 21 21 ), respectively, due to various nesting properties of the FS [43]. Indeed Q is the in-plane component of the most common AFM structure (Fig. 8.4) in RB6 , and Q is the HO wave vector in CeB6 .

8.2 Overview of RE Boride Compounds The RE borides, in particular hexaborides, have the advantageous property of existing for almost the whole series of rare-earth atoms within the same crystal structure, thus allowing for the study of systematic variations in physical properties. These change greatly due to the varying degree of localization or itineracy of 4f electrons, while the basic CEF multiplet states are the same due to the universal cubic Oh point symmetry of this class. One can roughly distinguish two cases: First, the stable moment compounds where the hybridization of well localized 4f electrons and conduction electrons can be treated perturbatively, leading to an on-site interaction with 4f multipoles in second order and to their intersite effective RKKY coupling in fourth order of the hybridization. These compounds are listed in the central part of Table 8.2 with their salient ordering characteristics. In the second case the hybridization is strong (as in CeB6 , YbB12 , and SmB6 compounds) and may destabilize the moment by screening and formation of a local singlet state that forms coherent heavy electron bands at low temperature, or, in the large hybridization case, may lead to a mixed-valent state with pronounced non-integer 4f-electron occupation. In both instances a hybridization (pseudo-)gap in the renormalized electron bands appears. Most frequently a metallic ground state with moderately

624 Multipolar Order and Excitations in Rare-Earth Boride Kondo Systems

(MV) or strongly (HF) enhanced effective m∗ electron mass is realized as, e.g., in CeB6 . The heavy electron liquid may become instable at the lowest temperatures leading to HF superconductivity as frequently happens in 122- and 115-type Ce compounds [44] or to multipolar order as in CeB6 . If the number of conduction electrons is suitable, the Fermi level may fall into the hybridization gap producing a rare “MV semiconductor” or “Kondo insulator” (Fig. 8.19). In the borides, two famous hybridization-gap insulators are known: YbB12 and SmB6 . In particular the latter has raised enormous attention due to its nontrivial topological state. On the other hand YbB6 is a conventional p-d semiconductor with nonmagnetic Yb2+ state. The strong hybridization compounds are the most investigated ones and will be the main focus of this review. Here, as an overview, we first briefly discuss some salient features of the stable moment compounds in Table 8.2. Among the light rare earth, PrB6 and NdB6 are the most studied. Also their CEF level schemes shown in Fig. 8.3 are well known [36, 45]. PrB6 with a 5 triplet ground state exhibits two consecutive firstorder transitions to an incommensurate (IC) phase at TNIC = 7 K and a further lock-in transition to a magnetic C-phase with wave vector Q = ( 14 41 21 ). The latter coexists with an induced AFQ order presumably with Q = ( 12 21 0) below TNC = 4.2 K. These wave vectors together with Q = ( 12 21 21 ) for HO characterize most of the ordered structures in the hexaboride compounds and are related to the nesting structure of the Fermi surface (Fig. 8.2) that translates into the preferred magnetic wave vectors via the RKKY mechanism (Section 8.1.2). The ordered arrangements of moments are illustrated in Fig. 8.4(a–c). In zero field the magnetic structure is of the noncollinear double-q type (a) which switches to single-q type in applied field H [110] (c). The incommensurate structure with wave vector ( 41 −δ 14 12 ) is shown in (b). A theoretical investigation for PrB6 has been presented in Ref. [47] based on the 5 CEF ground state which carries 3 4− dipoles and 5 (3+ , 5+ ) quadrupoles (here ± denotes even/odd behavior under time reversal). In distinction to CeB6 (Section 8.3) the non-Kramers triplet carries no octupoles. The presence of n.n. isotropic and nextnearest-neighbor (n.n.n.) pseudo-dipolar exchange interactions was

Overview of RE Boride Compounds 625

Figure 8.4 Magnetic structures of PrB6 and the H -T phase diagram. (a) Double-q structure with Q1 = ( 14 14 12 ), Q2 = ( 41 14¯ 12 ) (b) IC structure with QIC = ( 14 − δ 14 21 ) (δ = 0.05) (c) single-q structure with Q = ( 14 14 12 ) in external field H [110]. (d) H -T phase diagram for H along symmetry directions. Separate low- and high-field IC phases IC1 and IC2 are observed. The C-phase has coexisting easy-plane magnetic and Ox y quadrupolar order. Reproduced from Kobayashi et al. [46].

proposed to obtain the stability of the IC phase (although with a slightly different wave vector QIC = ( 14 −δ 14 −δ 12 ) with δ = 0.16. The transition to the primary IC phase is of second order, whereas the lower lock-in transition to the AFM C-phase is accompanied by a secondary Ox y quadrupole HO at wave vector ( 12 21 0) and therefore is of first order. The corresponding H -T phase diagram of PrB6 is shown in [Fig. 8.4(d)]. The transverse elastic constants c44 show a pronounced softening due to the Curie–Weiss type 5+ quadrupolar susceptibility resulting from the orbitally degenerate 5+ CEF ground state [48]. The softening is arrested, however, at the magnetic phase

626 Multipolar Order and Excitations in Rare-Earth Boride Kondo Systems

transition where the CEF ground state splits due to the molecular field. (2)

NdB6 has a quartet 8 ground state similar to CeB6 . The observed high-field magnetization anisotropy M[100] < M[110] < M[111] with [111] easy axis supports this CEF model [49]. But its ordered magnetic structure is much simpler than the noncollinear double-q structure of CeB6 (Section 8.4), corresponding to a collinear singleq type-I AFM below TN = 8 K with Q = (00 12 ), which implies three possible domains. The ordered moment is, however, oriented along the fourfold [001] axis instead of the CEF [111] easy axis. This has been attributed to a small ferro-type interaction of O20 quadrupoles [50]. In high fields (∼20 T) along [111] an unexpected metamagnetic transition to a noncollinear triple-q magnetic structure with Q1 = ( 12 00), Q2 = (0 12 0), Q3 = ( 21 00) occurs, which is stabilized by an interplay with induced triple-q AFQ order Q1 = (0 21 21 ), Q2 = ( 12 0 12 ), Q3 = ( 12 21 0) for Oyz , Ozx , Ox y quadrupoles, respectively (2) [49]. Therefore, although the 8 ground state of NdB6 does not lead to primary AFQ order as in phase II of CeB6 (Section 8.3), (2) the presence of 8 -sustained quadrupoles and their interactions shows a subtle influence on the low- and high-field magnetic order of (2) this compound. Similar to PrB6 , the orbitally degenerate 8 causes a softening of c44 elastic constants which is again arrested by the magnetic phase transition [48]. EuB6 is an outlier of the series in two aspects: Firstly it contains europium in the half-filled 4f 7 Eu2+ S-state ionic configuration with J = 72 and hence has no CEF splitting. Secondly the main interest in EuB6 does not stem from the 4f electrons which show simple bulk ferromagnetism below TC = 12.5 K due to their Sstate but rather from the peculiar transport properties of conduction electrons. Since this is not the central focus here, we only comment briefly on it. The compound is a ferromagnetic, partly spin-polarized semimetal with a small valence/conduction band overlap for the majority band of order 0.5 eV at zero temperature, while the spin minority band is gapped by a similar amount. This leads to small majority-spin electron pockets at the X point [51]. When approaching TC from below the overlap of majority bands and concomitantly the carrier density and plasma frequency decrease

Overview of RE Boride Compounds 627

[52]. This behavior can be described by a two-band Kondo lattice type model with FM/AFM coupling [53]. Above TC ferromagnetic features remain due to a phase separation into paramagnetic regime and percolating magnetic polarons [54] that become isolated at the percolation temperature TM = 15.5 K. At this temperature a cusp in the resistivity and giant magnetoresistance are observed [55]. GdB6 and TbB6 : In the center of the RB6 series we have again a half-filled S-state ion Gd3+ with J = 72 as for Eu2+ and therefore no CEF splitting. The Tb3+ (J = 6) CEF ground state should be a singlet (Table 8.2) as concluded from elastic constants measurements [48]. Both compounds show a first-order transition to an AFM state with Q = ( 14 41 21 ). For GdB6 (TN = 15 K) the moments are parallel to the 1 component of the ordering vector, while they are perpendicular 2 for TbB6 (TN = 21 K). The moments are large and correspond to the expected value of 3+ ions (Table 8.2). Therefore, magnetoelastic effects are noticeable [56, 57] and lead to various lattice distortions and concomitant superlattice reflections. Recently is was observed that TbB6 shows an additional ordering vector Q = ( 14 41 0) [57]. DyB6 and HoB6 : These heavy RE hexaborides are least investigated. ) and Therefore, in both cases, the CEF ground states of Dy3+ (J = 15 2 Ho3+ (J = 8) can only be conjectured (Table 8.2) from specific heat, magnetization, and elastic constant measurements [58, 59]. In the latter a huge softening of transverse c44 elastic constants is observed for both compounds. This means, the CEF ground state has orbital degeneracy leading to a Curie–Weiss-type quadrupolar (q = 0) susceptibility for  = 5+ -type quadrupoles (Oyz , Ozx , Ox y ) [44]. At the ordering temperature (Table 8.2) these quadrupoles acquire an expectation value as signature of a ferroquadrupolar HO which splits the orbital-degenerate ground state and distorts the lattice. This is the well-known cooperative Jahn–Teller (JT) effect. In this case we can denote THO = TQ in Table 8.2 also as TJT because the driving mechanism is not primarily the intersite coupling (g ) of quadrupoles but rather the linear magnetoelastic coupling (g ) of  = 5+ quadrupoles to the trigonal 5+ -type homogeneous (q = 0) lattice strains 5 = ( yz , zx , x y ). The softening of the symmetry elastic constants c = c44 in HoB6 as a precursor to the JT transition is shown in Fig. 8.5. Unlike in PrB6 and NdB6 it is not arrested

628 Multipolar Order and Excitations in Rare-Earth Boride Kondo Systems

Figure 8.5 (a) Transverse c44 elastic constant softening in HoB6 above the JT transition at TQ (H ). Reproduced from Goto et al. [58]. (b) Comparison for H = 0 data with results from Eq. (8.2). Best fit for γ > 1, therefore HoB6 is a cooperative Jahn–Teller compound. The c44 (t) for FQ case (γ < 1) behaves qualitatively different, AFQ case with γ < 0 (similar to CeB6 ) is shown for comparison. Here t = T /TQ∗ with TQ∗ = 5.1 K; c44 is normalized at tm = 9.8.

by a preceding magnetic transition. It is determined by the TQ dependence of the (q = 0) quadrupolar susceptibility χ according to [44, 60, 61] c (T ) 1 − (g˜ 2 + g )χ (T ) = Q c0 1 − g χ (T ) T − TQ∗ t−1 = ,  Q t − α T −  Q

(8.2)

where g˜ 2 = g2 /(c0 vc ) with c0 and vc denoting the background elastic constant and volume per RE ion, respectively. The monotonic behavior in Fig. 8.5 is dominated by the Curie contribution Q Q ∼(m )2 /T to the quadrupolar χ (T ) which can only come from a degenerate 5 ground state. Then the approximation in Eq. (8.2) Q holds, where TQ∗ = (m )2 (g˜ 2 + g ) is the quadrupolar transition Q temperature, neglecting the effect of excited CEF states (m is the quadrupolar ground state matrix element). It is treated as a fitting parameter for elastic constants and may differ somewhat from the

Overview of RE Boride Compounds 629

Table 8.3 Magnetoelastic JT (g5 ) and quadrupolar (g 5 ) coupling constants from c44 (5 symmetry) elastic constant measurements [48, 58]. Here γ < 0 corresponds to AFQ coupling, 0 < γ < 1 to FQ case, and γ > 1 to dominant JT coupling Compound

g5 [K]

g˜ 25 =

g2

5

0 c44 c

[K]

g 5 [K]∗

γ =

g˜ 2 5 g

Type

5

CeB6

190

0.078

–2.1

–0.037

AFQ

PrB6

200

0.093

–0.16

–0.58

AFQ

NdB6

83

0.016

0.032

0.50

FQ

DyB6

—

—

—

0.70

FQ

HoB6

—

—

—

1.43

JT

In the convention of experimental literature g < 0 and g > 0 correspond to AFQ and FQ intersite coupling, respectively. This is opposite to the conventions for zD = g in Sections 8.3– 5 8.5. ∗

Q

real TQ where the specific-heat jump occurs. Furthermore  = α TQ∗ with α = (1+γ )−1 , γ = g˜ 2 /g , and t = T /TQ∗ denotes the reduced temperature. Here γ is the ratio of magnetoelastic to intersite quadrupolar coupling which determines the qualitative temperature dependence of c (T ) in Fig. 8.5(b). For γ < 1 when quadrupole interactions dominate, the T-dependence is mostly flat and then a sudden softening occurs (FQ). When γ > 1 and magnetoelastic JT interaction dominates, the softening occurs over a large temperature range above TQ∗ . Figure 8.5(b) demonstrates that HoB6 is in the JT driven regime of softening, whereas all other hexaborides (Table 8.3) are in the quadrupolar interaction dominated regime, in particular CeB6 . Thus elastic constant measurements can identify the driving mechanism of quadrupolar order. In the AFQ case (g˜ 2 + g < 0), no softening occurs (in the approximate Eq. (8.2) minus signs will be replaced by plus signs). Under applied field the softening around TQ turns into a minimum that shifts to higher temperature in Fig. 8.5(a). Therefore, TQ (H ) increases with applied field. For DyB6 the zero field behavior is similar (although still γ < 1), but no field dependence of c44 is observed, indicating field-independent TQ up to 8 T. Furthermore an additional AFM phase transition appears at TN = 23 K < TQ , again with the canonical Q = ( 14 41 21 ) ordering wave vector. From the interpretation of thermodynamic measurements [59] it was concluded that the order parameters are carried by a

630 Multipolar Order and Excitations in Rare-Earth Boride Kondo Systems

(1)

8 (0) quartet ground state and a closeby 7 (9 K) doublet. A total splitting of the five multiplets of ∼160 K was proposed although no spectroscopic confirmation of the level scheme exists to date.

ErB6 and TmB6 heavy-rare-earth hexaborides have not been successfully synthesized [23] and may not be stable in the bulk form, presumably due to the small radius of their heavy 3+ ions. However, Yoshizawa et al. recently presented the evidence that they could stabilize the non-epitaxial TmB6 phase in thin films deposited by molecular beam epitaxy [62].

YbB6 is the last in the hexaboride series and has been one of the most controversial. For a while it was thought it might be a topologically nontrivial material similar to SmB6 , but recent ARPES experiments [63] for the nonpolar [110] surface have established a different picture: The binding energy of the 4f 7/2 state is quite large, about 1 eV (Fig. 8.2). Therefore, the stable purely divalent Yb2+ 4f ground state is realized, like Eu2+ in EuB6 . Hence there is no band crossing with 5d states. The latter exhibit a semiconducting gap ∼0.3 eV with respect to the lower B 2p states. Therefore, the electronic structure (schematically shown in Fig. 8.2) is reminiscent of EuB6 except that there is no magnetic order in YbB6 and hence no overlap of spin-split bands, consequently it stays semiconducting. Under pressure, however, the 2p and 5d bands overlap, transforming YbB6 into a slightly mixed-valent semimetal [63]. The ambient-pressure semiconductor may exhibit band bending effects and therefore 2D confined surface states can exist, which has led to previous misguided conclusions on the electronic structure. After this brief survey of RB6 materials with stable magnetic moments, we turn now to species with larger 5d-4f hybridization which show either Kondo lattice heavy-fermion behavior with hidden multipolar order like CeB6 and its La-diluted alloys or are Kondo insulators like YbB12 or strongly non-integer mixed-valent semiconductors with topological order like SmB6 .

Multipolar Hidden Order in CeB6 in the Localized 4f Scenario 631

8.3 Multipolar Hidden Order in CeB6 in the Localized 4f Scenario The high degeneracy and strong Coulomb repulsion of f electrons in lanthanide and actinide compounds can lead to exotic quantum matter states at low temperatures [1,17,65,66]. The hybridization of valence (conduction) electrons with strongly correlated f electrons may result in the formation of heavy-fermion metals with quasiparticles that have large effective masses and opening of hybridization gaps that can lead to a Kondo insulator state. This is most frequently observed in intermetallic Ce or Yb compounds which commonly have one 4f electron or hole state with orbital energy f not too far below the Fermi level and having considerable hybridization with conduction bands. Furthermore, at even lower temperatures, broken symmetry phases (usually magnetic or superconducting) appear,

Figure 8.6 Low-field phase diagram of CeB6 as obtained by tracking elastic constant anomalies. Ordering wave vectors are AFQ (II): Q = ( 21 21 12 ); AFM double-q (III) Q1 = ( 14 14 12 ); Q2 = ( 41 14 12 ) [structure of Fig. 8.4(a)]. Below the lowest line H < 5 kOe, domain formation occurs (10 kOe = 1 T). Adapted from Nakamura et al. [64].

632 Multipolar Order and Excitations in Rare-Earth Boride Kondo Systems

driven by residual quasiparticle interactions [66, 67]. Of particular interest are the HO phases introduced before. By various combinations of experimental methods (Section 8.1), the most detailed understanding of HO has been achieved for the cubic heavy-fermion compound CeB6 . It exhibits a second order transition at TQ = 3.3 K from paramagnetic phase I into a HO ´ phase II and then into coexisting AFM phase III at the lower Neel temperature TN = 2.3 K (Fig. 8.6). In zero field the HO phase II does not lead to any new Bragg peaks in x-ray or neutron diffraction. The field dependence of TQ (H ) has a large positive slope leading to an exceptional increase up to TQ = 10 K at H = 35 T. The compound is a prominent model system for HO because of the simplicity of 4f states in that case. The Oh symmetry (Fig. 8.1) leads to a cubic CEF which splits the six J = 52 Ce3+ (4f 1 ) states into a 8 ground state quartet and highly excited 7 doublet at ∼530 K [36, 68–70] (see Figs. 8.3 and 10.4), which may be neglected for all low-temperature phenomena. The quartet ground state is explicitly given by |+ ↑ = |+ ↓ =

 

5 | 6

+ 52  +

5 | 6

5  2

−

+

 

1 | 6

− 32 ;

|− ↑ = |+ 12 ,

1 | 6

3 ; 2

|− 12 .

(8.3) +

|− ↓ =

This may be thought of consisting of two orbitally inequivalent Kramers doublets with symmetry 7 (left part) and 6 (right part) that are forced into one quartet representation by the cubic symmetry. Therefore, it is suggestive to interpret σz = ↑, ↓ as Kramers pseudo-spin of each doublet and τz = ± as orbital pseudo-spin that distinguishes the two doublets [71]. This may be formalized by introducing the representations for the two pseudospins τ and σ defined by τ=

1 † 1 † fτ σ ρτ, τ  fτ  σ ; σ = f ρσ, σ  fτ σ  . 2 τ τ σ 2 τσσ τσ

(8.4)

Here ρ = (ρx , ρ y , ρz ) denotes the set of Pauli matrices. The fτ†σ one-electron fermion operators create the CEF states in Eq. (8.3) according to |τ σ  = fτ†σ |0 with |0 denoting the empty 4f 0 state.

Multipolar Hidden Order in CeB6 in the Localized 4f Scenario 633

8.3.1 Pseudospin Representation of 8 -Quartet Multipoles The 4f electrons with orbital angular momentum l = 3 are arranged in a shell with orbital, spin and total angular momenta (L, S, J ) according to Hund’s rules. For Ce3+ (4f 1 ) and Yb3+ (4f 13 ) they are equal to single electron or hole quantum numbers, respectively. The charge density and moment density operators of the 4f shell may be expanded in terms of multipole basis functions consisting of polynomials with rank p ≤ 2l associated with specific representations of the RE site symmetry. Their expectation values in a given 4f state (e.g., the CEF ground state) correspond to the classical electrostatic and magnetostatic multipoles, as discussed extensively in [1]. With the help of the Wigner-Eckhard theorem for the states with total angular momentum J, the multipole operators of rank p may be expressed as combinations of polynomials in P p (J x , J y , J z ) of rank p belonging to cubic representation (γ ) by using the Stevens operator technique [31]. Some of these operators up to rank 3 (octupoles) are listed in Table 8.4 and the symmetry

Table 8.4 Representations of 9 of the 15 multipoles of 8 quartet: Stevens notation using total angular momentum J components or components of pseudospins σ , τ. The components are the  n of the total angular momentum n λ X where coefficients λ can be read off linear combination J α = n α α n in the last column. In the last row symmetrization (summation over all permutations of x, y, z) is denoted by a bar Oh multipole∗ (degeneracy) 4− (3)

Rank p 1 (d)

3+ (2)

2 (q)

5+ (3)

2 (q)

2− (1)

3 (o)

∗

Stevens notation J α , (α = x, y, z) Jx Jy Jz 2 O20 = 12 (2J − J x2 − J y2 ) z √ 3 2 O2 =√ 2 (J x2 − J y2 ) Oyz = √23 (J y J z + J z J y ) Ozx = √23 (J z J x + J x J z ) Ox y = 23 (J√x J y + J y J x ) Tx yz = 615 J x J y J z

d = dipole, q = quadrupole, o = octupole.

Pseudospin form σ α , τα √ 7 [σ + 27 (−τz σx + √3τx σx )] 6 x 2 7 [σ y + 7 (−τz σ y − 3τx σ y )] 6 7 [σ + 27 (2τz σz )] 6 z τz = 18 O20 τx = 81 O22 τ y σx = 14 Oyz τ y σ y = 14 Ozx τ y σz =√ 14 Ox y τ y = 455 Tx yz

634 Multipolar Order and Excitations in Rare-Earth Boride Kondo Systems

Jz

Txyz

O20

O xy

Figure 8.7 Selected multipoles of 8 quartet (degeneracy in parentheses): Dipolar J z (4− (3), rank 1), quadrupolar O20 (3+ (2), rank 2), Ox y (5+ (3), rank 2) and octupolar Tx yz (2− (1), rank 3). Red/blue denote opposite signs of 4f spin density and green/magenta opposite signs of 4f charge densities.

of the corresponding real-space tesseral harmonics [31] in (x, y, z) cartesian coordinates is shown in Fig. 8.7. If we restrict further to the 8 ground state in CeB6 , the polynomials P p (J x , J y , J z ) may be mapped to the pseudospin algebra defined above by comparing their matrix elements within the quartet [11, 71]. The complete set of 8 multipole operators in pseudospin basis is given by {X n } = {σα , τα , σα τβ }

(8.5)

with n = 1 − 15 and α, β = x, y, z. They constitute a basis set for the fifteen multipole moments, acting in the space of 8 quartet states. The multipoles transforming as cubic representations in Table 8.4  are then generally linear combinations of the X n , e.g., J α = n λnα X n where the λnα may be read off from Table 8.4.

8.3.2 Multipole Interaction Model and Symmetry Breakings In the intermetallic RE compounds, the small hybridization with conduction electrons leads to effective intersite interactions between the multipoles on neighboring lattice sites. Those between the dipoles (rank 1) are commonly known as RKKY interactions, but the concept may also be extended to higher-rank multipoles [37, 38, 42] (Section 8.1.2). In the same way as the RKKY terms lead to magnetic ordering of dipolar moments J at low temperature, they may also induce HO of multipoles with rank p > 1. The pseudo-spin

Multipolar Hidden Order in CeB6 in the Localized 4f Scenario 635

representation for CeB6 has been proposed [71] and investigated in detail with respect to possible multipolar HO and excitations [11, 72, 73]. A model Hamiltonian describing the effective intersite coupling of 8 multipoles may be written as  [(τi · τ j ) + (σi · σ j ) + 4(τi · τ j )(σi · σ j ) + (8.6) H=D

i j  y y

y y

+ Q 4τi τ j (σi · σ j ) + O τi τ j ] −

 7 2 gμB (σi + ηi ) · H, 6 7 i

where we√introduced the √ multipole vector (cf. Table 8.4) η = (−τz σx + 3τx σx , −τz σ y − 3τx σ y , 2τz σz ) as an abbreviation. The first three terms describe a SU(4) ‘supersymmetric’ n.n. intersite interaction on the simple cubic lattice (sites i, j ) of strength D (coordination z = 6) which has no bias for any of the fifteen multipoles as primary order parameter. The following two terms express the symmetry breaking that favors quadrupolar or octupolar order depending on the size of the parameters  = (Q , O ). They correspond to 5+ -type quadrupoles and 2− type octupole, respectively. This preference is concluded from the experimental evidence discussed below and from derivation of the effective H from a more fundamental Anderson-type Hamiltonian [41, 42]. The ± sign denotes even/odd behavior under time reversal, i.e., the quadrupole corresponds to a charge and the octupole to a magnetic moment distribution (Fig. 8.7), in both cases with zero net charge or moment and therefore “hidden”. The last term is the Zeeman energy in pseudospin representation.

8.3.3 Experimental Identification of Multipolar Order Parameters In this section we present to some detail various experimental evidence to unravel the nature of HO in CeB6 , i.e., to identify which multipoles appear as order parameters below the transition temperature TQ . Firstly it was observed that on approaching TQ from above certain elastic constants exhibit typical small anomalies, although no real softening [48] (cf. Fig. 8.5). This already indicated that the primary order should be of the quadrupolar kind but not of the ferro-type. Therefore, the starting point of the model in Eq. (8.7)

636 Multipolar Order and Excitations in Rare-Earth Boride Kondo Systems

is appropriate. More direct evidence for the nature of HO comes from the following investigations:

a) Anisotropy and slope of critical field curves The most striking property of TQ (H ) is a very large positive slope of (dTQ (H )/dH )0 = 0.84 K/T (Fig. 8.8). To understand this exceptional B-T HO phase boundary, a mean-field analysis of the model is required [11]. Since we restrict to n.n. interactions and the quadrupolar order should not be of ferro type, we start from a twosublattice s = A, B, (¯s = B, A) structure for the mean-field version of Eq. (8.7):  hns Xˆ ins¯ − N E (h), (8.7) H mf = − s, i ∈¯s

where hns = hn − 2zD(n)xsn ;

E (h) = 2Dzxa · · xb ,

and xs = Xs  is the mean-field value of the multipole vector X = {X n }. It is also useful to define staggered xs = 12 (x A − x B ) and uniform x f = 12 (x A + x B ) order parameters. Furthermore we

Figure 8.8 (a) High-field anisotropy of critical fields of AFQ phase II for symmetry directions using 1/d expansion and compared to moderate field experiments (circles). At low fields critical field curves become isotropic in this method. Reproduced from Shiina et al. [74]. (b) Experimental [001] high-field results. Inset shows quadratic fit that suggests a maximum critical field H c (0) = 80 T. Reproduced from Goodrich et al. [75].

Multipolar Hidden Order in CeB6 in the Localized 4f Scenario 637

define the interaction model by setting (n, n ) = (n)δn, n with (5) = 1 + O (2− octupole), (8 − 10) = 1 + Q (5+ quadrupole) and (n) = 1 otherwise (all other multipoles). This singles out 5+ and 2− as preferred HO parameters in accordance with experiments discussed below. The components of the field vector h = (hn , n = 1 − 15) can be read off by comparison with the last term in Eq. (8.7). The mean-field Hamiltonian for A, B is a 4 × 4 matrix in the quartet space. Its diagonalization leads to new split eigenstates (Fig. 8.15) from which the free energy may be obtained and minimized. For any field strength and direction, the order parameters are then obtained. By the above choice of interaction, the primary hidden order parameter is the threefold degenerate 5+ (Oyz , Ozx , Ox y )s ≡ 4(τ y σx , τ y σ y , τ y σz )s at the staggered wave vector Q = ( 12 21 21 ) r.l.u. The field dependent transition temperature TQ (H ) is then obtained from the vanishing AFQ order, it is shown in Fig. 8.9. The transition temperature has a reentrant field dependence for all field directions. The strong increase of TQ (H ) in Fig. 8.9 up to intermediate fields has a simple origin in this model: For finite field, the octupolar staggered order is rapidly induced as secondary order parameter

Figure 8.9 Mean-field TQ with (Q , O ) = (0.2, 0.0) for magnetic fields along high-symmetry axes (from [76]). The field direction along [110] is given by the dashed line. The field and temperature are scaled by T0 (= 2Dz = 4.74 K), the zero-field TQ for Q = O = 0.

