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Principles of inorganic materials design, [third edition.]
 9781119486831, 1119486831

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Principles of Inorganic Materials Design

Principles of Inorganic Materials Design Third Edition John N. Lalena Physical Scientist

David A. Cleary Gonzaga University

Olivier B.M. Hardouin Duparc École polytechnique Institut Polytechnique Paris

This edition first published 2020 © 2020 John Wiley & Sons Inc. Edition history: “John Wiley & Sons Inc. (1e, 2005)” “John Wiley & Sons Inc. (2e, 2010)” All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, except as permitted by law. Advice on how to obtain permission to reuse material from this title is available at http://www.wiley.com/go/permissions. The right of John N. Lalena, David A. Cleary, and Olivier B. M. Hardouin Duparc to be identified as the authors of this work has been asserted in accordance with law. Registered Office John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, USA Editorial Office 111 River Street, Hoboken, NJ 07030, USA For details of our global editorial offices, customer services, and more information about Wiley products visit us at www.wiley.com. Wiley also publishes its books in a variety of electronic formats and by print-on-demand. Some content that appears in standard print versions of this book may not be available in other formats. Limit of Liability/Disclaimer of Warranty In view of ongoing research, equipment modifications, changes in governmental regulations, and the constant flow of information relating to the use of experimental reagents, equipment, and devices, the reader is urged to review and evaluate the information provided in the package insert or instructions for each chemical, piece of equipment, reagent, or device for, among other things, any changes in the instructions or indication of usage and for added warnings and precautions. While the publisher and authors have used their best efforts in preparing this work, they make no representations or warranties with respect to the accuracy or completeness of the contents of this work and specifically disclaim all warranties, including without limitation any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives, written sales materials or promotional statements for this work. The fact that an organization, website, or product is referred to in this work as a citation and/or potential source of further information does not mean that the publisher and authors endorse the information or services the organization, website, or product may provide or recommendations it may make. This work is sold with the understanding that the publisher is not engaged in rendering professional services. The advice and strategies contained herein may not be suitable for your situation. You should consult with a specialist where appropriate. Further, readers should be aware that websites listed in this work may have changed or disappeared between when this work was written and when it is read. Neither the publisher nor authors shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages. Library of Congress Cataloging-in-Publication Data Names: Lalena, John N. author. | Cleary, David A., 1957– author. | Duparc, Olivier B. M. Hardouin, author. Title: Principles of inorganic materials design / John N. Lalena, U.S. Department of Energy, David A. Cleary, Gonzaga University, Olivier B. M. Hardouin Duparc, E’cole Polytechnique. Description: Third edition. | Hoboken, NJ, USA : Wiley, 2020. | Includes bibliographical references and index. Identifiers: LCCN 2019051901 (print) | LCCN 2019051902 (ebook) | ISBN 9781119486831 (hardback) | ISBN 9781119486916 (adobe pdf) | ISBN 9781119486763 (epub) Subjects: LCSH: Chemistry, Inorganic–Materials. | Chemistry, Technical–Materials. Classification: LCC QD151.3 .L35 2020 (print) | LCC QD151.3 (ebook) | DDC 546–dc23 LC record available at https://lccn.loc.gov/2019051901 LC ebook record available at https://lccn.loc.gov/2019051902 Cover Design: Wiley Cover Images: Molecular model, illustration © VICTOR HABBICK VISIONS/SCIENCE PHOTO LIBRARY/Getty Images, Blue crystals © assistantua/Getty Images Set in 9.5/12.5pt STIXTwoText by SPi Global, Pondicherry, India Printed in United States of America 10 9 8 7 6 5 4 3 2 1

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Contents Foreword to Second Edition xiii Foreword to First Edition xv Preface to Third Edition xix Preface to Second Edition xx Preface to First Edition xxi Acronyms xxiii 1 1.1 1.1.1 1.1.2 1.1.3 1.1.4 1.1.5 1.2 1.2.1 1.2.1.1 1.2.1.2 1.2.1.3 1.3 1.4 1.4.1 1.4.2 1.4.2.1 1.4.2.2 1.5 1.5.1 1.5.2 1.5.3

Crystallographic Considerations 1 Degrees of Crystallinity 1 Monocrystalline Solids 2 Quasicrystalline Solids 3 Polycrystalline Solids 4 Semicrystalline Solids 5 Amorphous Solids 8 Basic Crystallography 8 Crystal Geometry 8 Types of Crystallographic Symmetry 12 Space Group Symmetry 17 Lattice Planes and Directions 27 Single-Crystal Morphology and Its Relationship to Lattice Symmetry 32 Twinned Crystals, Grain Boundaries, and Bicrystallography 37 Twinned Crystals and Twinning 37 Crystallographic Orientation Relationships in Bicrystals 39 The Coincidence Site Lattice 39 Equivalent Axis–Angle Pairs 44 Amorphous Solids and Glasses 46 Oxide Glasses 49 Metallic Glasses and Metal–Organic Framework Glasses 51 Aerogels 53 Practice Problems 53 References 55

2 2.1 2.1.1 2.2

Microstructural Considerations 57 Materials Length Scales 57 Experimental Resolution of Material Features 61 Grain Boundaries in Polycrystalline Materials 63

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2.2.1 2.2.2 2.2.3 2.2.4 2.2.5 2.2.6 2.3 2.3.1 2.3.1.1 2.3.1.2 2.3.1.3 2.3.2 2.3.3 2.3.4 2.3.4.1 2.3.4.2 2.3.4.3 2.4 2.4.1 2.4.2 2.4.3 2.4.4 2.5

Grain Boundary Orientations 63 Dislocation Model of Low Angle Grain Boundaries 65 Grain Boundary Energy 66 Special Types of “Low-Energy” Boundaries 68 Grain Boundary Dynamics 69 Representing Orientation Distributions in Polycrystalline Aggregates Materials Processing and Microstructure 72 Conventional Solidification 72 Grain Homogeneity 74 Grain Morphology 76 Zone Melting Techniques 78 Deformation Processing 79 Consolidation Processing 79 Thin-Film Formation 80 Epitaxy 81 Polycrystalline PVD Thin Films 81 Polycrystalline CVD Thin Films 83 Microstructure and Materials Properties 83 Mechanical Properties 83 Transport Properties 86 Magnetic and Dielectric Properties 90 Chemical Properties 92 Microstructure Control and Design 93 Practice Problems 96 References 96

3 3.1 3.1.1 3.1.2 3.1.3 3.1.4 3.1.5 3.1.6 3.2 3.2.1 3.2.2 3.2.3 3.2.4 3.2.5 3.3 3.3.1 3.3.2 3.3.3 3.3.4 3.3.5 3.4

Crystal Structures and Binding Forces 99 Structure Description Methods 99 Close Packing 99 Polyhedra 103 The (Primitive) Unit Cell 103 Space Groups and Wyckoff Positions 104 Strukturbericht Symbols 104 Pearson Symbols 105 Cohesive Forces in Solids 106 Ionic Bonding 106 Covalent Bonding 108 Dative Bonds 110 Metallic Bonding 111 Atoms and Bonds as Electron Charge Density 112 Chemical Potential Energy 113 Lattice Energy for Ionic Crystals 114 The Born–Haber Cycle 119 Goldschmidt’s Rules and Pauling’s Rules 120 Total Energy 122 Electronic Origin of Coordination Polyhedra in Covalent Crystals Common Structure Types 127

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3.4.1 3.4.1.1 3.4.1.2 3.4.1.3 3.4.1.4 3.4.1.5 3.4.1.6 3.4.1.7 3.4.1.8 3.4.1.9 3.4.1.10 3.4.1.11 3.4.2 3.4.3 3.4.3.1 3.4.3.2 3.4.3.3 3.5 3.5.1 3.5.2 3.5.3 3.5.4 3.6

Iono-covalent Solids 128 AX Compounds 128 AX2 Compounds 130 AX6 Compounds 132 ABX2 Compounds 132 AB2X4 Compounds (Spinel and Olivine Structures) 134 ABX3 Compounds (Perovskite and Related Phases) 135 A2B2O5 (ABO2.5) Compounds (Oxygen-Deficient Perovskites) 137 AxByOz Compounds (Bronzes) 139 A2B2X7 Compounds (Pyrochlores) 139 Silicate Compounds 140 Porous Structures 141 Metal Carbides, Silicides, Borides, Hydrides, and Nitrides 144 Metallic Alloys and Intermetallic Compounds 144 Zintl Phases 147 Nonpolar Binary Intermetallic Phases 149 Ternary Intermetallic Phases 151 Structural Disturbances 153 Intrinsic Point Defects 154 Extrinsic Point Defects 155 Structural Distortions 156 Bond Valence Sum Calculations 158 Structure Control and Synthetic Strategies 162 Practice Problems 165 References 167

4 4.1 4.2 4.3

The Electronic Level I: An Overview of Band Theory 171 The Many-Body Schrödinger Equation and Hartree–Fock 171 Choice of Boundary Conditions: Born’s Conditions 177 Free-Electron Model for Metals: From Drude (Classical) to Sommerfeld (Fermi–Dirac) 179 Bloch’s Theorem, Bloch Waves, Energy Bands, and Fermi Energy 180 Reciprocal Space and Brillouin Zones 182 Choices of Basis Sets and Band Structure with Applicative Examples 188 From the Free-Electron Model to the Plane Wave Expansion 189 Fermi Surface, Brillouin Zone Boundaries, and Alkali Metals versus Copper 191 Understanding Metallic Phase Stability in Alloys 193 The Localized Orbital Basis Set Method 195 Understanding Band Structure Diagram with Rhenium Trioxide 196 Probing DOS Band Structure in Metallic Alloys 199 Breakdown of the Independent-Electron Approximation 200 Density Functional Theory: The Successor to the Hartree–Fock Approach in Materials Science 202 The Continuous Quest for Better DFT XC Functionals 205 Van der Waals Forces and DFT 208 Practice Problems 210 References 210

4.4 4.5 4.6 4.6.1 4.6.2 4.6.3 4.6.4 4.6.5 4.6.6 4.7 4.8 4.9 4.10

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5 5.1 5.2 5.3 5.3.1 5.3.2 5.4 5.5 5.5.1 5.5.2 5.5.3 5.5.4 5.6

The Electronic Level II: The Tight-Binding Electronic Structure Approximation 213 The General LCAO Method 214 Extension of the LCAO Treatment to Crystalline Solids 219 Orbital Interactions in Monatomic Solids 221 σ-Bonding Interactions 221 π-Bonding Interactions 225 Tight-Binding Assumptions 229 Qualitative LCAO Band Structures 232 Illustration 1: Transition Metal Oxides with Vertex-Sharing Octahedra 236 Illustration 2: Reduced Dimensional Systems 238 Illustration 3: Transition Metal Monoxides with Edge-Sharing Octahedra 240 Corollary 243 Total Energy Tight-Binding Calculations 244 Practice Problems 246 References 246

6 6.1 6.2 6.2.1 6.2.2 6.2.3 6.2.4 6.3 6.3.1 6.3.2 6.4 6.4.1 6.4.1.1 6.4.1.2 6.4.1.3 6.4.1.4 6.4.2 6.4.2.1 6.4.2.2 6.4.2.3 6.4.3 6.4.4

Transport Properties 249 An Introduction to Tensors 249 Microscopic Theory of Electrical Transport in Ceramics: The Role of Point Defects Oxygen-Deficient/Metal Excess and Metal-Deficient/Oxygen Excess Oxides 256 Substitutions by Aliovalent Cations with Valence Isoelectronicity 261 Substitutions by Isovalent Cations That Are Not Valence Isoelectronic 263 Nitrogen Vacancies in Nitrides 266 Thermal Conductivity 268 The Free Electron Contribution 269 The Phonon Contribution 271 Electrical Conductivity 274 Band Structure Considerations 278 Conductors 278 Insulators 279 Semiconductors 281 Semimetals 290 Thermoelectric, Photovoltaic, and Magnetotransport Properties 292 Thermoelectrics 292 Photovoltaics 298 Galvanomagnetic Effects and Magnetotransport Properties 301 Superconductors 303 Improving Bulk Electrical Conduction in Polycrystalline, Multiphasic, and Composite Materials 307 Mass Transport 308 Atomic Diffusion 309 Ionic Conduction 316 Practice Problems 321 References 322

6.5 6.5.1 6.5.2

7 7.1 7.1.1

Hopping Conduction and Metal–Insulator Transitions Correlated Systems 327 The Mott–Hubbard Insulating State 329

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7.1.2 7.1.3 7.2 7.3 7.4 7.5

Charge-Transfer Insulators 334 Marginal Metals 334 Anderson Localization 336 Experimentally Distinguishing Disorder from Electron Correlation Tuning the M–I Transition 343 Other Types of Electronic Transitions 345 Practice Problems 347 References 347

8 8.1 8.1.1 8.1.2 8.2 8.3 8.3.1 8.3.2 8.3.3 8.3.4 8.3.5 8.3.5.1 8.3.5.2 8.3.5.3 8.3.6 8.4 8.5 8.5.1 8.5.1.1 8.5.1.2 8.5.2 8.5.3 8.5.3.1 8.5.3.2 8.5.3.3 8.5.4 8.6 8.6.1 8.6.2 8.7 8.8 8.8.1 8.8.2 8.8.3 8.8.4

Magnetic and Dielectric Properties 349 Phenomenological Description of Magnetic Behavior 351 Magnetization Curves 354 Susceptibility Curves 355 Atomic States and Term Symbols of Free Ions 359 Atomic Origin of Paramagnetism 365 Orbital Angular Momentum Contribution: The Free Ion Case 366 Spin Angular Momentum Contribution: The Free Ion Case 367 Total Magnetic Moment: The Free Ion Case 368 Spin–Orbit Coupling: The Free Ion Case 368 Single Ions in Crystals 371 Orbital Momentum Quenching 371 Spin Momentum Quenching 373 The Effect of JT Distortions 373 Solids 374 Diamagnetism 376 Spontaneous Magnetic Ordering 377 Exchange Interactions 379 Direct Exchange and Superexchange Interactions in Magnetic Insulators 382 Indirect Exchange Interactions 387 Itinerant Ferromagnetism 390 Noncollinear Spin Configurations and Magnetocrystalline Anisotropy 394 Geometric Frustration 394 Magnetic Anisotropy 397 Magnetic Domains 398 Ferromagnetic Properties of Amorphous Metals 401 Magnetotransport Properties 401 The Double Exchange Mechanism 402 The Half-Metallic Ferromagnet Model 403 Magnetostriction 404 Dielectric Properties 405 The Microscopic Equations 407 Piezoelectricity 408 Pyroelectricity 414 Ferroelectricity 416 Practice Problems 421 References 422

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9 9.1 9.2 9.3 9.4 9.5

Optical Properties of Materials Maxwell’s Equations 425 Refractive Index 428 Absorption 436 Nonlinear Effects 441 Summary 446 Practice Problems 446 References 447

10 10.1 10.2 10.2.1 10.2.2 10.2.3 10.2.3.1 10.2.3.2 10.2.3.3 10.2.4 10.3 10.3.1 10.3.2 10.3.3 10.3.4

Mechanical Properties 449 Stress and Strain 449 Elasticity 452 The Elasticity Tensors 455 Elastically Isotropic and Anisotropic Solids 459 The Relation Between Elasticity and the Cohesive Forces in a Solid 465 Bulk Modulus 466 Rigidity (Shear) Modulus 467 Young’s Modulus 470 Superelasticity, Pseudoelasticity, and the Shape Memory Effect 473 Plasticity 475 The Dislocation-Based Mechanism to Plastic Deformation 481 Polycrystalline Metals 487 Brittle and Semi-brittle Solids 489 The Correlation Between the Electronic Structure and the Plasticity of Materials 490 Fracture 491 Practice Problems 494 References 495

10.4

425

11 11.1 11.1.1 11.2 11.3 11.3.1 11.3.2 11.3.3 11.3.4 11.4 11.5 11.5.1 11.5.2 11.5.3 11.5.4

Phase Equilibria, Phase Diagrams, and Phase Modeling 499 Thermodynamic Systems and Equilibrium 500 Equilibrium Thermodynamics 504 Thermodynamic Potentials and the Laws 507 Understanding Phase Diagrams 510 Unary Systems 510 Binary Systems 511 Ternary Systems 518 Metastable Equilibria 522 Experimental Phase Diagram Determinations 522 Phase Diagram Modeling 523 Gibbs Energy Expressions for Mixtures and Solid Solutions 524 Gibbs Energy Expressions for Phases with Long-Range Order 527 Other Contributions to the Gibbs Energy 530 Phase Diagram Extrapolations: The CALPHAD Method 531 Practice Problems 534 References 535

12 12.1 12.1.1

Synthetic Strategies 537 Synthetic Strategies 538 Direct Combination 538

Contents

12.1.2 12.1.2.1 12.1.2.2 12.1.2.3 12.1.3 12.1.4 12.1.5 12.1.6 12.1.7 12.1.8 12.1.8.1 12.1.8.2 12.1.8.3 12.2

Low Temperature 540 Sol–Gel 540 Solvothermal 543 Intercalation 544 Defects 546 Combinatorial Synthesis 548 Spinodal Decomposition 548 Thin Films 550 Photonic Materials 552 Nanosynthesis 553 Liquid Phase Techniques 554 Vapor/Aerosol Methods 556 Combined Strategies 556 Summary 558 Practice Problems 559 References 559

13 13.1 13.2 13.2.1 13.2.2 13.2.3 13.2.4 13.2.5 13.2.6 13.3 13.3.1 13.3.1.1 13.3.1.2 13.3.2 13.3.2.1 13.3.2.2 13.3.2.3 13.3.2.4 13.3.2.5 13.3.2.6 13.3.2.7 13.3.2.8

An Introduction to Nanomaterials 563 History of Nanotechnology 564 Nanomaterials Properties 565 Electrical Properties 566 Magnetic Properties 567 Optical Properties 567 Thermal Properties 568 Mechanical Properties 569 Chemical Reactivity 570 More on Nanomaterials Preparative Techniques 572 Top-Down Methods for the Fabrication of Nanocrystalline Materials 572 Nanostructured Thin Films 572 Nanocrystalline Bulk Phases 573 Bottom-Up Methods for the Synthesis of Nanostructured Solids 574 Precipitation 575 Hydrothermal Techniques 576 Micelle-Assisted Routes 577 Thermolysis, Photolysis, and Sonolysis 580 Sol–Gel Methods 581 Polyol Method 582 High-Temperature Organic Polyol Reactions (IBM Nanoparticle Synthesis) Additive Manufacturing (3D Printing) 584 References 586

14 14.1 14.1.1 14.1.2

Introduction to Computational Materials Science 589 A Short History of Computational Materials Science 590 1945–1965: The Dawn of Computational Materials Science 591 1965–2000: Steady Progress Through Continued Advances in Hardware and Software 595 2000–Present: High-Performance and Cloud Computing 598 Spatial and Temporal Scales, Computational Expense, and Reliability of Solid-State Calculations 600

14.1.3 14.2

584

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14.3 14.3.1 14.3.2

14.3.3

Illustrative Examples 604 Exploration of the Local Atomic Structure in Multi-principal Element Alloys by Quantum Molecular Dynamics 604 Magnetic Properties of a Series of Double Perovskite Oxides A2BCO6 (A = Sr, Ca; B = Cr; C = Mo, Re, W) by Monte Carlo Simulations in the Framework of the Ising Model 606 Crystal Plasticity Finite Element Method (CPFEM) Analysis for Modeling Plasticity in Polycrystalline Alloys 613 References 617

15 15.1 15.2 15.3 15.4 15.5 15.6 15.7 15.8 15.9 15.10 15.11 15.12

Case Study I: TiO2 619 Crystallography 619 Microstructure 623 Bonding 626 Electronic Structure 627 Transport 628 Metal–Insulator Transitions 632 Magnetic and Dielectric Properties Optical Properties 634 Mechanical Properties 635 Phase Equilibria 636 Synthesis 638 Nanomaterial 639 Practice Questions 639 References 640

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16 16.1 16.2 16.3 16.4 16.5 16.6 16.7 16.8 16.9 16.10 16.11 16.12

Case Study II: GaN 643 Crystallography 643 Microstructure 646 Bonding 647 Electronic Structure 647 Transport 648 Metal–Insulator Transitions 650 Magnetic and Dielectric Properties Optical Properties 652 Mechanical Properties 653 Phase Equilibria 654 Synthesis 654 Nanomaterial 656 Practice Questions 657 References 658

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Appendix A: List of the 230 Space Groups 659 Appendix B: The 32 Crystal Systems and the 47 Possible Forms Appendix C: Principles of Tensors 667 Appendix D: Solutions to Practice Problems 679 Index

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Foreword to Second Edition Materials science is one of the broadest of the applied science and engineering fields since it uses concepts from so many different subject areas. Chemistry is one of the key fields of study, and in many materials science programs, students must take general chemistry as a prerequisite for all but the most basic of survey courses. However, that is typically the last true chemistry course that they take. The remainder of their chemistry training is accomplished in their materials classes. This has served the field well for many years, but over the past couple of decades, new materials development has become more heavily dependent upon synthetic chemistry. This second edition of Principles of Inorganic Materials Design serves as a fine text to introduce the materials student to the fundamentals of designing materials through synthetic chemistry and the chemist to some of the issues involved in materials design. When I obtained my BS in ceramic engineering in 1981, the primary fields of materials science – ceramics, metals, polymers, and semiconductors – were generally taught in separate departments, although there was frequently some overlap. This was particularly true at the undergraduate level, although graduate programs frequently had more subject overlap. During the 1980s, many of these departments merged to form materials science and engineering departments that began to take a more integrated approach to the field, although chemical and electrical engineering programs tended to cover polymers and semiconductors in more depth. This trend continued in the 1990s and included the writing of texts such as The Production of Inorganic Materials by Evans and De Jonghe (Prentice Hall College Division, 1991), which focused on traditional production methods. Synthetic chemical approaches became more important as the decade progressed and academia began to address this in the classroom, particularly at the graduate level. The first edition of Principles of Inorganic Materials Design strove to make this material available to the upper division undergraduate student. The second edition of Principles of Inorganic Materials Design corrects several gaps in the first edition to convert it from a very good compilation of the field into a text that is very usable in the undergraduate classroom. Perhaps the biggest of these is the addition of practice problems at the end of every chapter since the second best way to learn a subject is to apply it to problems (the best is to teach it), and this removes the burden of creating the problems from the instructor. Chapter 1, Crystallographic Considerations, is new and both reviews the basic information in most introductory materials courses and clearly presents the more advanced concepts such as the mathematical description of crystal symmetry that are typically covered in courses on crystallography of physical chemistry. Chapter 10, Mechanical Properties, has also been expanded significantly to provide both the basic concepts needed by those approaching the topic for the first time and the solid mathematical treatment needed to relate the mechanical properties to atomic bonding,

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crystallography, and other material properties treated in previous chapters. This is particularly important as devices use smaller active volumes of material, since this seldom results in the materials being in a stress-free state. In summary, the second edition of Principles of Inorganic Materials Design is a very good text for several applications: a first materials course for chemistry and physics students, a consolidated materials chemistry course for materials science students, and a second materials course for other engineering and applied science students. It also serves as the background material to pursue the chemical routes to make these new materials described in texts such as Inorganic Materials Synthesis and Fabrication by Lalena and Cleary (Wiley, 2008). Such courses are critical to insure that students from different disciplines can communicate as they move into industry and face the need to design new materials or reduce costs through synthetic chemical routes. Martin W. Weiser

Martin earned his BS in ceramic engineering from Ohio State University and MS and PhD in Materials Science and Mineral Engineering from the University of California, Berkeley. At Berkeley he conducted fundamental research on sintering of powder compacts and ceramic matrix composites. After graduation he joined the University of New Mexico (UNM) where he was a visiting assistant professor in chemical engineering and then assistant professor in mechanical engineering. At UNM he taught introductory and advanced materials science classes to students from all branches of engineering. He continued his research in ceramic fabrication as part of the Center for MicroEngineered Ceramics and also branched out into solder metallurgy and biomechanics in collaboration with colleagues from Sandia National Laboratories and the UNM School of Medicine, respectively. Martin joined Johnson Matthey Electronics in a technical service role supporting the Discrete Power Products Group (DPPG). In this role he also initiated JME’s efforts to develop Pb-free solders for power die attach that came to fruition in collaboration with John N. Lalena several years later after JME was acquired by Honeywell. Martin spent several years as the product manager for the DPPG and then joined the Six Sigma Plus organization after earning his Six Sigma Black Belt working on polymer/metal composite thermal interface materials (TIMs). He spent the last several years in the R&D group as both a group manager and principle scientist where he lead the development of improved Pb-free solders and new TIMs.

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Foreword to First Edition Whereas solid-state physics is concerned with the mathematical description of the varied physical phenomena that solids exhibit and the solid-state chemist is interested in probing the relationships between structural chemistry and physical phenomena, the materials scientist has the task of using these descriptions and relationships to design materials that will perform specified engineering functions. However, the physicist and the chemist are often called upon to act as material designers, and the practice of materials design commonly requires the exploration of novel chemistry that may lead to the discovery of physical phenomena of fundamental importance for the body of solid-state physics. I cite three illustrations where an engineering need has led to new physics and chemistry in the course of materials design. In 1952, I joined a group at the MIT Lincoln Laboratory that had been charged with the task of developing a square B–H hysteresis loop in a ceramic ferrospinel that could have its magnetization reversed in less than 1 μs by an applied magnetic field strength less than twice the coercive field strength. At that time, the phenomenon of a square B–H loop had been obtained in a few iron alloys by rolling them into tapes so as to align the grains, and hence the easy magnetization directions, along the axis of the tape. The observation of a square B–H loop led Jay Forrester, an electrical engineer, to invent the coincident-current, random-access magnetic memory for the digital computer since, at that time, the only memory available was a 16 × 16 byte electrostatic storage tube. Unfortunately, the alloy tapes gave too slow a switching speed. As an electrical engineer, Jay Forrester assumed the problem was eddy-current losses in the tapes, so he had turned to the ferrimagnetic ferrospinels that were known to be magnetic insulators. However, the polycrystalline ferrospinels are ceramics that cannot be rolled! Nevertheless, the Air Force had financed the MIT Lincoln Laboratory to develop an air defense system of which the digital computer was to be a key component. Therefore, Jay Forrester was able to put together an interdisciplinary team of electrical engineers, ceramists, and physicists to realize his random-access magnetic memory with ceramic ferrospinels. The magnetic memory was achieved by a combination of systematic empiricism, careful materials characterization, theoretical analysis, and the emergence of an unanticipated phenomenon that proved to be a stroke of good fortune. A systematic mapping of the structural, magnetic, and switching properties of the Mg–Mn–Fe ferrospinels as a function of their heat treatments revealed that the spinels, in one part of the phase diagram, were tetragonal rather than cubic and that compositions, just on the cubic side of the cubic–tetragonal phase boundary, yield sufficiently square B–H loops if given a carefully controlled heat treatment. This observation led me to propose that the tetragonal distortion was due to a cooperative orbital ordering on the Mn3+ ions that would lift the cubic-field orbital degeneracy; cooperativity of the site distortions minimizes the cost in elastic energy and leads to a distortion of the entire structure. This phenomenon is now known as a cooperative

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Jahn–Teller distortion since Jahn and Teller had earlier pointed out that a molecule or molecular complex, having an orbital degeneracy, would lower its energy by deforming its configuration to a lower symmetry that removed the degeneracy. Armed with this concept, I was able almost immediately to apply it to interpret the structure and the anisotropic magnetic interactions that had been found in the manganese–oxide perovskites since the orbital order revealed the basis for specifying the rules for the sign of a magnetic interaction in terms of the occupancies of the overlapping orbitals responsible for the interatomic interactions. These rules are now known as the Goodenough– Kanamori rules for the sign of a superexchange interaction. Thus an engineering problem prompted the discovery and description of two fundamental phenomena in solids that ever since have been used by chemists and physicists to interpret structural and magnetic phenomena in transition metal compounds and to design new magnetic materials. Moreover, the discovery of cooperative orbital ordering fed back to an understanding of our empirical solution to the engineering problem. By annealing at the optimum temperature for a specified time, the Mn3+ ions of a cubic spinel would migrate to form Mn-rich regions where their energy is lowered through cooperative, dynamic orbital ordering. The resulting chemical inhomogeneities acted as nucleating centers for domains of reverse magnetization that, once nucleated, grew away from the nucleating center. We also showed that eddy currents were not responsible for the slow switching of the tapes, but a small coercive field strength and an intrinsic damping factor for spin rotation. In the early 1970s, an oil shortage focused worldwide attention on the need to develop alternative energy sources, and it soon became apparent that these sources would benefit from energy storage. Moreover, replacing the internal combustion engine with electric-powered vehicles, or at least the introduction of hybrid vehicles, would improve the air quality, particularly in big cities. Therefore, a proposal by the Ford Motor Company to develop a sodium–sulfur battery operating at 3008 C with molten electrodes and a ceramic Na+-ion electrolyte stimulated interest in the design of fast alkali ion conductors. More significant was interest in a battery in which Li+ rather than H+ is the working ion, since the energy density that can be achieved with an aqueous electrolyte is lower than what, in principle, can be obtained with a nonaqueous Li+-ion electrolyte. However, realization of a Li+-ion rechargeable battery would require identification of a cathode material into/from which Li+ ions can be inserted/extracted reversibly. Brian Steele of Imperial College, London, first suggested the use of TiS2, which contains TiS2 layers held together only by van der Waals S2––S2– bonding; lithium can be inserted reversibly between the TiS2 layers. M. Stanley Whittingham’s demonstration was the first to reduce this suggestion to practice while he was at the Exxon Corporation. Whittingham’s demonstration of a rechargeable Li–TiS2 battery was commercially nonviable because the lithium anode proved unsafe. Nevertheless, his demonstration focused attention on the work of the chemists Jean Rouxel of Nantes and R. Schöllhorn of Berlin on insertion compounds that provide a convenient means of continuously changing the mixed valency of a fixed transition metal array across a redox couple. Although work at Exxon was halted, their demonstration had shown that if an insertion compound, such as graphite, was used as the anode, a viable lithium battery could be achieved, but the use of a less electropositive anode would require an alternative insertion-compound cathode material that provided a higher voltage versus a lithium anode than TiS2. I was able to deduce that no sulfide would give a significantly higher voltage than that obtained with TiS2 and therefore that it would be necessary to go to a transition metal oxide. Although oxides other than V2O5 and MoO3, which contain vandyl or molybdyl ions, do not form layered structures analogous to TiS2, I knew that LiMO2 compounds exist that have a layered structure similar to that of LiTiS2. It was only necessary to choose the correct M3+ cation and to determine how much Li could be extracted before the structure collapsed. That was how the Li1−xCoO2 cathode material was developed, which now powers the cell telephones and laptop computers. The

Foreword to First Edition

choice of M = Co, Ni, or Ni0.5+δMn0.5−δ was dictated by the position of the redox energies and an octahedral site preference energy strong enough to inhibit migration of the M atom to the Li layers on the removal of Li. Electrochemical studies of these cathode materials, and particularly of Li1−x Ni0.5+δMn0.5−δO2, have provided a demonstration of the pinning of a redox couple at the top of the valence band. This is a concept of singular importance for interpretation of metallic oxides having only M–O–M interactions, of the reason for oxygen evolution at critical Co(IV)/Co(III) or Ni(IV)/Ni (III) ratios in Li1−xMO2 studies, and of why Cu(III) in an oxide has a low-spin configuration. Moreover, exploration of other oxide structures that can act as hosts for insertion of Li as a guest species has provided a means of quantitatively determining the influence of a counter cation on the energy of a transition metal redox couple. This determination allows tuning of the energy of a redox couple, which may prove important for the design of heterogenous catalysts. As a third example, I turn to the discovery of high-temperature superconductivity in the copper oxides, first announced by Bednorz and Müller of IBM Zürich in the summer of 1986. Karl A. Müller, the physicist of the pair, had been thinking that a dynamic Jahn–Teller ordering might provide an enhanced electron–phonon coupling that would raise the superconductive critical temperature TC. He turned to his chemist colleague Bednorz to make a mixed-valence Cu3+/Cu2+ compound since Cu2+ has an orbital degeneracy in an octahedral site. This speculation led to the discovery of the family of high-TC copper oxides; however, the enhanced electron–phonon coupling is not due to a conventional dynamic Jahn–Teller orbital ordering, but rather to the first-order character of the transition from localized to itinerant electronic behavior of σ-bonding Cu: 3d electrons of (x2 − y2) symmetry in CuO2 planes. In this case, the search for an improved engineering material has led to a demonstration that the celebrated Mott–Hubbard transition is generally not as smooth as originally assumed, and it has introduced an unanticipated new physics associated with bond length fluctuations and vibronic electronic properties. It has challenged the theorist to develop new theories of the crossover regime that can describe the mechanism of superconductive pair formation in the copper oxides, quantum critical-point behavior at low temperatures, and an anomalous temperature dependence of the resistivity at higher temperatures as a result of strong electron–phonon interactions. These examples show how the challenge of materials design from the engineer may lead to new physics as well as to new chemistry. Sorting out of the physical and chemical origins of the new phenomena fed back to the range of concepts available to the designer of new engineering materials. In recognition of the critical role in materials design of interdisciplinary cooperation between physicists, chemists, ceramists, metallurgists, and engineers that is practiced in industry and government research laboratories, John N. Lalena and David A. Cleary have initiated, with their book, what should prove to be a growing trend toward greater interdisciplinarity in the education of those who will be engaged in the design and characterization of tomorrow’s engineering materials. John B. Goodenough

John received the Nobel Prize in Chemistry in 2019. See his biography page 386.

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Preface to Third Edition I contacted John Nick Lalena almost ten years ago, electronically by e-mail to inform him of my enthusiasm for a review paper he had written on crystallography and share some comments. Nick responded immediately, and we started an intense e-mail correspondence. I then discovered the second edition of this book, and I loved it very much, as a kind of multifacetted diamond. Of course, diamonds always have some defects, and I commented on them with Nick again, besides my admiration for the broad scope of the book and its most pleasant style to my taste. Somewhat later, in August 2018, Nick let me know about the preparation of a third edition and asked me if I would accept to review some of the new and rewritten chapters. As I started the job, I naturally made comments and even suggested new lines that could be added here and there – so much so that Nick and Dave proposed me to become a third co-author for the third edition. Principles of Inorganic Materials Design now has an added French touch in some places, even if science is naturally universal. The third edition is thicker than the second, which was thicker than the first. Quite a few sections in existing chapters have been rewritten and/or expanded. Three chapters have been added, two of them being case study chapters. Thanks to the skills of its three authors, PIMD3, as we like to call it, is very versatile. It has fundamental aspects presented pedagogically as appropriate for a textbook, with additional historical focus and biographical presentations of some of our heroes in crystallography, solid-state physics, and materials science. It has some sections on rather recent developments in electron density functional theory as well as a new appendix presenting tensors from different physical points of view. It also touches on a plethora of topics of relevance to the practitioners of materials science such as mechanical engineers, including a new section about additive manufacturing, Diamonds must be sculpted so that they can shine from whatever angle they are looked at, just as with the French Blue Diamond of the Crown now visible at the Smithsonian Institution. At the same time the best natural diamonds contain defects, and, to quote a phrase attributed to Sir Charles Frank, “crystals are like people: it is the defects in them that make them interesting.” I thus believe that readers, either students, teachers, researchers or practitioners, will learn a lot from this book and enjoy it even more than I enjoyed PIMD2, while, at the same time, they should hopefully find it not absolutely perfect in places from their own point of view. Science is not a closed book, and no science book should be a closed book either: a good science book should stay open and be read and annotated. Olivier B. M. Hardouin Duparc

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Preface to Second Edition In our first attempt at writing a textbook on the highly interdisciplinary subject of inorganic materials design, we recognized the requirement that the book needed to appeal to a very broad-based audience. Indeed, practitioners of materials science and engineering come from many different educational backgrounds, each emphasizing different aspects. These include solid-state chemistry, condensed-matter physics, metallurgy, ceramics, mechanical engineering, and materials science and engineering (MS&E). Unfortunately, we did not adequately anticipate the level of difficulty that would be associated with successfully implementing the task of attracting readers from so many disciplines that, though distinct, possess the common threads of elucidating and utilizing structure/property correlation in the design of new materials. As a result, the first edition had a number of shortfalls. First and foremost, owing to a variety of circumstances, there were many errors that, regrettably, made it into the printed book. Great care has been taken to correct each of these. In addition to simply revising the first edition, however, the content has been updated and expanded as well. As was true with the first edition, this book is concerned, by and large, with theoretical structure/property correlation as it applies to materials design. Nevertheless, a small amount of space is dedicated to the empirical practice of synthesis and fabrication. Much more discussion is devoted to these specialized topics concerned with the preparation of materials, as opposed to their design, in numerous other books, one of which is our companion textbook, Inorganic Materials Synthesis and Fabrication. Some features added to this second edition include an expanded number of worked examples and an appendix containing solutions to selected end-of-chapter problems. The overall goal of our second edition is, quite simply, to rectify the problems we encountered earlier, thereby producing a work that is much better suited as a tool to the working professionals, educators, and students of this fascinating field. John N. Lalena and David A. Cleary

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Preface to First Edition Inorganic solid-state chemistry has matured into its own distinct subdiscipline. The reader may wonder why we have decided to add another textbook to the plethora of books already published. Our response is that we see a need for a single source presentation that recognizes the interdisciplinary nature of the field. Solid-state chemists typically receive a small amount of training in condensed-matter physics, and none in materials science or engineering, and yet all of these traditional fields are inextricably part of inorganic solid-state chemistry. Materials scientists and engineers have traditionally been primarily concerned with the fabrication and utilization of materials already synthesized by the chemist and identified by the physicist as having the appropriate intrinsic properties for a particular engineering function. Although the demarcation between the three disciplines remains in an academic sense, the separate job distinctions for those working in the field is fading. This is especially obvious in the private sector where one must ensure that materials used in real commercial devices not only perform their primary function but also meet a variety of secondary requirements. Individuals involved with these multidisciplinary and multitask projects must be prepared to work independently or to collaborate with other specialists in facing design challenges. In the latter case, communication is enhanced if each individual is able to speak the “language” of the other. Therefore, in this book we introduce a number of concepts that are not usually covered in standard solid-state chemistry textbooks. When this occurs, we try to follow the introduction of the concept with an appropriate worked example to demonstrate its use. Two areas that have lacked thorough coverage in most solid-state chemistry texts in the past, namely, microstructure and mechanical properties, are treated extensively in this book. We have kept the mathematics to a minimum – but adequate – level, suitable for a descriptive treatment. Appropriate citations are included for those needing the quantitative details. It is assumed that the reader has sufficient knowledge of calculus and elementary linear algebra, particularly matrix manipulations, and some prior exposure to thermodynamics, quantum theory, and group theory. The book should be satisfactory for senior level undergraduate or beginning graduate students in chemistry. One will recognize from the Table of Contents that the entire textbooks have been devoted to each of the chapters in this book, and this limits the depth of coverage out of necessity. Along with their chemistry colleagues, physics and engineering students should also find the book to be informative and useful. Every attempt has been made to extensively cite all the original and pertinent research in a fashion similar to that found in a review article. Students are encouraged to seek out this work. We have also included biographies of several individuals who have made significant fundamental contributions to inorganic materials science in the twentieth century. Limiting these to the small number we have room for was, of course, difficult. The reader should be warned that some topics have been left

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out. In this book, we only cover nonmolecular inorganic materials. Polymers and macromolecules are not discussed. Nor are the other extreme, for example, molecular electronics. Also omitted are coverages of surface science, self-assembly, and composite materials. We are grateful to Professor John B. Wiley, Dr. Nancy F. Dean, Dr. Martin W. Weiser, Professor Everett E. Carpenter, and Dr. Thomas K. Kodenkandath for reviewing various chapters in this book. We are grateful to Professor John F. Nye, Professor John B. Goodenough, Dr. Frans Spaepen, Dr. Larry Kaufman, and Dr. Bert Chamberland for providing biographical information. We would also like to thank Professor Philip Anderson, Professor Mats H. Hillert, Professor Nye, Dr. Kaufman, Dr. Terrell Vanderah, Dr. Barbara Sewall, and Mrs Jennifer Moss for allowing us to use photographs from their personal collections. Finally, we acknowledge the inevitable neglect our families must have felt during the period taken to write this book. We are grateful for their understanding and tolerance. John N. Lalena and David A. Cleary

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Acronyms AC AFMs AIMD AOT APW BCC BM BMGs BOs BVS BZ CA CALPHAD CB CCP CCSL CDW CFSE CI CMR COs CRSS CSL CTAB CTE CVD CVM DE DFT DMFT DOS DR1 DSC (lattice) DSC DTA

alternating current antiferromagnets Ab initio molecular dynamics aerosol OT (sodium dioctylsulfosuccinate) augmented plane wave body-centered cubic Bohr magneton bulk metallic glasses Bloch orbitals - also referred as Bloch sums bond valence sums Brillouin zone cellular automation CALculation of PHAse Diagrams carbazole-9-carbonyl chloride cubic-close package constrained coincidence site lattice charge density wave crystal field stabilization energy configuration interaction colossal magnetoresistance crystal orbitals critical resolved shear stress coincidence site lattice cetyltrimethylammonium bromide coefficient of thermal expansion chemical vapor deposition cluster variation method double exchange density functional theory dynamical mean field theory density-of-states disperse red 1 displacement shift complete (lattice) differential scanning calorimetry differential thermal analysis

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Acronyms

EAM EBS EBSD ECAE ECAP EDTA EMF EOS EPMA FC FCC FDM FEM FVM GB GGA GLAD GMR GTOs HCP HeIM HF HOMO HRTEM IC IR JT LCAO LCAO-MO LCOAO LDA LHB LRO LSDA LUMO MC MCE MD M-I MMC M–NM MO MOCVD MP MPB MRO

embedded atom method electrostatic bond strength electron backscatter diffraction equal-channel angular extrusion equal-channel angular pressing ethylenediamine tetraacetate electromagnetic field equation of state electron probe microanalysis field cooled face-centered cubic finite difference method finite element method finite volume method grain boundary generalized gradient approximation glancing angle deposition giant magnetoresistance Gaussian-type orbitals hexagonal close packed helium ion microscope Hartree Fock highest occupied molecular orbital high-resolution transmission electron microscopy integrated circuits infrared radiation Jahn–Teller linear combination of atomic orbitals linear combination of atomic orbitals - molecular orbitals linear combination of orthogonalized atomic orbitals local density approximation lower Hubbard band long-range (translational) order local spin-density approximation lowest unoccupied molecular orbital Monte Carlo magnetocaloric effect molecular dynamics metal-insulator metal matrix composite metallic–nonmetallic molecular orbital metal–organic chemical vapor deposition Møller–Plesset morphotropic phase boundary medium-range order

Acronyms

MS&E MSD MWNT NA ND NFE NMR ODF PFCM PFM PIMD PIMD PCF PLD PV PVD PVP PZT QMC QMD RD RKKY RP RPA SALC SANS SAXS SC SCF SDS SDW SEM SHS SK SMA SP SPD SRO STM STOs SWNT TB TD TE TEM TEOS

materials science and engineering microstructure sensitive design multiwalled carbon nanotubes numerical aperture normal directions nearly free electron nuclear magnetic resonance orientation distribution function phase field crystal modeling phase field modeling principles of inorganic materials design path integral molecular dynamics (single-mode) photonic crystal fiber pulsed laser deposition photovoltaic physical vapor deposition poly(vinylpyrrolidone) Pb(Zr,Ti)O3 quantum Monte Carlo quantum molecular dynamics radial directions Rudderman–Kittel–Kasuya–Yoshida Ruddlesden–Popper random-phase approximation symmetry-adapted linear combination small-angle Newton scattering small-angle X-ray scattering simple cubic self-consistent field sodium dodecylsulfate spin-density wave scanning electron microscope self-propagating high-temperature synthesis Slater-Koster shape memory alloys spin-polarized severe plastic deformation short-range order scanning tunneling microscope Slater-type orbitals single-walled carbon nanotubes tight binding transverse directions thermoelectric transmission electron microscope tetraethyl orthosilicate

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Acronyms

TGG TIM TOPO TSSG UFG UHB UTS VEC VRH VSEPR XC XRD ZFC

templated grain growth thermal interface material trioctylphosphine oxide top-seeded solution growth ultrafine-grained upper Hubbard band ultimate tensile strength valence electron concentration variable range hopping valence shell electron pair repulsion exchange and correlation X-ray diffraction zero-field cooled

1

1 Crystallographic Considerations There are many possible classification schemes for solids that can be envisioned. We can categorize a material based solely on its chemical composition (inorganic, organic, or hybrid), the primary bonding type (ionic, covalent, metallic), its structure type (catenation polymer, extended threedimensional network), or its crystallinity (crystalline or noncrystalline). It is the latter scheme that is the focus of this chapter. A crystalline material exhibits a large degree of structural order in the arrangement of its constituent particles, be they atoms, ions, or molecules, over a large length scale whereas a noncrystalline material exhibits structural order only over the very short-range length scale corresponding to the first coordination sphere. It is structural order – the existence of a methodical arrangement among the component particles – that makes the systematic study and design of materials with prescribed properties possible. A crystal may be explicitly defined as a homogeneous solid consisting of a periodically repeating three-dimensional pattern of particles. There are three key structural features to crystals, between physics and mathematics: 1) The motif, which is the group of atoms or molecules repeated at each lattice point. 2) Symmetry, the geometric arrangement of the lattice points, defined by a repeating unit cell. 3) Long-range (translational) order (LRO), referring to the periodicity, or regularity in the arrangement of the material’s atomic or molecular constituents on a length scale at least a few times larger than the size of the unit cell. The presence of a long-range order allows crystals to scatter incoming waves, of appropriate wavelengths, so as to produce discrete diffraction patterns, which, in turn, ultimately enables ascertainment of the actual atomic positions and, hence, crystalline structure. The periodic LRO can be extended to quasiperiodic LRO to encompass quasicrystals; see Section 1.1.2.

1.1

Degrees of Crystallinity

Crystallinity, like most things, can vary in degree. Even single crystals (a.k.a. monocrystals) have intrinsic point defects (e.g. lattice site vacancies) and extrinsic point defects (e.g. impurities), as well as extended defects such as dislocations. Defects are critical to the physical properties of crystals and will be extensively covered in later chapters. What we are referring to here with the degree of crystallinity is not the simple proportion of defects present in the solid, but rather the spectrum

Principles of Inorganic Materials Design, Third Edition. John N. Lalena, David A. Cleary, and Olivier B. M. Hardouin Duparc. © 2020 John Wiley & Sons, Inc. Published 2020 by John Wiley & Sons, Inc.

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1 Crystallographic Considerations

Table 1.1

Degrees of crystallinity.

Type

Defining features

Monocrystalline

LRO

Quasicrystalline

Noncrystallographic rotational symmetry, no LRO

Polycrystalline

Crystallites separated by grain boundaries

Semicrystalline

Crystalline regions separated by amorphous regions

Amorphous and glassy state

No LRO, no rotational symmetry, does possess short-range order

of crystallinity that encompasses the entire range from really crystalline to fully disordered amorphous solids. Table 1.1 lists the various classes. Let us take each of them in the order shown.

1.1.1

Monocrystalline Solids

At the top of the list is the single crystal, or monocrystal, which has the highest degree of order. Several crystalline materials of enormous technological or commercial importance are used in monocrystalline form. Figure 1.1a shows a drawing of a highly symmetrical quartz crystal, such as might be grown freely suspended in a fluid. For a crystal, the entire macroscopic body can be regarded as a monolithic three-dimensional space-filling repetition of the fundamental crystallographic unit cell. Typically, the external morphology of a single crystal is faceted (consisting of faces), as in Figure 1.1a, although this need not be the case. The word habit is used to describe the overall external shape of a crystal specimen. Habits, which can be polyhedral or non-polyhedral, may be described as cubic, octahedral, fibrous, acicular, prismatic, dendritic (treelike), platy,

(a)

(b)

(c)

Figure 1.1 (a) A drawing of a quartz monocrystal. The morphology exhibits the true point symmetry of the lattice. (b) A portion of a Penrose tiling with fivefold rotational symmetry based on two rhombuses. A Penrose tiling is a nonperiodic tiling of the plane and is a two-dimensional analog of a quasicrystal. (c) A micrograph of a polycrystalline sample of aluminum plastically deformed under uniaxial tension.

1.1 Degrees of Crystallinity

blocky, or bladelike, among many others. The point symmetry of the crystal’s morphological form cannot exceed the point symmetry of the lattice. In industry, exceedingly pure single crystals are typically grown with specific crystallographic orientations and subsequently sliced or cut in a way that enables the maximum number of units to be obtained of that particular orientation. This process is applied in the manufacturing of semiconductors (e.g. silicon), optical materials (e.g. potassium titanyl phosphate), and piezoelectric materials (e.g. quartz). No other crystalline material manufactured today matches the very low impurity and defect levels of silicon crystals produced for the microelectronics industry. Dislocation-free silicon crystals were produced as far back as the early 1960s. However, the elimination of these dislocations allowed intrinsic point defects to agglomerate into microdefect voids. Although the voids were of a very low density (106 cm−3) and size (150 nm), the drive toward increasingly higher density integrated circuits has made their further reduction the biggest challenge facing single-crystal silicon producers (Falster and Voronkov 2000).

1.1.2 Quasicrystalline Solids It will be seen later in this chapter (Section 1.2.1.1.2) that a crystal may or may not possess nontrivial rotational symmetry, but if it does, the rotational symmetry can only be of specific orders (namely, 2, 3, 4, or 6, as n in rotational angle ϕ = 2π/n). By contrast, quasicrystals possess a noncrystallographic rotational symmetry, and, as a result, they do not possess translational order, or periodicity, in our pedestrian three-dimensional space. Hence, they are termed aperiodic or, preferably, quasiperiodic. They own a quasi-order that is coherent enough to scatter incoming waves, thereby producing sharp spots in a diffraction pattern, which may actually be a general definition of crystals, periodic (the usual crystals) or quasiperiodic (quasicrystals). The most common quasicrystals are ternary intermetallic phases. Like most other intermetallic phases, they are brittle, yet hard, solids. Some are currently being investigated as candidates for surface coatings and as nanoparticle reinforcements in alloys, owing to a variety of interesting properties, including reduced wetting, low thermal and electrical conductivities, low friction coefficients, high hardness, excellent, corrosion resistance, ductile–brittle transition, and high-temperature superplasticity (Jazbec 2009). Nonclassical crystallography rotational symmetry was observed in April 1982 by Daniel Shechtman (b. 1941) when he was at the National Bureau of Standard (NBS) (pre-NIST) at Gaithersburg, Maryland, USA. Dan, or Danny, was a good electron microscopist, and his observation of an electron diffraction pattern exhibiting fivefold symmetry could not be wrong. The problem was that it could be due to multiple twinning in the rapidly solidified Al–Mn alloy he was observing, following a suggestion first made by David Brandon, rather than to a genuine, nonclassical, fivefold symmetry obviously inconsistent with lattice translations. With his numerous microscopic observations at different scales, Danny had some arguments to think it was genuine and not due to multiple twinning. The observation was first reported and cautiously argued in a well written paper in November 1984 by Shechtman, Ian Blech (b. 1936), Denis Gratias (b. 1947), and John Cahn (1928–2016) who headed the NBS Center for Materials Research. Paul Steinhard (b. 1952) and his PhD student Dov Levine (b. 1958) had a paper published the following month, where they baptized as “quasicrystals” a new class of ordered crystals including the Al–Mn observed by Shechtman. Apart for Linus Pauling (1901–1994) who kept on arguing in favor of multiple twinning, sometimes in an excessively ridiculing manner, together with some of his followers in the United States, the existence of quasicrystals easily became rather well accepted. The Nobel Prize in Chemistry was awarded to Shechtman in 2011.

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1 Crystallographic Considerations

The first quasicrystals exhibited fivefold rotational symmetry. A Penrose tiling (a twodimensional tiling of the plane analogous to three-dimensional quasicrystals) with this order of rotational symmetry is illustrated in Figure 1.1b. The initial discovery was followed by the discovery of quasicrystals with 8-, 10-, and 12-fold rotational symmetry. All of these early samples were obtained by rapidly solidifying liquid phases whose equilibrium crystal structures contained icosahedrally packed groups of atoms. Sir Frederick Charles Frank (1911–1998) of the University of Bristol and John S. Kasper (1915–2005) of the General Electric Research Laboratories showed in 1958 that icosahedral coordination (Z = 12), as well as other coordination polytetrahedra with coordination numbers Z = 14, 15, and 16, is a major structural component of some melts. Such tetrahedrally close-packed structures, in which atoms are located at the vertices and centers of various space-filling arrangements of polytetrahedra, are now called Frank–Kasper phases (Frank and Kasper 1958a, b). Quasicrystalline phases form at compositions close to the related crystalline phases. When solidified, the resultant structure has icosahedra threaded by a network of wedge disclinations, having resisted reconstruction into crystalline units with three-dimensional translational periodicity. The most well-known examples of quasicrystals are inorganic phases from the ternary intermetallic systems: Al–Li–Cu, Al–Pd–Mn, Zn–Mg–Ln, Al–Ni–Co, Al–Cu–Co, and Al–Mn–Pd. In 2007, certain blends of polyisoprene, polystyrene, and poly(2-vinylpyridine) were found to form star-shaped copolymers that assemble into the first known organic quasicrystals (Hayashida et al. 2007). As the first reported quasicrystals were metastable phases at room temperature produced by rapid solidification, they were consequently of poor quality. Stable quasicrystals have since been discovered that have revealed very high structural perfection and can form macroscopic crystals. This discovery made it possible to apply conventional solidification techniques. The preferred method appears to be system-specific, as it depends on the temperature stability of the quasicrystalline phase. If the quasicrystal is only stable at elevated temperatures, for example, it can decompose into a crystalline phase if the melt is solidified slowly. If the phase is thermodynamically stable down to room temperature, as is the case for Al–Pd–Mn, quasicrystals can be grown with conventional cooling rates (e.g. 10 C/h). A relationship actually exists between periodic and quasiperiodic patterns such that any quasilattice may be formed from a periodic lattice in some higher dimension (Cahn 2001; John Werner Cahn 1928–2016). The points that are projected to the physical three-dimensional space are usually selected by cutting out a slice from the higher-dimensional lattice. Therefore, this method of constructing a quasiperiodic lattice is known as the cut-and-project method. In fact, the pattern for any three-dimensional quasilattice (e.g. icosahedral symmetry) can be obtained by a suitable projection of points from some six-dimensional periodic space lattice into a three-dimensional subspace. The idea is to project part of the lattice points of the higher-dimensional lattice to three-dimensional space, choosing the projection such that one preserves the rotational symmetry. The set of points so obtained are called a Meyer set after French mathematician Yves Meyer (b. 1939). The mathematics of quasicrystals is too involved to be adequately described in this book.

1.1.3

Polycrystalline Solids

The vast majority of inorganic materials used in commerce fit into the third class of Table 1.1, the polycrystalline form. A polycrystal may be a compacted and sintered (densified) powder, a solidification product, or some other dense aggregate of small crystallites or grains. Figure 1.1c shows the cross section of a polycrystalline aluminum sample, with a grain size of 90 μm, deformed under uniaxial tension. The crystallites of a polycrystal are made from the same unit cell as a monocrystal

1.1 Degrees of Crystallinity

is, but the grains of a polycrystalline body are separated from one another by grain boundaries. A polycrystal can be considered to consist of small crystalline regions separated by regions of disorder, although it must be stressed that the grain boundaries are not totally incoherent or amorphous. Three very important points should be remembered: 1) Depending on their size, the individual grains may or may not be visible to the unaided eye. Grains can range in size from nanometers to centimeters. 2) The grain boundaries are solid–solid interfacial regions a few nanometers thick. 3) Rarely do all the grains comprising a polycrystal have the same size, orientation, or even shape. In fact, polycrystalline grains are morphologically dissimilar to their monocrystalline counterparts.

1.1.4 Semicrystalline Solids Another category of inorganic solid is the inorganic polymer. All polymers – organic, inorganic, and organometallic – are special types of covalently bonded substances in which the entire solid may be considered a macromolecule composed of identical molecular units, called the monomer, which are linked together. One-dimensional chains and two-dimensional layers of atoms are often found in the structures of inorganic crystals. Therefore, any solid in which there exists extended covalent bonding in one or more directions could be classified as polymeric. For example, α- and β-Si3N4, as well as B2O3, contain layers of Si/N and B/O atoms, respectively, and could thus be considered twodimensional polymers, while ReO3, with its vertex-sharing network of ReO6 octahedra, might be thought of as a three-dimensional polymer. The majority of chemists reserve the term polymer for solids that retain their macromolecular structure and properties after a physical change (i.e. melting or solution behavior). Such inorganic materials, which include the polysilanes, polygermanes, and polystannanes, consist of a catenation (long chain) backbone made of one type of main group element other than carbon (but usually with organic substituents) or, as in the case of silicones and polyphosphazenes, a pseudo-catenation backbone made of two different non-carbon elements. These are invariably one-dimensional polymers. Of course, these chains pack and fold together to form three-dimensional solids. As with organic polymers, inorganic polymers can be crystalline, amorphous, or glassy. The ease with which macromolecules can pack together into a regular array will depend on the stereochemical sequence of the backbone’s monomeric units. This is worth looking at a little more closely with the more familiar organic polymers. In an organic molecule, a carbon–carbon single covalent bond is cylindrically symmetrical and thus exhibits free rotation about the bond. The different spatial arrangements of atoms attached to each of the carbons of the single bond are called conformations, while two different configurations of the same molecule are called conformational isomers. The most common method of illustrating the different possible arrangements is through Newman projections, named after Melvin S. Newman (1908–1993), an Ohio State University chemistry professor. An example using butane (CH3CH2CH2CH3) is pictured in Figure 1.2. In the figure, the circle represents the third carbon atom in the chain, C3, which is behind the plane of the page, while the second carbon atom in the chain, C2, is above the plane of the page, lying at the intersection of the three lines representing the two C–H bonds and the C–C bond to the first carbon atom in the chain, C1. The C2–C3 bond itself is in the plane of the page. There are three types of conformations brought about by the rotation of the C2–C3 bond anti (where the methyl groups are furthest apart), which is the lowest-energy conformation; eclipsed (where the methyl groups are closest), which is the highest-energy conformation; and gauche,

5

6

1 Crystallographic Considerations

A

A

A A

A

Figure 1.2 The possible conformations around adjacent carbon atoms in chain.

A Anti

(a)

Gauche

Eclipsed

(b) A

A C–C A B Erythro B C–C A Threo

C–C–C–C–C–C

Figure 1.3 (a) Relative configurations at two contiguous carbon atoms in a chain. (b) Relative configurations of consecutive, but not necessarily contiguous, constitutionally equivalent carbon atoms in a chain.

B

B Meso A

B

C–C–C–C–C–C B

A Racemo

which is the intermediate energy. Each C–C bond of a macromolecule can have its own conformation. The relative configurations at two contiguous carbon atoms in the main chain are designated by the prefix erythro or threo, as illustrated in Figure 1.3a. Stereosequences within a polymeric chain terminating in stereoisomeric centers at both ends of the segment, which comprise two, three, four, or five consecutive (but not necessarily contiguous) centers of that type, called diads, triads, tetrads, and pentads, respectively. Relative configurations of consecutive, but again not necessarily contiguous, constitutionally equivalent carbon atoms that have a symmetrically constituted connecting group (if any) are designated meso. Opposite configurations are called racemo. These are illustrated in Figure 1.3b. Polymers with long meso sequences are termed isotactic, while polymers with long racemo sequences are syndiotactic. The majority of C–C bonds in amorphous organic polymers, which are typically branched or contain large pendant groups (not part of the main chain), are generally found with random conformations. As a result, the chains are arranged randomly throughout the material. Some polymers may also form a brittle glass if rapidly cooled, particularly those containing chains that can easily become tangled or viscous during cooling. Glassy polymers include polystyrene, e.g. Styrofoam® (Tg = 100 C), and polyethylene terephthalate, e.g. Dacron® (Tg = 70 C). Unlike amorphous polymers, the C–C bonds in crystalline polymers are predominantly in the all-anti conformation, and the chains are arranged into lamellae (platelike configurations), but these well-packed regions can be separated by amorphous regions where the chains are entangled. Hence, even a crystalline polymer contains an amorphous fraction, as illustrated in the sketch of Figure 1.4. A percent crystallinity can therefore be specified. This situation has given rise to the term semicrystalline. In fact, most crystalline polymers are semicrystalline, the amorphous fraction typically accounting

1.1 Degrees of Crystallinity

Figure 1.4 A sketch depicting the semicrystalline nature of a polymer. The lamellae are crystalline regions while the entangled regions between them are amorphous.

for ~60 wt% of the total polymer (Cheremisinoff 2001). Semicrystalline polymers are also sometimes referred to as polycrystalline polymers, even though they are really single crystalline (they have no grain boundaries). The crystalline regions, or domains, may have differing texture or crystallographic orientation. Monocrystalline polymers are transparent, whereas polycrystalline polymers are translucent, and totally amorphous polymers are opaque. Likewise, crystalline polymers are harder, stiffer, and denser than amorphous polymers. With inorganic polymers, a similar situation is found. Both meso polymers and racemo polymers are capable of crystallizing, but polymers in which meso and racemo placement occur randomly along the backbone are amorphous. Other structural features that preclude crystallizability include any defects that introduce chain irregularity. Even when the conditions conducive to crystallinity are met, the resultant polymers still contain a significant fraction of amorphous material. This is owing to the length of the polymer chain; different segments become incorporated into different crystalline orientations (Mark et al. 2005). A relatively new field called supramolecular chemistry has been developed and continuously studied over the last four decades. Supramolecular assemblies and supramolecular polymers differ from macromolecules, where the monomeric units are covalently linked. In a supramolecular polymer, the monomeric units self-assemble via reversible, highly directional, noncovalent interactions. These types of bonding forces are sometimes called secondary interactions. Hydrogen bonding is the secondary force most utilized in supramolecular chemistry, but metal coordination

7

8

1 Crystallographic Considerations

and aromatic π–π electronic interactions have also been used. From a materials standpoint, supramolecular assemblies are promising because of the reversibility stemming from the secondary interactions. The goal is to build materials whose architectural and dynamical properties can respond reversibly to external stimuli. Solid phases are prepared by self-assembly from solution. In the solid state, supramolecular polymers can be either crystalline or amorphous.

1.1.5

Amorphous Solids

The final category in Table 1.1 is the amorphous solid, which includes, as a subset, the glassy or vitreous state that is further discussed in Section 1.5. These phases are totally noncrystalline. All glasses are monolithic and amorphous, but only amorphous materials prepared by rapidly cooling, or quenching, a molten state through its glass transition temperature (Tg), are glasses. Both silicabased glasses and metallic glasses have a variety of uses in commerce. Non-glassy amorphous solids are normally prepared by severely mechanochemically damaging a crystalline starting material, for example, via ion implantation or ball milling. Although amorphous and glassy substances do not possess the long-range translational order characteristic of crystals, they do usually exhibit shortrange structural order, for example, the first coordination sphere about a cation.

1.2

Basic Crystallography

Geometric crystallography is the scientific field concerned with the different possible ways atoms, or groups of atoms, which we term the structural motif, can fit together to form the periodic patterns observed in crystalline substances. Through crystallography, we may establish the internal arrangement of atoms within a crystal, as well as the possible types of morphological, or external, symmetry that can be observed. Specific crystal structures are presented in Chapter 3. Additionally, methods have been developed for ascertaining surface and interfacial configurations, for example, at grain boundaries. There is a fundamental postulate that pervades modern crystalline physics, and to which we can associate the names of Franz Ernst Neumann (1798–1895), Bernhard Minnigerode (1837–1896), Woldemar Voigt (1850–1918), and Pierre Curie (1859–1906), which asserts that the symmetries of the physical properties of crystals are related to the symmetries of their crystalline form, the symmetry of the physical properties exhibited by a crystal is at least as high as the crystallographic symmetry. It is apparent that crystallographic structure and symmetry are of great importance in studying and designing solids at every length scale, which makes geometric crystallography a fitting start for this book.

1.2.1

Crystal Geometry

A crystal is a physical object – it can be touched or at least physically observed. However, an abstract construction in Euclidean space may be envisioned, known as a direct-space lattice (also referred to as the real space lattice, space lattice, or just lattice for short), which is composed of equidistant lattice points representing the geometric centers of the structural motifs. Any two of these lattice points are connected by a primitive translation vector, R (which can also be noted T or t, but not τ), given by R = n1 a + n2 b

11

in two dimensions or, in three dimensions, by R = n1 a + n2 b + n3 c

12

1.2 Basic Crystallography

where n1, n2, n3 are integers that may be positive, negative, or equal to zero and a, b, c are the basis vectors. It is also possible to write the lattice vector R in terms of the components of the direction index, in which case n1, n2, and n3 are usually replaced by u, v, and w, respectively. Whereas direction indices are written in square brackets without commas, that is, [u v w] (note, however, that a family of directions is written as u v w , see Section 1.2.1.3), we shall convene to note a vector R by separating its components by commas: [n1, n2, n3]. So that, normally, [1 1 1] indicates a direction while [1, 1, 1] is a vector (which is in the [1 1 1] direction and can just as legitimately be expressed using the unit vectors as [1 i, 1 j, 1k]). A general spatial vector r = x a + y b + z c will be noted [x, y, z]. In three-dimensional space, the parallelepiped defined by the lengths of the basis vectors and the angles between them (α = angle between vectors b and c; β = angle between vectors a and c; γ = angle between vectors a and b) contains the smallest volume that can be stacked repeatedly to produce the entire crystal. It is called a primitive unit cell of the lattice. Hence, the lengths of the basis vectors are called unit cell parameters. Because the unit cell joins eight lattice points, each one shared between eight neighboring cells, a primitive unit cell contains exactly one lattice point 8 × 12 = 1 . In the crystal, associated with each lattice point is a copy of an atomic motif, which may be a single atom, a collection of atoms, an entire molecule (in molecular crystals, like fullerite made of C60 fullerenes), or an assembly of molecules. The atomic motif (from Georges Friedel 1895) is also referred to as the atomic basis (from Max Born 1915), the lattice complex (from Paul Niggli 1919), or the asymmetric unit (from Patterson 1928), since it exhibits no symmetry of the crystal of its own. The atomic basis must not be confused with the mathematical basis made of three vectors defining a unit cell as discussed above. The chemical/electronic surroundings of each lattice point are identical with those of each and every other lattice point. The crystal then looks the same when viewed from any of the lattice points. For example, in the rock salt (halite when considered as a mineral) cubic cell shown in Figure 1.5, the lattice points are on the edges of the cube and the centers of the faces (beware that in diamond, by contrast, the asymmetric unit is made of two carbon atoms, which look symmetric but are not because of their differently oriented neighborhoods). A sodium and chloride ion pair constitutes the asymmetric unit in NaCl, the ion pair being systematically repeated, using point symmetry and translational symmetry operations to form the crystal structure of the crystal. The atomic motif is therefore the minimum unit from which the structure can be generated by a combination of point symmetry and translational operations. In NaCl, the 1 : 1 stoichiometry means that the cell will look the same regardless of whether we start with the anions or the cations on the corner. For polyatomic motifs, in general, different but equivalent descriptions of the motif can generate the same crystal.

Figure 1.5 The face-centered cubic unit cell (lattice parameter a) of rock salt (halite) with a two-atom motif. = Chloride = Sodium α α α

9

10

1 Crystallographic Considerations

The motif may be an achiral molecule or a chiral molecule (non-superimposable on any of its mirror images, like a hand), but as stated above the motif need not be a molecule. The motif of trigonal α-quartz (a.k.a. low quartz) is a chiral motif made of three SiO2 chemical units. Chiral crystals occur in left- and right-handed enantiomorphs (see Section 1.2.1.2). Like low quartz, hexagonal β-quartz (a.k.a. high quartz) also occurs in left- and right-handed forms. There are 65 chiral, or enantiomorphic, space groups. Let us insist on vocabulary in order to avoid confusion: a crystal is a lattice plus an atomic motif. It is a mathematical lattice, as defined by Eq. (1.2), plus a physical motif, made of atoms. The lattice has its mathematical basis, a, b, c. The atomic motif is also many times simply called a basis since the introduction by Max Born. Another usual confusion is to call a crystal a lattice. For instance, diamond has the diamond structure that is not a lattice, the corresponding lattice being FCC. Similarly, HCP is a structure, not a lattice. It is a hexagonal lattice with a two-atom motif. Yet, in the phrases “lattice dynamics” and “lattice energy” (studied in Section 3.3.1), the word lattice corresponds to a crystal structure, with atoms and interatomic interactions, not just abstract lattice points. Due to historical developments of the word lattice in different languages (réseau, Gitter) and different scientific communities (see Hardouin Duparc 2014), there is no way to do better than to warn the reader to be especially careful that the meaning of words many times depends on their context. Example 1.1 How many atoms (ions) are shown in the cubic unit cell in Figure 1.5? How many should really be counted per cubic cell? Solution There are 14 sodium cations and 13 chloride anions. Among the sodium ions, eight located at the edges are each shared between eight cells so that they contribute for only 8 ⅛ = 1 sodium ion. The six ion atoms in the face-centered positions are shared between two unit cells, contributing 6 ½ = 3 sodium ions. The 12 chloride ions at the midpoints of each edge are shared between 4 cells and thus contribute 12 ¼= 3 chloride ions, whereas the chloride anion located at the center of the cell is wholly owned by it. The cubic cell thus actually owns four sodium cations and four chloride anions that are grouped into four sodium/chloride ion pairs. These four pairs have an absolutely equivalent environment, which means only one is necessary as primitive. Rock salt is a face-centered cubic lattice with a two ion motif. The two ion motif can be chosen as {(0,0,0), (0.5,0,0)} or {(0,0,0),(0,0.5,0)} or {(0,0,0),(0,0,0.5)}, which are equivalent with respect to the symmetry of the cubic cell. One can also choose the pair {(0,0,0),(0.5,0.5,0.5)}. The halite crystal can also be considered as being composed of two interpenetrating face-centered cubic sublattices, one of sodium ions and one of chloride ions. The two sublattices are displaced relative to one another by a/2 along a cube edge direction where a is the unit cell dimension, i.e. by a vector [0.5, 0, 0] or [0, 0.5, 0] or [0, 0, 0.5]. Still another pictured is proposed in Figure 3.3, with share NaCl6 octahedra. Copper, diamond, and rock salt are all FCC and are usually represented by their cubic unit cells. Yet, their primitive unit cell is not cubic. It is rhombohedral, so as to contain only one motif. A usual choice of primitive vectors is [0, 0.5, 0.5], [0.5, 0, 0.5], and [0.5, 0.5, 0], which join a lattice point to its three nearest neighbors (and see Table 4.1 in Section 4.5 for BCC). This rhombohedral unit cell is harder to visualize and does not clearly exhibit the cubic symmetry, which is actually implied by the fact that its rhombohedral angles are equal to 60 . This means that it is often convenient to choose a unit cell larger than the primitive unit cell.

1.2 Basic Crystallography

Non-primitive unit cells contain extra lattice points, not at the vertices. For example, in three dimensions, non-primitive unit cells may be of three kinds: 1) Face centered, where a lattice point resides at the center of each face of the unit cell. 2) Body centered, where a lattice point resides at the center of the unit cell. 3) Side centered, where an extra lattice point resides at each of two opposing faces of the unit cell. These are denoted as F, I, and A, B, or C, respectively, while primitive cells are denoted as P and rhombohedral as R. The I stands for Innenzentriert in German (internally centered). Several symmetry-related copies of the asymmetric unit may be contained in the non-primitive unit cell (a.k.a. conventional cell), which can also generate the entire crystal structure by means of translation in three dimensions. Although primitive unit cells are smaller than non-primitive unit cells, the nonprimitive unit cell may be preferred if it possesses higher symmetry. In general, the unit cell used is the smallest one with the highest symmetry. In two dimensions, there are only five unique ways of choosing translation vectors for a plane lattice, or net. These are called the 5 two-dimensional Bravais lattices (Figure 1.6a), after the French physicist and mineralogist Auguste Bravais who derived them in 1850 (Bravais 1850). The unit cells for each plane lattice may be described by three parameters: two translation vectors (a, b) and one interaxial angle, usually symbolized as γ. The five lattices are oblique, rectangular, centered

(a)

α

(b)

α

γ

P Triclinic

A

P

γ

Monoclinic

b α ≠ b; γ = 90°

b α = b; γ = 90° αʹ α

γʹ bʹ P

γ

I F Orthorhombic

C

b α ≠ b; γ = 90° αʹ = bʹ; γʹ ≠ 90°

P α

α b α = b; γ = 120°

P

I Tetragonal

R

Hexagonal Rhombohedral

γ b α ≠ b; γ ≠ 90°, 120° P

I Cubic

F

Figure 1.6 (a) The 5 two-dimensional Bravais lattices. Clockwise from upper left: square, rectangular, oblique, hexagonal, and centered rectangular (center). (b) The 14 three-dimensional Bravais lattices.

11

12

1 Crystallographic Considerations

Table 1.2

The volumes of the unit cells for each crystal system.

Cubic

V = a3

Tetragonal

V = a2c

Hexagonal

V = a2c sin(60 )

Trigonal

V = a2c sin(60 )

Orthorhombic

V = abc

Monoclinic

V = abc sin(β)

Triclinic

V = abc{(1 − cos2 α − cos2 β − cos2 γ) + 2(cos α cos β cos γ)}0.5

rectangular, square, and hexagonal. In three dimensions, there are 14 unique ways of connecting lattice points to define a unit cell. They are called the 14 three-dimensional Bravais lattices (Figure 1.6b), which represent the possible types of crystal symmetry called crystal classes (also called symmetry classes) based on their symmetry groups and the ways in which these groups act on the lattice points. The unit cells may be described by the six parameters mentioned earlier: the lengths of the three translation vectors (a, b, c) and the three interaxial angles (α, β, γ), from which seven crystal systems may be differentiated such as cubic, tetragonal, hexagonal, trigonal, orthorhombic, monoclinic, and triclinic. The volumes of each unit cell may be calculated from their unit cell parameters, as shown in Table 1.2.

1.2.1.1 Types of Crystallographic Symmetry

The most succinct and mathematically precise way to discuss crystal geometry or symmetry is via matrix algebra. Each point in a Euclidean space is uniquely described by a column of coordinates. However, the coordinates of a point are meaningful only when referred to some other point, for example, an arbitrarily chosen origin O with coordinates (0, 0, 0) and the choice of three basis vectors. The distance between the origin and, for instance, any other lattice point, R, is given by a primitive translation vector, such as Eq. (1.2). Hence, the numbers in the column of coordinates may be either the coefficients of the vector R, [n1, n2, n3], or the coordinates of the point R, (n1, n2, n3). It is customary to use boldface to represent vectors, as for a, b, c, R, r, V, etc. Once an origin O has been chosen, there is a one-to-one correspondence between a point R and a vector R, with R = OR, or. The point space definition is pursued here. Reference is now taken as a Cartesian reference frame, with three mutually orthogonal unitary axes, with basis vectors usually labeled i, j, k. A point x corresponding to a vector x with respect to the origin O as x1 i + x2 j + x3 k will be noted (x1, x2, x3) in the text or as a column in an equation: x1 x=

x2 x3

13

The right-hand side of Eq. (1.3) is a (3 × 1) column matrix. It is equivalent to a column vector x. The choice of parentheses, (), to enclose a matrix in Eq. (1.3) and the followings complies with the choice made by the International Tables for Crystallography (Hahn 2005 for vol. A).

1.2 Basic Crystallography

Now, any type of motion or symmetry operation that leaves a lattice invariant may be written in matrix notation. For example, if a lattice point is moved from point x in Euclidean space with coordinates (x1, x2, x3) to point x with coordinates (x 1 , x 2 , x 3 ), this can be written as x1 x2

=

x3

W 11 W 21

W 12 W 22

W 13 W 23

x1 x2

W 31

W 32

W 33

x3

+

w1 w2

14

w3

In this expression, the (3 × 3) square matrix W is a transformation matrix describing a rotation, proper or improper (its determinant is equal to +1 or to −1; see below), while the lone (3 × 1) column matrix on the right-hand side is a vector, w, describing a translation part. Following the ITCs, the operation at play in Eq. (1.4) is noted W or (W,w). Seitz’s original notation would be {W|w}, with {W|w} x = W x + w. Equation (1.4) is valid for any lattice type or crystal system. Its form indicates that the elements of the column vector x , representing the coordinates of point x , are given by 3

xi =

W ij x j + wi ,

i = 1, 2, 3

15

k =1

It is also possible to construct a (4 × 4) square augmented matrix for the system, which has the advantage that the motion is described by a single matrix W, rather than the pair (W,w). Successive applications of motions are then described by the product of the augmented matrices. This product is naturally non-commutative (non-Abelian), since already Wa Wb Wb Wa in general. It should be stressed that each type of motion, or symmetry operation, has corresponding values for the matrix elements in W and w, which are summarized in Table 1.3. The elements of W depend on our choice for the coordinate system. Conventionally, the symmetry directions are chosen as coordinate axes, along with the shortest compatible basis vectors, which will be explained shortly for each type of symmetry operation. In this way, W will be in the simplest possible form, six or five of the nine matrix elements being zeros and the remaining elements consisting of the integers +1 and/or −1. When the translation vector, w, is equal to zero (i.e. if the column consists entirely of zeros), the symmetry operation is reduced to W and has at least one fixed point. It is called a point symmetry Table 1.3 Types of symmetry operations based on the values of the translation vector w, the transformation matrix W, and its determinant. I is the identity matrix. w

W

det(W)

Symmetry

0

W

+1

Rotation

0

I

+1

Identity

0

W

−1

Rotoinversion

0

−I

−1

Inversion

0

W2 = I

−1

Reflection

T

I

+1

Translation

τ

W

+1

Screw rotation

τ

W

−1

Glide reflection

13

14

1 Crystallographic Considerations

operation. If its determinant, det(W) (given by W11W22W33 + W12W23W31 + W13W21W32 − W31W22W13 − W32W23W11 − W33W21W12), is equal to +1, it is called a proper rotation. If the rotation angle ϕ is null, it is simply the neutral transformation or identity operation, I. If det(W) is equal to −1, the motion is called an improper rotation, or rotoinversion, which is equivalent to the combination of a rotation about an axis and an inversion through a point on that axis (these operations commute). If the rotation part of the rotoinversion is null, then the rotoinversion is termed an inversion, W itself is equal to −I. If W2 = I, the improper motion is termed a reflection, and if W −I, the improper motion is a rotoinversion. If w 0 and W = I, the motion is a translation, the translation vector being w. If w 0 and det(W) = +1, the motion is termed a screw rotation. It is the combination of a rotation and a translation (the type of motion you do when you use a cork-screw). If w 0 and det(W) = −1, the motion is termed a glide reflection. For screw rotations and glide reflections, the exact value of w depends on the choice of the origin for the application of the corresponding W rotation. In any case, w should be smaller than a unit cell dimension, hence the choice of the symbol τ rather than T or t or R as used in Eq. (1.2). Note that only glide reflections can occur in two-dimensional space and neither glide nor screw motions are possible in one-dimensional space.

1.2.1.1.1 Inversion

An inversion center, also called a center of symmetry, is a point such that inversion through the point produces an identical arrangement. In a lattice, all lattice points are centers of symmetry of the lattice. Inversion moves a point from a position with coordinates (x, y, z) to the position (−x, −y, −z). The corresponding Seitz operator is (−I,0) where −I is the negative unit matrix: −1 0 0 0

−1 0

0 0 −1

16

With a chiral molecule (i.e. a molecule that is non-superimposable on its mirror image), the operation of inversion produces an enantiomer, or molecule with a reversal of sense. Pairs of chiral molecules (e.g. sodium ammonium tartrate), or of nonmolecular structural motifs (e.g. the helical arrangement of SiO4 tetrahedra in quartz), may crystallize as separate enantiomorphs, which are pairs of chiral crystals. These are left-handed and right-handed crystals, consisting exclusively of left- and right-handed units, respectively. This situation is actually quite rare. Morphologically, an enantiomorph exhibits hemihedry or mirror-image hemihedral faces. The adjective hemihedral refers to the fact that only half of the symmetry-related facets are modified (e.g. inclined) simultaneously and in the same manner. Hemihedral faces should not be confused with the term hemihedral crystal, which refers to a form exhibiting only half the number of faces of the holohedral form (those point groups with the highest possible symmetry of the crystal class). By far, the commonest situation is that pairs of chiral partners crystallize in an orthomorphic form or a racemic monocrystal with equally many left- and right-handed molecules in the fundamental body and hence in the macroscopic volume unit. In other words, pairs of enantiomers usually produce a crystal with holohedral morphology even though the molecules themselves possess chiral centers. It should be noted that some achiral molecules in solution (e.g. the aqueous solution of Na+ and [ClO3]−) can crystallize into enantiomorphic crystals (NaClO3, sodium chlorate, a strong herbicide).

1.2 Basic Crystallography

1.2.1.1.2

Rotational Symmetry

A crystal possesses an n-fold rotation axis if it coincides with itself upon rotation about the axes by 360 /n. Unlike inversion, rotation leaves handedness unchanged, that is, det(W) = +1. For any geometric rotation of a Cartesian coordinate or vector about a fixed origin in threedimensional Euclidean space, a matrix describing the rotation can be written. The rotation matrix is an n × n square matrix for which the transpose is its inverse and for which the determinant is +1. In general, a rotation need not be along a coordinate axis. If the rotation axis is given by the unit vector u = [ux, uy, uz], then a rotation by an angle ϕ about that fixed axis is given by the following expression Ru(ϕ) (Rodrigues 1840) that we provide here without demonstration (note that we follow the standard convention that a clockwise rotation by a vector in a fixed coordinate system makes a negative angle and a counterclockwise rotation, a positive angle): ux uy 1 − cos ϕ − uz sin ϕ cos ϕ + u2x 1 − cos ϕ ux uy 1 − cos ϕ + uz sin ϕ cos ϕ + u2y 1 − cos ϕ

ux uz 1 − cos ϕ + uy sin ϕ uy uz 1 − cos ϕ − ux sin ϕ

ux uz 1 − cos ϕ − uy sin ϕ

cos ϕ + u2z 1 − cos ϕ

uy uz 1 − cos ϕ + ux sin ϕ

17 The columns of the matrix Eq. (1.7) form a right-handed orthonormal set: if one choses to call them vectors, then v1, v2, v3, one has vi vj = δij (tedious to check but true), and “right-handed” because they correspond to the final rotated values of our standard “right-handed” basis vectors [1, 0, 0], [0, 1, 0], and [0, 0, 1], in that order. This rotation matrix is also called a special orthogonal matrix. The components ux, uy, and uz in Eq. (1.7) correspond to a unit vector u. They directly correspond to the directions cosines of u (the cosines of the angles formed by u with the respective Cartesian basis vectors i, j, and k). If the rotation axis were indicated by a larger vector, l, one simply has u=l

l =l

l2x + l2y + l2z

18

For example, for a rotation about the body diagonal [1, 1, 1] of a cube, u = [1, 1, 1]/ 3, and the values to be used in Eq. (1.7) are ux = uy = uz = 1/ 3. Let us give a very simple example of a rotation Ru(ϕ). We recall that the direction indices of the Cartesian coordinate axes are the x-axis = [1 0 0], the y-axis = [0 1 0], and the z-axis = [0 0 1]. These indices are identical with the i, j, and k unit vectors that are codirectional with the x-, y-, and z-axes, respectively. For example, if the rotation axis is the z-axis, described by the direction indices [u v w] = [0 0 1], these indices are numerically equivalent to the direction cosines: cos α = ux = 0, cos β = uy = 0, and cos γ = uz = 1. Therefore, with a counterclockwise rotation of angle ϕ about the vector u = [0, 0, 1], the z-axis, Eq. (1.7) reduces to

Rz ϕ =

cos ϕ sin ϕ

− sin ϕ 0 cos ϕ 0

0

0

19

1

Similar expressions are found to represent counterclockwise rotations about the x-axis and y-axis. These are left as an exercise for the reader. In a crystal lattice, with bases vectors a, b, and c, the rotation described above by Eq. (1.9) corresponds to a counterclockwise rotation around the c coordinate axis. As indicated in Eq. (1.9), the c coordinate will be unchanged by

15

16

1 Crystallographic Considerations

any rotation about c. Consider a fourfold counterclockwise rotation around the c-axis. Here, ϕ = −90 and Rc(ϕ) takes the form: Rc −90

=

0 −1

1 0

0 0

0

0

1

1 10

The lattice structure of a crystal, however, restricts the possible values for ϕ. In a symmetry operation, the lattice is mapped onto itself. Hence, each matrix element – and thus the trace of W (W11 + W22 + W33) – must be an integer. From matrix theory, we know that the trace of a matrix, as well as its determinant, is an invariant. Hence, the trace of a rotation matrix is, from Eq. (1.9), equal to (1 + 2 cos ϕ). It is then obvious that for a lattice symmetry rotation, 2 cos ϕ must be an integer. Thus, only onefold (360 ), twofold (180 ), threefold (120 ), fourfold (90 ), and sixfold (60 ) rotational symmetry are allowed. The corresponding axes are termed, respectively, monad, diad, triad, tetrad, and hexad. The onefold, 360 or 2π, case is actually trivial and corresponds to ϕ = 0 modulo 2π. Only triclinic crystals are in this case. A famous example is the gemstone turquoise. The limitation on the types of n-fold rotational axes can be easily visualized by considering the analogous task of completely tiling a two-dimensional plane with polygon tiles, a process called tessellation. Congruent regular polygons (equilateral and equiangular polygons) such as squares, equilateral triangles, and hexagons can be used, as can irregular polygons like simple quadrilaterals (e.g. isosceles trapezoids, rhombuses, rectangles, parallelograms) and even herringbones. In fact, a trip to the garden/tile shop will reveal that it is possible to produce a tessellation using plane figures with curved boundaries, as well. However, a tessellation of the plane cannot be produced with pentagons, heptagons, or higher regular polygons. In three dimensions, space-filling polyhedra include the cube, the rhombic dodecahedron, and the truncated octahedron. However, combinations of tetrahedra and octahedra, as well as of octahedra, truncated octahedra, and cubes, also fill space. Example 1.2 Derive the rotation matrix for a clockwise rotation about the z-axis, given by the vector u = [0, 0, 1]. Derive the matrix for the counterclockwise rotoinversion about the z-axis. Solution Our convention is that a clockwise rotation is given by a negative angle. From trigonometry, we know that cos(−ϕ) = cos(ϕ) and that sin(−ϕ) = −sin(ϕ). Hence, our matrix must be Rz −ϕ =

cos ϕ

sin ϕ

0

− sin ϕ 0

cos ϕ 0

0 1

In the conventional counterclockwise sense for ϕ but for a rotoinversion, which is a rotation followed by an inversion, the product −I Rz(ϕ) gives (see Eqs. 1.6 and 1.9) Wz ϕ =

− cos ϕ

− sin ϕ

sin ϕ 0

− cos ϕ 0 0 −1

0

It is easy to check that its determinant is equal to −1 (see Table 1.3). Note that since −I is purely diagonal, the product −I Rz(ϕ) is commutative: −I Rz(ϕ) = Rz(ϕ) (−I). The reader should check that, in general, Rz(ϕa) Rx(ϕb) Rx(ϕb) Rz(ϕa) (see Pratice Problem 9).