638 Multipolar Order and Excitations in Rare-Earth Boride Kondo Systems

Figure 8.10 (a) Staggered primary (quadrupolar qs ) order parameter at a fixed temperature T = 0.5 as function of the magnetic fields along highsymmetry axes. qs is scaled with the saturated value. The same scale for the field and the temperature is used as in Fig. 8.9. (b) Secondary (octupolar os and dipolar ds ) order parameter as a function of the magnetic fields (from [76]). os is scaled with the saturated value, whereas ds is not scaled. The dashed lines in both figures are given by [110] field. For [001] field ds is not induced. See also Table 8.5 for the detailed components.

to its saturation value (Fig. 8.10). This stabilizes the AFQ phase and therefore leads to a considerable increase in TQ (H ). It should be noted that TQ is affected also by the direction of the field; TQ [111] and TQ [110] are much more enhanced than TQ [001] in high fields, originating from the anisotropic magnetization in the 8 basis. This characteristic anisotropy is actually observed in the La-diluted system [77, 78]. On the other hand, the mean-field treatment has a deficiency as well: At zero field, due to the threefold degeneracy of the 5+ order parameter, fluctuations will be important and suppress exp TQmf (0) to the value TQ (0) = 3.3 K. Therefore, the observed exp increase of TQ (H ) starting from the zero-field value up to the exp maximum TQ (H max ) = 10 K at H max = 35 T will be even larger than predicted by the mean-field theory. This is shown in the experimental high-field phase diagram of Fig. 8.8(b). In fact, one can easily show that there is no chance for the mean-field TQ (H ) to go beyond twice of the zero-field TQ , irrespective of (Q , O ). To improve the situation the contribution of thermal multipole fluctuations have been considered in low-field [79] and high-field case [74] using a 1/d expansion in the spatial dimension d. The results in Fig. 8.8(a) which globally compares better with the experimental curve in

Multipolar Hidden Order in CeB6 in the Localized 4f Scenario 639

Fig. 8.8(b). However, in such expansion around the isotropic infinitedimensional limit the low-field anisotropy information of TQ (H) is lost.

b) Neutron diffraction in an applied magnetic field This method gave the first clue on the symmetry of the quadrupolar HO [3, 80]. Application of magnetic field along a symmetry direction, i.e., [001], [110], [111] reduces the symmetry group to lower than cubic. This has two effects: (i) It selects a coherent superposition of the threefold degenerate primary HO representation 5+ that depends on field direction (the third row in Table 8.5 corresponds ¯ to the experimental choice [110]). For a general field direction with unit vector (αβγ ), the primary AFQ HO corresponds to α Oyz + β Ozx + γ Ox y (see also the discussion in Section 7.3.1 on page 584). (ii) Previously different (Oh ) representations may become mixed for finite field, leading to an “induced” secondary order parameter with the same wave vector Q whose amplitude becomes nonzero for finite field, in particular the dipole J and octupole Tx yz . The homogeneous applied field breaks time reversal but preserves translational symmetry. Therefore, although the order parameter representations become mixed and further secondary components with opposite time-reversal symmetry are induced, their wave vector is identical to that of the  primary order. Starting from the  primary AFQ 5+ , Q = ( 12 21 21 ) , Table 8.5 lists the possible induced order for field along the high-symmetry directions. The octupole Table 8.5 Primary quadrupolar (q) 5+ order induces secondary (d,o) order parameters of odd time-reversal symmetry, depending on the direction of H. All order parameters are staggered with wave vector Q ( 21 12 12 ). Secondary induced moments (d) appear as AFM Bragg peaks at Q H direction [001] [110] ¯ [110] [111]

Primary∗ 5+ (q) Ox y Oyz + Ozx Oyz − Ozx Oyz + Ozx + Ox y

Induced 4− (d) — Jz Jz Jx + Jy + Jz

Induced 2− (o) Tx yz Tx yz Tx yz Tx yz

Symmetry C 4v C 2v C 2v C 3v

∗ The field selects a combination from the triply degenerate 5+ manifold. For general field direction with unit vector (αβγ ) the linear combination α Oyz + β Ozx + γ Ox y is selected.

640 Multipolar Order and Excitations in Rare-Earth Boride Kondo Systems

Tx yz is induced for all directions which explains the near isotropic enhancement of TQ (H). In particular it is seen that for H along ¯ [110] a staggered dipolar moment along [001] will be induced. This configuration corresponds to the ND experiment in [3, 80] (third row in Table 8.5) and leads to the observation of induced magnetic Bragg peaks from J z at Q . Then, in reverse this observation may be interpreted as evidence for the underlying primary quadrupolar Oyz − Ozx order.

c) NMR experiments The dependence of NMR resonance lines of nuclear moments on applied field strength and direction contains important information on the underlying polarization of electronic magnetic moments, imprinted by the hyperfine interaction between the two types of moments. In this way analysis of NMR splittings as function of field strength and angle was used to infer the magnetic structure of induced moments in phase II of CeB6 [10]. Surprisingly, the deduced structure did not agree with the structure obtained from ND results in a magnetic field [3, 80]. In particular NMR lines of the 11 B nucleus (site 3 in Figs. 8.1 and 8.11) show a clear splitting even when the field is oriented along the [001] direction. Now from Table 8.5 one observes that no dipolar moment J z is induced in this case. Assuming the standard hyperfine interaction where nuclear moments I z interact only with 4f magnetic dipole moments J z , one must conclude that there should be no NMR splitting for H [001] under the assumption of an underlying AFQ structure as determined by ND, in clear contradiction to the observation [10]. This discrepancy was solved by Shiina et al. [11, 76] who showed that the local symmetry at the boron sites allows for a more general hyperfine interaction that couples the nuclear spin I not only to the 4f dipolar moment but also to the octupolar moments. A simplified version1 of the hyperfine Hamiltonian at the inequivalent 11 B sites (1, 3 in Fig. 8.1) is then given by   H 1hf = a1 I x T˜ x yz − b1 I y J˜ z (Q ) + I z J˜ y (Q ) ,   (8.8) H 3hf = a3 I z T˜ x yz − b3 I x J˜ y (Q ) + I y J˜ x (Q ) . 1 There are further contributions due to other induced octupoles [11, 76].

Multipolar Hidden Order in CeB6 in the Localized 4f Scenario 641

11 Figure 8.11 NMR splittings hf B sites 1, 3 (see 1, 3 (θ) at inequivalent unit cell). Circles are experimental data from Ref. [10]. Solid line: model calculation with Eq. (8.8). Dashed line: octupolar part only. The values are normalized to hf 3 (0) (from [76]).

Here the tilde denotes operators normalized to their maximum value, and a1, 3 , b1, 3 are hyperfine coupling constants of 4f octupole and dipole moments for the two 11 B sites, respectively. The field is rotated in the diagonal plane containing [001], [111], [110] axes with θ denoting the angle from [001]. Since the 4f Zeeman energy scale is much larger than the hyperfine energies, i.e., g J μB H  a1, 3 , b1, 3 , the nuclear spins may simply be replaced by a classical vector that rotates with the field: √ I I = √ (sin θ, sin θ, 2 cos θ). (8.9) 2 Inserting this in Eq. (8.8) and using the mean-field solution for

J x , J y , and Tx yz  leads to field-angle dependent hyperfine 11 B sites that are shown in Fig. 8.11. splittings hf 1, 3 of inequivalent does not vanish for θ = 0 due to the Most importantly the hf 3 octupolar contribution in Eq. (8.8), in agreement with experiment. This resolves the discrepancy with ND results. In fact the two methods are complementary: while ND determined the underlying AFQ structure via the induced magnetic dipoles, NMR identifies the existence of an induced strong octupolar component that was indirectly also inferred from the large positive slope of the critical field of phase II.

642 Multipolar Order and Excitations in Rare-Earth Boride Kondo Systems

d) Resonant x-ray diffraction (RXD) results Although the previous methods concluded the existence and symmetry of HO from indirect evidence and its analysis, it would be reassuring to find direct evidence for quadrupoles and octupoles. In fact the more recent method of resonant x-ray diffraction is a useful new method to observe multipoles up to fourth rank [81] directly. In CeB6 this has been carried out [6, 82] using the signals from (optical) dipolar E 1 (2 p3/2 → 5d) and (optical) quadrupolar E 2 (2 p3/2 → 4f) resonance transitions around the L3 absorption edge (see also Chapter 7). The transitions at ω1 = 5724 eV and at ω2 = 5718 eV differ by ω = 6 eV due to the larger binding energy of 4f states and their line shapes overlap. The total intensity is given by I (ω, H ) = |F E 1 (ω, H ) + F E 2 (ω, H )|2 , where F E 1 , F E 2 are the complex amplitudes for each process and H is the applied ¯ magnetic field (along the [111] direction). These amplitudes contain contributions from electronic multipoles up to rank 2 (E 1) and up to rank 4 (E 2). To disentangle them, the magnetic field reversal is

Figure 8.12 Magnetic field dependence of the primary quadrupole HO ¯

Oyz − Ozx  and induced secondary dipolar J z  and octupolar Tx yz  for [110] field direction. Symbols: RXD, dashed line: from ND experiments, solid lines: guide to the eye. The momentum transfer is ( 23 23 12 ) corresponding to the HO vector Q . Reproduced from Matsumura et al. [82].

Multipolar Hidden Order in CeB6 in the Localized 4f Scenario 643

an essential tool because the even and odd rank contributions in F E 1 , F E 2 behave even and odd under field reversal. Let us define the average and difference intensities with respect to field reversal H → −H by Iav (ω, H ) = 12 [I (ω, H ) + I (ω, −H )] and I (ω, H ) = 1 [I (ω, H ) − I (ω, −H )]. From this approximate multipole order 2 parameters may be extracted as [82] primary (q): induced (d): induced (o):

Oyz − Ozx  H ∼



Iav (ω1 , H ); 

J z  H ∼ I (ω1 )/ Iav (ω1 , H ); 

Tx yz  H ∼ I (ω2 )/ Iav (ω1 , H ).

(8.10)

They are shown in Fig. 8.12. The induced dipole agrees with the ND results (dashed line), and the field dependencies of all moments are qualitatively consistent with the theoretical results in Fig. 8.10. The primary quadrupole Oyz − Ozx  H obtained from RXD in Fig. 8.12 (upper curve) shows considerable H dependence much larger than predicted by the mean-field calculation (Fig. 8.10). Again this is due to the neglect of fluctuations. Just as they suppress the experimental exp value of TQ (0) by a factor of two, they also suppress the size of the primary order parameter by a large factor (∼1.5) as compared to the mean-field prediction. At considerably larger field they approach each other. The octupole has a pronounced convex bending which is a signature of the strong octupolar interaction. In fact, it is confirmed experimentally by investigating the dependence of the octupole-quadrupole ratio on the uniform magnetization. According to the theoretical study, this ratio is quite sensitive to the octupole interaction strength, irrespective of the fluctuation effect [79]. In summary, the conclusion from critical field anisotropy [11], field-induced neutron diffraction [3], NMR results [76], and resonant x-ray scattering [6, 82] indicate that HO may be well described as a primary antiferroquadrupole 5+ order with wave vector Q = ( 12 21 21 ) and a secondary strongly field-induced octupolar 2− order parameter in addition to a smaller induced dipole component, both at the same wave vector Q . Semiquantitative agreement with experiments may be achieved by choosing  = (0.5, 0.5) [73] in the localized multipolar model of Eq. (8.7) and this should be considered as an appropriate set for CeB6 .

644 Multipolar Order and Excitations in Rare-Earth Boride Kondo Systems

8.4 Octupolar HO Phase IV in Diluted Ce1−x Lax B6 In the stoichiometric RB6 compounds, the 4f element may easily be replaced by other rare-earth species. The most interesting case is perhaps the series Ce1−x Lax B6 (0 ≤ x ≤ 1) where the magnetic Ce3+ sublattice is progressively diluted with nonmagnetic La3+ which has no 4f electrons. This will have two main consequences: Firstly the intersite multipole interactions will be progressively weakened [83] suppressing the tendency to multipolar order. Secondly the coherent heavy-fermion quasiparticle bands that exist for x  1 will gradually become site-incoherent and turn into narrow localized Kondo resonance states. This can in fact be directly inferred from the change of resistivity ρ(T ) from correlated metal A + B T 2 behavior to saturated unitary Kondo impurity resistivity at low temperatures [84]. In the present context we focus on the evolution of the hidden order phase diagram with x. When the distance between the 4f multipoles of Ce becomes larger and their interactions are reduced one has to ask how long the AFQ order will last. Simply extrapolating the mean-field solution of the concentrated compound does not give the correct answer. Because of the additional non-Kramers degeneracy of the 8 ground state (expressed by τ) the single-ion Q quadrupolar susceptibility χ+ (T ) [Eq. (8.2)] has a Curie divergence 5 ∼ 1/T for low T [44]. Therefore, the mean-field approach would lead to a quadrupolar phase transition for an arbitrary dilute compound, although TQmf (x) would approach zero for vanishing x. This is not the case and something rather more interesting is observed: The (zero-field) AFQ order vanishes rather rapidly with doping, and by x ≤ 0.8 is already replaced by a different phase IV. Its nature has been investigated as intensely as that of the parent compound.

8.4.1 Phase Diagram and Evidence for Primary Octupolar Order An example of the global low-field phase diagram of Ce1−x Lax B6 (x = 0.23) as obtained from magnetocaloric investigations [85] is shown in Fig. 8.13 together with the x-T phase diagram [86].

Octupolar HO Phase IV in Diluted Ce1−x Lax B6

Similar results were obtained from transport and magnetization experiments [87, 88] and earlier from ultrasonic measurements [89, 90]. These and other macroscopic analyses suggested the following basic properties of phase IV below TIV  1.6 K: Contrary to phase II of CeB6 at TQ , a large specific-heat jump is observed at TIV . This indicates that the degeneracy of localized quartet states is completely lifted, different from AFQ order in phase II where a twofold Kramers degeneracy remains in zero field (Fig. 8.15). Furthermore RXS gave clear evidence for a homogeneous (q = 0) trigonal lattice distortion in phase IV along [111] direction while none was observed in phase II (because the quadrupoles have a staggered order). In addition NMR and μSR experiments show the existence of an internal field below TIV . Therefore, phase IV breaks cubic crystal symmetry as well as time-reversal symmetry. This requires a primary octupole order parameter belonging to 5− as the only plausible candidate [17, 91]. Irrespective of its translational character it will always induce a ferro-type quadrupole as secondary order parameter, already at zero field. This explains nicely the strong softening of elastic constants due to strain-quadrupole coupling

Figure 8.13 (a) B-T phase diagram of Ce1−x Lax B6 for x = 0.23 which shows the new antiferro-octupolar phase IV corner below TIV = 1.6 K. AFQ phase II is no longer present for zero field. Reproduced from Jang et al. [85]. (b) x-T phase diagram for phases I (para), phase II (AFQ), phase III (AFM) and phase IV (AFO). Reproduced from Tayama et al. [86].

645

646 Multipolar Order and Excitations in Rare-Earth Boride Kondo Systems

Table 8.6 Octupolar order parameters for the phase IV, which is of antiferro-type with wave vector Q . Secondary ferro-type quadrupolar order is induced already in zero field. The bar denotes symmetrization (summation over all permutations of x, y, z)

Oh multipole Rank (degeneracy) p 5− (3)

Stevens notation J α (α = x, y, z)

Pseudospin form σα , τα √ √ 15 30 β 2 2 √ 3 (o) Tx = 6 (J x J y − J z J x ) 5 (− 3τz σx − τx σx ) √ √ T yβ = 615 (J y J z2 − J x2 J y ) √305 (− 3τz σ y − τx σ y ) Tzβ

=

√ 15 (J z J x2 6

− J y2 J z )

30 √ 2τx σz 5

Induced quadrupole rank 2, 5+ (3) Oyz Ozx Ox y

immediately below TIV [91]. The octupole order parameter of this symmetry and its induced quadrupoles are given in Table 8.6. In the supersymmetric part of the multipole intersite interaction, Eq. (8.7) implicitly includes an isotropic term ∼DTβ · Tβ . One simple way to reproduce the octupolar order in Ce1−x Lax B6 is an appropriate enhancement of the octupole term in the interaction as D → D(1+ O ). Some variants of the model have been studied in the literature and provided a consistent mean-field picture to interpret the complex experimental results in phase IV [88, 91]. However, it is not quite clear why the octupole interaction is selectively enhanced in the La-doped system. Another possible origin is the effect of random distribution of La that produces an additional CEF potential lower than cubic at Ce sites. Since the potential removes the nonKramers degeneracy in the 8 state, the randomness is expected to suppress the AFQ order more seriously. As a result, the AFO state surviving within the Kramers degeneracy gains a chance to overcome the quenched AFQ state [88, 92]. In fact, the existence of strong spatial disorder in phase IV of Ce1−x Lax B6 is inferred from broadening in the NMR spectra [93]. Anyway the AFO mean-field ground state will be Tβ  = T5− (±1, ±1, ±1) corresponding to different domains. One of them, (1, 1, 1), is illustrated in Fig. 8.14(a). If we pick this domain, the corresponding homogeneous induced 5+ (ferro-)quadrupole will be O = O5+ (1, 1, 1). Due to the coupling to homogeneous strains 5 = ( yz , zx , x y ), the crystal will distort

Octupolar HO Phase IV in Diluted Ce1−x Lax B6

Figure 8.14 (a) A schematic plot of the symmetry of T β octupole for the (111) domain. (b) Domain averaged magnetic form factor in the octupolarordered phase as function of the momentum transfer |κ|/4π = sin θ/λ (from [12]). Symbols (lmn) corresponds to κ = (l, m, n)/2 and solid lines to high-symmetry directions. Underbars in the indices are defined as n = 10 + n.

with a trigonal strain 5 (1, 1, 1) in accordance with conventional XD results [94]. While these macroscopic symmetry considerations are consistent, a direct proof by microscopic probes seems necessary. This was provided by analysis of angular dependent RXS [7, 95, 96] and large momentum transfer neutron diffraction [4, 12]. Firstly, the former shows clearly that in phase IV the resonant scattering occurs at Bragg points corresponding to an AFO propagation vector Q = ( 12 21 21 ), the same as in the AFQ phase II. Furthermore, the dependence of scattered intensity I (φ) on the azimuthal angle in the x-ray scattering plane shows a sixfold and threefold oscillation with φ in the full circle for E 2(σ σ ) and E 2(σ π) scattering channels, respectively. This can consistently be interpreted with an underlying octupolar-type hidden order [95,97]. Secondly, although conventional neutron diffraction at low momentum transfer can only identify dipolar order, the scattering at very large momentum transfer is sensitive to higher-order (odd rank) multipoles. Shiina et al. [12] have calculated the expected form factor for large momentum transfer κ = k − k of scattered neutrons for an underlying 5− AFO order. It is shown in Fig. 8.14(b). The fact that the form factor vanishes for low κ and shows a

647

648 Multipolar Order and Excitations in Rare-Earth Boride Kondo Systems

maximum at large κ is a typical signature of multipolar order. The experiments have been performed [4] and it was indeed observed on a few reflections that intensity increases with momentum transfer, in accordance with theoretical predictions. The anisotropy of the intensity for the momentum transfer can provide further information on the form of the octupole, and its experimental identification is left for a future study. Thus Ce1−x Lax B6 is one of the few confirmed cases of primary higher-rank r ≥ 3 multipolar order. The octupolar order in Ce1−x Lax B6 persists down to almost x ≈ 0.5 [85]. For even lower x the compounds are disordered at all temperature in zero fields and exhibit a Kondo impurity behavior. Another example for octupole ordering is NpO2 [2, 98], where the same component of octupole T β is believed to order with the triple-q ordering vector. Note also that higher magnetic and nonmagnetic multipole states are proposed to resolve the nature of the famous HO phase of URu2 Si2 [13–15], but still remain controversial in that case.

8.5 Collective Excitations in the AFQ Hidden-Order Phase II of CeB6 Materials with CEF-split 4f states show collective dispersive magnetic excitations, termed “magnetic excitons” [99] already in the paramagnetic phase. Their analysis leads to important information to build an exchange model [100]. A magnetic phase transition may be preceded by a softening of excitons at the ordering wave vector [99]. Below the transition their dispersion is modified due to the molecular field and additional collective Goldstone spin-wave modes appear describing the order parameter dynamics. In CeB6 we restrict ourselves to the fully degenerate 8 (above TQ ) and ignore the very high-energy 8 − 7 excitation. Then we have only to consider the quasielastic excitations of the 8 quartet. For T < TQ in the AFQ phase and by application of external fields, the quartet splits as schematically shown in Fig. 8.15, and the intersite coupling will then again lead to collective dispersive modes, this time on the energy scale of the 8 splittings. However, in the

Collective Excitations in the AFQ Hidden-Order Phase II of CeB6

Δij

E4 E2

Γ8

E1

P

E3

AFQ

AFQ(H)

Figure 8.15 Schematic splitting of 8 multiplet in phase II into two Kramers doublets by AFQ molecular field and into four nondegenerate states due to action of AFQ, field H, and induced dipole and octupole molecular fields. Splittings are sublattice (s = A, B) dependent. For T > 0, H = 0 all six excitations are possible. For low T only three excitations from the ground state (solid arrows) are important, leading to six dispersive modes in the interacting two-sublattice AFQ state (adapted from [72]).

present case their dispersion will not only be influenced by the dipolar exchange but by all multipolar interactions included in the intersite term of the Hamiltonian in Eq. (8.7). On the other hand their intensity appearing in INS is again be determined only by the dipolar dynamic structure function of these modes because neutrons (at low momentum transfer) do not directly couple to higher multipoles. The magnetic excitation spectrum of the AFQ phase II of the model in Eq. (8.7) has been calculated with the complementary generalized Holstein-Primakoff approach [73] or multipolar response function formalism in the RPA approach [72, 101]. Both include the full multipolar basis for calculation of the mode dispersions. For brevity we describe only the latter in this review but give a comparison of results from both methods for a typical case [Fig. 8.16(b)]. As a first step (Sec 8.3.3) one has to calculate the effective molecular fields X nA  ± X nB  (uniform and staggered) of each multipole basis operator which leads to the 8 splitting (Fig. 8.15). Here s = A, B denote the simple cubic sublattices of the antiferro-type HO defined by ordering vector Q .

649

650 Multipolar Order and Excitations in Rare-Earth Boride Kondo Systems

8.5.1 Generalized Multipolar RPA Method The CEF states are mixed by the molecular fields into new eigenstates |νsi with energies E νs at every sublattice site (s, i ) as si shown in Fig. 8.15. In terms of their standard basis operators aνμ = |νsi μs|i (ν, μ = 1, 4), the mean-field approximation to Eq. (8.7) is given by   1    si si s j E νs aνν − (Msνμ · Dss  · Msν  μ )aνμ aν  μ , (8.11) H = 2 i j ss  νν  , μμ ν, si where M is an n-component vector (n = 1 − 15) of matrix elements ns for the multipole operators defined by Mνμ = ν, s|X ins |μ, s, and the inter-sublattice multipole n × n diagonal interaction matrix is D A B (q) = D B A (q) = −2zDγq . Here gives the relative interaction strengths of multipoles (Section 8.3.3), furthermore γq = 13 (cos qx + cos qy + cos qz ). With the thermal occupations of molecular field eigenstates given by nsν = Z s−1 exp(−E νs /T ) and  Z s = μ exp(−E μs /T ), the bare n × n multipolar susceptibility for each sublattice s = A, B may be written as ns ls  Mνμ Mμν s χ0nl (ω) = (nsν − nsμ ). (8.12) s − Es − ω + iγ E μν μ ν νμ The γμν line widths of 8 transitions result from Landau damping due to conduction electrons [101]. The collective response of all 15 8 multipoles for the 2 sublattices is then described by the 30 × 30 RPA susceptibility matrix χ(q, ω) = [1 − χ0 (ω)D(q)]−1 χ0 (ω),

(8.13)

where D consists of two antidiagonal blocks D A B = D B A . The elements of Eq. (8.13) may be used to construct the dipolar moment (J) cartesian (3 × 3) susceptibility matrix according to  ss  λnα λm (8.14) χαβ (q, ω) = β χnm (q, ω), ss  , nm

where λnα

are the coefficients of J α in the pseudo-spin representation (Table 8.4). Then the dynamical dipolar structure function, which is the only one observable in INS, may be written as 1 S(Q, ω; H) = [1 − eω/kT ] π  × [δαβ − Qˆ α Qˆ β ]Imχαβ (q, ω; H). (8.15) αβ

Collective Excitations in the AFQ Hidden-Order Phase II of CeB6

Figure 8.16 Spectral function of magnetic excitations for zero field (a) and field along [001] axis (b) with h = 1 and T = 0.5T0 from HP [73] (dots) and RPA [101] (shading) approach. The momentum vector q is moving along the Brillouin zone (BZ) path ΓXMΓR (here q is normalized to the length of each path segment to achieve equal intervals). In this and all following similar figures the unit of mode energy ω is T0 = 2zD = 0.41 meV (4.74 K). Adapted from Ref. [101].

It depends parametrically on field strength and direction. Here Q = q + K is the total momentum transfer in INS with K denoting a reciprocal lattice vector, and Qˆ = Q/|Q|. The structure function is proportional to the INS intensity and will be discussed for various field strengths and directions below. The calculated dynamical RPA structure function S(q, ω) for CeB6 is shown in Fig. 8.16 for zero and finite field. Here we use model parameters kB T0 = 0.41 meV or T0 = 4.74 K and  = (0.5, 0.5) for CeB6 , also employed in Refs. [73, 101]. Furthermore a dimensionless field strength is defined by h = 0.672H[T]/T0 [K] with physical units for H and T0 (we will also use h = h T0 ). Then h = 1 corresponds to H = 6.97 T. There are locally six excitations between the molecular field- and Zeeman-split 8 states — three from the ground state and three from the thermally excited states (Fig. 8.15), the latter are thermally suppressed except close to TQ . Since there are two sublattices in the AFQ/AFO-type ordered phase, six excitation branches will appear prominently that are mostly visible, e.g., in Fig. 8.16(b) (for h = 0 there are additional degeneracies). Roughly speaking, the six branches can be arranged

651

652 Multipolar Order and Excitations in Rare-Earth Boride Kondo Systems

in two groups: Firstly, two high-energy branches (H) and secondly, four low-energy branches (L), two of them almost degenerate and largely flat. The former are mostly stabilized at higher energy by the octupolar molecular field, while the latter are stabilized by the Zeeman term. When temperature approaches T Qmf (h) from below, while the field remains constant, the high-energy modes collapse due to the reduced octupolar order while the low-energy modes are less affected. Complementary, when temperature is kept constant much below the transition and the field is reduced to zero, the highenergy modes change little and the low-energy modes are shifted downwards [cf. Fig. 8.16(a,b)]. Due to the threefold degenerate 5+ order, a Goldstone mode then appears at the Γ point [Fig. 8.16(a)]. It should be mentioned that the relative field independence of the higher modes is a consequence of the accidental degeneracy Q = O assumed in the model [101]. If we chose O somewhat less than Q , the octupolar order parameter τz Q would rapidly collapse at small fields and the two high-energy modes with it, similar to the behavior when temperature approaches the transition from below at zero field. The mode dispersions in the AFQ phase II of CeB6 from RPA calculations are in excellent agreement with the results from the HP approach, shown as dots in (b), including the intensities of modes as a function of momentum [73, 101]. The latter method was later also extended to the AFM phase III [102]. Experimental evidence for the multipolar mode dispersions in finite fields was found in [103] and [104–106].

8.5.2 Dependence of Mode Energies on Field Strength and Field Angular Rotation The field dependence at constant q is complementary to the standard INS method where the full q-dispersion is determined for a fixed field. In reality the latter may be difficult to carry out due to strong variation of intensity in the BZ. In fact some of the excitations were mostly identified at the symmetry points of the BZ, in particular at Γ and R, but also X , M. Therefore, for a comparison with theoretical results it may be a better strategy to keep the momentum transfer fixed at these symmetry points and vary the field strength and field direction. The mode frequencies are then

Collective Excitations in the AFQ Hidden-Order Phase II of CeB6

w

w

recorded in radial plots in the field rotation plane. This is a change of viewpoint as compared to the previous theoretical investigations [73, 101] which we discuss now in detail. Without magnetic field, the mean-field solution of Eq. (8.7) leads to a transition at Tmf = (1 + Q )T0 with the primary AFQ order parameter. At finite fields a secondary dipole and octupole staggered order will be induced, depending on field direction (Fig. 8.10). Their associated molecular fields split and mix the local CEF energies and states. This information is encoded in the energy-denominator and matrix elements of the RPA susceptibility in Eqs. (8.12, 8.13), and hence the excitation spectrum of Eq. (8.15) depends on order parameters, field strength, and direction. First, we consider the continuous field strength dependence of mode frequencies at two symmetry points shown in Fig. 8.17 for field along the [001] direction. In this case Ox y is selected from the 5+ quadrupolar manifold as the primary order. As mentioned before, the high-energy modes ω  4T0 stabilized by the octupolar molecular field are hardly influenced by the field. On the other hand, the low-energy modes around ω  T0 that are stabilized directly by the Zeeman term show an approximately linear increase of frequency and splitting with the field. At the zone center Γ , the mode with the highest intensity is the Goldstone mode starting at zero frequency, which then increases roughly linearly with field

h

h

Figure 8.17 H [001] field dependence of mode frequencies for the two BZ symmetry points Γ (000) and R( 21 12 12 ). Low-energy modes exhibit roughly linear Zeeman splitting. Field-independent high-energy modes are stabilized by the induced octupolar order parameter [106].

653

654 Multipolar Order and Excitations in Rare-Earth Boride Kondo Systems

Figure 8.18 Field angular anisotropies of multipolar excitation branches at Γ and R points. A polar presentation is used where angle and radius correspond to field direction and mode energy, respectively (ω = 6T0 at black circle). Field strength is h = 2T0 (14 T). For such large-field angular anisotropy is pronounced and mode frequencies (radii) are more distinct. Some mode intensities are interchanged with Γ ↔ R [106].

strength. Experimentally such linear field dependence of excitations at the Γ point has been found both in neutron scattering [107] and electron spin resonance [88, 111, 120, 121] (see also Chapter 10), although the situation is complicated due to the additional AFM order at a different wave vector, which is not included in the present model. The dependence of multipolar excitations on continuous field rotation at the Γ and R points [106] is presented in the polar graphs of Fig. 8.18 for field strength h = 2 (H  14 T). In these figures the radial coordinate represents the frequency of the excitation modes in the dipolar INS structure function of Eq. (8.15). The angular coordinate defines the field rotation angle between the various cubic axes [100], [110], [111] and their equivalents indicated at the outer boundary of the polar plots. The rotation is continuous and closed, however, in general not coplanar. The rotation sequence is chosen to facilitate comparison with present experiments on CeB6 carried out in the same geometry [106]. The field angular mode anisotropies show various signatures worthwhile to look for in experiment. Firstly large fields increase the anisotropy in the field angular variation of mode frequencies

Resonant Magnetic Excitations in the Itinerant CeB6 Kondo Lattice 655

(Fig. 8.18). Furthermore the low-energy (L) mode frequencies (small polar radii) change more rapidly (expand) with increasing field strength than for the high-energy modes (outer radii) in accordance with the previous discussion. We also note in both figures that the anisotropy pattern at the Γ and R points are quite similar. This is expected since both points have full cubic symmetry preserved. However, remarkably the relative intensity of low-energy (small radii) and high-energy (large radii) modes is interchanged when going from Γ to R and vice versa. At X and M the analogous behavior is observed. These polar anisotropy plots of multipolar excitations in CeB6 at BZ symmetry points present a compiled information on mode positions and intensities that may be very useful for comparing with experimental results and give guidance on where to look for the modes with largest intensity. If the model so far accepted for CeB6 with  = (0.5, 0.5) is reasonable, some features of the field anisotropy plots described above should be identified in future experiments.

8.6 Resonant Magnetic Excitations in the Itinerant CeB6 Kondo Lattice The 4f states in CeB6 , its ordered phases and excitations have so far been treated in a completely localized 4f picture. At first look this seems justified because the valence of CeB6 (2.95+) obtained from photoemission [112] is very close to integer 3+ corresponding to stable 4f 1 configuration with a binding energy |f | = 2.1 eV. The small deficiency is due to the Kondo resonance state formation above the Fermi level (F = 0) with a narrow width corresponding to a small Kondo temperature TK = 4.5 K [113]. In the lattice this temperature marks the onset of coherent heavy quasiparticle bands with a width and (indirect) hybridization gap of order TK and a corresponding large mass enhancement m∗ /m  20 (γ = 250 mJ/mol K2 ) (Table 8.1). However, the quasiparticle band width given by TK is about the same as the ordering temperature TQ for AFQ hidden order. Therefore it is questionable whether the fully localized approach to the HO phase and its excitations is sufficient. Indeed INS

656 Multipolar Order and Excitations in Rare-Earth Boride Kondo Systems

experiments [104, 114] justify this question. They suggest that CeB6 exhibits a magnetic low-energy mode in zero field that has all the basic features of a collective itinerant spin exciton resonance within the hybridization and hidden-order gaps: It sharply peaks at an energy ωr = 0.5 meV, it is narrowly confined at the simple cubic R point with AFQ wave vector Q , and the temperature dependence of ωr and resonance intensity increase in an order-parameter-like fashion with decreasing temperature. Simultaneously the intensity for ω < ωr is depressed, characteristic for a spin gap formation. Such collective spin exciton modes are ubiquitous within the gap of unconventional superconductors, including high-Tc [115], Fepnictide [116–118], and heavy-fermion [119–121] superconductors. In this case the sign change of the unconventional superconducting gap k+Q = − k at the SC resonance position Q is necessary. It ensures a finite coherence factor (matrix element of moment operator) at the gap threshold which results in a pronounced bound state peak in the collective magnetic response at an energy ωr < 2 0 , where 0 is the amplitude of the sign-changing gap function k .

8.6.1 Heavy Quasiparticle Band Properties in the PAM Since CeB6 is in the normal state one must conjecture that the hybridization gap and the additional gaps introduced by the orderings lead to the necessary singular behavior of the bare magnetic susceptibility to allow for a bound state [121]. This may be described by the mean-field hybridization model of Eq. (8.16) supplemented by the effect of the molecular fields due to AFQ and AFM order which lead to the additional gapping of the mean-field quasiparticle spectrum [122]. First, we briefly outline the constrained meanfield theory of heavy electron bands in the conventional SU(N f ) Anderson model with N f = 4 – fold degenerate conduction band and 4f quartet ground state. The strong on-site Coulomb repulsion of f electrons U f f eliminates double occupancy of f electrons (Ce) or holes (Yb). This constraint is implemented using the auxiliary slave-boson field bi at each site that represents the empty (Ce) or full (Yb) 4f shell. It requires the introduction of a Lagrangian  † † term λ( m fi m fi m +bi bi ) with 1 ≤ m ≤ N f . However, the constraint

Resonant Magnetic Excitations in the Itinerant CeB6 Kondo Lattice 657

is enforced only on the average by performing the mean-field approximation b = bi . The resulting mean-field Hamiltonian is described by    f † † † H= εkc ckm ckm + ε˜ k fkm fkm + V˜ k ckm fkm + h.c. + λ(r 2 − 1). km

(8.16) For the conduction band a simple n.n. tight-binding (TB) model is used which has the same main nesting vector Q as the true Fermi surface of Fig. 8.2. Furthermore λ is a Lagrange parameter introduced to enforce the occupation constraint. It moves the f effective f-electron level ε˜ km close to the Fermi level. Likewise the effective hybridization V˜ k is strongly reduced by the slave-boson mean-field amplitude bi  = r. Together we obtain εkc ,

f f V˜ k2 = r 2 Vk2 = Vk2 (1 − n f ); ε˜ km = εkm + λ.

(8.17)

In the following discussion of magnetic response we use a simplified form of the Anderson model that neglects the k dependence (but not necessarily orbital dependence) of hybridization. This means we are setting V˜ k = V˜ . For our purpose this simplification is adequate, but it cannot always be used, e.g., for the derivation of electronic structure in the pseudogap Kondo insulator CeNiSn [123] or the topological insulators like SmB6 (Section 8.8.2) [124–126]. The single particle type mean-field Hamiltonian may be diagonalized (b)

(a)

KI

( )

E+ (k

E- (k

-

HF

-

Figure 8.19 (a) The hybridized quasiparticle bands around the renormalized f-level ˜ f and (b) corresponding DOS for typical model parameters. Fermi level position for heavy-fermion metal (HF, like CeB6 ) and Kondo insulator (KI, like YbB12 and SmB6 ) is indicated. For KI F is inside the hybridization charge gap c (from [121]).

658 Multipolar Order and Excitations in Rare-Earth Boride Kondo Systems

and then leads to the quasiparticle Hamiltonian  † εkα aα, km aα, km + λ(r 2 − 1), Hmf = i, k, m, α

(8.18)   1 c f f 2 c 2 ˜ = k + ˜ k ± (k − ˜ k ) + 4Vk . 2 Here εk± are the pair (α = ±) of hybridized quasiparticle (aα, km ) bands, each N f -fold degenerate. The indirect gap in Fig. 8.19 has + − the size  of the Kondo temperature: ε0 − εQ  TK with TK = W exp −2/[N f J Nc (0)] . Here W, Nc (0) are conduction band width and DOS, respectively, J = 2V 2 /|ε f | is the on-site exchange constant. The transformation to quasiparticle states is given by εk±

fkm = u+, k a+, km + u−, k a−, km ; ckm = u−, k a+, km − u+, k a−, km . (8.19) with the coefficients defined by

 f f 2u2±, k = 1 ± (kc − ˜ k )/ (kc − ˜ k )2 + 4V˜ k2 .

(8.20)

They appear in the matrix elements in the numerator of the expression for the bare magnetic susceptibilities, possibly together with coherence factors of broken symmetry states (hidden order, magnetic or superconducting).

8.6.2 Collective Spin Exciton Modes In a model for the AFQ and/or AFM ordered phases II and III of CeB6 , further terms must be included which describe schematically the molecular fields due to orbital and (Kramers pseudo-)spin symmetry breaking due to AFQ and AFM order:  † †

Q ( fk, +σ fk, +Q −σ + fk, −σ fk+Q , +σ ) (8.21) H AFQ = k, σ =↑↓

and H AFM =



†

†

Q ( fk, τ ↑ fk+Q, τ ↓ + fk, τ ↓ fk+Q, τ ↑ ).

(8.22)

k, τ =±

Here τ = ± is the pseudo-orbital and σ = ↑↓ the pseudo-spin index of the 8 quartet. For finding the magnetic excitations,  we first require the bare dipolar susceptibility χ0ll (q, t) =

Resonant Magnetic Excitations in the Itinerant CeB6 Kondo Lattice 659



l −θ(t) T jql (t) j−q (0), where jql

=

 kmm 

† l ˆ mm fk+qm M  f km are the

ˆz = physical magnetic dipole operators (l, l = x, y, z) with M zz (7/6)τˆ0 ⊗ σˆ z . Due to cubic symmetry we can restrict to χ0 (q, ω) given by (i ν → ω + i 0+ )  zz α α 2 dω Gˆ 0ss (i ν + ω )Gˆ 0s  s  (ω ), (ρˆ k, q ) (8.23) χ0 (q, ω) ∝ αα  km1 m2

where we abbreviate s = (α, k + q, m1 ) and s  = (α  , k, m2 ). It contains the effect of the modified quasiparticle energies in Green’s  functions Gˆ 0ss and matrix elements ρˆ k,α qα containing the coefficients in Eq. (8.20) reconstructed by the molecular fields of Eqs. (8.21, 8.22). Due to the AFQ and AFM gap opening, the bare magnetic response described by χ0zz (q, ω) is pushed to higher frequencies and the real part is considerably enhanced. The collective RPA susceptibility, due to residual quasiparticle interactions described by Jq , is given by χRPA (q, ω) = [1 − Jq χ0zz (q, ω)]−1 χ0zz (q, ω).

(8.24)

χ0zz

Once the enhanced is large enough due to the influence of order parameters, the denominator of χRPA (q, ω) vanishes and a spin exciton bound state pole develops. This can be seen from Fig. 8.20(a), where below TQ and in particular TN a sharp resonance appears around the AFQ ordering vector q ≈ Q with an energy ωr / c = 0.64 (T → 0). Here c is the indirect hybridization charge gap (Table 8.7) determined by point-contact spectroscopy [127] and apparent in the DOS of Fig. 8.19(b). The momentum dependence Table 8.7 Overview of experimental spin resonance characteristics in heavy-fermion metals and Kondo insulators. Here c denotes the quasiparticle charge gap equal to the hidden-order or hybridization gap (for finite or vanishing THO ), respectively. The resonance appears inside the charge gap (ωr / c < 1) around the HO propagation vector or the characteristic FS vector Q THO [K]

c [meV]

ωr [meV]

ωr

c

Compound CeB6 SmB6 YbB12 URu2 Si2

3.2 — — 17.8

1.3 20 15 4.1

0.5 14 15 1.86

0.39 0.7 ∼1 0.45

Q [r.l.u.] ( 12 12 21 ) ( 12 00) ( 12 12 21 ) (001)

Refs. [114, 122, 127] [35, 128, 129] [130–132] [16, 133, 134]

660 Multipolar Order and Excitations in Rare-Earth Boride Kondo Systems

b (a) Q'

Jq JQ'

Im cRPA t

30

1 AFM

0.6

R AFQ PM

Q

20

0.2 0

0

0.5 TN

TQ

10

0 0.

1 AFM AFQ PM

0.1

0.2

0.3

0.4

0.5

w/t

Figure 8.20 (a) Spin exciton resonance peak in the RPA spectrum in the ordered phase developing at the R point Q = ( 12 12 12 ) (simple cubic zone corner) due to AFQ and AFM gaps (inset shows the temperature dependence). Jq is chosen as Lorentzian around R point (t = 22.4 meV). (b) Contour plot of ImχRPA (q, ω) for CeB6 along the ΓR direction. Localized resonance peak appears at R for ω = ωr and a spin gap develops below (adapted from Ref. [122]). (c) INS results at the experimental resonance frequency in (hhl) plane. Intensity is narrowly confined at the R-point resonance. Reproduced from Friemel et al. [114].

of the spectrum along [111] diagonal ΓR line is presented in Fig. 8.20(b). It demonstrates the confined resonance excitation at ωr and the signature of the spin gap (ω  ωr ), both at the R point. Away from the R point the low-energy spin fluctuations of the metallic state are still present. The complementary experimental CeB6 constant-ω INS intensity plot [114] for q in the (hhl) scattering plane at ω = ωr (0.5 meV) is shown in Fig. 8.20(c). It exhibits the pronounced accumulation of intensity confined narrowly at

Dispersive Doublet Spin Exciton Mode in the Kondo Semiconductor YbB12

the resonance location R corresponding to the peak formation in Fig. 8.20(a). The above discussion of the dynamic response uses the RPA approach for the ordered phases of CeB6 . A theory beyond this approximation for the static response in the paramagnetic phase has been proposed in Ref. [135].

8.7 Dispersive Doublet Spin Exciton Mode in the Kondo Semiconductor YbB12 In CeB6 the resonance is tied to the presence of hidden and AFM order that enhance χ0zz (q, ω) (in experiment it appears only below TN ). This enables the existence of a pole in Eq. (8.24). One might, however, expect that this is not always necessary and that under favorable conditions the resonance may appear already without the support of additional gapping due to order parameters. This case is realized in cubic YbB12 [132]. The compound is a model Kondo semiconductor with equal spin and charge gap of c ∼ 15 meV [130] and without any symmetry breaking. The 4f hole in Yb3+ has a (1) lowest J = 7/2 multiplet which is split by the CEF into a 8 ground state and two closeby doublets. The latter will be treated as another (2) pseudo-quartet 8 . Then the model in Eq. (8.16) must be slightly generalized replacing  f →  f +  and V → V to include the CEF splitting

2 − 1 of the two quartets and in particular their different average  1 2 2 . Therefore we obtain two sets of hybridization V = 12 m |Vm | quasiparticle bands with different size of the hybridization gap. The associated bare susceptibility is then given by     ∓  ±  f E (k + q) − f E  (k)  . (8.25) u±k+q u∓k χ0 (q, ω) = ∓ ± E (k) − E (k + q) −ω   k, ± This implies that the collective RPA susceptibility has contributions from the two sets of quasiparticle bands according to  χRPA (q, ω) = [1 − J (q)χ0 (q, ω)]−1 χ0 (q, ω). (8.26) 

Therefore one obtains two split collective modes with different energies if the resonance condition is fulfilled for each Γ . This

661

662 Multipolar Order and Excitations in Rare-Earth Boride Kondo Systems

Figure 8.21 Calculated dispersion of split resonance in YbB12 along Γ L direction where Lis the fcc ( 12 12 12 ) point (t = 320 meV). Dispersion stretches considerably into the BZ because ωr  c (Table 8.7), from Ref. [132].

is shown in Fig. 8.21 where two distinct resonance peaks appear at the Q = ( 12 21 21 ) fcc L point right on top of the single-particle hybridization gap. (YbB12 has a different cubic structure with a fcc Yb sublattice). This wave vector corresponds to the low-energy indirect interband (±) excitations across the hybridization gap (see illustration in Fig. 8.19). When |q| decreases away from the L point, the interband excitation energies increase from the indirect band gaps ∼TK at Q to the direct band gap ∼2V˜   TK for q → 0. This results in a decrease of Reχ0 (q, ω) for fixed ω and therefore the resonance condition becomes harder to fulfill for both modes. Ultimately at roughly one third into the BZ the intensity of the spin excitons vanishes (Fig. 8.21). The upward dispersion of the split modes is again due to the behavior of Reχ0 (q, ω) whose maximum in q, ω plane shifts to larger energies with decreasing |q|, this also results in larger energies of the two resonances (Fig. 8.21). The model parameters have been adjusted to obtain the hybridization gap, the observed resonance energies [130] (Table 8.7), and the dispersive features. It is interesting to speculate what would happen if they could be tuned by pressure. A decrease of the gap or an increase in J Γ (q) might lead to a soft spin exciton mode at the L

Dispersive Doublet Spin Exciton Mode in the Kondo Semiconductor YbB12

point and result in an antiferromagnetic Kondo insulator. Although the Kondo semiconductor gap in YbB12 is well documented, it shows anomalous transport properties that are not understood to date. Despite the insulating ground state, bulk Shubnikov–de Haas (SdH) oscillations with a large effective mass are observed [136]. Furthermore, thermal conductivity exhibits a linear term in κ/T that is not of electronic origin [137].

Figure 8.22 (a) Temperature dependence of Sm valence from HAXPES experiments, from Ref. [138]. (b) Dichroic spectrum Iq 100 − Iq 111 (NIXS data: black dots) and comparison with simulations for 8 and 7 ground states. A scale factor 0.6 for the Sm3+ part of the MV ground state is applied. Reproduced from Sundermann et al. [33].

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664 Multipolar Order and Excitations in Rare-Earth Boride Kondo Systems

8.8 Magnetic Excitations: Topological State in the Mixed-Valent Semiconductor SmB6 Among the RB6 series, SmB6 is the only strongly mixed-valent (MV) compound due to large hybridization of 4f states and 5d conduction states. This has been known for a long time, and the temperature and pressure dependence of the valence has been determined with various means, in particular XAS [139, 140] and HAXPES [138] [see Fig. 8.22(b)]. At low temperature ( 5 K) the valence is v = 2.55, almost intermediate between the 2+ and 3+ configurations which have ground states J = 0 and J = 52 , respectively. The strong hybridization with conduction electrons which leads to this noninteger valence also generates a hybridization gap of the order

c = 20 meV schematically shown in Fig. 8.19 within the Anderson model description. Because the resistance shows an activated-type increase when temperature decreases below T  50 K, somewhat below the gap temperature scale, it was concluded that SmB6 is a “Kondo insulator” where the Fermi energy F in Fig. 8.19 lies inside the hybridization gap. This designation is now generally used although ‘strongly correlated mixed-valent semiconductor’ would be more precise for SmB6 . Away from the hybridization region, the 5d-like bulk conduction bands have dispersions centered around the X points and are associated with constant-energy surfaces that qualitatively resemble much the 5d Fermi surface in the reference compound LaB6 (Fig. 8.2).

8.8.1 CEF and Collective Magnetic Excitations For describing the 5d-4f hybridization in detail an idea about the Sm3+ J = 52 states and their CEF splitting is necessary. This question was settled by inelastic x-ray investigations [33, 35], which have identified the Sm3+ level scheme 8 (0)−7 (20 meV). The symmetry of the ground state was concluded from x-ray dichroism spectra and a comparison with simulation from a full multiplet calculation [33]. As can be seen in Fig. 8.22, the 8 ground state simulation fits very well to the data. The CEF splitting = 20 meV of the upper 7 level has been determined by RIXS experiments in an indirect manner

Magnetic Excitations 665

via the CEF splitting of an excited 4 G∗5/2 term [35]. Interestingly it is equal to the observed hybridization gap obtained, e.g, from optical conductivity [141]. This explains also why the CEF splitting has not been found in INS experiments. The width of the quasielastic √ line in MV or Kondo compounds is given by [113] (T ) = T ∗ + A T , where T ∗ is of the order of the hybridization gap c . Then naturally when the quasielastic line width is of the same order as the CEF splitting, the latter cannot be identified in INS. However, INS did observe a pronounced spin exciton resonance located narrowly at the X-point at ωr = 14 meV [128, 129] and inside the hybridization gap. This collective magnetic mode is similar to the one found in the Kondo insulator YbB12 and in the HO/AFM state of the heavyfermion metal CeB6 (Table 8.7) discussed in the preceding sections.

8.8.2 SmB6 as a Strongly Correlated Topological Insulator Early experiments on SmB6 had shown a puzzling feature: According to optical conductivity, the hybridization gap should be well developed and lead to a complete suppression of dc transport. Although the resistance shows activated temperature behavior below T  50 K, at even lower temperatures T ≤ 4 K it abruptly saturates at a large but finite value [142, 143]. This behavior was originally attributed to the formation of in-gap impurity bands although the saturation value does not increase with sample quality. However, with the advent of ideas on topological insulators [144, 145] it was realized [125, 126, 146] that the resistance saturation in SmB6 may have a more profound origin, namely being due to conducting topologically protected surface states. The possibility of such states in noninteracting insulators was proposed in the ground-breaking work of Fu and Kane [144, 145]. For a 3D band insulator with spin–orbit coupling in the presence of inversion (I ) and time reversal () symmetry, the energy bands must be twofold (Kramers) degenerate at the time reversal invariant (TRI) points k∗m that are characterized by k∗m = −k∗m + G. Here G is a reciprocal lattice vector. There are m = 1–8 such points in the simple cubic lattice of Fig. 8.23 (appropriate also for SmB6 when we restrict to Sm 4f and 5d states). Due to inversion symmetry the n-th n = ±1. Bloch state at k∗m may be classified by its parity eigenvalue δm

666 Multipolar Order and Excitations in Rare-Earth Boride Kondo Systems

Figure 8.23 3D BZ with TRI points Γ (000), X ( 21 00) etc., M( 21 12 0) etc. and R( 12 12 21 ). Projected 2 BZ [along (001)] TRI points are denoted by bars [125]. The band inversion along Γ X leads to an odd number (3) of 2D isotropic Dirac cones at Γ¯ , X¯ in 2D surface projected BZ. Their dispersion ±v sF |k| is sketched and helical spin polarization with σ × kˆ = ±1 indicated.

The set of products for all occupied bulk states at a given TRI point m δm =

n (εnk∗m BQ = 1.7 T, the AFQ phase is established and stabilized up to very high fields, and for Bc < B < BQ , an intermediate magnetic phase III persists. Substitution with nonmagnetic La in Ce1−x Lax B6 also leads to a suppression of the AFM phase with a critical doping level xc = 0.3. This offers an alternative way to investigate the nature of the resonant peak at the R point by following its behavior with an increase in the La concentration.

Spin Dynamics in Ce1−x Lax B6 and Ce1−x Ndx B6

It has already been shown that the resonance at the R point within the AFM phase exhibits gradual broadening and shifts to lower energies upon warming, until it is transformed into a quasielastic line as soon as the AFM phase is suppressed [63]. It is natural to expect a similar behavior upon the suppression of the AFM order with La doping. The distribution of magnetic spectral weight in momentum space, on the other hand, is much less sensitive to the thermodynamic state of the sample and is mainly determined by the Fermi-surface geometry at the particular doping level. Notable changes in the quasielastic magnetic scattering have been observed within the paramagnetic phase even for the 50% and 75% La-doped samples, as discussed in Section 9.5.2. However, here we will be mostly interested in the magnetic excitation spectrum at low doping levels, x < xc , where the momentum-space redistribution of spectral weight can be neglected.

Figure 9.30 Color map of the raw INS intensity, measured by TOF spectroscopy on the Ce0.77 La0.23 B6 single crystal at T = 60 mK in zero magnetic field. The spectrum is plotted along a polygonal path in momentum space that contains all high-symmetry directions of the cubic Brillouin zone, adapted from Ref. [168]. The horizontal line around 1.25 meV is an experimental artifact.

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752 Neutron-Scattering Studies of Spin Dynamics in Pure and Doped CeB6

We start with presenting the spectrum of collective magnetic excitations in the absence of an external magnetic field. Figure 9.30 shows the low-temperature INS spectrum of Ce0.77 La0.23 B6 , measured by cold-neutron TOF neutron spectroscopy at T = 60 mK (within phase III), to be compared with the respective data in Fig. 9.6 for the parent CeB6 compound. The spectra are qualitatively similar, and the intensity maxima at the R and Γ points are still preserved. The energy of the exciton mode at the R point, ω R , is slightly reduced from 0.48 meV in CeB6 to 0.38 meV in Ce0.77 La0.23 B6 , while

TN 1.2 K TN 2.4 K

T = 0.07 K

T = 0. 4 K

TN

0.32 K

TN 1.66 K

T = 0.42 K

T = 0. 1 K

Figure 9.31 Color maps of the background-corrected INS intensity, S(QAFQ , ω), measured at the R point for (a) CeB6 and (b)–(d) Ce1−x Lax B6 with x = 0.18, 0.23, and 0.28. The doping level is indicated in every ´ temperature of the sample (TN ) and the panel together with the Neel corresponding measurement temperature (T ). The plotted intensity has been smoothed in order to reduce the statistical noise and enhance readability. The symbols denote peak positions derived from Lorentzian fits.

Spin Dynamics in Ce1−x Lax B6 and Ce1−x Ndx B6

the energy of the ferromagnetic resonance at the zone center, ωΓ = 0.25 meV, remains practically unchanged. This result is consistent with the expectation that the spin gap at the R point (which is the propagation vector of the AFO phase) should close at the quantumcritical phase transition to phase IV. It also implies that the behavior of the two resonant modes is qualitatively different or even opposite to the one observed as a function of magnetic field, where the Γpoint mode softens to zero with the suppression of the AFM phase (Fig. 9.10), while the R-point mode stays at a constant energy until a second field-induced mode emerges below it (Fig. 9.14). Friemel et al. [23] compared the evolution of magnetic excitations at the R point in Ce1−x Lax B6 samples with x = 0, 0.18, 0.23, and 0.28 upon applying the magnetic field along the [110] crystal direction. These results are summarized in Fig. 9.31, where the scattering function6 S(QAFQ , ω) at the wave vector QAFQ = ( 12 21 21 ) is plotted vs. energy transfer and magnetic field. The field ranges of phases III, III , and II are indicated at the bottom of each panel. These excitations were fitted to a Lorentzian line shape given by Eq. (9.4), and the mode energy ω0 vs. B is overlaid in Fig. 9.31 as black data points. In zero field, the x = 0 (TN = 2.4 K), x = 0.18 (TN = 1.66 K), x = 0.23 (TN = 1.2 K), and x = 0.28 (TN, onset ≈ 0.32 K) doped samples exhibit the exciton at ω R = 0.48, 0.41, 0.25, and  0.1 meV, respectively.7 In addition to the decrease in energy, the peak also broadens upon doping. Consequently, for x = 0.28, only a quasielastic line can be observed at low fields. Another, much weaker and broader peak can also be seen near ω2 = 0.94 meV in the x = 0 and x = 0.18 samples. In the AFM phase, spectra of the parent and 18% La-substituted compounds show that the exciton energy stays nearly constant vs. B, see Figs. 9.14 and 9.31(a, b), while its amplitude shows a gradual suppression. This contrasts with the resonant mode in the SC state of CeCoIn5 , whose energy splits in magnetic field with the main part S(Q, ω), the background intensity has been subtracted from the data. Details of this procedure can be found in Ref. [23]. 7 The energy of the R-point exciton in the x = 0.23 sample is somewhat lower than in Fig. 9.30 because of the higher measurement temperature. The TOF data in Fig. 9.30 were measured at 60 mK, whereas the triple-axis data in Fig. 9.31 were taken at 420 mK, which is only about 2.8 times lower than TN of this particular sample. 6 In order to calculate

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754 Neutron-Scattering Studies of Spin Dynamics in Pure and Doped CeB6

of the spectral weight carried by the lower Zeeman branch [173]. For neither of the modes do we observe any splitting in magnetic field, which agrees with the complete lifting of the degeneracy within the 8 quartet ground state by the consecutive AFQ and AFM orderings. However, the energy of the high-energy mode ω2 , shown with empty circles in Figs. 9.31(a, b), diminishes and gets sharper with field with a varying slope between the x = 0 and x = 0.18 compounds and a rather concave order-parameter-like field dependence. These facts together with the vanishing of the mode above TN let us conclude that it might correspond to the onset of the particle-hole continuum at twice the AFM charge gap. Its magnitude of ω2 = (0.94 ± 0.07) meV in zero field for CeB6 agrees with the Qaveraged gap size of 2 AFM ≈ 1.2 meV determined by point-contact spectroscopy [82]. The integrated spectral weight of the exciton, corresponding to the area of the peak, remains nearly constant with field below TN for x = 0, 0.18, and 0.23. As the system enters the aforementioned phase III above Bc , the amplitude increases. The peak position in energy is changing abruptly [Fig. 9.31(a, b)] or continuously [Fig. 9.31(c)]. Upon eventually entering the AFQ phase, the excitation starts shifting to higher energies, as seen in the high-field spectra for B > 2 T. Even for the heavily doped x = 0.28 sample, a rather broad mode emerges for fields B > 6 T. This mode (we will denote it here as AFQ1 ) is dominating the spectrum in the AFQ phase for all samples. Its peak intensity changes rather continuously when crossing the III -II phase boundary at BQ and remains nearly constant in the AFQ regime. Moreover, upon entering phase III at Bc , we observe the appearance of the second low-energy mode, which can be seen for the x = 0 and x = 0.18 compounds at approximately 0.2 meV in Fig. 9.31(a, b). This excitation, denoted here as AFQ2 , is very sharp and evolves monotonically and continuously into phase II, its energy increasing in parallel to that of the AFQ1 mode. The discussion about the nature of this mode in pure CeB6 can be found in Section 9.4; here we only note that it also persists in La-doped samples, demonstrating that the AFM and AFQ phases have clearly distinct spin dynamics. The linear monotonic increase of both the AFQ1 and AFQ2 mode energies with magnetic field in phase II is characterized

Spin Dynamics in Ce1−x Lax B6 and Ce1−x Ndx B6

by a common slope g = (0.11 ± 0.004) meV/T = (1.90 ± 0.07)μB , which is doping independent. This can be qualitatively explained by a transition between two Zeeman-split energy levels, consistent with the purely localized description of the spin dynamics in a meanfield model of ordered multipoles in magnetic field [15, 118], see also Section 9.4. The localized model would also naturally explain the increasing line width Γ of the AFQ1 mode with La doping, as the La-substitution randomly alters the environment of the Ce3+ ion, composed of six nearest neighbors. It still remains to be clarified how the exciton and the AFQ1 mode are related. One possible scenario [85] describes the exciton as a collective mode below the onset of the particle-hole continuum at 2 AFM . An alternative approach would understand the exciton as a multipolar excitation, which is overdamped by the coupling to the conduction electrons in the AFQ state T > TN , but emerges as a sharp peak in the AFM state where the damping is removed by the opening of a partial charge gap [63, 73]. On the one hand, it would be an oversimplification to identify the exciton with the AFQ1 mode, according to the second scenario, since the field dependence of the energy and the amplitude is completely different for both excitations. On the other hand, the zero-field extrapolation of the AFQ1 mode energy E 0 almost coincides with the exciton energy ω R , both following the suppression of the magnetic energy scale, kB TN , as shown in Fig. 9.32(a). Another piece of information is given by the doping and field dependencies of the exciton line width, Γ . Figure 9.32(b) shows that it increases with the ratio of the exciton energy to the AFM ordering temperature, ω R /kB TN , which can be considered as a rough measure of the relative distance between the exciton and the onset of the particle-hole continuum under the assumption that the charge gap AFM is proportional to TN . The points for all samples in which the exciton has been observed appear to fall on the same line, indicating that proximity to the continuum dominates the mode damping. A similar picture is given in Fig. 9.32(c), where the line width Γ is plotted directly vs. TN , whose dependence on the magnetic field has been taken into account. The universality of these dependencies among all the measured samples suggests that the suppression of the AFM order and the associated closing

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756 Neutron-Scattering Studies of Spin Dynamics in Pure and Doped CeB6

Figure 9.32 (a) Zero-field exciton energy, ω R , and B → 0 extrapolation of the AFQ1 mode, E 0 , as a function of TN . (b) Half width at half maximum of the exciton, Γ , plotted vs. ω R /kB TN . (c) The same vs. TN (B) in the AFM phase for all doping levels. Note the inverted direction of the horizontal axis. (d) Γ vs. TQ for the AFQ1 mode in the AFQ phase for all doping levels. The fielddependent transition temperatures TN (B) and TQ (B) were determined from measurements of the specific heat or from interpolation of the published phase diagrams (x = 0, x = 0.2, x = 0.25) [21, 22, 27, 33]. TN (B) for the x = 0.18 sample was estimated from the AFM charge gap ω2 . All lines are guides to the eyes.

of the partial charge gap lead to a broadening of the exciton, rather than the chemical disorder from the La substitution. This ultimately leads to a quasielastic line shape in the limit of the absent phase III in zero field, reached either by temperature for T > TN (point indicated by an arrow) or by doping (for x = 0.28), resulting in identical line widths for both cases within the experimental uncertainty. In contrast, the line width of the AFQ1 mode in phase II is independent of the respective AFQ energy scale, kB TQ , as shown in Fig. 9.32(d). The line widths for x = 0.18 and x = 0.23 are comparable, which can be explained with the similar disorder effect because of chemical substitution. Were the AFQ1 mode and the exciton of the same origin, we would expect a more similar response to disorder for both. Therefore, the exciton must be derived from itinerant HF quasiparticles that are not as sensitive to the randomized local molecular field of the Ce3+ ion as the localized AFQ1 mode. The contrasting field dependencies for the energies for the exciton and the AFQ1 mode in Fig. 9.31 further substantiate this conclusion.