1.2 Basic Crystallography

1.2.1.1.3

Reflection

Reflection is also called mirror symmetry since the operation is that of a mirror plane in three dimensions, or an axis in two-dimensions, which reflects an object into another indistinguishable one. Consider a reflection in a plane parallel to b and c. The reflection essentially changes the algebraic sign of the coordinate measured perpendicular to the plane while leaving the two coordinates whose axes define the plane unchanged. Hence, W for a mirror reflection in the bc (yz) plane takes the form −1 0 0 1 0 0

0 0 1

1 11

Like inversion through a center of symmetry, the operation of reflection produces a right-handed object from a left-handed one. 1.2.1.1.4

Fixed-Point-Free Motions

These include translations, screw rotations, and glide reflections. Because the primitive translation vector (Eq. 1.2) joins any two lattice points, an equivalent statement is that Eq. (1.2) represents the operation of translational symmetry bringing one lattice point into coincidence with another. However, we must choose the basis vectors (a, b, c) so as to include all lattice points, thus defining a primitive unit cell of the lattice containing only one lattice point. The basis vectors are then lattice translation vectors. The operation of translation requires that W in Eq. (1.4) be equal to the identity matrix and that the translation vector, w, be nonzero. If w 0 and det(W) = +1, the motion is termed a screw rotation. If w 0 and det(W) = −1, the motion is termed a glide reflection, which moves every lattice point and changes the handedness. 1.2.1.2

Space Group Symmetry

If one replaces each face of a crystal by its face normal (a vector), the point symmetry group of a macroscopic crystal is seen to be determined by the group of linear mappings of this vector space. Although morphological symmetry is determined by this group of linear mappings in vector space, the possible types of morphological symmetry are one and the same with a specific group of motions in direct space, which is the physical three-dimensional space in which we live. These types of point symmetries are known as the crystallographic point groups. They are the point groups that map a space lattice onto itself or the sets of point symmetry operations (i.e. rotations and reflections, but not translations) that may be performed on a crystal, which leave at least one point fixed in space while moving each atom in the crystal to a position of an atom of the same kind. It has been seen previously that the lattice structure of a crystal restricts the types of rotational symmetry. Only onefold (360 ), twofold (180 ), threefold (120 ), fourfold (90 ), and sixfold (60 ) rotational symmetry are allowed. If reflections are included as well as the different possible types of rotations, it is found that there are, in totality, two crystallographic point groups for one dimension, 10 for two dimensions, and 32 for three dimensions. These latter are listed in Table 1.4. By contrast, there are an infinite number of noncrystallographic point groups for dimensions greater than or equal to 2. When the 32 crystallographic point groups are arranged in the patterns allowed by the 14 threedimensional Bravais lattices, it is found that there exist only 230 three-dimensional space groups. These are listed in Appendix A, subdivided into the 32 crystallographic point groups. The notation system developed by Schoenflies for designating point group symmetry is still widely used for space

17

Table 1.4

The 32 crystallographic point groups and their symmetry elements.

Point group

Symmetry operations and/or elementsa

Triclinic 1

E

1

E, i

Monoclinic 2

E, C2

m

E, σ h

2/m

E, C2, i, σ h

Orthorhombic 222

E, C2, C 2 , C 2

mmm

E, C2, C 2 , C 2 , i, σ h, σ v, σ v

mm2

E, C2, σ v, σ v

Tetragonal 4

E, 2C4, C2

4

E, 2S4, C2

4/m

E, 2C4, C2, i, 2S4, σ h

4mm

E, 2C4, C2, 2σ v, 2σ d

422

E, 2C4, C2, 2C2 , 2C2

4/mmm

E, 2C4, C2, 2C2 , 2C2 , i, 2S4, σ h, 2σ v, 2σ d

42m

E, C2, 2C2 , 2σ d, 2S4

Trigonal 3

E, 2C3

3

E, 2C3, i, 2S6

32

E, 2C3, 3C 2

3m

E, 2C3, 3σ v

3m

E, 2C3, 3C 2 , i, 2S6, 3σ v

Hexagonal 6

E, 2C6, 2C3, C2

6

E, 2C3, σ h, 2S3

6/m

E, 2C6, 2C3, C2, i, 2S3, 2S6, σ h

622

E, 2C6, 2C3, C2, 3C 2 , 3C2

6mm

E, 2C6, 2C3, C2, 3σ v, 3σ d

62m

E, 2C6, 3C 2 , σ h, 2S3, 3σ v

6/mmm

E, 2C6, 2C3, C2, 3C 2 , 3C2 , i, 2S3, 2S6, σ h, 3σ v, 3σ d

Cubic

a

23

E, 8C3, 3C2

m3

E, 8C3, 3C2, i, 8S6, 3σ h

432

E, 8C3, 3C2, 6C2, 6C4

43m

E, 8C3, 3C2, 6σ d, 6S4

m3m

E, 8C3, 3C2, 6C2, 6C4, i, 8S6, 3σ h, 6σ d, 6S4

E = identity operation, Cn = n-fold proper rotation, Sn = n-fold improper rotation, σ h = horizontal plane reflection, σ v = vertical plane reflection, σ d = dihedral plane reflection, i = inversion.

1.2 Basic Crystallography

group symmetry by spectroscopists. However, crystallographers use the Hermann–Mauguin, or “International,” notation. This system was developed by Carl Hermann (1898–1961) and Charles Mauguin (1878–1958) (Hermann 1931; Mauguin 1931). Each space group is isogonal with one of the 32 crystallographic point groups. However, space group symbols reveal the presence of two additional symmetry elements, formed by the combination of point group symmetry (proper rotations, improper rotations, and reflection), with the translational symmetries of the Bravais lattices. The two types of combinational symmetry are the glide plane and screw axis. No glide planes or screw axes are necessary to describe 73 of the space groups. These are called symmorphic space groups. The others are called non-symmorphic space groups. The first character of an international space group symbol is a capital letter designating the Bravais lattice centering type (primitive = P; all-face-centered = F; body-centered = I; sidecentered = C, A; rhombohedral = R). This is followed by a modified point group symbol giving the symmetry elements (axes and planes) that occur for each of the lattice symmetry directions for the space group. The following symbols are used: m (reflection plane); a, b, c (axial glide planes); d (diamond glide plane); e (double glide plane for centered cells); g (glide line in two dimensions), n (diagonal glide line); 1 (inversion center); 2, 3, 4, 5 (rotoinversion axes); 2, 3, 4, 6 (n-fold rotation axes); and 2p, 3p, 4p, 6p (n-fold screw axes, np). Both short and full symbols are used for the space groups. In the latter, both symmetry axes and symmetry planes for each symmetry direction are explicitly designated, whereas in the former symmetry axes are suppressed. For example, the short symbol I4/mmm designates a body-centered tetragonal space lattice with three perpendicular mirror planes. One of these mirror planes is also perpendicular to the rotation axis, which is denoted by the slash between the 4 and the first m, while the other two mirror planes are parallel with, or contain, the rotation axis. The full symbol for this space group is I4/m 2/m 2/m 42/n 21/n 21/c, which reveals the additional presence of screw axes and a diagonal glide line. By convention, the coordinates of points giving the locations of atoms or molecules inside the unit cell are given by (x, y, z) relative to the origin (0, 0, 0), where x is a fraction of the a unit cell parameter, y is a fraction of b unit cell parameter, and z is a fraction of the c unit cell parameter (see Example 1.3). The selection of the origin depends on the symmetry of the space lattice. For centrosymmetric space groups, the inversion center is chosen as the origin. For noncentrosymmetric space groups, the point with the highest site symmetry and lowest multiplicity is chosen as the origin. If no site symmetry higher than 1 (no point symmetry) is present, a screw axis or glide plane is chosen as the origin. The site symmetry of the origin is often, but not always, identical with the short space group symbol. In contrast to a special position, a general position is left invariant only by the identity operation. Each space group has only one general position, but the position may have multiple equivalent coordinates. If an atom resides at a general position, it resides at all the equivalent coordinates. For example, for a phase crystallizing in the space group Im3m, an atom located in the general position (x, y, z) will, by symmetry, also be found at 96 other coordinates. In Im3m, the general position has a multiplicity of 96. In any space group, the general position always has the highest multiplicity of all the positions in the group. For primitive cells, the multiplicity of the general position is equal to the order of the point group of the space group; for centered cells, the multiplicity is equal to the product of the order of the point group and the number of lattice points per cell. Example 1.3 List the point coordinates, relative to the origin (0, 0, 0), for all the atoms of the cubic unit cell of a pure metal with the body-centered cubic (BCC) structure (space group Im3m).

19

20

1 Crystallographic Considerations

Solution This is the case of, for instance, alpha iron. By convention, we take the origin to be at the center of the (centrosymmetric) cube. The point coordinates for each of the nine atoms are then the origin (0, 0, 0) and the eight cube corners (±0.5, ±0.5, ±0.5) that each count for one eighth of the cube, being shared between eight equivalent cubes. The cubic unit cell really contains only two atoms. However, all the lattice points in any type of lattice are equivalent since they are related by simple translation. Therefore, any one of these nine equivalent atomic sites in the BCC structure could be chosen as the origin. They are all said to be the same “position” and that is (0, 0, 0). The primitive unit cell contains only one site and one atom in the pure metal case with BCC structure (such as iron and the VB and VIB periodic elements, for instance, at room pressure and temperature). Compounds, with forced fully polyatomic motifs, provide a wealth of structures, some of which will be presented in Section 3.4. We already introduced rock salt, NaCl, in Figure 1.5 and Example 1.1. Perovskite, with its CaTiO3, is analyzed in Example 1.4. The crystallographer will wonder at the fact not only that different chemical ABO3 compounds crystallize in the perovskite structure (see Section 3.4.1.6) but also that some others, which chemically look similar, crystallize in different structures, less symmetric. CaCO3, for instance, crystallizes as calcite in normal temperature and pressure conditions. Its crystal system is trigonal, its crystal class is hexagonal scalenohedral, 3m, and its space group R3c (#167), with a double chemical formula per primitive (rhombohedral) unit cell, is Ca2C2O6. Its rhombohedral cell angle is 46 6 , far from the 60 that would give it a cubic symmetry. Magnesite MgCO3, rhodochrosite MnCO3, sidérite (chalybite) FeCO3, and smithsonite (calamine) ZnCO3 also crystallize like calcite, with rather similar rhombohedral cell angles. This is a case of isomorphism, a notion due to the German chemist Eilhard Mitscherlich (1794–1863). But dolomite, Ca0.5Mg0.5CO3, is rhombohedral 3, with the R3 space group (#148). Even more surprising is CaSiO3, where C is just replaced by its chemical analog Si in the CaCO3: it crystallizes as wollastonite, which is only triclinic, in the pinacoidal, 1, crystal class with the P1 space group and six chemical formulae per primitive unit cell. As far as sodium chlorate NaClO3, already alluded to at the end of Section 1.2.1.1.1, is concerned, it crystallizes in the P212121 space group (#19), a chiral compatible non-enantiomorphous space group. René-Just Haüy (1743–1822) disagreed with Mitscherlich about the possibility of isomorphism. Haüy was wrong. Yet, Mitscherlich was certainly far from realizing all the possibilities just described. Mitscherlich was also puzzled by the properties of sodium ammonium salt of tartaric and paratartaric acids. A solution of tartrate rotates polarized light, while a solution of paratartrate has no effect. Mitscherlich, however, could not see any differences in the external form of their crystal salts, and he transmitted the puzzle to the French physicist Jean-Baptiste Biot (1774–1862), a specialist in the study of optical activity, as a note to be published by the French Academy of Science. The young Louis Pasteur (1822–1895), then a student in chemistry and optical crystallography, solved the enigma in the first months of 1848. Convinced that the paratartrate sodium ammonium salt had to be a mixture of two dissymmetric (we now say chiral) crystal forms with right- or leftleaning hemihedral facets, respectively, he proved able, thanks to his good eyes (and good temperature conditions for crystallization of these salts), to see the two forms and manually separate them, finding that in solution the two salts had opposite rotations, as did the liberated acids. Pasteur thus identified, for the first time, the two enantiomers of a chiral substance and recognized the existence of molecular chirality. Pasteur later beautifully illustrated the difference between the chirality of

1.2 Basic Crystallography

quartz and the chirality of some natural organic products in the second of his 1860 lectures before the Chemical Society of Paris: Permit me, in a crude but fundamentally correct way, to illustrate to you the structure of quartz and the natural organic products. Consider a spiral staircase the steps of which are cubes or other objects superimposable with their reflected image. If you destroy the staircase the asymmetry disappears. The asymmetry of the staircase was due to the method of placing together the single steps. So it is with quartz. The quartz crystal is the completed staircase. It is hemihedral, and in consequence it acts upon polarized light. If, however, the crystal is dissolved, fused, or in any other way has its physical structure destroyed, its asymmetry is no longer present, at the same time all action upon polarized light disappears; as is the case, for example, with an alum solution. Consider again the same spiral staircase, the steps of which are formed by irregular tetrahedrons. You may destroy the staircase, but the asymmetry remains, because you have to deal with a collection of irregular tetrahedra. It is important to distinguish between the chirality of a molecular component to a crystal (its point group), the chirality of the crystal structure itself, and the chirality of the space group to which the crystal structure belongs (a good discussion may be found in Flack 2003). Among the 230 crystallographic space groups, 65 groups are compatible with chiral crystal structures (meaning chiral structures crystallize in these 65 space groups). The space group a chiral crystal structure belongs to is not necessarily chiral. There are only 22 space groups that are chiral helical in and of themselves. In other words, there exist 11 right/left enantiomorphic space group pairs, such as P3121 (#152, right) and P3221 (#154, left) for α-quartz and P6222 (#180, right) and P6422 (#181, left) for β-quartz. These 22 groups contain at least one screw axis that is not the 21-screw axis. All of the remaining 208 (230 − 22) crystallographic space groups are achiral. Of these 208 achiral space groups, there are 165 groups containing at least one improper rotation (rotoreflection or rotoinversion, conventionally termed symmetry operations of the second kind), and these are the space groups in which achiral structures crystallize. The remaining 43 (208 − 165) achiral space groups contain only symmetry operations of proper rotations and the 21-screw rotation, conventionally termed symmetry operations of the first kind. These can transform into themselves through an inversion operation and are therefore proper achiral, but they are chiral compatible; their lattices can be associated with chiral motifs/molecules of uniform handedness, such as P212121 (#19) with NaClO3. Moreover, the crystals that pack these 43 space groups are always chiral. Hence, these 43 groups plus the 22 chiral space groups taken together constitute the 65 Sohncke groups. Pure tellurium, element 52 in the periodic table, also crystallizes as α-quartz, with a helical primitive three-atom motif, and is a semiconductor whose band structure is currently attracting a lot of attention because of its topological properties (Tsirkin et al. 2017). Example 1.4 The perovskite CaTiO3 crystallizes in the space group Pm3m. The unit cell contains oxygen atoms at the midpoints of every edge of a cube. There is a calcium atom in the center of the cube, and titanium atoms are located at each corner. The reader may want to refer to Figure 3.18. Where is the origin? What are the point coordinates for each type of atom? What are the site multiplicities?

21

22

1 Crystallographic Considerations

Solution There are three oxygen atoms in the unit cell. The O atom must reside on a site with threefold multiplicity. Likewise, there are Ti atoms on each corner, accounting for one Ti atom per unit cell; it is on a site that has a multiplicity of one, as is the Ca atom in the center of the cube. For noncentrosymmetric space groups, the point with the highest site symmetry and lowest multiplicity is chosen as the origin. This means that either the Ti atom or the Ca atom could be chosen as the origin since they both have the same multiplicity (onefold) and site symmetry m3m. However, it is important to note that Ca and Ti atoms are not at the same position. The Bravais lattice is primitive, as indicated by the space group symbol. Therefore, if we allow the Ti atom positions to coincide with lattice points, the Ca atoms and O atoms cannot. Choosing to place the Ti atom on a lattice point as our origin, we can assign it coordinates (0, 0, 0). Relative to this origin, the Ca atom is located at (0.5, 0.5, 0.5). The O atoms are at (0, 0.5, 0.5), (0.5, 0, 0.5), and (0.5, 0.5, 0), which are all equivalent. A “position” is defined as a set of symmetrically equivalent coordinate points. Within the unit cell, atoms or molecules may be located at general positions that do not lie on any symmetry element or at special positions. If they do lie either on a symmetry element (inversion center, rotation axis, or mirror plane) or at the intersection of several symmetry elements, each point is mapped onto itself by a symmetry operation of the space group. Thus, because the origin in the BCC metal of Example 1.3 is at a lattice point, this is obviously a special position and, in this case, one that has a multiplicity of two. The multiplicity of a special position is always a divisor of the multiplicity of the general position of the space group. If the center of a molecule happens to reside at a special position, the molecule must have at least as high a point symmetry as the site symmetry of the special position. With alloys and substitutional solid solutions, it is possible that a mixture of atoms (of similar size, valence, etc.) may reside at a general or special position and all its equivalent coordinates. The fraction of atoms of one type residing at that position is given by the site occupancy or site occupation factor. The sum of the site occupation factors for that site must equal unity. The distribution of two or more types of atoms over a single site is completely random. Where two atoms are distributed over all the equivalent coordinates of different sites with similar local coordination environments (but not identical site symmetry), electronic, or other, effects can result in partial site preferences. That is, there can be a nonstatistical distribution over the two sites. Both general and special positions are also called Wyckoff positions, in honor of the American crystallographer Ralph Walter Graystone Wyckoff (1897–1994). Wyckoff’s 1922 book, The Analytical Expression of the Results of the Theory of Space Groups, contained tables with the general and special positional coordinates permitted by the symmetry elements. This book was the forerunner of International Tables for Crystallography, which first appeared as International Tables for X-ray Crystallography in 1935. The arrangement of a set of symmetrically equivalent points of the general position in a space group is illustrated with space group diagrams, which also serve to show the relative locations and orientations of the space group’s symmetry elements. Space group diagrams are orthogonal projections, that is, the projection direction is perpendicular to the plane of the figure and is almost always a cell axis (exceptions include rhombohedral space groups with rhombohedral axes). The graphical symbols used for symmetry elements are shown in Figures 1.7 and 1.8. Representative examples of space group diagrams, as they are to be found in the International Tables for Crystallography, are illustrated in Figure 1.9 for both a space group of low symmetry and for a space group

1.2 Basic Crystallography

Symmetry planes perpendicular to the plane of projection

Symmetry planes parallel to the plane of projection

Symmetry planes inclined to the plane of projection

Reflection plane Reflection plane Reflection plane Axial glide plane Axial glide plane

Axial glide plane

Axial glide plane

Axial glide plane

Double glide plane Double glide plane Double glide plane Diagonal glide plane Diagonal glide plane Diagonal glide plane Diamond glide plane Diamond glide plane

Diamond glide plane

Diamond glide plane Diamond glide plane

Figure 1.7 Some graphical symbols used for symmetry planes in space group diagrams.

of high symmetry. The triclinic space group P1is shown in the upper left corner of the figure. In this diagram, there are eight distinct inversion centers represented by open circles. The +/− sign next to some of these circles indicates that those particular inversion centers are above/below the ab plane, which is the plane of the page. The comma inside the circle with the −z coordinate, which is located within the lower right quadrant of the cell, signifies that the point with +z coordinate, represented by the circle inside the upper left quadrant of the cell, is also turned upside down upon inversion through 0.5, 0.5, 0. As examination of the space group diagram for Pm3m in the lower right corner of Figure 1.9 will reveal however, picking out the symmetry elements of a high-symmetry space group can be a considerably more bewildering task. Indeed, it is quite remarkable that a systematic mathematical derivation of the 230 space groups was performed about a dozen years before the discovery of X-ray diffraction through crystals. The reader may be wondering how one might go about experimentally determining the space group of some crystal. This process relies on a specific diffraction effect to wavelengths of the order of the internuclear distances (e.g. X-rays), brought about by the periodic electron density in a

23

24

1 Crystallographic Considerations

Center of symmetry (inversion center): Rotation axes: twofold

threefold

fourfold

fourfold

sixfold

sixfold

Inversion axes: threefold

Rotation axes with center of symmetry: twofold

fourfold

sixfold

Screw axes:

Not pictured: symmetry axes parallel to the plane of projection; symmetry axes inclined to the plane of projection; and screw axes with center of symmetry

Figure 1.8

Some graphical symbols used for symmetry axes in space group diagrams.

, O+

, O+

i , O+

, O+ i

i

o

o

i o

o

o

i

o

o

i

i

o

o

i

o

i

o

i

i

o

i

Figure 1.9 Space group diagrams for the low-symmetry triclinic space group P1 (top) and the high-symmetry cubic space group Pm3m (bottom).

crystal. Although X-ray diffraction is similar to ordinary reflection, in reality, X-rays are not reflected but scattered in all directions by the electrons of the atoms in the crystal. The complete crystal structure, that is, ascertainment of the identities and locations of all the constituent atoms, is determined from knowledge of the amplitudes and phases of the scattered waves.

1.2 Basic Crystallography

1 Incident beam 2

1′

λ

λ′ = λ λ

Bragg case n = 1 Gl

gle

an

cin

ng

le

θ

P

θ θ S

B

n da

te

ga

A

Diffracted beam 2′

c fle Re

θ dhkI T

Q C

Figure 1.10

Bragg’s law illustrated. SQ = QT = dhkl sin θ(case n = 1 in Eq. 1.12).

Bragg’s law is seminal in W.L. Bragg (1912) Nature paper. With the schematic Figure 1.10 in view, consider a coherent soft X-ray plane wave of wavelength λ impinging with an angle θ (glancing angle), two exactly similar atomic planes of a crystal distant from each other by a distance d (note that the angle θ will remain the same through many atomic planes in depth because X-rays almost do not refract, contrary to light). These two similar atomic planes can be labeled according to their Miller indices h,k,l (see Section 1.2.1.3) so that d and θ can also be labeled d(hkl) and θ(hkl) when one needs to be explicit. Assuming a simple elastic diffusion process, the two waves specularly reflected by the two planes have a length lag δr equal to 2d sin θ (SQ + QT in Figure 1.10). This length lag gives a phase shift δφ = kδr, where k = 2π/λ; hence δφ = 2πδr/λ that will be constructive for the two waves 1 and 2 if and only if (iff) it is null, modulo 2π; i.e. iff δr a multiple of the wavelength λ: iff there is an integer n such that 2d sin θ = nλ

1 12a

which is Bragg’s famous law and should more precisely read as 2d hkl sin θ hkl ,n = nλ, n = 1, 2, …

1 12b

Numerous people reworked/re-derived Bragg’s law, including Wulff, Laue, Ewald, and Friedel, among others. Laue could have found it first, had he realized that a set of three integers in one of his equations could be interpreted as the Miller indices of a plane (see Authier 2013). For similar (hkl) atomic planes, one gets a series of diffracting angles θ(hkl),n such that θ(hkl),n = arcsin[nλ/2d(hkl)], n = 1, 2, …. Different (hkl) generate different series. The experimentalist works conversely. He/she observes peaks at various θ and must try to deduce a crystallographic structure with atomic planes consistent with all the θ values. This is done by making hypotheses about the crystallographic structure and choosing the best (most consistent) hypothesis in the usual hypothetico-deductive way. Experimental radiocrystallography requires a lot of care. Nowadays, there exist good instruments and software that can do the job routinely, at least for not too complicated structures (there are intensity measurement

25

26

1 Crystallographic Considerations

difficulties: the intensity of a peak decreases with increasing θ, due to the intrinsic atomic scattering factor, to temperature [the Debye–Waller factor], difficulties due to inaccuracy in angular measurements, width δθ due to wavelength spread in the incident wave packets, etc.). Knowing the structure, the series of n = 1, 2, … diffracting angles θ(hkl),n for (hkl) atomic planes can be called diffracting angles of first, second, nth order. One sometimes, rather improperly, call a second-order diffraction due to (hkl) planes a “first-order diffraction due to (2h2k2l) planes.” From a mathematical point of view, d(nh,nk,nl) = d(hkl)/n. (nh,nk,nl) may or may not correspond to existing atomic planes. Even if it does, the atomic planes are not strictly similar for n > 1 in the sense that the atomic translation vector between them is not d(nh,nk,nl). For instance, in the simple cubic structure (the only known realization is for polonium, α-Po – an unstable realization because polonium atoms are radioactive, with a half-life time of 138 days for its most stable isotope), (n00) atomic planes do not exist except for n = 1, yet (n00) reflections exist (until they become too feeble to be detected). Subtleties occur for elementary non primitive cubic structures such as elementary FCC structures (e.g. aluminum, copper, nickel, silver, gold, γ-Fe) and elementary BCC structures (e.g. α-Fe, β-Ti), due to the conventional cubic cell choice for the definition of the crystallographic axes and Miller indices for these cubic structures. An elementary FCC structure corresponds to a cubic cell with a four-atom motif (see Figure 1.6, lowest right figure, F cubic, assuming one atom per lattice point). This complicates the cubic (hkl) diffraction conditions that now additionally (besides Bragg’s condition) demand that h + k, k + l, h + l be all even. This explains why, for aluminum or copper, the (100) reflection does not exist. (200) reflection exists, as well as (400). The supplementary condition for an elementary BCC structure is h + k + l be even. Ferrite (α-Fe) can have (110) and (200) reflections, for instance, but not (100). The cubic diamond (CD) structure is accordingly a cubic cell with an eight (similar)-atom motif that entails that it diffracts if h, k, l are all even and h + k + l = 4p or if h, k, l are all odd. That is why there is no (100), no (200) (2 + 0 + 0 = 2 4p), and no (222) (2 + 2 + 2 = 6 4p) peaks for CD. Yet, the (222) peak (or spot in a Laue diagram) may appear in very sensitive X-ray, or in electron diffraction, for diamond (W.L. Bragg 1912) or silicon or germanium, because of the covalent electrons that make bond charges and add four-week diffraction centers per cubic cell. It cannot appear in neutron diffraction as neutrons only “see” nuclei. The (200) exists for the cubic diamond-related GaAs or ZnS structure (sphalerite, or zincblende or zinc blende, or ZnS β), because ZnS is a chemically asymmetric unit (Zn S). Note that both copper, diamond, and sphalerite pertain to the same FCC Bravais lattice, but with different space groups: Fm3m, Fd3m, and F4d3m, respectively. For KCl (sylvite, same structure as halite, i.e. rock salt; see Figure 1.5) by X-rays or electrons, because K+ and Cl− are isoelectronic, (111) does not exist (is absent) and the first strong reflection is (200), which could easily be mistaken for the (100) reflection of a primitive cubic cell half the actual FCC cell. Note that both copper and rock salt (and sylvite) pertain to the same FCC Bravais lattice and to the same space group Fm3m. In rock salt, the two ions occupy sites that are compatible with Fm3m, at two special positions among the Fm3m Wyckoff positions (vide supra, and also see Section 3.1.4 and Example 3.2). The absent reflections are termed systematic absences or characteristic extinctions. It may be owing to the presence of space symmetry elements (glide planes, screw axes), a non-primitive Bravais lattice type, or to atoms located at special positions. Equivalently, it may be stated that in order for a reflection to be observed, it must obey reflection conditions. Of the 230 space groups, only 27 have no reflection conditions at all. The systematic absences uniquely determine many of

1.2 Basic Crystallography

the remaining space groups. However, for a large number of space groups, the absences are common to other groups. In these cases, the possible space groups are usually tried, beginning with the highest symmetry space group, until the structure is determined. The reader should by now be convinced there are some subtleties in the application of the apparently so simple but so useful Bragg’s law. Less important in daily X-ray practice, but certainly more surprising from a conceptual point of view, is to learn that Bragg’s law works because crystals are usually ideally imperfect. It would not work for absolutely perfect crystals (as if we were to multiply planes A, B, C, with perfect regularity in Figure 1.10). Knowing that “the formulae developed by Laue are competent to show the positions in which the interference maxima occur, but do not give the intensities at the maxima,” Charles Galton Darwin (1887–1962, Darwin’s grandson) worked out a more realistic and complex formula that accounted for the fact that planes do not totally reflect waves but absorb part of the incident energy (Darwin 1914). He then tried to quantitatively compare his formula with respect to his previous measurements with Henry Moseley (1887–1915) with crystals of rock salt and found a disagreement by a factor of more than 10. Darwin found that the discrepancy could be attributed to the fact that in all crystals there is a non-negligible amount of distortion, so a crystal is actually made of a great many small regions in which perfect reflection takes place. For diffraction, a sum of small perfect regions, slightly disoriented with respect to one another, is much better than one large perfect crystal. Paul Peter Ewald (1888–1985) later called this kind of imperfect crystal a mosaic crystal and also imposed the appellation dynamical theory of diffraction to the diffraction theory initiated by Darwin as well as in his own 1912 dissertation. The Darwin–Ewald’s mosaic crystal corresponds to an aggregate of small perfect blocks having orientations that are locally very similar (on a large scale there is no need of a preferred orientation). The blocks must be small enough (typically less than a micron thick) to be considered as “kinematically” perfect with respect to the dynamical diffraction theory. In order for the kinematical approximation to be valid, atomic absorption must be negligible, as well as interaction between incident and diffracted waves negligible, and there should be no multiple diffraction. The diffracted intensity is then proportional to the square of the structure factor (due to the atomic motif ). If these conditions are not valid, the Darwin–Ewald dynamical theory of diffraction (Darwin 1914; Ewald 1917, independently) must be applied, and the diffraction intensity is only proportional to the magnitude of the structure factor. For electrons in high-resolution transmission electron microscopy (HRTEM), effects due to multiple diffractions occur almost immediately, and the use of dynamical theory is in order. Two main techniques exist in HRTEM literature to simulate the dynamical interactions of the incident electrons within an observed sample prepared as thin as possible: the Bloch wave method, derived from Hans Bethe’s original theoretical treatment of the Davisson–Germer experiment, and the multi-slice method. In multi-slice simulation the thickness of each slice over which the iteration is performed is taken as small enough so that within a slice the potential field can be approximated to be constant. 1.2.1.3

Lattice Planes and Directions

Often, it is necessary to refer to a specific family of lattice planes or a direction in a crystal. A family of lattice planes is a set of imaginary parallel planes that intersect the unit cell edges. Each family of planes is identified by its Miller indices. The Miller indices are obtained from the reciprocals of the reduced coordinates of the three points where one of its planes away from the origin intercepts each of the three axes, converting these reciprocals to smallest integers in the same ratio: (iaa, iab, iac) (1/ia, 1/ib, 1/ic,) (ibic iaic iaib) ≡ (h k l). For instance, a plane

27

28

1 Crystallographic Considerations

(111)

(011)

(100) c

a3

b a

Figure 1.11

(1010)

Examples of lattice planes and their Miller indices.

with reduced intersections (2,3,5) will have (15 10 6) as Miller indices (these indices are admittedly unusually large: this example was chosen only to unambiguously illustrate the procedure). The reason of considering reciprocals is that it allows to get rid of ∞ values: a plane parallel to the a-axis will never intersect it or will intersect it at ∞. Taking the reciprocal 1/∞ leads a corresponding simple zero Miller index. Some simple examples are illustrated in Figure 1.11. These indices were first introduced by the British polymath William Whewell (1794–1866), during a crystallography fellowship period in 1825. Whewell’s notation system was subsequently incorporated into an 1839 book by his student William Hallowes Miller (1801–1880) and now bears the latter’s name. When referring to a specific plane in a family, the numbers are grouped together in parentheses, (h k l). Any family of planes always has one member that passes through the origin of the unit cell. The first lattice plane of the (h k l) family away from the origin intercepts the axes at a/h, b/k, c/l, which are not lattice points if h, k or l, are not equal to 1. The lattice points of this first lattice planes are given by (1 + kln1, 1 + hln2, (1 − h − k)/l − hk(n1 + n2)), where n1 and n2 are integers. If one is given a representative plane with general fractional intercepts, the procedure to get its Miller indices is (pa/qa)a, (pb/qb)b, (pc/qc)c qa/ pa, qb/pb, qc/pc qapbpc, qbpapc, qcpapb h, k, l. A complete set of equivalent planes is denoted by enclosing the Miller indices in curly brackets as {h k l}. For example, in cubic systems (1 0 0), 1 0 0 , (0 1 0), 0 1 0 , (0 0 1), and 0 0 1 are equivalent, and the set is denoted in braces as {1 0 0}. Cubic axes are equivalent in cubic systems, and the maximum possible number of (h k l) combinations that are equivalent occurs for cubic symmetry and is equal to 48. It is not the case in other systems. In hexagonal systems, for instance, the (100) and (001) planes are not equivalent, but (100) and 1 1 0 planes are, as well as 2 1 0 and (110). Now, it is clear in Figure 1.11 that the a3-axis is equivalent to axes a and b. It might thus be of interest to consider the index i, the reciprocal of the fractional intercept of the plane on the a3-axis. It is

1.2 Basic Crystallography

Figure 1.12 The vector product [b × c] defines a new vector whose magnitude is given by the area of the parallelogram forming the base of the parallelepiped and whose direction is perpendicular to the plane of b and c. The scalar triple product is thus the area of the parallelogram multiplied by the projection of the slant height of a on the vector [b × c].

b×c

|a*|–1

a c b

derived in exactly the same way as the others. The relation i = −(h + k) always holds. Following Bravais, one can then use four Miller indices for hexagonal systems: (h k i l). The two previously mentioned equivalences now become clear by circular permutation over the first three equivalent indices, provided i = −(h + k): 1 0 1 0 and 1 1 0 0 and 2 1 1 0 and 1 1 2 0 . Sometimes, hexagonal indices are written with the i index as a dot, which is rather unfortunate since it also serves as an error correction code (ECC). In calculating the interplanar spacing, or perpendicular distance between adjacent planes of given indices, dhkl, in the direct lattice (whether or not these planes coincide with lattice points), it is helpful to consider the reciprocal lattice, which defines a crystal in terms of the vectors that are the normals to sets of planes in the direct lattice and whose lengths are the inverse of dhkl (see Section 4.5). The relationship between the interplanar spacing and the magnitude of the reciprocal lattice vectors, a∗, b∗, c∗, is given by d2hkl =

h2 a∗2

2 ∗2

+k b

+

l2 c∗2

+

2klb∗ c∗

1 cos α∗ + 2lhc∗ a∗ cos β∗ + 2hka∗ b∗ cos γ ∗

1 13

The magnitudes of the reciprocal lattice vectors may be obtained from a∗ =

b×c c×a a×b b∗ = c∗ = a b×c b c×a c a×b

1 14

where the numerators are cross products that define new vectors, which are perpendicular to the planes. For example, [b × c] is a vector that is perpendicular to the plane of b and c. Its magnitude is equal to the area of the parallelogram forming the base of the parallelepiped. The denominators in Eq. (1.14) are scalar triple products, whose magnitudes are equal to the volume of the parallelepiped formed by the bases vectors (the unit cell volume). Hence, each of the denominators in Eq. (1.14) is equal in magnitude. This is illustrated in Figure 1.12. The scalar triple product is equal to the area of the base (shaded gray) multiplied by the projection of the slant height of a on the vector [b × c] (i.e. the perpendicular distance between the base and its opposite face), which is the volume of the parallelepiped. The cosine terms in Eq. (1.13) are given by cos α∗ =

cos β cos γ − cos α cos α cos γ − cos β cos α cos β − cos γ , cos β∗ = , cos γ ∗ = sin β sin γ sin α sin γ sin α sin β 1 15

Example 1.5 Derive the expression for d2hkl , in terms of the direct lattice, for each of the crystal systems with orthogonal axes.

29

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1 Crystallographic Considerations

Solution For the cubic system, we have α∗ = β∗ = γ ∗ = 90 ; a∗ = b∗ = c∗. Directly from Eqs. (1.13) and (1.15), we obtain d2hkl = 1 h2 + k2 + l2 a∗2 . Because α = 90 , the cross product [b × c], which is given by bc sin α, is equal to 1. The scalar triple product is given by a [b × c] = a[bc sin α] cos θ = a[bc sin α] cos(0 ), where 0 is the angle between the cross product vector and a. Hence, from Eq. (1.14), it is seen that a∗ = 1/a. Finally, d2hkl = a2 h2 + k2 + l2 . For the tetragonal system, we have α∗ = β∗ = γ ∗ = 90 ; a∗ = b∗ c∗. Directly from Eqs. (1.13) and (1.15), we obtain d2hkl = 1 h2 + k2 a∗2 + l2 c∗2 . For the tetragonal h2 + k2 a2 + l2 c∗2 . system, a∗ = b∗ = 1/a and c∗ = 1/c. Hence, d2hkl = 1 For the orthorhombic system, we have α∗ = β∗ = γ∗ = 90 ; a∗ b∗ c∗. Directly from Eqs. (1.13) and (1.15), we obtain d2hkl = 1 h2 a∗2 + k2 b∗2 + l2 c∗2 . In the orthorhombic system, a∗ = 1/a, b∗ = 1/b, and c∗ = 1/c. Hence, d2hkl = 1

h2 a2 + k 2 b2 + l2 c2 .

In order to specify a crystal direction, a vector is drawn from the origin to some point P. This vector will have projections u on the a-axis, v on the b-axis, and w on the c-axis. The three numbers are divided by the highest common denominator to give the set of smallest integers, u, v, and w. The direction is then denoted in brackets as [u v w]. This process is illustrated in Example 1.6. For cubic systems, the [h k l] direction is always orthogonal to the (h k l) plane of the same indices. With the other crystal systems, this simple relationship does not hold. For example, in the hexagonal lattice, the normal to the (1 0 0) plane is in the [2 1 0] direction, the [1 0 0] direction being 120 to the (1 0 0) plane (see Practice Problem 8). It is one of the (almost magical) virtues of the reciprocal lattice to restore some Cartesian-like relationships in all cases: [u v w] ⊥ Shkl hu + kv + lw = 0. This is the hu + kv + lw = 0, after the German so-called Weiss zone law, or Weiss rule: [u v w] (h k l) mineralogist-crystallographer Christian Samuel Weiss (1780–1856). Just as a set of equivalent planes is denoted by enclosing the Miller indices in curly brackets as {h k l}, sets of related directions are enclosed in angled brackets u v w , such as 1 0 0 signifying the [1 0 0], [0 1 0], [0 0 1], [1 0 0], [0 1 0], and [0 0 1] directions in the cubic lattice. The term “related directions” means those directions for which their vectors are related by a symmetry operation, such as an n-fold rotation or reflection. It is perhaps easier to envision this in two-dimensional space, so, for example, if one draws a square grid (x and y dimensions are the same) with the equal magnitude vectors [1 i,0 j] and [0 i,1 j] representing, respectively, the [1 0] and [0 1] directions, it is quickly seen that the [0 i,1 j] is transformed into the [1 i,0 j] by a fourfold (90 ) rotation about the perpendicular axis passing through the origin. Therefore, the [1 0] and [0 1] directions belong in the same family, 1 0 , which also includes [1 0] and [0 1]. However, in a rectangular 2D lattice (x and y dimensions are not the same), the [1 0] and [0 1] do NOT belong to the same family since there is no n-fold rotation or any other symmetry operation that can take the [0 i,1 j] vector into the [1 i,0 j] vector, as the vectors are of different magnitude. As a mental exercise, the reader is next encouraged to sketch the [0 i,1 j] and [0 i,1 j] vectors, as well as the [1 i, 1 j] and [1i, 1 j] vectors on both a square and rectangular 2D grid in order to convince themselves that these pairs do belong, respectively, to the 0 1 and 1 1 families in both the square and rectangular 2D lattices, as each vector in both pairs are interchanged by a reflection. In the case of the rectangular lattice, 0 1 includes only [0 1] and [0 1], while the [1 0] and [1 0] directions constitute the 1 0 family. For the family 1 1 , the members are [1 1], [1 1], [1 1], and [1 1] in both the square and rectangular lattice. Note that it is also important to consider whether one is

1.2 Basic Crystallography

Figure 1.13 Starting from the origin, we move along the x-axis (parallel to a) 0.5 units of a. Next, we move from this point along a line parallel to the y-axis (and parallel to b) until we reach the point where the vector crosses the a edge of the cell. This is seen to be at y = 1b. There is no z component to the vector, since the z projection is zero. Reduction of 0.5, 1, and 0 to the lowest set of integers is accompanied by multiplication by a factor of 2. The direction is [1 2 0].

z

c

O

y a

2b P b x The vector OP points along the [120] direction in the orthorhombic unit cell defined by abc.

examining the abstract lattice or the physical crystal. Crystals can have a point group symmetry equal to or lower than that of the lattice. Two members belonging to the same family of directions in a lattice do not necessarily belong to the same family in the crystal. Example 1.6 A direction vector, passing through the origin of an orthorhombic unit cell, crosses the a edge (parallel with x-axis) of the cell at a/2 and the b edge (parallel with the y-axis) at 2b and is orthogonal to the c edge (parallel with the z-axis). What are the direction indices? Solution Sketch the three-dimensional unit cell with the vector drawn in Figure 1.13. Let us note r ∗hkl the reciprocal lattice vector normal to the planes of Miller indices (h k l). For example, a∗ is normal to the direct lattice plane containing b and c, and its length is equal to 1/d100 (see Eqs. 1.13 and 1.14 as well as Section 4.5). This orthogonality between a direct lattice plane and its reciprocal lattice vector is very useful computationally. For instance, the dot product affords us a means of determining the angle between two sets of planes with Miller indices (h1 k1 l1) and (h2 k2 l2) in the direct lattice since that angle is equal to the angle between the two corresponding reciprocal lattice vectors. The angle between any two directions, specified by their Cartesian direction indices [u1 v1 w1] and [u2 v2 w2] in a Cartesian frame, is given by the dot product θ = cos −1

u1 u2 + v1 v2 + w1 w2 u21

+ v21 + w21 u22 + v22 + w22

1 16

The general expression for the cosine of an angle between crystal planes (h1 k1 l1) and (h2 k2 l2) in any lattice is (Kelly and Groves 1970) cos θ = dh1 k1 l1 dh2 k2 l2

h1 h2 a∗2 + k1 k 2 b∗2 + l1 l2 c∗2 + k 1 l2 + l1 + k 2 b∗ c∗ cos α∗ + h1 l2 + l1 h2 a∗ c∗ cos β∗ + h1 k 2 + k1 h2 a∗ b∗ cos γ ∗ 1 17

Example 1.7 Derive the expression for the angle between a plane normal and any direction in an orthorhombic lattice.

31

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1 Crystallographic Considerations

Solution Denote an (hkl) plane normal as the vector r ∗hkl and an arbitrary direction [u v w] as vector u. The angle between r and u is given by the normalized dot product cos θ =

r ∗hkl r ∗hkl

u u

For an orthorhombic lattice, the plane normal vector ha∗ + kb∗ + la∗ is simply r ∗hkl =

h k l i+ j+ k a b c

The arbitrary direction vector u is u = ua + vb + wc = uai + vbj + wck Making these substitutions into the expression for the dot product, we get cos θ =

hu + kv + lw 2

2

h a2 + k b

2

+ l2 c2

u2 a2 + v2 b2 + w2 c2

The orthorhombic is actually a simple case because its axes are orthogonal. Nonorthogonal lattice would imply more complex formulae. A general formula exists, with a compact and practical form at least apparently independent of the lattice. It simply implies the use of what is known as the metric tensor. Tensors are introduced in Section 6.1 and in Chapter 10, with the metric tensor in Example 10.7 (Eqs. (10.59)–(10.62)) and Example 10.8. A whole appendix is devoted to tensors, Appendix C, with Section C.3 specifically about the metric tensor.

1.3 Single-Crystal Morphology and Its Relationship to Lattice Symmetry A crystal is similar to three-dimensional wallpaper, in that it is an endless repetition of some motif (a group of atoms or molecules). The motif is the fundamental part of a symmetric design that, when repeated, generates the “wallpaper” we call the crystal lattice, which is created by translation of the motif. The symmetry of the motif is the true point group symmetry of the crystal, and this point group symmetry applied to a motif can be deduced from the external symmetry of a perfect crystal. The symmetry of the crystal’s external (morphological) form can be no higher than the point symmetry of the lattice. The morphological symmetry of a crystal, as a whole, must belong to one of the 32 crystallographic point groups. In a perfectly formed crystal, the symmetry of the forms conforms to the angular components of the space group symmetry operations minus the translational components. Alternatively, the morphology symmetry may be said to be that of the point group isogonal with (possessing the same angular relations as) its space group. Interestingly, however, a crystal’s morphological appearance may imply a higher symmetry than its actual crystallographic point group. The American mineralogist Martin Julian Buerger (1903–1986) has illustrated this with the mineral nepheline, (Na, K)AlSiO4 (Buerger 1978). The true symmetry of the nepheline crystal lattice, the symmetry of the unit cell, consists merely of a sixfold rotation axis (class 6) as would be exhibited by a hexagonal prism with nonequivalent halves. That is, there is no mirror plane

1.3 Single-Crystal Morphology and Its Relationship to Lattice Symmetry

perpendicular to the rotation axis. However, the absence of this mirror plane is obviously not macroscopically visible in the hexagonal prism form development of nepheline, implying a higher apparent symmetry (6/mmm). The situation with crystallized iodoform, CHI3, is similar. The molecules are strictly pyramidal, but the crystal contains complementary positive and negative pyramids capping a hexagonal prism, as do the minerals zinkenite (Pb9Sb22S42) and finnemanite (Pb5(AsO3)3Cl). In fact, no crystals showing form development consistent with class 6 symmetry have been observed. It is observed, rather, that form developments tend to follow the holohedral, or holosymmetric symmetry, of the crystal class, that is, the point group with the highest symmetry of its crystal system. This is most commonly manifested by equal development of complementary forms in the merosymmetric classes (i.e. those with less symmetry than the lattice). There are four commonly used terms for describing morphology that should be understood. These are zone, form, habit, and twin. A zone is a volume enclosed by a set of faces that intersect one another along parallel edges. The zone axis is the common edge direction. For example, the crystallographic axes and the edges of a crystal are all zone axes. A crystal form is a collection of equivalent faces related by symmetry (e.g. a polyhedron). One can choose the directions of three edges of a crystal as coordinate axes (x, y, z) and define unit lengths (a, b, c) along these axes by choosing a plane parallel to a crystal face that cuts all three axes. For any other crystal face, integers (h, k, l) can be found such that the intercepts the face makes on the three axes are in the ratios a : h, b : k, and c : l. Together, these three integers describe the orientation of a crystal face. The integers are prime and simple (small), and they may be positive or negative in sign. In a cube (a regular hexahedron), all the faces are equivalent. The six faces have indices (1 0 0), 1 0 0 , (0 1 0), 0 1 0 , (0 0 1), and 0 0 1 , but the set is denoted as {1 0 0}, signifying the entire cube, whereas (1 0 0) signifies just one face. In a similar fashion, an octahedron has the form symbol {1 1 1} and consists of the following eight faces: (1 1 1), 1 1 1 , 1 1 1 , 1 1 1 , 1 1 1 , 1 1 1 , 1 1 1 , and 1 1 1 . One or more crystal forms are usually apparent in the crystal morphology, and these may be consistent with the point group symmetry of the lattice. A crystal of α-quartz (low quartz), for instance, may display five external forms showing trigonal point group symmetry (see Figure 1.14). Symmetry considerations limit the number of possible types of crystal forms to 47. However, when we look at crystals from the lattice-based viewpoint, there are only seven crystal systems. This is because there are 15 different forms, for example, in the cubic (isometric) crystal system alone. The 47 forms are listed in Appendix B, grouped by the crystal systems to which they belong. Included in the table are representative examples of minerals exhibiting these form developments. The cube and octahedron are both referred to as closed forms because they are composed of a set of equivalent faces that completely enclose space. All 15 forms in the isometric system, which include the cube and octahedron, are closed. One of the isometric forms, the hexoctahedron, has 48 faces. Six have 24 faces (tetrahexahedron, trisoctahedron, trapezohedron, hextetrahedron, gyroid, and diploid). Five isometric forms have 12 faces (dodecahedron, tristetrahedron, pyritohedron, deltahedron, and tetartoid). The final three isometric and closed forms are, perhaps, more familiar to the chemist. These are the tetrahedron (4 faces), cube (6 faces), and octahedron (8 faces). Nonisometric closed forms include the dipyramids (6, 8, 12, 16, or 24 faces), scalenohedrons (8 or 12 faces), the rhombic and tetragonal disphenoids (4 faces), the rhombohedron (6 faces), the ditrigonal prism (6 faces), the tetragonal trapezohedron (8 faces), and hexagonal trapezohedron (12 faces). Open forms do not enclose space. These include the prisms with 3, 4, 6, 8, or 12 faces, parallel to the rotation axis. These parallel faces are equivalent but do not enclose space. Other open (and

33

34

1 Crystallographic Considerations

Figure 1.14 Top: A “right-handed” (enantiomorph) quartz crystal exhibiting the true symmetry of the crystal class to which quartz belongs. Some authors use the convention that the principle axes are a, b, c rather than X, Y, Z. Bottom: The forms comprising such a quartz crystal. From left to right, the hexagonal prism, trigonal dipyramid, rhombohedron (there are two of these present, a positive form and a negative form), and trigonal trapezohedron.

Z

r

s

z

x m

X

Y

m

s

r, z

x

nonisometric) forms include the pyramid (3, 4, 6, 8, or 12 faces), domes (2 faces), sphenoids (2 faces), pinacoids (2 faces), and pedions (1 face). Open forms may only exist in combination with a closed form or another open form. As mentioned earlier, a crystal’s external morphology may not be consistent with the true point group symmetry of the space lattice. This may be owing to (i) one or more forms being absent or showing anisotropic development of equivalent faces or (ii) the symmetry of the unit cell simply not being manifested macroscopically upon infinite translation in three dimensions. First consider anisotropic development of faces. The growth rates of faces – even crystallographically equivalent faces and faces of crystals belong to the isotropic (cubic) crystal class – need not be identical. This may be owing to kinetic or thermodynamic factors. One possible reason is a nonsymmetrical growth environment. For example, the nutrient supply may be blocked from reaching certain crystal faces by foreign objects or by the presence of habitmodifying impurities. Since visible crystal faces correspond to the slow-growing faces, the unblocked faces may grow so much faster that only the blocked faces are left visible, while the fast-growing faces transform into vertices. Consider pyrite, which belongs to the cubic system. Crystal growth relies on a layer-by-layer deposition on a nucleus via an external flux of adatom species, which may very well be anisotropic. Hence, unequal development of crystallographically equivalent {1 0 0} faces can lead to pyrite crystals exhibiting acicular and platelike morphologies, instead of the anticipated cube shape. In fact, not all crystals exhibit distinct polyhedral shapes. Those that do not are termed non-faceted crystals. The word habit is used to describe the overall external shape of a crystal specimen. Habits, which can be polyhedral or non-polyhedral, may be described as cubic, octahedral, fibrous, acicular, prismatic, dendritic (treelike), platy, blocky, or bladelike, among many others. As a second example of a mineral with several possible form developments, let us consider, in a little more detail, quartz at room temperature, viz. α-quartz (low quartz, Figure 1.14). Quartz belongs to the symmetry class 32, which has two threefold rotation axes and three twofold axes. Although not hexagonal but trigonal, it is most common to use the hexagonal four Bravais–Miller

1.3 Single-Crystal Morphology and Its Relationship to Lattice Symmetry

indices for quartz. Five forms must necessarily be present to reveal its 32 symmetry: 1 0 1 0 , 1 0 1 1 , 0 1 1 1 , 1 1 2 1 , and 5 1 6 1 . These correspond, respectively, to a hexagonal prism; a dominant, or positive, rhombohedron; a subordinate, or negative, rhombohedron; a trigonal (triangular) dipyramid; and a trigonal trapezohedron. In mineralogy, these are labeled, in the order given, with the lowercase letters m, r, z, s, and x. The three orthogonal crystallographic axes are defined as X, bisecting the angle between adjacent hexagonal prism faces; Y, which runs through the prism face at right angles to X; and Z, an axis of threefold symmetry. As illustrated in Figure 1.14, both rhombohedra (r and z) cap, or terminate, the quartz crystal on each end. Each rhombohedron has a set of three faces. By convention, the larger set of three faces is considered the positive rhombohedron. When present, the trigonal trapezohedron (x) is seen at the junction of two prism faces (m) and the positive rhombohedron, and it displays a trapezohedral planar shape. The trigonal pyramid (s) is at the junction of the positive rhombohedron and the prism, which is in line vertically with the negative rhombohedron. It typically forms an elongated rhombus-shaped face. However, in some specimens, one or more of the aforementioned forms are missing or show a development inconsistent with the true point group symmetry of quartz. In fact, most quartz crystals do not display the trigonal dipyramid or trapezohedron faces, the former being especially rare. With these two forms absent, the rhombohedra may exhibit either equal or unequal development. The latter case implies the highest apparent (but still false) crystal symmetry, as the hexagonal prism appears to be terminated at both ends with hexagonal pyramids. It is also possible for the hexagonal prism to be absent, in which case the combination of the two rhombohedra results in a hexagonal dipyramid (or bipyramid), termed a quartzoid. Building on the work of French physicist Jean-Baptiste Biot (1774–1862), the British scientist Sir John Herschel (1792–1871) in 1820 specifically attributed the optical activity in quartz crystals to the hemihedry or mirror-image hemihedral faces, thereby connecting Biot’s physical findings with René-Just Haüy’s earlier crystallographic discovery of hemihedry (see Concise Timeline). Nearly another three decades were to pass before Pasteur’s work on molecular substances was to reveal that the fundamental reason for optical activity is due to arrangements of atoms with an opposite sense (for example, mirror-image molecules). Molecules displaying chirality are said to be enantiomers, while crystals of different sense of chirality, due to the presence of hemihedral faces that distinguish left from right forms, are termed enantiomorphs. By 1819, Sir David Brewster (1781–1868) had determined that all crystals could be divided optically into three classes, namely, isotropic, uniaxial, and biaxial. Higher-order axial crystals do not exist, as they are geometrically impossible. Isotropic crystals (i.e. those belonging to the cubic class), and amorphous solids, do not reorient a light ray. Optically uniaxial and biaxial crystals are anisotropic and, therefore, show birefringence, or double refraction, meaning that they are capable of decomposing a light beam propagating in a direction perpendicular to the optical axis (a.k.a. optic axis) into two rays. Uniaxial crystals (i.e. hexagonal, tetragonal, and trigonal, or rhomboid, minerals) possess but a single axis of anisotropy, which is the optical axis. Quartz belongs to the trigonal system and is thus an example of an optically uniaxial crystal. The optical axis corresponds to the direction of the c-axis, or “long” axis, defined in terms of crystal shape, running parallel to the sixfold, fourfold, or threefold rotational symmetry axes. The speed of light in the material, and therefore the material’s refractive index, depends on both the light’s direction of propagation and its polarization. The optical axis imposes constraints upon the propagation of light beams within the crystal. It is the dependence on polarization that causes birefringence, where a ray of light splits into an ordinary beam polarized in a plane normal to the optical axis and into an extraordinary beam polarized in a plane containing the optical axis (i.e. the

35

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1 Crystallographic Considerations

ray is split into two rays with orthogonal polarizations). For light passing along an optical axis, the speed does not depend on the polarization, so there is no birefringence although there can be optical activity (a rotation of the plane of polarization). As just discussed, optical activity occurs only in chiral materials. The symmetry properties associated with chirality were discussed at length in Section 1.2.1.2. It is important to note an optical axis in a uniaxial or biaxial crystal is a direction (not a single axis or line) in which a ray of transmitted light suffers no birefringence. Any light ray parallel to this direction will exhibit the same lack of birefringence. In summary, quartz, being an optically uniaxial crystal, will reorient the plane of polarization of a monochromatic plane-polarized light wave propagating parallel to its c-axis (our z-axis in Figure 1.14) from tip to tip; quartz is optically active. The direction of rotation is in the same direction as the orientation of the helicoidal SiO4 chains. Although there will be no birefringence when a light wave travels parallel to the c-axis, quartz will show birefringence when a light wave passes through the crystal perpendicular to the c-axis, albeit the birefringence in quartz is very low. Optically biaxial crystals, which are considerably more complex than uniaxial crystals, were shown by Johann Hessel and by Franz Ernst Neumann (1798–1895) in 1820 and 1823, respectively, to be of three kinds, corresponding with the orthorhombic, monoclinic, and anorthic (triclinic) systems of Weiss (see Concise Timeline). In the first two types, there are twofold symmetry axes, while in the latter type, there is no axis of rotational symmetry. In each of these three classes, however, the polarizability along three perpendicular directions is different. As a result, there are three refractive indices for light polarized in these three directions. Biaxial crystals can be cut perpendicular to the optic axis in two and only two directions that give circular sections (where light does not split into two rays and light retains its direction of polarization). There are two optical axes perpendicular to these disks, together defining an optical axial plane. Optical properties will be discussed more in Chapter 9. A Concise Timeline of the History of Geometric Crystallography 1669 1772 1784 1784

1816–1824

1825

1826 1830

1835

Niels Stensen (1638–1686) – more widely known by his Latinized name Nicolas Steno – reports on the morphology of quartz crystals; proposed the law of constancy of interfacial angles Jean-Baptiste Louis Romé de L’Isle (1736–1790) publishes his Essai de cristallographie, with over a hundred descriptions of crystal forms De L’Isle expands Cristallographie to 450 substances; demonstrates use of Steno’s law of constant angles to distinguish between different minerals French abbé and mineralogist René-Just Haüy (1743–1822) publishes his Théorie sur la Structure des Cristaux, in which he proposed the law of rational intercepts, showed that all varieties of crystal forms could be reduced to a few types of morphological symmetry, and conceived his idea of a fundamental building block, which he called molécules integrantes (integral molecules); observed hemihedry (enantiomorphism) in quartz crystals Christian Samuel Weiss (1780–1856) and Friedrich Karl Mohs (1773–1839) derive the six different crystal form classes: cubic, orthorhombic, rhombohedral, hexagonal, monoclinic, and triclinic William Whewell (1794–1866) introduces a notation system relating crystal faces to coordinate axes (Miller indices). These were published in a book by Whewell’s student William Hallowes Miller (1801–1880) in 1839 Mauritius Ludovicus (Moritz Ludwig) Frankenheim (1801–1869) employs two-, three-, four-, and sixfold rotation operations on axes of symmetry The German physician and mineralogist Johann Friedrich Christian Hessel (1796–1872) determines the finite number of morphological symmetry types a three-dimensional crystal can have, as a whole, to be 32 Frankenheim determines that there are 15 crystal systems

1.4 Twinned Crystals, Grain Boundaries, and Bicrystallography

1850–1851

1879 1884 1891 1919 1921 1922

1931 1962 1984 1991

1.4

Auguste Bravais (1811–1863) corrects Frankenheim’s number of crystal systems, noting that 2 were equivalent and that the remaining 14 coalesced by pairs, thus proving that there are really 7 distinct crystal systems; derives the 5 two-dimensional and 14 three-dimensional space lattices German physicist Leonhard Sohncke (1842–1897) derives the 65 rotational (chiral) space groups by considering screw axes and glide planes Pierre Curie (1859–1906) points out Sohncke’s omission of indirect symmetries (rotoreflection, rotoinversion) Evgraf Stepanovich Fedorov (1853–1919) and Arthur Moritz Schoenflies (1853–1928) independently derive the 230 space groups by including indirect symmetries Paul Niggli (1888–1953) publishes Geometrische Kristallographie des Diskontinuums, greatly refining space group theory into its present-day form Peter Paul Ewald (1888–1985) uses reciprocal lattice vectors to interpret the diffraction patterns by orthorhombic crystals and later generalized the approach to any crystal class Ralph Walter Graystone Wyckoff (1897–1994) publishes The Analytical Expression of the Results of the Theory of Space Groups, containing positional coordinates for general and special positions permitted by the symmetry elements Carl Hermann (1898–1961) and Charles Mauguin (1878–1958) develop the space group symmetry notation system now in use Ewald and Arthur Bienenstock (b. 1935) derived the 230 space groups in reciprocal space Quasicrystals discovered by D. Shechtman, I. Blech, D. Gratias, and J.W. Cahn. Rabson, Mermin, Rokhsar, and Wright compute all the three-dimensional quasicrystallographic space groups.

Twinned Crystals, Grain Boundaries, and Bicrystallography

1.4.1 Twinned Crystals and Twinning A twin is a symmetrical intergrowth of two or more crystals, which are more commonly referred to as individuals of the same substance (in which case the specimen as a whole may be referred to as “the crystal”). A simple twin contains two individuals; a multiple twin contains more than two individuals. The twin element is the geometric element about which a twin operation is performed, relating the different individuals in the twin. The twin element may be a reflection plane (contact twins) or a rotation axis (penetration twins). The twin operation is a symmetry operation for the twinned edifice only, not for the individuals. In twinning by merohedry, the twin and the individual lattice point group, as well as their translational symmetry, coincide. If both the point group and translational symmetries of the twin and individual differ, it is referred to as twinning by reticular merohedry. Most commonly, twinning is by syngonic merohedry, in which the twin operation is a symmetry element of the holohedral point group (one of the seven point groups exhibiting the complete symmetry of the seven crystal systems), while the point group of the individual crystals is a subgroup, exhibiting less than complete (holohedral) symmetry. With metric merohedry, the individual lattice has an accidently specialized metric corresponding to a higher holohedry, and a twin operation exists only belonging to the higher holohedry. For example, a twin operation belonging to an orthorhombic lattice may exist for a twinned edifice composed of two monoclinic crystals. The empirical rule of merohedral twinning was originally developed by Auguste Bravais, FrançoisErnest Mallard (1833–1894), and Georges Friedel (1865–1933). Like a grain boundary, the twin boundary is a higher energy state, relative to the crystal. However, because a twin boundary is highly ordered, it is of lower energy than a typical non-twin grain boundary. Recognizing this, Buerger later proposed that “if the crystal structure is of such a nature

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1 Crystallographic Considerations

Figure 1.15 Drawings of (left) a Dauphiné law penetration twin quartz crystal composed of two “left-handed” individuals (the trigonal trapezohedron face, x, is on the left side, compared with Figure 1.14) and (right) a Japan law contact twin quartz crystal. Dauphiné is a former province in south eastern France, where the fist penetration twins of quartz of this type were observed. Japan law twining was first discovered in 1829 by C.S. Weiss in quartz crystal from the La Gardette Mine in France. However, because of the abundance of these specimens in Japan, they are now known as Japanese twins.

that, in detail, it permits a continuation of itself in alternative twin junction configuration, without involving violation of the immediate coordination requirements (the first coordination sphere) of its atoms, the junction has low energy and the twin is energetically possible (Buerger 1945).” Twins may also be divided into the three following categories based on their origin: growth twins, gliding (or mechanical, or deformation) twins, and transformation twins. Growth twins originate at the nucleation stage under conditions of supersaturation, where there is greater likelihood for the arrival of clusters of atoms, already coordinated, at the twin position. Such twins persist and grow if subsequent clusters of adatoms continue to arrive in that fashion. Growth twins may be subdivided into penetration twins and contact twins. Of the two types, the contact twin is the simplest to visualize, in which the two individuals appear to have been united along a common plane and appear as mirror images across the twin boundary. A particular type of contact twin (known as the Japan law twin) and a particular type of penetration twin (known as a Dauphiné law twin) are illustrated in Figure 1.15. Penetration twins (which may also result from transformation as well as growth) are complete crystals that pass through or interpenetrate one another and share a common volume of space. In the case of the Dauphiné law, either only right-handed structures or only left-handed enantiomorphic structures are combined. In a Dauphiné law quartz twin, for example, the r and z rhombohedra (see Figure 1.14) are indistinguishable. We will see in Chapter 8 that this type of quartz twin is non-piezoelectric as mechanical pressure along the z-axis does not cause an electrostatic polarization. However, optically, they are similar to an untwinned crystal in that a monochromatic plane-polarized light ray passing through the specimen along the optical axis (in quartz, the optical axis is the z-axis, from tip to tip) changes the orientation of its oscillation plane. Another common type of a penetration twin found in quartz is the Brazil law twin, in which both a right-handed enantiomorphic individual and a left-handed enantiomorphic individual are combined within a single specimen. In this case, if the right- and left-handed individuals are equal, a light ray will pass through the twin unaltered. Glide (mechanical) twinning is caused by a specific type of structural shear in plastic deformation. The lower-energy non-twined crystals absorb part of the energy supplied in the plastic deformation process, and, if the crystal structure permits it, a layer of atoms glide into a twin position. With continued stress, gliding takes place in the next layer. Because gliding in all the parallel layers does not take place simultaneously, twin lamella forms. Calcite is an example of a crystal that readily forms glide twins at low differential stresses (~10 MPa). Because plastic deformation is so important in industrial and everyday life, the study of the microscopic mechanisms that may be at work

1.4 Twinned Crystals, Grain Boundaries, and Bicrystallography

during deformation twinning is very much active. The first descriptions of mechanical twinning started in Germany, with a first rationalization by Otto Mügge (1858–1933) (see Hardouin Duparc 2017). Transformation twinning occurs during the transformation from a high-temperature phase to a lower-symmetry low-temperature phase, for example, when sanidine (monoclinic KAlSi3O8) is cooled to form microcline (triclinic KAlSi3O8). In such a process, there is spontaneous formation of nuclei in different orientations, which subsequently grow into one another. Each member of the aggregate is either in parallel or in twinned orientation, with respect to one another. This follows from the fact that they could be brought into coincidence by one of the possible symmetry operations of the high-temperature phase that vanished in the formation of the low-temperature phase, which has a symmetry that is a subgroup of the high-temperature phase (Buerger 1945).

1.4.2 Crystallographic Orientation Relationships in Bicrystals The orientation relationship between a pair of grains of the same substance (the only kind we will consider in this book), a bicrystal, is often expressed by an axis–angle description, since one of the crystals can always be generated from the other by a rigid body rotation about a suitable axis. More precisely, their lattices can be made to coincide by turning one of the two crystals about a suitable rotation axis. Actually, the modern method for quantifying the goodness of fit between two adjacent grains in a pure polycrystalline substance (i.e. homophase interfaces), and even in a multiphase solid (i.e. heterophase, or interphase, interfaces), is based upon the number of lattice points (not atomic positions) in each grain that coincide. In the case of intergranular boundaries (homophase interfaces), this information is directly obtainable from the rotation matrix representing the rigid body rotation describing the misorientation. 1.4.2.1

The Coincidence Site Lattice

Specialists from many different fields contributed to the evolution of our modern picture of the crystalline interface, which may be summarized as follows. The geometric model of the inverse density of Coincidence Site Lattice that is very much used nowadays actually hacks back to Georges Friedel in 1904 (Études sur les groupements cristallins). Georges Friedel, who was the son of the Chemist and Mineralogist Charles Friedel, called this parameter the twin (macle) index and defined it as the ratio of the total number of nodes of the primitive lattice to the number of coincident nodes restored by the twin operation. Friedel’s 1904 “multiple lattice” is our Coincidence Site Lattice (Hardouin Duparc 2011). Friedel’s work, however, was apparently largely forgotten or unnoticed by metallurgists, and Walter Rosenhain (1875–1934) of the United Kingdom’s National Physics Laboratory (Rosenhain 1925) championed an amorphous or at best “irregular inter-crystalline cement layer” to explain grain boundary properties, and subsequent models by others focused more on the actual “material lattice” or crystalline structure across the interface, rather than the abstract mathematical lattice construct. This “amorphous cement layer” postulate was disputed by F. Hargreaves and R.J. Hill, who proposed a transitional lattice connecting the grains on either side, with drawings showing coincident atoms at the boundary plane itself (Hargreaves and Hill 1929). The physicist Nevill Francis Mott (1905–1996) first suggested that grain boundaries should contain regions of fit and misfit (Mott 1948). Merritt Lionel Kronberg and Francis Howard Wilson then pointed out the importance of the coincidence of atom positions across the grain boundaries they could observe in recrystallized copper (Kronberg and Wilson 1949). Srinivasa Ranganthan (b. 1941) presented a general procedure for obtaining coincidence relationships between lattices about rotation axes in cubic crystals (Ranganathan 1966), and this was later generalized to other lattice systems.

39

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1 Crystallographic Considerations

The modern method for quantifying the goodness of fit between two adjacent grains examines the number of lattice points from each grain that coincide. Beware that lattice points are not atoms, but atomic motifs are associated to lattice points, e.g. one copper atom per lattice point in FCC copper, one NaCl motif in FCC rock salt. In special cases, for example, when the grain boundary plane is a twin plane in an FCC metal, the atomic sites for each of the adjacent crystals coincide in the boundary, as it will be seen below. These are called coherent boundaries. It has long since been experimentally verified that coherent grain boundaries possess special properties. For example, coherent boundaries migrate faster than random boundaries during recrystallization (Aust and Rutter 1959). The main physical reason is that coherent boundaries are less prone to impurity segregation than random boundaries. Physics is more complicated than geometry. Yet, or henceforth, one must start with geometry and stay aware of the pitfalls of simplicity (relatively speaking). Consider a pair of adjacent crystals. We mentally expand the two neighboring crystal lattices until they interpenetrate and fill all the space. Without loss of generality, it is assumed that the two lattices possess a common origin. If we now hold one crystal fixed and rotate the other, it is found that a number of lattice sites for each crystal, in addition to the origin, coincide with certain relative orientations. The set of coinciding points form a coincidence site lattice (CSL), which is a sublattice for both the individual crystals. In order to quantify the lattice coincidence between the two grains, A and B, the symbol Σ customarily designates the reciprocal of the fraction of A (or B) lattice sites that are common to both A and B: Σ=

Number of crystal lattice sites Number of coincidence lattice sites

1 18

For example, if one third of the A (or B) crystal lattice sites are coincidence points belonging to both the A and B lattices, then Σ = 1/(1/3) = 3. The value of Σ also gives the ratio between the areas enclosed by the CSL unit cell and crystal unit cell. Georges Friedel introduced the Σ symbol in 1920 (Contribution à l’étude géométrique des macles) as the ratio of the volume of a (not necessarily primitive) multiple cell to the volume of the primitive cell. He provided his reader with several formulae that, in the cubic case, give Σ = h2 + k2 + l2 (h, k, and l being the indices of the twin plane; see Eq. 1.19) and a twin index I equal to Σ if Σ is odd or equal to Σ/2 if Σ is even (Hardouin Duparc 2011). The value of Σ is a function of the lattice types and grain misorientation. The two grains need not have the same crystal structure or unit cell parameters. Hence, they need not be related by a rigid body rotation. The boundary plane intersects the CSL and will have the same periodicity as that portion of the CSL along which the intersection occurs. The simple CSL model is directly applicable to the cubic crystal class. The lower symmetry of the other crystal classes necessitates the more sophisticated formalism known as the constrained coincidence site lattice (CCSL) (Chen and King 1988). In this book, only cubic systems will be treated. Interestingly, whenever an even value is obtained for Σ in a cubic system, it will always be found that an additional lattice point lies in the center of the CSL unit cell. The true area ratio is then half the apparent value. This operation can always be applied in succession, until an odd value is obtained – thus Σ is always odd in the cubic system. A rigorous mathematical proof of this would require that we invoke what is known as O-lattice theory, formulated by Walter Bollman (1920–2009) at the Battelle Memorial Institute in Geneva (Bollman 1967). The O-lattice accounts for all equivalence points between two neighboring crystal lattices. It includes as a subset, not only coinciding lattice points (the CSL) but also all nonlattice sites of identical internal coordinates. However, expanding on that topic would be well

1.4 Twinned Crystals, Grain Boundaries, and Bicrystallography

Figure 1.16 A view down the [0 0 1] direction of a tilt boundary between two crystals (A, B) with a misorientation angle of 36.9 about [0 0 1]. The grain boundary is perpendicular to the plane of the page. Every fifth atom in the [0 1 0] direction in B is a coincidence point (shaded). The area enclosed by the CSL unit cell (bold lines) is five times that of the crystal unit cell, so Σ = 5.

36.9°

[010]B

A

B

beyond the scope of this book. The interested reader is referred to Bollman (1970), Bhadeshia (1987), or Sutton and Balluffi (1995). Single crystals and bicrystals with no misorientation (i.e. θ = 0), by convention, are denoted Σ1. In practice, small- or low-angle grain boundaries with a misorientation angle of less than 10–15 are also included under the Σ1 term. Since Σ is always odd, the coincidence orientation for high-angle boundaries with the largest fraction of coinciding lattice points is Σ3 (signifying that 1/3 of the lattice sites coincide). Next in line would be Σ5, then Σ7, and so on. Figure 1.16 shows a tilt boundary between two cubic crystals. The grain boundary plane is perpendicular to the plane of the page. In the figure, we are looking down one of the 1 0 0 directions, and the [1 0 0] axis about which grain B is rotated is also perpendicular to the page and passes through the origin. At the precise misorientation angle of 36.9 , one fifth of the B crystal lattice sites are coincidence points, which also belong to the expanded lattice of crystal A; this is a Σ5 CSL misorientation. The set of coincidence points forms the CSL, the unit cell of which is outlined. Note that the area enclosed by the CSL unit cell is five times that enclosed by the crystal unit cell. Fortunately, there is an easy, although somewhat tedious, way to determine the value for Σ from the rotation matrix representing the rigid body rotation describing the misorientation between two crystals, A and B. Note that such a rotation is not a symmetry operation mapping a lattice onto itself. Rather, it takes the crystal A orientation into the crystal B orientation, which are distinguishable. Now, if an integer, N, can be found such that all the elements of the rotation matrix become integers when multiplied by N, then that integer will be the Σ value. The value of N is found simply by multiplying all the matrix elements by integers, in increments of one beginning with the number 1, until the matrix elements are all integers. If the value of Σ turns out be even using this procedure, then the true value is obtained by successively dividing N by two until the result is an odd integer. This method can be used to compute the value of Σ for any general rotation matrix. Example 1.8 Use Eq. (1.7) to compute the matrix corresponding to a rotation of 180 about the [1 1 2] direction in a cubic bicrystal. Then, calculate the value of Σ.

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Solution In order to use Eq. (1.7), we need to first convert the [1 1 2] direction indices into direction cosines. This is accomplished via Eq. (1.8), with (x1, y1, z1) = (0, 0, 0) and (x2, y2, z2) = (1, 1, 2): cos α = ux = cos β = uy = cos γ = uz =

1−0 12 + 1 2 + 22 1−0 2

2

1 +1 +2

2

2−0 2

2

1 +1 +2

2

=

1 6

=

1 6

=

2 6

Hence, the rotation axis is given by [1/ 6, 1/ 6, 2/ 6]. Substituting these values for ux, uy, and uz in Eq. (1.7), where ϕ = 180 (hence cos ϕ = −1 and sin ϕ = 0), gives

R 112 180

=

−1 + 2 6 2 6 4 6

2 6 −1 + 2 6 4 6

4 6 4 6 −1 + 8 6

=

1 3

−2 1 1 −2 2 2

2 2 1

It is safe to check that the obtained matrix is indeed orthonormal: its columns seen as vectors have norms equal to 1 and are mutually orthogonal, idem for its rows. Hence, Σ = 1/(1/3) = 3, meaning that one third of the crystal lattice sites are coincidence points belonging to both crystals. Alternatively, the area enclosed by the CSL unit cell is three times larger than the area enclosed by the crystal unit cell. This is illustrated in Figure 1.17 for the special case of a {111} boundary plane for an FCC metallic bicrystal.

{111} Boundary plane 111

Figure 1.17 The twin boundary (perpendicular to the plane of the page) is a Σ3 CSL misorientation, with a {111} boundary plane for an elementary FCC metal (e.g. copper). There is complete coincidence of the atoms in the boundary plane itself. Note that it would not be the case for a BCC metal (e.g. α-iron).

1.4 Twinned Crystals, Grain Boundaries, and Bicrystallography

For tilt grain boundaries, the value of Σ can also be calculated if the plane of the boundary is specified in the coordinate systems for both adjoining grains. This method is called the interface-plane scheme (Wolf and Lutsko 1989). In a crystal, lattice planes are imaginary sets of planes that intersect the unit cell edges. These planes are denoted by Miller indices, a group of integers that are the reciprocals of the fractional coordinates of the points where the planes intercept each of the unit cell edges. In cubic crystals, the (h k l) planes are orthogonal to the [h k l] direction. The tilt and twist boundaries (see Section 2.2.2 with Figure 2.3a and b) can be defined in terms of the Miller indices for each of the adjoining lattices and the twist angle, θtw, of both plane stacks normal to the boundary plane as follows: (h1 (h1 (h1 (h1

k1 k1 k1 k1

l1) = (h2 k2 l2); θtw = 0 symmetric tilt grain boundary (STGB) l1) (h2 k2 l2); θtw = 0 asymmetric tilt grain boundary (ATGB) l1) = (h2 k2 l2); θtw 0 twist boundary l1) (h2 k2 l2); θtw 0 mixed general grain boundary

The CSL-Σ value is obtained for symmetric tilt boundaries between cubic crystals as follows (Georges Friedel 1920, 1926): Σ = h2 + k 2 + l 2 =

for h2 + k2 + l2 = odd

h2 + k 2 + l 2 2

for h2 + k 2 + l2 = even

1 19

The STGB shown in Figure 1.17 corresponds to (hkl) = (111). From Eq. (1.19), (112) also corresponds to Σ = 3 since (12 + 12 + 22)/2 = 3. A given Σ always corresponds to at least two STGBs (even considering the sole [100] tilt axis, Σ = 65 mathematically corresponds to 8 different (0hl) STGBs). To just say that a symmetric interface has a certain Σ is not a sufficient information if the Miller indices of the boundary plane are not given. Two further caveats are that (i) Eq. (1.19) is for cubic lattices, be they primitive, i.e. simple cubic, or BCC or FCC, and (ii) in the same way that a lattice is not a crystal, a bilattice is not a bicrystal. For asymmetric tilt boundaries between cubic crystals, Σ is calculated from (see Randle 1993) Σ=

h21 + k21 + l21 h22 + k22 + l22

1 20

For example, if we mentally expand the lattices of both A and B in Figure 1.16, it will be seen that the grain boundary plane cuts the B unit cell at (3 4 0) in the B coordinate system and the A unit cell at (0 1 0) in the A coordinate system. Thus, Eq. (1.20) yields Σ = (25/1)1/2 = 5. In actual polycrystals, misorientation angles rarely correspond to exact CSL configurations. There are ways of dealing with this deviation, which set criteria for the proximity to an exact CSL orientation that an interface must have in order to be classified as belonging to the class Σ = n. The Brandon criterion (Brandon et al. 1964), named after the metallurgical and materials engineer David G. Brandon (b. 1935), asserts that the maximum permitted deviation is v0Σ−1/2. For example, the maximum deviation that a Σ3 CSL configuration with a misorientation angle of 15 is allowed to have and still be classified as Σ3 is 15 (3)−1/2 ~ 8.7 . Even restricted to FCC metals and to the lower in energy of the STGBs for a given Σ, this is only a proposed criterion, too tolerant, as numerous HRTEM observations backed by atomistic simulations on real bicrystals have shown. Some facetting occurs, depending on the FCC metal (copper is not aluminum). The coarsest lattice characterizing the deviation from an exact CSL orientation, which contains the lattice points for each of the adjacent crystals, is referred to as the displacement shift complete lattice or, better,

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44

1 Crystallographic Considerations

displacement symmetry conserving lattice, the set of displacement vectors that leave the CSL invariant (DSCL). This lattice is a very useful guide even if, again, it cannot be a deterministic guide since it does not contain the energy interactions between the atoms. Despite the difficulties associated with characterizing inexact CSL orientations, the CSL concept is useful because grain boundary structure, which depends on the orientation relationship between the grains and, hence, the CSL, directly influences intragranular properties like chemical reactivity (e.g. corrosion resistance), segregation, and fracture resistance. Grain boundary engineering is a relatively new field that concentrates on controlling the intragranular structure, or CSL geometry, to improve these properties, in turn improving bulk materials performance (Watanabe 1984, 1993; Priester 2013). For the most part, this means introducing a large fraction of low-Σ boundaries, particularly twin boundaries. It is believed, however, that optimal grain boundary properties may be restricted to narrow regions (small deviations) about exact CSL orientations. Some microstructural considerations will be found in Section 2.2.

The Friedel Family. Georges Friedel (1865–1933) was brilliantly admitted at the French École Polytechnique, completed his studies still being “major” (first rank) and entered the École des Mines in Paris. He was born to Charles Friedel (1832–1899), chemist and mineralogist who had as co-workers James Mason Crafts (Friedel–Crafts reactions) and Jacques Curie on pyroelectricity and twins. G. Friedel taught and did his research at the École des Mines in Paris in Saint-Étienne, becoming the director of this School until 1918 when he accepted to go and teach at the University of Strasbourg, just reconquered (Charles Friedel, of Alsacian origin, had chosen to move to Paris in 1870 in order to stay French). G.F. established the first reliable classification of grain boundaries together with the CSL (and its corresponding coincidence index). He is also famous for his pioneering research in liquid crystals (he named them nematics, smectics, cholesterics), disclinations, and X-ray diffraction (Friedel’s law). He also discovered the spongy nature of zeolites, and Friedel’s salt, known in the cement and concrete industry, is named after him (he first synthetized this salt as a twinning toy mineral in 1897). His son Edmond Friedel (1895–1972) was director of the École des Mines in Paris and dedicated his career to the research and exploitation of mineral resources. His grandson Jacques Friedel (1921–2014) did his PhD work under Nevill Mott in Bristol where he also became good friend with Charles Frank. J.F. worked both in solid-state physics (Friedel’s sum rule, Friedel’s oscillations, Friedel’s d-band model), in plasticity (he wrote a famous book on dislocations in 1956 in French, extended and translated in 1964), in fracture theory, about disclinations, and about quasicrystals. Photos courtesy of Jean Friedel, Georges’ great-grandson. From left to right: Charles, Georges, Edmond, and Jacques.