Spin Dynamics in Ce1−x Lax B6 and Ce1−x Ndx B6

Acknowledgments The authors of this chapter thank Peter Thalmeier, Alireza Akbari, Gerd Friemel, Hoyoung Jang, Yuan Li, Stanislav Nikitin, Bernhard Keimer, Vladimir Hinkov, George Jackeli, Andreas Koitzsch, Nikolay Sluchanko, Sergey Demishev, Alexey Semeno, Takeshi Matsumura, ¨ Takemi Yamada, Silke Buhler-Paschen, Vladislav Kataev, Oliver Stockert, Dongjin Jang, and Manuel Brando for many stimulating discussions and fruitful collaborations. Most of the results presented here would be impossible without the high-quality single crystal provided by Natalya Shitsevalova, Anatoliy Dukhnenko, and Volodymyr Filipov at the I. M. Frantsevich Institute for Problems of Materials Sciences of NAS, Kyiv, Ukraine, and the assistance of the instrument scientists at various neutron facilities: Astrid ˇ ´ and Igor Radelytskyi at the Julich ¨ Schneidewind, Petr Cerm ak, Centre for Neutron Science, JCNS-MLZ, Germany; Alexandre Ivanov, ¨ Jacques Ollivier, Arno Hiess, Paul Steffens, and Martin Bohm at the Institute Laue-Langevin (ILL), Grenoble, France; Jean-Michel Mignot, Yvan Sidis, Sylvain Petit, and Philippe Bourges from Laboratoire ´ Leon Brillouin (LLB), Saclay, France; Robert Bewley and Tatiana Guidi at the ISIS Neutron and Muon Source, Rutherford Appleton Laboratory, United Kingdom; Andrey Podlesnyak and Tao Hong at the Spallation Neutron Source, Oak Ridge National Laboratory, USA; Jose Rodriguez-Rivera, Nicholas Butch, and Yiming Qiu at the National Institute of Standards and Technology (NIST), Maryland, ¨ USA; Zita Husges, Zhilun Lu, Jianhui Xu, Michael Tovar, Karel Prokes, Ilya Glavatskyy, Diana Lucia Quintero-Castro, and Konrad Siemensmeyer at the Helmholtz-Zentrum Berlin, Germany; as well as Leonid Lev and Vladimir Strocov at the ADRESS-Beamline of the Swiss Light Source (SLS), Paul Scherrer Institute, Switzerland. We acknowledge financial support by the German Research Foundation (DFG) under individual research grants IN 209/3-2, ¨ IN 209/4-1, and PO 2621/1-1, as well as the Wurzburg-Dresden Cluster of Excellence on Complexity and Topology in Quantum Matter — ct.qmat (EXC 2147, project-id 39085490) and project C03 of the Collaborative Research Center SFB 1143 in Dresden (projectid 247310070).

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“Field-angle-resolved magnetic excitations as a probe of hidden-order symmetry in CeB6 ”; Phys. Rev. X 10, 021010 (2020). 119. Goremychkin, E. A., Osborn, R., Rainford, B. D., and Murani, A. P.; “Evidence for anisotropic Kondo behavior in Ce0.8 La0.2 Al3 ”; Phys. Rev. Lett. 84, 2211–2214 (2000). 120. Demishev, S. V., Semeno, A. V., Bogach, A. V., Paderno, Y. B., Shitsevalova, N. Y., and Sluchanko, N. E.; “Magnetic resonance in cerium hexaboride caused by quadrupolar ordering”; J. Magn. Magn. Mater. 300, e534– e537 (2006). 121. Demishev, S. V., Semeno, A. V., Ohta, H., Okubo, S., Paderno, Y. B., Shitsevalova, N. Y., and Sluchanko, N. E.; “High-frequency study of the orbital ordering resonance in the strongly correlated heavy fermion metal CeB6 ”; Appl. Magn. Reson. 35, 319–326 (2008). 122. Schlottmann, P.; “Electron spin resonance in antiferro-quadrupolarordered CeB6 ”; Phys. Rev. B 86, 075135 (2012). 123. Schlottmann, P.; “Electron spin resonance in CeB6 ”; J. Appl. Phys. 113, 17E109 (2013). 124. Schlottmann, P.; “Theory of electron spin resonance in ferromagnetically correlated heavy fermion compounds”; Magnetochemistry 4, 27 (2018). 125. Le, M., Quintero-Castro, D., Toft-Petersen, R., Groitl, F., Skoulatos, M., Rule, K., and Habicht, K.; “Gains from the upgrade of the cold neutron triple-axis spectrometer FLEXX at the BER-II reactor”; Nucl. Instrum. Methods Phys. Res. Sect. A 729, 220–226 (2013). 126. Semeno, A. V., Gilmanov, M. I., Bogach, A. V., Krasnorussky, V. N., Samarin, A. N., Samarin, N. A., Sluchanko, N. E., Shitsevalova, N. Y., Filipov, V. B., Glushkov, V. V., and Demishev, S. V.; “Magnetic resonance anisotropy in CeB6 : an entangled state of the art”; Sci. Rep. 6, 39196 (2016). 127. Semeno, A., Gilmanov, M., Sluchanko, N., Krasnorussky, V., Shitsevalova, N., Filipov, V., Flachbart, K., and Demishev, S.; “Angular dependences of ESR parameters in antiferroquadrupolar phase of CeB6 ”; Acta Phys. Pol. A 131, 1060–1062 (2017). 128. Gilmanov, M. I.; “Electron spin resonance in rare-earth hexaborides RB6 (R = Gd, Ce, Sm)” (in Russian); Ph.D. thesis; A. M. Prokhorov General Physics Institute of RAS, Moscow (2019). 129. Sera, M., Sato, N., and Kasuya, T.; “Magnetoelastic studies on Ce1−x Lax B6 single crystals”; J. Magn. Magn. Mater. 63–64, 64–66 (1987).

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130. Yoshino, Y., Kobayashi, S., Tsuji, S., Tou, H., Sera, M., Iga, F., Zenitani, Y., and Akimitsu, J.; “Nd ion doping effects on the multipolar interactions in CeB6 ”; J. Phys. Soc. Jpn. 73, 29–32 (2004). 131. Schenck, A., Gygax, F. N., and Solt, G.; “Complex phase diagram of Ce0.75 La0.25 B6 studied by muon spin rotation and relaxation in zero and nonzero external fields”; Phys. Rev. B 75, 024428 (2007). ´ G., Sera, M., and Iga, F.; “Effect of Nd 132. Mignot, J.-M., Robert, J., Andre, substitution on the magnetic order in Cex Nd1−x B6 solid solutions”; Phys. Rev. B 79, 224426 (2009). 133. Arko, A. J., Crabtree, G., Ketterson, J. B., Mueller, F. M., Walch, P. F., Windmiller, L. R., Fisk, Z., Hoyt, R. F., Mota, A. C., Viswanathan, R., Ellis, D. E., Freeman, A. J., and Rath, J.; “Large electron–phonon interaction but low-temperature superconductivity in LaB6 ”; Int. J. Quant. Chem. 9, 569–578 (1975). ´ M., Flachbart, K., Filippov, V., Paderno, Y., Shiceval134. Bat’ko, I., Bat’kova, ova, N., and Wagner, T.; “Electrical resistivity and superconductivity of LaB6 and LuB12 ”; J. Alloy. Compd. 217, L1–L3 (1995). 135. Inosov, D. S., Evtushinsky, D. V., Koitzsch, A., Zabolotnyy, V. B., Borisenko, ¨ S. V., Kordyuk, A. A., Frontzek, M., Loewenhaupt, M., Loser, W., Mazilu, ¨ I., Bitterlich, H., Behr, G., Hoffmann, J.-U., Follath, R., and Buchner, B.; “Electronic structure and nesting-driven enhancement of the RKKY interaction at the magnetic ordering propagation vector in Gd2 PdSi3 and Tb2 PdSi3 ”; Phys. Rev. Lett. 102, 046401 (2009). 136. Butch, N. P., Manley, M. E., Jeffries, J. R., Janoschek, M., Huang, K., Maple, M. B., Said, A. H., Leu, B. M., and Lynn, J. W.; “Symmetry and correlations underlying hidden order in URu2 Si2 ”; Phys. Rev. B 91, 035128 (2015). 137. Arko, A. J., Crabtree, G., Karim, D., Mueller, F. M., Windmiller, L. R., Ketterson, J. B., and Fisk, Z.; “de Haas – van Alphen effect and the Fermi surface of LaB6 ”; Phys. Rev. B 13, 5240–5247 (1976). 138. Chan, S. K., and Heine, V.; “Spin density wave and soft phonon mode from nesting Fermi surfaces”; J. Phys. F: Metal Phys. 3, 795 (1973). 139. Fawcett, E.; “Spin-density-wave antiferromagnetism in chromium”; Rev. Mod. Phys. 60, 209–283 (1988). 140. Borisenko, S. V., Kordyuk, A. A., Yaresko, A. N., Zabolotnyy, V. B., Inosov, ¨ D. S., Schuster, R., Buchner, B., Weber, R., Follath, R., Patthey, L., and Berger, H.; “Pseudogap and charge density waves in two dimensions”; Phys. Rev. Lett. 100, 196402 (2008).

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141. Ruderman, M. A., and Kittel, C.; “Indirect exchange coupling of nuclear magnetic moments by conduction electrons”; Phys. Rev. 96, 99–102 (1954). 142. Kasuya, T.; “A theory of metallic ferro- and antiferromagnetism on Zener’s model”; Prog. Theor. Phys. 16, 45–57 (1956). 143. Yafet, Y.; “Ruderman–Kittel–Kasuya–Yosida range function of a onedimensional free-electron gas”; Phys. Rev. B 36, 3948–3949 (1987). 144. Kim, J. G., Lee, E. K., and Lee, S.; “One-dimensional free-electron spin susceptibility at finite temperature”; Phys. Rev. B 54, 6077–6080 (1996). 145. Aristov, D. N.; “Indirect RKKY interaction in any dimensionality”; Phys. Rev. B 55, 8064–8066 (1997). 146. Litvinov, V. I., and Dugaev, V. K.; “RKKY interaction in one- and twodimensional electron gases”; Phys. Rev. B 58, 3584–3585 (1998). 147. Brown, P., Caudron, R., Fert, A., Givord, D., and Pureur, P.; “Helimagnetic structure in diluted Y-Gd alloys”; J. Physique Lett. 46, 1139–1141 (1985). 148. Fretwell, H. M., Dugdale, S. B., Alam, M. A., Hedley, D. C. R., RodriguezGonzalez, A., and Palmer, S. B.; “Fermi surface as the driving mechanism for helical antiferromagnetic ordering in Gd-Y alloys”; Phys. Rev. Lett. 82, 3867–3870 (1999). 149. Robinson, R. A.; Magnetism in Heavy Fermion Systems (World Scientific, Singapore, 2000). 150. Rossat-Mignod, J., Regnault, L., Jacoud, J., Vettier, C., Lejay, P., Flouquet, J., Walker, E., Jaccard, D., and Amato, A.; “Inelastic neutron scattering study of cerium heavy fermion compounds”; J. Magn. Magn. Mater. 76– 77, 376–384 (1988). 151. Regnault, L. P., Erkelens, W. A. C., Rossat-Mignod, J., Lejay, P., and Flouquet, J.; “Neutron scattering study of the heavy-fermion compound CeRu2 Si2 ”; Phys. Rev. B 38, 4481–4487 (1988). ¨ 152. Schroder, A., Aeppli, G., Bucher, E., Ramazashvili, R., and Coleman, P.; “Scaling of magnetic fluctuations near a quantum phase transition”; Phys. Rev. Lett. 80, 5623–5626 (1998). ¨ 153. Stockert, O., Lohneysen, H. V., Rosch, A., Pyka, N., and Loewenhaupt, M.; “Two-dimensional fluctuations at the quantum-critical point of CeCu6−x Aux ”; Phys. Rev. Lett. 80, 5627–5630 (1998). 154. Kadowaki, H., Sato, M., and Kawarazaki, S.; “Spin fluctuation in heavy fermion CeRu2 Si2 ”; Phys. Rev. Lett. 92, 097204 (2004).

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169. Eremin, I., Zwicknagl, G., Thalmeier, P., and Fulde, P.; “Feedback spin resonance in superconducting CeCu2 Si2 and CeCoIn5 ”; Phys. Rev. Lett. 101, 187001 (2008). 170. Sato, N. K., Aso, N., Miyake, K., Shiina, R., Thalmeier, P., Varelogiannis, G., Geibel, C., Steglich, F., Fulde, P., and Komatsubara, T.; “Strong coupling between local moments and superconducting ‘heavy’ electrons in UPd2 Al3 ”; Nature (London) 410, 340–343 (2001). 171. Blackburn, E., Hiess, A., Bernhoeft, N., and Lander, G. H.; “Inelastic neutron scattering from UPd2 Al3 under high magnetic fields”; Phys. Rev. B 74, 024406 (2006). 172. Chang, J., Eremin, I., Thalmeier, P., and Fulde, P.; “Theory of magnetic excitons in the heavy-fermion superconductor UPd2 Al3 ”; Phys. Rev. B 75, 024503 (2007). 173. Stock, C., Broholm, C., Demmel, F., Van Duijn, J., Taylor, J. W., Kang, H. J., Hu, R., and Petrovic, C.; “From incommensurate correlations to mesoscopic spin resonance in YbRh2 Si2 ”; Phys. Rev. Lett. 109, 127201 (2012).

Chapter 10

Theory of Electron Spin Resonance in Strongly Correlated CeB6 Pedro Schlottmann Department of Physics, Florida State University, Tallahassee, Florida 32306, USA [email protected]

In this chapter, we study the electron spin resonance (ESR) line width for localized moments within the framework of the Kondo lattice model. Only for a sufficiently small Kondo temperature can an ESR signal be observed for a Kondo impurity. On the other hand, for a Kondo lattice representing a heavy-fermion compound, short-range ferromagnetic (FM) correlations between the localized moments are crucial to observe a signal. The spin relaxation rate (line width) and the static magnetic susceptibility are inversely proportional to each other. The FM correlations enhance the susceptibility and hence reduce the line width. For most of the heavy-fermion systems displaying an ESR signal the FM order arises in the ab-plane from the strong lattice anisotropy. CeB6 is a heavy-fermion compound with cubic symmetry having a 8 ground-state quartet. Four transitions are expected for individual Ce ions with a 8 ground-state multiplet, but only one has been observed. Antiferro-quadrupolar order (AFQ) arises below 4 K due to the orbital content of the 8 -quartet. We address the effects

Rare-Earth Borides Edited by Dmytro S. Inosov c 2022 Jenny Stanford Publishing Pte. Ltd. Copyright  ISBN 978-981-4877-56-5 (Hardcover), 978-1-003-14648-3 (eBook) www.jennystanford.com

774 Theory of Electron Spin Resonance in Strongly Correlated CeB6

of the interplay of AFQ and FM correlations on the ESR line width and the phase diagram. It is usually difficult to distinguish among ESR resonances due to individual localized moments and conducting heavy electron spins, especially for anisotropic Ce and Yb compounds. However, for CeB6 an itinerant picture within the AFQ phase is necessary to explain the electron spin resonances, ruling out the individual localized moments (impurities) picture. The longitudinal magnetic susceptibility has a quasielastic central peak of line width 1/T1 and inelastic peaks for the absorption/emission of excitation. The latter are measured by inelastic neutron scattering (INS) and provide insights into the magnetic order. We briefly summarize some of the INS results for CeB6 in the context of the picture that emerged from the ESR experiments.

10.1 Introduction The precession of magnetic moments in a constant external magnetic field B along the z-axis is governed by Bloch’s equations [1] d Mx Mx d My My d Mz , , = γ B My − = −γ B Mx − = 0, dt T2 dt T2 dt

(10.1)

where γ = gμB / is the gyromagnetic ratio, and T2 is the transversal relaxation time. The Bloch equations, when solved in the presence of a small magnetic field of amplitude B1 rotating in the x y-plane with frequency ω, yields for the power absorption (proportional to B12 ) P (ω) =

ωγ Mz /T2 B 2. (ω − γ B)2 + 1/T22 1

(10.2)

The proportionality constant is the imaginary part of the transversal susceptibility χ T (z) = [z − γ B + i /T2 ]−1 [−γ B + i /T2 ]χ0T ,

(10.3)

where χ0T is the static transverse susceptibility and z = ω + i 0. The resonance at γ B is dephased by the relaxation time T2 arising from the interaction with other magnetic moments, conduction

Kondo Impurity Model 775

electrons, and fields in the neighborhood of the moment. Manybody effects can introduce a frequency dependence to the relaxation (retardation), and it is appropriate to define a relaxation function N T (z) [2], so that "−1 ! " ! −Beff + N T (z)/χ0T χ0T , χ T (z) = z − Beff + N T (z)/χ0T

(10.4)

where Beff is the effective Zeeman splitting.

10.2 Kondo Impurity Model We consider the Kondo impurity model which consists of a spin S = 1/2 placed at the origin, interacting with conduction electrons via the spin exchange J. The Hamiltonian is then H=



†

kσ ckσ ckσ − gf μB B S z +

J  † c S · s σ σ  c k σ  , N   kσ

(10.5)

kk σ σ

kσ

where S are the impurity spin operators and sσ σ  represent 1/2 times the Pauli matrices for the conduction states. For the Zeeman splitting of the conduction electrons we have kσ = k − σ Bc /2, where Bc = gc μB B. It is convenient to work with the Hartree–Fock factorization of H, i.e., we replace B → B  = B − JρF gc μB B/2,

(10.6)

which corresponds to the “Knight shift” of the magnetic resonance. Here ρF denotes the conduction-electronic density of states per spin component at the Fermi level. The ESR line is given by the transverse dynamical susceptibility, χ T (z) = −(gf μB )2

1

S + ; S − z , 2N

(10.7)

where S ± = S x ± i S y are the spin-flip operators, and the factor 1/2 is introduced to normalize the susceptibility (note that the S x and S y correlation functions are equal). The spin operators satisfy the standard commutation relations, [S + , S − ] = 2S z and [S z , S ± ] = ±S ± .

776 Theory of Electron Spin Resonance in Strongly Correlated CeB6

To second-order perturbation in J, the relaxation function N T (z) defined in Eq. (10.4) is [2, 3] ! " 2 π N T (z) = (gf μB )2 (JρF )2 i + S z φ(z) , 4 π z − gf μB B   z − gf μB B   φ(z) = ln(D/2π T ) − ψ 1−i z 2π T gf μB B   gf μB B   ψ 1+i , (10.8) − z 2π T where ψ is the digamma function and D is the band cut-off. In the zero-field limit we obtain the well-established Korringa relaxation rate 1 = π(JρF )2 T , (10.9) Trel which is proportional to T . Here the factor T arises from the inverse of χ0T , while, alternatively and equivalently, in textbook calculations the T originates from the integration over the Fermi functions for the conduction electrons. For gf μB B   T, on the other hand, the relaxation rate is gf μB B  1 π , = (JρF )2 Trel 4 1 + 12 (JρF )2 ln(gf μB B /2π T )

(10.10)

which shows an almost linear increase in the line width with the external field. Consequently, at low temperatures the residual line width in Q-band ESR (∼34 GHz) should be larger than the corresponding one for X-band (∼9 GHz), as found in many experiments. Terms of higher order in J introduce the Kondo effect. Both the static susceptibility χ0T and the relaxation function N T (z) are affected by the logarithmic corrections due to the Kondo effect. For the case T  TK the Kondo terms appear as logarithmic corrections as a consequence of the renormalization of the interaction vertex. Hence, the dressed vertex function J enhances N T (z) on a logarithmic scale [4, 5]. Similarly χ0T acquires logarithmic terms that reduce the susceptibility. Both effects are the precursor of the compensation into a singlet of the localized spin at low temperatures and fields by the spin-density of the conduction electrons. At high T the relaxation rate resembles a Korringa behavior, i.e., linear in

Kondo Impurity Model 777

T, with an enhanced exchange coupling and logarithmic corrections. The relaxation rate is approximately given by   1 π (gf μB B /2T ) = T 1+ T2 2 tanh(gf μB B /2T ) −2  1 1 , (10.11) × ln(T /TK ) + ψ(1) + Reψ(1 − igf μB B /2π T ) 2 2 and the g-shift is  −1 g = g7 − ln(T /TK ) − ψ(1) + Reψ[1−i (gf μB B /2π T )] . (10.12) Application of these results to 171 Yb and 174 Yb impurities in Au can be found in Refs. [6, 7]. Some data are shown in Fig. 10.1 for a slightly different fit than in [6] with TK = 0.5 × 10−8 K. The Kondo logarithms are necessary for a reasonable agreement. For |ω|, T , gf μB B   TK , on the other hand, a singlet spin state forms as a consequence of the Abrikosov resonance. The susceptibility is finite (in contrast to the Curie law), and the maximum value of N T (z = T = B  = 0) is determined by the unitarity bound for the Kondo scattering, i.e., i (2/π)(gf μB )2, giving rise to Fermi liquid properties [8]. Note that as a consequence of the unitarity bound the relaxation function in this limit is independent of J [9, 10]. The imaginary part of N T decreases as a function of ω and T as (ω/TK )2 and (T /TK )2 . The ground state is a spin singlet, so that the susceptibility χ0T is a constant (the singlet is coupled to the excited spin triplet via Van-Vleck-like terms) of the order of 1/TK . The relaxation rate, 1/T2 , is then proportional to TK rather than linear in the temperature (Korringa). Hence, only if TK is less than 100 mK, it would be possible to observe an X-band ESR resonance [6]. In summary, in order for an ESR resonance to be seen for a Kondo impurity, TK has to be very small. Otherwise the width of the resonance is going to be too broad to be observed. This difficulty can be overcome if the measurement is carried out in rather high magnetic fields with a correspondingly larger frequency of the microwave field. The above considerations lead to the commonly accepted statement that ESR of a Kondo ion cannot be observed.

778 Theory of Electron Spin Resonance in Strongly Correlated CeB6

Figure 10.1 (a) Relaxation rate over T of Au171 Yb for 9 GHz (X-band) as a function of T . Open symbols denote the mI = + 12 transitions and closed symbols the mI = − 12 transitions. Triangles correspond to a sample with 280 ppm and squares to a sample with 670 ppm. (b) g values for Au174 Yb samples at 9 GHz as a function of T : Triangles correspond to 70 ppm; squares to 200 ppm; and circles to 260 ppm. The solid curves are Eqs. (10.11) and (10.12) for TK = 0.5 × 10−8 K and g7 = 3.370. Adapted with permission from Spalden et al. [6], copyright by the American Physical Society.

10.3 Kondo Lattice Model The Kondo lattice consists of conduction electrons and a spin S = 1/2 “impurity” at every site, i.e., H = H0 + Hsd ,  lattice † kσ ckσ ckσ − gf μB B S zj , H0 = kσ

Hsd

j

J  i (k−k )·R j † = e ckσ S j · sσ σ  ck σ  , N kk σ σ  j

(10.13)

where j labels the lattice sites and R j denotes the position of the site j . As before the Hartree–Fock factorization of Hsd replaces B by

Kondo Lattice Model 779

B  [see Eq. (10.6)] corresponding to the Knight shift of the magnetic resonance. As before we express the transverse dynamical susceptibility in terms of a relaxation function N T (z) [2]. We are mainly interested in  the imaginary part of N T (z), i.e., N T (ω). Second-order perturbation theory in J leads to a sum over k and k . For a parabolic band the angular integrals of the momenta can be carried out and the result can be decomposed into two parts, one involving single site terms,  NsT (ω), and the other part corresponds to terms with multiple sites, T Nm (ω). Using that S z  = (1/2) tanh[(gf μB B  )/(2T )], the single-site contribution is identical to the impurity relaxation, Eq. (10.8), i.e., the Kondo impurity relaxation kernel. For the multi-site terms we obtain [11] 

NmT (ω) =

 sin(kF Ri j )2 π e−Ri j /l (gf μB JρF )2 2 4N (k R ) F i j i j, i = j   ω − gf μB B  + − × 2 Siz S zj  +

Si S j  , ω

(10.14)

where l is the spin mean-free path due to the spin-lattice relaxation. Our spin S is actually a pseudospin arising from the crystal-field splittings of the rare-earth total angular momentum. As such it has a large orbital component due to the strong spin–orbit interaction. This way, when the conduction electrons travel, their “spin” relaxes into the lattice (spin-lattice relaxation). The resonance takes place at the field B  , and the spin relaxation rate is now given by 

N T (ω = B  ) 1 = , T2 χ0T 



(10.15)



where N T = NsT + NmT . Here T2 depends on the magnetic order  of the phase, and the spin correlations in N T and χ0T have to be  evaluated accordingly. To simplify, below we consider N T (ω = 0)  rather than N T (ω = B  ), which yields qualitatively the same result.

10.3.1 Paramagnetic Kondo Lattice We now evaluate the multi-site terms in Eq. (10.14) for the paramagnetic phase. To second order in J we have for the intersite

780 Theory of Electron Spin Resonance in Strongly Correlated CeB6

terms, i.e., for i = j , Siz S zj  ≈ S z 2 and Si+ S −j  ≈ Si+  S −j  ≈ 0. The imaginary part of the relaxation function is then   π gf μB B /T T 2 1 + N (ω → 0) = (gf μB JρF ) 4 2 sinh(gf μB B /T ) ! "2 π + (gf μB JρF )2 tanh(gf μB B /2T ) 8  sin(kF Ri j )2 × e−Ri j /l , (10.16) 2 (k R ) F i j i j, i = j and the relaxation rate is approximately given by 

1/Trel

1 N T (ω → 0) C = , χ0T = , (10.17) T Trel T +θ χ0 where C is the Curie constant and θ ∝ TK the Curie–Weiss temperature. The proportionality constant of θ with TK is slightly larger than one. If the system has substantial spin frustration, θ can be much larger than TK .

Kondo, AFM Korringa

FM T Figure 10.2 Sketch of the relaxation rate as a function of T for AFM or Kondo correlations, FM fluctuations and no interactions (Korringa). Close to a magnetic transition (either AFM or FM) there are additional relaxation mechanisms due to collective excitations (spin waves or magnons) which have not been taken into account here. The horizontal dashed line schematically indicates the resonance energy. Only below that line is a signal observable. Reproduced from Ref. [11].

Kondo Lattice Model 781



The two terms of N T behave differently as a function of gf μB B /T. For large gf μB B /T the single site term decreases by 50% of its zero-field value, while the intersite term increases from zero. It now depends on the value of the spin mean-free path l on how many sites are correlated and hence which term dominates. In any case,  N T (0) is finite (and positive) and consequently 1/Trel ∝ TK at low T. For the same reason as in the case of the Kondo impurity the width of the resonance line is too large to be observed, unless the heavyfermion band is extremely narrow with an effective mass as large as 105 times the free electron mass. For T  TK , on the other hand, we recover a Korringa-like behavior with a renormalized Korringa constant [1/(Trel T )]. This is schematically shown in Fig. 10.2.

10.3.2 Antiferromagnetic Kondo Lattice For simplicity we assume that the antiferromagnetic order consists ´ state with two sublattices, so that nearest neighbor spins of a Neel have opposite ordered magnetic moments. The single site results are the same ones as for the paramagnetic phase, while for the intersite terms to second order in J we obtain

Siz S zj  ≈ (−1)i − j S z 2 , where i − j is either even or odd, giving rise to a sign oscillation. At low T the term Si+ S −j  is zero if we neglect collective modes. If we include spin waves this expression gives rise to a positive contribution. Without considering collective modes we have, similarly to the paramagnetic case   π gf μB B /T T 2 1 (ω → 0) = μ Jρ ) (g + N f B F 4 2 sinh(gf μB B /T ) ! "2 π + (gf μB JρF )2 tanh(gf μB B /2T ) 8  sin(kF Ri j )2 −Ri j /l × (−1)i − j e . (10.18) (kF Ri j )2 i j, i = j 

Note that N T in an antiferromagnet is reduced as compared to the paramagnetic phase. In heavy-fermion systems the spins of the rare-earth ions are usually antiferromagnetically correlated, even if the compound displays no transition to a long-range ordered phase. The shortrange correlations together with the Kondo screening gives rise

782 Theory of Electron Spin Resonance in Strongly Correlated CeB6

to a Curie–Weiss susceptibility with antiferromagnetic Weisstemperature θAF , χ0 = C /(T + θAF ). Here θAF is determined by the ´ temperature, TN , the frustration in the system (enhancing θAF ) Neel and TK . The relaxation rate is 

1 N T (ω → 0) = , Trel χ0T

(10.19)

so that the width of the resonance follows approximately a Korringa law, with a residual T = 0 line width proportional to θAF . Since θAF is still considerable, the same conclusions as for the paramagnetic phase hold, and a resonance line can only be observed if θAF is very small.

10.3.3 Kondo Lattice with Ferromagnetic Order If, on the other hand, the spins of the rare-earth ions are correlated ferromagnetically, the single site results are the same ones as for the paramagnetic phase, while for the intersite terms we have Siz S zj  ≈

S z 2 and Si+ S −j  is again zero if we neglect collective modes. However, if collective modes such as magnons are considered then these terms yield a positive contribution. The static susceptibility in this case is χ0T = C /(T − TC ) for T > TC , where C is the Curie constant and TC (> 0) is the Curie temperature of the ferromagnet. For T → TC , χ0T becomes very large (diverges) and, according to Eq. (10.19), the ESR line width turns very narrow, and hence becomes observable (see Fig. 10.2). In general, close to the critical point, we would have to approximate χ0T by |t|−γ , where t = (T − TC )/TC is known as the reduced temperature and γ the critical exponent, which is larger than 1. Consequently, the line narrows  even faster as t → 0. Here we have neglected in N T the relaxation through magnons (collective excitations), so that this result has to be taken with caution. Spin-resonance of ferromagnetic collective excitations has been studied by Huber [12, 13]. For weaker ferromagnetic correlations the system will not have long-range order at any T. The correlations are then short-ranged and the susceptibility is proportional to T −γ . In this case again the relaxation rate is strongly reduced at low T. The relaxation through magnons would only play a secondary role in this case, because

Kondo Lattice Model 783

for low energy excitations the wavelength would be larger than the range of the correlations. On the other hand, shorter wavelength magnons cannot be excited because their energy is larger than what the thermal bath can provide. Hence, there is the possibility that the electron spin resonance can be observed.

3.60

(a) g-factor

3.58 3.56 3.54 3.52

YbRh2Si2

3.50 3.48 0

2

4

6

8

10

12

14

T [K]

Figure 10.3 (a) ESR g-factor and (b) relaxation rate of YbRh2 Si2 for 9.4 GHz (X-band, triangles) and 34.1 GHz (Q-band, squares) as a function of T for B ⊥ c. The solid curve in (a) corresponds to a fit with g = 3.86 + 1.67/ ln(T˜ K /T ) with T˜ K = 20 mK. In (b) the dashed lines are fits to a line width with three contributions: a constant residual line width, a Korringa term proportional to T and an exponential activation representing the Orbach relaxation due to the phonon modulation of the crystal field. Courtesy of J. Sichelschmidt, adapted from Ref. [14], copyright by the American Physical Society.

784 Theory of Electron Spin Resonance in Strongly Correlated CeB6

The ferromagnetic correlations for YbRh2 Si2 are predominantly in the ab-plane. The width of the resonance then depends on the orientation of the magnetic field, i.e., in the plane or along the caxis. Hence, if the field lies in the ab-plane domains with short-range order tend to align and the static in-plane susceptibility, χ0T , will be strongly enhanced. If, on the other hand, the field is oriented along the c-direction, the tilting of the spin out of the plane gives rise to a smaller susceptibility. The line width of the resonance is then expected to be less for the field in the ab-plane, which agrees with the experimental observations [15, 16]. Since the g-factor in the ab-plane has a much smaller value than that along the c-axis, there is in addition to the above-mentioned line width enhancement, a broadening of the resonance due to the magnetic field for B c, see Eq. (10.10). The g-factor and line width for YbRh2 Si2 for B ⊥ c are shown in Fig. 10.3. The g-value has a similar logarithmic Tdependence as given by Eq. (10.12).

10.3.4 Summary Short-range correlations among the localized moments in a Kondo lattice play a fundamental role. Antiferromagnetic interactions between the spins lead to a resonance with a broad line width of the order of the Weiss temperature of the susceptibility. Consequently ESR can in general not be observed. If, on the other hand, the spins are correlated ferromagnetically the line width gets strongly suppressed and the possibility of an ESR signal is real. In other words, the ferromagnetic short-range correlations prevent the spinflips from being passed on from site to site. This situation can to some extent be interpreted as a narrowing of the signal due to bottleneck [15].

10.3.5 Other Theoretical Approaches and Experiments Within the framework of the Anderson lattice, Abrahams and ¨ Wolfle [17] investigated the line width of the ESR signal for a heavy-fermion compound. They concluded that the heavy mass in conjunction with ferromagnetic fluctuations can lead to observable narrow resonances. The heavy mass is equivalent to arguing with a

Kondo Lattice Model 785

small Kondo temperature for the lattice, but this alone is not enough to produce an observable ESR signal. The line width of the signal is further reduced by the ferromagnetic correlations. There is good agreement of this theory with experimental data in the Fermi liquid regime for YbRh2 Si2 [18]. Further extensions of the theory to the non-Fermi liquid regime of this material lead to a close relation of the T dependence of the specific heat and spin susceptibility with the observed T dependence of the g-shift and the line width [19,20]. There are several other proposals to explain the ESR in heavyfermion systems. Zvyagin et al. [21] showed that strong local anisotropic electron–electron interactions in the system together with a hybridization between localized and itinerant electrons can give rise to a g-shift of the ESR signal and cause a change in the line width. Huber [22], with main emphasis on the anisotropy of the gshift, studied the low-field ESR in YbRh2 Si2 and YbIr2 Si2 taking into account the effects of anisotropy and the Yb–Yb interactions. Finally, Kochelaev et al. [23] investigated the relaxation of a collective spin mode considering the anisotropic Kondo model with anisotropic Ruderman–Kittel–Kasuya–Yosida (RKKY) interaction, assuming that the Kondo coupling and the g-factor have the same anisotropy ratio. An ESR signal was found in single crystals of numerous heavyfermion compounds, e.g., YbRh2 Si2 [14, 15, 24], YbIr2 Si2 [25], YbRh [26], YbCo2 Zn20 [27], CeRuPO [28], and CeB6 [29–31]. The resonances are attributed to the Yb3+ and Ce3+ ions despite of their rather large TK . Common to all these compounds is a strong magnetic anisotropy (except for CeB6 ) with the easy-axis in the abplane, as well as ferromagnetic correlations among the rare-earth spins [26]. Due to the skin depth and spin diffusion in a metallic environment, all the observed resonances have a Dysonian line shape [32–34]. The resonance of local magnetic moments in a metal [36–38] and the ESR of conduction electrons are very difficult to distinguish since both have the same Dysonian line shape [32–34]. The data for the heavy-fermion compounds were analyzed within the known framework of ESR of magnetic impurities in metals [36–38], i.e., single ions with localized spins resonating independently. The gfactor and the line width are anisotropic as expected for a tetragonal crystal-electric field. An estimate indicates that in YbRh2 Si2 more

786 Theory of Electron Spin Resonance in Strongly Correlated CeB6

than 60% of the Yb3+ ions contribute to the ESR signal [16, 25]. In the case of a band of conducting heavy electrons, on the other hand, the g-shift is dominated by one of the f orbitals via hybridization and is going to have the crystal-field anisotropies of the rare-earth sites. Hence, based solely on ESR it is then difficult to decide if the resonances arise from the carriers in a heavy-electron band or localized moments [39].

10.4 Antiferroquadrupolar Ordered CeB6 In contrast to the other compounds discussed above, which have tetragonal symmetry and strong anisotropy, CeB6 is a cubic crystal, where the rare-earth ions sit at the center of a rigid cage formed by 8 surrounding B6 octahedra, as shown, for instance, in Fig. 8.1. The crystal field of trivalent Ce in eightfold coordination splits Hund’s rule J = 5/2 into a 8 ground state quartet and an excited 7 doublet, as illustrated schematically in Fig. 10.4. The

5d band bottom

Spin-Orbital Coupling

Cubic Crystal Field 7* 2-fold

2 eV

8* 4-fold

2F7/2

8-fold

64 meV 17 meV

6* 2-fold

4f orbital

~250 meV

14-fold 7 2-fold

46 meV

2F5/2

6-fold 8 4-fold

Figure 10.4 Schematic energy diagram illustrating the splitting of 4f orbital in CeB6 by the spin–orbit coupling and cubic crystal-electric field. Adapted from Ye et al. [35], copyright by the American Physical Society.

Antiferroquadrupolar Ordered CeB6

20

CeB6 H//[111]

H [kOe]

15

III’ 10

II 5

0

I

III

1

2

3

4

T [K] Figure 10.5 The low-temperature H vs. T phase diagram for CeB6 displays four phases. Phase I is the paramagnetic Kondo phase. Before the Kondo effect can compensate the internal degrees of freedom, there is a secondorder transition to the antiferro-quadrupolar phase (II) with the QAFQ at the R point of the Brillouin zone. At lower T, in the phases III and III , antiferromagnetism kicks in. Adapted from Kunimori et al. [40], copyright by the Physical Society of Japan.

crystal-field excitation energy is approximately 530 K (46 meV or 372 cm−1 ) [35, 41–43], so that for all practical purposes we can ignore the 7 doublet. The 8 quartet has simultaneously spin and quadrupolar content, which gives rise to a rich phase diagram consisting of four phases, as shown in Fig. 10.5. The hightemperature phase is the paramagnetic Kondo-like state (phase I) above TQ = 3.3 K (assuming zero magnetic field). The Kondo temperature is approximately 3 K. Phase II corresponds to antiferroquadrupolar (AFQ) order with QAFQ = R( 12 21 21 ) in the temperature range between TN = 2.3 K and TQ (again assuming zero magnetic field). Below TN the phase diagram displays two antiferromagnetic phases, phase III with QAFM, 1 = Σ( 14 41 0) and as a function of field above H ≈ 1.2 T phase III with QAFM, 2 = S( 14 41 21 ). In this chapter we mainly refer to phase II.

787

788 Theory of Electron Spin Resonance in Strongly Correlated CeB6

10.4.1 ESR in a 8 Quartet There are numerous examples of ESR studies of systems with 8 ground quartets, for instance, Dy3+ impurities in the insulator [45] CaF2 and the metal [46] Au, as well as Er3+ ions in the low-carrier heavy-fermion compound YBiPt [44]. The total angular momentum of Dy3+ and Er3+ ions is J = 15/2, which requires two crystalfield parameters, namely B4 and B6 , in cubic symmetry to describe the splittings or, equivalently, in terms of the Lea, Leask and Wolf

Figure 10.6 Angular dependence of the resonance field (a) and line width (b) for the four observed resonances of Er3+ in YBiPt. The groundmultiplet has 8 symmetry. The external field is rotated in the (110)-plane. Reproduced from Martins et al. [44], copyright by the American Physical Society.

Antiferroquadrupolar Ordered CeB6

parameters x and W. The wave functions and the energy levels depend on the ratio B4 /B6 [47] and are then not universal. Since the 8 is a linear combination of many of the 16 states of the J = 15/2, there are several possible transitions. Figure 10.6 shows the angular dependence of the resonance fields and line widths of the four observed transitions of Er3+ in YBiPt [44]. 4 4 --> 3 and 2 --> 1

(a)

μBH / hν

3

2

1 3 --> 2

4 --> 2 and 3 -->1 4 --> 1

0

0

10

20

30

40

50

θ (deg)

60

70

80

90

1.8 3 --> 2 4 --> 3 and 2 --> 1

1.4

(b)

1.2 1.0

4 --> 2 and 3 --> 1

2

|| (arb. units)

1.6

4 --> 3 and 2 --> 1

0.8 0.6

4 --> 1

4 --> 1

0.4 0.2 0.0

3 --> 2

0

10

20

30

40

50

θ (deg)

60

70

80

90

Figure 10.7 (a) ESR transition fields normalized to the microwave energy hν and (b) transition probabilities for a noninteracting Ce3+ in a 8 quartet with Zeeman splitting for a magnetic field rotating in the (110) plane (angle θ). The four eigenstates are labeled with decreasing energy in a Zeeman field. States 1 with 4 and 2 with 3 form two spin Kramers doublets. There are no level crossings as a function of θ. These transitions are universal for Ce3+ ions. Reproduced from Ref. [48], copyright by the American Physical Society.

789

790 Theory of Electron Spin Resonance in Strongly Correlated CeB6

10.4.2 ESR for Ce3+ Ions with 8 Ground State In contrast, the Ce3+ ions need only one crystal-field parameter, namely B4 , since J = 5/2. The wave functions of the 8 ground quartet, written in terms of the J z eigenstates, |J z  [right-hand side of Eq. (10.20)] are [47]   |+ ↑ = 56 | + 52  + 16 | − 32  ,   |+ ↓ = 56 | − 52  + 16 | + 32  , |− ↑ = | + 12  , |− ↓ = | − 12  .

(10.20)

Here the spin degrees of freedom, σ , are denoted with ↑ and ↓ and + and − refers to the quadrupolar (orbital) states. It is customary in ESR experiments to rotate the magnetic field in the (110)-plane. We √ parameterize the magnetic field as B = √ B(sin θ, sin θ, 2 cos θ)/ 2, so that for θ = 0 the field is B [001] axis, while if θ√ = π/2 the field is along the [110] axis, and for θ = arctan( 2) ≈ 54.7◦ the magnetic field points into the [111] direction. The theoretical positions of the resonances strongly depend on the angle θ, i.e., the relative angle of the magnetic field with the crystallographic axis, as shown in Fig. 10.7. There are then six possible microwave transitions within the quartet; however, two are doubly degenerate, so there are actually only four lines [48]. Experimentally, however, only one ESR signal was observed at 60 GHz for 1.8 K < T < 3.8 K for B parallel to the [110] direction with g = 1.59 [29, 30], rather than the four expected lines. Later experiments show that the angular dependence of this resonance is also different from the predicted ones [49]. All the ESR measurements were performed in phase II. A possible explanation for the observation of only one line is then that the remaining transitions are suppressed by the AFQ order. This will be analyzed in the next subsection.

10.4.3 g-Factor for ESR in Phase II of CeB6 As shown in Fig. 10.8, the resonance [50] has a Dysonian-like line shape, characteristic of a metallic environment, and a g-factor that depends of the direction of the applied magnetic field [49]. CeB6

Antiferroquadrupolar Ordered CeB6

Figure 10.8 (a) ESR spectra at 60 GHz calibrated in terms of the magnetic permeability μR for B [100] at different temperatures. DPPH marks the reference signal of diphenylpicrylhydrazyl. The inset is a sketch of the experimental geometry. (b) Line shape of the resonance at 60 GHz at T = 1.8 K for the three crystallographic directions. The circles are experimental points and the solid lines fits within a model of localized magnetic moments. Reproduced from Semeno et al. [49].

displays AFQ order [51] in the field and temperature range where the resonance is observed. The quadrupolar degrees of freedom drive the long-range order with QAFQ = R( 12 21 21 ), breaking the translational invariance of the lattice and forming two interpenetrating sublattices. The problem is conveniently studied using the Anderson lattice model which leads to hybridized localized and conduction states [48]. Within the reduced Brillouin zone there are now twice as many bands, i.e., four, and we need to place four electrons into these bands. The f-electron energies for the two sublattices due to the AFQ order are ε1 and ε2 and the corresponding g-factors g1 and g2 (they depend also on the direction of the applied field). Within the mean-field slave-boson formulation this leads to a 4 × 4matrix Hamiltonian for each k-value, with the single site energies in the diagonal and the hybridization terms on the off-diagonal entries. The diagonalization of the Hamiltonian yields the band dispersions [48]. The lower hybridized band is intersected by the

791

792 Theory of Electron Spin Resonance in Strongly Correlated CeB6

Fermi level close to the band-gap giving rise to heavy fermions. This leads to a single resonance with effective g-factor geff = (g1 + g2 )/2. This result depends on two angles, namely, the angle θ of the magnetic field with the crystal axis and an angle ϕ defining the AFQ long-range order [48]. The operators for the magnetization depend only on τx and τz , but not on τ y . The quantization axis for the τ -matrices defines the orientation of the orbital order. To determine the direction of the order we rotate the τ -matrices in the xz-plane, i.e., [48], τ˜ x = cosϕ τx − sinϕ τz , τ˜z = sinϕ τx + cosϕ τz .

(10.21)

Without loss of generality we can now choose the direction of the quadrupolar order along τ˜z . The effective g-factor was found to be [52] ! "1/2 geff (θ, ϕ) = cos2 θ a1 (+) + sin2 θ a2 (−) ! "1/2 + cos2 θ a1 (−) + sin2 θ a2 (+) +K + corr , (10.22)   2 4 a1 (±) = 1 ± cos ϕ , 7  2 12 2 (10.23) sin2 ϕ . a2 (±) = 1 ± cos ϕ + 7 49 Here “corr” is a small negative correction term of the order of 1 to 2 percent due to the Zeeman splitting of the conduction electrons and the dispersions of the bands [48]. Here K > 0 is the “Knight-shift” correction arising from the Hartree–Fock term of the exchange interaction, see Eq. (10.6). The magnitude of the Knightshift correction can be estimated from the Kondo temperature and is expected to be of the order of 0.15 to 0.20 of geff . The angular dependence of the resonance at 60 GHz has recently be measured in Ref. [49]. The data by Semeno et al. with error bars for 1.8 K (open circles) and 2.65 K (dark circles) are shown in Fig. 10.9. The solid curve is our fit to the data for 2.65 K using Eq. (10.22) with ϕ = 0.18π and K ∗ = K − |corr| = 0.48. The constant shift K ∗ is slightly larger than expected from our simple considerations. This fit is quite different from the one attempted in

Antiferroquadrupolar Ordered CeB6

1.9 [111]

[100]

1.8

[110]

1.8 K

geff

1.7

1.6

1.5

1.4 -0.2

ϕ=0.18π K*=0.48

2.65 K

-0.1

0.0

0.1

0.2

0.3 θ/π

0.4

0.5

0.6

0.7

Figure 10.9 geff as a function of θ for the rotation of the magnetic field in the

011-plane for two temperatures (T = 1.8 and 2.65 K) and 60 GHz from Ref. [49]. The solid curve is the fit to theory [48, 52] for T = 2.65 K with ϕ = 0.18π and K ∗ = 0.48. Stronger AFM fluctuations are expected at 1.8 K due to the proximity of phase III . Reproduced from Ref. [11].

Ref. [49] and would change the conclusions by Semeno et al. All data are taken in the AFQ phase II. For 2.65 K the system is in phase II for all fields, while for 1.8 K the resonance positions vary between 2.4 and 2.7 T, i.e., they are not far away from the boundary with phase III . In this region antiferromagnetic spin fluctuations are expected to interfere, and the present theory is no longer applicable without caution. Note that a pure AFQ phase in the absence of magnetic field is a “magnetically hidden order” phase which cannot be observed by neutron diffraction. The fact that only one resonance is observed and the single-ion picture yields two lines, one for each sublattice, shows that the signal in CeB6 can only be interpreted within the itinerant electron picture. This is in contrast to the compounds discussed before, e.g., YbRh2 Si2 , which allow an analysis as a single-site or a collective resonance. CeB6 clearly shows that ESR in heavy-fermion systems is a collective phenomenon.

793

794 Theory of Electron Spin Resonance in Strongly Correlated CeB6

10.4.4 Ferromagnetic Correlations in Phase II of CeB6 We have argued that the line width of the resonance in a paramagnetic Kondo lattice is of the order of TK and too broad to be observed, unless there are ferromagnetic correlations among the moments. ESR in compounds with ferromagnetic correlations were found in single crystals of, e.g., YbRh2 Si2 [14, 15, 24], YbIr2 Si2 [25], YbRh [26], YbCo2 Zn20 [27], and CeRuPO [28]. Common to all these compounds is a strong magnetic anisotropy (except for CeB6 ) with the easy-axis in the ab-plane, which favors ferromagnetic correlations among the rare-earth spins [26]. Hence, the resonances can be attributed to the Yb3+ and Ce3+ ions despite of their rather large TK . On the other hand, the isotropy of the cubic CeB6 lattice is, in principle, not favorable for ferromagnetic correlations. However, ferromagnetic fluctuations have to suppress the Curie–Weiss temperature arising from the heavy-fermion band (or the Kondo effect). Below we discuss the mechanism leading to ferromagnetic spin correlations in CeB6 and hence to an observable ESR signal within the AFQ ordered phase. A minimal model to study the correlations in CeB6 requires to consider the states of two f electrons on neighboring sites considering all internal degrees of freedom, i.e., coordinates, orbital (quadrupolar) and spin. The two-particle wave functions consist then of the product of three factors, namely, a coordinate wave function, a factor involving only quadrupolar degrees of freedom, and a wave function of the spin indices [48, 52]  ∼ ψcoor (r1 , r2 )ψorb (m1 , m2 )ψspin (σ1 , σ2 ) .

(10.24)

According to Pauli’s principle, fermion wave functions, such as , are antisymmetric under the interchange of the electrons, i.e., the indices 1 and 2, implying that either one or all three of the three factors in Eq. (10.24) are antisymmetric and the remaining ones symmetric. Since CeB6 has no charge density wave, the coordinate wave function at each site must be the same, i.e., ψcoor (r1 , r2 ) = ϕ(r1 )ϕ(r2 ), giving rise to a homogeneous charge distribution. Since ψcoor is necessarily a symmetric function, then either ψorb (m1 , m2 ) or ψspin (σ1 , σ2 ) must be antisymmetric and consequently the other function symmetric. We now suppose that the effective interaction

Antiferroquadrupolar Ordered CeB6

between the two sites is a quadrupolar exchange, i.e., Hint = aτ1 · τ2 . Here a is the quadrupolar exchange, which necessarily has to be positive to be able to generate AFQ order. Consequently, ψorb (m1 , m2 ) represents an orbital singlet which has odd parity. Hence, the wave function of the spins has to have even parity and must be a triplet. It follows that the spins are then ferromagnetically correlated. On the lattice the quadrupolar singlet state cannot be satisfied for all bonds (pairs of neighboring sites) simultaneously. This generates a resonant valence bond lattice for the quadrupolar degrees of freedom. On the other hand, the spins are aligned by an external magnetic field, enhancing this way the ferromagnetic alignment (order) and hence also the antiferro-orbital correlations. Consequently it stabilizes the orbital order. Hence, the Tc of the phase boundary between phase I (para-quadrupolar disordered Kondo) and phase II (AFQ long-range order) increases with magnetic field, as shown in Fig. 10.5. It is interesting to notice that the rate of increase of Tc with field decreases at higher fields [53,54], but does not saturate up to 35 T. This reduced increment of Tc with field is due to the orbital resonant valence bond lattice, which cannot satisfy all bonds simultaneously, but tends to be quenched when the magnetic field is very large.

Figure 10.10 (a) Inverse magnetic susceptibility of CeB6 as a function of temperature. The solid line is a guide to the eye. (b) Magnetization as a function of external field parallel to the [100] crystallographic direction for several different temperatures. The kink at lower T is the transition between phases I and II. Reproduced from Terzioglu et al. [55], copyright by the American Physical Society.

795

796 Theory of Electron Spin Resonance in Strongly Correlated CeB6

The left panel of Fig. 10.10 shows the inverse susceptibility of CeB6 over a large temperature interval. At high T both the ground-state 8 and the excited 7 multiplets contribute. This is the spin-fluctuation regime. As T is lowered, the 7 freezes out and χ increases gradually giving rise to a strongly reduced Weiss temperature. This is the Kondo regime or phase I. In the right panel we observe the magnetization as a function of magnetic field. The initial slope is the susceptibility which increases when T decreases, as already seen in the left panel. At low T the magnetization has a kink; fields below the kink correspond to phase I, while fields above the kink belong to with phase II. At the phase boundary Tc there is then an abrupt increase of χ associated with the ferromagnetic correlations among the spins. Hence, the magnetic susceptibility increases quite dramatically, as a consequence of a reduction of the Weiss temperature. In phase I at low T the Weiss temperature θ is of the order of TK , and in phase II it appears to have changed sign, i.e., the correlations are clearly ferromagnetic [50] (see also Fig. 5 of Ref. [56]). The magnetization measurements yield the longitudinal susceptibility, but we expect χ0T to behave analogously. The inverse proportionality of the static transversal susceptibility and the relaxation rate given by Eq. (10.19) then have the consequence that an increase in χ0T reduces the ESR line width, and the resonance becomes observable. A similar conclusion, although with different arguments, has been presented in Refs. [49] and [50]. In systems with Ce3+ and Nd3+ ions with 8 ground state, the quadrupolar degrees of freedom play an important role. In the first place they manifest themselves through interactions among the sites. However, there is no consensus about the origin of the interactions. Using nearest-neighbor intersite exchange and quadrupolar interactions, Kubo and Kuramoto [57] were successful in describing the excitation spectrum of NdB6 . Uimin and Brenig [58], on the other hand, proposed a different approach emphasizing crystal fields. For CeB6 , quadrupolar interactions between sites [59], the RKKY interaction arising from the Coqblin-Schrieffer model [60, 61], and a detailed group-theoretical study [62] have been presented. (2) The quadrupolar degrees of freedom of the 8 of Nd impurities in the metallic host LaB6 also give rise to interesting effects [63]. The

Antiferroquadrupolar Ordered CeB6

Figure 10.11 (a) Low-temperature magnetic susceptibility for the La1−x Ndx B6 (x = 0.0095) single crystal. A χ ∝ − ln(T ) variation is shown (2) for T ≤ 1 K. The ground multiplet of the Nd3+ ions is the 8 quartet. (b) Temperature dependence of the magnetic specific heat (C m ) per Nd ion for La1−x Ndx B6 single crystals with x = 0.0095 and 0.0295. The same data, plotted as C m /T vs. log(T ) are shown in the inset. Note again the − ln(T ) dependence at low T, a behavior that is closely analogous to the spin 1/2 two-channel Kondo problem. Reproduced from Stankiewicz et al. [63], copyright by the American Physical Society.

magnetic susceptibility and the magnetic specific heat over T both show a divergent logarithmic dependence at low T (see Fig. 10.11), which is characteristic of the n = 2 multichannel Kondo problem for S = 1/2 (quadrupolar Kondo effect). Note that the Kondo screening is incomplete and the screening length rather large. On the other hand, the Nd sites are not sufficiently far apart to be considered as isolated impurities, so that they interact with each other via local lattice distortions which couple to the quadrupolar (but not spin) degrees of freedom of the f electrons. This interaction, together with correlations mediated by the conduction electrons, could lift the (2) degeneracy of the 8 , in analogy to the effect of a magnetic field in the case of the S = 1/2 two-channel Kondo problem.

10.4.5 Line Width of ESR in Phase II of CeB6 The T-dependence of the line width of the Dysonian resonance for the three principal crystallographic directions (data from Ref. [49]) is shown in Fig. 10.12. The solid straight line corresponds to a Korringa relaxation for a small interval at intermediate T . The

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798 Theory of Electron Spin Resonance in Strongly Correlated CeB6

1.5

W [T]

[100]

1.0 [111] [110]

0.5

0.0 1.5

2.0

2.5

3.0 T [K]

3.5

4.0

Figure 10.12 Line width W of CeB6 at 60 GHz as a function of temperature for the three principal axes: The black triangles correspond to [100], the open squares to [110], and the open circles to [111]. The straight line represents a Korringa relaxation for intermediate T and the [110] and [111] directions. The dashed line is a parabolic fit. Reproduced from Semeno et al. [49].

figure clearly shows that the line width W (or 1/Trel ) does not have a Korringa-like T-dependence, but a more complicated one. In Section 10.3 we argued that 1/Trel is inversely proportional to the susceptibility. It is 1/χ0T that provides the dominant T-dependence for 1/Trel . We can conclude that the susceptibility is smaller in the [100] direction than in the other two main directions. The anomalous χ0T (T )-dependence is due to magnetic fluctuations in phase II for sufficiently large fields, e.g., the difference between the straight line (Korringa) and the dashed curve. At the highest temperatures the slight increase of the relaxation rate could also be the beginning of the onset of the Orbach spin-lattice relaxation mechanism [64] into the excited crystal-field 7 doublet, i.e., the modulation of the crystal field through the phonons. Due to the excitation into the 7 doublet the increase of 1/Trel shows only the tail of an exponential activation.

Antiferroquadrupolar Ordered CeB6

Figure 10.13 Magnetic field dependence of the magnetization of CeB6 along the three principal crystallographic axes at T = 1.4 K [40]. The kinks indicate the magnetic transitions from phases III to III to II. At high fields the magnetization along [100] is smaller than for the other two main directions. Reproduced from Kunimori et al. [40], copyright by the Physical Society of Japan.