1.4.2.2 Equivalent Axis–Angle Pairs

It turns out that a bicrystal can be represented by a number of equivalent axis–angle pairs determined by the crystal class to which it belongs. The rotational degeneracy of the seven crystal classes can be obtained from the character tables for their respective point groups, which we simply give here without demonstration (see Burns and Glazer 2013): cubic, Oh (24); hexagonal, D6h (12);

1.4 Twinned Crystals, Grain Boundaries, and Bicrystallography

hexagonal close packed, D3d (6); tetragonal, D4h (8); trigonal, D3d (6); orthorhombic, D2h (4); monoclinic, C2h (2); and triclinic, Ci (1). In order to calculate the equivalent axis–angles, the first rotation matrix, let us call it R, is “operated” on by matrices representing the various proper rotations, which are symmetry operations, let us call any of them C. The product matrix R = C R is also a symmetry operation of the crystal class. The axis–angle pairs are obtained directly from the product matrices, R . The new rotation angle ϕ is given from the knowledge of the trace of R , as discussed earlier in Section 1.2.1.1.2: Trace R = R11 + R22 + R33 = 1 + 2 cos ϕ If ϕ

1 21

π, the new rotation axis [u1 , u2 , u3 ] is obtained from the equations: ux =

R32 − R23 R − R31 R − R12 , uy = 13 , uz = 21 2 sin ϕ 2 sin ϕ 2 sin ϕ

1 22

where u 2x + u 2y + u 2z should be equal to 1. If ϕ = π (180 ), the following equations, simply obtained from Eq. (1.7) with cos ϕ = −1 and sin ϕ = 0, are needed to determine the rotation axis: 2

R11 = −1 + 2u x R21 = 2ux uy

1 23

R31 = 2ux uz One must extract the maximum component from the diagonal elements of the matrix: If R11 is of maximum magnitude, compute ux = R11 + 1

1 2

2, uy = R12 2ux , uz = R13 2ux

1 24a

If R22 is of maximum magnitude, compute uy = R22 + 1

1 2

2, ux = R12 2uy , uz = R23 2uy

1 24b

If R33 is of maximum magnitude, compute uz = R33 + 1

1 2

2, ux = R13 2uz , uz = R23 2uz

1 24c

Example 1.9 Compute the product matrix obtained by operating on the rotation matrix from Example 1.8 (a 180 rotation about the [1 1 2] direction in a cubic bicrystal) with the matrix representing a 90 clockwise rotation about the [1 0 0] plane normal. Then determine the equivalent axis–angle pair. Solution First, obtain the matrix describing the −90 (clockwise) rotation about [1 0 0]. This is accomplished via Eq. (1.7) with ux = 1, uy = uz = 0 and θ = −90 (for the cubic class, the (1 0 0) plane normal is the vector [1, 0, 0]):

R 100 90

=

1 0

0 0

0 1

0

−1

0

45

46

1 Crystallographic Considerations

Next we take the product, R , between this matrix and the matrix for R[1 Example 1.8, that is, R = R[1 0 0](90 ) R[1 1 2](180 ):

R =

1 3

1

0

0

−2

1

2

0 0

0 −1

1 0

1 2

−2 2

2 1

=

1 3

−2

1

2

2 −1

2 2

1 −2

1 2](180

) from

We can now use R to extract an equivalent axis–angle pair. The new rotation angle, ϕ , is obtained from the relation (1.21): Trace R = R11 + R22 + R33 = 1 + 2 cos ϕ . Hence, cos ϕ = [(−2 + 2 − 2)/3) − 1]/2 = [−2/3 − 1]/2 = −5/6, and ϕ ~ 146.44 . As sin ϕ = (1 − 25/36)1/2 = ( 11)/6 0, we can use Eq. (1.22) to compute the components of the rotation axis: ux =

R32 − R23 2 3−1 3 = 2 sin ϕ 2 11 6

=1

11

uy =

R13 − R31 2 3+1 3 = 2 sin ϕ 2 11 6

=3

11

uz =

R21 − R12 2 3−1 3 = 2 sin ϕ 2 11 6

=1

11

It is satisfying to observe that the norm of this axis is correctly equal to 1. The corresponding indices are [1, 3, 1]. Therefore, an equivalent axis–angle pair of a 180 rotation about [1 1 2] in a cubic bicrystal is a 146.44 rotation about [1 3 1].

1.5

Amorphous Solids and Glasses

Amorphous solids do not possess the long-range translational order characteristic of crystals, although they do usually exhibit short-range structural order. The glassy or vitreous state is a subset of the amorphous state. All glasses are monolithic and amorphous, but only amorphous materials prepared by rapidly cooling, or quenching, a molten state through its glass transition temperature (Tg) are glasses. Non-glassy amorphous solids are normally prepared by severely mechanochemically damaging a crystalline starting material, for example, via ion implantation or ball milling. Glasses are amorphous materials that are brittle (i.e. subject to fracture). It is often stated that glasses are not solids, but rather supercooled liquids. This is misleading. It is more accurate to say they are metastable, amorphous solids obtained from supercooled liquids. The persistent person may wish to argue that some vertically oriented window panes show evidence of viscous flow like a liquid, as they are thicker at the bottom. However, glassy viscous flow does not explain why some antique windows are observed to be thicker at the bottom. It is true that glasses can flow but this process, which is due to structural relaxation, occurs exceedingly slowly. At temperatures significantly below the glass transition temperature, the rate at which the liquid-like glass structure can change is inversely proportional to the viscosity. Alternatively stated, the further away from (i.e. below) the glass transition temperature, the slower the constituent atoms of the glass move, and the more solid-like the glass behaves. Simulations on some cathedral glass panes have indicated

1.5 Amorphous Solids and Glasses

that their room-temperature flow would require a span of time longer than the universe has existed since they are hundreds of degrees below their glass transition temperature and have viscosities as high as 1012–1013 Pa s (Zanatto 1998). Note that often viscosity is reported in Poise and 1 Poise equals 0.1 Pa s. It has been claimed that a more reasonable hypothesis of why some old European glass is thicker at one end probably depends on how the glass was made. At that time, glassblowers created glass cylinders that were flattened to make the window panes. Some of these resulting pieces may not have been entirely or uniformly flat, and workers installing the windows for some reason preferred to put the thicker sides of the pane at the bottom. This gives them a melted look, but does not signify that glass is truly liquid (Curtin 2007). Glasses are absent of long-range structural order as well as grain boundaries and other crystalline defects. A glass does, however, possess short-range order (SRO) and medium-range order (MRO). Hence, a glass can be defined as a rigid brittle noncrystalline solid with less structural order than a crystal but more than a liquid. Natural glasses form when certain types of rock melt as a result of volcanic activity, lightning strikes, or meteorite impacts, followed by very rapid cooling and solidification. Stone-age men are believed to have used some of these naturally formed amorphous materials as tools for cutting. An example is fused quartz, or quartz glass, formed by the melting and rapid cooling of pure silica. Upon rapid solidification, the SRO (geometry) of the SiO4 tetrahedra is preserved, but not the long-range crystalline order of quartz. The structure of crystalline quartz involves corkscrewing (helical) chains of SiO4. The corkscrew takes four tetrahedra, or three turns, to repeat, each tetrahedron essentially being rotated 120 . The chains are aligned along one axis of the crystal and interconnected to two other chains at each tetrahedron. The Si–O–Si bond angle between interconnected tetrahedra is nominally about 145 . As in quartz, every oxygen atom in fused quartz is also bonded to two silicon atoms (each SiO4 tetrahedron is connected to four other tetrahedra), but the tetrahedra are polymerized into a network of rings of different sizes, occurring in a wide range of geometries. Hence, in quartz glass there is a distribution of Si–O–Si bond and Si– O–Si–O torsion angles. The very high temperatures required to melt quartz were not attainable by early craftsmen. Hence, they prepared sodium silicate glass by mixing together and melting sodium carbonate with sand. The structure of this so-called water glass is similar to that of quartz glass, except that, with the random insertion of sodium ions within the network, nonbridging oxygen atoms (i.e. oxygen atoms bonded to a single silicon atom) are produced. Although cooling rates as high as 1011 K/s have been obtained on solid surfaces with pulsed laser melting, for bulk phases, rapid solidification typically refers to cooling rates in the range of 102–107 K/s. Broadly speaking, on the low end of this range, very fine-grained crystalline substances can be produced. On the high end, the formation of either a glassy phase or quasicrystal is favored over a crystalline phase, depending on the viscosity at the melting point temperature. There is no hard and fast rule on the critical cooling rate required for glass formation. For example, with cooling rates of around 107 K/s, liquid water can form a glass, while SiO2 can form a glass with cooling rates at least as low as 10−4 K/s. Whether a liquid crystallizes or forms a glass is a kinetic problem involving the competition between the rate of nucleation and crystal growth and the rate at which thermal energy can be extracted. Glass formation, or vitrification, can be compared with crystallization by referring to Figure 1.18, which is applicable to both metallic and nonmetallic systems. Crystallization follows path abcd. As the temperature of a non-glass-forming melt is lowered, the molar volume of the alloy decreases continuously until it reaches the melting point where it changes discontinuously, that is, where it experiences a first-order phase transition. The enthalpy and entropy behave similarly. In glass formation, rapid cooling forces the melt to follow path abef with decreasing temperature. The liquid

47

1 Crystallographic Considerations

a id

qu

Li

b Volume

48

f

e

Glass Glass

h

led oo c r de uid Un liq

g

L

Crystals d

Tg

Tm

Temperature

Figure 1.18 A comparison of glass formation (curve abef) and crystallization (curve abcd). The point Tg is the glass transition temperature and Tm is the melting temperature. (Source: After West (1985), Solid State Chemistry and its Applications. © 1984, John Wiley & Sons, Inc. Reproduced with permission.)

remains undercooled (it does not solidify) in the region be below the melting point. The molar volume continuously decreases in the undercooled region, and the viscosity increases rapidly. At the point Tg, called the glass transition temperature, the atomic arrangement becomes frozen into a rigid mass that is so viscous it behaves like a solid. Rapid cooling reduces the mobility of the material’s atoms before they can pack into a more thermodynamically favorable crystalline state. The state of our current scientific understanding of glass formation is founded, to a large extent, on the theoretical work of Harvard researcher David Turnbull (1915–2007). Turnbull’s criterion for the ease of glass formation in supercooled melts predicts that a liquid will form a glass, if rapidly solidified, as the ratio of the glass transition temperature, Tg, to the liquidus temperature, Tl, becomes equal to or greater than 2/3 (Turnbull 1950). The Tg/Tl ratio is referred to as the reduced glass transition temperature, Trg. The rate of homogeneous nucleation is dependent on the ease with which atomic rearrangement can occur (commonly taken as the atomic diffusion coefficient), which scales inversely with fluidity or viscosity. Easy glass-forming substances form highly viscous melts (e.g. >102 P), compared to non-glass-forming ones (e.g. water, with η ~ 10−2 P). In highly viscous melts, the atomic mobility is substantially reduced, which suppresses the homogeneous nucleation rate and, hence, crystallization. In fact, Igor Evgenevich Tammann (1861–1938) pointed out, as early as 1904, that the higher the viscosity of a melt, the lower its crystallizability (Tammann 1904). The homogeneous nucleation rate is, therefore, highly dependent on Trg. The Trg > 2/3 criterion successfully predicts glass formation in metallic and nonmetallic liquids. It must be noted,

1.5 Amorphous Solids and Glasses

however, that heterogeneous nucleation (e.g. on seed particles present inadvertently) may prevent glass formation. Indeed, crystallization is usually initiated in this manner. The above arguments are based on kinetics. It may also be shown on thermodynamic grounds that a high value for Trg (and therefore the tendency to form a glass at lower cooling rates) is obtained for deep eutectic systems, that is, where the melting point of some alloy composition is substantially lowered compared with the melting points of the pure components. These systems tend to be those with very little solid solubility between the components. When atoms do not “fit” together in the lattice (owing to mismatches in size, valence, etc.), the tendency for crystallization diminishes. This is owing to both a large negative heat of mixing and entropy of mixing for the liquid compared with the competing crystalline phase.

1.5.1 Oxide Glasses In theory, any substance can be made into a glass if it were cooled from the liquid state sufficiently fast to prevent crystallization. Indeed, in addition to oxides, other chalcogenides (e.g. As2S3, GeS2, and 70Ga2S3:30La2S3) as well as some fluorides (namely, beryllium, zirconium, barium, thorium, lanthanum, and lutetium) are known to also form glasses, but only oxide and metallic glasses will be discussed here. Oxygen is especially suited as an anion to link structural units having coordination numbers of 3, 4, or 6, and this makes possible continuous cross-linking that leads to highly polymeric network structures. Although, in common parlance, the term glass has come to refer to silicate glass of one kind or another, other binary oxides can form glass. Some common examples include boric oxide (B2O3), germanium oxide (GeO2), arsenic oxide (As2O3), antimony oxide (Sb2O3), and phosphorous pentoxide (P2O5). There are also ternary and quaternary glass oxides, such as Li2O–TiO2–P2O5 and PbO–TiO2–K2O–SiO2. Many possible combinations exist of network-forming oxides together with one or several network modifier alkali oxides RO (R = Li, Na, or K) whose incorporation alters glass properties due to the varying degrees of cross-linking. These include the molar volume, optical density, and the glass transition temperature. Several models have been put forth that attempt to account for the ease of glass-forming ability (GFA) of a substance based on aspects of the internal structure. However, all fall short of being entirely satisfactory, and there are exceptions to each of these “rules.” The various models are premised on coordination number (Goldschmidt’s radius ratio; Zachariasen’s random network theory), field strength (Dietzel’s field strength), bond strength (Sun’s single bond strength criterion), bond type (Smekal’s mixed bonding rule; Stanworth’s electronegativity rule), and topological constraints (Phillips theory). Most of these concepts will be covered in more detail in subsequent chapters. However, it is worth presenting a few of these very briefly here in the context of their relevance to glass formation. The Swiss-born Norwegian mineralogist Victor Moritz Goldschmidt (1888–1947) postulated in 1929 that for MmOn oxides, glass formation is only possible if the anion radius to cation radius ratio (ra/rc) falls between 0.2 and 0.4. However, William Houlder Zachariasen (1906–1979) pointed out that some oxides fulfill this requirement but do not form a glass. Zachariasen published a seminal paper in 1932 in which he suggested the random network structure of inorganic glass. He postulated that each oxygen atom be linked to no more than two cations, that cation coordination numbers be 3 or 4, that all coordination polyhedra share at least three corners in 3D networks, and that the polyhedra cannot share edges or faces. However, his rules only apply to oxides, and some oxides that are not predicted to form a glass by Zachariasen actually can vitrify. The crystallographer Bertram Eugene Warren (1902–1991) extended Zachariasen’s theory to modified oxide glasses in 1941.

49

50

1 Crystallographic Considerations

The German chemist and ceramist Adolf Hugo Dietzel (1902–1993) attempted to extend Goldschmidt’s rule in 1942 to include cation charge. Small cations with a high charge (Zc/[rc + ra]2 > 1.3) are glass formers, while large cations with small charge (0.4 ≤ Zc/[rc + ra]2 ≤ 1.3) are network modifiers that cannot form a glass alone and cations of medium size and charge (Zc/[rc + ra]2 < 0.4) are intermediate. In 1947, the physicist Kuan-Han Sun of Kodak Research Laboratories took a different approach in proposing a criterion for the glass-forming ability of simple oxides. He hypothesized that high-strength oxide bonds (>80 kcal/bond) are difficult to break/reform upon cooling. They thus have high viscosity melts since the stronger the M–O bonds, the more difficult are the structural rearrangements necessary for crystallization. These oxides therefore make good glass formers, while oxides with low-strength bonds (> Lx v

27

where Ds is the solute diffusivity in the solid, Lx is the system length scale in one dimension, and v is the solidification speed (Phanikumar and Chattopadhyay 2001). The self-diffusivities of most pure metals at their freezing points (Tf) are in the range 10−9–10−6 cm2/s. For a system length scale of 1 cm, the solidification rate (cm/s) must be lower than these numerical values for the diffusivities, which are very slow rates indeed. In other words, equilibrium solidification occurs only when the melt is cooled extremely slowly! The second limiting case approximates conventional metallurgical casting processes, in which the cooling rate is on the order of 10−3–100 K/s. As a result, the solidification rate is several orders of magnitude too fast to maintain equilibrium. The most widely used classical treatment of nonequilibrium solidification is by Erich Scheil (1942) who was at the Max Planck Institute for Metals

75

76

2 Microstructural Considerations

Research in Stuttgart. The model assumes negligible solute diffusion in the solid phase, complete diffusion in the liquid phase, and equilibrium at the solid–liquid interface. In this nonequilibrium solidification, Eq. (2.6) is invalid, and the following relation for the composition of the solid during derived by Scheil and as early as 1913 by G.H. Gulliver (1913) holds Cs = kC0 1 − f s

k−1

28

When the solid–liquid interface moves too fast to maintain equilibrium, it results in a chemical composition gradient within each grain, a condition known as coring. This is illustrated in Figure 2.10b. Without solid-state diffusion of the solute atoms in the material solidified at T1 into the layers subsequently freezing out at T2, the average composition of the crystals does not follow the solidus line, from α1 to α4, but rather follows the line α1 to α5, which is shifted to the left of the equilibrium solidus line. The faster the cooling rate, the greater will be the magnitude of the shift. Note also that final freezing does not occur until a lower temperature, T5 in Figure 2.10b, so that nonequilibrium solidification happens over a greater temperature range than equilibrium solidification. As the time scale is too short for solid-state diffusion to homogenize the grains, their centers are enriched in the higher freezing component, while the lowest freezing material gets segregated to the edges (recall how grain boundaries are formed from the last liquid to solidify). Grain boundary melting, liquation, can occur when subsequently heating such an alloy to temperatures below the equilibrium solidus line, which can have devastating consequences for metals used in structural applications.

2.3.1.2 Grain Morphology

In addition to controlling the compositional profile of the grains, the solidification velocity also determines the shape of the solidification front (the solid–melt interface), which, in turn, determines grain shape. The resulting structure arises from the competition between two effects. Undercooling of the liquid adjacent to the interface favors protrusions of the growing solid along energetically favorable crystallographic directions, giving rise to dendrites with a characteristic treelike shape. Simultaneously, surface tension tends to restore the minimum surface configuration – a planar interface. Consider the case of a molten pure metal cooling to its freezing point. When the temperature gradient across the interface is positive (the solid is below the freezing temperature, the interface is at the freezing temperature, and the liquid is above the freezing temperature), a planar solidification front is most stable. However, with only a very small number of impurities present in a pure melt on which nuclei can form, the bulk liquid becomes kinetically undercooled. Diffusion of the latent heat away from the solid–liquid interface, via the liquid phase, favors the formation of protrusions of the growing solid into the undercooled liquid (the undercooled liquid is a very effective medium for heat conduction). Ivantsov first mathematically modeled this for paraboloidal dendrites over half a century ago (Ivantsov 1947). It is now known that this is true so long as the solidification velocity is not too fast. At the high velocities observed in some rapid quenching processes (e.g. >10 m/s), dendritic growth becomes unstable, as the perturbation wavelengths become small enough that surface tension can act to restore planarity (Mullins and Sekerka 1963; Langer 1989; Hoglund et al. 1998). Owing to the small number of impurities in a pure metal, the undercooling can be quite large. There aren’t many nuclei for dendrites to form on. For those dendrites that do form, growth is a function of the rate of latent heat removal from the interface. Hence, for a pure metal, one would expect a small number of large dendrites. The shape of the dendrite is such that it maximizes its

2.3 Materials Processing and Microstructure

surface area for dissipating the latent heat to the undercooled liquid. The shape of the dendrite is also governed by the solidification rate; a rapid solidification time will yield dendrites with tightly packed secondary branches. However, when attention is turned to alloys, a slightly different situation can be seen. In alloys, it is generally expected that a larger number of smaller dendrites will be found than would be in a pure metal. Here, in addition to heat flow, mass transport must be considered as well. In fact, the planar interface-destabilizing event primarily responsible for dendritic morphology in conventional alloy casting is termed constitutional undercooling, to distinguish it from kinetic undercooling. The kinetic undercooling contribution can still be significant in some cases. In most models for two-component melts, it is assumed that the solid–liquid interface is in local equilibrium, even under nonequilibrium solidification conditions based on the concept that interfaces will equilibrate much more rapidly than bulk phases. Solute atoms thus partition into a liquid boundary layer a few micrometers thick adjacent to the interface, slightly depressing the freezing point in that region. As in the case for pure meals, the positive temperature gradient criterion for planar interface stability still holds. However, although the bulk liquid is above the freezing point, once the boundary layer becomes undercooled, there is a large driving force for solidification ahead of the interface into the thin boundary layer. The critical growth velocity, v, above which the planar interface in a two-component melt becomes unstable, is related to the undercooling, ΔTc, by an equation given by Tiller as GL ΔT c ≥ v DL

29

where ΔTc = mLC0(1 − k)/k and GL is the thermal gradient in the liquid ahead of the interface, v is the solidification speed, mL is the liquidus slope, C0 is the initial liquid composition, k is the partition coefficient (previously defined), and DL is the solute diffusivity in the liquid (Tiller et al. 1953). Constitutional undercooling is difficult to avoid except with very slow growth rates. With moderate undercooling, a cellular structure, resembling arrays of parallel prisms, results. As the undercooling grows stronger, the interface breaks down completely as anisotropies in the surface tension, and crystal structure leads to side branches at the growing tip of the cells along the easy-growth directions ( 1 0 0 for FCC and BCC, 1 0 1 0 for HCP), marking a transition from cellular to dendritic. In a polycrystalline substance, the dendrites grow until they impinge on one another. The resulting microstructure of the solidification product may not reveal the original dendritic growth upon a simple visual examination. Nevertheless, the dendritic pattern strongly influences the material’s mechanical, physical, and chemical properties. Over the last 50 years, a large amount of work has gone into obtaining accurate mathematical descriptions of dendrite morphologies as functions of the solidification and materials parameters. Dendritic growth is well understood at a basic level; however, most solidification models fail to accurately predict exact dendrite morphology without considering effects like melt flow. In the presence of gravity, density gradients owing to solute partitioning produce a convective stirring in the lower undercooling range corresponding to typical conditions encountered in the solidification of industrial alloys (Huang and Glicksman 1981). Melt flow is a very effective heat transport mechanism during dendritic growth that may result in variations in the dendrite morphology, as well as spatially varying composition (termed macrosegregation). The above discussion of solidification applies to pure substances and single-phase solid solutions (alloys). Although this book is concerned primarily with single-phase materials, it would now be beneficial to briefly describe the microstructures of a special type of multiphase alloy known as a eutectic. Eutectics are generally fine grained and uniformly dispersed. Like a pure single phase, a

77

78

2 Microstructural Considerations

eutectic composition melts sharply at a constant temperature to form a liquid of the same composition. The simplest type of eutectic is the binary eutectic system containing no solid-phase miscibility and complete liquid-phase (melt) miscibility between the two components (e.g. the Ag–Si system). On cooling, both components form nuclei and solidify simultaneously at the eutectic composition as two separate pure phases. This is termed coupled growth, and it leads to a periodic concentration profile in the liquid close to the interface that decays, in the direction perpendicular to the interface, much faster than in single-phase solidification. There are several methods for solidifying eutectics at conventional cooling rates, including the laser floating-zone method, the edge-defined film-fed growth technique, the Bridgman method, and the micro-pulling-down method. Generally, high volume fractions of both phases will tend to promote lamellar structures. If one phase is present in a small volume fraction, that phase tends to solidify as fibers. However, some eutectic growths show no regularity in the distribution of the phases. Eutectic microstructures normally exhibit small interphase spacing, and the phases tend to grow with distinctly shaped particles of one phase in a matrix of the other phase. The microstructure will be affected by the cooling rate; it is possible for a eutectic alloy to contain some dendritic morphology, especially if it is cooled relatively rapidly. The microstructures of hypo- or hypereutectic compositions normally consist of large particles of the primary phase (the component that begins to freeze first) surrounded by a fine eutectic structure. Often times the primary particles will show a dendritic origin, but they can transform into idiomorphic grains (having their own characteristic shape) reflecting the phases’ crystal structure (Baker 1992). Metal–metal eutectics have been studied for many years as a result of their excellent mechanical properties. Recently, oxide–oxide eutectics were identified as materials with potential use in photonic crystals. For example, rodlike micrometer-scaled microstructures of terbium–scandium– aluminum garnet: terbium–scandium perovskite eutectics have been solidified by the micro-pulling-down method (Pawlak et al. 2006). If the phases are etched away, a pseudohexagonally packed dielectric periodic array of pillars or periodic array of pseudohexagonally packed holes in the dielectric materials is left. 2.3.1.3 Zone Melting Techniques

Once an ingot is produced, it can be purified by any of a group of techniques known as zone melting. The basis of these methods is founded on the tendency of impurities to concentrate in the liquid phase. Hence, if a zone of the ingot is melted and this liquified zone is then made to slowly traverse the length of the ingot by moving either the heating system or the ingot itself, the impurities are carried in the liquid phase to one end of the ingot. In the semiconductor industry, this technique is known as the float zone process. A rod of polycrystalline material (e.g. silicon) is held vertically inside a furnace by clamps at each end. On one end of the rod, a short single-crystal seed is placed. As the rod is rotated, a narrow region, partially in the seed and partially in the rod, is melted initially. The molten zone is freely suspended and touched only by the ambient gas in order to avoid contact with any container material. It is then moved slowly over the length of the rod by moving the induction coils, thus the name floating zone. As the floating zone is moved to the opposite end of the rod, a high purity single crystal is obtained in which the impurities are concentrated in one end. The impurities travel with, or against, the direction of the floating zone, depending on whether they lower or raise the melting point of the rod, respectively. Typically, growth rates are of the order of millimeters per hour. The float zone technique has been used to grow crystals of several oxides, as well. To control the escape of volatile constituents, encapsulants, with a slightly lower melting point than the crystal, are inserted between the feed rod and container wall so that the float zone is concentrically surrounded by an immiscible liquid.

2.3 Materials Processing and Microstructure

In the related zone refining technique, a solid is refined by multiple floating zones, or multiple passes, as opposed to a single pass, in a single direction. Each zone carries a fraction of the impurities to the end of the solid charge, thereby purifying the remainder. Zone refining was first devised by the American metallurgist William Gardner Pfann (1917–1982) at Bell Labs in 1939 to grow lead single crystals, when he was given permission by his boss to do whatever he wanted half his time. He reused his idea in 1952 to purify germanium to the parts-per-billion impurity level for producing transistors (Pfann 1952).

2.3.2 Deformation Processing The most common shapes into which ductile metals and alloys are cast are cylindrical billets and rectangular slabs. Such castings are termed ingots, and they serve as the starting materials for subsequent deformation processing. For example, an ingot can be rolled to make ribbon or drawn/ extruded to make wire. These subsequent processes rely on plastically deforming the ingot through the application of compressive and shear forces and are thus termed deformation processes. In addition to rolling, drawing, and extrusion, deformation processes include forging (hammering), swaging, stamping, blanking, and coining. Development of texture during the deformation process is difficult to avoid. Often, one wishes to develop a strong texture, but in other cases, a weak or random texture is desired. Texture tends to become more pronounced as the degree of deformation increases, and sometimes more than one texture can coexist in a component. The texture produced by different deformation processes will depend somewhat on the material structure, particularly, the number and symmetry of the primary slip systems. In plastic deformation, twinning may compete with slip when the latter is limited to deformation in a specific direction (e.g. with HCP metals) and can thus play an essential role in determining the texture. By virtue of twinning, even small deformation rates lead to large lattice rotations, which change the orientation of the grains. With rolling, FCC metals with low stacking faults exhibit the so-called brass-type texture, with high fraction {1 1 0} 1 1 2 (brass) and {1 1 0} 0 0 1 (Goss) texture components, while metals with medium and high stacking faults exhibit the so-called copper-type texture with high fraction of {1 2 3} 6 3 4 (S) texture component. In a {} texture component, a {} plane are in the rolling plane, and a direction is parallel to the rolling direction (RD). Goss is after the name of the American metallurgist N.P. Goss who patented a process for cold-rolling silicon steel sheets in 1934, and it also corresponds to grain orientation silicon steel, in which the Goss texture is realized to minimize magnetic losses in electrical transformers. In BCC metals, the predominant texture is {1 0 0} 1 1 0 with the cube plane in the rolling plane. Other textures, such as {1 1 2} 1 1 0 and {1 1 1} 1 1 2 , can also form. With HCP metals, the rolling texture tends to take the form {0 0 0 1} 1 1 2 0 . In wire drawing, FCC metals tend to exhibit a 1 1 1 or 1 0 0 fiber texture, while BCC metals exhibit a 1 1 0 drawing texture and HCP metals a 1 0 1 0 texture. The plastic deformation exhibited by nonductile crystalline ceramics is not significant enough for deformation processes to be of much use in the fabrication of bulk articles. Noncrystalline materials (e.g. glasses) deform by the same mechanism as liquids, viscous flow. However, in a glass, the effect is only pronounced at temperatures elevated enough to decrease the viscosity.

2.3.3 Consolidation Processing Solidification and deformation processes are very seldom used to fabricate bulk articles from ceramics and other materials with low ductility and malleability. These substances are brittle and suffer fracture before the onset of plastic deformation. Additionally, ceramics normally have exceedingly

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high melting points, decompose, or react with most crucible materials at their melting temperatures. Many ceramics are worked with in powder form since the products of most solid-state chemical syntheses are powders. Fabricating a bulk part from a powder requires a consolidation process, usually compaction followed by sintering. Compaction can be of various forms, including die pressing, compression molding, injection molding, and slip casting. Within the compacted product, which is termed a green body, the individual particles adhere together only loosely with the result that total porosity is often a large volume percent. To develop the desired properties in the fabricated article, densification must be achieved. This is accomplished through what is called sintering. Sintering refers to the heating of a polycrystalline aggregate at a temperature below its solidus (melting point) but high enough that grain coalescence occurs via solid-state diffusion. In the early stages of sintering, volatile species can be removed (e.g. moisture content), and at later stages, homogenization improved (e.g. the removal of macrosegregation), and densification increased as the particles bond together (decreasing surface area) and eliminated pores (solid–vapor interfaces). Sintering is thus a thermodynamically irreversible process whose driving force is the reduction of the total free energy of the system, which arises during the densification stage. In solid-state sintering, diffusion is the dominant materials transport mechanism. Diffusion is a thermally activated process whose rate is usually significant only when the homologous temperature (the fraction of a material’s melting point) is in excess of 0.5Tm. The products resulting from consolidation have higher strength and reliability. Consolidation processing is also used for refractory metals/alloys, and compounds formed between different metallic elements known as intermetallic phases (see Section 3.4.2) that do not exhibit much ductility. In fact, consolidation is even used to fabricate ductile materials (cf. powder metallurgy) more easily into complex shapes than would be required via deformation techniques. Mechanical forces act to orient particles during consolidation. However, electrical, magnetic, and thermal gradients can be applied to induce rotation of anisotropic particles, inducing texture to a fabricated article. This tends to work better for larger particles, but such materials are more difficult to densify. It has been found that texture can be enhanced through templated grain growth (TGG), in which seed particles, initially randomly oriented, develop a preferred orientation during processing and grow in this direction during sintering.

2.3.4

Thin-Film Formation

The cornerstone of the entire semiconductor microelectronics industry is thin-film technology. There are many methods available for growing thin films, including physical vapor deposition (PVD) (e.g. sputtering, evaporation, laser ablation), chemical vapor deposition (CVD), plating, and so-called soft chemical techniques (e.g. sol–gel coating). Most of these processes are well established in the microelectronics industry, but many also have important applications in the areas of advanced coatings and structural materials. An enormous variety of films can be prepared. The interested reader is encouraged to refer to any of numerous texts on thin films for details regarding the deposition processes. An exciting new area of materials research that has begun to evolve in recent years is the application of combinatorial chemistry to the creation of thin-film libraries. By using masks (grids with separate squares), thousands of distinct combinations of materials can, in principle, be deposited onto a single substrate in order to greatly accelerate the screening of the resultant compounds for certain properties. This is part of a broad approach being sought for the rapid discovery of new materials known as combinatorial materials science, inspired by the success of combinatorial chemistry in revolutionizing the pharmaceutical industry.

2.3 Materials Processing and Microstructure

2.3.4.1

Epitaxy

Thin films can be polycrystalline or single crystalline. The majority of thin films are polycrystalline. Single-crystal films can be grown with a particular crystallographic orientation, relative to a singlecrystal substrate that has a similar crystal structure, if the deposition conditions and substrate are very stringently selected. Epitaxy is a word coined by the French mineralogist Louis Royer during his PhD work under Georges Friedel on the mutual orientations of two adjacent crystals of different structures. It means “arrangement,” (taxis) upon (epi). Homoepitaxy refers to the growth of a thin film of the same material as the substrate (e.g. silicon on silicon). In this case, the crystallographic orientation of the film will be identical to that of the substrate. In heteroepitaxy, such as silicon on sapphire, the layer’s orientation may be different from that of the substrate. Both types of epitaxy often proceed by an island nucleation and growth mechanism referred to as Volmer–Weber growth. Island growth also occurs with polycrystalline films, but in epitaxy, the islands combine to form a continuous single-crystal film, that is, one with no grain boundaries. In reality, nucleation is much more complex in the case of heteroepitaxy. Nucleation errors may result in relatively large areas, or domains, with different crystallographic orientations. The interfaces between domains are regions of structural mismatch called subgrain boundaries and will be visible in the microstructure. Epitaxy has been used to stabilize films with crystal structures that are metastable in the bulk phases. Kinetic stabilization is obtained when the growth is performed under conditions of high surface diffusion, but low bulk diffusion. In this way, crystallographically oriented film growth occurs while phase transformations are prohibited. The circumstances under which thermodynamic stabilization can be achieved have also been enumerated. Namely, these are by minimizing the following: 1) The lattice mismatch, or structural incoherency, and the free energy difference between the growing film and the substrate. 2) The film thickness. 3) The shear and elastic moduli of the film. Additionally, the growing film should be able to form a periodic multiple-domain structure (Gorbenko et al. 2002). The high-temperature bulk FCC phase δ-Bi2O3 is observed on heating α-Bi2O3 (monoclinic structure) above 717 C, but heteroepitaxial thin films of δ-Bi2O3 have been deposited on an FCC gold substrate at temperatures as low as 65 C. Similarly, YMnO3 crystallizes in the hexagonal structure under atmospheric pressure at high temperatures, yet was grown by pulsed laser deposition (PLD) as a stable perovskite film on an NdGaO3 substrate (Salvador et al. 1998). It has also recently been shown how metal organic chemical vapor deposition (MOCVD) results in the formation of metastable phases of GaS and GaTe, irrespective of the structure of the substrate (Gillan and Barron 1997; Keys et al. 1999). Crystalline GaS was even grown on an amorphous substrate. It appears in cases where the precursor’s structure, or its decomposition mechanism, completely controls the structure of the thin film. Likewise, polycrystalline CVD diamond is commercially grown on a variety of substrates, including titanium, tungsten, molybdenum, SiO2, and Si3N4. Each of these materials is capable of forming a carbide layer upon which an adherent diamond film can nucleate, although it is not clear whether carbide formation is essential. 2.3.4.2

Polycrystalline PVD Thin Films

The commonest morphology exhibited by PVD thin films is the highly oriented columnar (fiber) grain growth. This is a consequence of the fact that PVD processes generally deposit thin films,

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2 Microstructural Considerations 1 μm

1 μm

Figure 2.11 A slanted morphology on a surface resulting from glancing angle deposition (left). A helical morphology resulting from glancing angle deposition (right). (Source: Courtesy of M.J. Brett. © 2006, M.J. Brett, University of Alberta. Reproduced with permission.)

atom by atom, in a line-of-site fashion, wherein the sputtered or evaporant adatoms travel from the source to substrate on a straight path. The resulting grains are thus aligned with their long axes perpendicular to the surface when the incident beam arrives at a normal angle of incidence. If the depositing atoms arrive at an angle away from normal incidence, the columns tilt into the oncoming beam, as illustrated in Figure 2.11. This gives rise to shadowing effects, which result in the columns being separated from one another by voids. This is actually an alternative approach, termed glancing angle deposition (GLAD), which is designed specifically to avoid the perpendicular geometry. It can lead to interesting surface morphologies. In either case, the grains can completely traverse the thickness of the film. Sputtered films generally tend to be denser, more amorphous in nature, and more adherent than evaporated films because of the higher energy of the arriving adatoms. Slight variations can occur with columnar morphology, however. For example, Movchan and Thornton used structure zone models to illustrate how temperature influences the morphology of metal films (Movchan and Demchishin 1969; Thornton 1977). Grain growth often begins with island formation (nucleation sites). When the homologous temperature, Th (for thin films this is the ratio of the substrate temperature to the melting point of the thin film), is 0.5, bulk diffusion increases substantially and allows for equiaxed grains. A major disadvantage of PVD thin films is its inability for conformal coating, that is, it produces poor step coverage. Residual stresses in PVD thin films are generally compressive. As with solidification microstructures, kinetic factors are as important as thermodynamics. For example, the (2 0 0) surface has the lowest energy in TiN. But the preferred orientation of TiN thin films can vary between (1 1 1) and (2 0 0), depending on deposition conditions. This means that the texture depends on kinetic factors as well as the energy minimization.

2.4 Microstructure and Materials Properties

2.3.4.3

Polycrystalline CVD Thin Films

CVD is a process in which a volatile species is transported in the vapor phase to the substrate to be coated. Typically, a chemical reduction then occurs at the substrate to deposit the desired film. For example, volatile ReCl5 will deposit rhenium on a substrate heated to 1200 C. Where adatom energy exerts the strongest influence on film morphology in PVD processes, CVD is dominated by chemical reaction kinetics. The islands grown in CVD processes tend to be larger in size and fewer in number. There also appears to be a marked temperature dependence to the growth rate. Kinetic Monte Carlo simulations have shown that grain morphology in CVD thin films is primarily owing to an autocatalytic process, in which precursor molecules dissociate preferentially at existing nuclei sites (Mayer et al. 1994). Another difference between PVD and CVD films is that, in PVD, the residual stress in the deposited film is normally compressive, whereas in CVD, it is generally tensile. Nevertheless, as with PVD, the most common microstructure produced from CVD is the highly oriented columnar structure.

2.4

Microstructure and Materials Properties

In response to various different kinds of driving forces (e.g. electric fields, thermal gradients, and chemical potential gradients), a flux may be generated in a material. A flux can involve the transport of electrons, phonons, ions, atoms, or dislocations. For anisotropic crystalline lattices, there may be an easy axis or direction along which migration can occur. Flux particles are scattered at defects such as impurities and, in the case of polycrystals, at grain boundaries. The scattering may be specular, in which case the particle momentum parallel to the surface is maintained, or it may be diffuse, meaning that the particles travel in a random direction after collision. Hence, momentum transfer across a grain boundary will be affected by the orientation relationship between a pair of adjoining grains. Obviously, the long-ranged orientation distribution (texture) across the sample will affect the ability of species to move through the material. As a crude analogy, we may think of a network (the microstructure) of elbow-shaped pipes, of various lengths (grain sizes). The volume of fluid transported through the network can only reach a maximum with an interconnectivity (texture) in which each pipe is rotated such that its two ends are connected to the ends of two other pipes.

2.4.1 Mechanical Properties Materials processing is generally aimed at achieving a microstructure that will maximize a specific property of interest. Approaches with polycrystalline metals and alloys usually involve controlled solidification, deformation processing, and annealing. The microstructures of all metals and alloys greatly influence strength, hardness, malleability, and ductility of the finished products. These global properties describe a body’s plasticity, or ability to withstand permanent deformation without rupture. Plastic deformation is owing to the gliding motion, or slip, of planes of atoms. Slip most readily occurs on close-packed planes of high atomic density in the close-packed directions. However, the stress required to move a dislocation (dislocations were introduced in Section 2.2.2) through a crystal is much smaller than that required to move a full plane of atoms. Therefore, dislocations actually govern the ability of a coarse-grained material to plastically deform. This will be discussed in much more detail in Chapter 10. Polycrystalline metals are stronger than single crystals. It is not easy for dislocations to move across grain boundaries because of changes in the direction of the slip planes. Generally, a smaller average grain size leads to a larger volume fraction of grain boundary, which, in turn, leads to

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greater resistance to dislocation motion. Indeed, one possible mechanism for strengthening a polycrystalline metal is via grain size reduction. Another method is known as solid solution strengthening where a small number of atoms in the lattice are replaced with substitutional impurities of a slightly different size. This creates strain in the crystal, inhibiting the movement of dislocations, and strengthening the material (Section 10.3.2). Dislocation density also affects the strength of polycrystalline metals. Deformation processes, such as rolling and extrusion, increase the dislocation density in a metal, which hinders dislocation motion and strengthens the material. This is known as strain hardening or work hardening. As the temperature at which deformation takes place is well below the recrystallization temperature, it is also called cold working. While there is no set rule as to what constitutes “cold” when deforming a sample, most materials will be considered to be cold worked if deformed at temperatures significantly below half their melting point on an absolute scale, often referred to as a homologous temperature of 0.5, or 0.5Tm in shorthand. Some metals (e.g. Pb and Sn) have recrystallization temperatures significantly lower than their melting point (Pb: Tm = 327 C, Tr = 25 C; Sn: Tm = 232 C, Tr = 25 C) and cannot be cold worked at room temperature. An example of increased strength and decreased ductility in cold worked materials is familiar to anyone who has ever tried to reconfigure a wire coat hanger to perform some task. While it is relatively easy to bend the straight sections of the coat hanger into a new shape, it is extremely difficult to straighten out the tight radius (i.e. highly deformed) bends that form the coat hanger’s original shape. This is work hardening in action. The steel alloy used to produce the coat hanger was chosen for its ability to work harden so that the coat hanger would maintain its shape under load, for example, the weight of a coat or pair of pants (Lalena et al. 2008). During cold working, polycrystalline metals deform by mechanisms involving slip and, where slip is restricted, rotation of the individual grains. Both processes, of course, must satisfy the condition that the interfaces, along which the grains are connected, remain intact during deformation. As the extent of deformation increases, some of the larger grains may break up into smaller grains of different orientations, giving rise to a wider orientation spread. It has been seen that dislocation density also increases with cold working. Conversely, plastically deforming materials above their recrystallization temperature, referred to as hot working, results in a lower free energy owing to the rearrangement of dislocations into lower energy configurations of decreased dislocation density. This is termed recovery. However, competition with this is recrystallization, in which the resulting grain structure and texture depends on the spatial distribution and orientation of recrystallization nuclei. Grains grow at different rates that are determined by their own morphological properties (orientation, shape) and their environment, which leads to a distribution of recrystallization grain growth rates. The recrystallization temperature is defined as that temperature at which recrystallization just reaches completion in one hour. For most pure metals, the recrystallization temperature is typically about one-third of the melting temperature. It is generally accepted that recrystallization proceeds more rapidly in pure metals than it does in alloys, since alloying raises the recrystallization temperature. Both recovery and recrystallization are driven by the difference between the higher-energy strained state and the lower-energy unstrained state. That is, the process of forming a new grain structure is driven by the stored energy of deformation. However, the competition between the two processes, as well as their effects on texture and grain boundary structure, is hotly debated in the fields of materials science and metallurgy. Prior to the twentieth century, steels were used exclusively for most applications calling for wear resistance, as well as thermal resistance and/or corrosion resistance. Increasingly demanding materials requirements for more extreme service conditions eventually necessitated the development of what are now referred to as “superalloys.” These alloys have been and continue to be used in

2.4 Microstructure and Materials Properties

turbine blades/vanes in the combustion zones of jet aircraft engines, heat shields in combustion chambers, furnace fixtures/components, valves and valve seats in internal combustion engines, oil well drilling bits, biomedical implants, and components of high temperature gas-cooled nuclear reactors. The first such application for a superalloy came in 1939, for Rex-78, a modified austenite stainless steel invented by Firth-Vickers Research Laboratories. This alloy was used in the turbine blades and nozzle blades in the Whittle W.I (1939) and W.I.X (1941) gas turbine engine (made by Power Jets, Ltd., where Frank Whittle was chief engineer) because it was capable of withstanding temperatures as high as 780 C, 100 higher than the other alloys being tested. However, well before the introduction of this alloy to the market was the development of some cobalt-based superalloys (Stellite) by Elwood Hayes, beginning in the early 1900s. At first, these superalloys were used in cutlery, lathes, and other machine/cutting tools owing to their superior wear-resistant and corrosion-resistant properties. Eventually, Stellite 8, a carbide dispersion strengthened Co–Cr–Mo–Ni alloy, became one of the first nonferrous superalloys targeted and selected for use as a turbine blade material. Owing to its corrosion resistance, this alloy was originally intended as a dental and surgical implant, an application for which its variants still find use. Superalloys calling for only modest strength may be solid-solution strengthened with Co, Cr, Mo, and/or W. In the field of metallurgy, examples abound of solid-solution strengthening, and this topic will be discussed further in Chapter 10. Higher strength superalloys are obtained by carbide dispersion strengthening or precipitation strengthening. For example, some nickel-based superalloys used in static and rotating components in land-based gas turbine engines and jet engines today are precipitation strengthened with the intermetallic phase Ni3Al. This phase is one of two aluminides of significance in the Ni–Al system, the other being NiAl. Both are intermediate phases with slightly variable compositions. Note that Ni3Al must not be confused with NiAl3, a true line compound with a fixed composition. Both Ni3Al and NiAl have also received considerable attention as potential structural alloys because of their low density, high melting temperature, good thermal conductivity, and excellent oxidation resistance. Unfortunately, as with most intermetallics possessing a limited number of slip systems and/or low grain boundary cohesion (grain boundaries are inherently weaker than intragranular regions), the nickel aluminides have inadequate ductility at ambient temperature and lack good high-temperature creep resistance. Another family of alloys, worthy of mention here, is the tungsten heavy alloys (WHAs). These alloys normally contain 90–98 wt% tungsten with the remainder being iron and nickel (e.g. 95W4Ni-1Fe). Copper may be used in place of iron if one wishes to minimize the electrical and magnetic properties of the alloy. The W-5Ni-5Cu alloy was first produced by powder metallurgical methods in 1935 by Canadian physicist John Cunningham McLennan (1867–1935) and metallurgist Colin James Smithells (d. 1977). For enhanced mechanical properties, cobalt may be substituted for iron. WHAs are characterized by high density (17–18.8 g/cm3), hardness, reasonable formability (WHAs are strain rate sensitive; ductility decreases with increasing strain rate), corrosion resistance, elasticity, high radiation absorptivity, high thermal conductivity, and low thermal expansion. Such properties make them ideally suited for applications like damping/balancing weights, kinetic energy penetrators for piercing heavy armor, radiation shielding, and gyroscope components. They have been used in some of these applications for at least 20 years. It is known that these alloys are inferior to uranium-235 depleted alloys (a.k.a. DU, D-38, depletalloy) in their mechanical properties. For example, the density of DU is 19.1 g/cm3. However, due to safety and environmental issues associated with the generation and storage of DU, WHA’s are preferentially used. The German metallurgist Rudolf B. Irmann reported in 1915 that alloys with near 50% W were formed by heating nickel and tungsten to 1800 C in an electric furnace. He found the tensile strength of these Ni–W alloys to decrease rapidly with increases in tungsten content, reaching a

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minimum around 25% tungsten, but then rising rapidly thereafter. High-tungsten Ni–W binary alloys consist of two phases, pure body-centered cubic (BCC) tungsten, since W has a negligible Ni solubility, and the stoichiometric face-centered cubic (FCC) intermetallic line compound NiW2. So is for high-tungsten Fe–W alloys, which consist of W and FCC FeW. Thus, the WHAs have microstructures containing BCC tungsten particles dispersed in an FCC matrix. However, because of the very high melting point of tungsten, powder metallurgical methods are used in place of fusion/solidification to prepare WHAs. The component metals are intimately ground together, compressed by cold isostatic pressing in a vacuum, and liquid-phase sintered above 1495 C (the melting point of nickel) and then cooled, metallurgically binding the particles together. Some purists would argue, then, that WHA’s are not “true” alloys, preferring to use the term metal matrix composite for pressed/sintered powder mixtures. There are many other materials known for their ability to withstand extreme service conditions, and some of these will be discussed further in Section 2.4.4. Of particular interest for their potential use as structural materials in aircraft engines and land-based gas turbine engines are some refractory multiphase metal-silicide alloys and silicon nitride-based ceramics. In comparison with the nickelbased superalloys currently used in aeronautical turbines, both of these two other classes of materials can withstand even higher temperatures and some have lower densities. Alloys in the Nb–Si system exhibit good creep resistance, but poor toughness and oxidation resistance. To improve their properties, the niobium silicides may be alloyed with titanium, hafnium, chromium, and aluminum. With high chromium content, the alloy contains the NbCr2 phase. This compound belongs to a group of intermetallics with AB2 stoichiometry known as the Laves phases, named after the German mineralogist and crystallographer Fritz Henning Laves (1906–1978). The Nb–Si multiphase alloys can be produced by techniques such as vacuum arc melting, investment casting, and Czochralski drawing for directional solidification. Unfortunately, ingots are substantially heterogeneous and tend to contain pores, micro-cracks, and other metallurgical defects. Therefore, powder metallurgical methods like hot pressing are under investigation as a possible means of better controlling the microstructure. Powder metallurgy techniques are generally capable of producing alloys that are suitable for higher temperature and strength applications than those produced by cast-and-wrought methods. The mechanical properties of most intermetallics and ceramics differ greatly from those of metals. Ceramics and intermetallics are typically brittle, very strong, hard, and resistant to deformation. It is found that dislocation motion is virtually impossible except at high temperatures. In ionic solids, slip is constrained because it requires bringing ions with like charges in contact. Furthermore, although many ionic solids are almost close packed, slip is very difficult on anything but truly close-packed planes. The strong and directional bonding present in covalent solids similarly impedes dislocation motion since it requires bond breaking and distortion. As ceramics are too brittle for plastic deformation, they do not experience recrystallization, since a prerequisite for recrystallization is that the material be plastically deformed first (Callister 2005). The absence of a lattice-based mechanism, such as slip planes, does not necessarily preclude all deformation in brittle materials. Plastic flow can proceed in other modes. For example, at temperatures of about 40–50% of their melting points, grain boundary sliding can become important. Grain boundary sliding is believed to be the major contributor to the superplasticity observed in some polycrystalline ceramics.

2.4.2

Transport Properties

In addition to mechanical properties, other physical properties of polycrystalline materials, such as electrical and thermal conduction, are also affected by microstructure. Although polycrystals are

2.4 Microstructure and Materials Properties

mechanically superior to single crystals, they have inferior transport properties. Point defects (vacancies, impurities) and extended defects (grain boundaries) scatter electrons and phonons, shortening their mean free paths. Owing to the phonon and electron scattering at grain boundaries, a polycrystal has a lower thermal and electronic conductivity than a single crystal. An approximation known as Matthiessen’s rule, from the nineteenth century University of Rostock physicist professor Heinrich Friedrich Ludwig Matthiessen (1830–1906), asserts that all the various electron scattering processes are independent. Matthiessen was the first to point out that, as the temperature decreases, a metal’s resistivity decreases and reaches a constant value with further decreases in temperature. Although Matthiessen’s rule is strictly true only so long as the electron scattering processes are isotropic, it is nevertheless a useful approximation to assume the contributions to the resistivity owing to conduction electron scattering from thermal vibrations, impurities, and dislocations (which increase in number with plastic deformation), can simply be added up. The electrical resistivity of a polycrystalline metal can similarly be considered the sum of the contributions owing to electron scattering in the bulk crystal and at the interfaces, the latter of which is a function of the grain boundary network. Thus, the resistivity of a finegrained metal is higher than that of a coarse-grained sample because the former has a larger volume fraction of grain boundaries. There are some exceptions found to this rule with mixed ionicelectronic conducting metal oxides, where enhanced chemical reduction along the grain boundaries results in a concomitant accumulation/depletion of electrons/mobile ions giving rise to an overall electrical resistivity that increases with decreasing grain size. This will be discussed in Section 6.4.4. Transport properties (electrical and thermal conductivity, mass transport) are second-rank tensors, and they are isotropic only for cubic crystals and polycrystalline aggregates with a completely random crystallite orientation. For noncubic monocrystals and textured polycrystals, the conductivity will be dependent on direction. Therefore, it is also important to understand the effect of grain orientation/interconnectivity on transport properties. Some ceramics are low-dimensional transport systems, in which intragranular conduction is much weaker or completely absent along one or more of the principal crystallographic axes. If the transport properties of the individual grains or crystallites in a polycrystalline aggregate are anisotropic in this manner, sample texture will influence the anisotropy to the conductivity of the polycrystal. When a polycrystal is free of preferred orientation, that is, when it has a completely random or perfectly disordered grain or crystallite orientation distribution, its bulk transport properties will be like that of a cubic single crystal, isotropic as a whole regardless of the anisotropy to the single grains/crystallites comprising it. If anisotropic grains are randomly oriented, the macroscopic sample loses any anisotropy in any property. Consider the low-dimensional metal Sr2RuO4, with the tetragonal crystal structure shown in Figure 2.12a. At high temperatures, metallic conduction occurs in the ab planes within the perovskite layers, parallel to the two principal axes 0 1 0 and 1 0 0 . Transport along the c axis, perpendicular to the ab plane, is semiconducting at high temperatures. For polycrystals, two extreme ODFs, which give the volume fraction f(x) of crystals in orientation x, can be envisioned. These are as follows: 1) Zero preferred orientation 2) Perfect alignment The absence of preferred orientation statistically averages out the parallel and perpendicular conduction effects in Sr2RuO4. The bidimensional metallic conductivity, which is clearly observed in a single crystal, is completely inhibited in free powders composed of millions or billions of randomly

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(a)

(b)

(c)

[0 0 1] [1 0 0]

[0 1 0]

Perovskite-like slab

Perovskite-like slab

Tm

σc σ Perovskite-like slab

c b

σab

(d)

a 0 = Oxygen

= Ruthenuim

T(K)

300

= Strontium

Figure 2.12 (a) The crystal structure of Sr2RuO4. (b) Below Tm, transport along the c axis is semiconducting, while transport along the ab axis is metallic across the entire temperature range. (c) Random orientation in a powder aggregate with different crystallite sizes. Percolation paths exist for both σ ab and σ c in all directions, resulting in an isotropic conductivity at any given temperature. (d) If identical crystallites are all aligned with their c axes parallel, the polycrystal will have the same anisotropic conductivity as a single crystal.

oriented crystallites. This is shown in Figure 2.12c. By contrast, it can be expected that the application of several thousand pounds of uniaxial pressure in forming a pellet (i.e. a cylindrical shaped specimen) will induce a high degree of texture. This is especially true for elongated platelet crystals, where the c axes of the crystallites align parallel to the pressure axis. It cannot be assumed that all the crystallites are identical in size and shape. Thus, with the application of uniaxial pressure alone, many crystallites will remain misaligned. Higher degrees of texture may sometimes be induced by magnetic fields because of the anisotropy to the magnetic susceptibility. If the Sr2RuO4 crystallites were all to be identical and perfectly aligned with their c axes parallel (somewhat like stacking children’s building blocks), the pellet’s resistivity would exhibit the same anisotropy as a tetragonal single crystal of Sr2RuO4. The resistivity along the ab axis of the pellet would follow the ρab curve in Figure 2.12b, while along the c axis the resistivity would follow the ρc curve. This is illustrated in Figure 2.12d. Nonetheless, the current density of the pellet would be lower than that of a single crystal of the same dimensions owing to the grain boundary resistance. Further enlightenment can be gained on electrical conduction in polycrystals by likening the phenomenon to a random resistor network with the bond percolation model. In this model, there is a lattice in which each mesh point corresponds to an individual crystallite of the polycrystalline aggregate. The simplifying assumption for now is that all sites, or crystallites, are equivalent electrical conductors. The grain boundaries are the bonds connecting the individual mesh points. Each grain boundary has an electrical resistance, RGB, which is dependent on the orientation relationships of the grains. For now, also assume a bimodal RGB, that is, 0 (superconducting) or ∞ (nonconducting) are the only possible value for RGB. Each bond is either open with probability p or closed with probability (1 − p). The open bonds allow passage of current and the closed bonds do not.

2.4 Microstructure and Materials Properties

The main concept of the bond percolation model is the existence of a percolation threshold, pc, corresponding to the point at which a cluster of open bonds (a conducting path) first extends across the entire sample. Such a cluster is called a spanning cluster. For all p < pc, no current can flow owing to the lack of a complete current path, but there may be non-spanning clusters, connecting a finite number of points, which exist for any nonzero p. At pc, the system abruptly transitions to the electrically conductive state and for all p > pc, the sample is electrically conducting because of the presence of spanning clusters. For p > pc, the conductivity, σ, is proportional to a power of (p − pc): σ

p − pc

μ

2 10

The exponent μ is called a scaling exponent. It depends on the dimensionality of the system. For two-dimensional transport, μ = 1.30, and for three-dimensional transport, μ = 2.0 (Stauffer and Aharony 1994). The numerical value of the bond percolation threshold is dependent on the geometry of the grain boundary network. For example, pc is equal to 0.500 for a simple two-dimensional square lattice and 0.388 for a three-dimensional diamond lattice. For the simple cubic, FCC, and BCC lattices, pc is equal to 0.2488, 0.1803, and 0.119, respectively. With nontextured polycrystals, the geometries (grain orientations and/or angles) are random, and hence, the exact value for pc may not be known. Furthermore, the original assumption that the grain boundaries are either superconducting or insulating is obviously a drastic one. In reality, the grain boundary resistance is not bimodal. It can have values other than zero or infinity, which are dependent on the grain orientations and/or angles, as can be inferred from Figure 2.12c. In fact, a broad distribution of grain boundary resistances may be observed. For sufficiently broad distributions, however, the resistance of the bulk sample is often close to the resistance of the grain boundary cluster with exactly the percolation threshold concentration (Stauffer and Aharony 1994). For a derivation of Eq. (2.10), as well as a detailed treatment of percolation theory, the interested reader is referred to the book by Stauffer and Aharony (1994). Example 2.3 The oxide Bi4V2O11 is similar in structure to the layered aurivillius phase Bi2MoO6 (tetragonal crystal class). The aliovalent exchange of Mo6+ by V5+ results in oxygen vacancies in the BiO6 octahedral layers, which are separated along the c axis by edge-sharing BiO4 pyramids. The oxygen ion conduction in Bi4V2O11 is highly bidimensional, being much stronger in the ab planes of the octahedral layers than in the perpendicular out-of-plane direction along the c axis. Speculate on what the (0 0 1) pole figures for each sample given below would look like. The data are from Muller et al. (1996). Conductivities (×103 S/cm):

Single crystal Pressed polycrystal Magnetically aligned polycrystal

σ

σ⊥

73.2 40.2 59.1

2.6 20.2 13.8

Solution The (0 0 1) pole figure for the single crystal must have a single spot in the middle of the circle since all (0 0 1) planes in a single crystal are parallel. The anisotropy in the conductivity would be expected to be the greatest. The (0 0 1) pole figure for the pressed polycrystal should show

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the greatest spread, or orientation distribution, to the grains. This is because the anisotropy to the conductivity of the polycrystal was observed to be the smallest. Its properties are closest to that of a randomly textured sample. The (0 0 1) pole figure for the magnetically aligned polycrystal should have an intermediate spread since its anisotropy to the conductivity was intermediate.

2.4.3

Magnetic and Dielectric Properties

It is well known that the magnetic and dielectric properties of polycrystalline materials are dependent on grain/crystallite size, although the effect of grain/crystallite size can be difficult to isolate from other factors. For example, while grain/crystallite size is dependent on the sample processing route or history, as well as annealing or sintering temperatures, so too are the specimen crystallinity (disorder) and density (porosity), both of which increase (decrease) with increasing temperature, and these structural features also influence magnetic and dielectric properties. It is thus imperative in carrying out investigations aimed at quantifying the correlation between grain/crystallite size and physical properties that one produces the largest possible variation in grain/crystallite size while keeping all the other microstructural variables (e.g. chemical composition, dislocation density, surface oxidation, and crystallographic texture) as constant as possible. The relationship between particle size and crystallite/grain size (a particle may be composed of several grains/crystallites) and properties is particularly pronounced with nanoparticles for which a transition is often observed from a single-domain to a multi-domain structure at some critical size. Grain size has a very strong effect on magnetic losses. As the grain size increases, hysteresis losses decrease. Polycrystalline materials also possess a texture, or grain orientation distribution, and each grain may contain several ferromagnetic or ferroelectric domains (see Figure 2.13). It might, therefore, be expected that neighboring grains, and the domains within them, would interact in very different ways, giving rise to a response that will be very dependent on the grain size and texture. Such behavior is expected in the plots of magnetization versus applied field, so-called hysteresis

Figure 2.13 Schematic drawing of subgrain magnetic domains. Each of the three domains shown has a different net magnetic moment.

2.4 Microstructure and Materials Properties

Figure 2.14 Hypothetical hysteresis loops for polycrystalline films with the same chemical composition but with different textures.

M/Ms

H/Ms

loops, which follow one complete cycle of magnetization and demagnetization in an alternating field. Simulated hysteresis loops for polycrystalline films with different textures are shown in Figure 2.14. The greater the misalignment among grains, the less the magnetization that remains after the field is removed (remanence), and the lower the magnitude of the reverse field required for demagnetization (coercivity) (Jin et al. 2002). If these parameters can be controlled by manipulating the microstructure, different magnetic properties can be designed. For example, if ultrasmall (i.e. nanosized) grains of a highly magnetocrystalline anisotropic thin-film material can be deposited with all their easy magnetization axes normal to the film, magnetic storage media with much greater recording densities can be obtained. The magnetic properties of a metal are actually one of the most sensitive to changes in microstructure. Hence, magnetic measurements can be utilized in conjunction with metallographic techniques in elucidating microstructural features of alloys. For example, John Hopkinson (1849–1898) observed in 1886 that steel with a few weight percent tungsten had the highest residual magnetization (a.k.a. remanence). The remanence is the magnetization remaining after the removal of the magnetizing field. This quantity is dependent on the material’s composition and microstructure, as well as the geometry or form of the magnet (e.g. bar, ring, etc.). A high remanence, of course, is one quality that made tungsten steel an ideal metal for making permanent magnets. In 1918, a study by Kotaro Honda (1870–1954) and T. Murakami found that in tungsten steel magnets under equilibrium conditions, that is, which had been annealed and then slowly cooled, there were three ferromagnetic-to-paramagnetic transformations, on subsequent heating, at 700–770 and 400 C (the [Pierre] Curie points for ferrite and double carbide, respectively), as well as one at 215 C (the Curie point of cementite), the latter which disappeared with increasing tungsten content. However, it was not the slowly cooled alloys but, rather, quenched (martensitic) steels that were normally employed as permanent magnets in the first quarter of the twentieth century. These steels contained about 5.5–7% tungsten and about 0.5% carbon. Martensite’s enhanced magnetism is, microscopically, due to martensite’s greater resistance to magnetic domain wall motion than the ferrite phase and, accordingly, its higher magnetic coercivity (a.k.a coercive force) since the degree of hindrance to wall motion determines the magnitude of coercivity. Macroscopically, the coercivity is simply the intensity of the reverse field required for demagnetization. This property is characteristic of the material/microstructure but independent of the geometric form of the magnet. Martensite is so brittle, unfortunately, that some magnetism may need to be sacrificed for the sake of

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workability, depending on the application. Permanent magnet designers must judiciously choose processing conditions during solidification and subsequent heat treatments to reach the best compromise between optimal magnetic behavior and desirable mechanical properties.

2.4.4

Chemical Properties

Microstructure also affects the chemical properties exhibited by a metal. Intergranular atomic diffusion processes occur more rapidly than intragranular ones, since the regions between grains are usually not as dense as the grains. Remember also, that grain boundaries have a higher free energy than the grains themselves. Because of these aspects, a metal will usually oxidize or corrode more quickly along a narrow path that tends to follow grain boundaries, a condition known as intergranular corrosion. A similar kind of corrosion is transgranular corrosion, in which attack is highly localized and follows a narrow path through the material. However, the paths cut across grains with no apparent dependence on grain boundary direction. The oxidation rate may be very dependent on the grain boundary structure, which gives rise to the possibility for grain boundary engineering. This essentially refers to the incorporation of a high percentage of a specific grain boundary type in a polycrystal for the purposes of optimizing the properties of interest. For example, it has been shown that Ni–Fe (Yamaura et al. 2000) alloys and Pb electrodes (Palumbo et al. 1998) that have been deformation processed, or thermally processed to have a high fraction of low-Σ boundaries, are much more resistant to intergranular corrosion. Furthermore, those with random high-angle boundaries, which are probably higher in energy, more easily oxidize. For conventionally grained solids, however, the grain boundary volume is a small fraction of the total volume of the sample. Hence, intragranular diffusion is usually the dominant mass transport process, except for very small grain sizes or at low temperatures. Nonetheless, because impurity atoms tend to segregate at grain boundaries, intergranular chemical reactions that may not occur intragranularly are possible. In Section 2.4.1, we mentioned several alloys that are used for harsh service conditions. By 1912, Haynes had formed his own company, Hayes Stellite Works, which was bought by Union Carbide in 1920 but became an independent company, Haynes International, Inc., in 1987. While under Union Carbide, the company developed a line of Ni-based alloys they trademarked as HASTELLOY®. Like the Co-based superalloys, the Ni-based superalloys were also capable of withstanding extreme service conditions. For example, for many years, HASTELLOY X alloy, a Ni–Cr– Fe–Mo alloy developed in 1954, was the premiere combustor alloy for jet aircraft engines. Other nickel-chromium superalloys, such as the Incoloy® and Inconel® series (usually containing small amounts of several other elements, such as Al, C, Co, Cu, Fe, Mn, S, Si, Ti, V, or W) by the Mega Mex Company, are widely used as materials for construction of equipment that must have high strength and which must also resist oxidation, carburization, and other harmful effects of high-temperature exposure and/or corrosive environments (e.g. chemical reactors and petrochemical processing vessels/piping; heat exchangers, etc.). The elements chromium, aluminum, and silicon are by far the most employed alloying agents, as they impart resistance to oxidation as well as various kinds of corrosion through the formation of continuous adherent passive oxide scales on the alloy surface (CrO2, Al2O3, SiO2). Such scales are often referred to as thermally grown oxides, or TGOs. Eventual spallation of these protective oxides can occur if there is insufficient passivating alloying agents (e.g. Al) present near the surface during high operating temperature service. Unfortunately, the alloying composition for corrosion and oxidation resistance often contradict those for strength, necessitating a compromise in alloy design to achieve the right balance. The high percentage of nickel, fortunately, maintains an austenitic-like

2.5 Microstructure Control and Design

structure so that the alloys remain somewhat ductile. Today, high-temperature alloy development is primarily driven by demands from gas turbines to increase inlet gas temperatures, in order to achieve a higher energy efficiency and to reduce CO2 emissions. The Ni-based superalloys are still widely used, but alloys with even higher melting points, or liquidus temperatures, such as the Co– Re- and Co–Re–Cr-based alloys, are currently being investigated as candidate materials for future use.

2.5

Microstructure Control and Design

It was seen earlier how deformation processing tends to introduce texture in ductile polycrystalline materials. It has also been mentioned that one method of producing a textured powder aggregate involves the application of uniaxial pressure. However, in this method, a large number of grains that are misoriented with respect to the majority usually remain. Furthermore, this simple approach is not applicable for every desired shape. There are other methods, which have been used to produce textured materials. One is the TGG method, in which anisotropic seed particles regulate crystal growth in specific growth directions. This process involves the addition of a small amount of the template particles to a powder matrix (usually less than 10 vol%), which is seed particles of the same substance. The template particles are next oriented, and the matrix is then sintered, causing the template particles to grow by consuming the randomly oriented matrix. This produces a highly oriented microstructure. Template particles must possess a variety of properties for successful TGG (Kwon and Sabolsky 2003). If mechanical forces are acting to align the seed particles, templates must have a high aspect ratio so they can be aligned during consolidation processes. Morphologically, most of these templates are platelets or whiskers, and the desired crystallographic orientation must be aligned along the template axis. The template grains must be large enough so that there is a thermodynamic driving force for their growth at the expense of the matrix grains. The template seeds must also be distributed uniformly throughout the matrix to be most effective (Lalena et al. 2008). TGG has been demonstrated to be effective for a number of applications. The amount of growth, hence the degree of texture, depends on the number, size, and distribution of the template particles. The quality of the texture produced depends upon how well the seed texture particles are oriented during processing. Grains impinging upon each limits the attainment of a 100% textured material; however, XRD studies have shown that obtaining the desired texture in more than 90% of the material is possible (Messing 2001). When the unit cell of a crystalline ceramic substance is anisotropic, this gives rise to the possibility that the accompanying anisotropy in its physical properties may aid the texture control of powder samples. For example, a crystal with an anisotropic magnetic susceptibility will rotate to an angle minimizing the system energy when placed in a magnetic field. The reduction in magnetic energy is the driving force for magnetic alignment. However, alignment may be difficult in undispersed powders because of strong particle–particle interactions, which prevent the particles from moving. Therefore, dispersion of the powder in a suspension is usually necessary for effective utilization of a magnetic field. This can be readily accomplished in slip casting, where a suspension, or slurry, called the slip, is poured into a porous mold of the desired shape that absorbs the fluid. Magnetic alignment, in conjunction with slip casting, has been used to produce textured microstructures in a variety of substances, including some with only small anisotropic diamagnetic susceptibility, such as ZnO and TiO2, if the magnetic field is high enough, for example, several tesla.

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In addition to specifying the texture or orientation distribution to the grains, the fraction of a particular CSL boundary type (Section 1.3) can be specified. This approach is also useful because the grain boundary structure often correlates with certain materials properties, particularly conductivity, creep (time-dependent deformation at constant load), and corrosion. For example, coherent twin boundaries are able to block dislocation motion and strengthen a metal. They also allow for a much more efficient transfer of current than conventional grain boundaries (Suton and Balluffi 2007). Hence, it has been observed that pure copper samples with a high percentage density of nanoscale coherent growth twins are strengthened considerably (10× tensile strength of conventional coarse-grained samples) without a loss to the electrical conductivity (Lu et al. 2004). Likewise, it has been shown that Ni–Cr–Fe alloys, with a high fraction of special CSL boundaries, possess higher creep resistances (lower strain rates) than those with general boundaries (Thaveepringsriporn and Was 1997; Was et al. 1998). Thus far, the discussion has assumed that the materials engineer has a desired microstructure in mind. But how does one determine which microstructure is optimal for a given application? While theoretical models for computing properties given the microstructure are known for some systems, the inverse problem of obtaining the microstructures that lead to the desired properties is much more challenging. Microstructure design is an emerging field that focuses on producing microstructures that meet, or exceed, specified macroscale properties or performance criteria. This requires a combination of empirical techniques, mathematical modeling, and numerical simulation. A complicating factor has always been the enormous number of possible microstructures that can evolve from a myriad of processing variables. In additional, owing to the complexities of treating anisotropy and non-homogeneity into design problems, drastic simplifying assumptions are often made during analysis. Fortunately, advancements have been made in these areas. A major key to the success of a promising approach championed by Drexel University professor Surya R. Kalidindi and Brigham Young University professor Brent L. Adams, termed microstructure sensitive design (MSD), is the representation of the various functions in Fourier (or spectral) space, in order to compress the large amount of data (Adams et al. 2002; Kalidindi et al. 2005). First, the essential microstructural characteristics that are the most influential in controlling the macroscale properties of interest are carefully identified. These salient features are parameterized into spectral functions, each defining a microstructure represented as an infinite series of harmonic functions, the coefficients of which indicate the weight of the particular harmonic with which they are associated. The important terms are generally found at the front of a series that is truncated at some point that balances efficiency with accuracy. Thus, each microstructure is represented by a set of Fourier coefficients. For example, a microstructure is defined through the microstructure function: M x, h dh =

dV h V

2 11

This equation represents the volume fraction of material possessing an orientation lying within invariant measure dh of the local state of interest h within an infinitesimal neighborhood of the material point at x. Equation (2.11) can be described with a spectral basis using the simplest formulation as S

N

Dsn χ n h χ s x

M x, h =

2 12

s=1n=1

with Fourier coefficients Dsn, where the representative volume element is divided into S cells and the local state space is divided into N cells; χ s(x) = 1 if x is in cell s, and zero otherwise. If the local

2.5 Microstructure Control and Design

state of interest h is the lattice orientation, g, the spectral representation for the ODF (texture) may be expressed as follows: ∞

f g = α=1

F αTα g

2 13

where T(g) is the generalized spherical harmonic function (Kalidindi and Fulwood 2007). In quantifying the different possible microstructures, it is important to consider not only the orientations of the constituent crystallites but also their phases, as well as other parameters such as dislocation density. The complete set of feasible statistical distributions that describe the relevant details of the microstructure are termed the microstructure hull (each point in the microstructure hull represents a unique microstructure), while the set of feasible combinations of macroscale properties is termed the property closure. Homogenization relations are used to link the microstructure hull to the property closure. Bounds on material properties are represented by one or more families of bounding hypersurfaces (often hyperplanes) of finite dimension that intersect the microstructure hull. Consideration of the full range of these hypersurfaces gives rise to properties closures, representing the full range of combined properties that are predicted to be possible by considering the entire microstructure hull. For a given point in the property closure, a hypersurface may be identified that traverses the microstructure hull; each microstructure on such a surface is predicted to relate to the given combination of properties (Lemmon et al. 2007). Thus, the property closure is tied to the physically realizable microstructures that can simultaneously exhibit the specific combination of properties. The MSD framework has been successfully applied to linking the crystallographic texture with the elastic and plastic properties of some cubic and hexagonal polycrystalline metals and to optimizing the design of a radio-frequency microelectromechanical systems (MEMS) switch. Unfortunately, the efficiency of mathematical search-based designs deteriorates quickly as the candidate space grows. So, a natural trade-off is often the reduction of the search space to a practical size. Hence, despite the achievements of the MSD approach, these analyses do suffer when a reduced or truncated spectral series representation is employed. Much information on the microstructure is lost in the reduced representation, and the optimization search is done rather manually, by numerically interpolating the microstructure space to find the location of best performance. However, a unique microstructural solution obtained with a nontruncated series is not always practical since most low-cost manufacturing processes are limited in what they are capable of generating and a single design solution may not be feasible. For this reason, a complete space of optimal microstructures is often preferred. Combinatorial approaches leading to multiple solutions have also been investigated in materials science, but they are frequently incapable of dealing with the sheer size of search spaces with high dimensions (Liu et al. 2015). To address this, modern machine learning (ML) techniques have also been proposed as a tool to explore high-dimensional microstructure design problems with a diminished search time, where the number of distinct design candidates is initially infinite. In machine learning, which is a branch of artificial intelligence, a computer learns from existing data without being explicitly programmed and makes predictions on new data via construction of a model. Thus, machine learning can be considered a methodology of the broader field of informatics. Materials informatics is the emerging field of study that applies the principles of informatics to better understand the use, selection, development, and discovery of materials. Its goal is the achievement of high-speed and robust data acquisition, management, analysis, in order to reduce the time and risk required to develop, produce, and deploy new materials.

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Like the MSD framework, ML methods also reduce the search space, but in a different way. In machine learning, the number of design variables in search patterns is limited to form a superior search strategy. Three steps are crucial to the success of this approach. The first is random data construction, where the best and worst cases are determined. The second step is search path refinement and search region reduction, which are designed to develop heuristics for guiding the subsequent search step in the most promising directions, for example, by determining which variables to search first, which areas are the most promising, and which directions are likely to lead to quicker convergence. The third step is the actual search for microstructures that yield the desired properties. Ruoqian Liu et al. (2015) developed such a machine learning method for predicting the optimized microstructure of an Fe–Ge alloy with enhanced elastic, plastic and magnetostrictive properties.

Practice Problems *1

For a single-phase polycrystal with cubic symmetry, assuming the angular parameters are distinguished with 10 of resolution, how many distinct grain boundaries are there?

2

Give one reason why a reduction in the grain size of a polycrystalline pure metal or alloy may be required and describe one method by which grain size reduction could be accomplished.

3

Describe the difference between hot working and cold working. Explain the effect of each method on the dislocation density in a sample and how this, in turn, affects the extent of plastic deformation possible with each technique.

*4

Explain the difference between the terms annealing and sintering.

5

What is the difference between the glass-transition temperature and melting temperature?

6

Define the term monolithic. Which of the following materials can be monolithic: a glass, a single crystal, a casting, and a powder?

*7

What are the three primary methods for strengthening a material?

8

How are ceramics and other nonductile polycrystals fabricated into bulk articles?

9

Describe what is meant by the term texture, as it is applied to polycrystals, and why it is important to materials properties.

10

What are dislocations? What are the different types of dislocations?



For solutions, see Appendix D.

References Adams, B.L., Lyon, M., Henrie, B., Garmestani, H., and Kalidindi, S.R. (2002). Mater. Sci. Forum 408–412: 493–498. Baker, H. (ed.) (1992). ASM Handbook, Vol. 3: Alloy Phase Diagrams. ASM International.

References

Bhadeshia, H.K.D.H. (1987). Worked Examples in the Geometry of Crystals. London: The Institute of Metals, 70. Callister, W.D. (2005). Fundamentals of Materials Science and Engineering: An Integrated Approach. Wiley. Elliot, S.R. (1998). The Physics and Chemistry of Solids. Chichester: Wiley. Friedel, G. (1926). Leçons de Cristallographie. Paris: Berger-Levrault. Gillan, E.G. and Barron, A.R. (1997). Chem. Mater. 9: 3037. Gorbenko, O.Y., Samoilenkov, S.V., Graboy, I.E., and Kaul, A.R. (2002). Chem. Mater. 14: 4026. Gulliver, G.H. (1913). J. Inst. Metals 9: 120. Hardouin Duparc, O.B.M. (2010). Metall. Mater. Trans. A 41A: 1873–1882. Hardouin Duparc, O.B.M. (2011). J. Mater. Sci. 46: 4116–4134. Hoglund, D.E., Thompson, M.O., and Aziz, M.J. (1998). Phys. Rev. B 58: 189. Huang, S.C. and Glicksman, M.E. (1981). Acta Metall. 29: 71. Ivantsov, G.P. (1947). Dokl. Akad. Nauk 58: 56. Jin, Y.M., Wang, Y.U., Kazaryan, A., Wang, Y., Laughlin, D.E., and Khachaturyan, A.G. (2002). J. Appl. Phys. 92: 6172. Johnson, W.L. (2000). The Science of Alloys for the 21st Century: A Hume-Rothery Symposium Celebration (eds. P.E.A. Turchi and R.D. Shull). Warrendale, PA: The Minerals, Metals & Materials Society. Kalidindi, S.R. and Fulwood, D.T. (2007). JOM 59: 26–31. Kalidindi, S.R., Houskamp, J., Proust, G., and Duvvuru, H. (2005). Mater. Sci. Forum 495–497: 23–30. Keys, A., Bott, S.G., and Barron, A.R. (1999). Chem. Mater. 11: 3578. Kwon, S. and Sabolsky, E.M. (2003). Control of ceramic microstructure by templated grain growth. In: Handbook of Advanced Ceramics (ed. S. Somiya), 459–469. Oxford: Elsevier. Lalena, J.N., Cleary, D.A., Carpenter, E.E., and Dean, N.F. (2008). Inorganic Materials Synthesis and Fabrication. New York: Wiley. Lamzatouar, A., El Kajbaji, M., Charaï, A., Benaissa, M., Hardouin Duparc, O.B.M., and Thibault, J. (2001). Scripta Metall. 45: 1171. Langer, J.S. (1989). Science 243: 1150. Lemmon, T.S., Homer, E.R., Fromm, B.S., Fullwood, D.T., Jensen, B.D., and Adams, B.L. (2007). JOM 59 (9): 43. Liu, R., Kumar, A., Chen, Z., Agrawal, A., Sundararaghavan, S., and Choudhary, A. (2015). Sci. Rep. 5: 11551. http://dx.doi.org/10.1038/srep11551. Lu, L., Shen, Y., Chen, X., Qian, L., and Lu, K. (2004). Science 304: 422. Mayer, T.M., Adams, D.P., and Swartzentruber, B.S. (1994). Nucleation and evolution of film structure in chemical vapor deposition processes. In: Physical and Chemical Sciences Center Research Briefs, vol. 1–96. Sandia National Laboratories. Messing, G.L. (2001). Texture ceramics. In: Encyclopedia of Materials: Science and Technology (eds. K.H.J. Buschow, R. Chan and M. Flemings), 9129. New York: Elsevier. Movchan, B.A. and Demchishin, A.V. (1969). Phys. Metall. Metallogr. 28: 83. Muller, C., Chateigner, D., Anne, M., Bacmann, M., Fouletier, J., and de Rango, P. (1996). J. Phys. D 29: 3106. Mullins, W.W. and Sekerka, R.F. (1963). J. Appl. Phys. 34: 323. Palumbo, G., Lehockey, E.M., and Lin, P. (1998). JOM 50 (2): 40. Pawlak, D.A., Kolodziejak, K., Turczynski, S., Kisielewski, J., Rożniatowski, K., Diduszko, R., Kaczkan, M., and Malinowski, M. (2006). Chem. Mater. 18: 2450–2457. Petkewich, R. (2008). Chem. Eng. News 86 (47): 38. Pfann, W.G. (1952). Trans. AIME 194: 747. Phanikumar, G. and Chattopadhyay, K. (2001). Sādhanā 26: 25.

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Randle, V. and Engler, O. (2000). Introduction to Texture Analysis. CRC Press, 2nd Edition by Engler and Randle in 2010. Rayleigh, L. (1872). On the Diffraction of Object Glasses, vol. 33, 59–63. Monthly Notes of the Royal Astronomical Society. Read, W.T. and Shockley, W. (1950). Phys. Rev. 78: 275. Rohrer, G.S. (2007). JOM 59 (9): 38. Salvador, P.A., Doan, T.D., Mercey, B., and Raveau, B. (1998). Chem. Mater. 10: 2592. Saylor, D.M., Morawiec, A., Adams, B.L., and Rohrer, G.S. (2000). Interface Sci. 8 (2/3): 131. Scheil, E.Z. (1942). Metallkunde 34: 70. Stauffer, D. and Aharony, A. (1994). Introduction to Percolation Theory, Revised 2e, 52. London: Taylor & Francis. Suton, A.P. and Balluffi, R.W. (2007). Interfaces in Crystalline Materials. New York: Oxford University Press. Thaveepringsriporn, V. and Was, G.S. (1997). Metall. Trans. A. 28: 2101. Thornton, J.A. (1977). Ann. Rev. Mater. Sci. 7: 239. Tiller, W.A., Jackson, K.A., Rutter, R.W., and Chalmers, B. (1953). Acta Metall. 1: 50. Weertman, J. and Weertman, J.R. (1992). Elementary Dislocation Theory. New York: Oxford University Press, 6. Yamaura, S., Igarashi, Y., Tsurekawa, S., and Watanabe, T. (2000). Properties of Complex Inorganic Solids 2 (eds. A. Meike, A. Gonis, P.E.A. Turchi and K. Rajan), 27–37. New York: Kluwer Academic/Plenum Publishers.

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3 Crystal Structures and Binding Forces In the previous chapters, the structures of polycrystalline materials on a microscope scale, in which the smallest discernible particles are the crystallites, or grains, were discussed. The average grain size, shape, and orientation markedly affect materials behavior, but, ultimately, intrinsic materials properties are determined by the atomic arrangement within the crystals. In the present chapter, the crystal structure will be explored by “zooming in” to this length scale. First, a brief review of some different methods of describing crystalline arrangements. Then, after discussing the different types of cohesive forces, some commonly adopted crystal structures will be presented.

3.1

Structure Description Methods

There are at least three methodologies for portraying crystal structures. Each of them has its advantages and disadvantages, but they all are based on the idea of a repeating pattern. The simplest method, the close-packed sphere description, is also the oldest. With advances in crystallography came more sophisticated descriptions, which lead to polyhedra and, finally, to the crystallographic unit cell. Our discussion will follow their order of development, but all of these methods still remain in use today. This is because it is often helpful to visualize a specific structure from different perspectives. For example, although the unit cell is the most precise description, both the short-range and long-range symmetry are often more readily apparent with (interconnected) polyhedra, as is the nearest-neighbor coordination around a particular atom. In a similar fashion, close-packed spheres perhaps leave a more realistic impression of the density of a material, since in a unit cell illustration the contents (i.e. atoms) are often drawn as atomic centers, which can give the impression that distances between atoms are greater than they actually are. It is important to keep in mind that all types of structure descriptions simply tell us about the locations of the atoms. They reveal little or nothing about the nature of the bonding forces between the atoms.

3.1.1 Close Packing The structures of many metals can be described rather simply if one assumes that the constituent atoms are close-packed spheres. In this arrangement, the maximum possible density is achieved. Close packing of spheres dates back to seventeenth century when Sir Walter Raleigh asked the mathematician Thomas Harriot to study the stacking of cannon balls. Harriot debated this topic with Johannes Kepler (1571–1630), who was a staunch believer in the hypothesis for the corpuscular nature of matter (the precursor of atomism), and had already given thought to how water particles stack themselves to form symmetrical snowflakes. In 1611, Kepler hypothesized from Principles of Inorganic Materials Design, Third Edition. John N. Lalena, David A. Cleary, and Olivier B. M. Hardouin Duparc. © 2020 John Wiley & Sons, Inc. Published 2020 by John Wiley & Sons, Inc.