The g-shift (Knight-shift) for larger fields is proportional to the susceptibility of the conduction states. Hence, the Knight-shift subtraction along the [100]-direction is expected to be smaller for this orientation, and consequently geff is larger at lower T as seen in Fig. 10.9. In Fig. 10.13 we show the magnetization of the Ce ions along the three principal axes as a function of the magnetic field for T = 1.4 K [40]. This would correspond to a field sweep along a vertical line at 1.4 K in Fig. 10.5. The kinks of the curves indicate the phase transitions from phases III to III to II. At high fields the magnetizations along the [110] and [111] directions are essentially the same while along [100] the magnetization is somewhat smaller, so that a smaller χ0T and hence larger line width W is expected in that direction (see Fig. 10.12).

799

800 Theory of Electron Spin Resonance in Strongly Correlated CeB6

10.4.6 Second Resonance at High Fields in Phase II of CeB6 So far we only considered resonances between the initial and final states belonging to the AFQ condensate in phase II, i.e., both states are part of the ordered phase. At high resonance frequencies, however, the final state of the transition may be outside the energy range of the ordered phase and corresponds to the Kondo phase I. As argued in Section 10.4.3, the number of final states is larger in the paramagnetic phase I, since all states of the 8 are in principle available and a second (and perhaps third) transition may arise. In other words, fewer transitions are quenched if the final state belongs to phase I. Assuming a mean-field BCS-type for the condensate, the energy of the AFQ order parameter is approximately = 1.75 × kB Tc . At the relevant fields, we have Tc ≈ 6 K and ≈ 10 K ≈ 200 GHz. This implies that for frequencies less than 200 GHz, the initial and the final states of the transition have to correspond to the AFQ ordered phase, and as argued in Section 10.4.3, only one transition will be observed. On the other hand, for frequencies larger than 200 GHz the initial (ground) state is in the condensate, while the final (excited) state should be a free ion (8 ) state. A second resonance was detected by Demishev et al. [65] in ESR of CeB6 for the field in the [110] crystallographic direction for frequencies exceeding 200 GHz. The g-factor of the secondary resonance is considerably smaller than that of the primary line. The intensity of the secondary line is also less than that of the primary resonance. The two resonance energies are shown in Fig. 10.14 as a function of magnetic field (figure adapted from Ref. [66]). The splitting is approximately linear in the magnetic field indicating that it is a Zeeman splitting. In Fig. 4 of Ref. [65], the g-values are plotted as a function of frequency showing a small field dependence. A calculation of these resonances is rather complicated since in the AFQ-ordered state, the 8 states and the coherence of the Anderson lattice need to be invoked. Furthermore, it is difficult to describe the transition region from the AFQ-phase to the free 8 state regime. It is interesting to note that the main ESR resonance was also observed via inelastic neutron scattering as a collective magnetic excitation at the zone center (Γ point). The magnetic

Antiferroquadrupolar Ordered CeB6

1.6 1.4

ΔE [meV]

1.2 1.0 0.8 0.6 0.4 0.2 0.0 0

5

10 B [T]

15

20

Figure 10.14 Magnetic field dependence of the two resonances. One is observed for all fields (open circles) and the other one (open squares) is only seen at high magnetic fields above 12 T [65]. The dashed straight lines correspond to a pure Zeeman splitting with geff ≈ 1.7 and geff ≈ 1.3, respectively. Data reproduced from Portnichenko et al. [66], copyright by the American Physical Society.

fields were too weak to reach the region of the second resonance. The agreement is perfect in phase II. INS measurements have been extended to lower fields, where in phase III the resonance deviates from a Zeeman splitting. While ESR only measures at the Γ point, INS is not restricted to q = 0. It is therefore remarkable that the resonance was also observed at the R( 12 21 21 ) point in both phases II and III with a similar energy proportional to the magnetic field (Zeeman effect).

10.4.7 Inelastic Neutron Scattering in CeB6 The ESR results for CeB6 are complementary and closely related to those of inelastic neutron scattering (INS), which are addressed in detail in Chapter 9. While standard ESR measures the transversal dynamical magnetic susceptibility for q = 0, INS realizes the longitudinal dynamical magnetic susceptibility (see next section) as a function of wave vector q and frequency ω. In the paramagnetic

801

802 Theory of Electron Spin Resonance in Strongly Correlated CeB6

Kondo phase, the dynamical susceptibility has a quasielastic peak (centered at ω = 0) of width 1/T1 given by the maximum of T and TK . In phase II at zero-field the neutrons do not couple to the ordered quadrupolar moments. This phase is frequently called a “hidden order” phase with QAFQ = R( 12 21 21 ). A finite magnetic field breaks the symmetry and induces magnetic moments which can be detected by neutrons. Phase III has antiferromagnetic long-range order with QAFM1 = Σ( 14 41 0) and in phase III the Q-vector changes to QAFQ2 = S( 14 41 21 ). Both phases lead to neutron signals. The phase diagram is shown in Fig. 10.5. There are at least four types of magnetic excitations in CeB6 that have been observed by INS: (1) A resonant magnetic exciton mode [67], similar to the ones found in unconventional superconductors [68], including heavyfermion superconductors (CeCu2 Si2 , CeCoIn5 , CeRu2 Al10 ) [69, 70], was observed at R( 12 21 21 ), but in phase III. The mode is nondispersive, sharply peaked and associated with the opening of a spin gap at low energies. The spin gap is the consequence of the magnetic order, since for T > TN the resonance peak shifts to ω = 0 and becomes the quasielastic peak of the paramagnetic state. A theoretical interpretation of the resonant exciton mode was provided by Akbari and Thalmeier [71] based on a fourfold degenerate Anderson lattice model with the AFQ and AFM order parameters phenomenologically incorporated. (2) At the Γ point (zone center), a strong FM soft mode was observed [72]. These ferromagnetic fluctuations are large in phase III but also present (although weaker) in phase II. However, no dispersive magnon excitations were found in the AFQ phase in zero field. The intensity of the magnetic excitations collapses into a broad central peak at zero energy (quasielastic peak) just above TN . The INS line width is smallest at the Γ point. The ferromagnetic fluctuations are expected to be enhanced in a magnetic field and are the reason for an accessible ESR signal in CeB6 . (3) Conventional spin-wave modes emanate from the AFM wave vectors QAFM1 and QAFM2 below TN . They display a spin gap of

Antiferroquadrupolar Ordered CeB6

about 0.3 to 0.4 meV and at the zone boundary (M point) the modes reach up to 0.7 meV. Hence, the spin gap and the band width are comparable. All the above excitations merge to form a continuous dispersive magnon band in a narrow energy range. The band has a broader band width in the AFQ phase, when it is stabilized by an applied magnetic field. (4) In some unconventional superconductors, such as CeCoIn5 , a strong magnetic field splits the resonant magnetic exciton mode into two components. This is not the case for CeB6 , where a second field-induced magnon mode emerges instead, whose energy increases with magnetic field [66]. At the FM zone center (Γ point) only a single mode is found with a non-monotonic field dependence in phase III. Inside the hidden-order phase it agrees well with the ESR resonance energy (Fig. 10.9). INS measurements in the field range of the second (high-field) ESR resonance have not been carried out. It is interesting to point out that this secondary INS response occurs also at the R point, which is not accessible by ESR. INS and ESR results are complementary, and there is still much work to be done to understand the magnetic correlations in CeB6 .

10.4.8 Summary The AFM correlations in heavy-fermion compounds lead to a broad ESR line, which generally cannot be observed. Exceptions are compounds with very strong magnetic anisotropy, where ferromagnetic correlations reduce the line width as discussed in Section 10.3. CeB6 represents an exception to the exceptions, since it is a cubic Kondo lattice (no significant magnetic anisotropy) and a resonance was observed in the AFQ phase. This situation needs a separate explanation. Each Ce ion has a fourfold degenerate 8 ground state, displaying spin and orbital degrees of freedom. We concluded from the antisymmetry of the wave functions of the electrons that there is a strong interplay between AFQ and ferromagnetic order of the

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804 Theory of Electron Spin Resonance in Strongly Correlated CeB6

spins at the Γ point. This state is favored by a magnetic field and consequently the Tc of the boundary between the Kondo and the AFQ phases increases with B, which agrees with the experiment. Furthermore, the susceptibility is increased by the ferromagnetic correlations as observed experimentally, noticed especially at the transition from phase I to phase II, where χ0 displays a kink [50, 56]. As a consequence the ferromagnetic correlations reduce the ESR line width, which this way is accessible to observation in phase II. Previous ESR experiments on heavy-fermion systems with Kramers doublets, e.g., YbRh2 Si2 [14, 15, 25, 26], have originally all been interpreted as if localized moments are resonating, i.e., as for ESR on individual impurities. The local dynamical susceptibility is the proper response function in that case, rather than the global dynamical susceptibility [17, 18]. However, it is difficult to distinguish between the two approaches [39], since more than 60% of the Yb ions participate in the resonance [16]. The fact that only one resonance was observed in CeB6 , in conjunction with the analysis in Ref. [18], is evidence that the ESR signal is a collective phenomenon involving all the sites of the lattice. The 8 ground quartet of a single Ce3+ site leads to expect four resonances. Three of these resonances are quenched by the AFQ order at each site and the coherence of the wave functions reduces the signal to one resonance. The first experimentally observed g-value [29, 30, 50] is about 1.6 for the field pointing into the [110] direction. In general the g-value depends on the angle θ of the magnetic field with the crystallographic axis. The theoretical effective g-factor depends then on two angles, θ and ϕ, and the Knight shift. Here ϕ is the angle of the quadrupolar order relative to the crystallographic axis. There is good agreement between theory and experiment at T = 2.65 K for a proper choice of parameters (in contrast to statements in Ref. [49]). For T = 1.8 K the system is too close to phase III and antiferromagnetic fluctuations are going to affect the measurement, rendering the theory inapplicable in that regime. Both, the 8 multiplet and AFQ order are crucial ingredients for the observability of an ESR signal in a cubic environment.

Longitudinal Dynamical Susceptibility 805

10.5 Longitudinal Dynamical Susceptibility In a configuration in which the oscillating magnetic field of the microwaves is parallel to the Zeeman field (assumed along the z axis) the resonance is given by the longitudinal dynamical susceptibility,  χ L (z) = −(gf μB )2 N1 i j

Siz ; S zj z . This correlation function, but also as a function of a wave vector q, contains the response to inelastic neutron scattering (INS). We consider the same Hamiltonian as Section 10.3, namely Eq. (10.13), together with the Knight-shifted magnetic field, Eq. (10.6). To higher order in the Kondo exchange J the model generates a Heisenberg exchange between the different sites (RKKY-interaction). In Section 10.3 this spin-exchange was incorporated into the expectation values of spin operators and the static susceptibility. The same procedure will be followed here. The longitudinal response for INS consists then of a quasielastic central peak of width 1/T1 and inelastic peaks arising from the transition into spin-excited states by emission or absorption of ω(q). To study the latter it is convenient to artificially include a Heisenberg Hamiltonian with the appropriate symmetries. Here we limit ourselves to study the quasielastic peak. As before for the transversal susceptibility, we apply the equation of motion to the first operator argument of the susceptibility [2, 3], z

Siz ; S zj z =

[H, Siz ]; S zj z =

jiL ; S zj z .

(10.25)

The above equation defines the spin-current jiL at site Ri , i.e., J iL =

J  i (k−k )·Ri + † − † e (Si ckσ sσ σ  ck σ  − Si− ckσ sσ+σ  ck σ  ) . (10.26) 2N kk σ σ 

Applying the equation of motion on the second argument of the correlation function yields z

jiL ; S zj z = [ jiL , S zj ] −

jiL ; j Lj z ,

(10.27)

which is now evaluated for the noninteracting system. The Bloch equation for the longitudinal component can be written in the form [2, 3] "−1 ! N L (z) , (10.28) χ L (z) = z + N L (z)/χ0L

806 Theory of Electron Spin Resonance in Strongly Correlated CeB6

where χ0L is the static longitudinal magnetic susceptibility and N L (z) is the longitudinal relaxation function. To second order in the exchange N L (z) is given by [2, 3] 1  N L (z) ≈ −(gf μB )2 z

Siz ; S zj z N ij $ 1  1 # ! L z" = −(gf μB )2

ji , S j  −

jiL ; j Lj z . (10.29) N ij z Evaluating the imaginary part of N L (z) for the noninteracting system we obtain π  N L (ω) = (gf μB JρF )2 S z {2 coth(gf μB B /2T ) 4 ω + gf μB B  coth[(ω + gf μB B  )/2T ] − ω ω − gf μB B  coth[(ω − gf μB B  )/2T ]} + ω  sin(kF Ri j )2 π e−Ri j /l (gf μB JρF )2 + 2N (kF Ri j )2 i j, i = j   × Si+ S −j  + Si− S +j  . (10.30) Here we considered the Zeeman field of the conduction electrons only via the Knight shift. The first three terms are the single site contributions, while the last two terms arise from the intersite interactions. We first analyze the single site terms, which correspond to the  single impurity case. For ω → 0 the function N L (ω) reduces to 

N L (ω → 0) =

π gf μB B /2T . (gf μB JρF )2 S z  2 sinh2 (gf μB B /2T )

(10.31)

For gf μB B   T this expression becomes the constant (π/4) × (gf μB JρF )2 , and with χ0L = (gf μB )2 /4T (Curie law) we obtain the Korringa relaxation rate 1/T1 = π(JρF )2 T. This is the same expression as for 1/T2 , since for gf μB B   T the dynamical susceptibility is isotropic, i.e., the longitudinal and transversal susceptibilities are  identical. In the limit gf μB B   T, on the other hand, N L (ω = 0) tends to zero exponentially as exp(−gf μB B /T ), because the spinflips are suppressed by the magnetic field, i.e., the energy of the thermal bath is too small for a spin-flip. The suppression

Conclusions 807

of spin-flips also reduces the static susceptibility exponentially, χ0L = (gf μB )2 /[4T cosh2 (gf μB B /2T )], so that the spin-flip relaxation rate in the high-field limit is 1/T1 = (π/2)(JρF )2 gf μB B  , i.e., T1 = T2 /2 [2]. The width of the quasielastic central peak of the longitudinal dynamical susceptibility is 1/T1 and its spectral weight is roughly χ0L . In addition to the central peak, the longitudinal dynamical response has a shoulder at low T as a function of ω close to ±gf μB B  . The origin of this shoulder is the following. At low T the thermal bath is unable to provide sufficient energy to flip the spin, unless the external frequency is larger than gf μB B  . The Kondo effect is introduced to higher-order perturbation in J and impacts both the relaxation kernel N L (z) and the static susceptibility χ0L [73]. Eventually, as T → 0 the impurity spin is compensated by the conduction electrons, leading to a finite susceptibility and a finite 1/T1 relaxation rate [4, 5], in analogy to the transverse response. In a small magnetic field, the Kondo effect smears the above-mentioned properties and the characteristic energy of the central peak is the larger of T and TK . The intersite terms play a substantial role only in the presence of collective excitations, so that expectation values of spin-flips at different sites are nonzero. As for the transversal response function most of the physics is dominated by the static response. Ferromagnetic intersite correlations therefore again narrow the central peak, while antiferromagnetic correlations tend to broaden the quasielastic peak.

10.6 Conclusions An ESR signal for a magnetic impurity in a metal is not observable unless TK is very small. This suggests that the line width in heavy-fermion compounds is too broad for measurements with conventional ESR techniques. This general belief had to be corrected, when a resonance was found in several heavy-fermion Ce and Yb compounds. Common to these compounds are ferromagnetic shortrange correlations among the rare-earth moments due to a strong magnetic anisotropy usually in the ab-plane. The observed signals

808 Theory of Electron Spin Resonance in Strongly Correlated CeB6

have the Dysonian line shape as expected from spin diffusion in a metallic environment. The static transversal susceptibility and the ESR relaxation rate are inversely proportional to each other. For noninteracting impurities χ0T is a Curie law and hence 1/T2 ∝ T, known as the Korringa relaxation rate, which alternatively can also be obtained from integrating over the Fermi–Dirac distribution functions of the conduction states. The Kondo exchange interaction of the resonating spin with the conduction electrons gives rise to the Kondo spin compensation and the susceptibility becomes finite. As a consequence, the ESR line width is proportional to TK and is too large for the resonance to be observed, unless TK is much less than the microwave frequency as for Yb impurities in Au [6]. In a compound there are in addition short-range interactions among the localized moments, which play a crucial role. For heavy-fermion compounds, involving Ce3+ and Yb3+ ions, χ0T is inversely proportional to the band width (TK ). For antiferromagnetically correlated spins, the width of the line is of the order of the Weiss temperature θ of the static transversal susceptibility, which is usually too large for conventional ESR to be measured. For ferromagnetically correlated moments, on the other hand, the width of the resonance is strongly reduced, and it becomes possible to measure an ESR line. This situation can be considered a bottleneck [15] since the FM short-range correlations inhibit the spin-flip from being passed on to other sites. Similar conclusions for Kondo impurities and the Kondo lattice (i.e., involving localized ¨ moments) have been derived by Abrahams and Wolfle for the Anderson impurity and Anderson lattice [17, 18]. Due to spin diffusion and the skin depth, the characteristic ESR line shape in a metal is Dysonian. This is the case for both, resonating localized moments [36] and conduction states [32–34]. For heavyfermion compounds, the shape of the line can then not distinguish between localized moments and conduction states in a narrow band (heavy mass). The crystal-electric field scheme determines the g-tensor for the localized spins, i.e., the g-values and their anisotropy. On the other hand, the hybridization of the f electrons with the conduction states gives rise to the heavy-fermion states. In the neighborhood of the Fermi level they are dominated by the f character and, consequently, the g-tensor is again predominantly

Conclusions 809

given by the crystal-field scheme of the f states. Hence, from the g-tensor we once again cannot distinguish between resonating localized and conducting states. CeB6 is a cubic Kondo lattice with no significant magnetic anisotropy in the AFQ phase. The observation of a signal in CeB6 is then an exception to the exceptions. The 8 ground state of each Ce ion is a quartet containing orbital and spin degrees of freedom. Naively, this should give rise to four resonance lines; however, only one was observed in the usual range of microwaves. The AFQ longrange order and the coherence of the electron states due to the hybridization are necessary to quench the remaining transitions. The agreement of the angular dependence of the spectrum in the AFQ phase with theory is remarkable. At low T there are differences because of the proximity of phase III , which induces AF correlations not included in the theory. The Pauli principle requires antisymmetric electron wave functions. It follows that the spins are ferromagnetically coupled to each other in order to have AFQ correlations. A magnetic field favors this state, and the Tc of the phase boundary between the Kondo and AFQ phases increases with field, in agreement with experiment. In addition, the ferromagnetic correlations enhance the magnetic susceptibility [50, 56] and hence reduce the ESR line width, which then becomes accessible to observation. Demishev et al. [65] detected a second ESR resonance line in ESR for frequencies larger than 200 GHz. The g-factor of the secondary resonance and its intensity are considerably less than that of the primary line. The second resonance can be explained as a transition from the AFQ-ordered resonance into a final state belonging to phase I (Kondo phase). The INS results for CeB6 have provided insights into the low T ordered phases and their spin correlations. More work is still needed to relate the INS experiments to the ESR results.

Acknowledgments The author acknowledges support by the U.S. Department of Energy under grant No. DE-FG02-98ER45707.

810 Theory of Electron Spin Resonance in Strongly Correlated CeB6

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812 Theory of Electron Spin Resonance in Strongly Correlated CeB6

Yb-based heavy-fermion compounds: new experimental data”; J. Alloys Compd. 480, 126–127 (2009). 28. Bruning, E. M., Krellner, C., Baenitz, M., Jesche, A., Steglich, F., and Geibel, C.; “CeFePO: a heavy fermion metal with ferromagnetic correlations”; Phys. Rev. Lett. 101, 117206 (2008). 29. Demishev, S. V., Semeno, A. V., Paderno, Y. B., Shitsevalova, N. Y., and Sluchanko, N. E.; “Experimental evidence for magnetic resonance in the antiferro-quadrupole phase”; Phys. Status Solidi B 242, R27–R29 (2005). 30. Demishev, S. V., Semeno, A. V., Bogach, A. V., Paderno, Y. B., Shitsevalova, N. Y., and Sluchanko, N. E.; “Magnetic resonance in cerium hexaborite caused by quadrupolar ordering”; J. Magn. Magn. Mater. 300, e-534–e537 (2006). 31. Demishev, S. V.; “Electron spin resonance in strongly correlated metals”; Appl. Magn. Reson. 51, 473–522 (2020). 32. Feher, G., and Kip, A. F.; “Electron spin resonance absorption in metals. I. Experimental”; Phys. Rev. 98, 337–348 (1955). 33. Dyson, F. J.; “Electron spin resonance absorption in metals. II. Theory of electron diffusion and the skin effect”; Phys. Rev. 98, 349–359 (1955). 34. Pake, G. E., and Purcell, E. M.; “Line shapes in nuclear paramagnetism”; Phys. Rev. 74, 1184–1188 (1948). 35. Ye, M., Kung, H.-H., Rosa, P. F. S., Bauer, E. D., Fisk, Z., and Blumberg, G.; “Raman spectroscopy of f-electron metals: an example of CeB6 ”; Phys. Rev. Mater. 3, 065003 (2019). 36. Rettori, C., Davidov, D., Orbach, R., Chock, E. P., and Ricks, B.; “Electron spin resonance of rare earths in aluminum”; Phys. Rev. B 7, 1–12 (1973). 37. Rettori, C., Davidov, D., and Kim, H. M.; “Crystalline field effects in the EPR of Er in various cubic metals”; Phys. Rev. B 8, 5335–5337 (1973). 38. Rettori, C., Kim, H. M., Chock, E. P., and Davidov, D.; “Dynamic behavior of paramagnetic ions and conduction electrons in intermetallic compounds: Gdx Lu1−x Al2 ”; Phys. Rev. B 10, 1826–1835 (1974). 39. Schlottmann, P.; “Electron spin resonance in heavy fermion systems”; Phys. Rev. B 79, 045104 (2009). 40. Kunimori, K., Kotani, M., Funaki, H., Tanida, H., Sera, M., Matsumura, T., and Iga, F.; “Existence region of phase III’ in CeB6 ”; J. Phys. Soc. Jpn. 80, SA056 (2011). 41. Fujita, T., Suzuki, M., Komatsubara, T., Kunii, S., Kasuya, T., and Ohtsuka, T.; “Anomalous specific heat of CeB6 ”; Solid State Commun. 35, 569–572 (1980).

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¨ ¨ 42. Zirngiebl, E., Hillebrands, B., Blumenroder, S., Guntherodt, G., Loewenhaupt, M., Carpenter, J. M., Winzer, K., and Fisk, Z.; “Crystal field excitations in CeB6 studied by Raman and neutron spectroscopy”; Phys. Rev. B 30, 4052–4054 (1984). ¨ ¨ 43. Zirngiebl, E., Hillebrands, B., Blumenroder, S., and Guntherodt, G.; “New crystal-field level scheme of CeB6 deduced from Raman and neutron spectroscopy”; J. Appl. Phys. 57, 3769 (1985). 44. Martins, G. B., Rao, D., Barberis, G. E., Rettori, C., Duro, R. J., Sarrao, J., Fisk, Z., Oseroff, S., and Thompson, J. D.; “Electron spin resonance of Er3+ in YBiPt”; Phys. Rev. B 52, 15062–15065 (1995). 45. Bierig, R. W., and Weber, M. J.; “Paramagnetic resonance of Dysprosium in cubic and axial fields in CaF2 ”; Phys. Rev. 132, 164–167 (1963). 46. Davidov, D., Orbach, R., Rettori, C., Tao, L. J., and Chock, E. P.; “Anisotropic behavior of dilute Au:Dy alloys: observation of the 8 (quartet) resonance”; Phys. Rev. Lett. 28, 490–493 (1972). 47. Lea, K. R., Leask, M. J. M., and Wolf, W. P.; “The raising of angular momentum degeneracy of f-electron term by cubic crystal fields”; J. Phys. Chem. Solids 23, 1381–1405 (1962). 48. Schlottmann, P.; “Electron spin resonance in antiferro-quadrupolarordered CeB6 ”; Phys. Rev. B 86, 075135 (2012). 49. Semeno, A. V., Gilmanov, M. I., Bogach, A. V., Krasnorussky, V. N., Samarin, A. N., Samarin, N. A., Sluchanko, N. E., Shitsevalova, N. Y., Filipov, V. B., Glushkov, V. V., and Demishev, S. V.; “Magnetic resonance anisotropy in CeB6 : an entangled state of the art”; Sci. Rep. 6, 39196 (2016). 50. Demishev, S. V., Semeno, A. V., Bogach, A. V., Samarin, N. A., Ishchenko, T. V., Filipov, V. B., Shitsevalova, N. Y., and Sluchanko, N. E.; “Magnetic spin resonance in CeB6 ”; Phys. Rev. B 80, 245106 (2009). 51. Effantin, J., Rossat-Mignod, J., Burlet, P., Bartholin, H., Kunii, S., and Kasuya, T.; “Magnetic phase diagram of CeB6 ”; J. Magn. Magn. Mater. 47– 48, 145–148 (1985). 52. Schlottmann, P.; “Electron spin resonance in CeB6 ”; J. Appl. Phys. 113, 17E109 (2013). 53. Hall, D., Fisk, Z., and Goodrich, R. G.; “Magnetic-field dependence of the paramagnetic to the high-temperature magnetically ordered phase transition in CeB6 ”; Phys. Rev. B 62, 84–86 (2000). 54. Goodrich, R. G., Young, D. P., Hall, D., Balicas, L., Fisk, Z., Harrison, N., Betts, J., Migliori, A., Woodward, F. M., and Lynn, J. W.; “Extension of the temperature-magnetic field phase diagram of CeB6 ”; Phys. Rev. B 69, 054415 (2004).

814 Theory of Electron Spin Resonance in Strongly Correlated CeB6

55. Terzioglu, C., Browne, D. A., Goodrich, R. G., Hassan, A., and Fisk, Z.; “EPR and magnetic susceptibility measurements on CeB6 ”; Phys. Rev. B 63, 235110 (2001). 56. Terzioglu, C., Ozturk, O., Kilic, A., Goodrich, R. G., and Fisk, Z.; “Magnetic and electronic measurements in CeB6 ”; J. Magn. Magn. Mater. 298, 33– 37 (2006). 57. Kubo, K., and Kuramoto, Y.; “Magnetic and quadrupolar interactions in NdB6 ”; J. Phys.: Condens. Matter 15, S2251–S2254 (2003). 58. Uimin, G., and Brenig, W.; “Crystal field, magnetic anisotropy, and excitations in rare-earth hexaborides”; Phys. Rev. B 61, 60–63 (2000). 59. Uimin, G., Kuramoto, Y., and Fukushima, N.; “Mode coupling effects on the quadrupolar ordering in CeB6 ”; Solid State Commun. 97, 595–598 (1996). 60. Ohkawa, F. J.; “Orbital antiferromagnetism in CeB6 ”; J. Phys. Soc. Jpn. 54, 3909–3914 (1985). 61. Schlottmann, P.; “RKKY interaction between Ce ions in Cex La1−x B6 ”; Phys. Rev. B 62, 10067–10075 (2000). 62. Shiina, R., Shiba, H., and Thalmeier, P.; “Magnetic-field effects on quadrupolar ordering in a 8 -quartet system CeB6 ”; J. Phys. Soc. Jpn. 66, 1741–1755 (1997). 63. Stankiewicz, J., Evangelisti, M., Fisk, Z., Schlottmann, P., and Gor’kov, L.; “Kondo physics in a rare earth ion with well localized 4f electrons”; Phys. Rev. Lett. 108, 257201 (2012). 64. Orbach, R.; “Spin-lattice relaxation in rare-earth salts”; Proc. Royal Soc. A 264, 458–484 (1961). 65. Demishev, S. V., Semeno, A. V., Ohta, H., Okubo, S., Paderno, Y. B., Shitsevalova, N. Y., and Sluchanko, N. E.; “High frequency study of the orbital ordering resonance in the strongly correlated heavy fermion metal CeB6 ”; Appl. Magn. Reson. 35, 319–326 (2008). 66. Portnichenko, P. Y., Demishev, S. V., Semeno, A. V., Ohta, H., Cameron, A. S., Surmach, M. A., Jang, H., Friemel, G., Dukhnenko, A. V., Shitsevalova, N. Y., Filipov, V. B., Schneidewind, A., Ollivier, J., Podlesnyak, A., and Inosov, D. S.; “Magnetic field dependence of the neutron spin resonance in CeB6 ”; Phys. Rev. B 94, 035114 (2016). 67. Friemel, G., Li, Y., Dukhnenko, A. V., Shitsevalova, N. Y., Sluchanko, N. E., Ivanov, A., Filipov, V. B., Keimer, B., and Inosov, D. S.; “Resonant magnetic exciton mode in the heavy-fermion antiferromagnet CeB6 ”; Nat. Commun. 3, 830 (2012).