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geometrical considerations that the densest possible arrangement of spheres, be the water particles or cannon balls, is obtained with CCP, in which 74.05% of the cube’s volume is occupied. This came to be known as the Kepler conjecture, and the problem of proving it was the Kepler problem. A rigorous mathematical proof eluded mathematicians until only recently. It was not until 1998 that the proof was accomplished by Professor Thomas Hales of M. I. T. with the help of computers (Lalena 2006). It is well known that in two dimensions close-packed spheres have a hexagonal arrangement in which each sphere is tangent to six others in the plane. In three dimensions, there are two possible ways to achieve close packing. They have to do with the way close-packed planes are stacked together. In both cases, however, each sphere is tangent to, or coordinated by, 12 others. Figure 3.1 shows each type of arrangement. In the first case, each sphere in the upper layer, of the set of three layers, is directly above one sphere in the lower layer. The spheres of the middle layer rest in the hollows/voids between three spheres in each of the adjacent layers. The staggered

A

A

A

A

A

A A

A

A A

A A

A A

Second layer atoms in “B” sites Third layer atoms in “C” sites Fourth layer directly over first layer

Second layer atoms in “B” sites Third layer directly above first layer

A A

C

B

B

A

A

HCP

2 atoms per primitive unit cell

Figure 3.1

CCP

4 atoms per cubic unit cell

Closest-packed arrangements in three dimensions.

3.1 Structure Description Methods

Table 3.1

Packing densities for various structures.

Structure

Density (%)

Density (exact)

CCP

74.05

π√2 /6

HCP

74.05

π√2 /6

BCC

68.02

π√3 /8

Cubic (P)

52.36

π /6

Hexagonal

60.46

π /3√3

CD

34.01

π√3 /16

Random

~64

close-packed layers described above are sometimes represented as (…ABABAB…), where each letter corresponds to a two-dimensional close-packed layer, and in which the sequence required to achieve three-dimensional close packing is clear. This is called hexagonal closest packed and is abbreviated HCP. There are two types of hollows in any closest-packed structure (vide infra): tetrahedral (a void coordinated by four atoms) and octahedral (a void coordinated by six atoms). These voids are referred to as interstitial sites and can be important in enabling the diffusion of solute/impurity atoms through the solid, as we will learn in Section 6.5.1. A word of warning is in order here. Throughout this book, the reader will find seemingly similar terms to “void” used to describe quite different things. Let us set the record straight in advance and make the following distinctions. “Interstitial site” is reserved for the volume of space between atoms in closest-packed structures and nearly closest-packed structures (see Table 3.1). Although we have just used the term “void” for this, we caution that this word is often used to describe structural features on longer length scales (e.g. microstructural gaps) and the reader should always exercise diligence in taking the meaning from the context used by the author. The term “hole” is reserved for a missing electron. Finally, the term “vacancy” is reserved for an atom missing from its normal position. The second case illustrated in Figure 3.1 corresponds to three close-packed layers staggered relative to each other. It is not until the fourth layer that the sequence is repeated. This is known as cubic closest packed, or CCP, and is represented as (…ABCABCABC…). Geometric considerations show that, for equal-sized spheres in both the CCP and HCP arrangements, 74.05% of the total volume is occupied by the spheres. The packing densities of other non-closest-packed structures are given in Table 3.1. As the atoms of an element are all equal sized, the structures of many elements correspond to the CCP or HCP array. By contrast, many ionic compounds can be described as a close-packed array of anions (large spheres), with cations (smaller spheres) located in the hollows/voids between the anions. The voids, which are called interstitial sites, come in two different sizes as described above. Tetrahedral sites are coordinated by four anions, and octahedral sites are coordinated by six anions, as shown in Figure 3.2. For every anion in two adjacent close-packed layers, there are two tetrahedral voids (an upper and a lower) and one octahedral void. The octahedral voids are larger and can thus accommodate larger cations. When the cations are large, they effectively push the anions further apart. The term eutactic is useful for describing such systems, where the arrangement of atoms is the same as in a close-packed structure, but in which the atoms are not touching. Some covalent compounds (e.g. diamond) are sometimes described in terms of close-packed arrays:

101

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3 Crystal Structures and Binding Forces

Tetrahedral site

Tetrahedral site

Octahedral site

Figure 3.2 The octahedral and tetrahedral sites (voids) contained within two adjacent close-packed layers. The darkly shaded spheres are behind the plane of the page. The lightly shaded spheres are above the plane of the page.

diamond can be considered as two interlocking CCP lattices displaced by a quarter of the body diagonal. Yet, it is important to realize that the diamond structure (cubic diamond, CD) is a very open structure, with a very low packing density. A primary drawback of visualizing metals simply as close-packed spheres is that this description might seem to imply that all such metals would have the same oft-stated nondirectional bonding. However, it has been observed that the bonding in certain HCP metals actually has some directional character. For example, in the 1970s, valence electron charge density maps (which may be obtained from observed or calculated structure factors) of beryllium revealed that in this HCP metal the predominate interaction is bonding through the tetrahedral voids, while the electron densities around the octahedral voids are much less than the average density of interatomic sites (Yang and Coppens 1978). A similar situation has been found in the HCP metal magnesium (Kubota et al. 1993). Consider next, two metals from the 4d series of the periodic table, ruthenium and zirconium, both of which have the HCP structure. In ruthenium, bond critical points (the saddle points) in the electronic charge density are located between each Ru atom and its six nearest neighbors in the plane. There exists, as well, bond paths to the six atoms in and out of the plane, which gives rise to a cage point in the tetrahedral void. In essence, each Ru atom is bound to all of its 12 nearest neighbors, constituting true nondirectional bonding. In zirconium, by contrast, there are bond paths (critical points in the charge density) between each tetrahedral void to its three closest Zr atoms in the plane and to the two atoms above and below. Five Zr atoms are thus bound to one another through the tetrahedral void (Eberhart et al. 2008). Example 3.1 Cobalt crystallizes in the HCP structure, which contains two atoms per unit cell. The atomic radius of the Co atom is 0.1253 nm. 1) Use this information and Table 3.1 to determine the unit cell volume. 2) With the help of Table 1.2, compare the result to that calculated from the lattice parameters, a = 0.2507, c = 0.4069. 3) Compute the density from your answer for 1. The tabulated value is 8.92 g/cm3.

3.1 Structure Description Methods

Solution 1) The HCP structure has a packing density of 74.05%. The volume of a single Co atom is (4/3)π r3 = (4/3)π(0.1253 nm)3 = 0.008240 nm3. There are two atoms in the unit cell occupying a volume of 0.008240 × 2 = 0.04944 nm3, which is 74.05% of 0.02225 nm3. 2) From Table 1.2, the volume may also be calculated from the lattice parameters as follows: 3 2 = 0 02214 nm3, which is in close agreeV = a2 c sin 60 = 0 2507 nm2 0 4069 nm ment, indicating that the close-packed sphere model is very reasonable for cobalt. 3) The unit cell volume is 0.02225 nm3 × [(10−7)3 cm3/13 nm3] = 2.225 × 10−23 cm3. There are two atoms in the unit cell, each with a mass of (58.93 g/1 mol) × [1 mol/(6.0223 × 1023 atoms)] = 9.785 × 10−23 g. The density is then (2 × 9.785 × 10−23 g)/(2.225 × 10−23 cm3) = 8.796 g/cm3.

3.1.2 Polyhedra A second approach to describing structures emphasizes the coordination around specific types of atoms in the structure. In this method, the atoms are omitted from the representation and replaced by coordination polyhedra. The vertices of a polyhedron represent the centers of the coordinating atoms, typically anions, and at the center of the polyhedron resides the metal atom that is coordinated. This method was originated in 1840 by Gabriel Delafosse (1796–1878) who was chair of mineralogy at the Museum of Natural History in Paris and who reconciled Haüy’s earlier integral molecules with Dalton’s atomic theory. In Haüy’s view, the building blocks of crystals were solid polyhedral molecules that could not be further subdivided, that were arranged contiguously along parallel lines, and that appeared externally in a crystal’s macroscopic cleavage. Haüy had adopted the parallelepiped, octahedron, tetrahedron, hexagonal prism, rhombic dodecahedron, and hexagonal bipyramid as the shapes of these integral molecules. Haüy also envisioned these building blocks as the smallest particle that could retain the chemical properties of a substance, hence the name integral molecule (the word molecule was derived from the Latin molecula, a diminutive of the Latin moles for mass). Haüy was able to convince the scientific community of his time that spherical atoms did not exist. However, with the coming of Dalton’s atomic theory, Delafosse proposed that Haüy’s building blocks (the solid polyhedral molecules) be replaced by abstract polyhedra with spherical atoms at the vertices (Lalena 2006). For example, sodium chloride may be regarded as an array of edgesharing NaCl6 octahedra, as illustrated in Figure 3.3. Polyhedra can share corners, edges, or faces. The advantage of this view is that the extended structure is readily apparent. The disadvantage is that, in reality, the planar faces of the polyhedra do not exist physically, only the atoms do. The faces merely show the relative placement of the atomic centers around the centrally coordinated atoms, which are not shown as part of the polyhedra. Also, in representing the atoms centered at the vertices as points, these atoms may be perceived as being far from the central atom and from one another when, in fact, they often form a close-packed or nearly closepacked array.

3.1.3 The (Primitive) Unit Cell The crystallographic description is what is referred to as a (primitive) unit cell. The (primitive) unit cell is essentially a parallelepiped, the contents of which constitute a small (the smallest) volume of particles and their arrangement that can be repeated solely by infinite translation in three dimensions to generate the entire crystal. The parallelepipeds (which include the cuboid [six rectangular

103

104

3 Crystal Structures and Binding Forces

T– 1

1 1

1 1 1

1 1 1 1

1 1 1

31

1 1 1

1 1 1

31

3

1

2 1 1

3 3

Figure 3.3 The rock salt (halite) structure depicted as an array of edge-sharing octahedra. (Source: After West (1984), Solid State Chemistry and Its Applications. © 1984. John Wiley & Sons, Inc. Reproduced with permission.)

T– 1

2 1

2 1 1

T+

2

B

1

2 1

2 1

A

1 1

C 1

A

3

CI CI CI ≡ Na CI CI CI

faces], the cube [six square faces], and the rhombohedron [six rhombus faces]) are among the few polyhedra that tessellate three-dimensional space, the others being the regular-right triangular prism, the regular-right hexagonal prism, and truncated octahedron. The reader is referred back to Section 1.2.1 for the details. Let us note again that, most unfortunately for a rigorous linguist, a unit cell needs not be unitary and contains only one lattice point with only one motif (atomic basis). Such a unitary unit cell is called a primitive unit cell. So-called unit cells usually correspond to conventional symmetry cells that reflect the symmetry of the lattice but are multiple of the primitive unit cell. For example, the cubic unit cell is a double primitive unit cell for the cubic-centered lattice and a quadruple primitive unit cell for the face-centered cubic lattice.

3.1.4

Space Groups and Wyckoff Positions

The best to describe a crystal structure is to give its space group number, together with its Hermann–Mauguin (short or full, full would be preferable but tedious) symbol, its Schoenflies symbol, for double/triple checking, as given in volume A of the International Tables of Crystallography (see Section 1.2.1) as well as the chemical unit (formula), the number of chemical units per unit cell, usually labeled Z, and the list of the occupied Wyckoff positions. The Wyckoff positions are labeled by a minuscule, a, b, etc., preceded by a number indicating its multiplicity (see Example 3.2).

3.1.5

Strukturbericht Symbols

Strukturbericht means structure reports in German. These reports were published as supplements to the Zeitschrift für Kristallographie journal, starting in 1919, under the supervision of Carl Hermann (1891–1961, the same Hermann as for the Hermann–Mauguin symbols) and Paul Peter Ewald (1888–1985). In order to describe chemical crystallographic structures, they introduced a specific notation that tries to distinguish between different chemical orders of various crystals, especially polyatomic ones. These designations obey certain rules: A for pure elements, B for equiatomic compounds, C for compounds with a 2/1 ratio, D for compounds with a 3/1 ratio provided they are not metallic alloys and with the exception of Cr3Si that bears the A15 symbol, E–K for more complex alloys, L for alloys, S for silicates, and O for organics. For example, Cu is A1, Fe-α is A2, Mg is A3, NaCl is B1, CsCl is B2, ZnS zinc blende is B3, ZnS wurtzite is B4, CaF2 is C1, ordered CuAu is L10,

3.1 Structure Description Methods

ordered Cu3Au is L12, disordered CuAu and Cu3Au are A1, ordered Fe3Al is D03, etc. These symbols are still very much used, but they are not intuitive in many cases. Not all compounds have a Strukturbericht symbol, for example, oddly enough, α-quartz does not (β-quartz is C8). This is why Pearson symbols are also helpful (but not sufficient).

3.1.6 Pearson Symbols William B. Pearson (1921–2005) developed a shorthand system for denoting alloy and intermetallic structure types (Pearson 1967). It is now widely used for ionic and covalent solids, as well. The Pearson symbol consists of a small letter that denotes the crystal system, followed by a capital letter to identify the space lattice. To these a number is added that is equal to the number of atoms in the unit cell. Thus, the Pearson symbol for wurtzite (hexagonal, space group P63mc), which has four atoms in the unit cell, is hP4. Similarly, the symbol for sodium chloride (cubic, space group Fm3m), with eight atoms in the unit cell, is cF8. Unfortunately, the Pearson symbol does not uniquely identify the space group of a crystal structure. For example, diamond, zinc blende, and NaCl are all cF8, although they each belong to different space groups. Diamond is space group #227, Fd3m, Strukturbericht designation A4; zinc blende is #216, F43m, B3; and NaCl is #225, Fm3m, B1. Note that diamond is cF8 but has Z = 2 per primitive unit cell, corresponding to two chemical units (formulae) and Z = 8 per cubic unit cell. The cubic zinc blende ZnS is also cF8, with Z = 1 per primitive unit cell, and Z = 4 per cubic unit cell because the chemical unit is ZnS, Zn being chemically different from S. Because different scientific communities tend to use different designations and because misprints are too easy to make, it is very much advised to give as much information as possible, even with explicit atomic figures when possible, as we tried to do in Section 3.4. There are now reliable websites that the use of which is recommended, such as the crystal lattice structure site of the University of Vienna (Wien Universität, Austria), and Materialsproject.org. Example 3.2 and α-Al2O3.

Find relevant crystallographic data for CsCl, Cu3Au, LaMnO3, AlCFe3, Fe4N,

Solution CsCl is Cu3Au is LaMnO3 is AlCFe3 is Fe4N is

#221, #221, #221, #221, #221,

Pm3m, Pm3m, Pm3m, Pm3m, Pm3m,

cP2, cP4, cP5, cP5, cP5,

Cs in 1a (0,0,0), Cl in 1b (½ ½ ½). B2 Au in 1a (0,0,0), Cu in 3c (½ ½ ½). L12 La in 1a (0,0,0), Mn in 1b (½ ½ ½), O in 3c (0 ½ ½). E21 Al in 1a (0,0,0), C in 1b (½ ½ ½), Fe in 3c (0 ½ ½). L 12 N in 1a (0,0,0), Fe in 1b (½ ½ ½) and 3d (½ 0 0). L 1.

The Wyckoff positions for space group #221 are 1a (0,0,0); 1b (½ ½ ½); 3c (0 ½ ½); 3d (½ 0 0), 6e; 6f; 8g; 12h; 12i; 12j; 28k; 24l; 24m; 48n (see volume A of the ITC’s). Note that the sum of the multiplicities of the occupied Wyckoff positions equals the number given in the Pearson symbol. α-Al2O3 is

#167, R3c, hR10, Al in 4c, and O in 6e, D51

The crystal system is trigonal, the lattice is rhombohedral, and the primitive rhombohedral cell has Z = 2, thus, 10 atoms. CaCO3 calcite is similar, with Ca in 2b, C in 2a, and O in 6e. SpringerMaterials gives hR30, using a triple hexagonal R cell. Note that R3c exists and is #161, different from R3c (which actually is a short Hermann–Mauguin symbol, the full symbol is R32/c). This is a reason why it is safer to give both the (short) Hermann– Mauguin symbol and the number for a space group.

105

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3 Crystal Structures and Binding Forces

Some authors prefer to use Schoenflies symbols, e.g. O1h for #221, much shorter than its full Hermann–Mauguin symbol P4/m3 2/m. Space group #167 is D63d , #161 is C 63v . Great care is recommended.

3.2

Cohesive Forces in Solids

The cohesive forces bonding the atoms together in a crystal can be ionic, covalent, or metallic, or even a mixture of these types. Although distinguishing compounds based on this classification scheme may seem like a straightforward matter, it is complicated by the fact that, typically, the bonding within an inorganic material is a mixture of bonding types. Not only does the attraction between any two atoms in a solid, with the exception of pure elements, contain both an ionic and covalent character, but also many solids possess some fraction of predominantly ionic bonds as well as some fraction of predominantly covalent ones. Coordination numbers of the constituent atoms are not always helpful for differentiating bonding types either. The two most common coordination geometries observed in the covalent compounds of p-block and d-block elements are tetrahedral and octahedral coordination, respectively. These happen to be the same coordination numbers around the interstitial sites in the close-packed structures of many metallic elements and ionic compounds.

3.2.1

Ionic Bonding

The first ionic bonding model was suggested by the German chemist Richard Abegg (1869–1910) shortly after the British physicist Joseph John Thomson (1856–1940) discovered the electron. Abegg was studying the inert gases and noticed that their electron configurations were particularly stable. His research led him to hypothesize that atoms combined with one another because they exchange electrons, such that both end up with a noble gas electron configuration. Abegg’s rule, formulated in 1904, states that the difference between the maximum positive and negative valence of an element is often eight, where the term valence refers to the excess or deficit of electrons relative to these stable configurations. Gilbert Lewis and Irving Langmuir later adopted this idea for the octet rule (Section 3.3.5). Today, an ionic bond is recognized as the nondirectional electrostatic attraction between oppositely charged ions. For an isolated ion pair, the Coulomb potential energy, U, is simply U ion

pair

= −

1 q + q − e2 r 4πε0

31

In this expression, the q+/− terms are the ion charges, r is the distance separating them, e is the electron charge (1.602 × 10−19 C), and (1/4πε0) is the free space permittivity (1.11265 × 10−10 C2/J/m). The units of U are Joules. There is also a repulsive force that, as we now know, is due to the quantum mechanical overlapping of inner electron orbitals of neighboring atoms (thus implying Schrödinger’s equation and Pauli’s principle). Born and Landé simply suggested the following form in 1918 (see Section 3.3.1): V=

B rn

32

where B is a constant that can be determined once the total energy of the crystal is established (Section 3.3.1). The exponent, n, known as the Born exponent, is normally between 5 and 12.

3.2 Cohesive Forces in Solids

Arrays of ions tend to maximize the net electrostatic attraction between ions while minimizing the repulsive interactions. The former ensures that cations are surrounded by anions and anions by cations, with the highest possible coordination numbers. In order to reduce repulsive forces, ionic solids maximize the distance between like charges. At the same time, unlike charges cannot be allowed to get too close, or short-range repulsive forces will destabilize the structure. The balance between these competing requirements means that ionic solids are highly symmetric structures with maximized coordination numbers and volumes. A purely ionic bond is an extreme case that is never actually attained in reality. Between any twobonded species, there is always some shared electron density, or partial covalence, however small. An anion has a larger radius than the neutral atom because of increased electron–electron repulsion. Its valence electron density extends out well beyond the nucleus, and it is thus more easily polarized by the positive charge on an adjacent cation. One can usually presume predominately ionic bonding when the (Pauling) electronegativity difference, Δχ, between the atoms in their ground-state configurations is greater than about 2. However, on polarizability grounds alone, we should anticipate circumstances that challenge the validity of such a simplistic viewpoint based on a single parameter. Despite the fact that atoms are frequently assigned oxidation numbers, as well as the quite common usage of terms like cation and anion in reference to solids, one should not take these to necessarily imply an ionic model. There are some complimentary concepts that allow us to predict when an ionic model is unlikely, even if the Δχ condition is met. First, rules by the Polish-born American chemist Kasimir Fajans (1887–1975) state that the polarizing power of a cation increases with increasing charge and decreasing size of the cation. Likewise, the electronic polarizability of an anion increases with increasing charge and increasing size of the anion (Fajans 1915a, b, 1924). The electronic polarizabilities of some ions in solids (Pauling 1927) are listed in Table 3.2. It is expected that valence electron density on an anion will be polarized toward a highly charged cation, leading to some degree of covalence. Exactly how much charge is highly charged? It has been suggested that no bond in which the formal charge of the cation exceeds about 3+ can possibly be considered ionic (Porterfield 1993). If this is accepted, then it must be conceded that many so-called ionic solids, in fact, may not be so ionic after all. For example, in rutile (TiO2) the nominal formal 4+ charge on titanium would ensure

Table 3.2 Li+

Electronic polarizability of some ions in solids (10−24 cm3). Be2+

B3+

C4+

O2−

F−

0.029

0.008

0.003

0.0013

3.88

1.04

Na+

Mg2+

Al3+

Si4+

S2−

Cl−

0.179

0.094

0.052

0.0165

10.2

3.66

K+

Ca2+

Sc3+

Ti4+

Se2−

Br−

0.83

0.47

0.286

0.185

10.5

4.77

Rb+

Sr2+

Y3+

Zr4+

Te2−

I−

14.0

7.1

1.40

0.86

0.55

0.37

Cs+

Ba2+

La3+

Ce4+

2.42

1.55

1.04

0.73

Source: Adapted from Frederikse, H.P.R. (2001). CRC Handbook of Chemistry and Physics, 82e (ed. D.R. Lide). CRC Press, Inc. and Pauling (1927). © CRC Press. Reproduced with permission.

107

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3 Crystal Structures and Binding Forces

a substantial amount of covalency in the Ti–O bonds, despite a Δχ of almost two. Indeed, band structure calculations using density functional theory (DFT) indicate that titanium bears a +1.2 charge and oxygen −0.6 (Thiên-Nga [actually T.N. Lê] and Paxton 1998), while Hartree–Fock calculations on the series TiO/Ti2O3/TiO2 have shown that Ti–O bond ionicity decreases and bond covalence increases with increasing titanium valency (Evarestov et al. 1997). In 1951, Robert Thomas Sanderson (1912–1989) introduced the principle of electronegativity equalization that proposes, when two or more atoms combine, the atoms adjust to the same intermediate Mulliken electronegativity (Sanderson 1951). DFT tells us that the Mulliken electronegativity is the negative of the chemical potential (Parr et al. 1978). Sanderson’s principle then becomes very appealing in that it can be considered analogous to a macroscopic thermodynamic phenomenon – the equalization of chemical potential. When atoms interact, the electronegativity, or chemical potential, must equalize. Sanderson used his theory to calculate partial charges and ionic radii of atoms in solids and molecules. For example, the partial charges in TiO2 are evaluated as +0.78 and −0.39 for titanium and oxygen, respectively. Although these values are not in exceptional agreement with those obtained from band structure calculations, both sets of results reinforce the idea that high formal oxidation states derived from simple valence rules (e.g. Ti4+, O2− in the case of TiO2) do not reflect true charge – or bond ionicity!

3.2.2

Covalent Bonding

Covalent bonds form between atoms with similar electronegativities. There is electron density concentrated in the region between covalently bonded atoms, both of which have ownership in the electron pair. The American chemist Gilbert Newton Lewis (1875–1946) is generally credited with first describing the covalent bond as a shared electron pair in 1916, before the advent of quantum theory. Whereas there is no such thing as a completely ionic bond, the same cannot be said for covalent bonds. In elemental (homonuclear) silicon, for example, the bonding must be purely covalent. Such bonds are comparable in strength with predominately ionic bonds. Heteronuclear bonds, on the other hand, have a degree of ionicity, or charge transfer that is dependent on the electronegativity difference. The additional ionic interaction further strengthens the bond. Of course, an adequate treatment of the covalent bond, in contrast to the purely electrostatic attraction of an ionic bond, requires that quantum mechanics be invoked, as electrons are described by wave functions. Originally, there were two competing covalent bonding theories developed concurrently. In one, known as the valence bond theory, the wave functions for the bonding electrons are considered overlapping atomic orbitals. Owing to energetic and symmetry constraints, only certain atomic orbitals on each atom can effectively overlap. In the second formalism, molecular orbital (MO) theory, the wave functions are considered as belonging to the molecule as a whole. When molecules pack together to form crystalline molecular solids, the molecules are held together by noncovalent, secondary forces such as van der Waals interactions and hydrogen bonding. In a nonmolecular solid, however, the wave functions are described by crystal orbitals (COs) that are analogous to MOs in molecules and that must be consistent with the translational symmetry, or periodicity, of the crystal. Valence bond theory was the first satisfactory explanation of the stability of the chemical bond and was due to Walter Heitler (1904–1981) and Fritz London (1900–1954) in 1927 (Heitler and London 1927). In valence bond theory, a chemical bond is essentially regarded as being due to the overlap of neighboring valence-level hydrogen-like atomic orbitals, each of which is singularly occupied by an electron with spin opposite to the other. Each electron belongs to its own individual atom, as

3.2 Cohesive Forces in Solids

Figure 3.4 (a) The overlap of two one-electron atomic wave functions, each centered on a different atom, constitutes the Heitler–London (valence bond) theory. (b) A one-electron molecular wave function, or molecular orbital, in the molecular orbital theory of Hund and Mulliken.

(a)

(b)

shown in Figure 3.4a. However, because they are identical except for spin, each electron could belong to either atom. Hence, the total wave function for the molecule is the linear combination of the wave function for both cases. Linus Carl Pauling (1901–1994) and John Clarke Slater (1900–1976) later showed independently that np atomic orbitals and one s atomic orbital could be combined mathematically to give a set of equivalent, or degenerate, singularly occupied spn-hybrid orbitals (Pauling 1931; Slater 1931). The directional effects of the hybridization on the set of orbitals were consistent with the known geometries of several polyatomic molecules of fluorine, oxygen, nitrogen, and carbon. For example, sp3 hybridization of the carbon atom in methane, CH4, explained that molecule’s tetrahedral geometry. Pauling then extended this scheme to transition metal compounds with the inclusion of d atomic orbitals (Pauling 1932). In 1957, Ronald James Gillespie (b. 1924) and Ronald Nyholm (1917–1971) showed that approximate molecular geometry can be predicted simply by consideration of the Lewis structure, which shows the locations of the electron pairs (both bonding and lone pairs). The premise is that the Pauli principle of repulsion of electron domains dictates that orbitals containing valence electron pairs will be oriented to be as far a part as possible. Hence, ligands will arrange themselves about a centrally coordinated atom so as to maximize spherical symmetry. This simplistic, yet extraordinarily accurate, model became known as the valence shell electron pair repulsion (VSEPR) theory (Gillespie and Nyholm 1957). The n electron pairs, which may be bonding pairs or lone nonbonding pairs, form the n vertices of a polyhedron. If n = 2, the geometry is linear; for n = 3, triangular planar; n = 4, tetrahedron; n = 5, triangular bipyramid; n = 6, octahedron; and so on. It is important to include lone pairs in determining approximate geometry. For example, the central oxygen atom of a water molecule is surrounded by two bonding pairs and two lone pairs of electrons. These four electron pairs form the vertices of a tetrahedron even though the three atomic nuclei obviously lie in a plane. The 105 H–O–H bond angle is a little smaller than the ideal 109.5 tetrahedral angle since the unshared pairs spread out over a larger volume of space than that occupied by the bonding pairs. Gillespie and Nyholm’s work actually refined and expanded earlier work by Oxford crystallographer Herbert Marcus Powell (1906–1991) and theoretical chemist Nevil Vincent Sidgwick (1873–1952), which showed the importance of lone pairs (Sidgwick and Powell 1940). Although hybridization (valence bond) and VSEPR theories both proved useful for rationalizing molecular geometry and formalizing Lewis’ idea of shared electron pairs, they are not entirely complete. One of the major shortcomings of the use of equivalent orbitals is the inconsistency with spectroscopic findings. For example, photoelectron spectroscopy provides direct evidence showing that the equivalent electron charge densities in the four bonds of CH4 actually arise from the presence of

109

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3 Crystal Structures and Binding Forces

electrons in nonequivalent orbitals. The approach regarded as being the most accurate explanation of covalent bonding is MO theory. In the MO approach, one-electron wave function, or MOs, extends over all the nuclei in the entire molecule, having equal amplitude at equivalent atoms (Figure 3.4b). In classical (semiempirical) MO theory, molecule formation is explained in terms of a lowering of the electronic energy, with the MOs reflecting the known geometry of the molecule. Through advances in computational methods, ab initio MO theory became possible, in which molecular geometry can be predicted via the principle of total energy minimization (Section 3.3.4). One essentially traverses the entire energy landscape of a system and finds the structure with the lowest energy, which is taken as the thermodynamically stable phase. MO theory originated from the theoretical work of German physicist Friedrich Hund (1896–1997) and its application to the interpretation of the spectra of diatomic molecules by American physical chemist Robert S. Mulliken (1896–1986) (Hund 1926, 1927a, b; Mulliken 1926, 1928a, b, 1932). Inspired by the success of Heitler and London’s approach, Finkelstein and Horowitz introduced the linear combination of atomic orbitals (LCAO) method for approximating the MOs (Finkelstein and Horowitz, 1928). The British physicist John Edward Lennard-Jones (1894–1954) later suggested that only valence electrons need be treated as delocalized; inner electrons could be considered as remaining in atomic orbitals (Lennard-Jones 1929). Simultaneous with, and independent of, the work to explain molecular electronic structure, Felix Bloch (1905–1983), who was a graduate student under Werner Heisenberg trying to explain the conductivity of metals, used Fourier analysis to obtain wave solutions to Schrödinger’s equation for a one-dimensional periodic potential. He showed that the wave functions have equal amplitude at equivalent positions. His thesis, Quantum Mechanics of Electrons in Crystal Lattices, was published in Zeitschrift für Physik (Bloch 1928). In a crystalline solid, therefore, COs can be thought of as extending over the entire crystal, in an analogous fashion to MOs. This is termed the Bloch method, or band theory, and it is the bonding theory most appropriate for nonmolecular crystalline solids. The valence bond and MO (band) models both have been used to explain the cohesive forces in solids. The two theories can be made equivalent in the limit of including additional ionic plus covalent terms in the former and excited-state configurations (configuration interaction) in the latter. Each has its advantages and disadvantages, and the choice of one over the other may be dictated by the type of experimental data (e.g. magnetic measurements) at hand. It seems reasonable to expect the valence bond approach to most accurately reflect the bonding in more ionic solids (Seitz 1940). Many have advocated the argument that, in solids with internuclear distances greater than a critical value, the valence d electrons are also best described with localized atomic orbitals instead of (delocalized) COs (Mott 1958; Goodenough, 1966, 1967). This is equivalent to saying that valence bond, or Heitler– London, theory is more appropriate than MO theory in these cases. For many transition metal compounds, this picture is supported by magnetic data consistent with localized magnetic moments. Localization is a particularly crucial concept for 3d electrons, since they do not range as far from the nucleus as the 4s or 4p electrons. Valence s and p orbitals, by contrast, are always best described by Bloch functions, while 4f electrons are localized and 5f are intermediate. At any rate, whether the valence bond method or delocalized orbitals are used, it is found that covalent bonding is directional since, owing to symmetry constraints, only certain atomic orbitals on neighboring atoms are appropriate for combination. This will be explored further in Section 3.3.5.

3.2.3

Dative Bonds

A dative bond is a two-center, two-electron coordinate covalent bond in which the two bonding electrons originate from the same atom, i.e. the Lewis base. This kind of interaction is most

3.2 Cohesive Forces in Solids

associated with the bonding of metal ions to ligands in a coordination complex or compound. Although most inorganic coordination chemists are focused on molecular complexes containing ligands bound through coordinate covalent bonds (a.k.a. dative bonds) to individual metal atoms or small ensembles of metal atoms, it can be said that certain coordination compounds can be considered as consisting of metal cations coordinated by infinite ligands. Such solid or crystalline coordination compounds are sometimes referred to as polymeric electrolytes. For example, the ether oxygen atoms of poly(ethylene oxide) are capable of coordinating metal cations to yield a solid that is essentially an infinitely long one-dimensional coordination compound (Bruce 1996). There also exist other more or less nonmolecular solid-state compounds with dative bonds present, such as the rare earth metal boro-carbides and boro-nitrides containing infinite two-dimensional layers, onedimensional chains, or finite groups of boron with a metal, and carbon/nitrogen with a rare earth. These extended covalently bonded networks or layers are interconnected by dative bonds with complex anions such as [B2C]4− or [BC]3−. The compounds have been studied for their intriguing properties, such as superconductivity and long-range magnetic order (Niewa et al. 2011). Certain biological materials have also been reported, whose desirable mechanical behavior has been attributed to specific metal coordination-based cross-linking between the constituent biopolymers (Xu 2013). It is impossible to distinguish a dative bond from a covalent bond in a compound once it is formed (the bonding electrons in a compound do not care where they came from), and the difference between the two types of bonds is primarily a matter of electron bookkeeping. For the boron atom to satisfy the octet rule in the anion [BF4]−, for example, a dative bond must be formed. Hence, this anion is customarily considered to be composed of three covalent B–F bonds and one dative B–F− bond, even though the four bonds in this anion are entirely equivalent! An alternative way to view a dative bond is as a highly polar covalent bond. The bond in the HF molecule, for example, could be viewed as a polar covalent H–F bond or as a dative H+–F− bond. Our conceptions of covalent, coordinate, and ionic bonds are idealizations. In reality, chemical bonding is most accurately treated in terms of wave functions and their distribution in space. Nevertheless, most heteronuclear bonds that are best viewed as the coordinate type are typically weaker (i.e. have lesser orbital overlap) than pure nonpolar covalent bonds (i.e. homonuclear bonds), but the same can be said for polar covalent bonds. Therefore, before making generalized assumptions about the comparison between covalent and coordinate bond strengths, it is best to first remember that the bond strength and the polarity/degree of ionicity of both types of bonds are governed by the same factors. Consequently, the physical properties of a material (e.g. its mechanical behavior) that originate chiefly from chemical bond strength and stiffness are, as to be expected, actually dependent on the electronic properties of the atoms involved rather than our preferred bond-type classification. For our purposes, we will consider dative bonds, nonpolar covalent bonds, and polar covalent bonds, all just variants of covalent bonding.

3.2.4 Metallic Bonding The simplest model for a metal is the Drude–Sommerfeld picture of ions occupying fixed positions in a sea of mobile electrons. This model originated in 1900 from Paul Karl Ludwig Drude (1863–1906), and it was initially quite successful at explaining the thermal, electrical, and optical properties of matter (Drude 1900). It was further advanced in 1933 by Arnold Sommerfeld (1868–1951) and Hans Bethe (1906–2005) (Sommerfeld and Bethe 1933). The cohesive forces in this model are readily understood in terms of the Coulomb interactions between the ion cores and the sea of electrons. The ion cores are screened from mutually repulsive interactions by the mobile

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electrons. On the other hand, there are repulsive forces between electrons – electron motion is correlated. Two electrons with parallel spin tend to avoid one another (the Pauli exclusion principle). The e–e repulsive interaction, however, is outweighed by the strong Coulomb attraction between the oppositely charged electrons and ions, which is what bonds the atoms together in the solid. Although the cohesive forces in such an idealized metal as described would be nondirectional (as in ionic solids), the orientation effects of d orbitals contribute a directional covalent component to the bonding in transition metals that requires a more sophisticated definition for metallic bonding. The internuclear distances in the close-packed, or nearly close-packed, structures of most metallic elements are small enough that the valence orbitals on the metal atoms can overlap (in the valence bond model) or combine to form COs (in the MO or Bloch model). The bonding COs are lower in energy and, hence, more stable than the atomic orbitals from which they originate. However, since metal atoms have fewer valence electrons than valence orbitals, there are not enough electrons for the number of two-electron bonds required. For a metallic solid containing an enormous number of orbitals, all the bonding COs (which comprise the valence band) are filled, and the antibonding orbitals (which comprise the conduction band) are only partially filled. The Fermi level (analogous to the highest occupied molecular orbital, or HOMO) thus lies in this partially filled conduction band. There are available states very close in energy (the separation between them is infinitesimal) into which the electrons can be excited and accelerated by an electric field. The electrons are said to be itinerant, which gives rise to a high electrical conductivity. Thus, it can be considered that the bonding in metallic solids is the special case of covalent bonding in which the highest energy electrons (the Fermi level) are in a partially filled conduction band, or delocalized Bloch orbitals. This was the original application of the Bloch scheme. Treating substances other than close-packed elements in the same manner is not restricted. The band theory of solids predicts that any substance, be it an element or compound, close packed or non-close packed, in which the Fermi level lies in a partially filled band of delocalized states, will be metallic.

3.2.5

Atoms and Bonds as Electron Charge Density

If the valence electrons exist in covalent solids as delocalized MOs or COs belonging to the molecule or solid as a whole, as MO theory suggests, how can individual localized atoms and electrons in a molecule or solid be delineated, and what are their properties? It turns out, atoms and chemical bonds, as well as their characteristic properties, can be determined from their electron charge density. This approach, championed by the Canadian quantum chemist Richard F. W. Bader (1935–2012), focuses entirely on the topology of the charge density. Recall that the square of the wave function is a probability density (also referred to as the electron density or charge density), which gives the shape of the orbital. The value of the wave function can be either positive or negative, but the probability of finding the electron near a particular point in space is always positive. The probability is never zero; rather the electron density distribution falls off exponentially toward infinity. Therefore, an orbital has no distinct size. However, an electron orbital is most commonly defined as the region of space that encloses 90% of the total electron probability density. The electron distribution is a physical observable. It is a scalar field in Euclidean three-dimensional space; at every point r in this space, the electron density ρ(r) assumes a specific scalar value that can be measured. Of prime importance in Bader’s method is examination of the gradient of the electron charge density distribution, ∇ρ(r), and its Laplacian, ∇2ρ(r), which is the divergence of the gradient, directly related to the curvatures of ρ(r) along three mutually perpendicular directions. The Laplacian of the electron density shows where ρ is locally concentrated (∇2ρ(r) < 0) or depleted (∇2ρ(r) > 0). The topology of the charge density and, hence, the locations of atoms and bonds in molecules are described by critical points. In general, a critical point for a function of three variables is defined

3.3 Chemical Potential Energy

z

0

y

x

Figure 3.5 A saddle point for a function of two variables.

as a point where all three partial derivatives are equal to zero: ∂/∂x = ∂/∂y = ∂/∂z = 0. In the case of ρ(r), a critical point is one for which ∇ρ(r) = 0 or in which the electron density has a zero gradient in all three directions. A critical point can be a local minimum, a local maximum, or a saddle point. A local maximum in ρ(r) corresponds to the position of the center of an atom. This is the position of maximum electron density, from which the density falls off in all three mutually perpendicular directions of space (Bader 1990); they are designated as (3, −3). Minima correspond to the cavities between the atoms, from which electron density rises in all three directions of space; they are symbolized as (3, +3). In addition, there are two types of saddle points, designated as (3, −1) and (3, +1), in which, respectively, the electron density falls down in two directions and rises in the third, or falls in one direction and rises in the two other perpendicular directions. An example of a saddle point in two dimensions is shown in Figure 3.5. A bond, which is an accumulation of charge density between nuclei, manifests itself in three dimensions as a ridge of maximum electron density connecting two nuclei. The density along this path is a maximum with respect to any neighboring path. The ridge’s existence is guaranteed by the presence between bound nuclei of a (3, −1) saddle point, appropriately called the bond critical point, rc, for which ∇ρ(r) vanishes (Bader 1990). The line running along the top of the ridge delineates the bond path. The absence of such a line indicates the absence of a chemical bond. As described in the preceding paragraph, the point rc is characterized by three curvatures of ρ(r) along the three mutually perpendicular directions. The Laplacian model is directly related to these curvatures and it also defines the regions where ρ(r) is locally greater or less than its average value in the vicinity of r (Gibbs et al. 1999). A fruitful application of this view of chemical bonding can be seen in Chapter 10 where the mechanical properties of materials are studied.

3.3

Chemical Potential Energy

Chemical potential energy is a form of potential energy related to the structural arrangement of atoms or molecules. Related terms, all of which will be discussed in this section, are lattice energy, internal energy, and total energy. In the absence of any kinetic constraints, the atoms in a solid

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arrange themselves into a crystalline or noncrystalline configuration that minimizes their overall energy. This basically equates to geometry optimization; in other words, regardless of the type of interatomic bonding present, optimal spatial arrangements will be adopted such that either a local minimum (thermodynamically metastable states) or the global minimum (the thermodynamically most stable state) in the overall energy will be reached. As binding energies, preferred partners, and coordination preferences are all highly chemically specific (dependent on the atoms involved), it is of no surprise this gives rise to the rich structural diversity in inorganic solid-state chemistry. If one further considers structural distortions and defects, the possibilities are enormous. In this section, some guidelines for predicting local coordination geometries are presented. This is a well-established field in inorganic chemistry. The challenging part is predicting the way these coordination polyhedra interconnect to form a crystalline solid with long-range translational order. The process is straightforward for simple solids if one knows the local coordination. A case in point is an array of sp3-hybridized carbon atoms, which must adopt the diamond structure where each vertex of every CC4 tetrahedron is shared with three other tetrahedra. The predictive power for polyhedral connectivity decreases, however, with an increase in the number of constituents, each with its own coordination preference. Advances have been made, and one approach will be discussed in Section 3.6. In some cases, a particular constituent may actually be found in more than one coordination environment. For example, it has been reported that the Ti4+ ion takes on both octahedral and tetrahedral coordination by O2− in Ba6Nd2Ti4O17 (Kuang et al. 2002); that is, this oxide exhibits layers of face-sharing TiO6 octahedra as well as TiO4 tetrahedral layers. The rational design of novel structures possessing predetermined physical properties thus remains an extremely daunting task. From the perspective of the synthetic chemist, it is often fruitful to take a compound with a known thermodynamically stable or kinetically inert crystal structure and alloy one of more of the sublattices to form a stable solid solution in which the product phase retains the same basic crystal structure of the parent phase. In this way, the properties of the parent phase can be tuned to meet the requirements for a particular engineering function. It is reasonable to assume that atoms of similar size, valence, and coordination preference can be exchanged on a sublattice. For now, this chapter will go ahead and examine some simple factors that are useful to the synthetic chemist and that do not require sophisticated or time-consuming computational methods, beginning with ionic solids. A similar spirit will be found in Sections 10.2.3.3.2 and 10.2.3.3.3 for metallic and covalent solids, respectively.

3.3.1

Lattice Energy for Ionic Crystals

The lattice energy, U, of an ionic crystal is defined as the potential energy per mole of compound associated with the particular geometric arrangement of ions forming the structure. It is equivalent to the heat of formation from 1 mol of its ionic constituents in the gas phase. Equation (3.1) does not give the total energy of attraction for a three-dimensional array of ions. Calculation of the total electrostatic energy must include summation of long-range attractions between oppositely charged ions and repulsions between like-charged ions, extending over the whole crystal, until a physically correct convergent mathematical series is obtained (because of the alternated signs in the series due to the oppositely charged ions, the series are only semi-convergent. Sums over concentric electrically neutral groups lead to the correct physical value). The pair potential approximation is presumed valid, whereby pair interactions are assumed to dominate and all higher interactions are considered to be negligible. As a result, only two-body terms are included.

3.3 Chemical Potential Energy

The sums may be carried out with respect to the atomic positions in direct (real) space or to lattice planes in reciprocal space, an approach introduced in 1913 by Paul Peter Ewald (1888–1985), a doctoral student under Arnold Sommerfeld (Ewald 1913). In reciprocal space, the structures of crystals are described using vectors that are defined as the reciprocals of the interplanar perpendicular distances between sets of lattice planes with Miller indices (h k l). In 1918, Erwin Rudolf Madelung (1881–1972) invoked both types of summations for calculating the electrostatic energy of NaCl (Madelung 1918). Although direct-space summations may be conceptually simpler, convergence can be time consuming, if not problematic in 3d. Bertaut developed a method that achieves good convergence, utilizing reciprocal space summations exclusively (Bertaut 1952; Félix Bertaut, French crystallographer born Erwin Lewy in the Polish part of the German Empire in 1913, died in Grenoble in 2003). The most generally accepted way, however, was presented by Ewald in 1921 (Ewald 1921). In this method, an array of point charges neutralized by Gaussian charge distributions are summed in direct space, and an array of oppositely charged Gaussian distributions neutralized by a uniform charge density are summed in reciprocal space. A derivation may be found in the book by Ohno et al. (1999). Little would be gained in reproducing this arduous procedure here. Suffice it to say, when these mathematical summations are carried out, over larger and larger crystal volumes until convergence is achieved, a number known as the Madelung constant is generated (see Example 3.4), such that the final expression for the long-range force on an ion is U ion = −

αqi qj e2 1 r ij 4πε0

33

in which ε0 is the permittivity of free space (8.854 × 10−12 C2/J/m), e is the electron charge (1.6022 × 10−19 C), q is the ion charge, and α is the Madelung constant. The Madelung constant is dependent only on the geometric arrangement of ions and the distance that r is defined in terms of nearestneighbor, unit cell parameter (beware of different conventions when using Madelung constants from the literature!). The constant has the same value for all compounds within any given structure type. If r is in meters, the units of Eq. (3.3) will be in Joules per cation. Johnson and Templeton calculated values of α for several structure types using the Bertaut method (Johnson and Templeton 1962). Their results are partially reproduced in Table 3.3, where the second column lists the Madelung constant based on the shortest interatomic distance in the structure. The third column gives reduced Madelung constants based on the average shortest distance. For less symmetric structures, that is, when there are several nearest neighbors at slightly different distances, as in the ZnS polymorphs, the reduced Madelung constant is the more significant value. In order to obtain the complete expression for the lattice energy of an ionic crystal, it is necessary to 1) add the term representing the short-range repulsive forces; 2) include Avogadro’s number, N (6.022 × 1023 mol−1); 3) make provisions for ensuring that pairs of interactions are not overcounted (multiplying the entire expression by one half ). In so doing, the final expression for the lattice energy of an ionic crystal containing 2N ions was given by Max Born and Alfred Landé (1918) as U lattice = −

N αqi qj e2 B + n r ij 4πε0 r ij

34

115

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3 Crystal Structures and Binding Forces

Table 3.3

Madelung constants for several structure types.

Compound

M(R0)

MR

Al2O3 (corundum)

24.242

1.68

BeO

6.368

1.64

a

CaCl2

4.730

1.601

CaF2 (fluorite)

5.03879

1.68

CaTiO3 (perovskite)

24.7550



CdCl2

4.489

1.50

CsCl

1.76268

1.76

Cu2O

4.44249

1.48

La2O3

24.179

1.63

LaOCl

10.923



MgAl2O4

31.475



MgF2

4.762

1.60

NaCl (rock salt – halite)

1.74756

1.75

SiO2 (quartz)

17.609

1.47

TiO2 (anatase)

19.0691

1.60

TiO2 (brookite)

18.066

1.60

TiO2 (rutile)

19.0803

1.60

V2O5

44.32

1.49

ZnO

5.99413

1.65

ZnS (zinc blende)

6.55222

1.638

ZnS (wurtzite)

6.56292

1.641

a

M R = M(R0) R /R0, where R0 is the shortest interatomic distance and R is the average shortest distance. Source: From Johnson, Q.C. and Templeton, D.H. (1962). J. Chem. Phys. 34: 2004.

Equation (3.4) gives the lattice energy in Joules per mole. We can avoid having to determine a value for the parameter B by using the equilibrium interatomic distance as the value of r for which U is a minimum. This gives dU/dr = 0 and the following expression for U: U lattice = −

N αqi qj e2 1 1− r ij 4πε0 n

35

where n is the Born exponent introduced in Section 3.2.1. If the value of n is not known, an approximate value may be obtained from Table 3.4. The interatomic potential between a pair of ions in the lattice is given by Eq. (3.4) minus Avogadro’s number. It will be seen in Section 10.2 how the elastic modulus for an ionic solid can be estimated from such an expression by taking the second derivative with respect to r. Example 3.3 Some transition metal oxides contain ion exchangeable layers. In many cases, the Mn+ cations in these layers are amenable to aliovalent ion exchange with M(n + 1)+ ions of similar size. No structural change, other than possibly a slight expansion or contraction of the unit

3.3 Chemical Potential Energy

Table 3.4

Values of the born exponent.

Cation–anion electron configurations

Example

1s2 – 1s2

LiH

5

1s22s2p6 – 1s22s2p6

NaF, MgO

7

n

[Ne]3s p – [Ne]3s p

KCl, CaS

[Ar]3d104s2p6 – [Ar]3d104s2p6

RbBr, AgBr

10

[Kr]4d105s2p6 – [Kr]4d105s2p6

CsI

12

2 6

2 6

9

For mixed-ion types, use the average (e.g. for NaCl, n = 8).

O2–

O2–

Na+

Na+

O2–

O2–

Na+

Na+

O2–

1

1 ... 6

Na+

O2–

Ca2+

O2–

r 2r 3r U=–

O2–

2q2 r

O2–

Ca2+

1–

1 2

O2–

+

1 3

Ca2+

1



4

+

5



O2–

O2–

r 2r 3r U=–

2q2 r

1–

1 2

+

1 3



1 4

+

1 5



1 ... 6

Figure 3.6 Calculation of the Madelung constant for a one-dimensional array of cations and anions, before and after aliovalent ion exchange.

cell, occurs. In order to maintain charge neutrality, aliovalent exchange requires the introduction of a vacancy for every Mn+ ion exchanged. Based on lattice energy considerations, would you expect the ion-exchanged product to be favored? Solution To simplify the solution, imagine an analogous one-dimensional array as shown in Figure 3.6. Carry out a direct-space summation of the long-range attractive and repulsive forces felt by any arbitrary ion, extending across this one-dimensional array, before and after ion exchange. Before aliovalent ion exchange,

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U=

−2q2 2q2 2q2 2q2 + − + + r 2r 3r 4r

which can be written as U=

−2q2 1 1 1 1− + − + r 2 3 4

The second term in parenthesis may be recognized as an alternating series: 1−

1 1 1 + − + 2 3 4



−1

=

n+1

n=1

1 n

This series converges to ln 2, or ~0.69. Thus, the Madelung constant for the array corresponds to α = 2 × 0 69 = 1 38 For the ion-exchanged array, each M+ ion is exchanged with one M2+ cation and one vacancy. If the M2+ and M+ ions are approximately the same size (e.g. Na+ and Ca2+), r has essentially the same value. However, the absence of some of the cation–cation repulsion terms (every fourth term 1 now forces the new alternating series to converge to ~2.2, resulting in a Madebeginning with 2 lung constant of (2 × 2.2) = 4.4. The aliovalent ion-exchanged product is thus favored because of both a larger Madelung constant and a larger product of ion charges in the lattice energy expression. A one-dimensional analogy was chosen simply to illustrate the general summation procedure. To extend the summation to a real three-dimensional crystal requires the inclusion of a considerable number of additional terms, in which case a hand calculation can become significantly less tractable. It should be noted that for complex (ternary and higher order) compounds, the structure can be considered the superposition of simpler structures. For example, with the cubic perovskite structure of formula AIIBIVO3, the Madelung constant can be expressed as a combination of those of CsCl and Cu2O (Fumi and Tosi 1960). This is possible because linear relationships exist between the Madelung constants of different structures based on the same Bravais lattice. In cases where there are significant contributions from covalent bonding, Eqs. (3.4) and (3.5) will not reflect the true binding energy of the crystal. Nevertheless, they are still useful in comparing relative energies for different compounds with the same structure. For example, even though the perovskites, ABO3, possess significant covalent character, the ionic model is very useful because the stability of this structure is largely owing to the Madelung energy. Example 3.4 For the ternary oxide CaTiO3 (perovskite structure), write (i) the general expression for the lattice energy and (ii) the expression for the electrostatic attraction. Solution In this lattice, there is one type of anion (i), O2−, and two types of cations (j, k), Ca2+ and Ti4+. The general expression for the lattice energy, including all the summations between the anions and each type of cation over a given volume, is given by U lattice =

1 2 1 e 1− 2 n

p

m

j

i

qi qi + 4πε0 r ij

s

m

k

i

qi qk 4πε0 r ik

3.3 Chemical Potential Energy

where m is the number of O2− anions, p is the number of Ca2+ cations, and s is the number of Ti4+ cations. This expression includes the long-range attractive (Coulombic) terms and the short-range repulsive terms. The expression for the electrostatic attraction is given by U lattice =

1 2 qi qj α qq α + i k e 2 4πε0 r ij 4πε0 r ik

where the overall Madelung constant is a linear combination of those of the Cu2O and CsCl lattices, namely, α CaTiO3 = 2α Cu2 O + 4α CsCl , for cubic perovskites of formula AIIBIVO3. The determination of the Madelung constant for a crystal structure requires the evaluation of electrostatic self-potentials of the structure. Any lattice energy can also be expressed using lattice site self-potentials (which is what Ewald actually calculated): U lattice = Ne2 j

qj pj φj 2κ

where qj is the total charge number for point j, pj is the frequency of occurrence of point j in the unit cell, φj is the electrostatic potential for that lattice site, N is Avogadro’s number, and κ is the total number of molecules in the unit cell. In this expression, the Madelung constant is defined as α = −a j

qj pj φj 2κ

where α is described relative to the length a of the unit cell. Using self-potentials, we have, for perovskite (ABO3), U lattice = Ne2 qA φA + qB φB + 3qO φO where pA = pB = 1 and pO = 3; κ = 1 and qA + qB + 3qO = 0.

3.3.2 The Born–Haber Cycle Lattice energies of ionic crystals cannot be measured experimentally since they represent hypothetical processes: Mn + g + X n − g

MX s

However, the following reaction sequence, relating the heat of formation, ΔHf of a crystal M s + 12 X 2 g MX s to U[M+(g) MX(s)] is thermochemically equivalent (and ΔHf can be measured). M0 g + X 0 g

M+ g + X− g

ΔH 0s

D 1 M s + X2 g 2

MX s

In this diagram, ΔH 0s gives the enthalpy of sublimation of the metal [M(s) M0(g)]; D gives the dissociation energy, or bond energy of the diatomic gas 12 X 2 g X 0 g ; IE gives the ionization 0 + energy of the gaseous metal [M (g) M (g)]; and EA gives the electron affinity for the formation of the gaseous anion [X0(g) X−(g)]. The lattice energy is obtained through the relation

119

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3 Crystal Structures and Binding Forces

U = ΔH f − ΔH s +

1 D + IE + EA 2

36

Max Born (1882–1970, see his biography in Chapter 4) invented the reaction sequence and could check his formula thanks to data provided to him by Fritz Haber who immediately developed a method of representing Born’s procedure. Kasimir Fajans, Polish German American chemical physicist (1887–1975), discovered the same idea at exactly the same time, in 1919 in Munich (Born and Haber were in Berlin). Fritz Haber (1868–1934) had just won the 1918 Nobel Prize in Chemistry for his discovery of an efficient way to synthesize ammonia from nitrogen and hydrogen, a most important discovery for large-scale synthesis of both fertilizers and explosives. One difficulty with using a Born–Haber cycle to find values for U is that heats of formation data are often unavailable. Perhaps the greatest limitation, however, is that electron affinities for multiply-charged anions (e.g. O2−) or polyanions (e.g. SiO44 − ) cannot be experimentally obtained. Such anions simply do not exist as gaseous species. No atom has a positive second electron affinity; energy must be added to a negatively charged gaseous species in order for it to accommodate additional electrons. In some cases, thermochemical estimates for second and third electron affinities are available from ab initio calculations. Even so, if there are large covalent forces in the crystal, poor agreement between the values of U obtained from a Born–Haber cycle and Madelung calculations can be expected.

3.3.3

Goldschmidt’s Rules and Pauling’s Rules

Some guiding principles, enunciated by the Swiss-born Norwegian geochemist Victor Moritz Goldschmidt (1888–1947) and Linus Pauling, make possible the rationalization and prediction of the polyhedral connectivity for simple ionic solids. Three rules were devised by Goldschmidt to explain element distributions in minerals (Goldschmidt et al. 1925, 1926a, b, c). The basis of these rules is that ionic substitution of one cation by another is governed by the size and charge of the cations. The first rule is that extensive substitution of one cation for another can only occur with cations of the same size and charge. The second and third rules are that cations of smaller size and same charge, or same size and higher charge will preferentially incorporate into a growing crystal. Pauling subsequently introduced three rules governing ionic structures (Pauling 1928, 1929). The first is known as the radius ratio rule. The idea is that the relative sizes of the ions determine the structure adopted by an ionic compound. Pauling proposed specific values for the ratios of the cation radius to the anion radius as lower limits for different coordination types. These values are given in Table 3.5.

Table 3.5

Radius ratio rules.

rc/ra < 0.16

Threefold

0.16 < rc/ra < 0.41

Fourfold

0.41 < rc/ra < 0.73

Sixfold

0.73 < rc/ra < 1.00

Eightfold

1.00 > rc/ra

Twelvefold

3.3 Chemical Potential Energy

Unfortunately, the radius ratio rules are incorrect in their prediction of coordination numbers about as often as they are correct. Usually, it overestimates the coordination number of the cation. This model essentially regards ions as hard, incompressible spheres, in which covalent bonding is not considered. The directionality, or overlap requirements, of the covalent bonding contribution probably plays a significant role as ion size fitting. Pauling’s second rule is the electrostatic valence rule. It states that the charge on an ion must be balanced by an equal and opposite charge on the surrounding ions. A cation, Mm+, coordinated by n anions, Xx−, has an electrostatic bond strength (EBS) for each bond defined as m 37 EBS = n Charge balance, then, is fulfilled if m =x n

38

The third rule by Pauling states that the presence of shared polyhedron edges and faces destabilizes a structure. Polyhedra tend to join at the vertices (corners). Cations strongly repel each other as edges and faces are shared because the cation–cation distance decreases. The smallest decrease occurs for octahedral edge sharing, followed by tetrahedral edge sharing and octahedral face sharing, and the largest decrease is for tetrahedral face sharing, which makes this particular configuration quite unfavorable. Vertex-, edge-, and face-sharing octahedra are all commonly observed stable arrangements, as are vertex-sharing tetrahedra. Edge-sharing tetrahedra are not very common, and the tetrahedra are usually distorted, particularly if the cations are highly charged. Facesharing tetrahedra are not generally observed. Linus Carl Pauling (1901–1994) obtained his BS in chemical engineering from Oregon Agricultural College, now Oregon State University, and subsequently earned a Ph.D. in physical chemistry from the California Institute of Technology in 1925 under Roscoe G. Dickinson. In 1926, Pauling was awarded a Guggenheim Fellowship, on which he traveled to Europe where he worked under physicist Arnold Sommerfeld at the University of Munich. It was here where he first applied quantum mechanics to chemical bonding. While in Europe, Pauling also worked with Niels Bohr, Erwin Schrödinger, Walter Heitler, and Fritz London. This period was the beginning of Pauling’s extraordinary career, which spanned nearly seventy years. He became an assistant professor at Caltech in 1927, climbing to the rank of professor in 1931. He stayed at Caltech for 40 years. Pauling was a scientist of great versatility, having carried out research in numerous areas including crystallography, inorganic and physical chemistry, the theory of ferromagnetism, and molecular biology. He was awarded the Nobel Prize in chemistry in 1954 for his work on chemical bonding and molecular structure. Solid-state chemists are also indebted to Pauling for rules predicting the structures of ionic solids and for his work on the structures of metals and intermetallic compounds. Pauling’s work was not without disputes. W.L. Bragg accused him of stealing ideas on chemical bonding. Pauling debated his contemporaries on the merits of both MO theory and band theory as opposed to the more simplistic valence bond theory, and he expressed his disbelief in the existence of the quasicrystalline state. Pauling also endured great controversy in his personal life. His opposition to nuclear weapons and outspokenness on other war-related issues were regarded with suspicion by the U.S. government. Twice he was forced to appear before a U.S. Senate subcommittee to defend his views. He was even temporarily denied the right to travel abroad. However, in 1963, Pauling was awarded the Nobel Peace Prize for his efforts to ban nuclear testing. Pauling also was criticized late in life for his belief in the health benefits of high doses of vitamin C, which was never established or confirmed in extensive clinical trials conducted by the Mayo Clinic. Pauling was elected to the United States National Academy of Sciences in 1933. (Source: My Memories and Impressions of Linus Pauling by David Shoemaker, 1996. Courtesy Ava Helen and Linus Pauling Papers, Oregon State University Libraries.) (Photo from The Big T, the yearbook of the California Institute of Technology, in the public domain.

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3.3.4

Total Energy

It has been seen that it is possible to estimate the total binding energy, or lattice energy, of an ionic crystal by summing the long-range electrostatic interactions. From this expression, the interatomic potential energy between a pair of ions in the lattice may be obtained. In covalent solids, the binding energy of the crystal corresponds to the difference between the total energy of the electrons in the crystal and the electrons in the same – but isolated – atoms. One reason might then be that the binding energy of a covalent crystal could somehow be estimated from tabulated bond energies. However, such an approach is inherently flawed because bond strength values obtained from bond energy tables generally represent an average evaluation for the bonds in gaseous diatomic molecules, in which case the bonding is substantially different from that in a solid. The calculation of interatomic potentials in covalent solids must account for many-body effects, since the interaction between a pair of atoms is modified by the surrounding atoms. The LCAO tight-binding (Hartree–Fock) method and its successors, the post-Hartree–Fock methods, and the DFT, were originally used to solve electronic structure problems. All have also been applied to the calculation of total energy, which includes contributions due to core–core, core– electron, and electron–electron interactions. By varying the coordinates of the core, the dependence of the total energy on the core coordinates can be examined. This is referred to as the potential surface (also called energy landscape) of the system. These types of calculations are called first principles, ab initio, or quantum mechanical methods. They are much less amenable to large system size calculations and simulations (molecular dynamics, MD, and Monte Carlo Metropolis, MC) than classical (nonquantum mechanical) modelization of atomic interactions, such as pairwise Lennard-Jones or more sophisticated n-body phenomenological interactions which provide no information about the electronic structure. In quantum MD simulations, the motion of the atomic cores is still treated classically, using Newton’s law (or Hamilton–Jacobi’s laws). The electrostatic interactions between cores are often approximated as a sum over pair potentials, including those Coulomb interactions between pairs of ions separated by long distances, which is very similar to the expression for the Madelung energy. However, the (classical) force on the core owing to the electron system is also included, while the electronic structure problem itself is solved quantum mechanically. Alternatively, by keeping the core positions fixed, the quantum-molecular-dynamics method can be restricted to just the electronic structure component of the total energy. The concept of total energy is very similar, but not identical, to internal energy. Both are the sum of the potential energy and kinetic energy contributions in a system. However, internal energy is an extensive thermodynamic state function for a macroscopic system (~1023 atoms), irrespective of atomic numbers and atomic geometry. Furthermore, absolute values of internal energy are not defined by laws of thermodynamics, only changes in U that accompany chemical reactions or physical (phase) transformations of macroscopic substances. In DFT, the total energy is decomposed into three contributions: the electron kinetic energy, the interactions between the cores, and a term called the exchange–correlation energy that captures all the many-body interactions. Note the similarity with the statistical mechanical interpretation for absolute internal energy given earlier. In fact, the total energy can be argued to accurately represent a sort of specific absolute internal energy, making it an intensive property. Total energy is an intrinsic physical property that may be calculated for a single molecule, or, using a finite-sized representative collection of atoms (usually on the order of 103), for a macroscopic solid. This is somewhat akin to the way the Madelung energy for an ionic crystal is calculated; only total energy also includes electronic contributions as well.

3.3 Chemical Potential Energy

Other thermodynamic functions, in addition to internal energy, can also be calculated from first principles. For example, at a finite temperature, the Helmholtz free energy, A, of a phase containing Ni atoms of the ith component, Nj atoms of the jth, and so on, is equal to the DFT total energy at 0 K (neglecting zero-point vibrations) plus the vibrational contribution (Reuter and Scheffler 2001): A T, V , N i , …, N j = E tot V , N i , …, N j + Avib T, V , N i , …, N j

39

The Helmholtz free energy, in turn, is related to the Gibbs free energy, G, via G T, P, N i , …, N j = A T, P, N i , …, N j + pV T, P, N i , …, N j

3 10

The DFT total energy at zero kelvin is thus a reference system for determining the free energy, corresponding to a simple Einstein solid with all the atoms vibrating around lattice points independently with the same frequency. The vibrational contribution, Avib(T, V, Ni, …, Nj), contains the vibrational energy (including the zero-point energy), Evib, and entropy, Svib. Each of these components can be calculated from the partition function for an N-component system, Z, and using the relation Avib = Evib − TSvib. The reader is referred to Ashcroft and Mermin (1981) for the partition function expression, as well as the vibrational and entropy defined in terms of Z. As with thermochemical techniques, the most stable structure can be predicted from DFT total energy calculations by the principle of energy minimization – global energy minimization. Metastable structures that are kinetically stable can also be predicted by locating local minima in the potential surface surrounded by sufficiently high-energy barriers. It becomes increasingly difficult to predict definitely the structure, stability, and properties of solids, as the system size increases. This is because approximations have to be employed for the energy calculations on large systems with multiple minima. Ab initio methods are generally very computationally expensive. That is, they require an enormous amount of computer resources (time, memory, and disk space). Hartree–Fock methods typically scale as N4 (where N is the number of basis functions), which means a calculation twice as large takes 16 times as long to converge. Post-Hartree–Fock methods generally scale less favorably (MP2 = N5), while DFT scales similarly to Hartree–Fock methods. In general, the systems readily treated by these methods are limited to relatively small numbers of atoms, say, a few hundred. In order to make accurate total energy calculations, one must investigate the entire energy landscape having an approximate crystal structure in mind, which is used as a starting point for optimizing the geometry to obtain the lowest energy structure. Investigating anything less than the entire energy landscape runs the risk of not revealing the true minimum. The structure predictions can sometimes be surprisingly different from what would be expected based on chemical intuition. One such case involves the group IV(B) nitrides, which adopt the spinel structure at high pressures and temperatures. There are two possible cation distributions in spinel: the normal spinel, AB2X4, in which there is no alloying distribution of A or B cations on either the four-coordinate site (A), or eight-coordinate site (B); and the inverse spinel, B(AB)X4, with equal amounts of A and B cations randomly distributed over the octahedral sites. Since it is well known that silicon tetrahedral bonds are stronger than germanium tetrahedral bonds and that the Si4+ ion is smaller than the Ge4+ ion, initial DFT investigations were restricted to the possible normal configurations. It was predicted that this nitride should prefer to adopt the structure SiGe2N4, in which the four-coordinate sites are exclusively occupied by silicon and the eight-coordinate sites by germanium, and that GeSi2N4 was unstable/metastable to decomposition into the binary nitrides Si3N4 and Ge3N4. However, the reverse cation site preference (SixGe1−x)3 N4 was subsequently observed by X-ray powder diffractometry

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experiments in polycrystalline samples with x ≈ 0.6 (Soignard et al. 2001). These silicon and germanium site preferences were later confirmed by density functional total energy calculations (Dong et al. 2003). The observed geometry apparently provides for a much more symmetrical distribution of bond lengths around the four-coordinate nitrogen atom (sp3 hybridized). In fact, the normal spinel, GeSi2N4, where the tetrahedral sites are exclusively occupied by silicon and the octahedral sites are exclusively occupied by germanium, is predicted to be the only stable crystalline phase in the Si3N4–Ge3N4 system, although kinetically stable metastable solid solutions may exist for Gerich and Si-rich compositions (Dong et al. 2003).

3.3.5

Electronic Origin of Coordination Polyhedra in Covalent Crystals

In valence bond theory, the coordination number of an atom in a molecule or covalent solid is generally limited to the number of valence orbitals on the atom. Likewise, only certain combinations of atomic orbitals on the atoms involved are suitable for forming MOs possessing the point group symmetry of the molecule, or COs with the proper space group symmetry of the crystal. Furthermore, molecules are most stable when the bonding MOs or, at most, bonding plus nonbonding MOs are filled with electrons and the antibonding MOs are empty. These principles form the quantum mechanical basis of G. N. Lewis’ and Irving Langmuir’s octet rule for compounds of the p-block elements and for the 18-electron rule (also known as the effective atomic number rule) for d-block elements by N.V. Sidgwick. In covalent solids, a pair of electrons with opposing spins occupies each twocenter bonding site in the CO. The geometries of molecules and coordination polyhedra in covalent solids are thus determined by the types of valence orbitals contributed by the atoms involved. Knowing the molecular or crystal geometry allows us to sketch the overlapping atomic orbitals. This, in turn, enables construction of a qualitative energy-level diagram, from which we can verify the principles just discussed. There is, of course, a prescribed group theoretical treatment to be followed. This procedure is amply covered in many specialized textbooks on MO theory (e.g. Cotton 1990). It will be beneficial for us to very briefly review the basic methodology applied to molecules for later comparison with solids. With molecules, one first finds the irreducible representations to which the central atom atomic orbitals belong and then construct ligand group orbitals, which are symmetry-adapted linear combinations (SALCs) of the ligand atomic orbitals. For example, if the coordinate system for the ligand atomic orbitals in a molecule of Td symmetry is as shown in Figure 3.7, group orbitals can be found that transform according to the same rows of the same irreducible representations as the central atom orbitals. Only orbitals that have the same symmetry around the bond axes can form MOs. Figure 3.8 shows a generalized energy-level diagram for a tetrahedral molecule with σ- and π-bonds involving only s and p atomic orbitals (e.g. CCl4, ClO4− ). The Mulliken symmetry labels are shown for the MOs, which are the same as the labels for the irreducible representations to which they belong. Hans Albrecht Bethe showed that the electron wave functions (the MOs) can be chosen to transform like, or belong to, irreducible representations of the point group of the molecule (Bethe 1929). Electrons in the MOs, beginning with the lowest energy orbital (the aufbau, “building-up,” principle), can then be placed as is done with placing electrons in atomic orbitals. Each MO can hold two electrons, of opposite spin. Hund showed that the state with maximum spin multiplicity is the lowest in energy (Hund 1926). Thus, in degenerate sets, electrons are added singularly, one to each orbital, before double occupancy occurs. In CCl4, for example, there are 32 electrons available for filling the MOs. It can be seen that all of the bonding and nonbonding MOs in Figure 3.8 will be filled with electrons, the antibonding MOs will be empty, and the octet rule is

3.3 Chemical Potential Energy

z z4

x4

y4 x1

z1

y1

y x

x2

x3

z2

y3

y2

z3

Figure 3.7 The coordinate system for the atomic orbitals in a tetrahedral molecule. MX4

M

4X

T2(σ*,π*) A1σ*

T1π

p

p SALCs s

T2σ A1σ T2π

s SALCs

Eπ T2σ A1σ

Figure 3.8 A generalized MO energy-level diagram for a tetrahedral molecule with σ- and π-bonding involving only s and p atomic orbitals.

obeyed around the central carbon atom. Note also, an energy gap separates the HOMO and the lowest unoccupied molecular orbital (LUMO). It is important to realize, however, that the symmetry considerations alone provide no quantitative information on the actual energy levels. Now consider the nonmolecular solid diamond, which may be considered built of two interlocking carbon FCC sublattices, displaced by a quarter of the body diagonal. The structure is shown in Figure 3.9a. There are two atoms associated with each diamond lattice point, or basis, corresponding to a carbon atom on one FCC sublattice and one of its nearest-neighbor carbon atoms on the

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(a)

(c) s BO + s BO

(d) Г′2 s BO + py BO

(b)

s BO + px BO

s BO + pz BO

Г′25

Г15 Г1

1 a along 4 1 1 1 . In this projection along the [0 0 1] direction, only the top face of each cube is shown. (b) The unit cell. (c) Some possible sign combinations of the basis atomic orbitals used to construct LCAO COs from two Bloch sums. (d) A qualitative CO energy-level diagram for the center of the Brillouin zone, Γ = k(0, 0, 0). Figure 3.9

(a) The diamond structure viewed as two interlocking FCC sublattices displaced by

other FCC sublattice. These are the atoms at the vertices of each square in Figure 3.9a, which shows the top face of each interpenetrating cube, looking down the [0 0 1] direction. Every carbon has four such nearest neighbors that form a tetrahedron (Td point group symmetry). Thus, diamond may also be described as a three-dimensional network of vertex-sharing CC4 tetrahedra. The tetrahedral coordination can be more easily seen in the three-dimensional representation of the unit cell shown in Figure 3.9b. The electronic structure of diamond is well described by considering both nearestneighbor and second-nearest-neighbor interactions of the following type: ssσ, ppσ, ppπ, and spσ. Compliance with the octet rule in diamond could be shown simply by using a valence bond approach in which each carbon atom is assumed sp3 hybridized. However, using the MO method will more clearly establish the connection with band theory. In solids, the extended electron wave functions analogous to MOs are called COs. COs must belong to an irreducible representation, not of a point group, but of the space group reflecting the translational periodicity of the lattice. COs are built by combining different Bloch orbitals (which we will henceforth refer to as Bloch sums), which themselves are linear combinations of the atomic orbitals. There is one Bloch sum for every type of valence atomic orbital contributed by each atom in the basis. Thus, the two-carbon atom basis in diamond will produce eight Bloch sums – one for each of the s and p atomic orbitals. From these eight Bloch sums, eight COs are obtained, four bonding and four antibonding. For example, a Bloch sum of s atomic orbitals at every site on one of the interlocking FCC sublattices in the diamond structure can combine in a symmetric or antisymmetric fashion with the Bloch sum of s atomic orbitals at every site of the other FCC sublattice. Alternatively, a Bloch sum of s atomic orbitals could combine with a Bloch sum of p atomic orbitals. The symmetric (bonding) combinations of the basis atomic orbitals for the latter case are illustrated for one CC4 subunit in Figure 3.9c. The actual COs are delocalized over all the atoms with the space group symmetry of the diamond lattice. A LCAO–CO construction from Bloch sums is thus completely analogous to a LCAO–MO construction from atomic orbitals. It should be pointed out that some authors refer to COs as Bloch orbitals too, since a linear combination of Bloch orbitals is also a Bloch orbital. It will be worthwhile to look at this in yet a little more detail. First, note that the coordinate system of the atomic orbitals does not rotate at the vertices of the tetrahedra (representing one of the FCC sublattices) in constructing Bloch sums for diamond, as it did for the CCl4 molecule. In fact, as

3.4 Common Structure Types

mentioned, the basis (a.k.a. asymmetric unit, see Section 1.2.1) for the primitive unit cell for the diamond structure consists of only two atoms per each lattice point, not the five atoms in the CC4 tetrahedron. The Bloch sums in diamond are SALCs adapted, not to a molecule with Td point group symmetry, but to the primitive unit cell. Conversely, MO theory is equivalent to the band scheme minus consideration of the lattice periodicity. Nevertheless, a qualitative MO-like treatment of the smallest chemical point group can often (but not always!) easily allow for obtaining the relative placement of the energy bands in diamond at one special k-point. A k-point corresponds to a specific value of the quantum number k(x, y, z), which gives the wavelength, or number of nodes, in the Bloch sums that combine to give a CO. In a band structure diagram, the CO energies are plotted as a function of the k-point. At the special point k(0, 0, 0) = Γ, known as the center of the Brillouin zone, there are no nodes in the Bloch sums. At this k-point, the lowest energy CO in diamond arises from totally symmetric ssσ and spσ interactions (symmetric with respect to the product JC24, where J is the inversion and C4 is a proper rotation about a fourfold rotation axis of the cubic lattice). This CO, of course, transforms as the totally symmetric irreducible representation of the cubic lattice. That irreducible representation is designated Γ1 (analogous to the Mulliken symbol A1 for the cubic point groups). Next lowest in energy is a triply degenerate set of COs with both ppσ and ppπ interactions. The set is of symmetry designation Γ15 (also symmetric to JC24). The reader should easily be able to sketch these. Immediately above this is the triply degenerate antibonding set of symmetry Γ25 (antisymmetric with respect to JC 24 ). The highest CO is the antibonding CO, which is labeled Γ2 (again, antisymmetric with respect to JC 24 ). The relative energy levels of the various COs at Γ are shown in Figure 3.9d. Now, every CO can hold two electrons (of opposite spin) per two-center bonding site, and every carbon contributes four electrons. If electrons are added to the aforementioned COs of diamond, in accordance with the procedure used for MOs, the four bonding COs (collectively termed the valence band) are found to be completely filled, and the four antibonding COs (collectively termed the conduction band) are empty. The octet rule is, therefore, obeyed around each carbon atom. As a sizeable band gap separates the full valence band and empty conduction band, the diamond is an insulator. This same ordering of the energy bands is also found in other elements that adopt the diamond lattice (e.g. Si). However, in some of these substances (e.g. Ge), as well as in some compounds with the isostructural zinc blende lattice (e.g. InSb), the reverse ordering is observed (i.e. the Γ2 band may be lower in energy than the Γ25 band). Nonetheless, the general picture of a full valence band and an empty conduction band still holds. Furthermore, our MO treatment correctly predicts the formation of a pair of singularly degenerate MOs and a pair of triply degenerate sets. However, the relative energy levels and degeneracies of COs will change in moving between k-points (giving rise to the band dispersion in a band structure diagram), which are not accounted for by the simple MO treatment. Furthermore, the crude MO approximation for the energy levels at the center of the Brillouin zone is not always so applicable. See, for example, the case with TiO2 in Chapter 15 and GaN in Chapter 16.

3.4

Common Structure Types

The Inorganic Crystal Structure Database (ICSD) assigns crystals to the same crystal structure type, or structure prototype, if they are isopointal and isoconfigurational. The term isopointal means that the crystals have the same space group type or belong to a pair of enantiomorphic space group types and that the atomic positions, occupied either fully or partially at random, are the same in the

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two structures. The term isoconfigurational means that the crystals are isopointal and that for all corresponding Wyckoff positions, both the crystallographic point configurations (crystallographic orbits) and their geometrical interrelationships are similar. As of 2017, the ICSD listed 4860 distinct structure prototypes. Another more recently compiled crystal structure prototype database (CSPD) listed 26 023 inorganic structure prototypes in 2017. Unfortunately, it is only possible for us to present a very limited number of these thousands of possible structure types. We have thus chosen to describe the structures of several nonmolecular solids that are of historical or pedagogical significance, or that are presently of significant technological interest. Even so, a large number of omissions are inevitable. There are examples of ionic, covalent, and metallic compounds that exist for almost every structure type. Thus, the common practice of classifying the structure types themselves as ionic, covalent, or metallic is not followed in this text. It should also be noted that many structure types are common to both iono-covalent and intermetallic compounds. The convention followed here is that iono-covalent compounds are those formed between a metal or metalloid with a nonmetal, that is halide, chalcogenide, light pnictide (N, P, As), silicon, carbon, or boron. Any compound formed between an element from this group and a metallic element falls in this category. Compounds formed between two or more nonmetals shall also be classified as iono-covalent. Included in the category of intermetallic compound are those compounds formed between different metals or metalloids. Any compound containing a nonmetallic element is excluded from this category. When referring to a generic stoichiometry, the convention used is that of denoting the more electropositive elements (metal or metalloid) as A, B, and C and the electronegative (nonmetal) elements as X and Y. For example, NaCl, SiC, and GaN have AX stoichiometry, TiO2 has AX2 stoichiometry, SrTiO3 is of ABX3 stoichiometry, and so on.

3.4.1

Iono-covalent Solids

3.4.1.1 AX Compounds

Many solids of AX stoichiometry possess the rock-salt structure (Pearson symbol cF8) including alkali halides (with the exception of cesium) and alkaline earth chalcogenides (e.g. BaO, CaO, MgO). The arrangement of atoms in the rock-salt structure is very favorable for ionic compounds. However, there are examples of more covalent and even metallic compounds that also adopt this structure. These include SnAs, TiC, and TiN. Titanium oxide, TiO, is metallic, and nickel oxide, NiO, is a p-type (hole) semiconductor, although the carrier mobility is extremely low. The rock-salt unit cell is again shown in Figure 3.10. It consists of two interlocking FCC sublattices (one of A cations, the other of X anions) displaced relative to one another by a/2 along 1 0 0 , where a is the cubic cell dimension. Both the cations and anions are situated at sites with full Oh point

= Anion = Cation a a a

Figure 3.10 The cubic rock-salt, or sodium chloride, unit cell consists of two interlocking FCC sublattices, one of sodium cations and one of chloride anions, displaced by (1/2 a) along 1 0 0 .

3.4 Common Structure Types

symmetry. That is, every ion is octahedrally coordinated. The rock-salt lattice is a Bravais lattice, since every lattice point, which consists of a cation–anion pair, is identical. Other AX structure types include cesium chloride, CsCl (Pearson symbol B2, Figure 3.11); two polymorphs of zinc sulfide, wurtzite (Pearson symbol hP4) and zinc blende (Pearson symbol cF8); and NiAs (Pearson symbol hP4). Although these structure types are often classified as ionic, many substantially covalent compounds adopt them as well. For example, in γ-CuI, which has the zinc blende structure, the radius rules for ionic solids do correctly predict that copper should be tetrahedrally coordinated (r+/r− = 0.60/2.20 = 0.273). However, the electronegativity difference between copper and iodine is less than one unit, and CuI is insoluble in water and dilute acids, which would be quite unexpected for an ionic 1+/1− salt. Zinc blende can be considered isostructural with diamond but with zinc cations residing at the centers of the same tetrahedra and the sulfide anions at the vertices. For zinc blende, an alternative description is in terms of a CCP-like array of S2− anions with one half of the tetrahedral sites occupied by Zn2+ cations. The polyhedral representation is depicted in Figure 3.12. Both diamond and zinc blende are best considered as two interlocking FCC sublattices displaced by a quarter of the body diagonal. Many compounds composed of main group p-block elements, with Δχ < 1, adopt the zinc blende structure. Some of these include β-SiC (α-SiC has the wurtzite structure), BeSe, and the majority of binary compounds between group 13 and group 15 elements (e.g. GaAs) and binary compounds of group 12 with group 16 (e.g. CdTe, ZnSe, HgSe). The other polymorph of ZnS is wurtzite (Figure 3.13). The zinc atoms are tetrahedrally coordinated as in zinc blende, but the anions in wurtzite form an HCP-like array instead of a CCP-like array. Indeed, the wurtzite structure is often thought of as an HCP-like array of S2− anions with one half the tetrahedral sites occupied by Zn2+ cations. Hence, the next nearest and third nearest Figure 3.11 The cubic cesium chloride unit cell is a body-centered simple cubic structure with nonequivalent atoms in the center and corners of the cell, rather than a body-centered cubic Bravais lattice since there is no lattice point in the center of the cell.

= Anion = Cation

a a a

Figure 3.12 The diamond and zinc blende structure depicted as a cubic array of vertexsharing tetrahedra. In ZnS, the zinc cations reside at the center of the tetrahedra and the sulfide anions at the vertices. (Source: After Elliot (1998), The Physics and Chemistry of Solids, 79. © 1998. John Wiley & Sons, Inc. Reproduced with permission.)

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Figure 3.13 The hexagonal ZnS or wurtzite unit cell. Cations are the dark-shaded circles.

coordination in the two polymorphs are quite different. Nevertheless, wurtzite and zinc blende are almost energetically degenerate. Given their nearly identical Madelung constants (see Table 3.3), the overall Coulombic forces must be roughly comparable – wurtzite being only very slightly favored. Band structure calculations have also shown that the two ZnS polymorphs are essentially energetically equivalent (Saitta 1997). The wurtzite structure is adopted by most of the remaining compounds composed of elements from the same groups as zinc blende, but which do not take the zinc blende structure, for example, GaN, AlN, InN, CdSe. The structure seems to be able to accommodate larger electronegativity differences between the constituent atoms, as in BeO, GaN, and ZnO. For these more ionic compounds, the wurtzite unit cell must be more stable than that of zinc blende, to an extent governed by the specific bonding forces in each case. In contrast to wurtzite, the structure of nickel arsenide, NiAs (Figure 3.14a), contains vacant tetrahedral sites but a completely occupied set of octahedral sites. In NiAs, the NiAs6 octahedra share edges in one direction (the ab plane) and faces in another (along the c direction). Many transition metal chalcogenides with a 1 : 1 cation to anion ratio have this structure, for example, NiS, FeS, FeTe, CoTe, and CrSe. Some of these cannot possibly be considered ionic. For example, below 260 K, NiS is a semimetal (the resistivity is 10−3 Ω cm and temperature independent) and metallic above 260 K (with a resistivity as low as 10−5 Ω cm that increases with temperature), the transition not being accompanied by a change in the symmetry of the crystal structure (Imada et al. 1998). 3.4.1.2 AX2 Compounds

One structure with AX2 stoichiometry is that of CdI2 (Pearson symbol hP3, Figure 3.14b), which can be considered an HCP-like array of anions with cations occupying one half of the octahedral sites. It is very similar to the NiAs structure with alternating layers of nickel atoms missing. The CdI2 structure is very commonly observed with transition metal halides. Another structure type is that of rutile, TiO2 (Figure 3.14c), in which chains of edge-sharing octahedra run parallel to the c-axis. The chains are linked at their vertices to form a three-dimensional network. Many ionic compounds of AX2 stoichiometry possess the CaF2 (fluorite, Pearson symbol cF12), or Na2O (antifluorite, Pearson symbol cF12) structures shown in Figure 3.15. Fluorite is similar to CsCl but with every other eight-coordinate cation removed. Each fluoride anion is tetrahedrally

3.4 Common Structure Types

(a)

(b)

(c)

Figure 3.14 Some simple structure types containing octahedrally coordinated metal atoms: (a) NiAs; (b) CdI2; (c) rutile (TiO2). The anions are the light circles; the cations at the center of the octahedra are the darkest-shaded circles.

coordinated by calcium ions. This structure is adopted by several fluorides and oxides. In the antifluorite structure, the coordination numbers are the inverse. Most oxides and other chalcogenides of the alkali metals (e.g. Na2Se, K2Se) possess the antifluorite structure, but so do some more covalent compounds, such as the silicides of Mg, Ge, Sn, and Pb. An important oxide with the fluorite structure is ZrO2. At room temperature, zirconia has a monoclinic structure in which zirconium is seven-coordinate. This transforms to a tetragonal structure at 1100 C and, at 2300 C, to the cubic fluorite structure. Aliovalent substitution of Zr4+ by the trivalent ion Y3+ stabilizes the fluorite structure at low temperatures by the creation of oxygen vacancies. One vacancy is required for every two yttrium atoms introduced. Yttria (Y2O3)-stabilized zirconia has the general formula Zr1−xYxO2−(x/2). The Y3+ cations are randomly distributed, and there is some experimental evidence that suggests they are next nearest neighbors to the vacancies (Fabris et al. 2002). The Y3+ cations thus have eightfold coordination, as in the ideal fluorite Figure 3.15 The fluorite (CaF2) structure. The cations are the dark-shaded circles in the octant centers.

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structure. The presence of oxygen vacancies around the Zr4+ ions reduces the average coordination number of zirconium to values closer to seven, as in the stable monoclinic structure. The oxygen vacancies not only stabilize the fluorite structure in Zr1−xYxO2−(x/2) but also give rise to a mechanism for oxide ion conduction. Thus, there is interest in this material for use as an oxide ion-conducting electrolyte in solid-oxide fuel cells. Several AX2 compounds have open structures that facilitate the insertion and extraction of foreign atoms or ions. For example, MoS2 (Pearson symbol hP6, space group P63/mmc) consists of stacked monolayers of MoS2, themselves containing planes of molybdenum atoms sandwiched between planes of sulfur atoms. This sulfide adopts a hexagonal crystalline structure. Every molybdenum atom is covalently bonded to six sulfur atoms with a trigonal prismatic coordination, while each sulfur is bonded to three molybdenum atoms in a pyramidal geometry. Other compounds can adopt this structure. For example, WSe2 is isostructural with MoS2. In these phases, the MoS2 or WSe2 “molecular” layers are held together by van der Waals interactions. Another important group of layered compounds with AX2 stoichiometry is the metal–metalloid diborides, AB2, where A = Mg, Al, Sc, Y, Ti, Zr, Hf, V, Nb, Ta, Cr, Mo, W, Mn, Tc, Re, Ru, Os, U, P, or Pu. These compounds form graphite-like hexagonal monolayers of boron atoms that alternate with monolayers of A atoms. Sometimes the boron layers are puckered. Many of these materials have the distinction of being among the hardest, most chemically inert, highest melting, and heat-resistant substances. Many are also better electrical conductors than the constituent elements. For example, TiB2 has five times the electrical conductivity of titanium metal (Holleman and Wiberg 2001). There are still other AX2 phases that, although non-layered, are porous. The oxide α-MnO2 (Pearson symbol tI24, space group I4/m), for example, has a tetragonal hollandite-type structure comprising double chains of MnO6 octahedra forming (2 × 2) open tunnels through which foreign atoms can be made to move. Such open-tunnel oxides are attractive for various industrial applications such as insertion electrodes for lithium-ion secondary batteries and ion or molecular sieves. 3.4.1.3 AX6 Compounds

There also exist several binary borides with the AB6 formula. The structure can be visualized as a body-centered CsCl lattice with the Cl− ions being replaced by B6 octahedra, while the bodycentered cation may be Na, K, Rb, Cs, Ca, Sr, Ba, Sc, Y, Zr, La, lanthanide, or actinide. The AIIIB6 and AIVB6 borides have a high metallic conductivity (104–105 Ω−1 cm−1) at room temperature, but the other borides are semiconductors. Other boron-rich binary borides include AB3, AB4, AB10, AB12, and AB66. The AB12 structure, like AB6, is comparatively simple, possessing the NaCl structure, in which the A atoms alternate in the lattice with B12 cuboctahedra. The other boron-rich compounds are often very complex, containing interconnected B12 icosahedra. The icosahedron is a special kind of polyhedron, sometimes called a deltahedron, whose faces are equilateral triangles. Other deltahedra include the octahedron, trigonal bipyramid, and tetrahedron. The particular deltahedron observed for a given compound can be predicted with Wade’s rules (Wade 1976). It can be stated as follows: The number of vertices the deltahedron must have is equal to the number of cluster bonding (skeletal) electron pairs minus one. Examples of the use of Wade’s rules will be given in Section 3.4.3.1. 3.4.1.4 ABX2 Compounds

Many oxides with the ABX2 delafossite structure (Pearson symbol hR12), in which A = Cu, Pd, Ag, or Pt; B = Al, Sc, Cr, Fe, Co, Ni, Rh, or Ln; and X = O (CuFeO2 is the mineral delafossite), have been studied by Shannon, Prewitt, and co-workers (Shannon et al. 1971) and others (Seshadri et al. 1998). These oxides contain BO2− layers of edge-connected BO6 octahedra tethered to one another through

3.4 Common Structure Types

two-coordinate A+ cations. This interlayer cohesion results from electrostatic forces. Every A+ cation has six neighboring A+ cations in the same plane, which can be considered a close-packed layer. The BO2− and A+ layers can show stacking variants, one of which is illustrated in Figure 3.16 where the similarity with the octahedra of CdI2 (Figure 3.14b) can be seen. Many delafossites possess interesting properties. For example, some of the oxides with palladium or platinum as A have very high in-plane electronic conductivities (only slightly smaller than copper metal). However, the same oxides with copper or silver as A are insulating. Delafossites are also of interest magnetically, since the two-dimensional triangular lattice enhances geometrical spin (magnetic) frustration. Hagenmuller and co-workers have investigated the synthesis, structure, and properties of many other layered oxides with ABO2 stoichiometry (Delmas et al. 1975; Fouassier et al. 1975; Olazcuaga et al. 1975). Like in the delafossites, the octahedral slabs in these oxides are separated by ionic A+ layers. However, they differ in that the A+ cation is octahedrally (or prismatically) coordinated by oxygen, whereas in delafossite the A+ cations are linearly coordinated by only two oxide anions. For example, in α-NaMnO2, each Na+ cation has six equidistant oxygen anions, because every edgesharing MnO6 octahedron in each single-layer MnO2 slab coordinates one face to a Na+ cation Figure 3.16 The hexagonal delafossite structure. Lightly shaded circles are oxygen atoms. The dark circles in the centers of the octahedra are the B atoms. The A atoms are the gray circles located between the slabs of edge-sharing BO6 octahedra.

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above it and one face to a Na+ cation below it. The conventional notation used for this structure of α-NaMnO2 is O 3, where the O signifies octahedral coordination around the alkali metal, the prime designates a monoclinic distortion, and the number three refers to the number of sheets in the unit cell. The similar O2-type layered ABO2 oxides, including α-NaFeO2, NaNiO2, LiFeO2, LiCoO2, and LiNiO2, can be considered ordered derivatives of rock salt, the ordering occurring along alternate 1 1 1 layers. LiNiO2 and LiCoO2 are mixed conductors (Section 6.5.2), exhibiting fast ionic (Li+) conductivity as well as electronic conduction. Thus, they find use as cathode materials in rechargeable lithium batteries. During cell charging, lithium ions are extracted from the cathode and inserted into the anode. In the discharge cycle, the reverse reactions occur. LiNiO2, however, suffers from severe capacity loss during recharging. Another oxide, LiMnO2, has also been under consideration as a cathode material. This phase is metastable on cycling, transforming to the spinel LiMn2O4. 3.4.1.5 AB2X4 Compounds (Spinel and Olivine Structures)

The AB2X4 spinels (Pearson symbol cF56), based on the mineral MgAl2O4, and the inverse spinels, B [AB]O4, are predominately ionic mixed oxides, containing a CCP-like array of X2− anions (Figure 3.17). Most of the chalcogenides (O, S, Se, Te) can serve as the anion. More than 30 different cations can be incorporated into the spinel structure, with various combinations of charges: IV II VI I AII BIII 2 X 4, A B2 X 4, and A B2 X 4. In spinel, the A cations reside in one eighth of the 64 tetrahedral sites and the B cations in one half of the 32 octahedral sites. In the other extreme, the inverse spinels, the A cations and half the B cations swap positions. Many intermediate cation distributions have been observed between these two extreme cases. Spinels have been studied intensively because of the sensitive dependence of their electronic and magnetic properties on the cation arrangement. For example, Fe3O4 is a mixed-valence (FeII/FeIII) oxide with the inverse spinel structure. It is highly ferrimagnetic and has a high electronic conductivity, which can be attributed to electron transfer between FeII and FeIII. It was recognized two decades ago that the spinel structure might also exhibit ion conduction (Thackery et al. 1982). In fact, owing to the high lithium-ion conductivity in the oxide spinel LiMn2O4, this material has been used as a cathode replacement for LiCoO2 in rechargeable lithium batteries. Unfortunately, Figure 3.17 The spinel structure. The A cations are the light gray circles located at the corner and face-centered positions of the unit cell. The B cations (dark gray circles) and the anions (lightly shaded circles) are located at the corners of the four cubes contained in the octants.

3.4 Common Structure Types

directly prepared LiMn2O4 exhibits less capacity and poorer cycling stability than LiCoO2, especially at elevated temperatures. This is believed to be owing, in part, to a cooperative Jahn–Teller (JT) distortion of Mn3+, which causes the material to undergo a cubic-to-tetragonal phase transition. Capacity fading appears to be circumvented when the LiMn2O4 electrode is prepared in situ by cycling LiMnO2, which phase transforms the layered structure to spinel (Armstrong et al. 2004). This is most likely because of an intermediate submicron intragranular domain structure that can accommodate the strain caused by the cooperative JT distortion (Armstrong et al. 2004). The mineral olivine ((Mg,Fe)2SiO4 Pearson symbol tP6) is named after its typically olive green color. It may be considered a solid solution A2BX4 with two end members: Fe2SiO4 (fayalite) and Mg2SiO4 (forsterite). The two minerals form a series with a structure consisting of a distorted HCP arrangement of oxygen anions. The A cations (Mg and Fe) reside in octahedral sites and the B cations (Si) in tetrahedral sites. The olivine and spinel structures are related, and in fact, (Mg,Fe)2SiO4 assumes the spinel structure under high pressure, at a depth of 300–400 km depth in the Earth’s mantle. The spinel version of (Mg,Fe)2SiO4 is about 12% more dense. Unlike spinel, in olivine there are two distinct octahedral sites. One of the octahedral sites is slightly smaller than the other and is the usual site for the smaller cations (e.g. Mg2+). Larger substituted cations (e.g. Ni2+) reside in the larger octahedral site, while cations of intermediate size (e.g. Fe2+) may be found in either octahedral site. Goodenough’s group has demonstrated that one compound with the olivine structure, orthorhombic LixFePO4, has the potential to reversibly cycle lithium at 3.4–3.5 V (Padhi et al. 1997). During cycling, Li+ ions are removed topotactically to yield heterosite, maintaining the FePO4 framework. In LixFePO4, Li+ ions occupy the smaller octahedral sites, while Fe2+ occupies the larger octahedral sites. The Li+ sites form linear chains of edge-sharing octahedra along the b-axis, while the Fe2+ sites form staggered lines of vertex-sharing octahedra along the b-axis. The metal atoms can be viewed as occupying b–c metal planes, where the planes are alternatively occupied by each type of metal, that is, there is a Li–Fe–Li–Fe ordering along the a-axis. Although the structure contains planes of Li+ chains, the interchain distance is large, and it is believed that the basic geometry of the olivine structure favors transport of Li+ along one-dimensional tunnels (Morgan et al. 2004). Owing to its poor intrinsic electronic conductivity (~10−9 S/cm) and its low ionic conductivity, however, conductive diluents (~2%), such as carbon black, must be added to LixFePO4 in order for it to be used as a cathode material. In fact, even commercialized lithium-ion batteries containing so-called mixed conductors are actually complex composites, containing carbon for this very reason, as well as polymeric binders to hold the powder structure together.

3.4.1.6

ABX3 Compounds (Perovskite and Related Phases)

The perovskites, ABX3, are three-dimensional cubic networks of vertex-sharing BX6 octahedra. The mineral perovskite itself is CaTiO3 (Pearson symbol cP5), named after the Russian mineralogist and statesman Lev von Perovski (1792–1856) by the German mineralogist Gustav Rose (1798–1873) who first discovered it in 1839. Other compounds with this structure include SrTiO3, PbTiO3, and PbZrO3. Cubic perovskites are actually high-temperature phases. The ground states are usually distorted perovskite structures (see Figure 3.18). In the ideal cubic structure, the more ionic 12coordinate A cation sits in the center of the cube defined by 8 vertex-sharing octahedra (Figure 3.18). It has been pointed out that the cubic perovskite structure can also be viewed as a four-sided octahedral channel structure in which the A cation resides in the channels (Rao and Raveau 1998). Various combinations of A and B cation valences can be accommodated in the perovskites, including AIBVX3, AIIBIVX3, and AIIIBIIIX3. By far, most perovskites contain the oxide or

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Figure 3.18 The cubic ABX3 perovskite structure. The B cations (dark gray circles) are at the vertices of the octahedra (the mid-positions of the cell edges), and the A cation (light gray circle) is located in the center of the cube. Anions are at the corners of the cube.

a

a a

fluoride anions, but the structure is also found for some compounds containing other anions as well, for example, S2−, Cl−, H−, and Br−. Many closely related structures exist that can be considered distorted variants of the cubic phase. For example, depending on the relative sizes of the A and B cations (Section 3.5.3) as well as deviations from the ideal composition and JT effects, octahedral tilting or face sharing may occur that lowers the symmetry from cubic to tetragonal (e.g. BaTiO3, Pearson symbol tP5) or orthorhombic (e.g. GdFeO3, Pearson symbol oP20). In Ba2MgWO6, a rock-salt-like ordering of the octahedral cations is found to accompany octahedral tilting. Perovskites can also tolerate vacancies, mixed valency, and/or oxygen deficiency. There are several important perovskite-related phases in which perovskite slabs are interleaved with other crystal structures. One is that of the Aurivillius phases, (Bi2O2)(An−1BnO3n+1), which result from the intergrowth of perovskite with Bi2O2 layers. Another intergrowth structure is the Ruddlesden–Popper (RP) series of oxides, Srn + 1TinO3n + 1. These can be considered SrTiO3 perovskite layers interleaved with SrO rock-salt-like layers. The RP phases have been of renewed interest since the discovery that ion-exchangeable cations can replace strontium in the rock-salt-like layers. The ion-exchangeable oxides can be represented as A2 An−1 Bn O3n + 1 , where A = alkali metal and A = alkali, alkaline earth, rare earth, or main group element. One such n = 3 phase is Na2La2Ti3O10, shown in Figure 3.19. A related series of oxides are the Dion–Jacobson phases, A [An−1BnO3n+1]. Owing to their great structural and compositional flexibility, perovskites and perovskite-related compounds, as a structure class, exhibit perhaps the richest variety of magnetic and electrical transport properties in solid-state chemistry. Fortunately, because of their relatively simple structures, perovskites are rather easily amenable to theoretical treatment. The pyroxenes are a large family of compounds with the formula AMB2O6, sometimes written as (A,M)BO3 (A = Ca2+, Na+, Fe2+, Mg2+; M = Cr3+, Mg2+, Fe2+, Fe3+; and B = Al, Si, or Ge). With magnesium silicate, MgSiO3, the high-pressure phase (>25 GPa) is perovskite, while the lowpressure phase is pyroxene. Here, the structure consists of isolated quasi-one-dimensional chains of MgO6 octahedra linked together by single chains of vertex-sharing SiO4 tetrahedra extending along the c-axis of the unit cell. The Mg sites lie between the apices of opposing tetrahedra. When a portion of Mg is substituted for larger A cations, these larger cations lie between the bases of the tetrahedra and are in more distorted six- or eightfold coordination. A pictorial device used for representing pyroxenes is called the I-beam, with each I-beam composed of two tetrahedral chains

3.4 Common Structure Types

Figure 3.19 Na2La2Ti3O10, an n = 3 member of the ion-exchangeable RP-like phases. The TiO6 octahedra are shown. The O2− anions are at the vertices of the octahedra and the Ti4+ cations in the centers of the octahedra. The La3+ cations are the large circles. The Na+ cations are the small circles in between the triple-layer slabs.

pointing in opposite directions and linked by the M octahedral sites. Some oxide phases with the pyroxene structure, such as NaCrSi2O6 and NaCrGe2O6, are quasi-one-dimensional metals at low temperatures.

3.4.1.7

A2B2O5 (ABO2.5) Compounds (Oxygen-Deficient Perovskites)

Brownmillerite is the name given to the mineral Ca2FeAlO5 (Hansen et al. 1928). Several A2B2O5 compounds (A = Ca, Ba, Sr; B = Fe, Al, Ga, Mn, In) isostructural with brownmillerite are known to exist. The brownmillerite structure (Figure 3.20) can be thought of as an oxygen-deficient perovskite with the oxygen vacancies ordered into defect chains along the 1 1 0 direction. The structure can be described as an array of alternating layers of BO3 octahedra and BO2 tetrahedra, with the A2+ cations occupying the spaces between. In order to optimize the coordination around A+, the tetrahedra are distorted, and the octahedra are tilted. In principle, any trivalent cation that can accept either tetrahedral or octahedral coordination can be incorporated into the brownmillerite structure. It is also possible to lightly dope brownmillerite with aliovalent cation pairs, for example, Mg2+/Si4+ in Ca2Fe0.95Al0.95Mg0.05Si0.05O5. The usual strategy, however, is to incorporate pairs of cations in which one of the members has a distinct preference for a particular type of site. For example, in Ca2FeAlO5, the Al3+ cations are found in the

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ABO3

ABO2.5

–O

Perovskite octahedral framework

Brownmillerite octahedral/tetrahedral framework

Figure 3.20 The brownmillerite phase can be thought of as an oxygen-deficient perovskite, in which the oxygen vacancies are ordered along the 1 1 0 direction of the perovskite cell. The tetrahedra are distorted, and the octahedra are tilted in order to optimize coordination around the A cations. The A cations are not shown in this figure for clarity.

tetrahedral sites and the Fe3+ in the octahedral sites. Similarly, in Ca2MnGaO5, Ga3+ cations exclusively occupy the tetrahedral sites, while Mn3+ occupies the octahedral sites. The ability to achieve this type of ordered cation arrangement offers the potential to realize two-dimensional layers (e.g. MnO2 and GaO in Ca2MnGaO5) that may have concomitant magnetic or electronic transport properties of interest. Interest in the brownmillerites as fast oxide ion conductors was first stimulated by Goodenough, who showed that Ba2In2O5 displayed an abrupt increase in electrical conductivity above a certain temperature (Goodenough et al. 1990). Oxide ion conductors must have a high concentration of oxygen vacancies (either intrinsically or from aliovalent doping as in Y-stabilized ZrO2) for O2− hopping to occur. However, high oxide ion diffusivity is generally associated with a disordered oxygen sublattice (Norby 2001). Accordingly, fast oxide ion conduction has been observed only at high temperatures (800–1000 C) in Sr2Fe2O5 (Holt et al. 1999) and Ba2In2O5 (Goodenough et al. 1990). Atomistic modeling has supported the theory that the abrupt conductivity change in Ba2In2O5 is owing to an order–disorder phase transition. The computer simulations suggest that O2− anions become displaced from their lattice sites into the open interstitial sites in the tetrahedral layer, forming Frenkel defect pairs (Section 3.5.1). As the number of Frenkel defects rises with increasing temperature, the anions at the equatorial positions in the octahedral and tetrahedral layers become indistinguishable, and the displaced O2− anions diffuse rapidly through the material (Fisher and Islam 1999). Some phases with the ABO2.5 composition do not have the brownmillerite structure. This is possible when the B cation is a transition metal with a tendency to adopt geometries other than tetrahedral and/or octahedral. For example, the oxygen vacancies in LaNiO2.5 order in such a way as to form alternating NiO6 octahedra and NiO4 square planes within the ab plane, which results in chains of octahedra along c (Vidyasagar et al. 1985). In still other (primarily mixedvalence) ABO3 − x phases with x < 0.5, other polyhedra are observed. The oxide CaMnO2.8, for instance, is built up of an ordered framework of Mn(III)O5 pyramids and Mn(IV)O6 octahedra.

3.4 Common Structure Types

3.4.1.8

AxByOz Compounds (Bronzes)

The bronzes are channel structures with an openness that allows for the transport of atoms or ions into the crystal. Bronzes have the general formula AxByOz, in which A is an alkali, alkaline earth, or rare earth metal and B can be Ti, V, Mn, Nb, Mo, Ta, W, or Re. The German chemist Friedrich Wohler (1800–1882), who discovered NaxWO3 in 1824, called these materials bronzes owing to their intense color and metallic luster. The introduction of sodium into the WO3 perovskite structure chemically reduces a portion of the W6+ ions to W5+. When x ≥ 0.28, this results in metallic conductivity (Goodenough 1965; Greenblatt 1996). Nonstoichiometric sodium tungsten oxide (x < 1) has a distorted perovskite structure with unequal W–O bond lengths and tilted WO6 octahedra. Some bronzes have lamellar structures. For example, NaxMnO2, which was already discussed, is a bronze. This phase consists of slabs of edge-sharing MnO6 octahedra separated by layers of Na+ cations. There are many structural variations among the bronzes. In fact, some bronzes with rather complex structures bare little or no resemblance to perovskite, although they are all generally built from vertex-sharing and/or edge-sharing octahedra. The open channels may have triangular, square, rectangular, diamond-shaped, pentagonal, hexagonal, or other polygonal cross sections. 3.4.1.9

A2B2X7 Compounds (Pyrochlores)

Another channel structure is that of the pyrochlores (Figure 3.21), with general formula A2B2X7. The mineral pyrochlore is (Ca, Na)2Nb2O6(O, OH, F). The A and B cations form an FCC array with the anions occupying tetrahedral interstitial sites. The A ion has eightfold anion coordination, and B has sixfold anion coordination. Thus, the pyrochlore lattice consists of two sublattices: (A2X)B2X6.

Figure 3.21 The A2B2X7 pyrochlore structure. The B2X6 sublattice is a tetrahedral channel-forming network of vertex-sharing BX6 octahedra. The channels are occupied by the anions of the A2X sublattice of vertex-sharing AX4 tetrahedra (not shown).

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It may also be thought of as an anion-deficient derivative of the fluorite structure, but with an ordered arrangement of anion vacancies and an ordered cation arrangement. The size of the A cation has a large effect on the stability of the pyrochlore structure. As the size difference between the A and B cations decreases, the fluorite structure becomes favored over pyrochlore. The B2X6 sublattice is a tetrahedral network of vertex-sharing BX6 octahedra containing channels with hexagonal cross sections. However, the channels are obstructed by the anions of the hexagonal A2X sublattice of vertex-sharing AX4 tetrahedra, which prohibit cationic mobility. In nonstoichiometric (anion-deficient) pyrochlores, Gd1.8Ca0.2Ti2O6.95, for example, cationic mobility and ion exchange of the A-site cations is possible at high temperatures. The stoichiometric pyrochlore transition metal oxides exhibit a wide range of magnetic and electronic transport properties. These properties are, of course, dependent on the d electron count of the B cation. Electrical conductivity may be insulating (e.g. Gd2Ti2O7, Tb2Ti2O7), semiconducting (e.g. Tl2Ru2O7), or metallic (e.g. Nd2Mo2O7, Cd2Re2O7). The oxide Cd2Os2O7 undergoes a transition from semiconducting to metallic conductivity at 226 K accompanied by a magnetic transition (paramagnetic to antiferromagnetic), and Tl2Mn2O7 exhibits colossal magnetoresistance, a dramatic drop in its electrical resistivity in the presence of a magnetic field. The antiferromagnetic ground state of the pyrochlore oxides can have very large spin degeneracy. This is because the A2O sublattice that the magnetic rare earth cations reside on is a geometrically frustrated system with competing magnetic interactions. Hence, spin glass behavior, as characterized by hysteresis and nonlinearity in the magnetic susceptibility, is observed in many pyrochlore oxides at low temperatures, even in the absence of chemical or bond disorder, including Y2Mo2O7 and Tb2Mo2O7 (Gardner et al. 2001).

3.4.1.10 Silicate Compounds

Silicon, like carbon, has a propensity for tetrahedral coordination. The SiX4 tetrahedron can be considered the building block of most silicon compounds. The two polymorphs of silicon carbide, for example, adopt the zinc blende and wurtzite structures. In a similar fashion, silicon dioxide is found in nature in both crystalline and amorphous forms, in which the SiO4 tetrahedra share all their vertices. There are eight different modifications of the crystalline form of SiO2. The most stable is α-quartz, consisting of interlinked helical chains, with three tetrahedra per turn. Quartz crystals can be either right-handed or left-handed, so that they are nonsuperimposable on their mirror image, that is, they exhibit enantiomorphism (Greenwood and Earnshaw 1997). The other crystalline forms contain interlinked sheets of six-membered rings of tetrahedra. It has been said that more is known about the chemical, structural, physical, and electrical properties of SiO2 than any other oxide. This is, no doubt, in part, owing to the importance of this material as a dielectric for silicon-based microelectronic devices. There is an immense variety of silicate mineral structure types. However, the connectivity of the tetrahedra in most silicates can be determined from their formulae (West 1985). The smaller the Si: O ratio, the fewer the number of SiO4 vertices shared with neighboring units. For example, Mg2SiO4 (Si : O = 1 : 4) contains no bridging oxygens and thus has discrete SiO4 tetrahedra. By contrast, SiO2 (Si : O = 1 : 2) is a three-dimensional network containing no nonbridging oxygens. In fact, one classification system for silicate structures is based on the number of oxygen atoms per tetrahedron that are shared. The notation scheme is as follows: neso- (Si : O = 1 : 4), soro- (Si : O = 1 : 3.5), cyclo(ino-) (Si : O = 1:3), phyllo- (Si : O = 1:2.5), and tecto- (Si : O = 1:2), corresponding to, respectively, 0, 1, 2 (closed ring or continuous chain), 3, and 4 shared oxygen atoms. Unfortunately, owing to space constraints, it is not possible to provide a more detailed study of the structures of silicates. However,

3.4 Common Structure Types

a discussion on another binary silicon compound that is also important to the semiconductor industry, Si3N4, as well as a few aspects of zeolites and other porous solids, is included in Section 3.4.2. There are many other binary and ternary silicon compounds of commercial importance. For example, silicon nitride (Si3N4) occurs in two hexagonal forms: the α-form and a denser β-form. The crystal structures are very complex but may be thought of as close-packed nitrogen atoms, with three eighths of the tetrahedral vacancies occupied by silicon atoms. In this respect it is like SiO2, being a threedimensional network of tetrahedral units. However, although the nitrogen atoms do arrange roughly into a tetrahedron around the silicon atoms, the silicon atoms are arranged into planar triangles (not pyramids) around the nitrogen atoms. The β-form contains small-diameter (~0.15 nm), onedimensional channels. Silicon nitride is very hard, strong, and chemically inert toward most agents up to 1300 C (Holleman and Wiberg 2001). It is relatively impermeable to sodium, oxygen, and other species (even hydrogen diffuses slowly through silicon nitride). Hence, (amorphous) Si3N4 films find wide use in silicon-based integrated circuits as diffusion barriers and passivation layers. 3.4.1.11

Porous Structures

Many nonmetal oxides (e.g. silicates and phosphates) possess open, three-dimensional framework structures, built from vertex-sharing polyhedra, containing large tunnels or cavities. The microporosity (pore diameters ≤20 Å) or mesoporosity (pore diameters = 20–500 Å), plus the large internal surface area in these materials, enable their use as small molecule sieves, adsorbents, ion-exchange media, and catalysts. As with the silicates, the structures and chemical formulae depend on the numbers of free and shared polyhedral vertices. Frameworks containing aliovalent substitutional ions, such as Al3+ substituting for Si4+ ion in SiO4 tetrahedra, bear net negative charges. Cations must be accommodated within the framework to balance this charge. Examples include the felspars NaAlSi3O8 and CaAl2Si2O8. The structures of inorganic frameworks can be classified into two basic categories: zeolite types containing exclusively tetrahedra and those types with mixed tetrahedral– octahedral (or bipyramidal) frameworks. Figure 3.22 shows how 6 groups of 4 corner-sharing SiO4 tetrahedra – 6 squares of tetrahedra giving 24 tetrahedra total – connect together in the zeolites to form a cuboctahedron (intermediate between a cube and octahedron), which unites with other cuboctahedra to form the large opennetwork structure characteristic of these compounds. Zeolites are commercially prepared by a hydrothermal technique (high temperature and pressure) involving crystallization from strongly alkaline solutions of sodium silicate and aluminum oxide. The aluminophosphates possess an (Al,P)O4 tetrahedra framework similar to that of the zeolites. Many transition metal silicates, germanates, and phosphates possess a mixed framework of different types of polyhedra, for example, tetrahedra and octahedra. These phases can be prepared by hydrothermal techniques, but it is still not entirely clear how the structures assemble at the molecular level. Nonetheless, inorganic and organic moieties have found use as templates to control the pore size and shape in many cases, but the large phosphate structures often collapse if the template is removed on heating. Again, the channels or cavities possessed by these frameworks make them interesting for certain applications. For example, the silicates Na5BSi4O12 (B = Fe, In, Sc, Y, La, Sm) and the NASICONS (an acronym for sodium [Na] superionic conductors), which are silicophosphates with formula Na1+xZr2P3−xSixO12, are famous for superionic conduction. In each of these types of oxides, the transition metal–oxygen octahedra share their six vertices with SiO4 (or PO4) tetrahedra, but in Na5BSi4O12 channels are formed, whereas in the NASICONS, cavities are formed (Rao and Raveau 1998). The first mixed-valence manganese gallium phosphate, MnGaPO4, in which the Mn atoms are in the divalent and trivalent states, has been prepared by hydrothermal crystallization (Hsu and Wang 2000). The crystal structure contains discrete bioctahedra of

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3 Crystal Structures and Binding Forces

Figure 3.22 A zeolite is constructed from linked cuboctahedra, which in turn are constructed from four-membered squares of tetrahedra.

Ga2O10(H2O) and GaO5 trigonal bipyramids, connected via MnIIIO4, MnIIO4, and PO4 tetrahedra to generate intersecting tunnels in which piperazinium cations C4 H12 N22 + can reside. These materials may aid in our understanding and designing of redox-catalytic or magnetic molecular sieves. Periodic mesoporous silicas were reported for the first time in the literature by researchers at the Mobil Oil Corporation in the early 1990s, although a synthetic process that yields very similar reaction products was patented twenty years prior (Moller and Bein 1998). The reactive internal surfaces of these solids have been used to attach functional groups that can act as complexing agents for metal cations. However, as well as complexation and similar uses like ion exchange and sorption, the channels have been used to grow metal clusters and wires. Another class of porous materials is the pillared layered structures. In these phases, the pillars separate inorganic layers of interconnected polyhedra, to which they are covalently bonded on both ends. The pillars themselves may be organic or inorganic; as the former, the material is referred to as a hybrid. A schematic illustration of a hybrid structure is shown in Figure 3.23a. Porosity is introduced by spacing the pillars apart, which is most easily accomplished by interposing smaller R groups between the pillaring groups (Figure 3.23b). The smaller R groups are bonded on only one end. The study of organically pillared compounds began about 25 years ago with the synthesis of one of the first zirconium phenylphosphonates, Zr(O3PC6H5)2 (Alberti et al. 1978). When the inserted organic moieties are not covalently bonded to the inorganic layers, the material is considered a nanocomposite (Figure 3.23c). The hybrid organic–inorganic materials field and the field of porous solids, in general, are vast and rapidly growing but are, unfortunately, outside the main scope of this book. Yet another related class of porous material, used in a variety of applications such as chemical absorption, catalysis, filtration, and thermal insulation is the aerogel. Materials in this category possess a spongelike structure and are characterized by a very large surface area and porosity and, thus, low density. The geometric feature that gives rise to their aforementioned uses is the large percentage of air pockets present, which makes these materials lightweight, friable, and

3.4 Common Structure Types

(a)

(b)

(c) R R

R R

R R

R R

Figure 3.23 Cross sections of hybrid organic–inorganic materials. (a) A pillared layered structure, in which organic moieties (ovals) are covalently bonded to the inorganic layers (rectangles). The separation between the layers can be controlled by changing the size of the organic molecules. (b) Porosity is introduced by interposing smaller R groups (e.g. OH, CH3) between the pillars. (c) A nanocomposite formed by incorporating an organic molecule or polymer between two already separated inorganic layers. The organic moieties are not covalently bonded to the inorganic layers.

hygroscopic, although these properties can be tuned over a wide range of behavior by materials engineering. Aerogels are amorphous solids with nanoscale-sized pores and therefore exhibit strong capillary action since the height a liquid column will rise in a tube is inversely proportional to its radius. Aerogels have the lowest known densities, indices of refraction, and thermal, electrical, and acoustical conductivities of any solid materials. The most important aerogels are those of silica, carbon, and certain metal oxides. Of these, silica is by far the most common type. They are prepared by a highly cross-linked polymerization reaction followed by a careful supercritical drying step. The polymerization process forms the solid silica network surrounded by a sol–gel liquid. The drying process then removes the liquid. Metal oxide aerogels can be prepared by a similar process involving hydrolyzing their respective metal alkoxide in a solvent. The alkoxide groups are converted to hydroxyl groups with neighboring hydroxyl groups condensing to form metal–oxygen–metal bridges. Extensive such cross-linking yields a metal oxide gel that can then be supercritically dried. Higher-quality metal oxide aerogels are made through epoxide-assisted gelation processes. Alexander Frank Wells (1912–1994) earned his BS and MA degrees in chemistry from Oxford University and his PhD from Cambridge University under H. M. Powell in 1937. He received his DSc in 1956, also from Cambridge University. From 1944 to 1968, he was director of the Crystallographic Laboratory at Imperial Chemical Industries. The first edition of Wells’, now classic, book Structural Inorganic Chemistry was published in 1945. Four subsequent editions were eventually published, the last in 1985. This work constitutes a substantial portion of the body of knowledge on structural principles and space-filling patterns of inorganic solids. For many practitioners of solid-state chemistry, this book remains the standard reference for inorganic crystal structures. Wells also authored four other well-known books: The Third Dimension in Chemistry, Models in Structural Inorganic Chemistry, Three-Dimensional Nets and Polyhedra, and Further Studies of Three-Dimensional Nets. He was also among the first editorial advisors for the Journal of Solid State Chemistry. In 1968, Wells accepted a position as professor of chemistry at the University of Connecticut, becoming emeritus in 1982. Wells was commonly called “Jumbo” by his close friends and colleagues. (Source: B. Chamberland, personal communication, March 5, 2004.) (Photo courtesy of Dr. Terrell A. Vanderah. Reproduced with permission.

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3.4.2

Metal Carbides, Silicides, Borides, Hydrides, and Nitrides

Many of the compounds formed between metals with several members of the second-row elements, as well as hydrogen and silicon, are often described as interstitial structures and named Hägg compounds, after the Swedish crystallographer Gunnar Hägg (1903–1986). This is owing to the small atomic radii of these nonmetallic elements, which enables them to fit inside the interstices in closest-packed structures (CCP, HCP), BCC, and simple hexagonal structures of metal atoms of relatively larger size. Despite the -ide ending to the names of these compounds, the reader should be careful not to assume anything about a bonding model. These interstitial compounds are at best only partially ionic. They are usually mostly covalent in nature or, more precisely, metallic and thus possess metallic and refractory (inert and high-melting) properties. Like the intermetallic phases discussed in the next section, the chemical compositions or empirical formulae of the interstitial compounds are not easily rationalized from a simple valence perspective. Different crystal structures are observed for different metal to nonmetal atomic ratios. For the metal-rich phases, the nonmetal atoms may be isolated, in which they do not form any nonmetal atom network. For others, they may be described as containing isolated pairs of nonmetal atoms, zigzag chains, or double chains. Nonstoichiometric or sub-stoichiometric phases are also found, particularly with the nitrides. We generally expect a non-Hägg structure to form under two circumstances. Specifically, if the nonmetal to metal atomic radius ratio is greater than some critical value (rX/rM > ~0.59) or when there is considerable ionicity to the bonding, the structure is usually more complicated than that of an interstitial compound, and the stoichiometry can sometimes be more easily predicted. As example of the latter, the empirical formulae for GaN and AlN are correctly ascertained by assuming the validity of a predominantly ionic bonding model. On account of the electronegativity differences between the constituent elements, and thus the presence of the two types of trivalent ions, we can deduce a one-to-one stoichiometry, nevertheless, the choice between the common polymorphs (zinc blende or wurtzite) is not so easy (see Section 3.4.1.1). Tungsten carbide, by contrast, is composed of elements with similar electronegativities. However, it is an example showing the applicability of Hägg’s atomic radius ratio rule. Hexagonal WC (rC/rW = 70 pm/210 pm = 0.33) is made up of a simple hexagonal array of tungsten atoms in non-close-packed layers lying directly over one another, with carbon atoms filling half the interstitial trigonal sites. Unfortunately, it is not always the case that we can correctly predict the empirical formula by a simple atomic radius ratio rule. There are, for instance, a number of known chromium carbides (rC/rCr = 70 pm/200 pm = 0.35) – h-CrC, c-CrC, Cr3C, Cr3C2, Cr7C3, and Cr23C6 – all of which have a combination of metallic, covalent, and ionic bonding character with Cr7C3 being the most metallic.

3.4.3

Metallic Alloys and Intermetallic Compounds

In many cases, there is a substantial degree of solid solubility of one metal or metalloid in another. For example, the binary systems Ag–Au, Ag–Pd, Bi–Sb, Ge–Si, Se–Te, and V–W exhibit complete liquid and solid miscibility, with no atomic ordering at any temperature. This type of mixture forms a particular class of alloy called a single-phase solid solution, in which the solute may reside in either a substitutional or interstitial site. In the other extreme, some systems exhibit negligible solid solubility, forming instead multiphase alloys that consist of grains of each pure component. Examples are As–Bi, Bi–Cu, Bi–Ge, Na–Rb, Sb–Si, and Ge–Zn. Phase equilibria in these as well as the intermediate cases (grains of two or more solutions with different compositions) are discussed in more detail in Section 15.3.

3.4 Common Structure Types

Solid solution alloy composition is normally expressed by listing the constituent elements in alphabetical order by chemical symbol, each preceded by a number representing its mass/weight percentage in the mixture. Sometimes, the solvent (the base constituent, that component present in the largest weight percentage) is listed first, followed by the solute, the constituent(s) present in smaller quantities. For example, a mixture of 89% silver and 11% gold, by weight, could be expressed as Ag–11Au or 11Au–89Ag. The same conventions apply regardless of the number of components. It is also possible to list the components in their atomic percentages. In fact, chemists, physicists, and crystallographers tend to prefer atomic percent, while metallurgists and industrialists often prefer weight percentages. Atomic percentage is obtained from weight percentage simply by dividing through by the atomic mass of each element. For example, for an AB alloy, the at% A is computed by (wt% A)/(at wt A)]/{[(wt% A)/(at wt A)] + [(wt% B)/(at wt B)]}. A relatively new class of alloy is known as the high entropy alloys (HEAs). It encompasses the multi-principal element alloys (MPEAs) and complex concentrated alloys (CCAs). This group of alloys was originally defined as all alloys containing five or more elements in approximately equimolar concentrations, which maximize the configurational entropy. The field of study is not very old, and there are a number of hypotheses concerning the description of HEAs, such as the importance of lattice distortion and sluggish diffusion. The compositional restriction was subsequently lifted by redefining the elemental concentrations between 5 and 35 at% (i.e. HEAs need not have components of equimolar concentrations). The entropy-based definition assumes the alloy microstructure can be represented by a single-phase solid solution or multiphase solid solutions, in which the different elements take up random atomic positions within the crystal structure. Therefore, today the working definition of an HEA is understood to mean a multicomponent solid solution (single-phase or multiphase), explicitly excluding all alloys containing any atomically ordered intermetallic compound. Most HEAs are produced by liquid phase processing such as induction melting or Bridgman solidification. HEAs tend to have high yield strength and fracture toughness but low ductility. In addition to mechanical behavior, they are being investigated for potential relevance to functional applications that rely on electrical, thermal, and magnetic properties. Combinatorics gives us an idea of the number of possible alloy systems expressed as [n!/m!(n − m)!] where n is the number of elements and m is the order of the system. Consider the 59 stable metallic elements on the periodic table (up to bismuth included, technetium and promethium excluded because not stable, and not considering metalloids where carbon could be disputed, as a semimetal in its graphite structure or even as a large gap semiconductor in its diamond structure). If we combine these 59 chemical elements into groups of two distinct elements (m = 2), there are found to be 1711 binary systems if one accounts only for the kinds of metals, paying no regards to the variable proportions. Alternatively, if we combine the 59 elements into groups of 3 (m = 3), there are 32 509 ternary systems and so on: 455 126 quaternary systems, almost 5 million quinary systems, and about 45 million senary ones. The study of all of these alloys falls in principle within the realm of metallurgy, which is quite vast. Furthermore, although the chemical composition is of obvious importance with respect to the properties exhibited by an alloy, an alloy’s microstructure is of equal or more significance. In Section 2.3.1, recall we discussed the evolution of microstructure during conventional alloy solidification processes. Likewise, in Section 2.4 the influence of microstructure on the physical, chemical, and mechanical properties of polycrystalline materials was introduced. These topics will be expanded on in subsequent chapters. In the past, metallurgists sometimes referred to intermetallics as a type of alloy. The crystal structures of these phases consist of ordered arrangements of atoms, as opposed to the disordered arrangement in solid solution alloys. Intermetallics are thus actually compounds formed between

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two or more metallic (or metalloid) elements. When specifying the composition of an intermetallic phase, it is customary to use subscripts to indicate the number of atoms of each type in the formula unit, as for any other type of chemical compound. An intermetallic can have a fixed stoichiometry (line compounds) or exist over a narrow compositional range (called an intermediate phase). Despite the fact that intermetallics are composed of metallic elements, they usually exhibit properties that are intermediate in nature between those of typical metals and nonmetals. Like ceramics, they are brittle at low temperatures. However, intermetallics generally exhibit higher thermal and electrical conductivities than ceramics, although less than those of their constituent elements. Often times, the phase equilibria and/or phase transformation kinetics in a multicomponent metal system are such that an intermetallic phase coexists with either a pure element or solid solution under specific conditions of temperature, pressure, and composition. In these alloys, grains of the intermetallic compound are interspersed with grains of the other phases, usually with the result that the alloy is mechanically strong but of low ductility. Pure intermetallics, on the other hand, have been prepared by the mechanical alloying process known as ball milling, combustion synthesis (self-propagating combustion of a mixture of metal powders), and rapid solidification techniques and grown as single crystals. In some cases, it is also possible to deposit intermetallic thin films by sputtering techniques. The circumstances under which intermetallics form were elucidated by the British metallurgist William Hume-Rothery (1899–1968) for compounds between the noble metals and the elements to their right in the periodic table (Hume-Rothery 1934; Reynolds and Hume-Rothery 1937). These are now applied to all intermetallic compounds, in general. The converse to an intermetallic, a solid solution, is only stable for certain valence electron count per atom ratios and with minimal differences in the atomic radii, electronegativities, and crystal structures (bonding preferences) of the pure components. For example, it is a rule of thumb that elements with atomic radii differing by more than 15% generally have very little solid phase miscibility. The correlation between the valence electron counts and the stabilities of intermetallic phases and structures were also espoused by others, like the physical chemists Neils N. Engel (b. 1904) and Leo Brewer (1919–2005), although Hume-Rothery found their result somewhat controversial. The Engel–Brewer theory asserts that the crystal structures of transition metals and their intermetallic compounds are determined solely by the number of valence s and p electrons. For example, Engel suggested in 1949 that the BCC structure correlated with dn − 1 sl (where n is the total number of valence electrons) and that the HCP and FCC structures corresponded to dn − 2 slp1 and dn − 3 slp2 electron configurations, respectively. Importantly, although all unpaired electrons, including d electrons, were thought to contribute greatly to the bonding and cohesion, the d electrons were suggested to have only an indirect role in determining the crystal structure. Brewer subsequently revised and extended this concept, allowing him to successfully predict multicomponent phase diagrams and thermodynamic activities for a very large number of transition metal alloys (Wang and Carter 1993). David Pettifor (1945–2017), while working as a Ph.D. student at Cambridge, later showed that the structural trends across the transition metal series were driven by variation in the number of valence d electrons, not the s and p electrons as predicted by the Engel–Brewer theory. Considering the fact that the majority of elements are metals, it would seem that there could be a vast number of intermetallic compounds and structures. Indeed, there are, and the rules of bonding and valence in these materials are still largely unknown. Many intermetallic compounds are characterized by unusual and complex stereochemistry. In fact, it is often impossible to rationalize the stoichiometry using simple chemical valence rules, a situation reminiscent for the inorganic chemist of the hydrides, borides, and silicides.

3.4 Common Structure Types

William Hume-Rothery (1899–1968) was left deaf from a bout of cerebrospinal meningitis in 1917, which ended his pursuit of a military career at the Royal Military Academy. He went on to obtain a B.S. degree in chemistry from Oxford in 1922 and subsequently earned a Ph.D. in metallurgy under Sir Harold Carpenter from the Royal School of Mines in London in 1925. Hume-Rothery returned to the inorganic chemistry department at Oxford to carry out research on intermetallic compounds, bordering between metallurgy and chemistry. In 1956, he became the first chair of the newly founded Oxford metallurgy department, which evolved into the materials science department. HumeRothery was keen to use his chemistry training to bring a theoretical basis to the field of metallurgy. He wrote the now famous textbooks The Structure of Metals and Alloys (1936) and Atomic Theory for Students of Metallurgy (1946). The well-known Hume-Rothery rules established that intermetallic compounds result from differences in the atomic radii and electronegativities of alloy constituents, as well as certain valence electron count per atom ratios. Hume-Rothery was elected a Fellow of the Royal Society in 1937. (Source: “William Hume-Rothery: His Life and Science” byD.G. Pettifor (2000). In: The Science of Alloys for the 21st Century: A Hume-Rothery Symposium Celebration (eds. P.E.A. Turchi and R.D. Shull). Warrendale, PA: The Minerals, Metals & Materials Society.)c) (Photo courtesy of The Royal Society. Copyright owned y the estate of B. Godfrey Argent. Reproduced with permission.

3.4.3.1

Zintl Phases

Invoking Lewis’ octet rule, Hume-Rothery published his 8 – N rule in 1930 to explain the crystal structures of the p-block elements (Hume-Rothery 1930, 1931). In this expression, N stands for the number of valence electrons on the p-block atom. An atom with four or more valence electrons forms 8 – N bonds with its nearest neighbors, thus completing its octet. The Bavarian chemist Eduard Zintl (1898–1941) later extended Hume-Rothery’s (8 – N) rule to ionic compounds (Zintl 1939). In studying the structure of NaTl, Zintl noted that the Tl1− anion has four valence electrons, and he, therefore, reasoned that this ion should bond to four neighboring ions. The term Zintl phase is applied to solids formed between either an alkali or alkaline earth metal with a main group p-block element from group 14, 15, or 16 in the periodic table. These phases are characterized by a network of homonuclear or heteronuclear polyatomic clusters (the Zintl ions), which carry a net negative charge, and that are neutralized by cations. Broader definitions of the Zintl phase are sometimes used. Group 13 elements have been included with the Zintl anions and an electropositive rare earth element or transition element with a filled d shell (e.g. Cu) or empty d shell (e.g. Ti) has replaced the alkali or alkaline earth element in some reports. Although the bonding between the Zintl ions and the cations in the Zintl phases is markedly polar, by our earlier definition, those compounds formed between the alkali or alkaline earth metals with the heavier anions (i.e. Sn, Pb, Bi) can be considered intermetallic phases. A diverse number of Zintl ion structures are formed, even among any same two elements. For example, the metal phosphides MxPy may contain discrete P3− anions (K3P), or negatively charged chains (K4P6), rings (KP), or cages (K4P26) of phosphorus atoms. Very commonly, the Zintl ions adopt structures consisting of polyhedra all of whose faces are equilateral triangles. These are sometimes called deltahedra. Examples include the tetrahedron, trigonal bipyramid, octahedron, pentagonal bipyramid, dodecahedron, tricapped trigonal prism, bicapped square antiprism, octadecahedron, and icosahedron. Triangular networks make the most efficient use of a limited number of cluster bonding (skeletal) electrons, and so are electronically advantageous (Porterfield 1993). The Zintl phases are often prepared by dissolving the constituent elements or alloys in liquid ammonia or AlCl3. The anion connectivity of many Zintl phases can be rationalized in terms of Hume-Rothery’s (8 − N) rule. For example, in BaSi2 (with Si44 − clusters), the Si1− anion is isoelectronic with the nitrogen

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group elements, that is, it has five valence electrons. The (8 − N) rule correctly predicts that each silicon atom will be bonded to three other silicon atoms. Similarly, in Ca2Si, Si4− is isoelectronic with the noble gas elements. Again, the 8 − N rule correctly predicts that silicon will occur as an isolated ion. Indeed, this compound has the anti-PbCl2 structure, in which the silicon is surrounded by nine calcium ions at the corners of a tricapped trigonal prism. A generalized 8 − N rule was derived by Pearson in order to address those cases in which fractional charges appear on the anion (Pearson 1964). Fractional charges are usually indicative of multiple structures or anion connectivity existing in the compound. The generalized 8 − N rule is formulated as 8 − VECA = AA − CC

3 11

where VECA is the total number of valence electrons per anion, AA is the average number of anion– anion bonds per anion, and CC is the average number of cation–cation bonds per cation. Unfortunately, neither Hume-Rothery’s original rule nor the generalized 8 − N rule is valid for nonpolar intermetallics, or when an octet configuration is unnecessary for stability of the compound. In order to increase the domain of structures for which one can make predictions, extensions have been made to the generalized 8 − N rule (Parthé 2000), but are not discussed here. Traditionally, the Zintl phases are semiconductors that either obey the 8 − N or generalized 8 − N rule. Unfortunately, there appears to be controversy over the range of applicability for the empirical rules guiding what constitutes a Zintl compound. The distinguished professor John D. Corbett, a leading researcher in the field of intermetallics, has argued that some weakly metallic compounds containing anion arrays with slightly delocalized electrons can also be classified structurally as Zintl phases. The R3In5 (R = Y, La) phases are one such example. In R3In5 (R = Y, La), the electropositive R is assumed to have a formal charge of +3. Each indium anion thus has a charge of −(3 × 3)/5 = −9/5 = −1.8. Since a neutral indium atom has a valence electron count of 3, the VECA for In−1.8 is equal to 3 + 1.8, or 4.8. The R3In5 phases contain distantly interconnected square pyramidal In5 clusters (Corbett and Zhao 1995). Hence, four indium anions have threefold coordination, and one indium has fourfold coordination, giving AA = 16/5 = 3.2. The generalized 8 − N rule correctly predicts that CC should be equal to zero, which is the case, as In anions are the nearest neighbors to each R cation. The approach developed by Durham University chemistry professor Kenneth Wade (1932–2014) for predicting the geometric structures of boranes and boron halides (Wade 1976) is also useful for many Zintl anions. For convenience, it is restated here: the number of vertices the deltahedron must have is equal to the number of cluster bonding (skeletal) electron pairs minus one. In counting the cluster bonding electrons in Zintl ions, one uses the formula v − 2 where v is the number of valence electrons on the element. A total of n cluster bonding electrons (n/2 pairs) require a deltahedron with n/2 – 1 vertices, even with a different number of atoms in the anion cluster. For example, in R3In5 (R = Y, La), each indium atom contributes 3–2 = 1 electron to the cluster, which has a charge of nine, In95 − . There is 9 + 5 = 14 cluster bonding electrons, or seven pairs. Wade’s rules correctly predict that the In95 − cluster will adopt an octahedral geometry (7 – 1 = 6) with one vertex missing – the square pyramid. Included among the interesting properties currently being pursued in some Zintl phases are the metal–nonmetal transition and colossal magnetoresistance. Zintl phases have also been used as precursors in the synthesis of novel solid-state materials. For example, a fullerene-type silicon clathrate compound, Na2Ba6Si46 (clathrates are covalent crystals, whereas fullerides are molecular crystals), with a superconducting transition at 4 K was prepared from the Zintl phases NaSi and BaSi2 (Yamanak et al. 1995). In the final product, Ba atoms are located in the center of tetrakaidecahedral

3.4 Common Structure Types

(Si24) cages, while the Na atoms are in the centers of pentagonal dodecahedral (Si20) cages. The Na2Ba6Si46 clathrate compound was the first superconductor found for a covalent sp3-hybridized silicon network. 3.4.3.2

Nonpolar Binary Intermetallic Phases

Zintl phases are characterized by the presence of markedly heteropolar bonding between the Zintl ions (electronegative polyatomic clusters) and the more electropositive metal atoms. By contrast, the bonding between heteronuclear atoms within other intermetallic compounds is primarily covalent or metallic. A number of different structure types exist for any given type of stoichiometry, as indicated in Table 3.6. An important AB structure type is the austenite phase, steel’s thermodynamically stable high-temperature FCC interstitial solid solution of carbon in iron. This phase has the CsCl (Pearson symbol cP2) structure. When austenite is quenched (rapidly cooled), a metastable body-centered tetragonal phase (Pearson symbol tI2) originally called martensite is obtained. This material is a hard solid solution strengthened phase. Today, the term martensitic transformation is applied to similar phase transformations in other systems that occur by a shearing or displacive motion of the lattice as opposed to diffusive motion. The transformation is responsible for the shape Table 3.6

Several binary intermetallic structure types.

Stoichiometry

Prototype

Pearson Symbol

AB

NiTi

mP4

ηAgZn

hP9

CoSn

hP6

AuCd

oP4

CoU

cI16

σCrFe

tP30

ωCrTi

hP3

Cu2Sb

tP6

PdSn2

oC24

Cu2Mg (Laves)

cF24

MgNi2 (Laves)

hP24

AB2

AxBy

MgZn2 (Laves)

hP12

Al2Cu

tI12

NiTi2

cF96

MoPt2

oI6

βCu3Ti

oP8

Ni3Sn

hP8

Ni3Al Al3Ni2 Al3Zr

cP4 hP5 tI16

AlFe3

cF16

Al4Ba

tI10

PtSn4

oC20

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memory effect and super elasticity, which is utilized in so-called smart materials, the most famous of which is NiTi (~51% Ni). Unlike the hard and brittle Fe–C martensite, the martensitic phase in NiTi is monoclinic (Pearson symbol mP4) and easily deformable. Smart materials are discussed further in Section 10.2.4. The largest single intermetallic structural class is commonly referred to as the Laves phases. They are named after the German mineralogist and crystallographer Fritz Henning Laves (1906–1978). These AB2 intermetallic compounds form dense tetrahedrally close-packed structures. In MgCu2 (Pearson symbol cF24), the Mg atoms are ordered as in CD. However, in MgZn2 (Pearson symbol hP12) the Mg atoms are located on sites corresponding to those in the hexagonal diamond (lonsdaleite) structure. Both are illustrated in Figure 3.24. The crystal structure of MgZn2 was first determined by the American electrical engineer James Byron Friauf (1896–1972), when he was at the Carnegie Institute of Technology (Friauf 1927). The phase is composed of intrapenetrating icosahedra (each one with 6 Mg atoms and 6 Zn atoms at the vertices) that coordinate Zn atoms and 16-vertex polyhedra that coordinate Mg atoms. The latter are actually interpenetrating tetrahedra (with Mg atoms at the 4 vertices) and 12-vertex truncated tetrahedra (with Zn atoms at the vertices). The 16-vertex polyhedron, so formed from the two smaller polyhedra, is called, appropriately, a Friauf polyhedron. In fact, the Laves phases are sometimes referred to as the Friauf phases or Laves–Friauf phases. A third Laves phase, with prototype MgNi2 (hP24), has a dihexagonal structure. The Laves phases are the most common structure type of binary intermetallic compound between a 3d transition element and a 4d, 5d, or 6f element. Compounds crystallizing in the hexagonal structure include TaFe2, ZrRe2, NbMn2, and UNi2. Those with the cubic structure include CsBi2 and RbBi2, while compounds with the dihexagonal structure include NbZn2, ScFe2, HfCr2, UPt2, and ThMg2. Owing to their high-temperature deformability and good oxidation resistance, some Laves phases (e.g. NbCr2) are being considered as structural materials in gas turbine engines. Others have been considered for functional applications. For example, the cubic ZrV2 exhibits a superconducting transition below 8 K. It also has been considered as a hydrogen storage material, in which hydrogen can be absorbed into interstitial sites and reversibly desorbed at high temperatures. The Laves

(a)

(b)

Figure 3.24 The AB2 Laves phases: (a) the hexagonal structure (the unit cell is shown by the dashed line) and (b) the cubic structure. The A atoms are the dark circles.

3.4 Common Structure Types

phases are a subset of the family of tetrahedrally close-packed structures known as the Frank– Kasper phases, in which atoms are located at the vertices and centers of various space-filling arrangements of polytetrahedra (Frank and Kasper 1958a, b). In Section 3.4.3.3, it will be seen that the Frank–Kasper phases have aided the understanding of quasicrystal and liquid structures. Other binary intermetallic phases, such as Ni3Al (Table 3.6), are also of interest for their potential use under harsh service conditions, owing to their excellent strength and their corrosion resistance (due to an Al2O3 protective scale) at elevated temperatures. In fact, Ni3Al has a yield strength that rises with increasing temperature. In Ni3Al, the Al atoms occupy the corner positions of an FCC unit cell with the Ni atoms located in the face-centered positions. Despite its great promise, the brittleness and low ductility of Ni3Al still limit its use as a strengthening constituent to fine dispersions of precipitation particles, known as precipitation hardening (Section 14.3.2). In its Quadrennial Technology Review of 2015, the US Department of Energy listed materials for harsh service conditions as one of its 14 advanced manufacturing-focused areas. There are other types of harsh, or aggressive, environments in addition to high temperatures and corrosive chemicals. These include thermal cycling, high pressures, dust and particulates, mechanical wear, neutron irradiation, and hydrogen attack. Many intermetallic phases can function under some of these conditions and are usually used in the form of a fine precipitate to strengthen certain alloys (a.k.a. superalloys, or high-performance alloys). 3.4.3.3

Ternary Intermetallic Phases

Many ternary intermetallics have interesting structures and properties. For example, of great theoretical interest to crystallographers in recent years are the ternary intermetallic systems that are among the few known stable solids with perfect long-range order but with no three-dimensional translational periodicity. The former is manifested by the occurrence of sharp electron diffraction spots and the latter by the presence of a non-crystallographic rotational symmetry. It will be recalled that three-dimensional crystals may only have one-, two-, three-, four-, or sixfold rotation axes; all other rotational symmetries are forbidden. However, the discovery of metastable icosahedral quasicrystals of an Al–Mn alloy exhibiting fivefold rotational symmetry was reported 20 years ago (Shechtman et al. 1984). Since that time, many thermodynamically stable quasicrystals of ternary intermetallic compounds have been found. These have mostly been obtained by rapidly solidifying phases with equilibrium crystal structures containing icosahedrally packed groups of atoms (i.e. phases containing icosahedral point group symmetry). The quasicrystalline phases form at compositions close to the related crystalline phases. The icosahedron is one of the five platonic solids, or regular polyhedra, and is shown in Figure 3.25. A regular polygon is one with equivalent vertices, equivalent edges, and equivalent faces. The icosahedron has 20 faces, 12 vertices, 30 edges, and 6 fivefold proper rotation axes (colinear with 6 tenfold improper rotation axes). It is possible for crystal twinning to produce disallowed diffraction patterns, but, in order to produce the fivefold symmetry, twinning would have to occur five times in succession. The possibility of twinning was, in fact, the main point of contention after the first report on the discovery of icosahedral quasicrystals. However, numerous attempts to disprove the true fivefold symmetry failed, and the icosahedral symmetry was confirmed as real. It turns out that icosahedral coordination (Z = 12) and other coordination polytetrahedra with coordination numbers (Z = 14, 15, and 16) are a major component of some liquid structures, more stable than a close-packed one, as was demonstrated by F.C. Frank and J. Kasper. When these liquid structures are rapidly solidified, the resultant structure has icoshedra threaded by a network of wedge disclinations, having resisted reconstruction into crystalline units with three-dimensional translational periodicity (Mackay and Kramer 1985; Turnbull and Aziz 2000). Stable ternary

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Figure 3.25 The dodecahedron (top) and icosahedron (bottom) are two of the five platonic solids. The others (not shown) are the tetrahedron, the cube, and the octahedron. 12 pentagons

20 equilateral triangles

intermetallic icosahedral quasicrystals are known from the systems Al–Li–Cu, Al–Pd–Mn, and Zn–Mg–Ln. Several other ternary systems yield metastable icosahedral quasicrystals. Some stable ternary intermetallic phases have been found that are quasiperiodic in two dimensions and periodic in the third. These are from the systems Al–Ni–Co, Al–Cu–Co, and Al–Mn–Pd. They contain decagonally packed groups of atoms (local 10-fold rotational symmetry). It should be noted that there are also known metastable quasicrystals with local 8-fold rotational symmetry (octagonal) and 12-fold rotational symmetry (dodecagonal) as well. The dodecahedron is also one of the five platonic solids. Several crystalline ternary intermetallic compounds are presently used in engineering applications. The ternary phase Ni2MnIn (Pearson symbol cF16) is illustrated in Figure 3.26a. Each atom is located on the site of a cesium chloride cubic lattice. The unit cell consists of a FCC arrangement of nickel atoms with one additional nickel atom located in the center of the unit cell, while a manganese or indium atom is in the center of each octant of the cell with eightfold nickel coordination. Other compounds crystallizing in this structure with A2BC stoichiometry include Ni2MnGa, Ni2MnAl, Cu2MnAl, Co2MnSi, Co2MnGe, Co2MnSn, and Fe2VAl. These compounds are known as Heusler alloys and are named after the German mining engineer and chemist Friedrich Fritz Heusler (1866–1947) who was at the University of Bonn where he studied the magnetic properties of Mn2CuAl and Mn2CuSn in the early 1900s (Heusler 1903). Heusler alloys have a rich variety of applications, owing to some of their unique properties. Some of these phases are half-metallic ferromagnets, exhibiting semiconductor properties for the majority spin electrons and normal metallic behavior for the minority spin electrons. Therefore, the conduction electrons are completely polarized. The Ni2MnGa phase is used as a magnetic shape memory alloy, and single crystals of Cu2MnAl are used to produce monochromatic beams of polarized neutrons. If half of the nickel atoms in Figure 3.26a are removed, the half-Heusler ABC structure of Figure 3.26b is obtained. In the half-Heusler structure, each atom still resides on a cesium chloride lattice site. The rock-salt component (B and C) remains intact, but the A atoms form a zinc blende lattice with B and C. Examples of compounds with this structure include MnNiSb, AuMgSn, BiMgNi, and RhSnTi. Some intermetallics containing rare earth elements are under consideration as magnetic refrigerants. The advantages of magnetic cooling over gas compression technology include the removal of environmentally hazardous coolants (i.e. chlorofluorocarbons) and energy-consuming

3.5 Structural Disturbances

(a)

(b)

Figure 3.26 The crystal structure of the A2BC Heusler alloys, (a), and ABC half-Heusler alloys, (b). The A atoms are the lightest-shaded circles.

compressors. In the magnetic cooling process, a strong magnetic field is applied to the refrigerant, aligning the spins of its unpaired electrons. That provides for a paramagnetic–ferromagnetic phase transition (a magnetic entropy reduction) upon cooling, which causes the refrigerant to warm up. Upon removal of the field, the spins randomize, and the refrigerant cools back down. This is termed the magnetocaloric effect (MCE) and is normally from 0.5 to 2 C/T change in the magnetic field. Other types of magnetic ordering (ferromagnetic, antiferromagnetic, and spin glass) absorb energy internally when the spins are aligned parallel by the applied field, thus reducing the MCE (Gschneidner et al. 2000). As at present, gadolinium and its alloys have been the most studied materials for this application, since the 4f orbitals provide for a comparatively large magnetic entropy change. Recently, a giant magnetocaloric effect (a 3–4 C/T change) was reported in the intermetallic compounds Gd5(SixGe1−x)4, where x ≤ 0.5 (Pecharsky and Gschneidner 1997a, b). In Gd5(Si1.8Ge2.2), the magnetocaloric effect has been associated with a field-induced first-order structural transition (Morellon et al. 1998). The structure of Gd5(SixGe1−x)4 can be described as a monoclinic distortion of the orthorhombic Gd5Si4 phase (Sm5Ge4 structure type), in which the Gd atoms occupy five independent fourfold sites and Si and Ge occupy four independent fourfold sites, in a random manner.

3.5

Structural Disturbances

Inhomogeneous structural disturbances, of course, can only be understood by comparison to a reference standard, or ideal structure. The types of disturbances discussed will include defects and bond length/angle distortions. Defects may be intrinsic or extrinsic. Intrinsic defects are the result of thermal activation in an otherwise perfect crystal, where there is no reaction between the substance and the environment or other substance. By contrast, extrinsic structural defects may be defined as those introduced in a substance through reaction with an external agent, which may be another substance, or the environment (e.g. irradiation or a mechanical force). If another substance, the agent may or may not be native to the lattice. Defects can be further classified into point defects and extended defects. Unassociated point defects are associated with a single atomic site and are thus zero-dimensional. These include vacancies, interstitials, and impurities, which can be intrinsic or extrinsic in nature. Extended defects are multidimensional in space and include dislocations, stacking faults, twins and grain boundaries. These tend to be metastable, resulting from materials processing. The mechanical properties of

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solids are intimately related to the presence and dynamics of extended defects. Extended defects are discussed elsewhere in this book. For now, only point defects are covered. Their importance in influencing the physical and chemical properties of materials cannot be overemphasized.

3.5.1

Intrinsic Point Defects

The concept of a zero-dimensional intrinsic point defect was first introduced in 1926 by the Russian physicist Jacov Il’ich Frenkel (1894–1952), who postulated the existence of vacancies, or unoccupied lattice sites, in alkali halide crystals (Frenkel 1926). Vacancies are predominant in ionic solids when the anions and cations are similar in size and in metals when there is very little room to accommodate interstitial atoms, as in close-packed structures. The interstitial is the second type of point defect. Interstitial sites are the small voids between lattice sites. These are more likely to be occupied by small atoms, or, if there is a pronounced polarization, to the lattice. In this way, there is little disruption to the structure. Another type of intrinsic point defect is the antisite atom (an atom residing on the wrong sublattice). In ionic crystals, there is a requirement that charge neutrality be maintained within the crystal. Hence, if a cation vacates its lattice site and ends up on the crystal surface, an anion must also vacate its sublattice and end up on the surface. The individual vacancies need not be located near one another in the crystal. Interestingly, as pointed out by Seitz (1940), the occurrence of cation–anion vacancy pairs was first postulated by Frenkel on the grounds of excessively high activation energy for diffusion by interchange on the ideal alkali halide lattice. Nevertheless, the cation–anion vacancy pair is called the Schottky defect, after Walter Haus Schottky (1886–1976), who studied the statistical thermodynamics of point defect formation (Schottky and Wagner 1930). The Frenkel defect, on the other hand, is the term applied when a displaced cation in an ionic solid ends up in an interstitial site, rather than at the surface. Note that, in this case, charge neutrality in the crystal is still maintained since the extra positive charge (the cation interstitial) is balanced by the negatively charged vacancy. The interstitial and vacancy defect pair constitute the Frenkel defect. A finite equilibrium concentration of intrinsic point defects can be found in any crystalline material because a small number of defects is thermodynamically favored. This can be seen by considering the configurational entropy, or the number of possible ways in which n defects can be distributed among N lattice sites. The number of ways, Ω, is given by combinatorics as Ω=

N n N −n

3 12

The configurational entropy is thus ΔS =

k ln N n N −n

3 13

As discussed above, in ionic solids, intrinsic point defects are really defect pairs owing to the charge neutrality requirement. There is a need to account for this in the configurational entropy term. For example, for the interstitial vacancy (Frenkel) defect pair, Eq. (3.13) would be ΔS = k ln

N ni N − n

N nv N − nv

3 14

The equality nv = ni can be used to obtain ΔS = k 2 ln N − 2 ln n N − n

3 15

3.5 Structural Disturbances

Using Stirling’s approximation, ΔS = 2k N ln N − N − n ln N − n − n ln n

3 16

Finally, this term can be inserted into an expression for the Gibbs energy to obtain G = G0 + nΔg − 2k N ln N − N − n ln N − n − n ln n

3 17 0

In Eq. (3.17), nΔg is the free energy change necessary to create n defects and G is the Gibbs energy of a perfect crystal. Equation (3.17) is a minimum with respect to n when (∂G/∂n)T,P = 0. By approximating N − n ≈ N, the equilibrium concentration is found to be n = exp N

− Δg 2kT

3 18

It is important to study point defects because they mediate mass transport properties, as will be discussed in Chapter 6. Although the influence of point defects on physical and chemical properties is well founded, relatively little has been reported concerning their effect on the mechanical behavior of solids. A mechanical force applied over a macroscopic area induces the displacement of a large numbers of atoms in a cooperative motion, mediated by dislocations and other extended defects. Recently, however, variations in the vacancy concentrations of a series of binary intermetallics have been associated with the differing hardening behavior of those phases. For example, the hardening of FeAl (CsCl structure) is extremely sensitive to thermal history, while that of the isostructural NiAl is not. Interestingly, this has been attributed to a high vacancy concentration in FeAl, which is indicative of structural instability (Chang 2000).

3.5.2 Extrinsic Point Defects Nonequilibrium concentrations of point defects can be introduced by materials processing (e.g. rapid quenching or irradiation treatment), in which case they are classified as extrinsic. Extrinsic defects can also be introduced chemically. Often times, nonstoichiometry results from extrinsic point defects, and its extent may be measured by the defect concentration. Many transition metal compounds are nonstoichiometric because the transition metal is present in more than one oxidation state. For example, some of the metal ions may be oxidized to a higher valence state. This requires either the introduction of cation vacancies or the creation of anion interstitials in order to maintain charge neutrality. The possibility for mixed valency is not a prerequisite for nonstoichiometry, however. In the alkali halides, extra alkali metal atoms can diffuse into the lattice, giving M1+δ X (δ 1). The extra alkali metal atoms ionize and force an equal number of anions to vacate their lattice sites. As a result, there are an equal number of cations and anions residing on the surface of the crystal, with the freed electrons being bound to the anion vacancies. Interestingly, the localized electrons (known as F-centers) have discrete quantized energy levels available to them, which fall within the visible region of the electromagnetic spectrum. The nonstoichiometric crystals thus become colored. Another type of extrinsic point defect is the impurity, which may substitute for an atom in the lattice or reside in an interstitial site. The impurity need not be of the same valency as the host, but overall charge neutrality must be maintained in the crystal. This type of extrinsic point defect can greatly influence electrical transport properties (electrical conductivity). If the impurities or dopants are electrically active, they can vary the charge carrier concentration and, hence, the electrical conductivity of the host. This topic will be expounded on for oxides and semiconductors in Sections 6.2 and 6.4, respectively. The siltation is quite complex for oxides, but for now a very brief superficial explanation for semiconductors assuming the band model is valid is appropriate. In a

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semiconductor, the magnitude of the energy gap separating the filled states (valence band) from the empty states (conduction band) is small enough that electrons may be thermally excited from the top of the valence band to the bottom of the conduction band. In an intrinsic semiconductor, the number of electrons that have been thermally excited to the conduction band equals the number of holes left behind in the valence band. Both the electrons and holes are charge carriers that contribute to electrical conduction. The concentration of either one of these carrier types can be increased by the addition of electrically active dopants (intentionally added impurities), to give an extrinsic semiconductor. Dopants that donate electrons are termed donors or n-type dopants (e.g. phosphorus atoms in silicon), since the negatively charged electrons become the majority carrier. By comparison, impurities that accept electrons (boron atoms in silicon), creating holes, are termed acceptors or p-type dopants, since the positively charged holes become the majority carrier. In silicon, and other semiconductors, the energy levels associated with the dopants are located within the band gap. However, n-type dopant levels are very close to the conduction band edge, while p-type dopant levels are very close to the valence band edge. Such dopants are called shallow impurities. Thermally excitation of charge carriers from these shallow impurities to one or other of the bands of the host is possible. Behavior of this type was discussed early on in semiconductor physics (see, e.g. Wilson 1931). Vacancies can also make possible the low-temperature manipulation of a solid’s structure. For example, aliovalent ion exchange (e.g. Ca2+ for Na+) has been used to create extrinsic vacancies in some transition metal oxides at relatively low temperatures (Lalena et al. 1998; McIntyre et al. 1998). These vacancies were then used to intercalate new species into the lattice, again at relatively low temperatures. Intercalation refers to the insertion of an extrinsic species into a host without a major rearrangement of the crystal structure. If the new species is a strong reducing agent, mixed valency of the transition metal may be introduced into the host.

Walter Haus Schottky (1886–1976) received his doctorate in physics under Max Planck from the Humboldt University in Berlin in 1912. Although his thesis was on the special theory of relativity, Schottky spent his life’s work in the area of semiconductor physics. He alternated between industrial and academic positions in Germany for several years. He was with Siemens AG until 1919 and the University of Wurzburg from 1920 to 1923. From 1923 to 1927, Schottky was professor of theoretical physics at the University of Rostock. He rejoined Siemens in 1927, where he finished out his career. Schottky’s inventions include the ribbon microphone, the superheterodyne radio receiver, and the tetrode vacuum tube. In 1929, he published Thermodynamik, a book on the thermodynamics of solids. Schottky and Carl Wagner studied the statistical thermodynamics of point defect formation. The cation–anion vacancy pair in ionic solids is named the Schottky defect. In 1938, he produced a barrier layer theory to explain the rectifying behavior of metal–semiconductor contacts. Metal–semiconductor diodes are now called Schottky barrier diodes. (Photo courtesy of Siemens Archives, Munich. Reproduced with permission.

3.5.3

Structural Distortions

The types of structural distortions observed in solids include bond length alterations and bond angle bending. Localized structural distortions occur around point defects such as vacancies and substitutional impurities owing to their differing mass and force constants. Such bond distortions occur when they can lower the overall energy of the system. For example, it can be seen how Pauling’s third rule predicts that edge- or face-sharing tetrahedra will be distorted owing to increased repulsion between the neighboring cations. Octahedral rotation and tilting, commonly observed in the

3.5 Structural Disturbances

perovskites and related phases (e.g. brownmillerites), are similar low-energy distortion modes driven primarily by ion–ion repulsion (electrostatic) forces. Tilting occurs when the A-site cation is too small for its 12-coordinate site in the perovskite structure. The distortion effectively lowers the coordination number from 12 to 8. Goldschmidt recognized many years ago that octahedral tilting in perovskites results from a need to optimize the coordination around the A-site cation (Goldschmidt 1926). He quantified the goodness of fit of the A cation with a tolerance factor, t: t=

R A + RO 2 R B + RO

3 19

where RA, RB, and RO are the ionic radii of the A and B cations and oxygen, respectively. Using the radii by Shannon and Prewitt, perovskites are found for values of t ranging from 0.8 to 1.1 (Shannon and Prewitt 1969). The ideal cubic perovskite has t = 1.00, RA = 1.44 Å, RB = 0.605 Å, and RO = 1.40 Å. If the A ion is smaller than the ideal value, then t becomes smaller than 1. As a result, the BO6 octahedra tilt in order to fill space, and this lowers the crystal symmetry from cubic to tetragonal to orthorhombic. With values less than 0.8, the ilmenite structure is more stable. Conversely, if t is larger than 1 because of a large A ion or a small B ion, this stabilizes a hexagonal distortion. More recently, Glazer developed a very influential notation for describing the 23 different tilt systems that have been observed (Glazer 1972). The description is based on the rotations of the octahedra about each of the three Cartesian axes. The direction of the rotation about any one axis, relative to the rotations about the other two axes, is specified by a letter with a superscript that indicates whether the rotations in adjacent layers, which are coupled to the other layers via the vertexsharing octahedra, are in the same (+) or opposite (−) directions. For example, a+a+a+ signifies an equivalent rotation about the three axes in each of the adjacent layers. Similarly, in the a0a0c+ tilt system, the octahedra are rotated only about the axes parallel to the z-axis, with the same direction in both layers. Woodward has shown that tilt systems in which all of the A-site cations remain crystallographically equivalent are strongly favored when there is a single ion on the A site (Woodward 1997). Of these systems, the lowest energy configuration depends on the competition between ionic bonding (A–O) and covalent (A–O σ, B–O σ, B–O π) bonding. The orthorhombic GdFeO3 structure (a+b−b−) is the one most frequently found when the Goldschmidt tolerance factor is smaller than 0.975, while the rhombohedral structure (a−a−a−) is most commonly observed with 0.975 ≤ t ≤ 1.02 and highly charged A cations. Undistorted cubic structures (a0a0a0) are only found with oversized A cations and/or B–O π bonding. Tilt systems with nonequivalent A-site environments are favored when more than one type of A cation (with different sizes and/or bonding preferences) are present, with the ratio of large to small cations dictating the most stable tilt system. Vertex-sharing polyhedra can also exhibit asymmetric M–X–M linkages (cation displacements), which arise from electronic driving forces, rather than electrostatic forces. H.A. Jahn (1907–1979) and Edward Teller (1908-2003) showed in 1937 that nonlinear symmetrical geometries, which have degenerate electronic states, are unstable with respect to distortion. That is, such systems can lower their electronic energy by becoming less symmetrical and removing the degeneracy. The first-order JT effect is important in complexes of transition metal cations that contain nonuniformly filled degenerate orbitals, if the mechanism is not quenched by spin–orbit (Russell– Saunders) coupling. Thus, the JT effect can be expected with octahedrally coordinated d9 and high-spin d4 cations and tetrahedrally coordinated d1 and d6 cations. The low-spin state is not observed in tetrahedral geometry because of the small crystal-field splitting. Also, spin–orbit coupling is usually the dominant effect in T states so that the JT effect is not observed with tetrahedrally coordinated d3, d4, d8, and d9 ions.

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In solids, the observance of JT distortions normally requires that the d electrons be localized. For example, the FeO6 octahedra in the cubic perovskite SrFeO3 might be expected to exhibit JT distortions. The oxide ion can serve as a good π donor, so an Fe4+ cation octahedrally coordinated by six oxygens should have a high-spin d4 t 32g e1g electron configuration. However, SrFeO3 is metallic. The eg d electrons are delocalized in a Bloch functions, and no localized JT distortion is observed. However, a so-called band JT effect has been confirmed as the cause of the structural distortions observed in some cases, such as the intermetallic phase Ni2MnGa, a Heusler alloy (Brown et al. 1999). An increase in the extent of valence d electron localization is expected for smaller principal quantum numbers and as one moves to the right in a period because of a contraction in the size of the d orbitals. For example, with compounds of the late 3d metals, a mixture of 4s bands and more or less localized 3d atomic orbitals may coexist, in which case, it becomes possible for cubic crystal fields to split the degenerate d orbitals and give rise to a localized JT distortion (e.g. a single octahedra), or small polaron in physics terminology. High concentrations of JT ions, where the polyhedra share structural elements, are subject to a cooperative JT effect, which can cause distortion to a lower crystalline symmetry. The case of nearly degenerate states is treated by a second-order correction in perturbation theory and gives rise to the second-order or pseudo-JT effect (Öpik and Pryce 1957). This theorem predicts that, in systems with a small HOMO–LUMO gap (say, 1.0 cannot be prepared, even though the isostructural triple-layer oxide Sr4Ru3O10 is known. In Sr4Ru3O10, there are no electrostatic or electronic driving forces for out-of-center distortion in the outer octahedral layers (Cao et al. 1997). Hence, although the origin for the cation displacement in the outer octahedral layers of Na2La2Ti3−xRuxO10 is electrostatic in nature, the electronic second-order JT effect places an upper limit on the ruthenium doping level that can be tolerated in these sites. In custom-designing materials with tailored properties, it is often necessary to synthesize metastable phases that will be kinetically stable under the temperature and conditions of use. These phases are obtainable only through kinetic (chemical) control. In many cases, kinetic control has been achieved via the soft chemical low-temperature (e.g. electrochemical synthesis, sol–gel method) and/or topochemical routes (e.g. intercalation, ion exchange, dehydration reactions), since these routes use mild synthetic conditions. It should be noted that not all soft chemical routes are topochemical. A reaction is said to be under topochemical control only if it follows the pathway of minimum atomic or molecular movement (Elizabé et al. 1997). Accordingly, topochemical reactions are those in which the lattice of the solid product shows one or a small number of crystallographically equivalent definite orientations relative to the lattice of the initial crystal, if the reaction has proceeded throughout the entire volume of the initial crystal (Günter 1972). Under the proper circumstances, most soft chemical processes can allow ready manipulation of the ionic component of many nonmolecular materials. Indeed, the solid-state literature has seen an enormous growth in the number of reports, wherein the utility of these synthetic strategies are exploited. However, these methods typically leave the covalent framework of iono-covalent structures intact. It would be extremely desirable to exercise kinetic control over the entire structure of a solid, thereby maximizing the ability to tune its properties. As alluded to at the beginning of Section 3.3, probably the most challenging task is predicting, a priori, the extended structure – not only the coordination preferences of all the atoms or ions but also the polyhedra connectivity. In this area, Tulsky and Long have taken a major step forward by formalizing the application of dimensional reduction to treat the formation of ternary phases from binary solids (Tulsky and Long 2001): MX x + nAa X

Ana MX x + n

3 22

In this reaction, the first reactant on the left is the binary parent phase, and the product, to the right of the arrow, is a ternary child. The second reactant on the left is an ionic reagent termed the dimensional reduction agent, in which X is a halide, oxide, or chalcogenide. The added X anions terminate M–X–M bridges in the parent yielding a framework in the child that retains the metal coordination geometry and polyhedra connectivity mode (i.e. corner-, edge-, or face-sharing) but which has a lower connectivity (e.g. fewer polyhedra corners shared). The A cations serve to balance charge, ideally, without influencing the M–X covalent framework. The utility of this approach lies in its ability to predict the framework connectivity of the child compound. Connectivity is defined as the average number of distinct M–X–M linkages around the metal centers or, alternatively, the average number of bonds that must be broken to liberate a discrete polyhedron. For frameworks with only one kind of metal and one kind of anion, the connectivity of the parent is given by CN M CN X − 1

3 23

Practice Problems

where CNM is the coordination number of the metal and CNX denotes the number of metal atoms coordinated to X. The framework connectivity of an AnaMXx + n child compound is predicted to be 2 CN M − x + n

3 24

The reaction schemes of Eqs. (3.25) and (3.26) illustrate how, as Eq. (3.24) predicts, connectivity is progressively lowered by reaction of a binary parent with increasing amounts of a dimensional reduction agent: VF3

6-connected

Aa F

Aa VF4 4-connected

Aa F

A2a VF5 2-connected

Aa F

A3a VF6 0-connected

3 25

where A = alkali metal or Tl; a = 1 MnCl2 12-connected

2 3 NaCl

Na2 MN3 Cl8 8-connected

1 3 NaCl

NaMNCl3 6-connected

NaCl

Na2 MnCl4 4-connected

3 26

Thus, solid-state reaction between VF3, with corner-sharing VF6 octahedra in three dimensions, and AaF in a 1 : 3 M ratio yields A3VF6, containing discrete octahedral anions. Similarly, reaction of MnCl2, which consists of MnCl6 octahedra sharing six edges, with NaCl in a 1 : 1 M ratio yields a two-dimensional framework with octahedra sharing only three edges. Solids with octahedral, tetrahedral, square planar, and linear metal coordination geometries, including many different types of polyhedra connectivity modes, are amenable to dimensional reduction. Tulsky and Long compiled an enormous database of over 300 different allowed combinations of M and X and over 10 000 combinations of A, M, and X. The formalism may be extendable to quaternary phases as well. However, frameworks containing anion–anion linkages; anions other than halides, oxide, or chalcogenides; nonstoichiometric phases; and mixed-valence compounds were excluded from their initial study. Other limitations of the approach include its inability to predict which of many possible isomers, each satisfying the criteria set by Eq. (3.24), will result. Furthermore, Eq. (3.24) is only valid for compounds derived from parents with one- and two-coordinate anions, albeit this is the majority of cases. Finally, the effect (most probably size and polarizing power) of the A counterion may be significant enough in some cases to destabilize the covalent MXx framework, diminishing the success of dimensional reduction. Despite these limitations, Tulsky and Long’s formalism is an exciting development in the evolution of what has become to be known as rational materials design. It is hoped that dimensional reduction can be coupled with other synthetic strategies to provide access to a wider range of tailored materials.