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68. Inosov, D. S., Park, J. T., Bourges, P., Sun, D. L., Sidis, Y., Schneidewind, A., Hradil, K., Haug, D., Lin, C. T., Keimer, B., and Hinkov, V.; “Normal-state spin dynamics and temperature-dependent spin-resonance energy in optimally doped BaFe1.85 Co0.15 As2 ”; Nature Phys. 6, 178–181 (2010). 69. Stock, C., Broholm, C., Hudis, J., Kang, H. J., and Petrovic, C.; “Spin resonance in the d-wave superconductor CeCoIn5 ”; Phys. Rev. Lett. 100, 087001 (2008). 70. Stockert, O., Arndt, J., Faulhaber, E., Geibel, C., Jeevan, H. S., Kirchner, S., Loewenhaupt, M., Schmalzl, K., Schmidt, W., Si, Q., and Steglich, F.; “Magnetically driven superconductivity in CeCu2 Si2 ”; Nature Phys. 7, 119–124 (2011). 71. Akbari, A., and Thalmeier, P.; “Spin exciton formation inside the hidden order phase of CeB6 ”; Phys. Rev. Lett. 108, 146403 (2012). 72. Jang, H., Friemel, G., Ollivier, J., Dukhnenko, A. V., Shitsevalova, N. Y., Filipov, V. B., Keimer, B., and Inosov, D. S.; “Intense low-energy ferromagnetic fluctuations in the antiferromagnetic heavy-fermion metal CeB6 ”; Nat. Mater. 13, 682 (2014). ¨ 73. Gotze, W., and Schlottmann, P.; “Retardation effects in the longitudinal spin relaxation in metals. II”; J. Low Temp. Phys. 12, 149–152 (1973).

Chapter 11

Bulk and Surface Properties of SmB6 Priscila F. S. Rosaa and Zachary Fiskb a Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA b University of California at Irvine, Irvine, California 92697, USA

[email protected]

Samarium hexaboride crystallizes in a simple cubic structure (space group #221, P m3m), but its properties are far from being straightforward. Initially classified as a Kondo insulator born out of its intriguing intermediate valence ground state, SmB6 has been recently predicted to be a strongly correlated topological insulator. The subsequent experimental discovery of surface states has revived the interest in SmB6 , and our purpose here is to review the extensive and in many aspects perplexing experimental record of this material. We will discuss both surface and bulk properties of SmB6 with an emphasis on the role of crystal growth and sample preparation. We will also highlight the remaining mysteries and open questions in the field.

11.1 Introduction First synthesized in polycrystalline form in 1932, SmB6 started to be more heavily studied only in the 1970s owing to the successful Rare-Earth Borides Edited by Dmytro S. Inosov c 2022 Jenny Stanford Publishing Pte. Ltd. Copyright  ISBN 978-981-4877-56-5 (Hardcover), 978-1-003-14648-3 (eBook) www.jennystanford.com

818 Bulk and Surface Properties of SmB6

growth of single crystals, the prospect of its use in thermoionic emission applications, and its intriguing mixed valence (Sm2.6+ ), which gives rise to a Kondo-insulating ground state. The properties of SmB6 have been investigated by a wide variety of experimental techniques and have been reported in more than 800 scientific articles, with about half of them being published just in the past decade. This flourish of research was motivated, to a great extent, by theoretical predictions invoking topological concepts recently extended to condensed-matter physics. SmB6 remains a puzzling material, and here we take the challenge of reviewing our current knowledge of its bulk and surface properties. This book chapter is organized as follows. First, we will discuss structural aspects and crystal growth methods typically used to synthesize single crystals of SmB6 . We will then present a very brief summary of proposed theoretical models. The third and fourth sections will review surface and bulk properties of SmB6 , respectively.

11.2 Crystal Growth and Structural Properties An important message of this chapter is that the physical properties of SmB6 are sensitive to growth conditions. It will become clear in the next sections that advances in materials characterization have made the investigation of single crystals the current accepted norm, but we note that SmB6 was first synthesized in polycrystalline form via borothermal reduction [1]. Typically, a homogeneous mixture of samarium oxide (Sm2 O3 ) and boron powder is pressed into a pellet and heated by induction to temperatures from 1500 to 1800 ◦ C in vacuum or in a hydrogen atmosphere [2]. For the growth of single-crystalline SmB6 , molten flux and floating-zone methods, which were introduced in Chapter 1, are the typical methods of choice. The molten-flux technique often allows the use of a suitable low-melting-point element that acts as a solvent out of which crystals can be grown at much lower temperatures than the melting point of the compound [3–5]. This is particularly true here for SmB6 , and aluminum is the flux of choice. Though the

Crystal Growth and Structural Properties 819

solubility of hexaborides in molten Al is quite low in general, melts with typical concentration of the hexaboride of order 10−3 molar make the growth of millimeter-sized single crystals possible. The synthesis is performed in alumina tube furnaces at temperatures from 1500 to 1150 ◦ C in a protective atmosphere of ultra-high-purity argon, which flows at a slow rate. Usually, 50 ml alumina crucibles hold the melts, and chemical etching with NaOH solution is used to remove the Al at room temperature. A more benign leaching can be accomplished by growing from an Al-Ga melt, which can be leached using H2 O. Independent of the etching procedure, the etchant does not attack the hexaboride; however, larger crystals often enclose Al lamellae which cannot be removed by etching. This is the main disadvantage of synthesizing SmB6 out of flux. As we will discuss below, x-ray computed tomography experiments on flux-grown crystals reveal the presence of co-crystallized, epitaxial aluminum. Nevertheless, aluminum inclusions can be mechanically removed by polishing. The traditional optical-image floating zone growth technique is based on the directional solidification of a crystal from a liquid floating zone heated by bulb lamps and parabolic mirrors [6]. The elimination of an alumina crucible and of the aluminum flux prevents some of the issues with the flux technique; however, crystals grown by the floating zone method require much higher temperatures (T > 2000 ◦ C), which lead to composition variations. Powder x-ray diffraction in consecutive cuts of a floating-zone grown crystal reveals a systematic change in lattice parameters [7]. Precisely identifying the origin of the composition change (e.g., Sm/B vacancies, crystallographic defects, etc.) is challenging and remains a major open question in the field. The simple CsCl-type arrangement of metal atoms and B6 octahedra found in the rare-earth and alkaline-earth hexaborides was known from the early x-ray studies [8, 9]. The electronic closed-shell configuration for a B6 octahedron in this structure was determined later by molecular cluster calculations [10]. Twenty electrons were found to be required for the closed-shell configuration of an octahedron. This finding suggested that divalent metals forming hexaborides should be semiconducting, whereas

820 Bulk and Surface Properties of SmB6

Figure 11.1 (a) Lattice parameters of the cubic hexaborides RB6 (R is a rare-earth element) at room temperature. Inset shows the cubic unit cell. (b) Temperature dependent magnetic susceptibility measured on an Al-flux grown single crystal as well as its powder. The scaling factor is 1.05 to account for a small loss of powder mass.

trivalent ones should be metallic. Subsequent de Haas–van Alphen studies of metallic LaB6 were consistent with this picture, finding a Fermi surface containing one electron per formula unit [11, 12]. Measurements of the ordinary Hall coefficient in PrB6 and NdB6 also found one electron per formula unit [13]. SmB6 , however, is neither divalent nor trivalent. The intermediate valence character of SmB6 is already seen in the hexaboride lattice parameter variation across the rare-earth sequence shown in Fig. 11.1(a). The majority of rare-earth hexaborides is trivalent, whereas EuB6 and YbB6 are strictly divalent. Hexaborides with Er, Tm, and Lu do not form, possibly because their metallic radii are too small for the structure to be stable. Further evidence for the mixed valence comes from magnetic susceptibility measurements, shown in Fig. 11.1(b). At high temperatures, the magnetic susceptibility of SmB6 falls between the values expected for Sm2+ (Van Vleck) and for Sm3+ (J = 5/2 moment). Below about 60 K, the magnetic susceptibility decreases with decreasing temperature, in agreement with the opening of a gap. At temperatures below about 14 K, a Curie-like tail is observed. This tail is sample dependent, and measurements on a powdered single crystal indicate that it is a bulk effect [Fig. 11.1(b)].

Theoretical Remarks 821

11.3 Theoretical Remarks Studies of other intermediate valence rare-earth compounds with features similar to those seen in SmB6 led to the suggestion of their classification as Kondo insulators, a group of semiconducting compounds containing f elements which do not order magnetically [14]. As a result, an isostructural semiconducting compound forming with non-f elements has the integral valence of one of the fconfigurations of the Kondo insulator. SmB6 is seen as a prototypical Kondo insulator, the non-f semiconducting counterparts being the alkaline-earth hexaborides. Though Kondo physics usually seems quite distinct from that of intermediate valence, in the case of SmB6 it was found that dilute Sm impurities in divalent SrB6 showed a Kondo scale of 3 K, whereas dilute Sm impurities in trivalent LaB6 appeared to be divalent, 4f 6 , with no Kondo behavior. This appears consistent with lattice parameter studies of SmB6 /LaB6 and SmB6 /YbB6 solid solutions, in which Sm appears to go in as divalent and trivalent ions, respectively [15]. Perhaps the most accepted picture that emerged for this class of small-gap semiconducting materials is that the narrow gap arises from hybridization between the 4f electrons and d conduction bands at the Fermi level [16, 17]. Other early scenarios include Wigner crystallization in the Kondo lattice [18] and exciton-polaron models of charge fluctuations [19]. In the former, carrier localization occurs owing to correlations. In the latter, an electronic instability leads to a mixed-valence state hosting a soft electronic exciton with mixed 4f and 5d wave functions. A more recent scenario argues for a mixed-valence state originated from an unrecognized dynamical bonding effect, i.e., the coexistence of two Sm-B bonding modes corresponding to two different oxidation states driven by the motion of boron [20, 21]. In 2010, SmB6 was proposed to be the first topological Kondo insulator, i.e., a material that hosts a topologically protected surface state surrounding an insulating bulk driven by hybridization between Sm f electrons and conduction d electrons [22]. Band structure calculations indicate that the parity of the hybridized bands is inverted at three symmetry-equivalent X-points in the cubic Brillouin zone. The resulting topological invariant therefore

822 Bulk and Surface Properties of SmB6

predicts a nontrivial topological insulating state [23–25]. Further, motivated by the unusual bulk properties we discuss below, more recent theories involve ingredients such as neutral Fermi surfaces, spin excitons, fractionalized quasiparticles, and disorder [26–37]. Our goal here is not to present a detailed overview of the numerous theories, but instead to present the ensemble of experimental data against which these theories need to be checked. For theoretical reviews, we refer the reader to Refs. [38–40].

11.4 Surface Properties 11.4.1 dc Electrical Conductivity Figure 11.2 displays a summary of representative dc electrical resistivity (panel a) and resistance (panel b) data of SmB6 [41– 45]. These measurements are typically performed in a four-point configuration using a low-frequency ac excitation and a lock-in amplifier for detection. At high temperatures, the electrical resistance of SmB6 increases with decreasing temperature, in agreement with an insulating response. At around 15 K, a broad feature usually emerges, which

a) 10

Nickerson et al, 1971 Polycrystal Allen et al, 1979 Flux Hatnean et al, 2013 Floating This chapter Flux

10

R (Ω)

1

ρ (Ω.cm)

Menth et al, 1969 Polycrystal Kim et al, 2014 Flux This chapter Flux

100

0.1

1

0.01

0.1

0.001

0.01 0.001

1E-4 1

10

100

T (K)

1

10

100

T (K)

Figure 11.2 Compilation of temperature dependent (a) electrical resistivity and (b) electrical resistance obtained in SmB6 samples grown by different methods. Reproduced from Menth et al. [41], Nickerson et al. [42], Allen et al. [43], Hatnean et al. [44], and Kim et al. [45].

Surface Properties 823

we will come to later. Finally, at temperatures below about 4 K, the resistance saturates. This saturation was initially thought to be extrinsic and attributed to impurity conduction (e.g., holes from Sm vacancies); however, this plateau has been recently revisited due to the prediction of surface states. In 2013, three independent groups reported clever ways of probing whether the resistance saturation is caused by surface states. The first way involves nonlocal measurements, as shown in Fig. 11.3(a) [46]. Eight coplanar electrical contacts were attached on the (100) and (100) surfaces of a polished flux-grown SmB6 single crystal. The standard configuration used in previous measurements, Rlat , cannot distinguish surface and bulk conduction; however, measurements using contacts on opposite sides of the samples could. For instance, a vertical measurement, R vert , is obtained by flowing current from one side of the sample to the opposite side and measuring the voltage drop on another pair of contacts also placed in opposite sides. Further, a hybrid measurement, Rhybrid , is obtained by flowing current on one side of the sample and measuring the voltage on the opposite side. Figure 11.3(b) shows the experimental resistance, which is in agreement with the simulations taking into account the low-temperature surface-dominated conduction. The second way of measuring the surface contribution with dc electrical resistance measurements is by polishing the sample to a well-defined wedge, as shown in the inset of Fig. 11.3(c) [47]. By attaching several Hall voltage leads along the length of the sample, one can directly measure the thickness-dependent Hall response, as shown in the main panel of Fig. 11.3(c). If dominated by the bulk, the Hall resistance should be inversely proportional to the thickness; however, if conduction occurs mainly on the surface, the Hall resistance should be independent of the thickness. In fact, the Hall resistance ratios between different thicknesses of a fluxgrown SmB6 sample become constant below about 4 K, as shown in Fig. 11.3(d). The authors are also able to fit the experimental Hall data with a simple two-channel conduction model containing a temperature-independent surface channel and an activated bulk channel. The third approach makes use of quasiparticle tunneling spectroscopy, which measures the bulk density of states [48]. The

824 Bulk and Surface Properties of SmB6

Figure 11.3 (a) Schematic diagram of the cross section of the sample and electrical contact configurations. Arrows indicate current direction, and lines indicate equipotentials. Reproduced from Wolgast et al. [46]. (b) Log-log plot of the experimentally determined electrical resistance vs. temperature in different configurations. Inset shows a linear plot of Rlat and Rhybrid . Reproduced from Wolgast et al. [46]. (c) Temperature dependence of Hall resistance of SmB6 at different thicknesses. Reproduced from Kim et al. [47]. (d) Hall resistance between different thicknesses. Reproduced from Kim et al. [47].

authors argue that, if the resistance saturation below 4 K were due to coherent transport from bulk in-gap states, the electronic structure would change and lead to a zero-bias peak. Their results, however, indicate that the bulk density of states is robust below 10 K, which is taken as evidence for a metallic surface state. Several resistivity studies followed the initial reports, including the use of ionic liquid gating, which supports the scenario of an insulating bulk at high temperatures and metallic surface states that dominate at low temperatures [49]. We note that all the measurements discussed so far were performed in flux-grown SmB6 . To our knowledge, there are fewer similar experimental investigations on floating-zone samples, and the community would benefit from systematic dc electrical

Surface Properties 825

resistivity measurements on these crystals. Here we mention two representative reports. In the first one, the authors performed Rvert and Rhybrid measurements on floating-zone samples and conclude that there is not only surface conduction at low temperatures but also an additional residual bulk conduction possibly arising from a valence-fluctuation induced hopping from bulk in-gap states [50, 51]. The second report performed a systematic resistivity measurement in different cuts of a floating-zone grown crystal, and no significant plateau was observed below about 4 K, only a “knee” [52]. Systematic growths in the presence of carbon were then performed because carbon is a common impurity in boron. Remarkably, a thickness-independent plateau emerged in C-doped SmB6 crystals. A sensible possibility suggested by the authors is that both topological and trivial surface states coexist, as nontopological surface states may arise from chemical variations at the surface. For instance, as we will discuss below, the valence of Sm is observed to change to 3+ at the surface (Section 11.5.6), and the polar (001) surface is prone to polarity-driven surface states [53]. It is worth noting that a number of alternative theoretical proposals for the surface conductivity of SmB6 have been put forward, including trivial surface states discussed above, impurity bands, phonon bound states due to magnetoelastic coupling, Wigner lattice, and Mott minimum conductivity. We therefore conclude that, though the community appears to agree that SmB6 hosts surface states, the topological nature of these states remains unsettled. In spite of the many possible scenarios for its origin, it is fair to state that surface conduction at low temperatures has been established via dc electrical resistivity. As a result, many more questions arise. Why is there a substantial variation in the low-temperature resistivity in Fig. 11.2? Can one probe the bulk resistance of SmB6 without the influence of the surface states? What is the origin of the feature at 15 K? And finally, are these surface states topologically protected? The answer to the last question is still disputed, and the rest of this chapter will overview experimental results to help the reader reach a conclusion. The answer to the first question likely lies on the experimental observations that the bulk-to-surface ratio influences the lowtemperature crossover and that small subsurface cracks increase

826 Bulk and Surface Properties of SmB6

the surface conductivity [55]. As a result, the surface saturation will depend on how the sample was prepared — in particular, whether it is as-grown, roughly polished, or finely polished. As the surface quality improves (i.e., no cracks), the conductivity of the surface is reduced. This mechanism further explains the apparent high carrier density obtained in Hall measurements. The evident consequence of the presence of two conduction channels, one of them with varying conductivity, is that one cannot use the typical inverse resistance ratio (i.e., IRR = R2 K /R300 K ) as a good measure of the quality of the sample. This brings us to our next question: Can one probe the bulk resistance of SmB6

Figure 11.4 (a) Schematic diagram (left) and photograph (right) of the transport geometry used in double-Corbino measurements. (b) Electrical resistance as a function of reciprocal temperature for flux-grown SmB6 crystals prepared for Corbino measurements (blue solid curve) and asgrown (red solid curve). (c) Bulk resistivity as a function of reciprocal temperature for a stoichiometric sample (S1) as well as samples grown with Sm vacancies (S2–S4). Reproduced from Eo et al. [54].

Surface Properties 827

without the influence of the surface states? The answer is yes, via Corbino-disk measurements [54]. The transport geometry for these measurements is shown in Fig. 11.4(a). The flux-grown samples are first finely polished with a final step of aluminum oxide slurry of 0.3 μm. The Corbino patterns are then fabricated on the sample via photolithography followed by e-beam evaporation of Ti/Au ˚ ˚ By preparing two Corbino disks on opposite surfaces, (20 A/1500 A). one can measure either the standard resistivity using just one of the Corbino disks or the so-called “inverted” resistivity by applying current on one Corbino and measuring voltage on the opposite one. Figure 11.4(b) shows the difference in standard resistance between a sample prepared by the Corbino method (blue curve) and an asgrown sample measured by the usual four-probe configuration and no surface preparation (red curve). The significant difference further confirms that the magnitude of the resistance plateau is greatly sensitive to sample preparation. In addition, the extracted gap value also changes from one measurement to another. Figure 11.4(c) shows the activated plot for the “inverted” resistivity measurements, which only probe the bulk of the sample. Remarkably, the bulk resistivity of flux-grown SmB6 (S1) rises by 10 orders of magnitude on cooling from 300 to 2 K with no saturation at low temperatures. Samples grown off-stoichiometrically, with Sm vacancies, do show saturation at low temperatures (S2–S4), indicating that the gap is robust against point defects and strikingly exponential over a range of temperature where one would normally expect some deviations due to temperature dependent scattering rate. An open question is whether floating-zone samples exhibit a similar behavior. The feature in dc electrical resistivity at about 15 K is not often discussed, but it is reproducible. A recent theoretical framework explains this feature by modeling SmB6 as an intrinsic semiconductor with an accumulation length that diverges at cryogenic temperatures [56]. The self-consistent solution to Poisson’s equation taking into account surface charges leads to band bending in the valence and conduction bands as well as to a crossover at about 12 K to bulk-dominated conduction dominated by surface effects. The authors argue that band bending effects explain why the activated gap obtained from dc electrical resistance measurements is smaller than the gap obtained from spectroscopy measurements:

828 Bulk and Surface Properties of SmB6

Spectroscopic methods measure the gap near the surface, whereas transport probes the average gap over the bulk. We end this section by mentioning dc electrical resistivity measurements on doped or irradiated SmB6 . Small amounts (∼3%) of Gd in flux-grown SmB6 were shown to destroy the surface conduction, whereas the resistance saturation remains intact for samples doped with nonmagnetic Y and Yb at the same concentration level [45]. This result was taken as evidence for topological surface states, which is destroyed by impurities that break time-reversal symmetry. Similarly, magnetic and nonmagnetic ion irradiation was used to damage the surface layers of flux-grown SmB6 [57, 58]. The dc resistance results suggest that the surface state is not destroyed by ion irradiation, but instead it is reconstructed below the poorly conducting damaged layer, whether the damage was caused by a magnetic or a nonmagnetic ion. A recent doping study on floating-zone samples has investigated the dc electrical resistivity of SmB6 doped with lanthanum, europium, ytterbium, and strontium, revealing a complex response [59].

11.4.2 Tunneling Spectroscopy and Thermopower The measurements presented in the previous section are an average over the whole surface of SmB6 , whereas scanning tunneling microscopy and spectroscopy measurements are able to probe the surface of SmB6 on an atomic level. SmB6 , however, lacks a natural cleavage plane. As a result, various surface terminations are possible, and surface domains might emerge. Independent groups have performed density functional theory calculations of the surface formation energy for different surface terminations, as exemplified in Fig. 11.5 [60–62]. The lower energy pentaboride termination is argued to form disordered regions [62], whereas the 2×1 Sm termination was argued to be ordered and nonpolar [63, 64], though this assignment has been questioned recently [65]. In fact, reports on flux-grown SmB6 observed several distinct surface terminations on the same surface [61, 63, 64, 66, 67]. The crystals are typically cleaved in situ, and four main categories of (001) surface terminations have been observed: (i) Sm-terminated surfaces; (ii) B-terminated surfaces; (iii) disordered reconstructed

Surface Properties 829

Figure 11.5 Surface formation energy of six different surface reconstructions in SmB6 . Reproduced from Matt et al. [62].

surfaces; and (iv) ordered reconstructed surfaces (e.g., 2×1), as shown in Fig. 11.6. Further, the temperature at which the crystals are cleaved appears to play a significant role in determining the surface termination [68]. At room temperature, Ruan et al. cleaved the samples through the B6 octahedra, exposing a donutshaped structure [66], whereas the cleavage planes appear between octahedra at lower temperatures [64]. Experimental reports on floating-zone samples, however, argued that only one topography is observed on a given surface, though the authors recognize they did not scan the entirety of the cleaved area [65]. For details on the surface termination assignment, we refer the reader to Ref. [69]. We recall that the bulk-truncated (001) surface of SmB6 is polar, which could give rise to band bending, charge puddles, and the surface reconstructions mentioned above. Further, surface reconstructions could generate metallic surface layers, making the observation of a topologically nontrivial surface state challenging. In spite of the outstanding issues with surface termination, there are similarities between different reports we would like to highlight. First, a decrease in the dI /dV spectrum is observed in scanning tunneling microscopy experiments at about 10–20 meV at temperatures below ∼35 K and attributed to the opening of the hybridization gap. The conductance, however, does not vanish at zero voltage, indicating a finite density of states at E F , which is consistent with both an incomplete gap and an additional conductance channel at the surface. At lower temperatures, several features emerge within the gap. In particular, a large sharp peak is observed at around

830 Bulk and Surface Properties of SmB6

Figure 11.6 Summary of representative topographies observed on cleaving ¨ et al. [69]. The upper left and a (001) SmB6 surface. Reproduced from Roßler center topographies display Sm-terminated surfaces, whereas the bottom left and center topographies display B-terminated surfaces. The upper right topography shows a reconstructed disordered surface, whereas the bottom right topography shows a 2×1 reconstruction.

−7 meV, even in reconstructed surfaces, though the origin of such a feature is not agreed on. Jiao et al. argue that this feature contains not only a bulk contribution but also a surface component below 7 K [67]. In Gd-doped SmB6 , this −7 meV feature is destroyed at the impurity site with a healing length of approximately 1 nm in the vicinity of a defect. In addition, a magnetic tip is found to have significant effect on the local electronic structure. These findings are consistent with the expectation that the protected nature of a topological surface conducting state can be destroyed by timereversal symmetry breaking [70]. Evidence for topologically nontrivial surface states is also presented by quasiparticle interference spectroscopy [63], which images the formation of linearly dispersing surface states with heavy effective masses (e.g., m∗ ≈ 400me at the X point). Thermopower and Nernst effect measurements on the (110) plane of SmB6 also indicate that the metallic surface state has a large effective mass [71]. Scanning tunneling spectroscopy measurements and analysis by Herrmann et al. [65], however, argue

Surface Properties 831

for a modification of the low-energy electronic structure at the surface, which would in turn give rise to topologically trivial surface conductivity due to the termination-dependent 4f-like intensity near the Fermi level. Planar tunneling spectroscopy measurements are also sensitive to the surface, and a Pb–SmB6 junction has been used to probe the spectroscopic properties of SmB6 [72, 73]. The differential conductance on both (100) and (110) surfaces displays a peak at about −21 mV attributed to the bulk hybridization gap, in agreement with scanning tunneling spectroscopy measurements discussed above. Below about 15 K, the conductance increases, taken as an indication of a stronger surface state contribution, in agreement with the feature in dc resistivity discussed above. A Vshaped linear conductance is then observed at low bias arguably to the expected Dirac fermion density of states. This linearity, however, ends at about 4 mV and is attributed to inelastic tunneling via emission and absorption of bosonic excitations, i.e., spin excitons on the surface. The authors conclude, enlightened by theory, that the topological protection of the surface states is incomplete owing to their strong interaction with bulk spin excitons. As a result, lowenergy protected surface states may only exist below 5–6 K when the interaction with spin excitons becomes negligible. Finally, we note that no quantum oscillations were observed in dc resistance, planar tunneling spectroscopy or thermopower measurements described so far, though torque magnetometry measurements initially revealed oscillatory patterns as we will describe in the next section.

11.4.3 Quantum Oscillations The first report of quantum oscillations in SmB6 used flux-grown crystals [74]. Li et al. used torque magnetometry to map the oscillation pattern as a function of temperature and angle, as exemplified in Fig. 11.7(a,b). Three pockets were observed at α = 29 T, β = 286 T, and γ = 385 T with unexpectedly light effective masses of m/me = 1.1, 0.8, and 0.36, respectively [Fig. 11.7(c,d)]. The presence of light quasiparticles was argued theoretically to be due to a reduction

832 Bulk and Surface Properties of SmB6

of the Kondo effect on the surface associated with Kondo breakdown [29]. Both the effective masses and the angular dependence of the observed quantum oscillations, however, resemble that of aluminum, the flux used to prepare the single crystalline samples used in the experiment. Li et al. argued that the aluminum pellets used in the growth are polycrystalline and, as a result, could not generate the observed angular dependence. The authors therefore concluded that the measured Fermi surface cross sections scaled as the inverse cosine function of the magnetic field tilt angles, demonstrating the two-dimensional nature of the conducting (surface) states. The same group later showed that, though the angular dependence of the frequency of the β branch is fourfold symmetric, the angular dependence of the amplitude of the same

Figure 11.7 (a) Magnetic torque of flux-grown SmB6 at 300 mK as a function of magnetic field. (b) Oscillatory torque as a function of reciprocal field. (c) Fast Fourier transform (FFT) of the oscillatory torque. (d) Temperature dependence of the FFT amplitude. Reproduced from Li et al. [74].

Surface Properties 833

branch is twofold symmetric. This result was taken as evidence of multiple light-mass surface states in SmB6 [75]. A recent report on this subject has revisited the quantum oscillation patterns of flux-grown SmB6 [76]. Thomas et al. were only able to observe quantum oscillations in about 50% of the samples, and these samples tended to have larger thickness. As mentioned in Section 11.2, the main disadvantage of the flux technique in this case is that flux could become an inclusion. As a result, a thicknessdependent study was performed on the thicker samples displaying quantum oscillations. Care was taken to polish only one side of the sample to maintain any quantum oscillations coming from the other surfaces intact. After polishing the samples, large unconnected aluminum deposits were revealed, as shown in Fig. 11.8(a), and these deposits could be easily removed with dilute hydrochloric acid. After their removal, the thickness of the samples was about 100–200 microns, and no quantum oscillations were observed to 45 T. This experiment demonstrates that quantum oscillations in fluxgrown SmB6 originates from flux inclusions; however, the original aluminum pellet used in the growth is polycrystalline. To solve this apparent contradiction, one needs to take into account the fact that aluminum also crystallizes in a cubic structure with a lattice parameter that is only 2% smaller than that of SmB6 . In fact, xray tomography performed previously by a third group showed the presence of several aluminum deposits co-crystallizing with the SmB6 host in typical millimeter-sized samples Fig. 11.8(a) [7]. Precisely because several aluminum deposits may exist within one large SmB6 single crystal, the amplitude of the quantum oscillation pattern may not be C4 symmetric if the deposits are not perfectly aligned. In Section 11.5.2, we will discuss bulk quantum oscillation results in floating-zone samples.

11.4.4 Angle-Resolved Photoelectron Spectroscopy A detailed overview of angle-resolved photoemission spectroscopy measurements on SmB6 is beyond the scope of this book chapter. In principle, spin- and angle-resolved photoemission spectroscopy

834 Bulk and Surface Properties of SmB6

Figure 11.8 (a) Thickness dependence of the oscillatory torque in a fluxgrown SmB6 crystal. (b) FFT of the oscillatory torque. The FFT was scaled by the sample mass. Inset shows the FFT of polycrystalline Al. Reproduced from Thomas et al. [76]. (b) Angular dependence of the oscillatory frequency compared with the 1/ cos θ behavior as well as with the early report on Al by Larson and Gordon [77]. (c) Diffraction pattern of a flux-grown SmB6 crystal and an x-ray computed tomography image showing the presence of aluminum inclusions. Reproduced from Phelan et al. [7].

Surface Properties 835

would be one of the most direct ways of probing the surface state dispersion and spin texture; however, the issues with surface terminations mentioned above as well as the small hybridization gap of SmB6 hinder a consensus. As in the case of scanning tunneling microscopy, there are reports in favor of either topologically protected surface states [78–81] or trivial surface conductivity [53, 82]. For recent reviews of the subject, we invite the reader to refer to Refs. [62, 65, 83, 84].