Practice Problems 1

Predict which of the following binary solids has a substantial covalent contribution to the interatomic bonding: NaCl, MnO2, Fe2O3, SiC, MgO, ZnS, NiAs.

2

Explain the primary difference between the Heitler–London and Hund–Mulliken theories of covalent bonding.

*3

What does the VSEPR theory tell us about the nature of the interatomic bonding?

165

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3 Crystal Structures and Binding Forces

4

True or False: Hybridization of atomic orbitals is best used for rationalizing known molecular geometries rather than for predicting molecular geometries.

5

What is the analogous entity to a MO in a nonmolecular crystal?

*6

Generally speaking, would you expect the lattice energies of ionic solids to increase or decrease with increasing charges on the anion and cation?

7

As the lattice energies of a series of ionic solids increase, what might you expect to happen to the water solubilities?

8

The reported bond lengths between the Mn and O atoms of the perovskite oxide series Ln1 − 3+ 2+ xAxMnO3 (Ln = La , A = Pb ; x = 0.1 – 0.5) are listed in the table below. Each Mn atom is octahedrally coordinated. The axial oxygen is denoted as O1 and the equatorial oxygen atoms as O2 and O3. The observed bond lengths are listed below. x = 0.1: Mn–O1 Mn–O2 Mn–O3

1.962 1.972 1.961

x = 0.2: Mn–O1 Mn–O2 Mn–O3

1.979 1.967 1.956

x = 0.3: Mn–O1 Mn–O2 Mn–O3

1.964 1.960 1.937

x = 0.4: Mn–O

1.943

x = 0.5: Mn–O

1.939

Use data from Table 3.7 to determine the average oxidation state of the manganese atom. *9

What is the difference between a solid solution alloy and an intermetallic compound?

10

In what ways do intermetallic phases differ from ceramics? In what ways are they alike?

11

What are the different kinds of point defects?

12

From a thermodynamic stability viewpoint, how are extrinsic point defects and extended defects similar?



For solutions, see Appendix D.

References

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171

4 The Electronic Level I An Overview of Band Theory

This chapter begins a series of chapters devoted to electronic structure and transport properties. In the present chapter, the foundation for understanding band structures of crystalline solids is laid. This chapter will start with some basics about the many-body Schrödinger equation and the mean field one-electron (Hartree–Fock) approximation. It will be seen that another set of basics needs to be introduced that are relevant for crystalline solids, viz. Bloch’s theorem for wave functions in a periodic potential and the reciprocal space. Two general approaches are then described for solving the extended electronic structure problem, the nearly free-electron model and its (opposite) counterpart, and the tight-binding approximation (akin to the LCAO method), both of which rely on the independent-electron approximation. Finally, the consequences of the independent-electron approximation are examined. Chapter 5 studies the tight-binding method in more detail. Chapter 6 focuses on electron and atomic dynamics (i.e. transport properties), and the metal– insulator transition is discussed in Chapter 7.

4.1

The Many-Body Schrödinger Equation and Hartree–Fock

As is well known from an historical point of view, the French physicist Louis de Broglie postulated the wave nature of the electron and suggested that all matter has wave properties in his PhD thesis he defended at the end of 1924 and which was published in 1925. Peter Debye, then in Zurich, Switzerland, suggested in November to Erwin Schrödinger that he give an informal seminar on de Broglie’s work. Debye then said that a formal three-dimensional wave equation should be written to substantiate the idea. Schrödinger came back a few weeks later with his time-independent equation, checked on the spectra of hydrogen, which he published in January 1926. Heisenberg, Bloch, Pauli, Fowler, Sommerfeld, and others immediately applied Schrödinger’s equation, together with Pauli’s exclusion principle, to an incredible wealth of physical problems. In 1929, the British theoretical physicist Paul Dirac could write the following statement, very famous except for its last part: The general theory of quantum mechanics is now almost complete (… Dirac was only worried of difficulties when high-speed particles are involved. He had written his own equation the year before, but had written it in order to have an equation for electrons mathematically consistent with Einstein’s special relativity requirements, not specially in order to treat high-speed particles). The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known, and the difficulty is only that the exact application of these laws leads to equations much too complicated to be soluble. It therefore becomes desirable that Principles of Inorganic Materials Design, Third Edition. John N. Lalena, David A. Cleary, and Olivier B. M. Hardouin Duparc. © 2020 John Wiley & Sons, Inc. Published 2020 by John Wiley & Sons, Inc.

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4 The Electronic Level I

approximate practical methods of applying quantum mechanics should be developed, which can lead to an explanation of the main features of complex atomic systems without too much computation. Although the computational means of today are incomparable with those of his time (when only logarithmic tables and slide rules were available!), we will see that Dirac’s last statement remains entirely true. The Schrödinger equation for a many-electron system is written as 41

HΨ = EΨ

where H is known as the Hamiltonian operator, E is the eigenvalue representing the total energy of the system, and Ψ is the eigenfunction (or eigenstate). The time-independent Schrödinger equation is thus an example of an eigenvalue equation (in German, eigen means characteristic; each H operator has its own characteristic set of eigenvalues E’s and eigenfunctions Ψ s). In words, Eq. (4.1) states that the Hamiltonian operates on the eigenfunction (Ψ) and gives a constant (E), which is the eigenvalue, times the same function. Ψ and E are a characteristic (eigen) solution of H. This formal equation has the appearance of being rather simple, which is quite deceptive. Except for the hydrogen atom, and the hydrogen-like ions, no exact solution is known for it. For a system containing N electrons and N nuclei (or ions), the Hamiltonian operator is expressed, in atomic units, as −

1 N 2 1 N 2 ∇ν − ∇ − 2 ν=1 2M n = 1 n

N

N

n=1 ν=1

Zn r ν − Rn

N

+ n,m > n

Zn Zm Rm − Rn

N

+ ν,μ > v

1 rν − rμ

42

The quantities shown in bold type are the spatial positions of the entities labeled by the subscript. The five summation terms represent, from left to right, electron kinetic energy, nuclear kinetic energy, electron–nuclear Coulomb attraction, nuclear–nuclear Coulomb repulsion, and electron–electron Coulomb repulsion. Zn is the electric charge number of the nth nucleus. Upon substitution of Eq. (4.2) for H in Eq. (4.1), the eigenvalue equation’s complicated form becomes more obvious, and in fact, unless the number of electrons and nuclei is extremely small, it is not possible to solve Eq. (4.1) directly on even the largest and fastest computers of today. The eigenfunctions of Eq. (4.1) depend on the positions of all the electrons, rν, and nuclei, Rn. The problem of solving for this many-body (N-electron) eigenfunction can be simplified if one can separate the degrees of freedom connected with the motion of the nuclei from those of the electrons. It fortunately happens to be the case in most situations: because the nuclei are so much heavier than the electrons, their kinetic energy (motion) can usually be neglected. This is called the Born– Oppenheimer approximation (Born and Oppenheimer 1927, extrapolating anew what Born and Heisenberg had written three years before, at a time when quantum mechanics was rapidly developing). The second term in Eq. (4.2) can then be set to zero and the last term treated as a parameter. One can alternatively say that the electrons almost immediately adapt their behavior, on the fly, with respect to the nuclei that move much more slowly and one thus also speaks of “adiabatic approximation.” The motions of the electrons and of the nuclei can be treated independently. Making the above approximation, the Schrödinger equation can now be written for the electrons in the system as −

1 N 2 ∇ − 2 ν=1 ν

N

N

n=1 ν=1

Zn r ν − Rn

N

+ ν,μ > v

1 rν − rμ

ψ r = Eψ r

43

4.1 The Many-Body Schrödinger Equation and Hartree–Fock

where Ψ is the many-body N-electron eigenfunction, which, in general, is a wave-like solution; E is the electronic energy. The total energy may be obtained by adding in the electrostatic energy between the nuclei. We do not of course consider in the total energy defined here the internal energy of the nuclei, let alone the internal energy of their constitutive hadrons. Unfortunately, even with the Born–Oppenheimer approximation, the problem of solving for the electronic wave functions using Eq. (4.3) remains intractable because of the large number of degrees of freedom and because of the third term representing the electron–electron interactions. One will see that this term implies the knowledge of the electron wave functions, i.e. the knowledge of the solution of the equation to be solved! This represents an unsolvable problem of selfconsistency, if one is after a perfect solution. The simplest approach to start working is a kind of mean field approximation: assume that the electrons mutually interact only via an averaged potential energy and kind of move independently of one another in such a mean field potential. This is known as the Hartree approximation and was derived by the British mathematician and physicist Douglas Rayner Hartree (1897–1958) in 1928 (Hartree 1928), under the name of self-consistent field (SCF, vide infra) as a method for solving a set of simultaneous equations obtained from the assumption that one can write the N-electron wave function as a product of N one-electron wave functions. Unfortunately, the Hartree one-electron functions turned out to be most unsatisfactory because their design missed some essential ingredient in quantum mechanics to start with, namely, the Pauli principle. Their product (the total wave function) is not antisymmetric under permutation of all electronic coordinates, including the spin variables. The Russian theoretician Vladimir Aleksandrovich Fock (1898–1974) extended the Hartree equation by considering the antisymmetry (Fock 1930), resulting in Eq. (4.5), which is now called the Fock, or Hartree– Fock equation (for a derivation, see Thijssen 1999). Equivalently, Slater showed that an antisymmetric total state function can be constructed as a sum of products of N one-electron functions having the property that it may be written as a (normalized) single determinant that will automatically satisfy the Pauli principle; Eq. (4.4) and see Example 4.1. Each one-electron wave function is given by a spin orbital that is a function depending on both the spatial coordinates (position) of the νth electron, rv, which, in turn, contains three variables, namely, xν, yν, zν, and its spin coordinate, sv. Thus, we will now change our notation to that for a spin orbital, denoted as ψ v (X), where X = (rv, sv). The one-electron function may be of the atomic, molecular, or Bloch type, depending on the problem at hand:

ψ Slater =

1 N

det ψ v =

1

ψ1 X1

ψ2 X1

ψN X1

ψ1 X2

ψ2 X2

ψN X2

N

∴ ψ1 XN

ψ2 XN

44

ψN XN

That single determinant is called a Slater determinant in quantum mechanics. It is antisymmetric because its sign is reversed upon interchange of two columns, which contain spin orbital functions (each being a product of spatial and spin functions). It is zero if two columns are identical: a manyfermion system cannot have two fermions in the same state at the same place. It is named for the American theoretician John Clarke Slater (1900–1976; see his bio. in Section 5.4) who published Slater determinants as a means of ensuring the antisymmetry of a wave function through the use of matrices. Example 4.1 Write down the wave function for a two-electron system and for a three-electron system using the Slater determinants. Directly check their required properties.

173

174

4 The Electronic Level I

Solution ψ= ψ=

1 2

ψ1 X1

ψ2 X1

ψ1 X2

ψ2 X2

1 2

=

ψ1 X1 ψ2 X2 − ψ2 X1 ψ1 X2

1 ψ X1 ψ2 X2 ψ3 X3 + ψ2 X1 ψ3 X2 ψ1 X3 + ψ3 X1 ψ1 X2 ψ2 X3 6 1 − ψ3 X1 ψ2 X2 ψ1 X3 − ψ2 X1 ψ1 X2 ψ3 X3 − ψ1 X1 − ψ3 X2 ψ2 X3

It is not difficult to directly check that these functions are null if two ψ labels are equal (e.g. ψ1 = ψ2), change of sign if two ψ labels are interchanged (e.g. ψ1 ψ2), and are normalized if the ψ v is normalized. Reversing to the (rv, sv) notations to emphasize the interchange in the spin orbitals, the antisymmetric wave functions can also be written as ψ(r1,s1; r2,s2) = ϕ(r1)ϕ(r2)[α(s1)β(s2) − α(s2)β(s1)]/√2. Because of the self-consistent subtleties already alluded to, it will turn out that the total energy of the N-electron system is not the simple sum of the eigenvalues of the effective one-electron eigenvalue equations that will be obtained. Using a single Slater determinant, Eq. (4.4), as a trial ground state wave function for the N-electron system in Eq. (4.3), it is found upon application of the variational principle that the one-electron spin orbitals must satisfy the following equation (actually a set of N-coupled nonlinear equations, ν = 1 – N), which also requires that the one-electron wave functions be orthonormal (normalized and orthogonal to one another), i.e. ψ ∗ν ψ μ dτ = δνμ, a stipulation met by the inclusion of Lagrangian multipliers λνμ: 1 − ∇1ν − 2 −

μ

n

Zn r ν − Rn

ψ ∗μ X ψ ν X rν − r

+ μ

ψμ X rν − r

dX ψ μ X =

μ

2

dX

ψν X 45

λνμ ψ μ X

A possible, although not mathematically compulsory choice for the λνμ when the spectrum is non-degenerate is λνμ = 0 for μ ν. The λνν are then renamed εν and become eigenvalues (often called one-electron HF orbital energies). In other words, εν are the eigenvalues of the matrix made of the λνμ, which simply implies a unitary transformation over the ψ ν. Let us note ΦKν and ΦNnν the first two operators within the bracket in Eq. (4.5). They come from the first two terms in Eq. (4.3) and simply correspond to the one-electron kinetic energy operator and to the attractive Coulomb interaction between the νth electron and the fixed N nuclei. The third term in Eq. (4.3) gives rise to two subtle terms in Eq. (4.5). The third term within the bracket gives a contribution known by various names, including the direct term, the Coulomb term, and the Hartree term (even though Hartree’s product of N one-electron wave functions is not antisymmetric). It represents the potential felt by the νth electron resulting from a charge distribution caused by all the other electrons. We shall note it ΦCν. Albeit already a complicated term, implying the knowledge of ψ μ, it deserves to be called a direct term, with respect to the next and last but conceptually not least term. The last term on the left-hand side of Eq. (4.5) is called the Exchange term, and its operator can be noted ΦXν. Specially notice that ψ ν is within the integral, in the mixed product ψ m ∗ ψ ν, as opposed ∗ ∗ to ψ μ ψ μ (=ψ μ ψ μ ) in the direct Coulomb (Hartree) term with ψ ν outside the integral. With these notations and with diagonalized ψ ν, Eq. (4.5) reads F ν ψ ν = εν ψ ν , ν = 1, N

4 6a

4.1 The Many-Body Schrödinger Equation and Hartree–Fock

F ν = ΦKν + ΦN ν + ΦCν + ΦXν

4 6b

The sum of operators Fν can be called the Fock, or Hartree–Fock operator, and is also sometimes referred to as the Hartree–Fock effective one-electron Hamiltonian. The symbolic form of Eq. (4.6) makes it look deceptively simple. It is actually a set of coupled integrodifferential equations and must be solved by the SCF, or SCF approach. The HF energy will be given, in the usual Dirac braket notation, by ΨHF HHF ΨHF (ΨHF, or ΨSlater, defined by Eq. 4.4, such as ΨHF ΨHF = 1). The pre-multiplication by L with such a box (antiperiodic condition, with a minus sign in Eqs. 4.10 and 4.11, exp(i k1 L1) = −1, gives λ = 2L/(2p + 1), which allows λ = 2L, but not λ = L). Although λ = 0.75L is smaller L, such a wave is not allowed in an L-box. It would be in a larger box with L = 3L, with λ = L /4. The k-distance between two successive kp, i.e. between kp and kp + 1, is δk = 2π/L. It does not depend on kp. The number of possible modes between k and k will be given by [floor(k /δk) − 1) and, again, does not depend ceil(k/δk)], which is about round (Δk/δk) (at least for Δk/δk on k: the distribution is said to be flat. Numerical applications: With L = 300 Å, the maximum wavelength is 300 Å, and the corresponding minimum wavenumber is k1 = δk ~ 0.020944 Å−1. There is no minimum wavelength and no maximum wavenumber. The number of possible waves between 5 and 6 Å−1 is floor (286.479) − ceil (238.74) = 287 – 237 = 47. The number of possible waves between 5000 and 5001 Å−1 is floor (238 780.16) − ceil(238 732.415) = 238 780 – 238 733 = 47, which checks the flatness of the mode distribution. Max Born (1882–1970) entered the University of Göttingen in 1904 where he became David Hilbert’s assistant, regularly met Hermann Minkowski, and was urgently invited by Felix Klein to work on an elasticity problem for which he won the 1906 Philosophy Faculty Prize. In 1907, he was admitted to Gonville and Caius College, Cambridge, and studied physics for six months at the Cavendish Laboratory under J.J. Thomson and Joseph Larmor. Back to Germany, Born worked on special relativity with Minkowski who died abruptly in 1909, and in the dynamics of crystals with Theodore von Kármán (1881–1963) with whom he developed the periodic boundary conditions that he explicited in a book, Dynamik der Kristallgitter, in 1915, a chance meeting with Fritz Haber in Berlin in November 1918 led to the description of the Born–Haber cycle (how an ionic compound is formed when a metal reacts with a halogen). With his former student Werner Heisenberg and Pascual Jordan, he developed the so-called matrix mechanics formulation of quantum mechanics in 1925 (since matrix product is not commutative, pq − qp 0). He proposed the now standard probabilistic interpretation of Schrödinger’s Wave function (through its square Ψ∗Ψ) the following year, as well as what became known as the Born approximation in scattering theory. With Heisenberg again and with Robert Oppenheimer, he developed the Born–Oppenheimer approximation. As many Jewish professors (Robert Wolfgang Cahn, John Werner Cahn, Erwin Panofsky, …), Born left Germany in 1933 (Göttingen’s Institute for Theoretical physics literally disappeared). C.V. Raman, already a Nobel prize winner, wanted to have him settled in Bangalore, but the Indian government did not create a corresponding tenure position. Pyotr Kapitsa offered a position in “his” Institute in Moscow when Charles Darwin, Charles Darwin’s grandson, asked him to take his succession at Edinburgh. A controversy lasted for more than two decades between Raman and Born about the validity of Born’s boundary conditions: “the Born postulate is in the clearest contradiction with experimental facts observed in crystals,” to which Born replied: “this contention is of course too absurd to be taken seriously.” Born got the Nobel prize in 1954. He remained at Edinburgh until 1952. He is buried at Göttingen with the pq − qp = h/2πi formula inscribed on his gravestone. Max Born also happens to be the singer Olivia Newton-John’s grandfather. (Photo Public domin, Wiki source.

4.3 Free-Electron Model for Metals: From Drude (Classical) to Sommerfeld (Fermi–Dirac)

4.3 Free-Electron Model for Metals: From Drude (Classical) to Sommerfeld (Fermi–Dirac) The free-electron gas (FEG) model is originated from the work of Paul Karl Ludwig Drude (1863–1906), who utilized the kinetic theory of gases to treat electrical conduction in metals (Drude 1900a, 1900b, 1902). In Drude’s model, a metal is regarded as a gas of free-valence electrons immersed in a homogeneous and positively charged sea, with a total charge just canceling the charge of the free electrons, for the sake of global neutrality. This “sea” is entirely permeable to the free electrons and can also be called an ethereal jelly. The model is referred to as a jellium. In this model, the free electrons experience a constant electrostatic potential everywhere in the metal, and all of them have the same average kinetic energy, 3/2 kBT, which corresponds to the 1/2 mv2 in Newton’s second law. Soon afterward, Hendrik Lorentz applied the Maxwell–Boltzmann statistics to these free electrons, and the model is sometimes called the Drude–Lorentz model (Lorentz 1904–1905). The Fermi–Dirac statistics was established soon (1926) after Pauli stated his exclusion principle for electrons (1925). Ralph Fowler (1889–1944), Dirac’s supervisor, immediately applied it to show that the electron gas in a star as dense as a white dwarf is degenerate, i.e. not classical and dominated by the exclusion principle, thus solving a thermodynamical paradox that had puzzled Arthur Eddington. Arnold Sommerfeld (1868–1951) applied the same statistics to the free electrons in a metal, showing that they must also be regarded as a degenerate Fermi gas (Sommerfeld 1928). He thereby solved the puzzles of the classical Drude–Lorentz model concerning both the electronic heat capacity Cel and the electronic heat conductivity κ el (both were wrong, by a large factor due to the same flaw in the classical treatment and with the same factor, so that their ratio was essentially correct and in rather good agreement with the experimental Wiedemann–Franz–Lorenz law). If free electrons were classical particles, every one of them would contribute 3/2 kBT to the kinetic and potential energies, hence, 3 kB to Cel so that the total heat capacity of a mole of monovalent metal such as sodium or copper would be 6 R (3 + 1 3), and 12 R (3 + 3 3) for aluminum, in obvious contradistinction with the experimental Dulong and Petit law. Actually, being fermions, they are, from an energetic point of view, simply piled up with no degree of liberty for the great majority of them until those whose energies are close to the Fermi energy EF. Because it turns out that EF is much larger than kBTamb, by about three orders of magnitude, only a negligible fraction of the free electrons have an energy close to EF and can contribute to Cel. More precise calculations imply the derivation of the Fermi–Dirac function, and the interested reader may refer to any of several books on solid-state physics for this (e.g. Seitz 1940; Ashcroft and Mermin 1976), knowing that Sommerfeld’s calculation, with his famous first-order integral developments, is actually both wrong and right, right by chance because the following orders happen to exactly sum up to zero (Weinstock 1969). There was still one conundrum, however, namely, the free electrons should scatter on the ions and have a mean free path of the order of the atomic distance. Yet, such a distance is much too short to explain the observed resistances. The answer actually arrived that same 1928 year, with Felix Bloch; see Section 4.4. To put it in a nutshell, Bloch showed that electrons move as free waves (rather than as particles) with amplitudes simply modulated by the lattice periodicity of the ions in the crystal. These free Bloch waves do not scatter on atoms, and, in fact, even do not have to scatter at all, except at surfaces of the finite crystal: their mean free path now seems much too large! The answer is that no crystal is internally perfect, and the scattering of the electrons is caused by the crystalline defects, due either to thermal fluctuations or to (thermal) vacancies and impurities. When the temperature is lowered, electron mean free paths in a metal should thus increase, up to a limit that is set up by impurity concentration, which is indeed what is experimentally observed.

179

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4 The Electronic Level I

In the spirit of Section 4.1, we must wonder about the eigenfunctions and eigenvalues of the Sommerfeld model. It corresponds to an empty lattice (no ions), the electronic Hamiltonian contains only the electron kinetic energy term (1/2)mev2 ≡ p2/2me (me is the electron rest mass), and there is no electron–electron interactions. With the well-known quantum mechanical – particle operator wave correspondence p ≡ −iℏ∇ (ħ is the reduced Planck constant), the time-independent Schrödinger equation is (ℏ2/2me)∇2ψ = Eψ. The eigenfunctions are simple plane waves of the general form ψk(r) = exp[ik r], and the corresponding eigenvalues are given by the well-known formula Ek = (ℏ2/2me)k2. This E(k) dependence is a parabola in 1d, a paraboloid in 2d, and cannot be visualized in 3d except for its iso-energetic k–surfaces, which are spherical, of radius k = (2me E)1/2/ℏ. E (k) is called a band structure (an electronic band structure). E(k) = (ℏ2/2me)k2 is the band structure of the free electron gas (FEG) model. Note that E is often equivalently labeled ε.

4.4

Bloch’s Theorem, Bloch Waves, Energy Bands, and Fermi Energy

The potential felt by an effective electron corresponds to the sum of the last three terms on the righthand side of Eq. (4.6b). Because of the electron–ion interaction, this potential has a perfect periodicity in a crystalline solid and may be represented as V r+R =V r

4 13

where r is an atomic position and R is a lattice vector. If Eq. (4.13) is true, a symmetry operation, such as translation, ℑR, that transforms a crystal into itself, does not change the Hamiltonian, H, nor does it change the Fock operator, F, that is, the Hamiltonian and Fock operators are translational invariant. Hence, ℑR commutes with H and F, for example, ℑRH − HℑR = 0. Therefore, a one-electron wave function satisfying the Schrödinger equation (or Hartree–Fock equation) is also an eigenstate of ℑR, or ᑣR ψj r = ψj r + R = f ψj r

4 14

where, for clarity, just the spatial coordinate of the electron, r, has been emphasized and the subscript μ dropped for generality. The eigenvalue satisfying this requirement and Eq. (4.14) is f = exp i k R

4 15

where R is a real-space lattice vector equal to n1a1 + n2a2 + n3a3 (n1, n2, and n3 are integers, and a1, a2, and a3 are primitive translation vectors). The quantity k is the wave vector that can be expressed in whatever basis one wants, the cleverest choice being the primitive reciprocal vectors a∗1 , a∗2 , a∗3 with respect to a1, a2, and a3 (see Section 4.5). If external Born–von Kármán condition are applied to the crystal, k = k1a∗1 + k2a∗2 + k3a∗3 with k1, k2, and k3 quantized as 2πp1/L1, 2πp2/L2, 2πp3/L3, L1, L2, and L3 characterize the crystal dimensions considered to be parallelepipedic along a1, a2, and a3, and p1, p2, and p3 are integers (see Section 4.2). (Beware that notations may differ from books to books, even from chapters to chapters. Notations are free, being conventional. They are context dependent, as many words are, so always check the context.) Considering the propagation of periodic waves in an inner periodic medium, an important theorem was derived from Eqs. (4.14) and (4.15) in 1928 by Felix Bloch (1905–1983) while he was a PhD student at the University of Leipzig working under Heisenberg. Bloch’s theorem states that, owing

4.4 Bloch’s Theorem, Bloch Waves, Energy Bands, and Fermi Energy

to the translational invariance of the Hamiltonian, any one-electron wave function can be represented by a modulated plane wave (Bloch 1928): ψ k r = uk r exp ik r

4 16

where exp(ik r) is the plane wave and uk(r) is a modulating function with the periodicity of the real-space Bravais lattice. (Paul Peter Ewald had had similar considerations when he considered in his own PhD thesis, more than 14 years before, the behavior of incoming plane waves of X-rays once they are in a crystal.) The modulating function uk(r) satisfies the following equation: uk r + R = uk r

4 17

Eqs. (4.16) and (4.17) imply the following relation: ψ k r + R = exp ik R ψ k r

4 18

This means that the Bloch wave function has the same amplitude at equivalent positions in each unit cell. Thus, the full electronic structure problem is reduced to a consideration of just the number of electrons in the unit cell (or half that number if the electronic orbitals are assumed to be doubly occupied) and applying boundary conditions to the cell as dictated by Bloch’s theorem (Eq. 4.16). Each unit cell face has a partner face that is found by translating the face over a lattice vector R. The solutions to the Schrödinger equation on both faces are equal up to the phase factor exp(ik R), determining the solutions inside the cell completely. Since the wave functions are subject to some external boundary conditions, here BvK, the energy eigenvalues are quantized through the quantization of k (see Section 4.2). There are N one-electron eigenstates, ψj, with corresponding eigenvalues, εj (there may be sets of energetically degenerate eigenstates for some values of k). Each of the unique eigenvalues is termed an energy level. These energy levels are distributed within energy bands. Because N is so large for a real crystal (much larger than the Avogadro number), the distribution is quasicontinuous within each band, quasicontinuous but not uniform, and characterized by a density of states (DOS, n(E)). Energy intervals between bands are called gaps or forbidden bands or band gaps. n(E) is zero in the gaps. The highest energy level occupied by electrons is termed the Fermi level or Fermi energy (symbolized EF). All eigenvalue levels below the Fermi level are occupied with electrons, and all levels above it are empty. If EF is inside a band, the crystal can conduct electricity and is a metal. The corresponding band is called the conduction band. If EF is at the top of a band, the crystal cannot conduct electricity and is called an insulator. EF is then placed in the middle between the top of the last occupied band, which is called the valence band, and the bottom of the nearest non-occupied (empty) band, called the conduction band. The reason of this middle position is because it can be more adequately seen as a chemical position; see Example 4.3. The energy gap in between is called the band gap of the insulator. If the band gap is just of the order of 1 eV or less, the corresponding crystal is called a semiconductor because it can be made relatively conducting by raising the temperature and by doping it with relevant atomic impurities (foreign atoms). The band gap of silicon is about 1.2 eV. It is about 5.5 eV for diamond. These energy bands we just talked about are simply defined by n(E) being nonzero or zero. They actually have an inner structure. The energy eigenvalues are functions of k, that is, εj(k). The eigenvalues change smoothly as k changes, forming curves as one moves from 1 k-point to another in E – k-space. The dispersions of the various curves are displayed in electronic band structure, or band dispersion, diagrams; see Figure 4.9 for an example.

181

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4 The Electronic Level I

Example 4.3 Considering the Fermi energy EF as a chemical potential for electrons, find out why EF is in the middle of the band gap of an (intrinsic) insulator (at temperature T = 0). Hint: use μ = dU/dN ~ [(U(N + 1) − U(N)) + (U(N) − U(N − 1))]/2), the average of the energy changes for adding or removing a particle. Solution For an insulator, the valence band being totally filled, a new added electron (the (N + 1)th electron) must occupy the Ec level (the bottom of the next available band, called the conduction band) so that U(N + 1) − U(N) = Ec, whereas the previously added electron was on the Ev level, with U(N) − U(N − 1) = Ev, hence μ = (Ec + Ev)/2, which is also equal to Ev + Eg/2 with Eg = (Ec − Ev), the width of the band gap. For a metal, as the energy Emax of the Nth electron is strictly below the top of the partially filled band where it sits, one has U(N + 1) − U(N) ~ U(N) − U(N − 1) ~ Emax, hence μ = Emax, a usual definition of EF in a metal. In a metal, the valence band and the conduction band are one and the same band. At nonzero temperature, U should be replaced by the free energy F. Felix Bloch (1905–1983) studied at the University of Leipzig where Schrödinger developed his equation because Debye had remarked that the new de Broglie’s wave for particles ought to be considered through a wave equation. Soon after attending these seminal colloquia, Bloch did his PhD thesis under Heisenberg. In it, he introduced the theorem that allows us to write the electron wave function in a periodic lattice as a modulated plane wave. This concept is fundamental to electronic structure calculations on crystalline solids. From 1928 to 1929, Bloch was an assistant to Wolfgang Pauli at the Swiss Federal Institute of Technology in Zurich. Bloch is no less famous for his contributions as an experimentalist. Upon Hitler’s ascent to power, he left Europe and accepted a position at Stanford University, where he carried out research on nuclear magnetic measurements. In 1939, he determined the magnetic moment of the neutron with an accuracy of about 1%. Bloch and E.M. Purcell were awarded the 1952 Nobel Prize in physics for the first successful nuclear magnetic resonance experiments, performed independently in 1946. Purcell measured the nuclear magnetic absorption while Bloch’s method utilized nuclear induction, the well-known Bloch equations describing the behavior of a nuclear spin in a magnetic field under the influence of radiofrequency pulses. Nuclear magnetic resonance has since been applied to the investigation of molecular structure and medical diagnostics, via magnetic resonance imaging. Bloch was elected to the United States National Academy of Sciences in 1948. (Primary source: Hofstadter, R. (1994). Biographical Memoirs of the U.S. National Academy of Sciences, Vol. 64, pp. 34–71.) (Photo courtesy of Stnford University Archives, Green Library. Reproduced with permission.

4.5

Reciprocal Space and Brillouin Zones

It has just been stated that a band structure diagram is a plot of the energies of the various bands in a periodic solid versus the value of the reciprocal space wave vector k. It is now necessary to recall the concept of the reciprocal space lattice and its relation to the real-space lattice. The crystal structure of a solid is ordinarily presented in terms of the real-space lattice composed of lattice points, which have an associated atom or group of atoms whose positions can be referred to them. Two real-space lattice points are connected by a primitive translation vector, R:

4.5 Reciprocal Space and Brillouin Zones

Table 4.1

Primitive translation vectors of the real-space cubic lattices: R = ua1 + va2 + wa3.

Lattice

Primitive vectors

Simple cubic (SC)

a1 = ax a2 = ay a3 = az

BCC

a1 =

FCC

a1 =

R = n1 a1 + n2 a2 + n3 a3

a 2 a 2

− x + y + z a2 = y + z a2 =

a 2

a 2

x+z

x−y + z a3 =

a 2

a3 =

a 2

x + y−z

x+y

4 19

where n1, n2, and n3 are integers and a1, a2, and a3 are called basis vectors. The primitive vectors for the cubic lattices, for example, are given in Table 4.1 in which a is the unit cell parameter. Besides lattice vectors, one also defines sets of parallel lattice planes, and it turns out that the best way to define these sets is to introduce reciprocal space notations with the so-called Miller indices; see Section 1.2.1.3 and what follows. A lattice point being chosen as an origin and the lattice plane that is closest to this origin, but not including it, intersect the unit cell axes at a1/h, a2/k, and a3/l (in case of no intersect, h, k, or l equals zero). The reciprocals of these coefficients are h, k, and l, which are used to denote the set (hkl) of parallel planes. The distance from the origin to the closest plane is the interplanar spacing, dhkl. Any such set of planes is thus completely specified by dhkl and the unit vector normal to the set, nhkl, since the former is given by the projection of, for example, a1/h onto nhkl, that is, dhkl = (a1/h) nhkl. If one divides nhkl by dhkl, one gets a vector Shkl, which is such that Shkl a1 = h, Shkl a2 = k, and Shkl a3 = l. It means that Shkl can be written as Shkl = ha∗1 + ka∗2 + la∗3

4 20

where a∗1 , a∗2 , a∗3 are the so-called reciprocal basis vectors obeying the following relationships: ai a∗j = δij

4 21

for all i and j from 1 to 3 and δij being the Kronecker delta. Shkl denotes the set of (hkl) planes. If one wants to study the propagation of physical waves (X-rays, free or nearly free electrons (NFE), phonons) in a crystal, it will be more convenient to include a “2π” factor in the definition of the reciprocal vectors and use Khkl = 2π Shkl: K hkl =

2πnhkl dhkl

4 22

This will allow to consider exp(ik r) formulae instead of exp(2i π s r), thus saving many “2π” (as is well known, for waves, ω = 2πν and k = 2πλ). A vector [uvw] belonging to a (hkl) plane is perpendicular to Shkl by definition (Shkl = nhkl/dhkl or Eq. 4.22), and the mathematical relationship is simply hu + kv + lw = 0 because of Eqs. (4.20) and (4.21). This is known as Weiss zone law (or Weiss zone rule, Christian Samuel Weiss; see Chapter 1). One of the great virtues of the reciprocal lattice vectors is to restore Cartesian-like relationships even for nonorthorhombic lattices. Each vector Shkl or Khkl of the reciprocal lattice corresponds to a set of planes in the real-space lattice with Miller indices (hkl). These vectors are perpendicular to the (hkl) planes, and their lengths are equal to the reciprocal of the plane spacing (without or with a 2π factor). Hence, each crystal structure has associated with it a real-space lattice and a reciprocal space lattice.

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4 The Electronic Level I

A reciprocal lattice vector of Eq. (4.22) can thus be defined in terms of basis vectors (b1, b2, b3): K hkl = hb1 + kb2 + lb3

4 23

The (b1, b2, b3) are defined from the (a1, a2, a3) via the relations bi aj = 2π δij, i.e. bi = 2πa∗i , with our notations. In practice, (b1, b2, b3) can be obtained from (a1, a2, a3) by a relation given by J.W. Gibbs (Wilson 1901; Margenau and Murphy 1956): b1 =

2π a2 × a3 a1 a2 × a3

4 24

b2 =

2π a1 × a3 a1 a2 × a3

4 25

b3 =

2π a1 × a2 a1 a2 × a3

4 26

In Eqs. (4.24)–(4.26), the numerators contain cross (vector) products and the denominators are scalar triple products equal to the volume of the real-space unit cell. It may be computed from its determinant (see Practice Problem 7). Labeling is conventional. One can also find (a, b, c) for the real-space basis vectors and (a∗, b∗, c∗) for the reciprocal space basis vectors, but one then loses the conveniency of using the Kronecker delta δij. Crystallographers like to use (a1, a2, a3) and a∗1 , a∗2 , a∗3 in order to keep using the δij and to comply with the ∗ convention for the reciprocal space. The reciprocal space is also called by mathematicians the dual space of the real space, and the covariant/contravariant formalism can also be invoked, although it will not be in this book. Geometrical crystallographers prefer to use the “not 2π” convention ai a∗j = δij so that the dual space of a dual space is the original space. As said above, X-ray crystallographers and solid-state physicists dealing with electrons or phonons prefer to use the “2π” convention. The reader should check that with this convention, the dual equivalent of Eq. (4.24) on b1, b2, b3, viz. 2π(b2 × b3)/ [b1 (b2 × b3)], gives 4π 2 a1, not a1. Taking the primitive translation vectors for one of the real-space cubic lattices from Table 4.1, Eqs. (4.24)–(4.26) can be used to obtain the primitive translation vectors for the corresponding reciprocal lattice, which are given in Table 4.2. By comparing Tables 4.1 and 4.2, it is seen that the primitive vectors of the reciprocal lattice for the real-space FCC lattice, for example, are the primitive vectors for a BCC lattice. In other words, the FCC real-space lattice has a BCC reciprocal lattice (and vice versa). Besides the reciprocal lattice vectors K defined by Eq. (4.23), one can introduce general vectors k in reciprocal space with k1, k2, and k3 being general real numbers: k = k1 b1 + k2 b2 + k3 b3

Table 4.2

4 27

Primitive translation vectors of the reciprocal lattices.

Real-space lattice

Corresponding reciprocal lattice primitive vectors

SC

b1 = b1 = (2π/a)x b2 = (2π/a)y b3 = (2π/a)z

BCC

b1 = b1 = (2π/a)(y + z) b2 = (2π/a)(x + z) b3 = (2π/a)(x + y)

FCC

b1 = b1 = (2π/a)(−x + y + z) b2 = (2π/a)(x − y + z) b3 = (2π/a)(x + y − z)

4.5 Reciprocal Space and Brillouin Zones

Note that this vector appears in the expression for the electronic wave function (e.g. Eqs. 4.10, 4.15 sqq.). Mutatis mutandis, k’s are to K’s what r’s are to R’s. Another usual labeling for k and K is g and G. It is used to define a special unit cell in reciprocal space, which may be derived in the following manner. First, reciprocal lattice vectors K are drawn between a given reciprocal lattice point (the origin) and all other points. Perpendicular planes are added at the midpoints to these K vectors. The smallest volume enclosed by the polyhedra defined by the intersection of these planes about the central lattice point is conventionally referred to as the first Brillouin zone (BZ), after French physicist Léon Brillouin (1889–1969) who showed that the surfaces of energy discontinuity à la Peierls (see Figure 4.2, in only 1d) form special polyhedra in 3d reciprocal space (Brillouin 1930). From a mathematical point of view, a Brillouin zone is a Voronoi construction in the reciprocal lattice, named after the Russian mathematician Georgy Feodosevich Voronoi (1868–1908). The Wigner–Seitz cell, by comparison, is a Voronoi construction in the real-space lattice. Voronoi constructions are also known by various other names: Voronoi diagram, Voronoi tessellation, Voronoi decomposition, Voronoi partition, or Voronoi cell. Voronoi constructions can be carried out with respect to any set of points, however, including atoms in an amorphous structure, cities in a geographical map (Thiessen polygons), etc. For our purposes here, the essential characteristic of a Voronoi cell is that, for a given lattice point, it encompasses the region or volume of space closer to that lattice point than to any other lattice point. In their physical sense, they correspond to regions of influence as originally defined by Descartes for his “solar systems” in his universe as made of star vortices in 1644. Voronoi defined them in 1908, the year of his death, when working on the solutions of positive quadratic forms in pure mathematics, after Gustav Peter Lejeune Dirichlet in 1850 so that one also finds the phrase “Dirichlet regions.” One more remark is that Brillouin zones, as well as Wigner–Seitz cells, are not like unit or primitive cells: they are uniquely determined for a given lattice structure, by a well-defined geometrical procedure. They are not defined by three basis vectors, which can always be chosen with some arbitrariness (even primitive cells can be chosen differently for a given lattice, provided they have the same minimum volume). Primitive cells could have been chosen instead, originated at K = 0 (called the Gamma point, Γ), or centered on Γ. In all cases, the k-volume would be the same. Brillouin’s choice gets the k’s that are the closest to the origin that to any other lattice points, hence the shortest k’s: this is why it is the adopted choice. The first Brillouin zone contains all the symmetry of the reciprocal lattice, that is, all of the reciprocal space is covered by periodic translation of this unit cell. There are higher-order zones that may be constructed similarly to the first BZ, by drawing vectors to the next nearest neighbors. The result is that the higher-order zones are fragmented pieces separated from each other by the lower zones. Each zone occupies an equal volume of k-space. The first Brillouin zones for the SC, BCC, and FCC lattices are shown in Figure 4.1. The inner symmetry elements for each BZ are, using the symbols proposed by Bouckaert, Smoluchowski, and Wigner (BSW) the following: the center, Γ; the threefold axis, Λ; the fourfold axis, Δ; and the twofold axis, Σ. The symmetry points on the BZ boundary (faces) (X, M, R, etc.) depend on the type of polyhedron. The reciprocal lattice of a real-space SC lattice is itself a SC lattice. The Wigner–Seitz cell is the cube shown in Figure 4.1a. Thus, the first BZ for the SC real-space lattice is a cube with the high-symmetry points shown in Table 4.3. The reciprocal lattice of a BCC real-space lattice is an FCC lattice. The first BZ zone of the FCC lattice is the rhombic dodecahedron in Figure 4.1b. The volume enclosed by this polyhedron is the first BZ for the BCC real-space lattice. The high-symmetry points are shown in Table 4.4.

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4 The Electronic Level I

(a)

(b)

(c) W

R A Γ Σ

T Δ

P S

M

X Z

D Γ

L

F

N

G

Q H

Γ K

U

W

Z

S X

Figure 4.1 The first Brillouin zone (BZ, Voronoi cell in the reciprocal space) for the SC real-space lattice is itself a simple cube (a regular hexahedron) (a). For the BCC real-space lattice, the first BZ is a rhombododecahedron (b). For the FCC real-space lattice, the first BZ is a truncated octahedron (c).

Table 4.3

High-symmetry points for the SC real-space lattice cube.

k-Point label

Cartesian coordinates

Γ

(0, 0, 0)

X

(π/a, 0, 0)

M

(π/a, π/a, 0)

R

(π/a, π/a, π/a)

Table 4.4

High-symmetry points for the BCC real-space lattice.

k-Point label

Cartesian coordinates

Γ

(0, 0, 0)

H

(2π/a, 0, 0)

N

(π/a, π/a, 0)

P

(π/a, π/a, π/a)

The reciprocal lattice for the FCC real-space lattice is a BCC lattice. Its first BZ is a truncated octahedron (Figure 4.1c). From the physical point of view of the electronic wave vectors, the shapes of the BZs for the SC and BCC real-space lattices are completely and uniquely determined by the condition that each inner vector, k, go over into another by all the symmetry operations. This is not the case for the first BZ of the FCC real-space lattice. The truncated octahedron satisfies the condition, but it is not uniquely determined by it (Bouckaert et al. 1936). The truncating planes (Figure 4.1c) correspond to symmetry planes perpendicular to vectors joining reciprocal lattice points, not the octahedral planes. The Cartesian coordinates of the special high-symmetry points are given in Table 4.5.

4.5 Reciprocal Space and Brillouin Zones

Table 4.5 High-symmetry points for the FCC real-space lattice. k-Point label

Cartesian coordinates

Γ

(0, 0, 0)

X

(2π/a, 0, 0)

W

(2π/a, π/a, 0)

K

(3π/2a, 3π/2a, 0)

L

(π/a, π/a, π/a)

Figure 4.2 The first Brillouin zone for the hexagonal real-space lattice has itself an hexagonal shape.

L H

R

S

A Δ

P K

H

L

M T

H P

U Γ

Σ M

H

S′

K

T′

A L

H

The BZ for the hexagonal real-space lattice is hexagonal, see Figure 4.2. Its symmetry elements were studied by the American physicist Conyers Herring (1914–2009) in 1942. Beware that the choice of basis vectors for hexagonal systems can be confusing. To comply with the International Union of Crystallography recommendations, the angle between the a (a1) and b (a2) vectors in real space should be 120 (and not 60 ); see Figures 1.6 and 1.10, as well as Practice Problem 5. This choice gives a 60 angle in the reciprocal lattice, but the shape of the BZ, and its symmetry elements, is uniquely determined by its geometrical construction and does not depend on the choice of mathematical basis vectors for subsequent calculations so that any choice can be made, including a convenient Cartesian frame. Given this freedom, great care must be paid to what choices are made by different authors in their papers or computer softwares. One can now join the Bloch Eq. (4.16) to the K-periodicity in the reciprocal lattice space. The Bloch wave functions thus also obey the following equation: ψk r = ψk + K r

4 28

Bloch’s theorem thus means that, in order to know all electronic properties of a crystal, it is sufficient, and necessary, to only study its Bloch wave functions inside its first Brillouin zone. For the same physicomathematical reasons, it is also true for the study of the lattice vibrational properties (known as phonons when quantized). It is “sufficient”: well, it, of course, still needs to be done…

187

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4 The Electronic Level I

Léon Brillouin (1889–1969) was born to Marcel Brillouin, one of the six French physicists invited to the first Solvay Conference, who always outlined in his teaching the history of the subject. Like his father, Léon entered the prestigious École normale supérieure in 1908 where he studied the Brownian motion under Jean Perrin in 1912, as well as the blue color of the sky, mainly due to inhomogeneities of the atmospheric gas. Just married, L.B. went to Munich to study theoretical physics under Arnold Sommerfeld in 1913 where he worked on the diffraction of electromagnetic radiation in a dispersive media such as real matter with its electrons. There, he also met Woldemar Voigt. Back to France, he started his PhD work and published a short note on the scattering of light in condensed matter. WWI broke out and he was asked to work for the emission and reception of radio waves, for radiotelegraphy. He patented several devices, including an amplifier tube. L.B. could then resume his 1914 work, in a more precise way, getting a formula giving the change of frequency of a light wave scattered by the acoustic waves of a solid, which is the Brillouin effect that could be experimentally verified only in 1932. It was two years before, in 1930, that L.B. published his famous two- and three-dimensional drawings of the zones within which wave vectors are permissible for (nearly) free electron propagation in metals, which are actually equally valid for the propagation of phonons. Being a specialist in radio wave propagation, L.B. was appointed as director general of the French state-run agency, Radiodiffusion Nationale in August 1939. He was then part of the “French” government that retired to Vichy in May 1940. Léon resigned six months later and went to the United States where he joined the “France libre” organization. He naturally worked in the field of radar, developing the electronic theory of the magnetron. At the end of the war, L.B. decided to stay in the United States with his Jewish wife Stéfa. Being a physicist so much implied in the transmission of information, and a teacher, he became involved in the then new field of the theory of information. Working on the negative entropy concept introduced by Schrödinger in 1944, L.B. established a negentropy principle of information in 1953, the year when he was elected to the US National Academy of Sciences in 1953. His 1938 French lecture book Tensors in Mechanics and Elasticity was translated in 1964. He died in New York in 1969. (Primry source: Mosseri, R. (1999. Léon Brillouin. À la croisée des ondes. Paris: Belin. Photo taken around 1930, coll. Sylvie Devers, granddaughter of Léon and Stéfa Brillouin.

4.6 Choices of Basis Sets and Band Structure with Applicative Examples The one-electron wave function in an extended solid can be represented with different types of basis sets. Discussed here are only two types, representing opposite extremes: the plane wave basis set (free-electron and nearly free-electron models) and the atomic orbitals basis set. In the latter case, a periodic solid is considered constructed from the coalescence (linear combination) of atoms (hence the acronym LCAO) into extended Bloch wave functions. The LCAO approach will be dealt with in Section 4.6.4 and in the next chapter. A brief presentation of the plane wave basis set is first in order owing to its historical significance and for certain insights it provides, particularly with metals. In the early years, the free-electron model and subsequently the plane wave expansion method came into favor with physicists and metallurgists, mainly because of the substantial efforts at the time to understand the conductivity of metals. In fact, a plane wave basis set holds several advantages over a localized atom-centered basis set: plane waves are orthonormal, independent of atomic positions, and do not suffer from superposition errors. For these reasons, the plane wave basis set is most commonly used in solid-state calculations. Unfortunately, a large number of plane waves are necessary to describe the fluctuation of electron charge density near the atomic cores, and this requires modifications such as the pseudopotential plane wave (PPW), orthogonalized plane wave (OPW), or linearized augmented plane wave (LAPW) methods.

4.6 Choices of Basis Sets and Band Structure with Applicative Examples

4.6.1 From the Free-Electron Model to the Plane Wave Expansion Again, in the FEG model, the εk = (ℏ2/2me)k2. The additive filling in the reciprocal case is of two electrons (spins up and down) per (2π/L)3 cube so that the number of free electrons N per real-space volume V (=L3) can be approximated by 2 (4/3)π k3/(2π/L)3. Labeling εk as E, this gives N = N E3/2. The DOS n(E) = dN/dE of the FEG model thus has the parabolic dependence n (E) (E) = α E1/2 with α = (2me)3/2 (4πV/h3). At EF, one thus has n E F = 2me

3 2

4πV h3

EF

4 29

From slightly more sophisticated formulae independent of FEG and implying the Fermi–Dirac function (see, e.g. Seitz 1940), it turns out that, for nonmagnetic metals, the constant volume specific heat C(T) [C(T) = (∂U(T)/∂T)V], obtained experimentally from calorimetric measurements at low temperatures, can give an estimation of n(EF), the DOS of the metal at the Fermi level. The lattice contribution to C(T) at ambient temperature is 3R, (R being the gas constant), in agreement with the classical Dulong and Petit experimental law. At low temperatures, the quantification of the lattice modes (phonons) let Clat(T) become proportional to T3. For metals, there is also an electronic contribution Cel(T), which is proportional to T. One thus have, at low temperatures for nonmagnetic metals, C(T) = aT3 + γT. The coefficient of proportionality γ is, quite reasonably (with kB the Boltzmann’s constant) γ = π 2 3 kB 2 n EF

4 30

Higher terms in T are negligible. Thus, when plotted as C(T)/T versus T2, low-temperature (typically below 20 K but of course above a possible superconducting transition temperature) measurements yield a straight line whose ordinate intercept is γ, from which n(EF) can be obtained. Of course, the heat capacity normally measured is CP, the heat capacity at constant pressure. However, for solids the difference between CP and C is typically less than 1% at low temperatures and thus can be neglected. If one compares the simple free-electron model value, Eq. (4.20), to experimental values obtained via C(T) measurements from Eq. (4.21), one gets the right order of magnitude but with discrepancies as high as a factor of two or three, which just means that the FEG model is only a good first level approximation but is too simple in many cases. Indeed, a crystal, even a metal, is not a box simply filled with a structureless smeared out positive background and free electrons (also see Sections 4.6.2 and 10.2.3.1 pointing out limits of FEG). It is necessary to go beyond the FEG model. The next step is to go to the so-called NFE model. The electronic properties of most main group s- and p-block elements are better described by introducing a periodic potential as a small perturbation. After all, Bloch construction is based on the existence of the geometrical atomic lattice periodicity. It would thus seem natural to go beyond geometry (the introduction of Bloch waves and the reciprocal lattice) and take into account a physical interaction between the nuclei (or ions, even as fixed) and the electrons. Historically indeed, Peierls showed in 1930 that, in the NFE limit, band gaps arise from electron diffraction, a natural consequence of wave propagation in a periodic structure (Peierls 1930); see infra and Section 7.5 with Figure 7.7. Brillouin generalized the result and showed that, in three dimensions, the surfaces of discontinuity form polyhedra in reciprocal space, the BZ (Brillouin 1930). In Sections 4.4 and 4.5, we considered the direct and reciprocal implications of the periodic V(r) of Eq. (4.11) on the properties of the wave function ψ(r), as ψk(r), and we did not care to consider Ek. It is clear that K-periodicity implies Ek + K = Ek, just as ψk + K(r) = ψk(r) (Eq. 4.1). But what about E(k)

189

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4 The Electronic Level I

itself? It might be hoped that for not too complicated V(r) (no strong electron–electron coupling for instance), since the Bloch waves are simply modulated free waves, their E(k) might not differ too much from the free (ℏ2/2me)k2 dependence. Periodicity allowed Bloch to show that the electronic waves are Bloch waves (with uk(r) having the periodicity of the lattice). In order to take further advantage of periodicity to go further in the resolution of the Schrödinger equation, one must Fourier expand both V(r) and uk(r): V r =

V K eiKr

4 31

ck K eiKr

4 32

ck K ei K + k r

4 33

K

uk r = K

Hence Ψk r = K

Assuming nearly free electronic ψk(r) solutions, only a few ck(K) coefficients are considered, and one gets several sets of EG(k) solutions, the simplest one, with the lowest energy, being for G = 0 (G is just another labeling for K; we use it here to avoid confusion with the K’s in the sum in the following formula): E = V0 +

ℏ2 k 2 + 2m K

V K 2 2m ℏ2 2 0k − K + k

4 34

2

which reduces to the free solution if the sum can be considered as negligible, which, since V(r) is supposed to be weak, is the case for all k such that k2 − (K + k)2 is not negligible. It is satisfying of course to get E(k) ~ ℏ2k2/2m for many k’s (we can get rid of V0, which is VK = 0, by changing the arbitrary origin of energy). And we now also know for which k’s it will not be the case. For these latter k’s, the Eq. (4.34) cannot help. Since we want to consider the k’s such k2 − (K + k)2 can be negligible, we shall note them k = ko + Δk with ko being solution k2o − (K − ko)2 = 0 which is the equation of a Brillouin zone boundary, then also usually called a Bragg plane; see Example 4.4. After some algebra and reasoning (see for instance Elliott 1998), one is led to a second-order equation in ε, whose solutions are E ± = k o + Δk =

ℏ2 k 2o ℏ2 + 2k o + K Δk + Δk 2m 2m

2

±

VK

2

+

ℏ2 KΔk 2m

2

4 35 which, at the boundary plane itself, defined by Δk = 0, reduces to E ± = ko +

ℏ2 k 2o ± VK 2m

4 36

E− is lower than the free-electron value by |VK|, and E+ is higher than the free-electron value by |VK|, thereby opening up a band gap of magnitude 2|VK|, for every K, as shown pictorially for a onedimensional crystal in Figure 4.3 (in 1d, ko = K/2).

4.6 Choices of Basis Sets and Band Structure with Applicative Examples

Figure 4.3 The electronic structure of a onedimensional crystal in the NFE approximation is based on a broken and distorted parabola with energy gaps of magnitudes 2|Vi| occurring at the BZ boundaries. Note that 2π/a periodicity should be superposed onto this picture. Source: After Elliott (1998), The Physics and Chemistry of Solids. © John Wiley & Sons, Inc. Reproduced with permission.

ε

2|V3|

2|V2|

42|V1| 2|V1|

k –3π/a –2π/a –π/a

Example 4.4

0

π/a

2π/a 3π/a

Show that a Brillouin zone boundary corresponds to a “Bragg plane.”

Solution On one side, a Brillouin zone boundary is a plane that is perpendicular to a reciprocal lattice vector K and contains the K/2 point (extremity). The k-equation of such a plane is given by kK = 12 K 2 because the k’s whose extremities belong to that plane are given by k – K/2 ⊥ K (k – K/2) kK = K2/2. K=0 On the other side, the Laue–Bragg reflection by a crystal lattice is defined, from an incident k wave, by the k waves elastically scattered such that k − k = K, K being a reciprocal lattice vector. One thus have k = k – K together with k = k, hence ||k – K|| = ||k||, which is equivalent to ( ) (k – K)2 = k2 K2 – 2kK = 0 (k – K/2)K = 0 k K = K2/2, which is the equation of a Brillouin zone boundary. Q.E.D. It should not be much of a surprise to observe similarities in the behavior of electronic waves and X-ray waves in a crystal.

4.6.2 Fermi Surface, Brillouin Zone Boundaries, and Alkali Metals versus Copper In Figure 4.3, the eigenvalues are concentrated in intervals separated by forbidden regions. In three dimensions, this corresponds to discontinuities in the energy contours at the zone boundaries. Iso-energetic k–surfaces for k vectors that terminate well short of the first BZ boundary are spherical (radius k = (2me E)1/2/ℏ). When k increases, k may approach a border of the BZ, and the corresponding ε(k) may be curved down (for symmetry borders), due to the presence of the periodically located ions in the crystal, with which the (not entirely) free electrons having these k vectors would interact. For structures such that the Fermi wave vector does not touch the BZ boundary, but rather is of a length appreciably less than the distance to the first BZ boundary (i.e. when the Fermi wave vector lies well within the first BZ), the Fermi surface ought to be spherical. This is the case for the monovalent (one conduction electron) BCC alkali metals (Figure 4.4a). This should also happen for copper if copper could be assumed to be monovalent, as its atomic electronic configuration 3d104s1 suggests. If the ratio for alkaline metals kF monovalent/kBCCBZ is 0.88, appreciably less than 1, the corresponding ratio for copper kF monovalent/kFCCBZ is 0.9 (see Example 4.5), also appreciably smaller than 1, and not much greater than 0.88. Yet, for copper, the

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4 The Electronic Level I

(a)

(b) W

P D Γ

N

L

F

Q

H G

Γ K

W

U

S

Z

X

Figure 4.4 For the monovalent BCC alkali metals (a) with a low electron density, the Fermi wave vector (the radius of the Fermi sphere) lies well below the first BZ boundary. The Fermi surface is unperturbed. For monovalent FCC copper (b), the shape of the first BZ boundary and the electron density forces the Fermi wave vector to terminate very near the L-point. The Fermi sphere necks outward toward that face of the BZ boundary and electron energy is lowered.

Fermi surface clearly necks toward the L-point on the first BZ boundary (Figure 4.4b). This was experimentally found by Brian Pippard in 1954–1957 thanks to the de Haas–van Halphen effect in its extreme animal limit, coupled to Lars Onsager’s theoretical developments to interpret the oscillations of the magnetoresistance of a metal in a magnetic field based on its Fermi surface (see Ziman 1962). We also recalled in Section 4.5 that the hexagonal face for FCC, which provides the shortest BZ boundary kL, is special in the sense that there is no mirror plane normal to the [111] direction in the FCC lattice. But the necking of the Fermi surface of copper also points out to the role of the d electrons in copper. Atomic 3d and 4s levels are very close in energy, and the rigid atomic level model does not work for copper whose atomic electronic configuration is 3d104s1 and not 3d94s2 (Co, Ni, and Zn are, as naively expected, 3d74s2, 3d84s2, and 3d104s2). Note the conceptual similarity between the rigid atomic level model for atoms and the rigid band model for metals and alloys (vide infra, spec. Section 4.6.6). Example 4.5 Find the free-electron model value kF for alkaline metals and for copper considered as a monovalent metal (z = 1). Get the shortest first BZ boundary k in both cases and compare the obtained ratios. Solution For cubic structures, the general formula to get the requested kF is kF(FE,na,z) = (3π 2 na z)1/3/a where na is the number of atoms per cubic cell of size a and z is the number of valence electrons provided by each atom. For monovalent metals, z = 1. The shortest boundary BZ k’s are obtained from inspection of first Brillouin zones, see Figure 4.1b and c and Tables 4.3–4.5. For BCC metals, na = 2 and kBZBmin = kN = π√2/a, hence kF(2,1)/kN = (6π 2)1/3/π√2 ~ 0.87731 ~ 0.88. For FCC metals, na = 4 and kBZBmin = kL = π√3/a, hence kF(4,1)/kL = (12π 2)1/3/π√3 ~ 0.902505 ~ 0.9. The two ratios are very close. But the choice of z = 1 for copper can be discussed, because of its d electrons. If one were to take all of them into account, the new z would be 11, which obviously sounds too large and would not fit Figure 4.4b either. See the discussion infra in the text between “e/a” and VEC.

4.6 Choices of Basis Sets and Band Structure with Applicative Examples

4.6.3 Understanding Metallic Phase Stability in Alloys An interesting area still under debate in the field of metallurgy is the consequences of Fermi surface topology on the phase equilibria in alloy systems between elements having different crystal structures. Elucidation of the connection between these two, seemingly unrelated, features started with the work of William Hume-Rothery (1899–1968, see his short biography in Section 3.2). HumeRothery observed (1926) that the critical valence electron to atom ratios, e/a, corresponding to structural phase transitions such as fcc to bcc, and bcc to hcp, in some copper–zinc alloys (brass), were 1.36 and 1.48 (Hume-Rothery et al. 1934). Harvey Jones later (1936) showed that the Fermi surface touches the BZ edge in those phases at precisely these same values, which strongly points to a connection between BZ touching and phase stability (Mott and Jones 1936). When the Fermi surface touches the Brillouin zone, for a critical e/a, the alloy changes the crystalline structure, in order to have a Brillouin zone and a Fermi surface so that the electronic energy is decreased. In reality, the change is not complete, and one has an intermediate two-phase, α–β or β–γ, region in the actual Cu–Zn phase diagram; see Figure 4.5. The naive Jones’ Fermi surface–Brillouin zone model gives a destabilizing e/a ratio of π√3/4 (~1.36) for fcc and π√2/3 (~1.48) for bcc; see Example 4.6. There is then a complex cubic γ–Cu–Zn (Cu5Zn8) structure with 52 atoms per unit cell for e/a in the

Atomic percent zinc 10

20

30

40

50

60

70

80

90

100

1100 1000

Liquid

α + liquid β + liquid

902°

900

834°

Temperature (°C)

800

γ + liquid 700°

700

α + β

α

600

β + γ

β

500

γ + δ + liq δ δ 600°

γ 560°

𝜖 + liq

468°

454°

𝜖

400 300 200

α + β′

0

10

20

30

40

β′ 50

β′ + γ

γ +𝜖

60

70

423°

𝜖+η

80

90

100

Weight percent zinc

Figure 4.5 The Cu–Zn phase diagram. Pure copper is FCC and so is the α phase. The β structure is BCC, with chemical disorder. β is chemically ordered BCC. Pure zinc is HCP and so are the η and ε phases (ε has a much lower lattice parameter c/a ratio than η). The γ structure is a complicated one. Except for the η and ε phases, these structures are called brasses, even in the two-phased regions (duplex brasses). See Chapter 8 for a general understanding of phase diagrams. The concentration x discussed in the text is the atomic percent zinc concentration.

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4 The Electronic Level I

1.58–1.66 range. Cu–Zn becomes hcp at e/a in the 1.58–1.66 range and beyond, as pure zinc is hcp. As already said, this simple model, which only considers s (or s–p) electrons cannot but be much too simple. Remarkably, however, for copper and silver alloys with s–p electron per atom ratios away from e/a = 1, the presence of the filled d band does not alter fundamentally Jones’ arguments (Pettifor 2000). Hume-Rothery’s original “e/a” concept only considers s (or s–p) electrons. It has since been proposed to include the d electrons when present and to call the new number the valence electron concentration, or VEC (concentration per atom, not per unit volume as N/V introduced at the beginning of Section 4.6.1). Copper’s VEC equals 11, yet an unreasonable value as said in Example 4.5. For 3d metals with s and p electrons, the “3d” electrons are rather core-like electrons from a spatial point of view. This is not the case for 4d and 5d metals, and VEC is rather popular in the high-entropy alloy (HEA) community; see Sections 10.2.3.2 and 10.3.4. Also see Mizutani (2010) for discussions. Example 4.6 Within the nearly free-electron model, it show that the Cu1 – xZnx alloy, which is FCC for very small zinc concentration, gets destabilized toward BCC for x ~ 36%, and BCC is destabilized for x ~ 48%. Solution Although we are not in the strict free-electron model, we must consider the free-electron Fermi kF equal to kBZBmin because it is the k value beyond which energy states above ℏ2k2/2m + |VK| will be occupied if the alloy does not change of structure (see Figure 4.3). The filling in valence electrons corresponding to this kF implies a critical average number of valence electrons per atom zc (e/a), which is such that zcnaN = (L3/3π 2)k 3F (N is the number of cubes of parameter a within L3); hence zc = k 3Fa3/(3π 2na). Writing kF = kBZBmin, one gets the critical zc beyond which the alloy will be forced to occupy energy states beyond ℏ2k2/2m + |VK| if it does change of structure (i.e. if no phase transition occurs). The crystalline structure should move to a new structure, less dense or having a larger first BZ so that the ε(k) can go down to ℏ2k2/2m or even to ~ℏ2k2/2m-|VK|. For the FCC structure kBZBmin = π√3/a and na = 4, which gives zc1 = (π√3)/4 ~ 1.36035. For the Cu–Zn system, the structure changes to the less dense BCC structure and the 3π 2zc1(naFCC/a3FCC )k3-value is lowered to 3π 2zc1(naBCC/a3BCC ). In the hard sphere model, the ratio (naBCC/a3BCC)/(naFCC/a3FCC) is (3√3)/(4√2) ~ 0.91856 (actually the ratio of their sphere packing densities, dBCC/dFCC, see Table 3.1). From the point of view of the first BZ boundaries and their shortest kBZBmin, it turns out that for hard sphere structures kN = kL. The ε(k) are thus indeed lowered to ℏ2k2/2m or even to ~ℏ2k2/2m − |VKN|, and the alloy will be content with this BCC structure until it reaches a new zc2 that is (π√2)/3 ~ 1.48096 (kN = (π√2)/aBCC and naBCC = 2. Note that zc2/ zc1 = dFCC/dBCC). Taking Cu with s1 and Zn with s2, one gets for a Cu1 – xZnx, alloy a z = 1(1 – x) + 2x = x + 1, hence xc1 ~ 0.36 and xc2 ~ 0.48. Note that for a bronze alloy, Cu1 – xSnx, Sn being with 5s2 5p2 brings four electrons, so that z = 1 (1 – x) + 4x = 3x + 1. The corresponding xc1 and xc2 will be 0.12 and 0.16, respectively. Of course, this simple calculation neglects again the role of the d electrons. Thus, as a polyvalent metal is dissolved in a monovalent metal, the electron density increases, as does the Fermi energy and Fermi wave vector. Eventually, the Fermi sphere touches the BZ boundary, and the crystal structure becomes unstable with respect to alternative structures (Raynor 1947; Pettifor 2000). Subsequent work has been carried out confirming that the structures of Hume-Rothery’s alloys (alloys composed of the noble metals with elements to the right on the periodic table) do

4.6 Choices of Basis Sets and Band Structure with Applicative Examples

indeed depend only on their electron per atom ratio (Stroud and Ashcroft 1971; Pettifor and Ward 1984; Pettifor 2000). Unfortunately, the importance of the e/a ratio on phase equilibria is much less clear when it does not correspond precisely to BZ touching. For low-dimensional (one-dimensional, two-dimensional) metals, the topology of the Fermi surface is especially significant, as this can lead to a charge density wave state. When a portion of the Fermi surface can be translated by a vector q and superimposed on another portion of the Fermi surface, the Fermi surface is nested. Geometrical instabilities can result when large sections of the Fermi surface are nested. For one-dimensional metals, if q = b/n, then a distortion, known as the Peierls distortion, leading to a unit cell n times as large as the original one is predicted (Burdett 1996). If the nesting is complete, the system will exhibit a metal–insulator transition after the modulation destroys the entire Fermi surface by opening a band gap (Canadell 1998). When the nesting is less than complete, the driving force for distortion is reduced, the gap opening being only partial. Charge density waves and the Peierls distortion are discussed in more detail in Section 7.5.

4.6.4 The Localized Orbital Basis Set Method Plane wave expansion in a periodic potential may require a very large number of plane waves to achieve convergence. In general, finite plane wave expansions are inadequate for describing the strong oscillations in the wave functions near the nuclei. To alleviate the problem, other approaches, such as the augmented plane wave (APW) method, which treats the atomic core and interstitial regions differently, and the pseudopotential method, which neglects the core electrons completely in the calculation scheme, have been introduced. Finally, because the conduction electron kinetic energy is the dominant attribute of metallic systems, metals have been the chief application of the NFE approach to band theory. Some believe it not as appropriate for systems where covalent energy is the dominant attribute, transition metal and rare-earth systems with tightly bound valence electrons (valence d and 4f orbitals do not extend as far from the nucleus as valence s and p orbitals), or for describing inner-shell core electrons in systems. For solids with more localized electrons, the LCAO approach is perhaps more suitable. Here, the starting point is the isolated atoms (for which it is assumed that the electron wave functions are already known). In this respect, the approach is the extreme opposite of the free-electron picture. A periodic solid is constructed by bringing together a large number of isolated atoms in a manner entirely analogous to the way one builds molecules with the LCAO approximation to molecular orbital (LCAO–MO) theory. The basic assumption is that overlap between atomic orbitals is small enough that the extra potential experienced by an electron in a solid can be treated as a perturbation to the potential in an atom. The extended-(Bloch) wave function is treated as a superposition of localized orbitals, χ, centered at each atom: ψ kμ r = N − 1

eik R χ μ r − R

2

4 37

R

One defines a Bloch sum (or BO) for each atomic orbital in the chemical point group (the atomic motif, or basis, associated to each lattice point), and crystal orbitals (COs) are then formed by taking linear combinations of the Bloch sums. In actuality, χ is not an atomic orbital (technically speaking, those are solutions of the Hartree–Fock equations for the wave function for a single atomic electron in a SCF), but rather a Slater-type orbital (STO) or Gaussian-type orbital (GTO). The latter two were introduced for, respectively, accuracy and computational expediency. It is important to first optimize a localized orbital basis set by varying the exponents or contraction coefficients to minimize the energy, a procedure unnecessary with a plane wave basis set.

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4 The Electronic Level I

At kx = 0, λ = ∞

At kx = π/d, λ = 2d

Figure 4.6 A one-dimensional periodic chain of p atomic orbitals. Shown at the top is the sign combinations corresponding to the k-point Γ (k = 0), where λ = ∞. At the bottom are the sign combinations for the k-point X (k = π/d), where λ = 2d.

First, take the simplest possible case, a monatomic solid with a primitive Bravais lattice containing one atomic orbital per lattice point. The COs are then equivalent to simple Bloch sums. The wavelength, λ, of a Bloch sum is given by the following relation: k =

2π λ

4 38

It can be seen how the phase of a Bloch sum changes in a periodic lattice by considering a simple one-dimensional lattice of (σ-bonded) p atomic orbitals, with a repeat distance d. Figure 4.6 shows such a chain and the sign combinations of the atomic orbitals for three special values of k (=kx in the one-dimensional case). Euler’s relation, e±iθ = cos θ ± i sin θ, can be used to evaluate the term exp(ikxnd). At Γ, where kx = 0 and exp(ikxnd) = (1)n = 1, so there is no phase change from one unit cell (atomic orbital) to the next and, from Eq. (4.37), λ = ∞. At X, kx = π/d and exp(ikxnd) = (−1)n, so the phase alternates as n, the signs of the atomic orbitals, and λ = 2π/(π/d) = 2d. For crystals with more than one atom per lattice point, combinations of Bloch sums have to be considered. In general, the LCAO approach requires that the result be the same number of MOs (COs in solids) as the number of atomic orbitals (Bloch sums in solids) with which was started. Thus, expressing the electron wave functions in a crystalline solid as linear combinations of atomic orbitals (Bloch sums) is really the same approach used in the 1930s by Hund, Mulliken, Hückel, and others to construct MOs for discrete molecules (the LCAO–MO theory). The orbitals used to construct Bloch sums are usually STOs or GTOs, since these types of orbitals well describe the electron density in molecules and solids, having the correct cusp behavior near the nucleus and the correct fall-off behavior far from the nucleus; this is called a minimal basis. The use of a minimal basis together with a semiempirical two-center fixing of the Hamiltonian matrix elements is known as the tight-binding method. Because of its wide applicability, Chapter 5 is dedicated to the application of the tight-binding method to crystalline solids. It should be noted that it is also possible to write a tight-binding Hamiltonian for an amorphous substance with random atomic positions (off-diagonal configurational disorder), but the wave functions in such a system cannot be written as Bloch sums. To simplify the calculations, geometrical mean models have been introduced that reduce the off-diagonal disorder in the Hamiltonian into diagonal disorder (diagonal Hamiltonian matrix elements) (Kakehashi et al. 1993). Electronic structure calculations on amorphous solids are not discussed in this book.

4.6.5

Understanding Band Structure Diagram with Rhenium Trioxide

The Bloch sums are linear combinations symmetry adapted, not to a molecule with some point group symmetry, but to the primitive unit cell. Nevertheless, a qualitative MO-like treatment of

4.6 Choices of Basis Sets and Band Structure with Applicative Examples

M

MO6

6O

t1u(σ*, π* ) a1g(σ* )

(n + 1)p (n + 1)s

eg(σ* ) t2g(π* )

nd 6px + 6py

t1g(π) t2u(π) t1u(π) t2g(π)

t1u(π)

6(s – pz)

eg(σ)

a1g(σ)

Figure 4.7 A MO energy-level diagram for octahedral transition metal complexes with metal–ligand σ and π bonding.

the smallest chemical point group can often easily allow for obtaining the relative placement of the energy bands at one special k-point at the center of the BZ (but, contrarily, see the more complex cases for TiO2 in Chapter 15 and GaN in Chapter 16). To reinforce this idea, let us consider the energy-level diagram for a hypothetical octahedral ReO6 molecule, shown in Figure 4.7, where the Re d–O p π-interactions and the Re d–O p σ-interactions are shown. By placing the electrons in the diagram in accordance with Hund’s rule and the aufbau (Aufbau in German) principle, there will be 36 electrons contributed by the six oxygens and one from the Re6+ (5d1) cation. Note that since the Re–O p π mixing is weaker than the Re–O p σ mixing, the bonding combinations resulting from the π interactions will be higher in energy than the σ interactions. Thus, the HOMOs have π symmetry. Furthermore, the π ∗-antibonding combinations are lower in energy than the σ ∗-antibonding combinations on the same grounds, so that the LUMO also has π symmetry. Shortening the Re–O bond distance will lower the bonding orbital energy and raise the antibonding orbital energy. The bonding and nonbonding MOs for our hypothetical ReO6 are all filled with electrons, and one of the antibonding orbitals is half filled (singularly occupied). This is the condition, in a solid, that would be expected to lead to metallic behavior. The particular splitting pattern of the d orbitals in Figure 4.7 is characteristic of cubic (octahedral type) crystal fields. In the tetrahedral type, the eg and t2g ordering would be reversed. In other symmetries, the d splitting is as shown in Figure 4.8. There is much more information contained in a band structure diagram than in a MO energylevel diagram, however. In the former, the band dispersion, or variation in electron energy, is plotted as one moves between high-symmetry k-points. The real solid most closely related to our hypothetical ReO6 molecule is the perovskite ReO3, which contains vertex-sharing ReO6 octahedra, linked through Re d–O p π bonds. The band structure and DOS for ReO3, as calculated by Mattheiss (1969), is shown in Figure 4.9.

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4 The Electronic Level I

2

2

2

2

4 10

Figure 4.8

d3z2–r2 dyz

2

6

Free atom

dx2–y2

2

4

Cubic (octahedral type)

dxz

2

2

Tetragonal

Orthorhombic

dxy

The crystal-field splitting of the d orbitals under cubic, tetragonal, and orthorhombic symmetries.

x1

M1

R25ʹ x2 x5

Γ12 Energy (Rydbergs)

198

Γ25ʹ

x3 R15ʹ x5ʹ x3ʹ x5ʹ x4ʹ x5 R1

Γ15 Γ25 Γ15

R25ʹ x1 R12 Γ Δ X

Z M

Σ

Γ

Δ

R

S

X

M3 M5

M4 M4ʹ M5ʹ M5 M3 M1 M2 R T M

0

10

20

30

40

50

60

States of one spin/Rydberg unit cell

Figure 4.9 The LCAO (tight-binding) band structure for ReO3. The dashed line represents the Fermi energy. To the far right is the density of states (DOS) curve for states of one spin. The occupied states (up to the Fermi level) are shaded gray. Note that the valence band is completely filled, while the conduction band is partially filled. Hence, ReO3 should be metallic.

Some important points to remember, when looking at any band structure diagram, are as follows: 1) A band structure diagram is a map of the variation in the energy, or dispersion, of the extended wave functions (called bands) for specific k-points within the first BZ, which is the unit cell of k-space.

4.6 Choices of Basis Sets and Band Structure with Applicative Examples

2) The total number of bands shown in a band structure diagram is equal to the number of atomic orbitals contributed by the chemical point group, which constitutes a lattice point. As the full crystal structure is generated by the repetition of the lattice point in space, it is also referred to as the basis of the structure. 3) There are 2NM electrons in the BZ (where N is the number of unit cells in the crystal and M is the number of atomic orbitals per cell and two accounts for the usual spin up/down possibility). 4) Band structure diagrams show k values of only one sign, positive or negative. Now look at the band structure of ReO3. On the left-hand side of Figure 4.7 is the band dispersion diagram, and on the right-hand side is shown the DOS curves, which give the DOS per ΔE interval. Accounting for the way the atoms are shared between neighboring unit cells, it can be seen that, in ReO3, there is one rhenium atom and three oxygen atoms contained within a single-unit cell. In fact, the Bravais lattice for perovskite is simple cubic, and this same combination of atoms is associated with each lattice point (its atomic motif/basis; ReO3 is space group #221, Pm3m, Pearson symbol cP4, Strukturbericht symbol D09; see Section 3.1.3 sqq.). In his calculations, Mattheiss considered combinations made up of Bloch sums formed from the five rhenium 5d orbitals and the 2p orbitals of the three oxygen atoms. Because there are 14 (5 + 3 3) atomic orbitals in his basis set of atomic orbitals, there are 14 different Bloch sums, which will combine to give 14 tight-binding basis functions or crystalline orbitals (COs). It should thus be expected to find 14 bands in the band structure diagram, which is the case. Starting from the bottom up, the lowest group of nine bands in the diagram, with a bandwidth spanning from about −0.65 up to −0.2 Ry, correspond to the oxygen 2p states. It is customary to refer to the entire set as the p band, even though they are really COs composed of linear combinations of metal d and oxygen p Bloch sums. The reason for this is that these COs have mostly p characters since they are closer in energy to the oxygen p Bloch sums. This set constitutes the top of the valence band in ReO3 (the oxygen 2s states, which were neglected in the calculations, lie lower in energy). The next highest in energy, spanning from about −0.15 to 0.15 Ry, correspond to the antibonding rhenium t ∗2g orbitals (xy, yz, zx) or, more precisely, the t ∗2g manifold of bands under the periodic potential. Finally, the two highest energy bands are the antibonding rhenium e∗g manifold (the 3z2 − r2, x2 − y2 bands). It is customary to refer to the two manifolds together as the d band, and it makes up the bottom of the conduction band (the rhenium 4s states lie higher in energy). The band filling is carried out in the same manner as in our earlier MO treatment using rule 3 above. Thus, the valence band is completely full, and the lowest lying t2g band is half filled. Because the Fermi level (HOMO) lies in a partially filled conduction band, ReO3 is predicted to be metallic, which, for an oxide, is rather exceptional. Rhenium trioxide has indeed a metallic luster and an electrical conductivity close to that of silver and copper, which increases with decreasing temperature.

4.6.6 Probing DOS Band Structure in Metallic Alloys Often times, the basic band structure of a compound does not change much on alloying. This sort of fixed band structure behavior is referred to as the rigid band model. It means that the band structure shape does not depend on how many electrons are filling it. It is, of course, only a first approximation. It is roughly applicable to many types of phases, including the perovskites and the tungsten bronzes. Thanks to it, the electronic properties (e.g. the Fermi level) can be varied in a controlled manner by alloying. A nice example is the Pd–Ag system. Palladium and silver are both FCC. Yet, the (isolated) palladium atom is 4d10. Since Mott 1935, it is assumed that in the bulk palladium metal, the 4d band overlaps the 5s band so that the 4d band is not full. One would have 4d9.4–9.45. The d band DOS is

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4 The Electronic Level I

Figure 4.10 Schematic picture of DOS n(E) for palladium, with respect to its Fermi level.

1.2 1 DOS n(E) (eV–1)

200

0.8 0.6 0.4 0.2

–5

0 E – EF (eV)

5

narrow in energy range and mainly has high n(E) values except of course at its top extremities, beyond peaks known as van Hove singularities (after the Belgian theoretician Léon van Hove 1924–1990); see Figure 4.10. Alloying palladium metal with some silver atoms, isolated Ag being 4d105s1, will first contribute to fill the almost full d band, thus first decreasing significantly the n(EF) value, which then stays relatively constant on the rather flat s band. This transition is expected at about 60% (55–60%) of added silver, which is about what is observed from electronic specific heat measurements. Thus, even if a photoemission study showed large differences of DOS n(E) profiles between Pd-20%Ag and Pd85%Ag, the Pd–Ag alloy is a case where the rigid band model, at least a rigid n(E) shape, independent of how much the band is filled, works rather well (see Rossiter 1987). Alloying palladium with rhodium (Rh) or with ruthenium (Ru) will entail more complex effects because Rh and Ru contain fewer electrons than Pd (Rh is 4d85s1 and Ru 4d75s1) so that their presence will let n(EF) scan the internal profile n(E) of the d band, which is rather complex. EF, being originally just right of a van Hove peak in the case of palladium, n(EF) will first raise a little then significantly decrease and oscillate following the oscillations of n(E). This remains true for measures made at very low temperatures, as probed by electronic specific heat measurements; see Section 4.6.1. Magnetic susceptibility measurements done in temperature imply a broadening of the electronic occupation around EF because of the Fermi–Dirac function, and these measurements depend very much on temperature. Closer analyses should consider in details the electronic shielding of the added impurities, in the way Nevil Mott and Jacques Friedel pioneered it, with Friedel oscillations around an impurity and the concept of virtual bound states, followed by the Anderson impurity model. This would led beyond the scope of this book, far among the beauties of impurities (Georges 2016).

4.7

Breakdown of the Independent-Electron Approximation

Both the LCAO and NFE methods are complementary approaches to one-electron band theory, in which electrons are allowed to move independently of one another, through an averaged potential generated by all the other electrons. The true Hamiltonian is a function of the position of all the

4.7 Breakdown of the Independent-Electron Approximation

electrons in the solid and contains terms for all the interactions between these electrons, that is, all of the electron–electron Coulombic repulsions. Electronic motion is correlated; the electrons tend to stay away from one another because of Coulombic repulsion. Note that the Pauli antisymmetry of the wave function (Fermi correlation) is accounted for at the single-determinant level in the Hartree–Fock theory (Eq. 4.8), and that correlation owing to the exchange hole (involving parallel spin electrons at different sites) is included in the form of the aforementioned average potential. However, the deviation from the exact energy caused by attempting to represent the wave function by a single determinant, called static correlation, as well as the classic Coulomb repulsion between electrons, dynamic correlation, are neglected. It is possible to account for static correlation in order to improve on the results from Hartree–Fock calculations. One of the post Hartree–Fock techniques involves the introduction of the configuration interaction (CI), which represents many-electron wave functions with linear combinations of several Slater determinants. The first determinant is the Hartree–Fock ground state; the second one is the first excited state, and so on. One adds to the HF wave function terms that represent promotion of electrons from occupied to virtual molecular orbitals. Another method, popular among chemists, is the Møller–Plesset (MP) perturbation theory (Møller and Plesset 1934, Christian Møller 1904–1980, Danish chemist and physicist, and Milton Spinoza Plesset 1908–1991, American physicist). In this method, the difference between the Fock operator and the true Hamiltonian, the correlation potential, is treated as a perturbation. A number is used to indicate the order of the perturbation theory, so MP2 is the second-order theory and so on. The perturbation expansion philosophy is quite different from the variational philosophy, which is at the basis of the Hartree–Fock approach, and the MP approach remained dormant for three or four decades, until the development of many-body perturbation theory to describe interactions between nuclear particles (see Cremer 2011). One then encountered up to MP5: fifth-order perturbation MP. Either intrinsically or because of analytical mistakes in perturbative expansions, MPn with n > 2 went out of fashion and chemists now prefer to concentrate on well mastered versions of MP2. MP2 is usually considered as of the same level as RPA that is popular among materials scientists. MP works by perturbation and CI by addition. The coupled-cluster (CC) method is related to both approaches. It involves excitations of single electrons (S, CCS), of pairs of electrons (double-D, CCSD), of triplets of electrons (CCSDT), etc. It was first developed in nuclear physics in the 1950s (Fritz Coester and Hermann Kümmel) and then adapted to atom and molecular chemistry (Jiří Čížek, Josef Paldus). CCSDT is more precise than MP2. The gold standard, however, remains the expansion to the full set of Slater determinants. This is called full configuration interaction (FCI), in which all electrons are distributed among all the orbitals and thus exactly solves the electronic Schrödinger equation. Unfortunately, it is the most difficult technique to implement on systems containing anything but a small number of atoms since the number of Slater determinants increases very rapidly with the number of electrons and orbitals (e.g. in FCI the number of determinants required grows factorially with the number of electrons and orbitals; by 2019, recent benchmark calculations on molecules with two, three, or four atoms involved billions, even trillions, of configuration state functions). FCI is, per construction, systematically improvable but rapidly prohibitively expensive. And yet, let us recall, it is nonrelativistic (the Schrödinger equation) and restricted to the Born– Oppenheimer approximation. Full CI leads to more accurate calculated energies. However, what about the dynamic correlation? The most detrimental consequence of neglecting dynamic correlation effects in solids is the possibility that the wrong type of electric transport behavior (i.e. metallic, semiconducting, or insulating) may be predicted. For example, one-electron band theory predicts metallic behavior whenever the bands are only half filled, regardless of the interatomic separations. However, this is incorrect and even counterintuitive; isolated atoms are electrically insulating. The excitation

201

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4 The Electronic Level I

energy, or electron transfer energy, for electronic conduction in a solid is essentially equal to the Coulomb repulsion between electrons of antiparallel spin at the same bonding site. In fact, the ratio of this on-site Coulomb repulsion to the one-electron bandwidth determines whether an electron is localized or itinerant. Unfortunately, as has been seen, the intrastate Coulomb repulsion is not accounted for in the Hartree–Fock approach. Consider a d-electron system, such as a transition metal compound. The valence d atomic orbitals do not range far from the nucleus, so COs composed of Bloch sums of d orbitals and, say, O 2p orbitals, tend to be narrow. As the interatomic distance increases, the bandwidth of the CO decreases because of poorer overlap between the d and p Bloch sums. In general, when the interatomic distance is greater than a critical value, the bandwidth is so small that the electron transfer energy becomes prohibitively large. Thus, the condition for metallic behavior is not met; insulating behavior is observed. The archetypal examples are the 3d transition metal monoxides NiO, CoO, FeO, MnO, VO, and TiO. All of these oxides possess the rock-salt structure (which makes both cation–cation and cation–anion–cation overlap important). One-electron band theory correctly predicts the metallic behavior observed in TiO, which is expected of a partially filled 3d band. However, except for VO, all the other monoxides are nonconducting. This can be explained by the presence of localized electrons. As the radial extent of the d atomic orbitals increases as one moves to the left in a period, the extent of electron localization decreases for the lighter transition elements in a row (Harrison 1989). By contrast, the heavier oxides in this group represent cases where the bandwidth is so narrow, that dynamic correlation effects dominate. A material that is insulating because of this type of electron localization is referred to as a Mott insulator or, sometimes, a Heitler–London type insulator, since it will be recalled that in the Heitler–London theory of chemical bonding, valence electrons are in orbitals localized on atoms, as opposed to delocalized MOs.

4.8 Density Functional Theory: The Successor to the Hartree–Fock Approach in Materials Science Today, electronic structure calculations on solids are performed with the DFT, based on the work of Hohenberg, Kohn, and Sham (Hohenberg and Kohn 1964; Kohn and Sham 1965). The main objective of the DFT is to replace the N-electron wave function ψ, which is dependent on 3N variables (three spatial variables for each of the N electrons), with the electronic density, which is only a function of three variables and thus sounds simpler to work with conceptually and practically. A precursor DFT model was developed in 1927 by Thomas and Fermi, but it was far from being satisfactory; see Section 14.1. Pierre Hohenberg (1934–2017) and Walter Kohn (1923–2016) first demonstrated the existence of a one-to-one mapping between the ground state electron density n(r) (also often noted ρ(r)) and ground state electron wave function in the absence of a magnetic field, which was later generalized. They further and mainly proved an exact theorem, viz. that the ground state density minimizes the total energy E, as E[n(r)], which is a function (called a functional) of the function n(r). DFT does not require the a priori knowledge of any electron wave function(s). The Hohenberg and Kohn (HK) theorem, however, is just an existence theorem and provides no practical way to get n(r) and E. The most common implementation of DFT is the Kohn–Sham (KS) method, after Kohn and Lu Jeu Sham (b. 1938) in which the properties of an N-electron system is mapped onto the properties of a system containing N noninteracting fictitious electrons, described by fictitious electron wave functions, under an effective, potential.

4.8 Density Functional Theory: The Successor to the Hartree–Fock Approach in Materials Science

In KS DFT, as in the Hartree–Fock approach, an effective independent-particle Hamiltonian is arrived at, and the electronic states are solved for self consistently. The many-electron wave function is still written as a Slater determinant. In KS DFT, however, the wave functions, used to construct the Slater determinant, are not the same effective one-electron wave functions of the Hartree–Fock approximation. These KS wave functions, usually called Kohn–Sham “orbitals,” have no physical interpretation a priori (so that the most unbiased and mathematically convenient basis set will be the set of plane waves). The KS wave functions are merely used to construct the total electron charge density n(r), which for a Slater determinant wave function is simply the sum of the square of the occupied “orbitals.” The difference between the Hartree–Fock and DFT approaches lies in the dependence of the Hamiltonian in DFT on the exchange–correlation potential, VXC(r), a functional derivative of the exchange–correlation energy, EXC: V XC r =

δEXC n r δn r

4 39

In KS DFT, the Schrödinger equation is expressed as a set of equations of the following form for every ψ μ(r) (compare with Eq. 4.5): 1 − ∇2 − 2

Zn r − Rn

n

+

n r dr + V XC r r −r

ψ μ r = εμ ψ μ r

4 40

The first three terms within brackets on the left side of Eq. (4.40) are the same as in the Hartree– Fock equation (Eq. 4.5). The kinetic energy functional of a system of noninteracting electrons is known exactly (but it is not the real kinetic energy of the interacting electron system). The fourth term in brackets is the exchange–correlation potential lumping together the many-body effects. This term contains all the contributions not accounted for by the first three terms. These include exchange (the exchange hole – electrons with parallel spin avoid each other) and all the unknown correlation effects, the latter of which are neglected in the Hartree–Fock theory. In essence, the exchange term in the Hartree–Fock expression has been replaced with the exchange–correlation potential in the DFT formalism. The main result is that a form of the exchange–correlation potential exists that depends only on the electron density, which yields the exact ground state energy and density. However, the exact form of this potential is unknown and herein lies the difficulty with DFT. More will be said about it in Section 4.9. The ground state electron density is given by the square of the occupied Kohn–Sham wave functions: 2N

nr = μ=1

ψμ r

2

4 41

The electronic ground state energy is given by (compare with Eq. 4.8) N

εμ −

E= μ=1

1 2

nr nr drdr + E XC n − r −r

V XC n r dr

4 42

Per Hohenberg–Kohn–Sham construction, these n(r) and E are those of the real system. By far, most DFT calculations on periodic solids use a plane wave basis set to express the Kohn– Sham orbitals, ψ μ(r) in Eq. (4.41). The advantages of using plane waves as the basis set are that they are orthonormal, independent of atomic positions, and do not suffer from superposition errors. A simple cutoff criterion can be applied for the size of the plane wave basis to be considered in order

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to achieve some desired numerical accuracy over E: an Ecut value will limit the summation over the Ks for the plane waves in Ψk(r) of Eq. (4.33) via k + K < 2me E cut ℏ2

4 43

Ecut is increased until desired numerical accuracy is achieved. The total energy expressions and the Hamiltonian are easy to implement and can be efficiently evaluated using the fast Fourier transform (FFT) technique. Yet, much too many plane waves are necessary to describe the fluctuation of electron charge density near the atomic cores, and this requires implementations of appropriate methods such as the PPW, OPW, or LAPW methods. Equation (4.40) is known as the Kohn–Sham equation, and the effective one-electron Hamiltonian associated with it is the Kohn–Sham Hamiltonian. Kohn and Sham were the first to evaluate Eq. (4.40) approximately. For the exchange–correlation potential, they started with the exchange– correlation energy of a homogeneous electron gas, evaluated from the electron density at the point r under consideration. Basically, the Hamiltonian then depends only on the local value of the density, even in the presence of strong inhomogeneity. This is called the local-density approximation (LDA), and it is the simplest approximation since the functional only depends on the exchange– correlation energy density εXC at the coordinate where the functional is evaluated: ELDA XC n =

n r εXC n dr

4 44

The process can be outlined as follows: one begins with an initial wave function (typically constructed from atomic orbitals for molecules or plane waves for periodic solids) to obtain an electron density that is used to find the effective single-particle potential of a noninteracting system of electrons. For any ground state density of an interacting system, there exists a noninteracting system with the same ground state density. Then, the Kohn–Sham equation is solved again to update the Kohn–Sham orbitals yielding a new electron density. The iterative procedure used to find the Kohn–Sham functions is a SCF calculation. It minimizes the total energy and finds the Kohn–Sham orbitals, allowing determination of the electron density (the fundamental physical observable), by continuing the Kohn– Sham calculations until consecutive computations agree to within an expected tolerance or precision. How well the DFT results reproduce ground state physical properties (e.g. lattice parameters, bulk moduli) that agree with experimental values will depend on the pseudopotential, optimization constraints, and exchange–correlation functionals used in the DFT calculations. The LDA has been adopted in most DFT electronic structure calculations on solids since the 1970s but was not considered accurate enough for quantum chemistry until the 1990s when refinements were made about the quality and use of pseudopotentials thanks to the projector augmented wave (PAW) method for instance. The local spin density approximation (LSDA) is a generalization of the LDA to account for electron spin: n ,n ELSDA XC

=

n r εXC n , n dr

4 45

The utility of this approximation will become evident in Chapter 8 where magnetotransport properties are discussed. LSDA functionals are more complicated than LDA functionals and usually imply gradients corrections, ∇n and ∇n . As usual, LSDA functionals are useful in solid-state physics but have been largely replaced by more complex functionals for molecular calculations. More details about the different types of functionals will be discussed in the next section. We will only quote two books: Lewars (2016) for a chemist point of view and Martin (2004) for a solid-state physicist point of view.

4.9 The Continuous Quest for Better DFT XC Functionals

We end this Section 4.8 with three more advanced remarks about DFT and its extensions. Just a few months after the publication of Hohenberg and Kohn’s theorem about the representation of the inhomogeneous electron gas in its ground state, (Nathaniel) David Mermin (b. 1935) proposed an extension of the theorem to the finite temperature case. Now, in pratice, as long as the required nonzero temperature remains much less than the “temperature” of the Fermi level of the system, which is of the order of 105 K for typical metals (copper, iron, aluminum), this extension is not necessary. For temperatures less than, say, 10 000 K, the electrons can safely be considered in their ground state (for instance, for copper, iron and aluminum, EF/kB ~ 0.82, 1.3 and 1.36 105 K, respectively; for the sake of comparison, their melting temperatures are 1358, 1811, and 936 K, respectively). At higher temperatures, the use of the Born–Oppenheimer approximation would certainly become questionable, except for systems maintained dense enough so as to remain solids at these temperatures, corresponding to very extreme conditions. At temperatures close to the Fermi temperature, the DFT techniques to be used must be specially capable of describing the electronic excited states, which brings us to the following point. Since Hohenberg and Kohn’s theorem deals with the ground state of the electronic system, the description of the electronic excitations seems to be ruled out by principle. It can be argued, however, that once a correct Hamiltonian is known, and the K–S Hamiltonian is exact, per principle, all physical properties of a system, including its excitation states, can, in principle, be determined. They can be determined, in principle, but one does not how in practice. One only knows that the eigenvalues of the KS equations, which are mathematically defined as Lagrange coefficients, do not correctly provide the experimentally known excitation energies, at least within the LDA, as well as within the generalized gradient approximation (GGA). Much better band gap values have been obtained with a new raising star among the rich realm of the DFT exchange and correlation functionals, viz. the HSE06 hybrid functional (HSE, for Jochen Heyd, Gustavo Scuseria and Matthias Ernzerhof, in 2006; see Section 4.9). One optimistic point of view might be the following: If the Kohn–Sham eigenvalue εKS have no reason to be good, being mainly Lagrangian multipliers, yet they are not totally bad, depending on the material, and the Kohn–Sham eigenfunctions φKS calculated with these εKS give, per construction, the good fundamental n(r). These φKS cannot be too bad either, although this is not a real proof. Formally, they can always be taken as a first start. Indeed, the forms of the KS and the many-body-quasiparticle equations look similar. So, why not consider the φKS as a first approximation and try to improve from them, with well guessed improvements, of course? This is one of the several paths that are followed. Finally, in order to study the time-evolution of electronic excitations, a time-dependent DFT theorem needs to be established. It has been established in 1984 by the two German solid-state theoreticians Erich Runge (b. 1959) and Eberhard Groß (b. 1953), and is known as the Runge–Gross theorem. It states that for a given initial state, the time-dependent density is a unique functional of the external potential. The use of this theorem is very difficult of course, and we only mention it here for the sake of (relative) completeness.

4.9

The Continuous Quest for Better DFT XC Functionals

As previously said, even if Hohenberg and Kohn’s DFT formalism is exact, the corresponding exact XC functional is unfortunately not known, so there are today many approximated-guessed XC functionals, already more than 200 in 2017 (see Mardirossian and Head-Gordon 2017). In the quest for better XC functionals, some theoreticians prefer to progress in a pragmatic and almost experimental way, proposing to include terms that are recognized to be missing, in an

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empirical way with coefficients that are first fitted versus data bases obtained by John Pople and coworkers with larger and larger Gaussian bases, G1, G2, G3… This is the way exemplified by the German–Canadian physical chemist Axel Becke (b. 1953). Other theoreticians prefer to progress more slowly, with more prescriptions, guidelines and constraints to respect, possibly rationalizing intuitions proposed by the former school. This way is exemplified by the American physicist John Perdew (b. 1943). For instance, Becke proposed various improvements/modifications of the XC functional in 1992, with three empirical parameters (hence the B3 in the famous B3PBE91 and B3LYP hybrid functionals). Three years later, Perdew et al. proposed a simpler hybrid functional with only one parameter with some rational for a 1/4 numerical value, the PBE0 hybrid functional. Both schools are clearly necessary, implying different approaches, as well as a bit of chivalrous rivalry and sometimes a sense of a holy grail quest. Just to give a few quotations: “For many of us, DFT is a useful tool. For some of us, it is a continuing obsession.” Solid-state physicists consider that “the uniform density limit is sacrosanct,” Chemically oriented physicists reply that “atoms are no less sacred than uniform or nearly uniform electron gases, or the sufficient-but-not-necessary local Lieb-Oxford bound.” As there are different levels of approximations, John Perdew compared these levels with rungs of a ladder, and compared this ladder with the ladder Jacob, a famous figure in Bible’s Book of Genesis, saw in a dream: “he [Jacob] dreamed that there was a ladder set up on the earth, the top of it reaching to heaven,” the message being that Perdew’s ladder should bring us to DFT heaven. Let us start with the first rung. To repeat, the “L” in LDA means local: the electron density is considered as locally homogeneous. This is true of course in the case of an everywhere homogeneous electron gas. It is a workable approximation in the case of a slowly varying electron density system, with no defect. It is less true in other systems. In cases when the electron density varies rapidly in some places, LDA is not adequate and one naturally has to go beyond. But just adding a first-order correction, without any further guidelines, may not work because the “expansion/perturbation series” is not guaranteed to be well behaved: the first-order correction might introduce a spurious effect that would be exactly canceled by the second order, so that staying at the zeroth order would be better than introducing just the first order. There are some physical constraints, exact in some physical limits, and hence called “exact constraints,” which may be respected by LDA but not by some GGA functionals. Nowadays, PAW-PBE pseudos are rather well designed, but of course still not perfect, still depending, albeit in a less crucial way, on how many electrons are explicitly considered as “valence” electrons, and with internal parameters in the PAW construction, which may be important for very high pressure studies. There is only one LDA, because the exact LDA was worked out by David Ceperley and Berne Alder in 1980 with an extensive quantum Monte Carlo simulation. But one must speak of GGAs, with plural. Not all GGAs are equal. Some GGAs were not well designed, introducing ill designed first-order corrections in a non-well-behaved density gradient expansion. Meta-GGAs try to add corrections coming from the kinetic energy density. GGAs εXC include ∇n (r). The next step is to include ∇∇n(r), but it turns out that, in the expansion, ∇n(r) ∇n(r) must also be involved; hence the often encountered phrase that the Meta-GGA functionals include is the kinetic functional |∇n(r)|2. The only nonempirical MGGA to date is TPSS (Tao et al. 2003), according to Perdew’s standards. The next rung of the ladder has Hyper-GGAs that include nonlocality (because inhomogeneity of the electron density is not the only problem in the XC functional) by considering the exact exchange energy density. In “XC,” X is for exchange, and C is for correlation. The X energy can be calculated

4.9 The Continuous Quest for Better DFT XC Functionals

exactly, in principle, via Hartree–Fock, known as the so-called exact EXHF. But HF does not have any C. C is only in the DFT XC functional that also contains some X, so that one cannot do a simple addition like EXHF + XC funct. As empirically proposed by Becke in 1993, one can try some mixing very schematically written here as λEXHF + (1 – λ)XC funct, with a mixing coefficient λ halfguessed, half-fitted on experimental data of interest, and later half-justified (this is an oversimplified presentation, and there may be much more than one mixing parameter). These mixed functionals are called “hybrid functionals.” The one developed by Becke used three parameters in a mix with the Perdew–Wang correlation functional proposed in 1991 so that it is called the B3PBE91 hybrid functional. A famous hybrid functional is the B3LYP because it uses the Lee–Yang–Parr correlation functional. John Perdew, Matthias Ernzerhof, and Kieron Burke proposed in 1996 a one-parameter hybrid functional in 1996, using their PBE XC functional. It is called PBE0, with some rational for λ = 1/4. Jochen Heyd, Gustavo Scuseria, and Matthias Ernzerhof extended the PEB considerations in 2003 and built hybrid functionals based on a screened Coulomb potentials. The Heyd–Scuseria–Ernzerhof (HSE) screened Coulomb hybrid density functional is designed to produce exchange energies comparable with traditional hybrids while only using the short range, screened HF exchange. The PBE0-based HSE hybrid functional is called the HSE06 hybrid functional. Note that some considered the introduction of hybrid functionals by Becke in 1993 as marking the end of pure DFT and later wrote a witty obituary (Gill 2001). The last step to heaven, according to Perdew’s (Jacob’s) Ladder, is to consider unoccupied orbitals. But Becke said that DFT is occupied orbitals only, OOO! DFT is usually believed to be for electronic ground states only. Unoccupied orbitals are not supposed to be describable within DFT, hence the band gap problem. Yet, in practice, hybrid HSE06 gives correct band gaps… So, why not be practical? But would our theoretician friend say, this has to be justified? This is the continuing obsession mentioned earlier, as if we were expecting some ab initio answer to anything. Toward the end of the previous section, it was mentioned that once a Hamiltonian is known, then everything is known, or can be known, in principle. “Give me one firm spot on which to stand with a long lever and I’ll move the world” said Archimedes 23 centuries ago, “Give me a firm Hamiltonian and a good computer and I’ll explain the world” is what the theoretician physicist wants to say today. The realm of the electronic excitations is naturally richer and still more complicated than the already difficult ground state realm. You can have one-particle excitations, as when one electron from a filled band is taken out far away, out of a material, leaving a “hole” around which the remaining band electrons reorganize. This quasihole excitation can be treated thanks to a oneparticle response function called the one-particle Green function (because of a mathematical resolving technique introduced by the British mathematical physicist George Green 1793–1841: if you cannot explicitly work out a function F(r), try to look at some properties of its “inverse,” the function G(r,r ), defined by F(r)G(r,r ) = δ(r − r ). As for Fourier transforms, it may sound as adding a complexity, but it actually opens new vistas. It corresponds to the response at point r of a driving function F at point r. The response can naturally be in both space and time, G(r,t,r ,t ). G(r,t,r ,t ) “says” how an F event at r and t propagates to r at t (normally > t). G(r,t,r ,t ) is thus sometimes called a propagator. The mathematics of these functions, where they are also called resolvents, is of course much more involved than what is said in this simple “in parentheses” illustration). The “hole” is screened by the other electrons, and it turns out that it suffices to replace in the KS equation the εXC functional by, in a first approximation, a GW product where G is the one-particle Green function and W represents a screened Coulomb potential introduced by Hubbard. This is the famous GW approximation (GWA) developed by the Swedish solid-state theoretician Lars Hedin (1930–2002) in 1965 (Hedin 1965). It was a great achievement, but it would not be worth to go

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beyond it by expanding further the one-particle Dyson equation (associated with the one-particle Green function) because two-particle phenomena rapidly come into play in the realm of electronic excitations: the electron that left a hole probably interacted with it, at least a little, and/or might still be interacting with it if it has not left the material (molecule, metal, semiconductor, …). One has thus to consider the two-particle, electron–hole (or quasielectron and quasihole) interaction, and consider the Bethe–Salpeter equation, which is the two-particle equivalent of the Dyson equation, unfortunately much more than twice more complicated. The Bethe–Salpeter equation was originally developed in 1951 by Hans Bethe (1906–2005) and Edwin Salpeter (1924–2008) as a relativistic equation in nuclear physics to describe bound states of systems of two interacting particles like a proton and a neutron in the deuteron (the nucleus of deuterium). It was rigorously proven the same year by Murray Gell-Mann and low in the framework of quantum field theory. Forgetting the relativistic aspect, the mathematics is the same in condensed matter physics as shown by Le Jeu Sham and Rice in 1966 to consider the effective mass approximation of the Wannier exciton (Sham and Rice 1966). One famous approach to these many-body problems is the so-called Feynman diagram approach, which helps to understand what kinds of terms one should consider in the approximations one has to do in order to approximate the infinite series involved in the many-body equations, when one has to solve them. Here again, different choices exist, which are well beyond the scope of this book.

4.10

Van der Waals Forces and DFT

This subject is admittedly outside the field of the band theory of solids, but in view of the aforementioned considerable success of DFT, it is worth mentioning a venerable interatomic, or intermolecular, force, of course due to interactions between electrons, which, in spite of its age and apparent simplicity, still partly resists DFT foundations, namely, the so-called van der Waals (vdW) force(s), at least its London part. With the advent of the first models of atoms at the beginning of the twentieth century, new inquiries interatomic forces could be made. Peter Debye (1884–1966) thought that the mutual interaction between electrons oscillating around their nuclei in two neutral atoms might induce an attraction between the atoms, but a classical electrostatic calculation showed that the net effect is zero. During a visit in New York in 1927, Debye persuaded a Chinese research student at Columbia University, Shou Chin Wang (1905–1984), to repeat the calculation using the new wave mechanics. Wang worked out a crude model with which he obtained an attractive force at atomically large distance, proportional to R−7, where R is the interatomic distance. The result was immediately published, in German in the famous Physikalische Zeitschrift, Physics Journal of which Debye was an editor. Fritz London (1900–1954) tackled the problem again three years later, in a more general way, and this force is now known as the London force. The neutral electronic charge around a nucleus cannot be perfectly at rest, because of Heisenberg’s principle, and presents instantaneous fluctuating dipoles, quadrupoles, etc. These fluctuations may interact with the electronic charge of a somewhat distant atom and the correlated interactions result in a lowering of the total energy, proportional to the distance between the atoms, with an R−6 dependence for the most important term. This is because the electronic clouds are mutually polarizable. The polarization of the electronic cloud of an atom under the sinusoidal electric field of light (as an electromagnetic wave) is also responsible for the change of the refractive index with the frequency of the light, i.e. the dispersion of light. London thus christened the R−6 energy term the “dispersion energy.”

4.10 Van der Waals Forces and DFT

The other terms, in R−8, R−10, …, are usually also called dispersion terms, and the associated forces are called dispersion forces, or London forces, or (one of the three) vdW forces because they are (partly) responsible for the capillary (or surface tension) forces Johannes Diderick van der Waals (1837–1923) was interested in, after many other scientists but at a time when new progress could be made (Laplace had preferred to dismiss speculations about the R-dependence form of these forces; these forces are also responsible for the pressure change P P + a/v2 where v is the molar volume in van der Waals’ famous equation of state for real gases). The London forces are weak; they are not responsible for the strong chemical bonds in covalent molecules or solids or for ionic bonds. Yet they exist and, and besides vdW equation of states for real gases, they explain the cohesion of solid argon for instance, the formation of dimers made of “inert” atoms, Ar2, Kr2, … interactions between molecular groups in large proteins or molecules in molecular crystals (as fullerite, made of C60 molecules, the fullerenes), the interlayer cohesion of graphite (though π–π interactions). They also play a more or less important role in all solids (up to 30% of the cohesive energy of gold) and liquids. Yet, since they originate from the dynamic interaction between electronic charges that do not have to overlap, actually mediated by the zero-point quantum fluctuations of vacuum (same origin as the Casimir effect), or screened “vacuum” in dense solids, they are not readily described by DFT. For instance, the B3LYP XC functional fails to bind Kr–Kr. PBE binds it but with an unrealistically small interaction energy with respect to the CCSD(T) value (by a factor of 3 and at an equilibrium distance slightly too large). Note that MP2, sometimes considered as reliable to describe London’s forces, systematically underestimates intermolecular distances for π–π interactions, with respect to CCSD(T). One can say that, in principle, DFT can do it all, and Kohn et al. (1998) designed a way to compute the vdW forces and got surprisingly good results for He–He. Their tour de force was never repeated, however, and is considered as numerically too expensive to be useful in practice. A most popular way currently seems to be to work pragmatically, “experimentally,” adding pairwise interatomic “C6/R6” terms in existing DFT codes; see Grimme (2011) for a review. Stefan Grimme called his successively refined 2004, 2006, and 2010 versions DFT-D1, DFT-D2, and DFT-D3, “D” standing for dispersion. The “−C6/R6” and the like terms naturally need to be damped at small values of R, where chemical interaction predominates, and this tuning requires a lot of care. In the DFT-D (or DFT + disp) scheme, one simply adds to the DFT energy the following contribution: E disp = −

n f n,damp RiA,jB C AB n RiA,jB

4 46

iA, jBn = 6, 8, 10, …

The sum is over all pairs of atoms. The CAB n , n = 6, 8, 10, … are the averaged nth-order dispersion coefficients for the chemical atom pair iA–jB, empirically determined, and RiA,jB is their internuclear distance. The damping functions can have different forms and can imply several semiempirical parameters, among which the so-called vdW radii RA and RB. Note that three-body terms, known to be important in the cohesive energy of rare gas solids, should be added but almost never are. Similarly, because a complete fitting is excessively complicated to achieve, only the sixth-order term is usually considered, although the 8th- and 10th-order terms are known to significantly contribute at equilibrium distances. In order to avoid the empirical damping tuning process in Eq. (4.46) necessary to properly “seam” the long-range Edisp and the local EDFT contributions, several authors look for a “van der Waals density functional,” vdW-DF, which would work at all R ranges, in a “seamless” way. This effort, which started with Bengt Lundqvist and David Langreth (1937–2011), from the Swedish Chalmers University of Technology, together with people from the US Rutgers University, hence the

209

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4 The Electronic Level I

nickname Rutgers-Chalmers vdW-DF, is in progress, with regularly improved vdW-DFs (see Berland et al. 2015, for a review).

Practice Problems 1

Explain the differences, with respect to the basis sets used, between the LCAO and freeelectron models.

2

What is the Hartree–Fock approximation?

3

What is Bloch’s theorem?

4

What is the Wigner–Seitz cell?

*5

The primitive translation vectors for the HCP lattice can be chosen as a=

a 3a x+ y b= − 2 2

a 3a x + y c = cz 2 2

What are the primitive translation vectors of the reciprocal lattice? 6

Show that the volume of the first BZ, ΩBZ, for a lattice in any dimension, d, is (2π)d times the reciprocal of the real-space primitive cell volume, Ωcell. Hint: Use the vector identity: A × B × C × D = B A C × D −A B C × D

*7 ∗

Show how to compute the scalar triple products in Eqs. (4.24)–(4.26) through the determinant.

For solutions, see Appendix D.

References Ashcroft, N.W. and Mermin, N.D. (1976). Solid State Physics. Orlando: Hartcourt, Inc. Becke, A.D. (2014). J. Chem. Phys. 140: 18A301. Berland, K., Cooper, V.R., Lee, K., Schröder, E., Thonhauser, T., Hyldgaard, P., and Lundqvist, B.I. (2015). Rep. Prog. Phys. 78: 065501. Bloch, F. (1928). Z. Phys. 52: 555. Born, M. and Lerdermann, W. (1943). Nature 151: 197. Born, M. and Oppenheimer, J.R. (1927). Ann. Phys. 84: 457. Born, M. and von Kármán, T. (1912). Z. Phys. 13: 297. Bouckaert, L.P., Smoluchowski, R., and Wigner, E. (1936). Phys. Rev. 50: 58. Brillouin, L. (1930). J. Phys. Radium 1: 377. Burdett, J.K. (1996). J. Phys. Chem. 100: 13263. Burke, K. (2012). J. Chem. Phys. 136: 150901. Canadell, E. (1998). Chem. Mater. 10: 2770.

References

Cremer, D. (2011). WIREs Comput. Mol. Sci. 1: 509. Dirac, P.A.M. (1929). Proc. R. Soc. A 23: 714. Drude, P.K.L. (1900a). Ann. Phys. 1: 556. Drude, P.K.L. (1900b). Ann. Phys. 3: 369. Drude, P.K.L. (1902). Ann. Phys. 7: 687. Elliott, S.R. (1998). The Physics and Chemistry of Solids. Chichester: Wiley. Fock, V. (1930). Z. Phys. 61: 126. Georges, A. (2016). CR Phys. 17: 430. Gill, P.M.W. (2001). Aust. J. Chem. 54: 661. Grimme, S. (2011). Comput. Mol. Sci. 1: 211. Harrison, W.A. (1989). Electronic Structure and Properties of Solids: The Physics of the Chemical Bond. New York: Dover Publications. Hartree, D.R. (1928). Proc. Camb. Philos. Soc. 24: 89, 111. Hedin, L. (1965). Phys. Rev. 139: A796. Herring, C. (1942). J. Franklin Inst. 233: 525. Hohenberg, P. and Kohn, W. (1964). Phys. Rev. B 136: 864. Hume-Rothery, W., Abbott, G.W., and Channel-Evans, K.M. (1934). Philos. Trans. R. Soc. A233: 1. Kakehashi, Y., Tanaka, H., and Yu, M. (1993). Phys. Rev. B 47: 7736. Kohn, W. and Sham, L.J. (1965). Phys. Rev. A 140: A1133. Kohn, W., Meir, Y., and Makarov, D.E. (1998). Phys. Rev. Lett. 80: 4153. Lewars, E.G. (2016). Computational Chemistry. Introduction to the Theory and Applications of Molecular and Quantum Mechanics, 3e. Switzerland: Springer International Publishing. Mardirossian, N. and Head-Gordon, M. (2017). Mol. Phys. 115: 2315–2372. Margenau, H. and Murphy, G.M. (1956). The Mathematics of Physics and Chemistry, 2e. Princeton: D. Van Nostrand Company, Inc. Martin, R. (2004). Electronic Structures: Basic Theory and Practical Methods. Cambridge: Cambridge University Press. Mattheiss, L.F. (1969). Phys. Rev. 181: 987. Mermin, N.W. (1965). Phys. Rev. A 137: 1441. Mizutani, U. (2010). Hume-Rothery for Structurally Complex Alloy Phases. CRC Press, Taylor & Francis Group. Møller, C. and Plesset, M.S. (1934). Phys. Rev. 46: 618. Mott, N.F. and Jones, H. (1936). The Theory of the Properties of Metals and Alloys. Oxford: Oxford University Press. Peierls, R. (1930). Ann. Phys. 4: 121. Perdew, J.P. and Schmidt, K. (2001). Density Functional Theorry and Its Application to Materials (eds. V. Van Doren et al.). AIP. Perdew, J.P., Ruzsinsky, A., Tao, J., Starokevov, V.N., Scuseria, G.E., and Csonka, G.I. (2005). J. Chem. Phys. 123: 062201. Pettifor, D.G. (2000). The Science of Alloys for the 21st Century: A Hume-Rothery Symposium Celebration (eds. P.E.A. Turchi and R.D. Shull). Warrendale, PA: The Minerals, Metals & Materials Society. Pettifor, D.G. and Ward, M.A. (1984). Sol. State Commun. 49: 291. Raynor, G.V. (1947). An Introduction to the Electron Theory of Metals. London: Institute of Metals. Rossiter, P.A. (1987). The Electrical Resistivity of Metals and Alloys. Cambridge, NY: Cambridge University Press. Runge, E.K. and Gross, E.K.U. (1984). Phys. Rev. Lett. 52: 997. Seitz, F. (1936). Ann. Math. 37: 17.

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Seitz, F. (1940). The Modern Theory of Solids. New York: McGraw-Hill Book Company. Sham, L.J. and Rice, T.M. (1966). Phys. Rev. 144: 708. Sommerfeld, A. (1928). Z. Phys. 47: 1. Stroud, D. and Ashcroft, N.W. (1971). J. Phys. F 1: 113. Tao, J., Perdew, J.P., Staroverov, V.N., and Scuseria, G.E. (2003). Phys. Rev. Lett. 91: 146401. Thijssen, J.M. (1999). Computational Physics. Cambridge: Cambridge University Press. Weinstock, R. (1969). Am. J. Phys. 37: 1267. Wigner, E. and Seitz, F. (1933). Phys. Rev. 43: 804. Wilson, E.A. (1901). Vector Analysis: A Textbook for the Use of Students of Mathematics and Physics Founded upon the Lectures of J. Willard Gibbs. New Haven, CT: Yale University Press. Ziman, J.M. (1962). Electrons in Metals: A Short Guide to the Fermi Surface. London: Taylor & Francis.

213

5 The Electronic Level II The Tight-Binding Electronic Structure Approximation

When the number of atoms and electrons is very small, it is possible to use an exact method like configuration interaction to calculate the true many-electron wave function. However, beyond about 10 electrons such calculations are no longer possible. For systems containing up to a few thousand atoms, we can use first principles (either Hartree–Fock or DFT) to solve for the electronic states in a self-consistent manner without explicitly calculating the many-electron wave function, which usually gives a reasonable approximation to the actual band structure of the crystal. In systems with around 10 000 or more atoms, the computational expense of ab initio methods is prohibitive, and we must calculate the band structure in a semiempirical way with what is known as the tight-binding (TB) method, which uses parameters that can be adjusted to match experimental results. Without a doubt, before the advent of density functional theory and advances in computer hardware, the semiempirical method based on the superposition of localized atom-centered functions was used to obtain approximate solutions to the many-body Schrödinger equation in solids. Most band structure calculations on solids today are made with the density functional theory. However, because of the need to treat large systems and, more importantly for our purposes, because of its similarity to the very familiar Hückel molecular orbital theory (a.k.a. LCAO-MO theory), the TB formalism of Bloch’s method for describing electrons in periodic crystals will serve well in this chapter. It is possible to use a TB approximation in Hartree–Fock or DFT computations, but the latter has the advantage of being more firmly rooted in first principles. The reader is referred to excellent reviews of this by Martin (2004) and Koskinen and Mäkinen (2009) and more recently by Foulkes (2016). However, our treatment will be purely qualitative; nowhere in this chapter will we actually be computing an energy value. The reason for this is twofold. First, a quantitative treatment would be too lengthy, and more of the computational details will be presented in Chapter 14. Second, what is of real value to the nonspecialist (i.e. for those who are not computational chemists or materials scientists) is the ability to make reasonable predictions without having to carry out time-consuming calculations that, in the end, are based on approximations themselves! In earlier chapters, it was seen how a qualitative energy-level diagram for the smallest repeating chemical point group, or lattice motif (also known as the basis, or asymmetric unit), can be used to approximate the relative placement of the energy bands in a solid at the center of the BZ. This is so because the LCAO-MO theory is equivalent to the LCAO band scheme, minus consideration of the lattice periodicity. The present chapter will investigate how the orbital interactions vary for different values of the wave vector over the BZ.

Principles of Inorganic Materials Design, Third Edition. John N. Lalena, David A. Cleary, and Olivier B. M. Hardouin Duparc. © 2020 John Wiley & Sons, Inc. Published 2020 by John Wiley & Sons, Inc.

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5 The Electronic Level II

5.1

The General LCAO Method

Most chemists are well acquainted with LCAO-MO theory. The numbers of atomic orbitals, even in large molecules, however, are miniscule compared to a nonmolecular solid, where the entire crystal can be considered one giant molecule. In a crystal there are in the order of 1023 atomic orbitals, which is, for all practical purposes, an infinite number. The principle difference between applying the LCAO approach to solids, versus molecules, is the number of orbitals involved. Fortunately, periodic boundary conditions allow us to study solids by evaluating the bonding between atoms associated with a single lattice point. Thus, the lattice point is to the solid-state scientist, what the molecule is to the chemist. As a prelude to our development of the LCAO treatment of solids, it will be beneficial to briefly review the LCAO-MO method. The cyclic π systems from organic chemistry are familiar and relatively simple and, more importantly, resemble Bloch functions of periodic solids. Thus, they will be used as the introductory examples. The independent electron approximation was discussed in the previous chapter. The molecular wave functions, ψ, are solutions of the Hartree–Fock equation, where the Fock operator operates on ψ, but the exact form of the operator is determined by the wave function itself. This kind of problem is solved by an iterative procedure, where convergence is taken to occur at the step in which the wave function and energy do not differ appreciably from the prior step. The effective independent electron Hamiltonian (the Fock operator) is denoted here simply as H. The wave functions are expressed as linear combinations of atomic functions, χ: N

ψ= μ=1

cμ χ μ

51

Approximating the one-electron orbitals as linear combinations of localized or atom-centered functions that can be solved self-consistently in this manner is called the LCAO self-consistent field (SCF) method, and it was developed independently by the Dutch physicist Clemens C. J. Roothaan (1918–2019) and the Irish mathematician George Garfield Hall (1925–2018). Also defined are the following one-electron integrals (all two-electron integrals are ignored in this chapter): H μμ =

χ ∗μ Hχ μ dτ

H μv = H vμ = Sμv = Svμ =

χ ∗μ H χ v dτ χ ∗μ χ v dτ

52 53 54

The latter two integrals can be represented as a square matrix, for which each matrix element corresponds to a particular combination for the values of μ and v. It is noted that because of the Hermitian properties of H, Hμv = Hvμ and Sμv = Svμ. Equation (5.2) represents the energy of an electron in any given orbital, relative to an electron at infinity. In Hückel theory, it is called the Coulomb integral (to denote the attraction between the electron and the nucleus) and given the symbol α. Equation (5.3) gives the energy of interaction between neighbors, and it is known as the hopping integral or transfer integral (termed the exchange or resonance integral in Hückel theory). First-nearest neighbor (adjacent) interactions are conventionally denoted as β in Hückel theory and b in solids. Equation (5.4) is the overlap integral, which is a measure of the extent of overlap of the orbitals centered on two adjacent atoms. The transfer integral and overlap integral are proportionally related.

5.1 The General LCAO Method

The variational method by Walter Ritz (1878–1909) indicates that ψ ∗ Hψ dτ E=

55

ψ ∗ ψ dτ

Insertion of Eq. (5.1) into Eq. (5.5) gives

E=

μ μ

v

c∗μ cv H μv

v

56

c∗μ cv Sμv

This expression shall be a minimum if ∂E/∂cμ and ∂E/∂cv are zero, which leads to a system of equations of the form N

cμ H μv − ESμv = 0

57

μ=1

To illustrate a simple case, let Eq. (5.7) express explicitly in terms of a two-term LCAO-MO (i.e. for the molecule ethylene): H μμ − ESμμ c1 + H μv − ESμv c2 = 0

58

H vμ − ESvμ c1 + H vv − ESvv c2 = 0

59

These two linear algebraic equations in c1 and c2 have a nontrivial solution if, and only if, the determinant of the coefficients vanishes, that is, H μμ − ESμμ H vμ − ESvμ

H μv − ESμv =0 H vv − ESvv

5 10

Equation (5.10) results in a quadratic equation in the energy, which has two roots corresponding to the energies of the two π-MOs of ethylene. It is known that Hμμ = Hvv and Hμv = Hvμ and that Sμμ = Svv = 1, if χ μ and χ v are normalized. Thus, the two roots of Eq. (5.10) are found to be E1 =

H μμ + H μv 1+S

5 11

E1 =

H μμ − H μv 1+S

5 12

By substitution of these energies into Eqs. (5.8) and (5.9), one may obtain the orbital coefficients, giving explicit expressions for the MOs. In general, a linear combination of N functions (in which N is the number of atomic orbitals in the basis set) obtains an N × N secular determinant: H 11 − ES11 H 21 − ES21 H 31 − ES31

H 12 − ES12 H 22 − ES22 H 32 − ES32

H 13 − ES13 H 23 − ES23 H 33 − ES33

H 14 − ES14 H 24 − ES24 H 34 − ES34

H 1N − ES1N H 2N − ES2N H 3N − ES3N

H 41 − ES41

H 42 − ES42

H 43 − ES43

H 44 − ES44

H 4N − ES4N

H N1 − ESN1

H N2 − ESN2

H N3 − ESN3

H N4 − ESN4

H NN − ESNN

=0

5 13

215

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5 The Electronic Level II

Equation (5.13) may be written as H μv − ESμv

=0

5 14

At this point, the Hückel approximations are often imposed to simplify Eq. (5.14). These were introduced by the German physicist Erich Armand Arthur Joseph Hückel (1896–1980). Even though atomic orbitals on neighboring atoms are nonorthogonal (they have nonzero overlap), it is possible to make the approximation that Sμv = 0 =1

if if

μ v μ=v

5 15

In other words, the overlap integrals, Sμv, including those between atomic orbitals on adjacent atoms, are neglected. With this, Eq. (5.13) becomes H 11 − E H 21 H 31

H 12 H 22 − E H 32

H 13 H 23 H 33 − E

H 14 H 24 H 34

H 1N H 2N H 3N

H 41

H 42

H 43

H 44 − E

H 4N

H N1

H N2

H N3

H N4

H NN − E

=0

5 16

Neglecting the overlap integral is a severe approximation. However, this approach is still useful because a general picture of the relative MO energy levels, utilizing primarily symmetry arguments, can be obtained. The secular determinant can be further simplified by considering only interactions between first-nearest neighbors. In this case, all the other Hμv matrix elements become equal to zero. Using the familiar Hückel notation, Eq. (5.16) then looks like α−E β

β α−E

0 β

0 0

0 0

0 0

β 0

α−E β

β α−E

0 0

0

0

0

0

α−E

=0

5 17

Even with these simplifications, an N × N secular equation must still be solved with nonzero off-diagonal matrix elements, which becomes a formidable task for large molecules. An N × N determinant will give an equation of the Nth degree in the energy, which has N roots. At this point, how the calculations can be simplified considerably by adapting the atomic orbitals to the symmetry of the molecule will be shown. This is most easily illustrated with the molecule benzene, a conjugated cyclic π system. Suppose that, instead of using an AO basis set, the secular equation for benzene is written as an N × N array of SALCs of the atomic orbitals. Each SALC corresponds to an irreducible representation of the point group for the molecule. Only orbitals of the same irreducible representation can interact; SALCs that have different irreducible representations are orthogonal. Thus, if the SALCs of a given representation are grouped together in the secular determinant, the only nonzero matrix elements will lie in blocks along the principal diagonal, thereby factoring the equation into smaller determinants, each of which is solved separately. The secular equation is said to be block-diagonalized. Since the entire equation is to have the value

5.1 The General LCAO Method

of zero, each of the smaller block factor determinants must also equal zero. It should be noted that the orbital coefficients are obtained by this process, rather than from solution of the secular equation as before. In general, the SALCs themselves are generated by the use of what are known as projection operators. A thorough description of this procedure can be found elsewhere (Cotton 1990). Fortunately, it will not be necessary to cover the details of this process, as the authors make use of the extremely useful simplification that, for cyclic π systems, CnHn, there will always be n π MOs, one belonging to each irreducible representation of the Cn pure rotation group. In other words, benzene belongs to the D6h point group, but the essential symmetry elements needed for constructing our SALCs are contained in the C6 pure rotation subgroup. Furthermore, the coefficients of the MOs are the characters of the irreducible representations of C6. Thus, from inspection of the character table for the C6 point group, the SALCs for benzene can immediately be written as (Cotton 1990) 1 χ1 + χ2 + χ3 + χ4 + χ5 + χ6 6 1 B = χ1 − χ2 + χ3 − χ4 + χ5 − χ6 6 1 χ 1 + εχ 2 − ε∗ χ 3 − χ 4 − εχ 5 + ε∗ χ 6 E1 = 6 1 E1 = χ 1 + ε∗ χ 2 − εχ 3 − χ 4 − ε∗ χ 5 + εχ 6 6 1 χ 1 − ε∗ χ 2 − εχ 3 + χ 4 − ε∗ χ 5 − εχ 6 E2 = 6 1 E2 = χ 1 − εχ 2 − ε∗ χ 3 + χ 4 − εχ 5 − ε∗ χ 6 6

ψ A = ψ ψ ψ ψ ψ

5 18

where ε is equal to exp(2πi/6); A, B, E1, and E2 are the Mulliken symmetry labels for the irreducible representations; and the 1/60.5 factor is a normalization constant such that the square of the coefficients on each atomic orbital in a given MO adds up to one. Each wave function in Eq. (5.18) can be written (see Example 5.1) as (Albright et al. 1985) N

ψj = μ=1

cjμ χ μ = N − 1

N 2

exp μ=1

2πij μ − 1 N

χμ

5 19

where j runs from 0, ±1, ±2, …, ±N/2 and i is the square root of −1. The term in brackets is the orbital coefficient for the μth atomic orbital. It is possible to write the wave functions in Eq. (5.19) as linear combinations that have real coefficients instead of imaginary ones. However, the authors will not do this as leaving them in this form will reveal their similarity to Bloch functions later. Equation (5.20) gives the resultant block-diagonalized secular equation where the subscripts of the matrix elements now refer to the SALCs: ψ A ψ A ψ B ψ E1 ψ E1 ψ E2 ψ E2

ψ B

ψ E1

ψ E1

ψ E2

ψ E2

H 11 − E H 22 − E H 33 − E

=0 H 44 − E H 55 − E H 66 − E

5 20

217

218

5 The Electronic Level II

The energy of each MO is obtained by solving its respective block factor in Eq. (5.20). Thus, each of the six MOs of benzene gives Ej =

ψ ∗j Hψ j dτ

5 21

In order to calculate the energy, the simple Hückel theory for π systems is used. That is, only the π interactions between adjacent p orbitals are considered; the overlap integrals are neglected. Equation (5.21) then becomes equal to Ej =

χ ∗μ Hχ μ dτ +

1 − 2πij 2πij exp + exp N N N

χ ∗μ H χ v dτ

5 22

The first integral is simply α and the second integral is β. The sum of the two terms in brackets is equal to 2 cos (2πj/N), and since there are N atoms in the molecule, the energy is finally found to be Ej = α + 2β cos

2πj N

5 23

where j runs from 0, ±1, ±2, …, ±N/2. Example 5.1

Show that Eq. (5.19) does indeed give the wave functions of Eq. (5.18).

Solution Since j runs from 0, ±1, ±2, …, ±N/2, and the wave functions of E1 and E2 symmetry are doubly degenerate, the expression in Eq. (5.19) for j = 0, ±1, ±2, and +3, and μ = 1 − 6 must be evaluated. Replacing j with 0, for example, results in cj = 1 for every value of j, leading to ψ 0 = 1/60.5(χ 1 + χ 2 + χ 3 + χ 4 + χ 5 + χ 6), which is the expression for ψ(A). When j = +1, then, for μ = 1 − 6, ψ 1 = 1 60 5 exp 0 χ 1 + exp 2πi 6 χ 2 + exp 4πi 6 χ 3 + exp 6πi 6 χ 4 + exp 8πi 6 χ 5 + exp 10πi 6 χ 6 Recognizing the following relations: eiθ = cos θ + i sin θ

e − iθ = cos θ − i sin θ,

the expression for ψ 1 is found to be equivalent to ψ 1 = 1 60 5 χ 1 + exp 2πi6 χ 2 + exp 4πi 6 χ 3 − χ 4 + exp 8πi 6 χ 5 + exp 10πi 6 χ 6 It can also be seen that exp 4πi 6 χ 3 = − exp −2πi 6 χ 3 = −ε∗ χ 3 exp 8πi 6 χ 5 = − exp 2πi 6 χ 5 = −εχ 5 exp 10πi 6 χ 6 = exp −2πi 6 χ 6 = ε∗ χ 6 Making the substitutions gives ψ 1 = 1 60 5 χ 1 + εχ 2 − ε∗ χ 3 − χ 4 − εχ 5 + ε∗ χ 6 which is ψ(E1). The reader should show that proceeding in an analogous fashion will give the remaining expressions in Eq. (5.18).

5.2 Extension of the LCAO Treatment to Crystalline Solids

5.2

Extension of the LCAO Treatment to Crystalline Solids

Bloch sums are composed of an enormous number (~1023) of atomic orbitals. Such a gigantic basis set is handled by making use of the crystalline periodicity. As the crystal structure is periodic, its electron density is periodic also, since the presence of structural periodicity imposes translational periodicity on the wave functions. Consider the one-dimensional chain of N atoms, with spacing a (see Figure 4.6 for instance) to be of finite length L, where L = Na. For running wave solutions to Schrödinger’s equation, describing the motion of an electron in the array, periodic boundary conditions are appropriate (see Section 4.2). An integral number of wavelengths must fit into L: ψ r =ψ r+L

5 24

In reality, the atoms at the ends (surfaces in three dimensions) experience different forces from those of the bulk. However, if one is not concerned with surface effects, the advantages of using periodic boundary conditions are enormous; all atoms become equivalent, and, combined with inner lattice periodicity, one can use Bloch theorem; see Section 4.4, partly repeated below. Certainly, as N ∞, the main local properties of the atoms deep in the bulk are unaffected by the surface conditions (this will obviously not be true for aggregates or nano-objects or for wave properties whose wavelength is not small before L). The most general expression for a periodic function is the plane wave, eiθ, in which θ is a parameter equal to the vector dot product k r. Hence, the wave function, ψ(r), of Eq. (5.24) is of the form ψ r = eik r u r

5 25

where u(r) has the periodicity of the lattice, u(r + R) = u(r), and modulates eik r. Equation (5.25) states that ψ(r), which is an eigenstate of the one-electron Hamiltonian, can be written in the form of a plane wave times a function periodic in the Bravais lattice of the solid. If Eq. (5.25) holds, then so does the following relation: ψ r + R = eik R ψ r

5 26

Equations (5.25) and (5.26) are equivalent expressions. If the physical location in real space is shifted by R, only the phase of the wave function will change. Because ψ(r) over any unit cell is known, it can be calculated for any other unit cell using ψ(r + R) = eik Rψ(r). Thus, eik R is an eigenfunction of the translation operator. Bloch’s theorem states that the eigenfunctions of the Hamiltonian have the same form – the Hamiltonian commutes with the translation operator. Therefore, the Hamiltonian only has to be solved for one unit cell. For each type of atomic orbital in the basis set, which is the chemical point group, or lattice point, one defines a Bloch sum (also known as Bloch orbital or Bloch function). A Bloch sum is simply a linear combination of all the atomic orbitals of that type under the action of the infinite translation group. These Bloch sums are of the exact same form in Eq. (5.19), but with χ μ replaced with the atomic orbital located on the atom in the nth unit cell χ(r–Rn): ϕ k = N −1

N

exp ik Rn χ r − Rn

2

5 27

n=1

In this equation, N is equal to the number of unit cells in the crystal. Note how the function in Eq. (5.27) is the same as that of Eq. (5.19) for cyclic π molecules, if a new index is defined as

219

220

5 The Electronic Level II

k = 2πj/Na. Bloch sums are simply symmetry-adapted linear combinations of atomic orbitals. However, whereas the exponential term in Eq. (5.19) is the character of the jth irreducible representation of the cyclic group to which the molecule belongs, in Eq. (5.27) the exponential term is related to the character of the kth irreducible representation of the cyclic group of infinite order (Albright et al. 1985). This, in turn, may be replaced with the infinite linear translation group because of the periodic boundary conditions. It turns out that SALCs for any system with translational symmetry are constructed in this same manner. Thus, as with cyclic π systems, there should never be a need to use the projection operators referred to earlier to generate a Bloch sum. Now, how Bloch sums combine to form COs must be considered, for example, like the σ- or π-combinations between the Bloch sums of metal d orbitals and oxygen p orbitals in a transition metal oxide. Bloch sums are used as the basis for such COs: ψm k =

cmμ k φμ k

μ

5 28

where it is noted that a linear combination of Bloch functions is also a Bloch function. The eigenvalue problem that is being solved here can be represented as Hψ m k = Eψ m k

5 29

It was seen earlier how using SALCs to construct MOs resulted in a block-diagonalized secular equation. Exactly the same thing happens with solids, resulting with an N × N determinant (where N is the number of unit cells; ~1023) diagonalized with n × n block factors (where n is the number of atomic orbitals in the basis set), each having a particular value of k. For example, a substance containing valence s, p, and d atomic orbitals and having a Bravais lattice with a one-atom basis (one atom per lattice point) gives a 9 × 9 block factor or an 18 × 18 with a two-atom basis. Within each block factor, the matrix elements can be written as (Canadell and Whangbo 1991) H μv k = =

ϕ∗μ k Hϕv k dτ χ ∗μ r Hχ v r dτ + ×

1 N

exp ik r m exp ik r n m

5 30

n

χ ∗μ r − r m Hχ v r − r n dτ

Sμv k =

φ∗μ k φv k dτ

=

χ ∗μ r χ v r dτ +

1 N

exp ik r m exp ik r n m

n

χ ∗μ r − r m χ v r − r n dτ 5 31

where the indices m and n denote the mth and nth unit cells and φμ and φv represent two different Bloch sums (e.g. one formed from, say, all the oxygen p orbitals and one formed from all the transition metal d orbitals in a metal oxide). If there is only one atom at each lattice point, the primitive lattice translation vectors, Rn = (rn − rm), are the basis vectors, which give the displacement from the atom on which the orbital χ μ is centered to the atom on which χ v is centered. It is also recognized that, since there is translational invariance in a Bravais lattice, the sum over m atoms is done N times. Hence, the factor N−1 is canceled by the sum over m. Equations (5.30) and (5.31) can now be written as

5.3 Orbital Interactions in Monatomic Solids

H μv k = Sμv k =

χ ∗μ r Hχ v r dτ + χ ∗μ r χ v r dτ +

χ ∗μ r Hχ v r − Rn dτ

exp ik Rn n

χ ∗μ r χ v r − Rn dτ

exp ik Rn n

5 32 5 33

The variational principle can be used to estimate the energy. If only the first-nearest neighbor interactions and an orthonormal (Eq. 5.15) set of atomic orbitals are considered, substitution of Eqs. (5.32) and (5.33) into Eq. (5.5) yields E k = E0 k +

exp ik Rn n

χ ∗μ r Hχ v r − Rn dτ

5 34

where E0(k) is χ ∗μ r Hχ v r dτ, which is the on-site, or Coulomb, integral or energy, α. The vector notation used to this point is concise, but it will be instructive to resolve the vectors into their components. Each atom is located at a position pai + qbj + rck where a, b, c are lattice parameters and i, j, k are unit vectors along the x, y, z axes. A Bloch sum may thus be written as ϕ kx , ky , kz = =

1 N 1 N

p

q

r

exp ik x pa + ik y qb + ikz rc χ μ r − pa − qb − rc exp ik x pa

p

q

5 35

exp ik y qb

r

× exp ik z rc χ μ r − pa − qb − rc At the risk of being superfluous, the corresponding expression for the energy is written here. Upon resolving the vectors in the matrix elements (Eqs. 5.32 and 5.33), Eq. (5.34) becomes χ ∗μ r Hχ v r dτ +

E kx , ky , kz =

× exp ik y pq

×

exp ik z rc r

q

r

exp ik z rc

χ ∗μ r Hχ v r dτ +

=

exp ik x pa p

χ ∗μ r H r − pa − qb − rc dτ 5 36

exp ik x pa p

exp ik y qb q

χ ∗μ r Hχ v r − pa − qb − rc dτ

The phase factor sum indicates that the amplitude of the plane wave at the lattice point in question is the sum of contributions from all the atoms. Indeed, in the LCAO method it is really just expressing the electronic wave function in a solid as a superposition of all the atomic wave functions.

5.3

Orbital Interactions in Monatomic Solids

5.3.1 σ-Bonding Interactions Let us consider a primitive Bravais lattice with one atomic orbital of spherical symmetry (1s atomic orbital) per lattice point. For example, in a SC lattice each atom has six first-nearest neighbors. Relative to the atom in question, the six neighbors are at coordinates (a, 0, 0),

221

222

5 The Electronic Level II

(−a, 0, 0), (0, a, 0), (0, −a, 0), (0, 0, a), and (0, 0, −a). The phase factor sums for each of the planes defined by these six points are ± a, 0, 0

eikx a + e − ikx a e0 e0 = 2 cos kx a

0, ± a, 0

e0 eiky a + e − iky a e0 = 2 cos k y a

0, 0, ± a

e0 e0 eikz a + e − ikz a = 2 cos k z a

Thus, the energy of a Bloch sum of s atomic orbitals in the SC lattice (with one atom per lattice point) is E = Es,s 000 + 2 cos kx a + 2 cos k y a + 2 cos kz a E s,s 100 = Es,s 000 + 2Es,s 100 cos k x a + cos k y a + cos k z a

5 37

The first term on the right, Es,s(0 0 0), represents the first integral on the right-hand side of Eq. (5.36), the energy of an isolated atomic orbital. In Hückel theory, Es,s(0 0 0) is given the symbol α. The term Es,s(1 0 0) represents the second integral on the right-hand side of Eq. (5.36), the interaction between an atomic orbital with its first-nearest neighbors. In the SC lattice, these are located at the (1, 0, 0), (−1, 0, 0), (0, 1, 0), (0, −1, 0), (0, 0, 1), and (0, 0, −1). In Hückel theory, the integral representing first-nearest neighbor interactions is given the symbol β, but it is often represented simply as b in solids. With lower-dimensional systems the results are, of course, analogous. For example, with a two-dimensional (2D) square lattice of s atomic orbitals, by considering only interactions between first-nearest neighbors, E(kx, ky) is given as Es,s(0 0) + 2Es,s(0 0) [cos(kxa) + cos(kya)] or, in Hückel theory, as α + 2β[cos(kxa) + cos(kya)]. For the one-dimensional chain as previously illustrated in Figure 4.4, E(kx) = Ex,x(0) + 2Ex,x(1) cos(kxa) ≡ α + 2β cos(kxa). Using this simple expression, a qualitative band structure diagram can be constructed corresponding to the one-dimensional array of Figure 4.4. As k changes from 0 to π/a, the energy of the CO changes from [Ex,x(0) + 2Ex,x(1)] to [Ex,x(0) − 2Ex,x(1)]. Since Ex,x(0) is the energy of an electron in a nonbonding atomic orbital, it can be used to set our zero of energy. Therefore, the two energies Ex,x(0) ± 2Ex,x(1) must correspond to the maximum bonding and maximum antibonding orbitals. Between these two energies, there is a quasi-continuum of levels, which gives rise to the curve shown in Figure 5.1. If one expands the cosine terms for small k-values to second order, the parabolic dependence of E on k from the NFE model (see Figure 4.2) is reproduced, with E(k) symmetric near k = 0 (Γ). The TB bandwidth, W, or band dispersion, is given by 5 38

W = 2zβ

where z is the coordination number (the number of first-nearest neighbors) and β = −Es,s(1). In the case of the one-dimensional chain, z = 2, W equals 4β. For the 2D square lattice, each atom has four Figure 5.1 The band dispersion diagram for a one-dimensional Bloch sum of σ-bonded p atomic orbitals.

Ex,x(0) – 2Ex,x(1)

Ex,x(0) + 2Ex,x(1) Γ (k = 0)

X (k = π/a)

5.3 Orbital Interactions in Monatomic Solids

nearest neighbors, and W equals 8β. In the SC lattice, each atom has six nearest neighbors, so the bandwidth is equal to 12β. The band dispersion depends on the atomic arrangement in the unit cell. Having discussed the SC system, the focus will now evaluate some other structure types. For the time being, consideration will be restricted to a one-atom basis of s atomic orbitals. The BCC lattice contains eight first-nearest neighbors located, relative to the atom in question, at the coordinates ± 12 a, ± 12 a, ± 12 a . Equation (5.36) gives E kx , ky , kz =

χ ∗μ r Hχ μ r dτ + eikx a × eikz a

=

2

+ e − ikz a

2

2

2

eiky a

2

+ e − iky a

kx a 2

2 cos

ky a 2

2 cos

±

±

1 1 a, ± a, 0 , 2 2

1 1 a, 0, ± a , the energy is given by (see Example 5.2) 2 2

E k x , ky , kz = Es,s 0 0 0 + 4Es,s 1 1 0 ky a kz a cos + cos 2 2 Example 5.2

5 39

kx a ky a kz a cos cos 2 2 2

For the FCC lattice, with 12 first-nearest neighbors located at coordinates 1 1 a, ± a , and 2 2

kz a 2

χ ∗μ r Hχ v r − pa − qb − rc dτ = Es,s 000 + 8Es,s 111

× cos

0, ±

2

χ ∗μ r Hχ v r − pa − qb − rc dτ

χ ∗μ r Hχ μ r dτ + 2 cos ×

+ e − ikx a

kx a ky a cos 2 2 kx a kz a + cos cos 2 2 cos

5 40

Verify that the sum of the phase factors given in Eq. (5.40) is correct.

Solution ± a 2, ± a 2, 0

exp ik x a 2 + exp ik x − a 2 exp ik y a 2 + exp iky − a 2 exp 0

2 cos k x a 2 2 cos k y a 2

± a 2, 0, ± a 2

exp ik x a 2 + exp ik x − a 2 exp 0 exp ik z a 2 + exp ik z − a 2

2 cos k x a 2 2 cos k z a 2

0, ± a 2, ± a 2

exp 0 exp ik y a 2 + exp iky − a 2

2 cos k y a 2 2 cos k z a 2

exp ik z a 2 + exp ik z − a 2 exp ik Rn = 4 cos k x a 2 k y a 2 + 4 cos k x a 2 k z a 2 + 4 cos k y a 2 k z a 2 n

= 4 cos kx a 2 cos k y a 2 + cos k y a 2 cos k z a 2 + cos k x a 2 cos k z a 2 It is possible to consider interactions between atoms separated by any distance, of course. For example, returning to the SC lattice, had it been chosen to consider second-nearest neighbor interactions as well, the result would have been

223

224

5 The Electronic Level II

E k = E s,s 0 0 0 + 2E s,s 1 0 0 cos kx a + cos k y a + cos k z a + 4E s,s 1 1 0 cos kx a cos k y a + cos k x a cos k z a + cos k y a cos kz a 5 41 where Es,s(1 1 0) is the contribution to the energy that arises from interaction of an atom with its 12 second-nearest neighbors at positions (±a, ±a, 0), (0, ±a, ±a), and (±a, 0, ±a). The inclusion of second (and often third terms) is particularly important for certain structure types. For example, in the BCC and CsCl lattices, the second-nearest neighbors are only 14% more distant than the first nearest (Harrison 1989). Accounting for the six second-nearest neighbor interactions in the FCC and BCC lattices, the energies of a Bloch sum of s atomic states are given by Eqs. (5.42) and (5.43), respectively: E k x , ky , k z = Es,s 0 0 0 + 4E s,s 1 1 0 + cos

ky a kz a cos 2 2

cos

kx a ky a cos 2 2

+ cos

kx a kz a cos 2 2

+ 2E s,s 2 0 0 cos k x a + cos k y b + cos kz c 5 42

E k x , k y , kz = Es,s 0 0 0 + 8E s,s 1 1 1

cos

kx a ky b kz c cos cos 2 2 2

5 43

+ 2E s,s 2 0 0 cos kx a + cos k y a + cos kz a Second nearest-neighbor interactions are also important for solids with nonprimitive lattices. Up to now, this evaluation has focused only on structures where there is a single atom associated with each lattice point. Next, consider diamond, which is equivalent to zinc blende, but with all the atoms of the same type. The diamond lattice is not a Bravais lattice, since the environment of each carbon differs from that of its nearest neighbors. Rather, diamond is an FCC lattice with a two-atom basis – four lattice points and eight atoms per unit cell. The structure consists of two interlocking FCC Bravais sublattices, displaced by a quarter of the body diagonal. The displacement vectors from 1 1 1 a given site to its four nearest neighbors, belonging to the other sublattice, are a, a, a , 4 4 4 1 1 1 1 1 1 1 1 1 a, − a, − a , − a, a, − a , and − a, − a, a , where a is the lattice 4 4 4 4 4 4 4 4 4 parameter. These four points are the corners of a tetrahedron whose center is taken as the origin (0, 0, 0). Since there are two atoms per primitive cell, or lattice point, and consideration is still on just s atomic orbitals, two separate Bloch sums are required. These combine to form two COs with energies given by the sum and difference in energy between the nearest neighbor and second-nearest neighbor interactions. The nearest-neighbor interactions are between atoms on the two different sublattices, that is, between an atom on one sublattice and its four neighbors on the other sublattice, which form a tetrahedron around it: 4Es,s 1 1 1

cos

kx a ky a kz a kx a ky a kz a cos cos − i sin sin sin 4 4 4 4 4 4

5 44

The 12 second-nearest neighbor interactions are between atoms all on the same sublattice: E s,s 0 0 0 + 4Es,s 1 1 0

kx a ky a ky a kz a cos + cos cos 2 2 2 2 kx a kz a + cos cos 2 2 cos

5 45

5.3 Orbital Interactions in Monatomic Solids

5.3.2 π-Bonding Interactions In the foregoing examples, it was not necessary to include π or δ interactions. This is not generally the case for atomic orbitals with a nonzero angular momentum quantum number. Consider the 2D square lattice of p atomic orbitals shown in Figure 5.2. The p orbitals bond in a σ fashion in one direction and in a π fashion in the perpendicular direction. As might be expected, these two interactions are not degenerate for every value of k. At Γ, k = (0, 0), the σ interactions are antibonding (overlapping lobes on neighboring sites have the opposite sign), and the π interactions are bonding (overlapping lobes have the same sign) for both the px and py orbitals. At M, k = (π/a, π/a), the reverse is true. At X, k = (π/a, 0), however, both σ and π interactions are bonding for the px, but both are antibonding in the py orbitals. For the SC lattice, the energy of a Bloch sum of px atomic states, including first- and second-nearest neighbor interactions, is E k = E x,x 0 0 0 + 2Ex,x 1 0 0 cos k x a + 2E y,y 1 0 0 cos k y a + cos k z a + 4E x,x 1 1 0 cos k x a cos k y a + cos k x a cos kz a + 4E x,x 0 1 1 cos k y a cos kz a 5 46 The energy of the py (or pz) band is obtained by cyclic permutation. The behavior of Eq. (5.46), for 2D square lattices of px and py atomic orbitals, using the first-nearest neighbor approximation, is shown later in Figure 5.5. A very noteworthy example involving π interactions is the single graphite sheet (graphene), with the honeycomb structure. This has been of renewed interest since the discovery of nanographites (graphene ribbons of finite width) and carbon nanotubes (slices of graphene rolled into cylinders). Each carbon atom is sp2 hybridized and bonded to its three first-nearest neighbors in a sheet via sp2 σ bonds. The fourth electron of each carbon atom is in a π(pz) orbital perpendicular to the sheet. The electronic properties are well described just from consideration of the π interactions. Like the diamond structure discussed earlier, the honeycomb structure is not itself a Bravais lattice. If the lattice is translated by one nearest-neighbor distance, the lattice does not go into itself. There are two nonequivalent, or distinct types of sites per unit cell, atoms a and b, separated by a distance a0, as shown later in Figure 5.4. However, a Bravais lattice can be created by taking this pair of distinct atoms to serve as the basis. Doing so shows that the vectors of the 2D hexagonal

Figure 5.2 A two-dimensional square array of px atomic orbitals. The bonding is by σ interactions in the horizontal direction and by π interactions in the vertical direction.

Γ : kx,y = (0, 0)

225

226

5 The Electronic Level II

lattice, a1 and a2, are primitive translation vectors. A given site on one sublattice with coordinates (0, 0) has three nearest neighbors on the other sublattice. They are located at (0, a2), (a1, 0), and (−a1, 0). As in the case of diamond, two different Bloch sums of pz atomic orbitals are needed, one for each distinct atomic site: ϕμ k =

1 N exp ik r μ χ r − r μ Nn=1

5 47

ϕv k =

1 N exp ik r v χ r − r v Nn=1

5 48

The secular equation, with the usual approximations, is H μμ − E

H μv

H vμ

H vv − E

=0

5 49

Noting the Hermitian properties of the Hamiltonian H vμ = H ∗μv , and setting our zero of energy at zero (Haa = Hbb = 0), Eq. (5.49) becomes E H ∗μv

H μv =0 E

5 50

Thus, for the energies, there is the very simple expression E ± = ± H ∗μv H μv

1 2

5 51

As expected, it has a phase factor sum and an integral representing the interactions between an atom and its three nearest neighbors, which are of the other type. Thus, Eq. (5.51) can be written as E k = ±

1 + e − ik

a1

+ e − ik

a2

1 + eik

a1

+ eik

a2

E 10

5 52

It can readily be shown that Eq. (5.52) for the band dispersion simplifies to E k = ± E 10

3 + 2 cos k a1 + 2 cos k a2 + 2 cos k

a2 − a1

The primitive lattice vectors in the (x, y) coordinate system are a1 = a2 =

3 2 a0 , −

5 53 3 2 a0 ,

1 2

and

1 , where a0 is the basis vector. Thus, Eq. (5.53) can also be written as 2

E kx , ky = ± E

3 1 , 2 2

1 + 4 cos 2

kx a 2

+ 4 cos

kx a cos 2

3k y a 2

5 54

In two dimensions, these functions are surfaces. Figure 5.3 shows the band structure diagram of graphene. In graphene, the top of the valence band just touches the bottom of the conduction band at the corner point of the hexagonal 2D Brillouin zone (BZ) where the energy goes to zero; the π and π∗ bands are degenerate at this point, as shown in Figure 5.3. The points where the energy bands touch are called Dirac points. The Fermi level, EF, which is analogous to the highest occupied MO, passes right through this intersection in graphene. There is thus a zero density of states at EF,

5.3 Orbital Interactions in Monatomic Solids

π* EF

π

M

Γ

K

Figure 5.3 The tight-binding band structure for graphene. The electronic properties are well described by the π and π∗ bands, which intersect at the point K, making graphene a semimetal.

but no band gap, which is the defining characteristic of a semimetal (see Section 6.4.1.4). In fact, graphene belongs to a class of materials known as Fermionic Dirac matter, which are described by the Dirac equation and thus are consistent with the principles of both quantum mechanics and special relativity. Additional substances in this category of condensed matter include topological insulators and other kinds of semimetals. For our purposes, semimetals differ operationally from semiconductors in that their resistivities have a metallic-like temperature dependency. However, graphene assemblies can be transformed into superconducting, metallic, semiconducting, and insulating states by either physical or chemical modification. For example, it has been shown that graphene can chemically react in a reversible manner with hydrogen to form crystalline graphane, which is an insulator whose lattice periodicity is markedly shorter than that of graphene. The original semimetallic graphene can be restored by simple annealing the hydrogenated product (Elias et al. 2009). Graphene laminates intercalated with calcium atoms, which are made to reside between the layers, have been shown to be superconducting (see Section 6.4.3) around a temperature of 6 K (Chapman et al. 2016). Superconducting behavior has also recently been observed at 1.7 K in a sandwich structure of atomically thin chemically pure graphene layers twisted, or offset, by an angle of 1.1 , albeit with just one-ten-thousandth of the electron density of conventional superconductors displaying this ability at the same temperature (Cao et al., 2018). Metallic and semiconducting single-walled carbon nanotubes (SWNTs) are single slices of graphene containing several hundred or more atoms rolled into seamless cylinders, usually with a removable polyhedral cap on each end. Multi-walled carbon nanotubes (MWNTs) are several slices rolled into concentric cylinders. The direction along which a graphene sheet is rolled up is related to the 2D hexagonal lattice translation vectors a1 and a2 via C = na1 + ma2 where C is called the chiral or wrapping vector. Thus, C defines the relative location of the two lattice points in the planar graphene sheet that are connected to form the tube in terms of the number of hexagons along the directions of the two translation vectors, as illustrated in Figure 5.4. Each pair of (n, m) indices corresponds to a specific chiral angle (helicity) and diameter, which give the bonding pattern along the circumference. The chiral angle is determined by the relation θ = tan −1

3n 2m + n

and the diameter by the relation d =

3π ac−c m2 + mn + n2

1 2

,

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5 The Electronic Level II

C = na1 + ma2

a2

a1

Figure 5.4 The chiral vector, C, defines the location of the two lattice points that are connected to form a nanotube.

where ac−c is the distance between carbon atoms. The indices (n, 0) or (0, m), θ = 0 , correspond to the “zigzag” tube (so-called because of the /\/\/ shape around the circumference, perpendicular to the tube axis). The indices (n, m) with n = m (θ = 30 ) correspond to the “armchair” tube (with a \_/−\_/ shape around the circumference, perpendicular to the tube axis). If one of the two indices n or m is zero, the tube is nonchiral; it is superimposable on its mirror image. A general chiral nanotube (nonsuperimposable on its mirror image) occurs for all other arbitrary angles. Common nanotubes are the armchair (5, 5) and the zigzag (9, 0). Unlike planar graphene, which is a 2D semimetal, carbon nanotubes are either metallic or semiconducting along the tubular axis. In planar graphene, which is regarded as an infinite sheet, artificial periodic boundary conditions are imposed on a macroscopic scale. This is also true for the nanotube axis. However, the periodic boundary condition is imposed for a finite period along the circumference. One revolution around the very small circumference introduces a phase shift of 2π. Electrons are thus confined to a discrete set of energy levels along the tubular axis. Only wave vectors satisfying the relation C k = 2πq (q is an integer) are allowed in the corresponding reciprocal space direction (see Eq. 5.27 and accompanying discussion; let C = N and q = n). This produces a set of one-dimensional subbands and what are known as van Hove singularities in the density of states (if the tube is very long) for a nanotube. Transport can only propagate along parallel channels down the tubular axis, making a nanotube a one-dimensional quantum wire. It is not semimetallic like graphene because the degenerate point is slightly shifted away from the K-point in the BZ owing to the curvature of the tube surface and the ensuing hybridization between the σ∗ and π∗ bands, which modify the dispersion of the conduction band (Blase et al. 1994; Hamada et al. 1992; Saito et al. 1992). These factors, in turn, are dependent on the diameter and helicity. It has been found that metallicity occurs whenever (2n + m) or (2 + 2 m) is an integer multiple of three. Hence, the armchair nanotube is metallic. Metallicity can only be exactly reached in the armchair nanotube. The zigzag

5.4 Tight-Binding Assumptions

nanotubes can be semimetallic or semiconducting with a narrow band gap that is approximately inversely proportional to the tube radius, typically between 0.5 and 1.0 eV. As the diameter of the nanotube increases, the band gap tends to zero, as in graphene. It should be pointed out that, theoretically, if sufficiently short nanotubes electrons are predicted to be confined to a discrete set of energy levels along all three orthogonal directions, such nanotubes could be classified as zero-dimensional quantum dots. Nanometer-width ribbons of graphene, the so-called nanographites, similarly may have armchair or zigzag edges. The edge shape and system size are critical to the electronic properties. Zigzag edges exhibit a localized nonbonding π electronic state with flat bands that result in a very sharp peak near EF in the density of states. This is not found in armchair nanographites. Interestingly, a very recent addition to the family of Dirac materials is honeycomb borophene (the boron analogue of graphene), a 2D polymorph of boron atoms arranged into a triangular crystalline structure. The cohesive energy is enhanced by removing a few boron atoms to create hexagonal holes in the honeycomb lattice. Importantly, however, borophene is found to exhibit novel metallic behavior (due to sp2 hybridization) rather than semimetallic behavior (Kistanov et al. 2018). Each boron atom forms three σ-bonds to its nearest neighbors through sp2 hybridization of the s, px, and py orbitals. The σ-bonds are highly localized between boron atoms, and there is a large energy gap separating the bonding and antibonding σ states. The remaining pz orbitals results in delocalized π-bonds. The bonding and antibonding π-states touch at six points in the Brillouin zone (the Dirac points), which might seem to indicate semimetallic behavior. What makes borophene differ from graphene is the electron deficiency, that is, the number of valence electrons is only three for each boron atom, which is less than the four available bonding orbitals. Because most of valence electrons occupy the three σ-bonding orbitals while the π-bonding states are mostly unoccupied, the Dirac point in honeycomb borophene is about 3.5 eV higher than the Fermi level, and borophene is metallic rather than semimetallic. Density functional calculations on the band structure and the DOS have been made, and optimized SK parameters have been obtained that show excellent agreement with the DFT results. In addition to its novel transport behavior, borophene is being studied for its outstanding physical, chemical, and mechanical properties (Zhu and Zhang 2018).

5.4

Tight-Binding Assumptions

The original LCAO method by Bloch (1928) is difficult to carry out with full rigor because of the large number of complicated integrals that must be computed. Consider perovskites of the general formula ABX3. The Hamiltonian matrix size for these phases is, in general, 45 × 45 since we have the possibility of s, p, and d valence orbital contributions from each of the five atoms of the unit cell. These matrix elements contain the integrals representing the possible types of orbital interactions, namely, σ-, π-, and δ-bonding. The common practice for Brillouin zone sampling is to diagonalize the Hamiltonian matrix using the basis functions for each k-point independently. In principle, an infinite number of k-points are needed. However, in practice, band paths are chosen as a set of specific line segments connecting distinctive BZ points, as well as those connecting a distinctive BZ point to the Γ point. The band path for a given crystal should reflect the symmetry of the crystal and not include redundant symmetrically equivalent paths. Points and line segments with high crystallographic symmetry are generally more important than those of low symmetry, so k-points with high symmetry should be preferentially included in the band path. The number of k-points needed to converge the calculation is system dependent. Smaller real-space unit cells require more

229

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5 The Electronic Level II

k-points than larger real-space unit cells (supercells) because the BZ in reciprocal space becomes smaller when a larger supercell in real space is used, such that the k-points appear less far apart, and less of them become needed. The number of k-points can range from >101 to >104. For example, it is not uncommon to use the high-symmetry points and 20 or 30 k-points along the path between each of these high-symmetry points, totaling, say, over 150 k-points in all, to obtain selfconsistency. In their landmark paper of 1954, John Clarke Slater and George Fred Koster (1926–2012) of the Massachusetts Institute of Technology (MIT) proposed a modified LCAO approach in which the on-site and two-center integrals are replaced by parameters obtained from ab initio calculations performed only at the high-symmetry points using plane waves. The parameters were then used to interpolate the band structures across the first Brillouin zone along the paths between these high-symmetry points. Moreover, a minimal basis set is chosen (see below) using only the valence atomic orbitals contributing to the bonding (Slater and Koster 1954). These two features drastically reduce, respectively, the number of matrices and the number of matrix elements to be evaluated. Additionally, the parameters are tabulated and, if properly chosen, are transferable to other structures. This scheme became known as the TB method (sometimes denoted the SK-TB or just TB method), and it has been applied to metals, semiconductors, and insulators. A major question is the transferability of the SK parameters to phases with differences in structural and chemical environments and their applicability over wide ranges of interatomic separations. For example, the TB formalism is well suited, in principle, for nanostructures because it can be parameterized to incorporate effects due to limited periodicity, strain, morphology, defects, and surfaces, all of which are important in nanosystems. Ab initio DFT is certainly also capable of modeling nanostructures, especially where there are bonding changes with local environments and geometries different from that of the bulk crystal, which are difficult to model with fitted empirical potentials. However, DFT methods are computationally more expensive because they spend a great deal of time forming and solving the Hamiltonian matrix self-consistently. This results in poor scaling with complex systems. The trade-off between the two methods as they are applied to large or complex systems and nanostructures is that while the TB approach using only near-neighbor interactions simplifies the problem to sparse Hamiltonian matrices that can be diagonalized very efficiently, developing a TB parameter set that is accurate and transferable across material systems can be extremely time consuming. Actually, what has been described all along is the TB method. Some important simplifications are made in the TB scheme, which are now stated explicitly. First, in the original SK method, it is assumed that the atomic orbitals and the Bloch sums formed from them are orthogonal. This is not generally true, but the atomic orbitals can be transformed into orthogonal orbitals. One such way this can be accomplished was owing to Swedish physicist Per-Olov Löwdin (1916–2000) (Löwdin 1950). Because of this simplification, the original SK method is sometimes referred to as the linear combination of orthogonalized atomic orbitals (LCOAO) method. Leonard Francis Mattheiss, of the AT&T Bell Laboratories, later made a modification that did away with the Löwdin orthogonalization, transforming the method into a generalized eigenvalue problem, involving overlap integrals in addition to the onsite and exchange integrals (Mattheiss 1972). Second, Slater and Koster treated the potential energy term in the Hamiltonian like that of a diatomic molecule, that is, as being the sum of spherical potentials located on the two atoms at which the atomic orbitals are located. In reality, the potential energy is a sum of potentials (they are not necessarily spherically symmetric) located at all the atoms in the crystal. Hence, there are three-center integrals present. The two-center approximation, however, allows one to formulate the problem using a smaller number of parameters. Matrix elements that are two-center integrals use

5.4 Tight-Binding Assumptions

orbitals that are space quantized with respect to the axis between them, so they have a form that is dependent only on the internuclear separation and the symmetry properties of the atomic orbitals (see below). Ideally, the SK parameters should be somewhat independent of the crystal structure and, thus, transferable from one structure type to another, which is a major advantage of the SK method. Two-center SK parameterizations have also been formulated within the density functional theory (Cohen et al. 1994; Mehl and Papaconstantopoulos 1996), which has become the predominant method of calculating band structures. Slater and Koster introduced notation that clearly distinguishes σ, π, and δ interactions. For example, referring back to Figure 5.2, it can be seen that in the y direction, the px orbitals bond in a π fashion. The third term on the right-hand side of Eq. (5.46) must correspond to the energy of π interactions to first-nearest neighbors. Hence, the integral Ey,y(1 0 0) is replaced with the two-center integral symbolized as (ppπ)1. Making similar substitutions throughout Eq. (5.46) allows that equation to be rewritten as E k = E p,p 0 + 2 ppσ 1 cos k x a + 2 ppπ 1 cos k y a + cos k z a + 2 ppσ 2 cos k x a cos k y a + cos k x a cos kz a + 2 ppπ

2

cos kx a cos k y a + cos kx a cos kz a + 2 cos k y a cos kz a 5 55

Of course, in an actual TB calculation, all such integrals are evaluated only at the high-symmetry points in the BZ. Fitted parameters are then used to interpolate the band structure between these points. In the TB scheme, one uses a minimal basis set of atomic-like functions, often consisting of only a few basis functions per atom, or a localized linear combination of exact wave functions. The latter are called Wannier functions by physicists or Foster-Boyd orbitals by chemists. In practice, the Hamiltonian matrix can be simplified (minimizing the basis set) through our knowledge of the participation of various atomic orbitals. For example, in ABO3 oxides, the O s orbital does not contribute to the valence states and can be omitted. In some perovskite oxides, only the s orbitals of A and the s and p orbitals of B and, in others, only the B d orbitals and the O p orbitals participate. These approximations significantly reduce the size of the matrix.

John Clarke Slater (1900–1976) received his PhD in physics from Harvard University in 1923 under P. W. Bridgeman. He then studied at Cambridge and Copenhagen, returning to Harvard in 1925. In 1929, he introduced the Slater determinant in a paper on the theory of complex spectra. Slater was appointed head of the physics department at MIT by Karl Taylor Compton. He remained at MIT from 1930 to 1966, during which time he started the school of solid-state physics. Among Slater’s PhD students were Nobel laureates Richard Feynman and William Shockley. In World War II, Slater was drawn into the war effort, having been involved in the development of the electromagnetic theory of microwaves, which eventually led to the development of radar. In 1953, Slater and Koster published their now famous simplified LCAO interpolation scheme for determining band structures. After retirement from MIT, Slater was research professor in physics and chemistry at the University of Florida. Slater authored 14 books on a variety of topics from chemical physics to microwaves to quantum theory. Slater is also widely recognized for calculating algorithms, known as STOs, which describe atomic orbitals. He was elected to the United States National Academy of Sciences in 1932 (Primary source: Morse, P.M. (1982). Biographical memoirs of the U.S. National Academy of Sciences. vol. 53, pp. 297–322). (Photo courtesy of AIP Emilio Segrè Visual Archives. Copyright owned by the Massachusetts Institute of Technology Museum. Reproduced with permission.