11.4.5 Thin Films and Nanowires Though this chapter focuses on bulk and surface properties of SmB6 single crystals, in this section we give a brief overview of recent efforts into the synthesis and characterization of SmB6 thin films and nanowires. Early reports on thin film synthesis can be found in Refs. [85, 86]. In 2014, polycrystalline SmB6 thin films were deposited by co-sputtering of SmB6 and boron target within a combinatorial composition-spread approach [87]. A similar approach was used in 2017 by Petrushevsky et al. [88]. Growth attempts using pulsed laser deposition, molecular beam epitaxy or sputtering using a single target often result in thin films with substantial boron deficiency [87, 89, 90]. Batkova et al., however, have recently reported the preparation of stoichiometric SmB6 thin films via pulsed laser deposition [91]. Electrical resistance measurements on SmB6 thin films typically display a semiconducting behavior with a resistance ratio, R2 K /R300 K , much smaller than that of the bulk, in agreement with the larger surface-to-bulk ratio [87, 91]. Initial point-contact spectroscopy using a superconducting tip revealed the Andreev reflection, suggesting the presence of surface states and of a transparent SmB6 /superconductor interface [87]. Further evidence for a transparent interface was obtained by in situ deposition of superconducting Nb layers [92]. On the one hand, magnetotransport as well as penetration depth measurements on SmB6 thin films were argued to be consistent with topological surface states [93, 94]. On the other hand, Li et al. observe that the electrical resistivity of SmB6 thin films is thickness dependent, in apparent contradiction with the surface conductivity scenario [95]. Li et al. also show that SmB6

836 Bulk and Surface Properties of SmB6

thin films display large spin–orbit torque. Nevertheless, a recent point-contact spectroscopy measurement reports the observation of perfect Andreev reflection in a heterostructure formed by insulating SmB6 and superconducting YB6 . This observation was understood as a manifestation of Klein tunneling due to the proximity-induced superconducting state in a topological insulator [96]. Rare-earth hexaboride nanowires have been synthesized by a variety of methods including vapor–liquid–solid mechanism [97, 98], chemical vapor deposition with [99] and without [100] a catalyst, and palladium-nanoparticle-assisted chemical vapor deposition [101]. Electrical resistance measurements are argued to be consistent with the presence of topological surface states [102–105].

11.5 Bulk Properties 11.5.1 ac Electrical Conductivity As discussed in Section 11.4.1, the bulk of flux-grown SmB6 has been shown to be remarkably insulating, with a 10-order-of-magnitude increase in dc resistivity on cooling from room temperature to 2 K. The bulk ac conductivity of SmB6 , however, is several orders of magnitude larger than that of any known impurity band, and this is the next puzzle we would like to address. Because of the large index of refraction of hexaborides and the particularly large ac conduction in SmB6 , previous optical measurements were typically performed in reflection mode, relying on the Kramers–Kronig transformation [106–110]. Further, temperatureand thickness-dependent transmission experiments were missing until the renewed interest in SmB6 . Early low-temperature ac conductivity measurements on floating-zone SmB6 in the range from 0.6 to 4.5 meV provided evidence for a 19 meV energy gap and an additional narrow donor-type band at 3 meV below the conduction band [111]. The properties of SmB6 below 8 K were first attributed to localized carriers in the narrow band responsible for hopping conductivity [111] and later to exciton-polaron complexes [112].

Bulk Properties 837

Figure 11.9 Frequency dependence of the complex transmission for (a) floating zone and (b) flux-grown SmB6 single crystals with thicknesses of 22 and 62 μm, respectively. Frequency dependence of the real part of the optical conductivity for (c) floating zone and (d) flux-grown SmB6 single crystals. Reproduced from Laurita et al. [113].

High-resolution optical measurements in the terahertz frequency range were recently performed to revisit these results. Laurita et al. investigated both floating zone and flux-grown samples with thicknesses smaller than 100 microns [113]. The experimental energy range, ω ≈ 1–8 meV, is smaller than the spectroscopic gap energy, which allows this technique to probe states within the bulk gap. Figure 11.9 shows the frequency dependence of the complex transmission at various temperatures for floating-zone (panel a) and flux-grown (panel b) SmB6 crystals, respectively. It is worth noting that flux-grown samples are smaller, and the required mosaic of samples displays worse signal to noise. Both real and imaginary parts of the complex optical conductivity are therefore extracted from the complex transmission, and the real part of the optical conductivity is shown in Fig. 11.9(c,d). At high temperatures, the optical conductivity displays a Drude-like response, i.e., σ1 increases

838 Bulk and Surface Properties of SmB6

with decreasing frequency, indicating the presence of free charge carriers at E F . Above 30 K, the sample becomes opaque in the THz range. The magnitude of the Drude-like response decreases with decreasing temperature, in agreement with the opening of the hybridization gap. Below 13 K, the ac conductivity increases linearly with frequency before saturating at about 1 THz. This additional conduction channel is termed “localized” because it does not contribute to dc transport. The pressing question here is whether this in-gap conduction is due to impurity states or to exotic neutral excitations, and answering this question is challenging. A recent analysis starts with the conjecture that the localized conductivity response is independent of temperature based on the weak temperature dependence of the conductivity at high frequencies [114]. The conductivity can be in turn modeled as a sum of the localized contribution, the Drude response, and a frequency-independent background. The latter term has been interpreted in previous measurements as a Mott minimum conductivity [111]. The dc conductivity can be extracted from the Drude response, providing an activated gap of 4.1 meV, in agreement with dc measurements discussed in Section 11.4.1. The frequencyindependent conductivity is finite only above 12 K and reaches 9.4 −1 at 17.5 K. The localized contribution within the gap dominates the conductivity response at 1.6 K. At about 1 THz (4 meV), the frequency dependence of the conductivity displays a crossover from σ1 ∝ ω0.8 to a linear dependence with frequency. The frequency dependence below the indirect gap is inconsistent with the weak Drude peak predicted by disordered Kondo insulator models, which take into account effects of substitution on the f-electron site above a percolation threshold [115, 116]. A sensible scenario, however, is that SmB6 is below the percolation threshold, in which impurity states are localized. In addition, substitutional (pointdefect) impurities might not be the only source of defects in SmB6 . A quantitative comparison between SmB6 and localization driven insulators (e.g., Si:P at 40% doping) reveals that the in-gap ac conductivity of SmB6 is about four orders of magnitude larger than the typical impurity band conduction, being comparable to completely amorphous alloys [117, 118]. This possibility brings up

Bulk Properties 839

again the lack of a deep understanding of disorder in this material. In fact, a recent theoretical effort revisits the donor impurity band mechanism in SmB6 taking into account its peculiar “Mexican-hatlike” band structure and shows that the resulting impurity band is in many ways distinct from the conventional semiconductor case [37]. In particular, the critical doping concentration necessary to drive an insulator–metal transition is much larger than in the conventional case. Estimates of the ac conductivity in this framework are in agreement with the experimental results discussed above and provide an explanation for why SmB6 is a robust dc insulator but an ac conductor. The second sensible scenario is the presence of charge-neutral quasiparticles within the Kondo gap, which could couple to an ac electric field [31]. Though this scenario has been proposed theoretically, a qualitative prediction that matches the experimental power law behavior with frequency is missing. This scenario has also been suggested by quantum oscillation and thermal conductivity measurements on floating-zone samples, as we will discuss in the next two sections. Finally, we note that the spectral weight of the in-gap conductivity provides further information on the density of charge carriers and effective mass via the conductivity sum rule relation. A rough estimate using the bare mass of the electron gives a charge density of about 1 electron per 1000 unit cells, which is roughly consistent with both an impurity contribution as well as a neutral Fermi surface. Other experimental probes are therefore required to answer this question.

11.5.2 Quantum Oscillations Shortly after the report of quantum oscillations in flux-grown SmB6 , which we discussed in Section 11.4.3, similar torque magnetometry experiments were performed on floating-zone crystals [119]. The first key remark is that the oscillatory behavior shown in Fig. 11.10(a) is not observed in every floating-zone sample. This is an outstanding question which calls for immediate attention, especially considering that flux-grown samples free of aluminum do not show quantum oscillations and that floating-zone rods host compositional changes that are not fully understood.

840 Bulk and Surface Properties of SmB6

When quantum oscillations are present in floating-zone samples, they are observed at high frequencies and are independent of the surface details. Further, the angular dependence of the oscillations resembles those of LaB6 and PrB6 , as shown in Fig. 11.10(c). Taken

Figure 11.10 (a) Torque signal of a floating-zone SmB6 crystal as a function of magnetic field. (b) Quantum oscillation amplitude as a function of temperature. (c) Angular dependence of the quantum oscillation frequency for SmB6 and PrB6 . Reproduced from Tan et al. [119].

Bulk Properties 841

together, these observations are inconsistent with surface-driven quantum oscillations and suggest a bulk origin. Quantum oscillations in floating-zone samples were termed “unconventional” owing to the temperature dependence of the quantum oscillation amplitude, shown in Fig. 11.10(b). Above 1 K, the amplitude follows the conventional Lifshitz–Kosevich formula [120] with a small effective mass (0.1–0.8 me ); however, below 1 K the amplitude increases greatly down to base temperature. Importantly, oscillations in the dc electrical resistivity are not observed. As in the case of ac conductivity measurements, we are again faced with (at least) two possibilities. The first one is that quantum oscillations may arise from spatially unconnected metallic patches from a secondary unknown phase, similar to the aluminum inclusions observed in flux-grown samples. Though Tan et al. state that this possibility appears to be unlikely, it cannot be ruled out at this point. The second possibility is the presence of low-energy neutral excitations within the charge gap of SmB6 . Hartstein et al. find that the density of states obtained from their quantum oscillation Fermi surface matches that of specific heat measurements (γ ≈ 4(1) mJ/mol·K2 ) [121]. The steep increase in the low-temperature quantum oscillation amplitude also resembles the increase in the specific heat. Further, the low-temperature entropy obtained from the oscillatory pattern remains finite below 1 K, indicating a finite density of states. These observations, combined with thermal conductivity measurements to be discussed in the next section, were taken as evidence for bulk itinerant low-energy excitations that couple to magnetic fields but not weak dc electric fields.

11.5.3 Thermal Conductivity Thermal conductivity (κ) measurements are a valuable way of probing possible fermionic charge-neutral excitations, which carry entropy and would contribute to a finite residual term, κ0 /T, in the T = 0 limit. Previous experiments performed almost 40 years ago concluded that the thermal conductivity of SmB6 above 1.5 K is dominated by phonons, i.e., κ/T ∝ T 2 [123], but this scenario

842 Bulk and Surface Properties of SmB6

Figure 11.11 (a) Thermal conductivity of a flux-grown SmB6 sample vs. temperature (top) and T 1.94 (bottom). Reproduced from Xu et al. [122]. (b) Thermal conductivity of a floating-zone SmB6 sample vs. T 2 (top) and magnetic field (bottom). Reproduced from Hartstein et al. [121].

has been revisited recently in experiments performed at much lower temperatures and much higher magnetic fields. The first contemporary report was performed on flux-grown SmB6 single crystals in a dilution refrigerator with fields to 14.5 T [122]. The thermal conductivity was measured on a (100) surface using a standard four-wire steady-state method with two RuO2 chip thermometers. Figure 11.11(a) displays thermal conductivity data down to 0.1 K in zero field and at 14.5 T. By fitting the zero-field data to κ/T = a + bT α−1 , one extracts a residual term κ0 /T = −0.003±0.004 mW K−2 cm−1 , which is zero within the experimental error bar. The contribution from surface states is estimated to be two orders of magnitude smaller than this error bar and therefore negligible. As shown in Fig. 11.11(a) (bottom panel), α = 2.94, in agreement with the expected phonon contribution. Finally, applied magnetic

Bulk Properties 843

fields have very little effect on the thermal conductivity, and the residual term κ0 /T remains negligible. Xu et al. therefore conclude that thermal conductivity measurements do not support fermionic charge-neutral quasiparticles. One possible scenario raised by Xu et al. to explain the bulk quantum oscillation results discussed above is spatial inhomogeneity below a percolation threshold. An independent report on floating-zone crystals, however, reveals a finite zero-field κ0 /T, which increases as a function of applied field as shown in Fig. 11.11(b) [121]. Hartstein et al. argue that phonons are unlikely to be the origin of this behavior because the phonon thermal conductivity is at its maximum in the boundary scattering limit. The similarity between the field dependent thermal conductivity of SmB6 and that of organic insulator κ-(BEDTTTF)2 Cu2 (CN)3 is argued to be evidence for a neutral Fermi surface hosting excitations that transport heat but not charge. A recent report by a third group sheds light on this controversy by measuring field-dependent thermal conductivity down to 70 mK on a variety of single crystals grown by both flux-grown and floating-zone techniques [124]. The authors confirm the absence of a residual term κ0 /T on six different samples in both zero field and at high magnetic fields, as summarized in Fig. 11.12. A large fieldinduced enhancement of κ, however, is observed for all floating zone samples as well as two out of three flux-grown samples. Importantly, this sample dependence points to an extrinsic origin. In addition, the field-induced enhancement of κ is systematically smaller in flux-grown samples, and the behavior of sample F3 is in agreement with the first report discussed above [122]. Generally speaking, the field dependence of the thermal conductivity is consistent with two distinct mechanisms: low-energy magnetic excitations (e.g., magnons or spinons) or phonons scattered by a field-dependent contribution (e.g., spin fluctuations or magnetic impurities). The authors conclude that their data is in line with the latter scenario as magnetic impurities may significantly affect the phonon thermal conductivity in insulators even at a 1% level. The application of magnetic field splits the energy levels of the magnetic impurities, which causes a mismatch between the phonon energy and the impurity level splitting. To test their interpretation, the authors reduced the cross-sectional area of one

844 Bulk and Surface Properties of SmB6

Figure 11.12 Temperature dependence of the thermal conductivity for (a) floating-zone and (b) flux-grown SmB6 crystals. Reproduced from Boulanger et al. [124]. The references to Hartstein et al. in (a) and Xu et al. in (b) correspond to Refs. [121] and [122], respectively.

of the floating zone samples, which makes the boundary-limited phonon conductivity smaller. According to their expectation, κ/T 2 decreases with decreasing cross section because the magnetic field gaps out the zero-field scattering mechanism, and the thermal conductivity is then set by the boundary limit. For more details on the analysis, including the high-temperature regime, please refer to [124]. It is also worth noting that, though there is no evidence for fermionic neutral quasiparticles, they may be thermally decoupled from the measurable, heat-carrying phonons.

11.5.4 Specific Heat Figure 11.13(a) shows a compilation of specific heat data on a variety of samples grown using different methods [76, 119, 121, 125–127]. Similar to the ac and thermal conductivity measurements discussed above, specific-heat measurements are sensitive to impurities that may not percolate in dc resistivity measurements. In fact, the large variation of the residual specific heat in Fig. 11.13(a) points to an extrinsic origin. The lowest specific-heat magnitude

Bulk Properties 845

Figure 11.13 (a) A compilation of zero-field specific heat data as a function of temperature for SmB6 crystals grown by different techniques [76, 119, 121, 125–127]. Reproduced from Thomas et al. [76]. (b) Zero-field specific heat data of carbon-doped floating-zone SmB6 . Reproduced from Phelan et al. [52]. (c) Zero-field specific heat data of floating-zone LaB6 and SmB6 . Sample a was grown with isotopically enriched elements, sample b with natural impurities and sample c with natural impurities and triply melted. Reproduced from Orendac et al. [127].

reported to date was obtained on a crystal grown with isotopically enriched 154 Sm and 11 B [127] providing further evidence that the broad feature centered near 1.5 K is caused by naturally occurring impurities. A residual feature, however, remains at about 7 K, and we will come back to this energy scale in Section 11.5.7. Rare-earth purification is challenging and expensive. Rare-earth elements obtained from Ames Laboratory are one of the purest available sources, but Ames samarium is still 99.99% pure at best. Typically, europium, erbium, lanthanum, and iron can be found at several ppm level. In agreement with this reasoning, enhanced low-temperature specific-heat values are observed in flux-grown samples intentionally doped with gadolinium. In addition, specificheat data from samples with distinct Gd concentrations can be scaled to follow the same power law at low temperatures [125]. This scaling was argued to be consistent with the Kondo impurity model.

846 Bulk and Surface Properties of SmB6

We note that enhanced values of the low-temperature specific heat are also observed in floating-zone samples doped with carbon [52]. Further, floating-zone samples that are doubly or triply melted display similar specific-heat enhancement, though in this case it is less clear what exact type of defects one would encounter [127]. Attempts to fit the specific heat above 2 K have been made by independent groups typically including a combination of the standard electronic (γ T ) and lattice (βT 3 ) contributions as well as an exchange-enhanced paramagnetic term, αT 3 ln(T /T ∗ ), and the Schottky anomaly, AT −2 [52, 127]. Phelan et al. showed that the phonon and exchange terms follow a universal curve with a characteristic temperature T ∗ = 17 K associated with Kondo hybridization. Below 2 K, however, additional contributions come into play. Previous specific-heat measurements performed at dilution refrigeration temperatures successfully explained the difference between in-field and zero-field measurements by calculating the contributions from nuclear magnetic moments of 147 Sm, 149 Sm, 10 B, and 11 B isotopes as well as 300 ppm of magnetic impurities [128]. A quantitative understanding of the anomalous bulk behavior of SmB6 even at zero field, however, remains unavailable [126, 129].

11.5.5 Raman Spectroscopy Now we turn our attention to spectroscopic techniques. Raman spectroscopy is based on the inelastic scattering of light in a material, providing a direct means of probing the development of an energy gap. This technique also provides energy and symmetry information about other low-frequency excitations including crystalfield excitations, excitons, and anomalous phonons. Evidence for an energy gap in SmB6 was observed in early reports of the temperature dependence of the Raman continuum scattering, as shown in Fig. 11.14(a) [130]. The electronic scattering intensity shifts abruptly below T ∗ ≈ 50–60 K, which corresponds to the temperature at which the magnetic susceptibility peaks. The shift in spectral weight is centered about an isosbestic point at 290 cm−1 , an energy substantially larger than the energy scale over which the gap appears in the scattering response. A number of theoretical models were proposed to account for electrical transport as well as Raman

Bulk Properties 847

Figure 11.14 Raman scattering response function of a flux-grown SmB6 single crystal at various temperatures (panel a). Reproduced from Nyhus et al. [130]. Raman scattering response function of both flux-grown and floating-zone crystals at room temperature (panel b). Reproduced from Valentine et al. [131].

848 Bulk and Surface Properties of SmB6

data [132]; however, the discovery of non-Ohmic behavior arising from a conducting surface state in SmB6 discussed above makes analysis not taking this into account of limited value. In the past few years, the Raman response of SmB6 was revisited, and a timely sample dependence investigation was performed [131]. Figure 11.14(b) shows the room-temperature Raman spectra of three distinct single crystals: one crystal grown by the flux technique (Al flux SmB6 ), a second crystal grown by the floating zone technique close to being stoichiometric (FZ SmB6 – Pure), and a third crystal also grown by the floating zone technique but less stoichiometric (FZ SmB6 – Defc). As we mentioned above, an increase in the number of Sm vacancies is observed along the length of a floating zone crystal rod due to the vaporization and has been characterized by a systematic decrease in lattice parameters. Three symmetry-allowed phonons are observed in Fig. 11.14(b), in agreement with previous reports [133]. These narrow peaks are related to distortions of the B6 octahedra as follows: T2g phonon at 89.6 meV (723 cm−1 ), E g phonon at 141.7 meV (1143 cm−1 ), and A 1g phonon at 158.2 meV (1277 cm−1 ). At much lower energies, however, two additional features appear at 10 and 21 meV, which are not allowed by the cubic space group in first-order Raman scattering. The latter feature has been previously assigned to a two-phonon scattering mode [130]. The former feature, however, has been recently attributed to a finite-momentum scattering from acoustic phonons due to local symmetry breaking induced by the presence of Sm defects [131]. Neutron scattering experiments show a relatively flat dispersion of the acoustic phonon branches, in agreement with the narrow line width at 10 meV [134]. Further evidence for this scenario comes from the 50% increase in spectral weight at 10 meV in the most deficient floating-zone sample, which is estimated to have only about 1% of Sm vacancies. A significant impurity-driven enhancement is also observed in the specific heat of these samples [135].

11.5.6 X-Ray, Neutron, and M¨ossbauer Spectroscopies Early L3 -edge x-ray absorption experiments [138] as well as 149 Sm ¨ Mossbauer resonance measurements [137, 139] estimated the

Bulk Properties 849

Figure 11.15 (a) Sm L3 -edge x-ray absorption spectra at different temperature. Reproduced from Mizumaki et al. [136]. (b) Average Sm valence as a function of temperature. Reproduced from Mizumaki et al. [136]. (c) Isomer shift as a function of temperature. Reproduced from Cohen et al. [137].

valence of Sm to be +2.6 at room temperature. X-ray absorption spectra display two separate peaks corresponding to Sm2+ and Sm3+ [136]. The spectra evolve with temperature as shown in Fig. 11.15(a,b), revealing that the valence of Sm decreases with decreasing temperature. The temperature dependence of the lattice parameters [140] combined with negative thermal expansion and anomalous elastic constants at low temperatures [141, 142] also ¨ indicate valence changes as a function of temperature. Mossbauer

850 Bulk and Surface Properties of SmB6

resonance, however, only contains a single-line spectrum, and no change in the Sm valence as a function of temperature is detected by isomer shift measurements, Fig. 11.15(c). This apparent contradiction can be resolved by taking into account the different time scales of these two measurements. X-ray absorption is a ¨ resonance (10−8 s), faster measurement (10−12 s) than Mossbauer indicating that the valence of Sm fluctuates at about 10−8 s. An alternative explanation, however, has been recently proposed based on the boron-dimer model discussed in Section 11.3 [20]. The authors predict a constant ratio of f-shell occupations, which is in ¨ turn probed by Mossbauer spectroscopy, and a variable d-orbital ¨ occupation, which cannot be probed by Mossbauer measurements. We note that the Sm valence obtained from the above Sm L3 x-ray absorption spectroscopy results has been reproduced by several independent groups as well as by distinct methods including x-ray ¨ photoelectron spectroscopy [143–146]. The Mossbauer isomer shift result has also been recently revisited and reproduced by Tsutsui et al. [147]. Resonant x-ray emission spectroscopy as well as x-ray absorption spectroscopy also have been employed to investigate the valence of SmB6 under applied pressure. Electrical resistivity measurements under hydrostatic conditions show that the insulating gap of SmB6 vanishes at Pc = 10 GPa, accompanied by magnetic order [148]. This critical pressure is dependent on the hydrostaticity of the pressure media and may be as low as ∼4 GPa in quasi-hydrostatic environments [149–151]. X-ray spectroscopy measurements report the increase in Sm valence under pressure towards the trivalent state as the systems goes towards the antiferromagnetic metallic ground state [152–156]. A discrepancy, however, is observed in the quantitative analysis by different groups. For example, Zhou et al. report a valence very close to 3+ at 20 GPa [154], whereas other groups argue for a finite divalent character to 35 GPa [153,155,157]. Sm M-edge x-ray absorption and x-ray magnetic circular dichroism measurements have been simultaneously performed on the surface and in the bulk of SmB6 , respectively. Phelan et al. confirm the presence of Sm2+ and Sm3+ in the bulk of floating-zone crystals; however, the polished surface is shown to contain mostly Sm3+ with a net magnetic moment of 0.09μB at T = 10 K. The authors

Bulk Properties 851

argue that the discrepancy between the valence at the surface and in the bulk could generate band bending at the interface. Though Chen et al. also report a higher Sm valence at the surface of powdered floating-zone SmB6 , the extracted value of ν = 2.7 is below the trivalent state [155]. Employing the same techniques on vacuum-cleaved samples reveal that both divalent and trivalent Sm are present on the surface of SmB6 , indicating that polishing has a significant effect on the surface valence [158]. In addition, Fuhrman et al. report that the Sm3+ magnetic dipole moment antialigns with an applied magnetic field below T = 75 K and suggest that Sm3+ couples antiferromagnetically to large-moment paramagnetic impurities known to be present in the samples. Sm N-edge x-ray absorption spectroscopy has also been performed on flux-grown SmB6 [159]. He et al. show that the Sm3+ contribution on a cleaved surface increases irreversibly with time as it ages in an ultra-high-vacuum chamber. In fact, soft x-ray absorption and reflectometry measurements on floating-zone crystals reveal that a cleaved (001) surface of SmB6 undergoes significant valence and chemical reconstruction as a function of time. The final surface is argued to be boron-terminated with a Sm3+ subsurface region. At room temperature, this reconstruction takes less than 2 h, whereas below 50 K it takes about a day [68]. These results shed light on discrepancies observed in scanning tunneling spectroscopy measurements discussed in Section 11.4.2. Recent nonresonant and resonant x-ray scattering experiments provide a powerful tool to determine the ground state and crystal field splitting of SmB6 , respectively. On one hand, corelevel nonresonant scattering in the hard x-ray region is a bulksensitive spectroscopic technique performed at large momentum transfer. The scattering function therefore provides information on higher multipole terms and allows the determination of the ground state wave function even in cubic compounds [160]. On the other hand, M4, 5 -edge (3d → 4f) resonant inelastic x-ray scattering is both element- and configuration-sensitive and allows the determination of the crystal field splitting of the Sm3+ Hund’s rule ground state [161]. These two spectroscopic techniques find that the J = 5/2 multiplet of Sm f 5 splits into a 8 quartet ground state and a 7 first excited doublet at 20 meV. It is worth noting that this

852 Bulk and Surface Properties of SmB6

Figure 11.16 (a) Experimental (top) and theoretical (bottom) energy integrated neutron scattering intensity of floating-zone 154 Sm11 B at highsymmetry planes. (b) Comparison between the squared magnetic form factor of different scattering centers (lines) and the experimental integrated neutron scattering intensity (symbols) as a function of momentum Q. Reproduced from Fuhrman et al. [166].

crystal field splitting agrees well with the expected splitting from the extrapolation of crystal field parameters obtained within the RB6 series (R is a rare-earth element). Inelastic neutron scattering has also been employed to shed light on this problem. The strong neutron absorption of both samarium and boron, combined with the presence of hybridization and two Sm configurations, makes these measurements challenging. Nevertheless, early inelastic neutron scattering experiments using double isotopic samples, 154 Sm11 B, identify broad intermultiplet transitions at about 36 and 130 meV as well as a narrow low-energy excitation at the R point ( 12 21 21 ) centered at 14 meV, only observed at temperatures below 100 K [134, 162–165]. This low-temperature feature does not follow the localized f-electron form factor and was therefore argued to be caused by the mixed-valence state of Sm and exciton states within the gap. Recent inelastic neutron scattering measurements have revisited the low-energy, low-temperature features of SmB6 in the entire Brillouin zone [166]. Fuhrman et al. show that the narrow (>2 meV) resonant mode at 14 meV is also intense near the X point

Bulk Properties 853

( 12 00), see Fig. 11.16(a), and, though much weaker, goes beyond the first zone. As a result, the form factor is mapped out and shown to unexpectedly follow the 5d electron form factor, Fig. 11.16(b), indicating a critical role of such orbitals in the exciton formation and providing strong constraints to realistic theories. Fuhrman et al. propose a minimal phenomenological model with third neighbor hopping, which allows for 5d-electron “pseudo-nesting” to enhance the generalized susceptibility that appears in the inelastic neutron scattering measurements through interband transitions. This phenomenological model generates a band structure with inversion pockets at X , in agreement with the topological Kondo insulator picture. The spin exciton may also shed light on the presence of impurities in SmB6 as coupling of the exciton to a fermionic density of states at E F would cause a finite relaxation rate in inelastic neutron scattering measurements. A nearly temperature-independent lifetime, however, is observed from 15 to 3.5 K, indicating no coupling to additional energy scales below the 14 meV spin-exciton mode. In addition, no indication of magnetism was observed in the low-energy spectrum, and an upper bound fluctuating moment of 0.05(2)μ2B was estimated. The authors conclude that magnetic impurities in SmB6 may be screened.

11.5.7 Nuclear, Electron, and Muon Spin Resonance Nuclear magnetic resonance (NMR) takes advantage of the nonzero nuclear spin moment of many stable isotopes to obtain local information on the internal field as well as spin dynamics in a material. Though it is challenging to detect Sm nuclei in SmB6 , resonance lines from 11 B are easy to detect and allow an indirect measure of the Sm mixed valence through the hyperfine interaction. ˜ et al. showed that the 11 B spinEarly NMR measurements by Pena lattice relaxation above 15 K is activated with an energy gap of about 6 meV, in rough agreement with the transport gap (i.e., 4 meV) [167, 170]. An anomalous peak, however, emerges at about 5 K as shown in Fig. 11.17(a). The authors state that evaluation of models for this anomaly is hindered by the lack of precise defect characterization in nearly pure SmB6 , and this statement remains accurate to date.

854 Bulk and Surface Properties of SmB6

Figure 11.17 (a) 11 B relaxation rate as a function of temperature in flux˜ et al. [167]. (b) NMR line grown powdered SmB6 . Reproduced from Pena width as a function of inverse temperature. Reproduced from Bose et al. [168]. (c) Spin-echo spectra at 4.2 K and 20.7 MHz (top) and 11 B relaxation rate (bottom) of a SmB6 single crystal. Reproduced from Takigawa et al. [169].

˜ et al. argue that the maximum may be related to a decrease in Pena fluctuation amplitude at low temperatures, which in turn could be caused by Sm3+ formation near Sm vacancies. A second 11 B NMR experiment reported constant Knight shift and line width data from 100 to 480 K [168]. Bose et al. ascribed this result to a temperature-independent valence fluctuation of the Sm ion owing to the fast fluctuation rate (