231

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5 The Electronic Level II

5.5

Qualitative LCAO Band Structures

Unlike the prior two examples – diamond and graphite – compounds have more than one atom type per lattice point. Accordingly, it is usually necessary to consider interactions between atomic orbitals with different angular momentum quantum number (e.g. p–dπ, s–pσ bonding). Slater and Koster gave two-center integrals for s–p, s–d, and p–d interactions expressed in terms of the direction cosines (l, m, n) of the interatomic vector. Examined here is a relatively simple structure that has been of enormous interest for decades – the transition metal oxide perovskite, ABO3. Figure 5.5a shows the real-space unit cell, which contains five atoms. Figure 4.1a shows the BZ for the body centered cubic lattice. It is important to note that the unit cell is not BCC, since the body-centered atom is different from the atoms at the corner positions. Perovskite can also be considered a threedimensional network of vertex-sharing octahedra. Many possible types of interactions between the various atomic orbitals can be deduced. Table 5.1 lists some of the first- and second-nearest neighbor two-center matrix elements that would have to be considered in a rigorous analysis. Nowhere in this chapter has a band structure calculation actually been made. Although a sophisticated mathematical formulation has been presented for some simple cases, it has yet to provide any numerical values for the energies. Since nothing to this point has been quantitative, rather than continue to develop the mathematical treatment, a switch will now be made to a conceptual treatment of the TB method for compounds, in which symmetry and overlap are considered qualitatively. This is a great simplification, which allows one to obtain approximate band structure diagrams, without becoming bogged down in tedious mathematics. In principle, the scheme is applicable to any periodic solid. However, in practice, the types of structures that are most conducive to the methodology tend to have high symmetry, a small number of atoms per unit cell, and/or a covalent framework with linear cation–anion–cation linkages (180 M–X–M bond angles). These are the only types considered here. The utility of the conceptual approach to low-dimensional systems (including layered perovskite-like oxides) has been exhaustively treated in several papers by Whangbo, Canadell, and co-workers (Whangbo and Canadell 1989; Canadell and Whangbo 1991; Rousseau et al. 1996). For transport properties, one is primarily interested in the nature of the bands near the Fermi energy, in which case only the dispersion at the bottom of the conduction band needs evaluation,

(a)

(b)

R A Γ

α

T Δ Σ

S Z M

X

α α

Figure 5.5 The perovskite structure (a) corresponds to a SC real-space Bravais lattice and SC reciprocal lattice. The first BZ for the SC reciprocal lattice is shown in (b).

5.5 Qualitative LCAO Band Structures

Table 5.1

Interatomic matrix elements for the transition metal perovskite oxides.

Atomic orbitals involved

A s, O s A x, A x

Integral

Es,s

Two-center approx.

1 1 , , 0 2 2

Ex,x(1, 0, 0)

(ssσ)1 (ppσ)2

A x, A x

Ex,x(0, 1, 0)

(ppπ)2

A y, A y

Ey,y(1, 0, 0)

(ppπ)2

A y, A y

Ey,y(0, 1, 0)

(ppσ)2

A y, A y

Ey,y(0, 0, 1)

(ppπ)2

A x, O x A y, O x A z, O z B xy, B xy

Ex,x

1 1 , , 0 2 2

1 ppσ 1 − ppπ 2

Ey,x

1 1 , , 0 2 2

1 ppσ 1 − ppπ 2

Ez,z

1 1 , , 0 2 2

(ppπ)1

Exy,xy(1, 0, 0)

(ddπ)2

B xy, B xy

Exy,xy(0, 0, 1)

(ddδ)2

Bz2, B z2

Ez2,z2(0, 0, 1)

(ddσ)2

B x2–y2, B x2–y2

Ez2 − y2,z2 − y2(0, 0, 1)

(ddδ)2

1 0, 0, 2

(sdσ)1

1 2

(pdσ)1

1 , 0 2

(pdπ)1

B z 2, O s B z 2, O z B xy, O x

Ez2,s

Ez2,z 0, 0, Exy,x 0,

1

1

at least for systems with a low electron count. How does one know where the Fermi level is? Well, the relative band energies at the center and corner of the BZ in a band structure diagram usually correspond reasonably well to the order of the localized states in the energy-level diagram for the geometric arrangement at a discrete lattice point (i.e. MO diagram). The Fermi level in the solid can be approximated by filling the orbitals in such a diagram with all the available electrons, two (each of opposite spin) per orbital, and noting where the HOMO falls. This is precisely what was done in Chapter 3 where it was shown how the energy-level diagram for a discrete MX6 octahedral unit approximated the relative band energies of a three-dimensional array of vertex-sharing octahedra, that is, perovskite. It must be stressed that lattice periodicity is not part of an MO picture – the band energies in a solid will vary over the BZ. A generic MO energy-level diagram is in no way equivalent to a band structure diagram! One simply tries to use the knowledge acquired from the MO diagram to determine which of the bands to assess. Returning to the perovskite example, it was revealed in the discussion accompanying Figures 4.7 and 5.5 that, for the transition metal oxides with this structure, the Fermi level lies within the t2g or eg block bands. The focus here will be on just these bands. The five d orbitals from the transition metal atom and the three p orbitals on each of the three oxygen atoms give the 14 TB basis functions listed in Table 5.2. Hence, there is still a 14 × 14 secular equation to solve at various k-points.

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5 The Electronic Level II

Table 5.2

Transition metal d and oxygen p basis functions for perovskite. No.

Metal d

Oxygen p

Position

Function

1

(0, 0, 0)

xy

2

(0, 0, 0)

yz

3

(0, 0, 0)

xz

4

(0, 0, 0)

z2

5

(0, 0, 0)

x2–y2

6 7 8 9 10 11 12 13 14 Metal d

1 a 1, 2 1 a 1, 2 1 a 1, 2 1 a 0, 2 1 a 0, 2 1 a 0, 2 1 a 0, 2 1 a 0, 2 1 a 0, 2 and oxygen p Hamiltonian

x

0, 0

y

0, 0

z

0, 0

x

1, 0

y

1, 0

z

1, 0

x

0, 1

y

0, 1

z

0, 1 matrix elements

H7,1 = 2i(pdπ)1 sin(kxa/2)

H9,1 = 2i(pdπ)1 sin(kya/2)

H11,2 = 2i(pdπ)1 sin(kya/2)

H13,2 = 2i(pdπ)1 sin(kza/2)

H8,3 = 2i(pdπ)1 sin(kxa/2)

H12,3 = 2i(pdπ)1 sin(kza/2)

H6,4 = −i(pdσ)1 sin(kxa/2)

H10,4 = −i(pdσ)1 sin(kya/2) H 6,5 = − 3i pdσ 1 sin k x a 2

H14,4 = 2i(pdσ)1 sin(kza/2) H 10,5 = − 3i pdσ 1 sin k y a 2

This equation contains diagonal transition metal d–d and oxygen p–p Hamiltonian matrix elements. The interest here is in the dispersion of the d bands resulting from σ and π interactions between the d and oxygen p atomic orbitals. These p–d interactions are off-diagonal matrix elements. The mathematical procedure is beyond the scope of the present discussion, but it is possible by means of a unitary transformation to reduce this matrix at certain high-symmetry points in the BZ to a group of 1 × 1 and 2 × 2 submatrices. Taking the B atom as the origin of a Cartesian coordinate system, along any axis there is a dimer unit consisting of a B atom and an O atom at each lattice point. Consequently, each dimer unit has a bonding and antibonding π level, given from Table 5.2: E k = E 0 0 0 ± 2i pdπ 1 sin

ka 2

5 56

5.5 Qualitative LCAO Band Structures

The bonding combinations, E(0 0 0) + 2i(pdπ)1 sin(ka/2), correspond to lower energy states located at the top of the valence band (the fully occupied group of states below the Fermi level). The dispersion of the oxygen p bands can be considered as being driven by the bonding p–d interactions (see Figure 4.7). The CO composed of these bonding combinations has a lower energy and longer wavelength than those of the antibonding combinations. The antibonding combinations, E(0 0 0) − 2i(pdπ)1 sin(ka/2), are higher energy (shorter wavelength) states at the bottom of the conduction band (the vacant or partially occupied bands above the Fermi level). In perovskite transition metal oxides, the antibonding p–d interactions cause the dispersion of the metal d bands, which is of interest here. Qualitative information on the d bandwidth in perovskite can be acquired without carrying out these mathematical operations. This is accomplished simply by evaluating the p–d orbital interactions for some of the special points in the BZ. For now, the main focus is on the π interactions between the metal t2g and oxygen p orbitals. From Figure 4.7, it is expected that the Fermi level will lie in one of the t2g block bands for AIIBIVO3 oxides if B is an early transition metal with six or fewer d electrons. First, Bloch sums of atomic orbitals are constructed for each of the atoms in the basis, for example, at Γ (kx = ky = kz = 0), λ = 2π/0 = ∞. Hence, in perovskite, the metal dxy atomic orbitals forming the Bloch sum at Γ, viewed down the [0 0 1] direction, must have the sign combination shown in Figure 5.6a, where the positive or negative sign of the electron wave function is indicated by the presence or absence of shading. By comparison, at X for the SC lattice, kx = π/a, ky = kz = 0, and λ = 2a. Thus, the Bloch sum is formed from dxy atomic orbitals with the sign combination like Figure 5.6b. Similarly, the Bloch sums of the py and px orbitals at Γ, viewed down the [0 0 1] direction, are constructed from atomic orbitals with the signs given in Figure 5.7a. Those at X are like Figure 5.7b. By simply knowing something about the nature of the interactions between given pairs of Bloch sums for a few k-points, a qualitative band structure diagram can be drawn! Now consider how Bloch sums interact in a symmetry-allowed manner. There are some guiding principles that aid the construction of LCAOs in solids, which are analogous to LCAO-MOs. The first is that the combining atomic orbitals must have the same symmetry about the internuclear axis. Second, the strength of the interactions generally decreases in going from σ to π to δ symmetry. Third, orbitals of very different energies give small interactions. The major principle, however, is from group theory and states that the many-electron wave function in a crystal forms a basis for some irreducible representation of the space group. Essentially, this principle means that the wave function, with a wave vector k, is left invariant under the symmetry (a)

(b)

Figure 5.6 The sign combination required of the dxy atomic orbitals for a Bloch sum at Γ (a) and X (b) viewed down the [0 0 1]. Positive or negative sign is indicated by the presence or absence of shading.

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5 The Electronic Level II

(a)

(b)

Figure 5.7 The sign combinations required of the px (top) and py (bottom) atomic orbitals for a Bloch sum at Γ (a) and at X (b).

elements of the crystal class (e.g. translations, rotations, reflections) or transformed into a new wave function with the same wave vector k.

5.5.1

Illustration 1: Transition Metal Oxides with Vertex-Sharing Octahedra

The aforementioned principles can be used to construct qualitative dispersion curves for the conduction band in transition metal oxides with vertex-sharing octahedra (e.g. perovskite, tungsten bronzes), rather easily. Figure 5.8 shows the metal t2g block and O 2p orbital interactions in transition metal oxides with the perovskite structure. At Γ (kx = ky = kz = 0), no p–d interactions are symmetry allowed, including those with the two axial oxygen atoms above and below the plane of figure, for any of the t2g block orbitals. The reader may want to expand the diagrams in Figure 5.8 to show the signs of the axial p orbitals at the various k-points. By contrast, at X (kx = π/a, ky = kz = 0), an atomic dxy orbital can interact with two equatorial oxygen py orbitals on either side, as can the dxz and oxygen pz orbitals. Another high-symmetry point of the first BZ for the SC lattice is M (kx = ky = π/a, kz = 0). At this point, the dxy orbitals can interact with all four equatorial px and py orbitals. The dxz and dyz orbitals each interact with only two of the equatorial p orbitals and no axial orbitals. The number of interactions per metal atom can be used to plot the band dispersion that results from moving between the high-symmetry points in the BZ, as shown in Figure 5.9. At Γ, the dxy, dxz, and dyz bands are all nonbonding (all three are degenerate) with an energy equal to α. In moving from Γ to X, the dxy has two antibonding interactions per metal atom, as does the dyz and the dxz orbitals. This amounts to a destabilization of each of these bands by an amount that can be

Γ

X

M

y x

y x

y

x

Figure 5.8 The oxygen p–metal d π∗ interactions in a transition metal oxide with vertex-sharing octahedra (e.g. perovskite) at the k-points Γ, X, and M, viewed down [0 0 1]. Top row: dxy. Middle row: dxz. Bottom row: dyz.

0.6 0.5 0.4 0.1 8β

0.0 –0.1

dxy dxy

dxz



dxz

dxy

dyz

dxz dyz dyz

α=0 Γ

X

M

Γ

Figure 5.9 Dispersion curves for the dxy, dxz, and dyz bands in a transition metal oxide with octahedra sharing vertices in three dimensions (e.g. perovskite). Top upper insert: Calculated band dispersion for SrTiO3. Bottom upper insert: Calculated band dispersion for ReO3. The dashed line is the Fermi level.

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5 The Electronic Level II

estimated from Eq. (5.38) as 4β. These two bands are thus degenerate at X. The dyz band is nondispersive as it still has no symmetry-allowed interactions, just like at Γ. In moving from X to M, the dxz and dyz bands now become degenerate, with two antibonding interactions (destabilization = 4β) per metal atom with the oxygen p orbitals. These two bands are lower in energy than the dxy band, which has four antibonding interactions per metal atom and a destabilization of 8β (Eq. 5.38). Figure 5.9 also shows two insets in the upper right corner: the top is the calculated band dispersion in SrTiO3, and the bottom is the calculated band dispersion in ReO3. Bearing in mind that the qualitative t2g dispersion behavior was obtained by only considering the symmetry and overlap between first-nearest neighbor metal and oxygen atoms, the agreement with the calculated dispersion curves is reasonable. Indeed, the d block band dispersion in transition metal perovskite oxides is due almost entirely to p–d hybridization effects. In a real calculation, however, second-nearest neighbor interactions as well as metal–metal and oxygen–oxygen contributions would be included. Second, although the t2g block bands are vacant in the insulating phase SrTiO3, they are partially occupied in the metallic phase ReO3 (note the location of EF in the latter). Nonetheless, the general shape is unchanged, although a wider bandwidth is observed in ReO3. The latter feature is expected for second and third row transition metals, however, because the larger radial extent of the 4d and 5d orbitals should result in a stronger overlap. It can be concluded that the dispersion is independent of the electron count, which is consistent with the earlier claims. Last, ReO3 has the octahedral framework of the SrTiO3 structure minus the 12-coordinate atom in the center of the unit cell. However, the orbitals on this atom are of such high energy (Sr electron configuration = 4s24p65s2) that they do not hybridize with the Ti 3d–O2p bands. In the perovskite structure, this atom simply provides electrons to the system that can occupy the valence or conduction bands. Hence, there is little change to the band dispersion directly resulting from the presence of the A cation.

5.5.2

Illustration 2: Reduced Dimensional Systems

Consider now a layered structure that consists of covalent vertex-sharing single-layer octahedral slabs alternating with ionic rock-salt-like slabs, such as the n = 1 Ruddlesden–Popper or Dion–Jacobsen phases described in Chapter 3. The unit cells are typically tetragonal with an elongated c axis. This constitutes a 2D system in that electronic transport can only occur within the ab plane of the vertex-sharing octahedral layers occurring along the c axis of the unit cell. Electronic conduction does not occur in the ionic rock-salt slabs. How does the reduced dimensionality affect the dispersion relations? Whangbo, Canadell, and co-workers have qualitatively evaluated the band dispersion in such systems in the same way as was done for the three-dimensional perovskite structure in the last section by counting the number of oxygen p orbital contributions present in the COs at certain points in the BZ. The Fermi surface of a low-dimensional transport system has a special topology. For example, because wave vectors in the direction perpendicular to the perovskite layer – the c direction in real space – do not cross the Fermi surface, there are no electrons at the Fermi level having momentum in that direction, and the system is nonmetallic along c. Nevertheless, the axial oxygen contributions still must be considered when constructing the band structure diagram. For a discrete 2D sheet of vertex-sharing octahedra, or for two such sheets separated by, say, nonconducting rock-salt-like slabs, the equatorial and axial oxygen atoms are in two nonequivalent positions, as they have different chemical environments. The equatorial oxygen atoms are in bridging

5.5 Qualitative LCAO Band Structures

Figure 5.10 A layer of octahedra sharing vertices in only two dimensions, as might be found, for example, in perovskite intergrowths like the Ruddlesden–Popper oxides. The axial and equatorial oxygen atoms are not equivalent in that the former belong to single octahedra, whereas the latter are in bridging positions, shared by two octahedra.





α=0 Γ

X

M

Γ

Figure 5.11 Dispersion curves for the dxy, dxz, and dyz bands from a single layer of octahedra sharing vertices in only two dimensions, as in Figure 5.10.

positions – shared by two octahedra – while the axial atoms belong to a single octahedron, as shown in Figure 5.10. It has been shown that the amount of destabilization that results from an axial contribution is one fourth that due to an equatorial contribution (Rousseau et al. 1996). For the dxy orbitals, there are no axial contributions. However, for the dxz and dyz orbitals, there are two axial contributions per metal atom at all three of the k-points of Figure 5.8, which are not shown in that figure. Thus, the band dispersion diagram for this case is obtained simply by shifting the dxz and dyz bands upward by E/2, or β, on the energy axis, as shown in Figure 5.11. The difference between the dispersion of these three d bands in a three-dimensional system, like perovskite, and a 2D system, like the Ruddlesden–Popper series, is that the dxy band in the latter is of the lowest energy (at Γ). Since the easy axis for electronic conduction is the ab plane of the unit cell, the Fermi level is expected to lie in the dxy band for low d electron counts.

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Example 5.3 If the lowest energy point lies at Γ, predict the lowest-lying t2g block band for the n = 3 member of the Ruddlesden–Popper phase, which exhibits out-of-center octahedral distortion in the outer layers (see Fig. 3.27). Discuss the implications on the electronic properties of oxides with low d electron counts. Solution The n = 3 member of the Ruddlesden–Popper phases has triple-layer vertex-sharing octahedral slabs separated by rock-salt-like layers. It is a low-dimensional transport system with nine t2g bands. These can be denoted xz , xz , xz , and so on, where denotes the d band of the central layer of an M–O–M–O–M linkage. Drawing the orbitals in the M–O–M–O–M linkage allows one to count the antibonding contributions in each of these bands. Compared with that of the single-layer octahedral slab, each xy band in the triple-layer slab will have three times as many O p orbital contributions. The xz and yz bands will have three times as many minus two axial contributions. The xz and yz will have two times as many plus one bridging and minus two axial contributions. The xz and yz bands are equal to those of the xz and yz plus two bridging contributions. Since the assumption is that the lowest energy states occur at Γ, only the energy at this one k-point needs to be estimated. Using Canadell’s relation, which states that the energy destabilization of an oxygen p contribution in a bridging position is four times that of an oxygen p orbital contribution in an axial position, it is found that the lowest-lying bands in the undistorted triple layer are three degenerate xy bands. However, shortened apical M–O bonds in the outer layers exhibiting out-of-center distortion raise the energies of those antibonding states, so the xy band of the inner octahedral layer is the lowest band state in the oxide of Figure 3.27. The implications are that conduction electrons confined to the inner layer slab, in oxides with low d electron counts, may be more spatially screened from electron localizing effects such as chemical or structural disorder in the rock-salt-like slabs, as compared with conduction electrons in single-layer slabs.

5.5.3

Illustration 3: Transition Metal Monoxides with Edge-Sharing Octahedra

Now look at the transition metal monoxides with the rock-salt structure. Since the rock-salt structure is a three-dimensional network of edge-sharing MX6 octahedra, in which the metal may possess an incomplete d shell, it can be concluded that the Fermi level should reside in the metal t2g or eg block bands. Considering the similarity of this structure to the transition metal perovskite oxides, one might expect to see the same dispersion curves for the d bands. However, because of the proximity of the metal atoms to each other, neglecting the metal d–d interactions in the monoxides produces dispersion curves that are grossly in error. The d–d interactions in the Bloch sums are important in affecting the dispersion of the d–p COs. To begin, the dispersion of one of the t2g block bands, the dxz band, is examined. The dxz and pz Bloch sums can be drawn at Γ to include the atomic orbitals in the face-centered positions, as shown in Figure 5.12, where the Cartesian coordinate system shown has been chosen. The rock-salt structure can be considered as two interpenetrating FCC lattices, one of cations and one of anions, displaced by one half of a unit cell dimension along the 1 0 0 direction. This can be done with the two Bloch sums of Figure 5.12 as shown in Figure 5.13. At Γ, the dxz and pz Bloch sum interactions are not symmetry allowed. Nor is the hybridization between the dxz and py Bloch sums. The dxz orbital at the origin does, however, have a π∗

5.5 Qualitative LCAO Band Structures

y x

Figure 5.12 The sign combinations required of the atomic orbitals at Γ (kx = ky = kz = 0, λ = ∞) for the dxz Bloch sum (left) and pz Bloch sum in a single plane viewed down [0 0 1].

y

x

Figure 5.13 A schematic illustrating the superposition of the dxz and pz Bloch sums at Γ viewed down [0 0 1]. The only symmetry-allowed dxz–pz interactions at Γ are those between the dxz orbitals and the axial oxygen pz orbitals directly above and below them (not shown). Thus, the dispersion at Γ is driven primarily by d–d interactions.

antibonding interaction with the axial px orbitals (not shown) directly above and below it. In other words, the dxz orbital at the origin only interacts with two of the six first-nearest neighbor oxygen atoms. In the rock-salt structure, there are d–d orbital interactions between the 12 second-nearest neighbor metal atoms that must also be included. Figure 5.13 shows four of these dxz–dxz interactions, two of which are π∗ antibonding and two are δ bonding. The eight remaining d–d interactions are much weaker because of the poorer overlap between the orbitals on these atoms with the orbital at the origin. Hence, at Γ there is a roughly nonbonding dxz–dxz net interaction plus an antibonding p–d contribution. The interactions at X are shown in Figure 5.14. The equatorial px and py orbitals do not have the correct symmetry for interaction with the dxz orbitals. However, there are π∗ antibonding interactions between the dxz and equatorial pz orbitals (shown) and axial px orbitals (not shown).

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5 The Electronic Level II

y x

Figure 5.14 A schematic illustration of the superposition of the dxz and pz Bloch sums at X viewed down [0 0 1]. The dispersion at X is driven primarily by π-type and δ-type d–d interactions.

y x

Figure 5.15 A schematic illustration of the superposition of the dxz and pz Bloch sums at W viewed down [0 0 1]. The net d–d interaction is π-bonding, while the p–d interactions are antibonding.

The bonding interactions cause the dispersion of the p bands and are not considered here. The overlap between the dxz and equatorial pz orbitals in Figure 5.14 is poor, so these orbitals interact weakly. The dxz–dxz interaction at X is both π and δ bonding. As at Γ, these are much stronger than the p–d interactions as well as the eight remaining d–d interactions. The dxz band is thus lower in energy relative to Γ. Another symmetry point in the first BZ for the FCC lattice is W (kx = 2π/a, ky = π/a, kz = 0). At this point, λ = a along x and 2a along y. The Bloch sums for this k-point are shown in Figure 5.15. The p–d interactions are the same as those at X. However, the net d–d interaction is less bonding than it is at X. The dxz band is higher in energy at W, relative to X. It is also slightly higher than at Γ, owing to a greater number of antibonding p–d interactions. This same procedure can be followed to investigate the dispersion of one of the eg block bands, for example, the dx2 − y2 band. The interactions at Γ, X, and W are shown in Figure 5.16. At point Γ (kx = ky = kz = 0), λ = 2π/0 = ∞, all d–d interactions are bonding. The net interaction between the dx2−y2 and py orbitals is nonbonding as is the dx2−y2 – px interaction (not shown). The pz orbitals are not of the correct symmetry for interaction.

5.5 Qualitative LCAO Band Structures

Γ

X

W

y

x

Figure 5.16 The dx2−y2 – dx2−y2 and p–d interactions at Γ, X, and W, in the transition metal monoxides with the rock-salt structure, viewed down [0 0 1].

At X (kx = 2π/a, ky = kz = 0), λ = a along the x axis and ∞ along the y axis. The net d–d interaction is roughly nonbonding, while the d–py and d–pz interactions are not symmetry allowed. The four d–px interactions, however, are antibonding. Thus, the dx2−y2 band is raised in energy at X, relative to Γ. At W (kx = 2π/a, ky = π/a, kz = 0), λ = a along the x axis and 2a along the y axis. The net d–d interaction is antibonding. None of the d–p interactions are symmetry allowed. In moving from W to X, the dx2−y2 band may be either nondispersive or slightly raised in energy. The actual bandwidth will depend on the strength of the M 3d–M 3d interactions in comparison with the M 3d–O 2p interactions. The top curve on the left-hand side of Figure 5.17 shows our qualitative dispersion curves for the dx2−y2 band and dxz band from Γ to X to W. Note that, although there are no numerical values for the energies, the dxz band has been placed lower than the dx2−y2 band, that is, the zero of energy for each band is not equal. A cubic crystal field would be expected to have this effect in a solid, just as in a molecule. The top curves in each diagram on the right-hand side show the corresponding calculated dispersion curves in TiO and VO.

5.5.4 Corollary With the conceptual approach just outlined, it is possible to acquire a reasonably accurate picture of the shape of the dispersion curves, which helps in the understanding of exactly what it is that is viewed when an actual band structure diagram is inspected. In the previous sections, the dispersion of the curves in the conduction band was examined. The width of the conduction band directly influences the magnitude of the electrical conductivity. Electrons in wide bands have higher velocities and mobilities than those in narrow bands. Hence, wide-band metals have higher electrical conductivities. Had it been chosen, the dispersion curves, comprising the valence band, could have been evaluated as well. The most striking feature of the band structures of ionic compounds (e.g. KCl) is the narrow bandwidth of the valence band. With the electron transfer from cation to anion comes contraction of the lower-lying filled cation orbitals (owing to a higher effective nuclear charge). For example, K+ (3s23p63d10) has spatially smaller orbitals than K (3s23p63d104s1). Thus, the overlap integral between orbitals on nearest neighbors (e.g. K+ 3d and Cl− 3s and 3p), β, is small, which leads to very narrow valence bands.

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0.9

dx2–y2

0.8 dx2–y2 0.7

dxz

0.6

dx2–y2

0.8 dxz

0.7 dxz 0.6 0.5

Γ

X

W

Figure 5.17 Dispersion curves for the dx2−y2 and dxz bands in the transition metal monoxides. Upper right: Calculated band dispersions for TiO. The Fermi level is indicated by the dashed line. Bottom right: Calculated band dispersions for VO.

Care should be taken not to overplay the significance of the results from the last three sections, however. This treatment has not provided any numerical estimates for the bandwidth, the extent of band filling (occupancy), or the magnitude of the band gap, all of which are extremely important properties with regard to electronic transport behavior. This will be discussed in Section 6.3.1. Knowledge of the bandwidth is especially critical with transition metal compounds. It will be demonstrated in Chapter 7 that nonmetallic behavior can be observed in a metal if the one-electron bandwidth, W, is less than the Coulomb interaction energy, U, between two electrons at the same bonding site of a CO. For that matter, in regard to predicting the type of electrical behavior, one has to be careful not to place excessive credence on actual electronic structure calculations that invoke the independent electron approximation. One-electron band theory predicts metallic behavior in all of the transition metal monoxides, although it is only observed in the case of TiO! The other oxides, NiO, CoO, MnO, FeO, and VO, are all insulating, despite the fact that the Fermi level falls in a partially filled band. In the insulating phases, the Coulomb interaction energy is over 4 eV, whereas the bandwidths have been found to be approximately 3 eV, that is, U > W.

5.6

Total Energy Tight-Binding Calculations

Although the TB method was originally intended for explaining electronic spectra and structure, Chadi showed it may also be applied to total energy calculations (Chadi 1978). If one wishes to describe total energies, calculating the electronic eigenvalues of the occupied valence states is insufficient. Total energy is composed of five contributions: the kinetic energy of the electrons, the

5.6 Total Energy Tight-Binding Calculations

attraction between the electrons and the nuclei, the repulsion between the electrons, the repulsion between the nuclei, and the kinetic energy of the nuclei. Due to the large masses of the nuclei, their kinetic energy is neglected with the Born–Oppenheimer approximation (in HF and DFT calculations), and only the electrostatic energy of the nuclei is considered. Thus, for a fixed nuclear geometry, the first four terms combined represent the total energy, while the first three of these terms taken together constitute the total electronic energy. Total energy calculations can be used for exploring bulk properties such as lattice constants, cohesive energies, elastic constants, bulk moduli, and phase transitions, as well as surface structure. Because of its ease of implementation, low computational workload for large systems, and relatively good reliability, the TB scheme is a good compromise between ab initio simulations and modelpotential calculations (Masuda-Jindo 2001). As discussed at the end of Chapter 4, the true ground state total energy is accessible, in principle, via density functional theory if one knows the exact exchange–correlation functional. The total energy of a solid with fixed or stationary nuclei can be written as: E tot = EKE + Eie + Eii + Eee

5 57

where EKE is kinetic energy of a system of electrons, Eie is the ion–electron Coulomb interaction energy, Eii is the ion–ion electrostatic interaction energy, and Eee represents the electron–electron interactions (exchange and correlation). In this context, the word “ion” is taken to mean either true cations and anions in an ionic crystal or the positively charged nuclei in a homogeneous negatively charged (electronic) background. In the Kohn–Sham DFT method, EKE is the kinetic energy of an artificial system of noninteracting electrons with density ρ. Since this is computed for a single Slater determinant wave function, we chose to symbolize this quantity as TS(ρ) (T signifies translational kinetic energy). Likewise, the Eee term in KS-DFT is broken into the Hartree energy (EH) and the exchange–correlation energy (Exc) as Eee = EH + Exc = [EKE − TS(ρ)] + [Eee – EH]. Although DFT calculations require iteration to self-consistency, in TB DFT total energy calculations, the Schrödinger equation only has to be solved once, and no self-consistency is required. The four individual contributions in Eq. (5.57) can be regrouped into two terms in a TB total energy expression. Particularly useful for total energy calculations is the DFT-TB method, although the parametrization process is quite difficult. We will name the first term the band energy, Eband, which is given by the sum over the occupied orbital energies (up to the Fermi level), εi, derived from the diagonalization of the electronic Hamiltonian. In the one-electron methodology, Eband is equal to EKE + Eie + 2Eee, where the factor 2 comes from double counting the Coulomb repulsion energy between electrons by summing over the eigenvalues. The band energy term is attractive, in that valence band energy levels are lower in energy than the atomic orbitals from which they are derived, and it is thus responsible for the cohesion of a solid. The second term in our TB total energy expression accounts for the short-range two-particle (pairwise) Coulomb potential energy between the nuclei (not contained in Eband), and it also contains a correction for double counting the electrons in the band energy. It is equal to Epair = Eii – Eee. It should be noted, however, that in DFT the ion–ion repulsion is often but not always assumed to be simple pairwise. Symbolically, the new total energy expression can, therefore, be written as E tot tb = E band + E pair

5 58

Some authors keep separate the ion–ion repulsion and the electron–electron interactions in our Epair, thus having three terms in the expression for the total energy. From the pairwise term some important vibrational data, which allow the estimation of certain mechanical properties of solids, can be determined. For example, in Chapter 10, it will be shown that the Young’s modulus

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(an elastic constant) of a solid can be approximated by treating the interatomic bonds as springs. The restoring force for small displacements of the atoms from their equilibrium positions (the stiffness of the spring) is given by the first derivative of the potential energy between a pair of ions, U, with respect to the interatomic distance, dU/dr. Of course, one should take the derivative of Eq. (5.58), the total energy, in order to include the contribution of the band term to the force, as well. Even IR and Raman spectra can be deduced for small clusters of atoms from TB total energy minimizations, as well as ground state geometries and binding energies per atom.

Practice Problems *1

Show that Eqs. (5.52) and (5.53) are equivalent expressions. Hint: eika = cos(ka) + i sin(ka).

2

In the LCAO scheme, which integral represents the energy of an isolated atomic orbital? Which integral gives the energy of interaction between neighboring atoms?

3

Why are projection operators not required to generate Bloch sums for crystalline solids?

*4

List the principles that aid in the construction of qualitative band structure diagrams, which are analogous to the construction of qualitative MO diagrams.

5

True or false? The tight-binding scheme is an interpolation method that requires one to perform actual ab initio calculations only at a few high-symmetry points.

6

What were the key assumptions made in the original SK method?



For solutions, see Appendix D.

References Albright, T.A., Burdett, J.K., and Whango, M.-H. (1985). Orbital Interactions in Chemistry. New York: Wiley Interscience. Blase, X., Benedict, L.X., Shirley, E.L., and Louie, S.G. (1994). Phys. Rev. Lett. 72: 1878. Bloch, F. (1928). Z. Phys. 52: 555. Canadell, E. and Whangbo, M.-H. (1991). Chem. Rev. 91: 965. Cao, Y., Fatemi, V., Fang, S., Watanabe, K., Taniguchi, T., Kaxiras, E., and Jarillo-Herrero, P. (2018). Nature 556: 43–50. Chadi, D. (1978). J. Phys. Rev. Lett. 41: 1062. Chapman, J., Su, Y., Howard, C.A., Kundys, D., Grigorenko, A.N., Guinea, F., Geim, A.K., Grigorieva, I.V., and Nair, R.R. (2016). Sci. Rep. 6: 23254. Cohen, R.E., Mehl, M.J., and Papaconstantopoulos, D.A. (1994). Phys. Rev. B50: 14694. Cotton, F.A. (1990). Chemical Applications of Group Theory, 3e. New York: Wiley. Elias, D.C., Nair, R.R., Mohiuddin, M.G., Morozov, S.V., Blake, P., Halsall, M.P., Ferrari, A.C., Boukhvalov, D.W., Katsnelso, M.I., Geim, A.K., and Novoselov, K.S. (2009). Science 323: 610. Foulkes, W. M.C., E. Pavarini, E. Koch, J. van den Brink, and G. Sawatzky (eds.) Quantum Materials: Experiments and Theory Modeling and Simulation Vol. 6 Forschungszentrum Jülich, 2016, ISBN 978-395806-159-0.

References

Hamada, N., Sawada, S., and Oshiyama, A. (1992). Phys. Rev. Lett. 68: 1579. Harrison, W.A. (1989). Electronic Structure and the Properties of Solids. New York: Dover Publications, Inc. Kistanov, A.A., Cai, Y., Zhou, K., Srikanth, N., Dmitriev, S.V., and Zhang, Y.-W. (2018). Nanoscale 10: 1403. Koskinen, P. and Mäkinen, V. (2009). Comput. Mater. Sci. 47: 237–253. Löwdin, P.O. (1950). J. Chem. Phys. 18: 365. Martin, R. (2004). Electronic Structures: Basic Theory and Practical Methods. Cambridge: Cambridge University Press. Masuda-Jindo, K. (2001). Mater. Trans. 42: 979. Mattheiss, L.F. (1972). Phys. Rev. B5: 290. Mehl, M.J. and Papaconstantopoulos, D.A. (1996). Phys. Rev. B54: 4519. Rousseau, R., Palacín, M.R., Gómez-Romero, P., and Canadell, E. (1996). Inorg. Chem. 35: 1179. Saito, R., Fujita, M., Dresselhaus, G., and Dresselhaus, M.S. (1992). Appl. Phys. Lett. 60: 2204. Slater, J.C. and Koster, G.F. (1954). Phys. Rev. 94: 1498. Whangbo, M.-H. and Canadell, E. (1989). Acc. Chem. Res. 22: 375. Zhu, L. and Zhang, T. (2018). Solid State Commun. 282: 50–54.

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6 Transport Properties The phenomena investigated in this chapter include thermal conduction, electrical conduction, and atomic/ionic transport in solids. These processes can be described by the set of linear phenomenological equations listed in Table 6.1. The equations relate a response or flux (diffusion of atoms, electrons, etc.) to a driving force (electric field, thermal gradients, etc.). Note that there is no need to rely on the details of any microscopic mechanism in order to describe macroscopic transport behavior although there will be a need to learn about mathematical quantities called tensors. The phenomenological equations can be regarded as macroscopic, or continuum-level, descriptions. An equation is said to be phenomenological if it describes a complex system in a simplified manner. Of primary interest in this chapter, however, will be the underlying microscopic, or atomistic-level, descriptions that explain the behavior by using principles of atomic and electron dynamics.

6.1

An Introduction to Tensors

Single crystals are generally not isotropic. It can be expected then that the physical properties of single crystals will be anisotropic or dependent on the direction in which they are measured. Table 6.1 reveals that fluxes and driving forces describing the transport properties are related. Their magnitudes are linearly proportional. However, only for a cubic monocrystal or polycrystalline aggregate with a random crystallite orientation are the directions of the flux and driving force parallel through the macroscopic sample. They are not in general, and κ, σ, and D are not simple scalars at all. They rather look like 3 × 3 matrices and are actually what one calls tensors, of rank 2, to be defined soon. They could be noted κ, σ, and D (the gradient symbols ∇ correspond to vectors: ∇ or ∇). They are necessary to properly explain anisotropic transport properties. The word “tensor” was first used, in the present sense of mathematical representations of physical properties (elastic, thermal, electric, magnetic), in 1898 by Woldemar Voigt (1850–1919), a professor of theoretical physics at Göttingen (Voigt 1898). Appendix C is devoted to an as simple as possible introduction to tensors, of rank zero, one, two, etc. Yet, a few words are in order here. The number of components necessary to describe a tensor is given by dn where d is the number of dimensions in space and n is called the rank of the tensor. For example, a zero-rank tensor is a scalar, which has a 30 = 1 component. A first-rank tensor is a vector; it has three components in three-dimensional (3D) space (31), which are the projections of the vector along the axes of some reference frame, e.g. the mutually perpendicular unitary axes of a Cartesian coordinate system. The magnitude and direction of a Principles of Inorganic Materials Design, Third Edition. John N. Lalena, David A. Cleary, and Olivier B. M. Hardouin Duparc. © 2020 John Wiley & Sons, Inc. Published 2020 by John Wiley & Sons, Inc.

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Table 6.1

Examples of physical properties relating vector fluxes and their driving forces.

Flux

Driving force

Physical property tensor

Equation

Heat flow, q

Temperature gradient, ∇T

Thermal conductivity, κ

q = −κ∇T

Current density, j

Electric field, E (= −∇V)

Electrical conductivity, σ

j = σE = σ∇V

Diffusional flux, J

Concentration gradient, ∇c

Diffusion constant, D

J = −D∇c

A flux is defined as the rate of flow of a property per unit area, having the dimensions (quantity) (time)−1 (area)−1.

physical quantity can obviously not depend on the arbitrary choice of such a reference frame. Yet, a vector is practically defined by specifying its components from projections onto the individual axes of such a chosen reference system. A vector can be defined by the way these components change, or transform, as the reference system is changed by a rotation or reflection. The coordinates transform in the inverse way with respect to the transformation of the axis vectors of the reference system. Noting the coordinate rotation matrix as (mij), corresponding to the coordinate rotation x i = mijxj, i = 1, 2, 3, a vector V is defined as a set of three quantities V1, V2, V3, that transform as the coordinates xj: V i = mijVj, i = 1, 2, 3 (Einstein’s summation convention on the repeated dummy index, here j, being understood). The rotation matrix can, for instance, be written as in Eq. (1.7), where one sees that the nine components mij are not independent of one another and mij mji. Woldemar Voigt (1850–1919) entered the University of Leipzig in 1869. His studies were interrupted in 1870 by service in the Franco-Prussian War. He resumed them in 1871, at Königsberg, and became the last doctoral student of Franz Ernst Neumann (1798–1895, Carl Samuel Weiss’ most gifted student in Berlin). WV defended his PhD thesis in 1874 (on the measure of the elastic constants of rock salt). Neumann was 76 and did not help much, so WV had to move back as a school teacher to Leipzig and accept an extraordinary (i.e. not paid) professorship in Königsberg before eventually getting in Göttingen in 1883 the status of ordinary professor for theoretical (mathematical) physics and director of the mathematical physics department. As William Thomson (Lord Kelvin) at about the same time, he was very much obsessed with trying to understand connections between light and crystals and the elastic properties of ether, the then believed medium for the propagation of light, without much direct success of course. Voigt always tried to draw mathematical consequences from general principles that were based on experience, and, just as W. Thomson and Pierre Curie, among others, he supported the fact that accurate measurements, struggling for more decimals, are essential for the advance of physics. Besides his influence via the Department of Mathematical Physics, he had significantly improved in Göttingen (where he was succeeded by Peter Debye then Max Born under whom the department was renamed Institute for Theoretical Physics) and was essentially successful in the study of crystalline properties and their relations with crystalline symmetries, in the spirit of his master F.E. Neumann. He originated the name tensor in its current meaning. He was also a specialist of Bach’s church cantatas and used to conduct a choir to perform them. He wrote: “In my opinion, the music of physical laws does not sound in richest chords as in crystal physics.” (Photo Public domain, Wiki source.)

A tensor, in general, is an object with many components that look and behave like components of ordinary vectors. An nth-rank tensor (or “tensor of rank n”) with n ≥ 2 cannot be drawn geometrically like a vector can, but it can be treated algebraically with the same principle as a vector. Tensors are defined by the transformation laws of their components. These transformation laws need not be considered here, but they are explicated in Appendix C.

6.1 An Introduction to Tensors

For us, being materials scientists, a second-rank tensor is a physical quantity, a physical property, like the transport properties studied in this chapter. It has nine components (32) written in (3 × 3) matrix-like notation that very much resembles that of a transformation matrix for a vector between two sets of reference axes described in the preceding paragraph. However, it does not relate two sets of axes the way a transformation matrix does. It usually relates two physical vectors: a driving force or stimulus vector F imposed to a crystal sample and the physical response vector Q. (Note that some physical second-rank tensors do not relate two vectors but relate another second-rank tensor with a scalar. For example, the coefficients of thermal expansion for an anisotropic crystal relate the response strain, a second-rank tensor, to a temperature change, a zero-rank tensor, or scalar). For a general second-rank tensor τ (or τ) that relates two vectors, F and Q, one has in a coordinate system with three orthogonal axes Q1 = τ11 F 1 + τ12 F 2 + τ13 F 3 Q2 = τ21 F 1 + τ22 F 2 + τ23 F 3 Q3 = τ31 F 1 + τ32 F 2 + τ33 F 3

61

The tensor array, with components τij, is thus written in matrix-like notation as τ11

τ12

τ13

τ21 τ31

τ22 τ32

τ23 τ33

62

Note that each component of Q, the physical response vector, in Eq. (6.1) is related to all three components of F, the driving force, which is an applied or stimulus vector. Thus, each component τij of the second-rank tensor τ in Eq. (6.1) is associated with two axes: for example, τ32 gives the component of Q parallel to the third axis when F is parallel to the second axis. Fortunately, several simplifications can be made (Nye 1957). Transport phenomena, for example, are processes whereby systems transition from a state of nonequilibrium to a state of equilibrium. Thus, they fall within the realm of irreversible or nonequilibrium thermodynamics. Onsager’s theorem, which is central to nonequilibrium thermodynamics, dictates that as a consequence of time-reversible symmetry, the off-diagonal elements of a transport property tensor are symmetrical, that is, τij = τji (for antisymmetric tensors, τij = −τji). This is known as a reciprocal relation (Onsager 1931); thus, transport properties are symmetrical second-rank tensors. The Norwegian physical chemist Lars Onsager (1903–1976) was awarded the 1968 Nobel Prize in Chemistry for reciprocal relations. Using the reciprocal relations, the tensor τ in Eq. (6.2) can be rewritten as (we tend to follow Frederick Nye, vide infra, and use square brackets, [ ], for tensors and round brackets, ( ), for matrices) τ11

τ12

τ13

τ12 τ13

τ22 τ23

τ23 τ33

63

Note the very important change in subscripts from Eqs. (6.2) to (6.3), leaving merely six independent components. Finally, symmetrical tensors can also be diagonalized. For second-rank tensors, three mutually perpendicular unit vectors can be found that define three principal axes such that if these axes are used as coordinate axes, the tensor arrays are diagonal. This gives τ11 0 0

0 τ22 0

0 0 τ33

64

251

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6 Transport Properties

Because of this further simplification, only three independent quantities in a symmetrical secondrank tensor are needed to define the magnitudes of the principal components. The other three components (from the initial six), however, are still needed to specify the directions of the axes with respect to the original coordinate system. In the case of physical properties, crystal symmetry imposes even more restrictions on the number of independent components (Nye 1957). A tensor representing a physical property must be invariant with regard to every symmetry operation of the given crystal class. Tensors that must conform to the crystal symmetry in this way are called matter tensors. The orientation of the principal axes of a matter tensor must also be consistent with the crystal symmetry. The principal axes of crystals with orthogonal crystallographic axes will be parallel to the crystallographic axes. In the monoclinic system, the x and z crystallographic axes are orthogonal to each other but nonorthogonal to y. For triclinic crystals, there are no fixed relations between either the principal axes or crystallographic axes and no restrictions on the directions of the principal axes. The effects of crystal symmetry on symmetrical second-rank matter tensors are given below. For cubic crystals and nontextured polycrystals: τ11 0 0

0 τ11 0

0 0 τ11

65

For tetragonal, trigonal, and hexagonal crystals: τ11 0

0 τ11

0 0

0

0

τ33

66

For orthorhombic crystals: τ11 0 0

0 τ22 0

0 0 τ33

67

For monoclinic crystals: τ11 0

0 τ22

τ13 0

τ13

0

τ33

68

For triclinic crystals: τ11 τ12 τ13

τ12 τ22 τ23

τ13 τ23 τ33

69

The diagonal elements in the above tensors follow from the indistinguishability of the axes in their respective crystal classes. For example, if the normalized unit length of the three crystallographic axes in each crystal class is denoted with the letters a, b, c and denote the angles between these three axes with the Greek letters α, β, γ, it can be seen that there are three indistinguishable orthonormal axes (orthogonal axes normalized to the same unit length) in the cubic class (a = b = c; α = β = γ = 90 ), two orthonormal axes in the tetragonal class (a = b c; α = β = γ = 90 ), two orthonormal axes in the trigonal class (a = b = c; α = β = γ 90 ), two orthonormal axes in the hexagonal class (a = b c;

6.1 An Introduction to Tensors

α = β = 90 , γ = 120 ), no orthonormal axes in the orthorhombic class (a b c; α = β = γ = 90 ), no orthonormal axes in the monoclinic class (a b c; α = γ = 90 , β 90 ), and no orthonormal axes in the triclinic class (a b c; α β γ 90 ). The off-diagonal elements in the monoclinic and triclinic crystals give the additional components necessary to specify the tensor. Notice that a cubic single crystal is isometric and so has isotropic properties. The same is also true for polycrystals with a random crystallite orientation (e.g. powders), regardless of the crystal class to which the substance belongs. If anisotropic grains are randomly oriented, the macroscopic sample loses any anisotropy in any property. Thus, a single scalar quantity is sufficient for describing the conductivity in monocrystals of the cubic class and nontextured polycrystalline materials. It is sometimes possible to use the anisotropy in certain physical properties advantageously during fabrication processes. For example, the magnetic susceptibility, which describes the magnetic response of a substance to an applied magnetic field, is a second-rank matter tensor. It is the proportionality constant between the magnetization of the substance and the applied field strength. When placed in a magnetic field, a crystal with an anisotropic magnetic susceptibility will rotate to an angle in order to minimize the magnetic free energy density. This magnetic alignment behavior can aid in texture control of ceramics and clays, if the particles are sufficiently dispersed in order to minimize the particle–particle interactions, which can be accomplished with slip casting or other powder suspension process. The route has been used to prepare many bulk substances and thin films, including some with only a small anisotropic paramagnetic or diamagnetic susceptibility, such as gadolinium barium copper oxide, zinc oxide, and titanium dioxide (anatase), with textured (grain-aligned) microstructures and correspondingly improved physical properties. The components of a symmetrical second-rank tensor transform like the coefficients of the general equation of a second-degree surface (a quadric). The quadric is thus a useful way of visualizing the symmetry of the tensor. In general, the length of any radius vector of the representation quadric is equal to the reciprocal of the square root of the magnitude of the tensor in that direction. Hence, if all three of the principal coefficients of a quadric are positive, an ellipsoid becomes the geometrical representation of a symmetrical second-rank tensor property. The ellipsoid is the quadric for the tensors representing electric polarization, thermal and electrical conductivity, and optical properties. The exact shape of the property and its quadric and the orientation of the quadric with respect to the crystal are restricted by the point group symmetry. For a cubic crystal, the ellipsoid is a sphere. For the tetragonal, hexagonal, and trigonal classes, the ellipsoid is uniaxial. For orthorhombic, monoclinic, and triclinic, the ellipsoid is triaxial. Example 6.1 Write the expression for the general equation of the quadric representing a symmetrical second-rank tensor in a triclinic crystal. Solution A representation quadric may be represented as Sij x i x j

1= i, j

where Sij are the coefficients of the representation quadric that transform like the symmetrical second-rank tensor coefficients. Since a symmetrical second-rank tensor has τij = τji, correspondingly, it can be written Sij = Sji. Using this knowledge and Eq. (6.9), the equation can be simply expanded by performing the summations to obtain 1 = S11 x 21 + S22 x 22 + S33 x 23 + 2S23 x 2 x 3 + 2S31 x 3 x 1 + 2S12 x 1 x 2

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This is the equation for an ellipsoid that, referred to an orthogonal system of coordinates (principal axes x1 = x, x2 = y, and x3 = z), may be written as 1 = S11 x 2 + S22 y2 + S33 z2 + 2S23 yz + 2S31 zx + 2S12 xy where Sij are six linearly independent coefficients. If one of the principal coefficients is negative, then the quadric is a hyperboloid of one sheet. If two of the principal coefficients are negative, the quadric is a hyperboloid of two sheets (Nye 1957). The quadric for the coefficient of the thermal expansion tensor, for example, can be a hyperboloid, if there is an expansion in one direction and a contraction in another. If all three of the principal coefficients are negative, the quadric is an imaginary ellipsoid, which is the case with many paramagnetic and diamagnetic susceptibilities. An ellipsoid has inherent symmetry m. The relevant features are as follows: 1) It is centrosymmetric. 2) It has three mirror planes perpendicular to the principal directions. 3) Its twofold rotation axes are parallel to the principal directions. Typically, one sees normalized principal axes, a mean shape factor, and a mean anisotropy degree specified for an ellipsoid. Often, the ellipsoid is characterized by its overall shape with respect to its three radii. Oblate ellipsoids (disk shaped) have a = b > c; prolate ellipsoids (cigar shaped) have a = b < c; and scalene ellipsoids have three unequal sides (a > b > c). In a sphere, of course, a = b = c.

John Frederick Nye (1923–2019) was a professor of physics at the University of Bristol, England. He earned his PhD from Cambridge in 1948 where, with Sir Lawrence Bragg and Egon Orowan, he helped to develop the bubble-raft model of a metal and showed how the photoelastic effect could be used to study arrays of dislocations in crystals. While in the mineralogy department at Cambridge University (1949–1951) and later at Bell Telephone Laboratories (1952–1953), Nye wrote Physical Properties of Crystals: Their Representation by Tensors and Matrices, which became the definitive textbook. He joined the Physics Department at the University of Bristol in 1953, becoming emeritus in 1988. Nye applied physics to glaciology, serving as president of both the International Glaciological Society and the International Commission of Snow and Ice. In 1974, Nye discovered the existence of dislocations in propagating wavefronts and in 1983 their analogues, polarization singularities, in electromagnetic waves. These discoveries, along with others, have developed into a new branch of optics, called singular optics, described in his book Natural Focusing and Fine Structure of Light (1999). Nye was elected a Fellow of the Royal Society in 1976. (Source: J. F. Nye, personal communication, July 17, 2003.) (Photo courtesy of H. H. Wills Physics Laboratory, University of Bristol. Reproduced with permission.)

6.2 Microscopic Theory of Electrical Transport in Ceramics: The Role of Point Defects In Section 6.1, we focused solely on the directionality aspects of transport behavior, relating the responses or fluxes (diffusion of atoms, electrons, etc.) to the driving forces (electric field, thermal gradients, etc.) along the various axes of a coordinate system. We did not rely on the details of any microscopic explanation in our description of macroscopic transport behavior. We shall now

6.2 Microscopic Theory of Electrical Transport in Ceramics: The Role of Point Defects

present an atomistic description specifically for electrical conduction in oxides and briefly for nitrides, deferring the analogous discussions for the other transport properties until their respective sections in this chapter. By delving into the microscopic theory of electrical transport in ceramics, the critical role that crystalline point defects play in influencing a macroscopic material property will be very effectively illustrated. Due to the critical nature of this topic, the reader is encouraged to not skip ahead and to read all of Section 6.2 from start to finish, perhaps more than once. Electrical conduction in both bulk ceramics and crystalline semiconductors profoundly depends on the concentrations of point defects, particularly dopants (intentionally added impurities) and vacancies, to such an extent that methods exist for determining impurity concentrations from electrical measurements. However, while electrical conductivity, Hall effect, thermoelectric power, and ionic and self-diffusion measurements are capable of detecting charge carrier (electron or hole) and atomic point defect densities down to 108/cm3, individually, they suffer from a lack of specificity. They must normally be used in conjunction with one another in order to unambiguously identify the type of defect. We will not be too concerned with that aspect here, but rather we simply wish to show the manner in which microscopic defects control macroscopic electrical transport and how manipulating them is thereby a powerful tool in tailoring materials behavior. There are two types of electrical conduction that may occur in materials – electronic (electrons and/or holes are the charge carriers) and ionic (ions carry the current). For the time being, we will not consider ionic conduction, which is a type of mass transport. Electronic conduction can be further divided based on the transport mechanism of the charge carriers. The first type is termed hopping conduction, which is the thermally assisted motion of a charge carrier (hole or electron) between localized sites. The hop may occur only between nearest neighbors or the hopping distance may be of a variable range. Hopping is important both for materials with structural disorder and for materials where the internuclear distances are above some critical value. Additionally, hopping is also found for materials in which the same transition metal ion is present in two oxidation states since this allows the transfer of charge from one ion to the other without any overall change in energy of the system. In some cases, it is possible to dope many of these materials and drive them to a metal–insulator transition, in which the hopping conduction (which is, in reality, semiconducting) converts to metallic-like behavior. Charge carrier localization and hopping conduction are the topics of Chapter 7. The second type of electronic conduction arises from the motion of electrons that are delocalized throughout the crystal in bands. In Chapters 4 and 5, we presented the band theory of electrical conduction in solids, where we defined itinerant electrons as those free to move throughout the crystal in delocalized orbitals. For an intrinsic metal (conductor) or semiconductor crystal, an electrical current is defined as the drift of charge carriers (electrons or holes) through the material, via the conduction band, in response to an applied electric field. With intrinsic ceramics and semiconductors, the conduction band is vacant, but an electric field can generate a current if charge carriers have been thermally activated. For extrinsic semiconductors, dopants play the crucial role of supplying the charge carriers, with the typical result that an increase in dopant/impurity density creates an increase in the electrical current. In metals, the opposite is normally observed. Increasing the defect densities decreases the electrical conductivity because rather than injecting charge carriers, defects more predominantly act as scattering centers or obstacles hindering the flow of current in metals. Let us now examine electrical conduction specifically in metal oxides whose components are in equilibrium with a gaseous species of the atmosphere. Electrical conduction in conventional semiconductors will be addressed in Section 6.4. In order to keep things as simple as possible, we will only consider cubic systems with isotropic structures and properties where the conductivity is the

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same in all directions. A very commercially important class of covalent oxide, the cubic perovskite of the generic formula ABO3, is such as phase. In what follows, we will follow the pedagogy of Donald M. Smyth (1930–2015), an inorganic solid-state chemist who spent a number of years as an electroceramics research and development (R&D) scientist at the Sprague Electric Company before becoming a professor of materials science and engineering at Lehigh University. If one were to possess a hypothetically perfect specimen of, for example, SrTiO3 in which no defects were present, they would observe it to be an insulator at room temperature and ambient pressure, as would be predicted by band theory since the band gap is around 3.25 eV. At this point, the reader may wish to review Sections 4.6 and 5.5. As long as the breakdown potential was not exceeded, a potential difference applied across the SrTiO3 specimen’s ends (monocrystalline or polycrystalline) would not produce an electrical current along any of the principal axis of the specimen, for example, the 100 , 110 , and 111 sets of directions. However, in any real oxide sample, monocrystalline or polycrystalline, extrinsic impurities or ion vacancies may be present or intentionally introduced, which act as either electron donor dopants or electron acceptor dopants. It will be recalled from Section 3.5 that intrinsic defects, such as the Schottky or Frenkel defect found in stoichiometric compounds, are a type of disorder formed as a result of internal equilibria in a pure crystal. These equilibrium or intrinsic defects do not involve reactions with the surroundings. What we are concerned with here are nonequilibrium extrinsic defects, resulting from the introduction of foreign atoms. Note that foreign atoms are called solutes if they are intentionally added to the material in relatively large amounts or dopants in small amounts, but they may also be referred to simply as impurities. Extrinsic point defects affect almost all engineering properties. For example, they are important in structural metals and alloys, where they are used to increase mechanical strength, and in semiconducting crystals and oxides, where they are used to control electrical transport properties, which is the focus here. Foreign atoms with a charge different from that of the lattice ions they replace are termed aliovalent ions and their presence results in nonstoichiometry, meaning simply that such compounds disobey the law of definite proportions. Their quantities are not in a simple fixed integral ratio or are not in the ratio expected from an ideal formula or equation. Nonstoichiometric compounds are also known as berthollides, as opposed to the stoichiometric compounds, which are called daltonides. Berthollides are ubiquitous with metal oxides. Aliovalent extrinsic defects create a charge imbalance in the solid, which may be neutralized by the generation of a charge-compensating species that, in turn, is capable of introducing electrons or holes that impart semiconducting (activated) transport behavior at some impurity concentration. For example, in SrTiO3, monovalent ions (e.g. K+) are commonly found as naturally occurring impurities substituting for the divalent Sr2+, and trivalent cation impurities (e.g. Al3+) are found substituted for the tetravalent Ti4+. These substitutions of host cations by cations of lower positive charge would result in a net negative charge in the crystal. However, the charge imbalance is neutralized by the creation of positively charged oxygen vacancies, which can act as electron donors and impart n-type semiconduction. One would thus measure a current along each of the 100 , 110 , and 111 axial directions of such a SrTiO3 specimen, roughly proportional to the impurity concentration, and this current will change in magnitude as a function of the oxygen partial pressure, PO2.

6.2.1

Oxygen-Deficient/Metal Excess and Metal-Deficient/Oxygen Excess Oxides

We can present our discussion in this section in the context of the chemical potential of oxygen, μO2, or the oxygen partial pressure, PO2, which are related to one another through μO2 = μO2 + RT ln(PO2), where μO2 is the chemical potential of oxygen in its standard state. We choose to describe

6.2 Microscopic Theory of Electrical Transport in Ceramics: The Role of Point Defects

the dependency to the electrical conductivity on the oxygen partial pressure, PO2, which is inescapable for metal oxides. Let us explore why this is the case. Firstly, oxygen partial pressure is obviously an important parameter for all oxides, as the equilibrium at any given temperature between a metal, its oxide, and the gas phase, or between two metal oxides containing the same element in different valence states, occurs at a unique PO2. Thus, from Le Chatelier’s principle, if the partial pressure of oxygen in such an equilibrium system is greater than the equilibrium value, there is a driving force for the oxidation reaction to take place in order to decrease the moles of O2 gas. As the metal is oxidized, the partial pressure of oxygen will drop until it reaches equilibrium. Adjustments in oxygen partial pressure and/or temperature are used to convert many transition metal oxides into lower or higher oxides (e.g. MnO/Mn3O4/Mn2O3/MnO2). In fact, even for oxides containing a metal cation that is not further oxidizable (e.g. Ti4+), under high PO2, there will be a tendency for consumption of gaseous oxygen through adsorption on the oxide surface, accompanied by the creation of cation vacancies and holes (a hole signifies the absence of an electron), and this imparts p-type conductivity. Conversely, for a metal oxide placed under low oxygen partial pressure, there will be a thermodynamic driving force toward reduction of the metal ions at the surface to a lower-valence state with release of oxygen gas. A low PO2 atmosphere can be created by replacement of air by a non-oxidizing or inert gas (e.g. N2, Ar) or via removal of all gases entirely (i.e. a vacuum). Indeed, high-temperature vacuum annealing is used to reduce surface oxides films and scale on metals. However, it is difficult to remove the last traces of oxygen to obtain ultralow oxygen partial pressures without the use of ultrahigh-vacuum systems equipped with ion pumps and sublimation pumps as well as cold traps. Very low PO2 can be obtained with more moderately priced systems using a reducing gas like H2 gas. It must be noted that for a metal oxide having a free energy of formation at the process temperature less negative in algebraic sign than the free energy of formation of H2O (metal oxides less thermodynamically stable than water), there will be strong driving force for the fully stoichiometric chemical reduction of the bulk oxide. For metal oxides whose free energy of formation is more negative than the free energy of formation of H2O (metal oxides more thermodynamically stable than water), oxygen vacancies will still be generated on the surface of the oxide if it is placed under low PO2 when the lower-valence oxide is more stable than the higher-valence oxide or when the structural changes occurring at such a reduced surface are more thermodynamically favorable (e.g. with increasing concentrations, oxygen vacancies can give rise to crystallographic shear planes and increased entropy). The formation of oxygen vacancies requires reduction of a corresponding small fraction of the metal ions in order to maintain charge neutrality. However, the reaction front may move very slowly through the depth of the solid and not effect a stoichiometric reduction of the bulk; in fact, it may not be possible to even remove all the surface oxygen atoms. For example, Ti2O3 is more thermodynamically stable than TiO2, but, due to sluggish kinetics, when heated at 1000 C under H2 gas for three months, TiO2 is only slightly reduced to TiO1.9986, in which some of the Ti4+ ions are reduced to Ti3+, as is evidenced by an increased sample conductivity. The presence of Ti3+ has been identified by e.s.r. spectra. The chemical stability and the change in an oxide’s oxygen content under reducing (low PO2) or oxidizing (high PO2) environments are typically referred to as its redox behavior and redox principles apply on the scale of microscopic point defects as well. From a thermodynamic point of view, a solid with point defects may be considered a solid solution in which the point defects are the solute. If we write a reaction involving point defects, it must be mass and charged balanced, which we will show shortly. For the moment, we wish to simply emphasize that, in analogy to the discussion in the preceding paragraph, sintering a metal oxide sample under low PO2 will result in a

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thermodynamic tendency toward the creation of oxygen vacancies, formed by the transfer of oxygen atoms to the gaseous state. The total number of lattice sites in such an oxygen-deficient phase may change, so long as the ratio of metal (cation) sublattice to oxygen (anion) sublattice sites remains constant. The two electrons of the O2− ion are associated with the oxygen vacancy, which thus acts as an electron donor if these electrons are delocalized into the conduction band. In reality, electrical conduction in a metal oxide is a combination of contributions by electrons, holes, and ions, although one type of charge carrier usually predominates. As discussed previously, charge transport (either electron or hole) may be classified as band conduction or hopping conduction. At low impurity levels we might suspect hopping, the process of small polarons (of electrons or holes) contributing predominantly to the current at high temperatures. For both band and hopping conduction, there may be an Arrhenius-type temperature dependency to the conductivity (σ(T) = σ aexp-Ea/kBT, with the activation energy Ea corresponding to the band-gap energy). As the temperature falls, non-Arrhenius conductivities appear. In undoped specimens, equilibrium may be established relatively quickly at lower sintering temperatures (e.g. as low as 800 C for SrTiO3). For doped specimens, due to slow kinetics of incorporating substitutional ions into the lattice, the time it takes to reach equilibrium at the same sintering temperature may be substantially longer. Hence, higher sintering temperatures are often used for the derivation of defect chemistry models, as well as for making actual devices (e.g. as high as 1400 C for SrTiO3-based dielectrics). Let us now formally define dopants in a metal oxide as follows: Donors in metal oxides are aliovalent ions with a higher valance than the cation they replace. Likewise, acceptors in metal oxides are aliovalent ions with a lower valance than the cation they replace. This may seem less intuitive than the more familiar way dopants are defined for semiconducting elements and compounds in terms of the number of valence electrons (Section 6.4.1). However, the definitions are equivalent from the viewpoint of the neutral atom, say, before it becomes a charged ion. For example, an ion with a greater positive charge (e.g. Nb5+), which has substituted for one of lesser positive charge (e.g. Ti4+), will have donated “extra” electron(s), thus achieving a higher valence state in the oxide, just as a phosphorus atom with five valence electrons donates an extra electron when substituting for a silicon, which only has four valence electrons. There are, unfortunately, complications that arise from the different possible charge-compensation mechanisms. In addition to free electrons, one may have cation vacancies, or anion interstitials, for example. The competition between the various mechanisms ultimately determines the resultant electrical conductivity. In order to proceed, it is necessary for us to introduce a shorthand notation, commonly used in the study of defect chemistry. It is called the Kröger–Vink notation after Ferdinand Anne Kröger (1915–2006) and Hendrik Jan Vink (1915–2009). The notation centers on symbols of the type M CS M may be a substitutional impurity atom (denoted by its atomic symbol), a vacancy (denoted by the uppercase letter V), an interstitial atom (denoted by the lowercase i), an electron (denoted by the lowercase e), or a hole (denoted by the lowercase h). S indicates the lattice site that the species occupies. For instance, in a sample of nickel metal, one might be referring to a copper atom impurity substituting for a nickel atom. In this case, the species and subscript would be CuNi.

6.2 Microscopic Theory of Electrical Transport in Ceramics: The Role of Point Defects

C corresponds to the effective electronic charge of the species relative to the site that it occupies. The effective charge of the species is calculated by the charge on the current site minus the charge on the original site. For example, Ni often has the same valency as Cu. So, for a copper atom impurity on a nickel metal lattice site, the relative charge is zero. To indicate null charge on a species, × is used. A single • indicates a net single positive charge, while two •’s would represent two net positive charges. Finally, signifies a net single negative charge, so two would indicate a net double negative charge. Here are some examples of single-site atomic/ionic defects that might be found in SrTiO3: BaxSr = A divalent barium ion (Ba2+) residing on a divalent strontium (Sr2+) lattice site (electrically neutral). Y•Sr = A trivalent yttrium ion (Y3+) residing on a divalent strontium (Sr2+) lattice site (a net positive charge). YTi = A trivalent yttrium ion (Y3+) residing on a tetravalent titanium (Ti4+) lattice site (a net negative charge). Nb•Ti = A pentavalent niobium ion (Nb5+) residing on a tetravalent titanium (Ti4+) lattice site (a net positive charge). V•• O = An oxygen ion site vacancy (a net double positive charge). VSr = A strontium ion site vacancy (a net double negative charge). However, with defect chemistry, we must always follow the rule of maintaining the overall electrical charge neutrality in the crystal. What this means is that a charged defect is compensated by another defect, in near proximity, of an equal and opposite charge. For example, the creation of an oxygen vacancy – as might be produced by firing a SrTiO3 sample at high temperatures under low PO2 – generates two free electrons that had been associated with the oxygen atom formally residing in that lattice site. This can be represented with a mass and charge balanced defect reaction using the Kröger–Vink notation as OO

1 O2 + V•• O + 2e 2

6 10

The main contribution to the oxygen vacancy formation in SrTiO3 is, in fact, the oxygen partial pressure, or chemical potential. However, in some perovskites, for example, (La,Sr)(Co,Fe)O3, it has been speculated that lattice vibrations or phonons (see Section 6.3.2) play a significant role (increasing with temperature) in lowering the defect formation energy that results in a high concentration of oxygen vacancies (Gryaznov et al. 2014). We can see from Eq. (6.10) how the creation of an oxygen vacancy with a double positive charge in SrTiO3 produces two charge-compensating electrons (e signifies a negatively charged electron), which act as donors, imparting n-type conductivity. In fact, the number of electrons is proportional to the oxygen vacancies as n 2[VO••]. It can be shown through a mass-action analysis that for undoped SrTiO3 n is approximately proportional to the oxygen activity, as n PO2−1/6 for PO2 less than that corresponding to the minimum conductivity and as n PO21/6 for PO2 greater than that corresponding to the minimum conductivity. For doped samples, the PO2 dependency to the electrical conductivity changes. In SrTiO3 fired under reducing conditions, these electrons may be donated to the conduction band made up of Ti 3d states, represented as 2Ti4 + + 2e−

2Ti3 +

6 11

Conceptually, we could represent the mixed-valence oxygen-deficient SrTiO3 phase chemically as + Ti3δ + O3 − 2δ SrTi41−δ

259

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6 Transport Properties

Representing the material this way unfortunately does not clearly indicate the usual finding that under these highly reducing conditions, dark to black semiconducting SrTiO3 samples are obtained with resistivities on the order of 0.1 Ω cm. The electrons that are the basis for the high-temperature equilibrium conductivity are not extensively trapped when the material is cooled to room temperature. Because of this, donor levels are said to be shallow, meaning they are very close to the bottom of conduction band. The ionization energy of the oxygen atom under low PO2 sintering is on the order of kT so it produces shallow impurity states that act as donors. In terms of the carrier electron wave function, we could alternatively state that these wave functions are not highly localized, but rather delocalized over at least a few atomic centers. A shallow trap refers to a small perturbation of the lattice potential, produced by Coulombic or electrostatic potential of a defect. It may be created by a substitutional doping atom, be able to bind an electron (if created by a donor atom) or a hole (if created by an acceptor atom), the same way an electron is bound by an H+ ion, i.e. the trapped carrier sees a long-range 1/r potential. A shallow level carrier is easily excited (e.g. thermally or optically) into a conductive state. In the opposite case of excess oxygen added to a lattice containing non-oxidizable cations (such as Ti4+), such as might be obtained by annealing or sintering SrTiO3 under high PO2, excess oxygen atoms from the atmosphere fill in a portion of the existing oxygen vacancies in the structure (even the most carefully prepared SrTiO3 samples contain oxygen vacancies due to naturally occurring acceptor impurities), with the concomitant creation of holes (positive charge carriers, denoted as h•): 1 O2 + V•• O 2

OO + 2h•

6 12

However, these holes are trapped or localized at the acceptor sites to give a low p-type conductivity. We should further point out that a hole is not a positron, which is the fundamental antiparticle or mathematical opposite of an electron. The electron–hole model is merely a simplistic way of viewing a hole as the absence of an electron at the atomic level in the shell model for atoms and also in the band structure for crystalline semiconductors, quasicrystals, and glasses. However, holes do not exist in metals/conductors. The valence band, where a hole would reside after an electron is promoted into a higher-energy conducting state, overlaps the conduction band and is fully occupied. The valence band lies well below the Fermi energy in a metal. Hence, only in materials with band gaps can a hole exist near the top of the valence band, once an electron has been excited from the valence band maximum (VBM) into the conduction band. We can summarize the trends for undoped SrTiO3 (which in actuality is governed by background acceptor impurities) by noting that in the Figure 6.1 plot of the high-temperature equilibrium conductivity (where PO2 is the abscissa and σ is the ordinate), n-type conductivity is observed at low PO2, and in this regime σ increases with decreasing PO2 since oxygen vacancies and free electrons are the dominant defects. The negative slope to this region of the curve eventually crosses over into a positive slope region of p-type conductivity in which σ increases with increasing PO2 since holes and cation vacancies begin to dominate. We thus have a wide, roughly V-shaped curve. The two slopes of this curve can be derived by treating the point defect equilibria in the system using the law of mass action, with the background acceptor impurities present giving rise to the transition between the slopes. Despite the strong experimental data supporting an oxygen vacancy mechanism for the observed conductivity in SrTiO3, care should be taken not to assume universal applicability. Charge transport and mass transport (ionic conductivity) mechanisms are determined by their defect disorder, and a proper derivation of the defect chemistry in any doped oxide must be based on experimental data corresponding to the annealing or sintering temperatures at which a gas/solid equilibrium and a

6.2 Microscopic Theory of Electrical Transport in Ceramics: The Role of Point Defects

Conductivity of ATiO3 (A = Ba, Sr) at 1000 °C

Increasing dopant concentration

Conductivicty (Ω–1 cm–1)

100

10–1

10–2 Donor-doped Acceptor-doped 10–3

Undoped

10–20

10–15

10–10

10–5

100

Oxygen partial pressure (atm)

Figure 6.1 A pictorial representation of the qualitative behavior to the high-temperature equilibrium semiconductivity in polycrystalline SrTiO3 and BaTiO3 samples as a function of PO2 and dopant concentration. It is assumed that all samples are at the same temperature and have the same A/B ratio as well as same grain size. The undoped phase (solid line) crosses over from n-type to p-type conductivity at a PO2 usually around 0.001 atm at 1000 C. With low donor dopant concentrations (dotted lines), the conductivity curve is shifted slightly upward (increased conductivity) and to the right with respect to the undoped phase but begins to flatten out with larger donor concentrations into a relatively PO2-independent region at high PO2. Conversely, with low acceptor dopant concentrations (dashed lines), the conductivity curve is shifted slightly upward and to the left and begins to flatten out at low PO2 with larger acceptor concentrations. See the text for an explanation. Similar behavior is observed for the temperature dependency to the conductivity (not shown) except that in the p-type region there is a smaller temperature dependence, that is, the isothermal lines are closer together than in the n-type region when T is the abscissa.

dense homogeneous solid solution with a stable crystalline structure and microstructure can be established. Even so, for many semiconducting metal oxides (e.g. ZnO, SnO2, TiO2), first principles studies have questioned conclusions based on indirect empirical evidence. In these oxides, oxygen vacancies are purported to be deep rather than shallow donors and are thus unlikely sources of conductivity. For example, although undoped ZnO almost always displays n-type conductivity, the oxygen vacancy long held and widely accepted to be responsible for this, may be more stable in the unionized, neutral charge state and therefore incapable of liberating electrons to the conduction band. It has been hypothesized that, alternatively, in these oxides unintentional interstitial and substitutional hydrogen may act as shallow donors and impart n-type conductivity at low PO2 (Lyons et al. 2017).

6.2.2 Substitutions by Aliovalent Cations with Valence Isoelectronicity We have just seen that for an oxygen-deficient oxide, as would be produced by sintering under low PO2, the predominating defect is generally accepted to be the oxygen vacancy compensated by electrons (Eq. 6.10). Aliovalent cation substitutions follow the same rule requiring mass and charge

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6 Transport Properties

balance. For example, with electron donors (a dopant with a higher valence state than the cation it replaces) sintered into SrTiO3 at some PO2, there are three possible charge-compensation mechanisms, namely, cation vacancies, anion interstitials, and electrons, which can give rise to both electronic and ionic contributions to the conductivity. As was discussed extensively for this class of perovskite phase in Chapters 4 and 5, the conduction band is made up of fairly low-lying reducible Ti 3d states. Thus, we find that electron compensation (in which the electrons are liberated from the donor centers) is typically observed with donor dopants in titanate perovskites. For example, if we substitute a pentavalent ion (e.g. Nb5+) on the tetravalent titanium lattice site in SrTiO3 under reducing conditions, we would find that two electrons are produced and the oxygen vacancies generated by the low oxygen partial pressure (Eq. 6.10) are annihilated by filling them with oxide ions via Nb2 O5

2Nb•Ti + 4OO +

1 O2 + 2e 2

6 13

The Fermi level of Nb-doped SrTiO3 moves into the bottom of the conduction band, and the system becomes an n-type semiconductor. We could write this Nb-doped SrTiO3 phase conceptually (but quite non-conventionally) as SrTi41 +− x Nb5x + ex O3 The conductivity will depend on the oxygen partial pressure as well as the donor concentration, since the introduction of donors increases the concentration of electrons but decreases the concentration of intrinsic oxygen vacancies. With small concentrations of donors (up to few tenths of an atom %) in SrTiO3, donor compensation is by electrons in both reducing and oxidizing conditions. These donor-doped SrTiO3 phases absorb/attenuate all wavelengths of visible light and are thus dark-colored semiconducting materials. However, it turns out that under oxidizing conditions with high concentrations of donor dopants in SrTiO3, compensation is by titanium or strontium cation vacancies (depending on the Sr/Ti ratio) rather than electrons: Sr2 + Ti41 +− 5x Nb5x + O3 or Sr21 +− 0 5x Ti41 +− x Nb5x + O3 4

These materials are light-colored insulators. Thus, we find our entire V-shaped curve of σ vs. PO2 for slightly donor-doped SrTiO3 to be shifted to the right, toward higher PO2, as compared with the undoped SrTiO3 (Figure 6.1). However, for larger donor concentrations, the conductivity begins to plateau into a relatively PO2-independent regime at high PO2 where the ionic contribution becomes less significant and the free electron concentration becomes fixed by the donor concentration. Turning now to acceptor dopants, of the possible charge-compensation mechanisms (interstitial cations, holes, or oxygen vacancies), oxygen vacancies are the preferred compensation choice in the titanate perovskite structure. A predominant semiconductivity via the mobile oxide ion vacancies (ionic transport) is observed. Thus, the addition of an acceptor ion, such as Al supplied as the oxide Al2O3, in SrTiO3 under air can be represented as Al2 O3 +

1 O2 2

2ATi + 4OO + V•• O

6 14

One might have predicted the creation of holes in the valence band in analogy with electrons in the conduction band for donors. However, those are equivalent to oxidizing O2−, which is not energetically favorable. Under oxidizing conditions, i.e. high PO2, a small portion of the oxygen vacancies

6.2 Microscopic Theory of Electrical Transport in Ceramics: The Role of Point Defects

will be filled with oxygen atoms from the atmosphere to create holes, via Eq. (6.12) repeated here again for convenience: V•• O +

1 O2 2

OO + 2h•

6 15

The holes are trapped during the cooling process, as evidenced by a drop in conductivity, indicating that acceptors are rather deep level in the band gap, or high above the top of the valence band and far below the bottom of the conduction band, toward the mid-gap region. This is consistent with experimental findings in which low p-type conductivity is observed at high PO2, especially at low temperatures where most of the holes are trapped at acceptor centers. Acceptor centers in SrTiO3 could be also be provided by vacant Sr2+ sites in addition to impurity ions with a lower charge than the ions they replace. To summarize the picture in acceptor-doped SrTiO3, we find a small p-type contribution to the electrical conductivity in addition to the oxygen ion conductivity, but these holes are mobile via a hopping mechanism. Because acceptors are such effective hole traps, we expect acceptor-doped SrTiO3 to be a light-colored insulator. However, when fired under reducing conditions, the oxygen vacancies that are created act as electron donors so that n-type conductivity is observed at low PO2, per Eq. (6.10). Overall, our entire V-shaped curve of σ vs. PO2 for slightly acceptor-doped SrTiO3 (i.e. concentrations above the background acceptor impurity level) is shifted to the left towards lower PO2, as compared with the undoped SrTiO3 (Figure 6.1). For higher concentrations of acceptor ions, the equilibrium conductivity begins to broaden and flatten out as it does with large donor concentrations, but this time the PO2-independent behavior is due to a very large ionic contribution from the extrinsic oxygen vacancies (see Eq. 6.14) rather than by a relatively fixed free electron concentration. In comparison to shallow impurities, deep acceptor impurities require energies larger than the thermal energy to ionize so that only a fraction of the impurities present in the semiconductor contribute to free carriers. Deep impurities that are more than five times the thermal energy away from either band edge are very unlikely to ionize. Such impurities are effective recombination centers in which electrons and holes fall and annihilate each other. Electrons will not pass into the conduction band if unfilled acceptor sites are available at lower energies. Such acceptors are also capable of immobilizing electrons that would contribute to hopping transport as well. In general, it is possible that electrons can be bound to positively charged donors and holes can be bound to negatively charged acceptors. However, in titanates, Ti4+ is not oxidizable, the valence band is made up of filled O 2p states, and it is not receptive of holes. As just discussed in the preceding paragraph, any holes that might be present are trapped at acceptor centers, deep in the band gap: ATi + h•

AxTi

6 16

Contrarily, electrons in reduced titanates are not trapped at donor centers (symbolically noted as D) since the conduction band is receptive to electrons and the donors are at shallow levels just below the conduction band: DxTi

D•Ti + e

6 17

6.2.3 Substitutions by Isovalent Cations That Are Not Valence Isoelectronic For the aliovalent dopants discussed in the previous section, the impurity ions were valence isoelectronic with the host ions they replaced. That is, the dopants contained different ionic charges, but the same valence electron counts as compared to the host ion. For example, both

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6 Transport Properties

the donor ion Nb5+ and the acceptor ion Al3+ had zero valence electrons, as did the Ti4+ ion they replaced. As a final scenario, we shall now discuss the special case of an isovalent impurity with a different number of valence electrons as compared to the host atom it replaces. That is, the dopant has a different number of valence electrons but the same ionic charge as the host atom. A dilemma arises in this situation because isovalent impurities make no significant contribution to charge neutrality and thus have no significant effect on defect concentrations. The effects of adding these types of dopants are thus primarily electronic. As a consequence, it is of little help to write an equation to represent a defect reaction in this case. Consider Ru4+-doped SrTiO3. For simplification, we will neglect any natural impurity that may be present and assume hypothetically pure SrTiO3 and SrRuO3 phases. Both titanium and ruthenium are tetravalent, but this is where the similarity ends. Nominally pure SrTiO3 is a wide-bandgap (WBG) semiconductor with a 3.57 eV band gap, which is insulating at room temperature. Its conduction band is made up of empty Ti 3d states (Ti4+ has zero 3d electrons), and the valence band is composed of filled O 2p states. Pure SrRuO3, on the other hand, has partially filled 4d orbitals with the four 4d electrons on Ru4+ (Ru4+ has a 4d electron count of four valence electrons) delocalized in a partially filled conduction band made up of Ru 4d states. The Fermi energy at 0 K lies in the middle of this partially filled conduction band, and we would thus expect this oxide to be metallic, showing an increasing resistivity with increases in temperature. Electrons have been established experimentally as the dominant charge carrier in SrRuO3 at low temperatures. In fact, because it is a band itinerant ferromagnet, SrRuO3 thin films find use as ferromagnetic electrodes in magnetic tunnel junctions and are of interest as metallic electrodes in thin-film perovskite ferroelectric capacitors for nonvolatile memory applications (resistive switching random access memory [ReRAM]). The Ru4+ and Ti4+ ions have similar ionic radii (62 vs. 60.5 pm), and it has been shown that the two oxides have a large solid solubility limit in each other. Therefore, if Ru4+ can be made to substitute for some small fraction of Ti4+ ions in the parent SrTiO3 phase, it should be possible for Ru4+ to either (i) donate its four 4d electrons to the conduction band of SrTiO3 if the Ru4+ states are high enough in energy or (ii) act as a shallow dopant with states inside the band gap at least near to one of the band edges. In the former case, metallic band conduction would be observed. In the latter case, band or hopping semiconduction would be expected. For example, excitation of an electron from the valence band to Ru4+ states lying just above the VBM would leave a hole in the valence band via Ru4+ Ru3+ + h•. Alternatively, if the Ru4+ states are just below the conduction band minimum (CBM), they could function as shallow donors via Ru4+ Ru5+ + e . It turns out either (i) or (ii) may be observed. At a doping concentration somewhere between about x = 0.4 and x = 0.5, SrTi1 − xRuxO3 undergoes a metal–insulator transition. More precisely stated, for x above about 0.5, SrTi1 − xRuxO3 is metallic, while for x below 0.4 SrTi1 − xRuxO3 is nonmetallic. In the latter case, the four 4d electrons simply remain localized on the Ru4+ sites, and the phase is found to be a p-type semiconductor, rather than a n-type. In fact, even the pure SrRuO3 phase shows anomalous metallic behavior. It is a marginal (“dirty”) metal with a high room temperature resistivity (~2 × 10−6 Ω m, compared with Cu metal with ρ = 1.7 × 10−8 Ω m). In these ruthenium oxide systems, one must account for the combined effects of disorder and electron correlation, which will be discussed in Chapter 7. We have actually painted a somewhat simplified and perhaps unrealistic picture for SrTi1 − xRuxO3 by further neglecting the existence of any point defects other than the ruthenium dopant. SrRuO3, for example, tends to be ruthenium cation deficient, while oxygen vacancies are minority defects. This arises from the disproportionation of RuO2 via 2 RuO2 (s) Ru (s) + RuO4 (g). Under oxidizing condition, the elemental Ru is immediately reoxidized. The negatively charged ruthenium vacancies (V ''''Ru) produced by this disproportionation, however,

6.2 Microscopic Theory of Electrical Transport in Ceramics: The Role of Point Defects

are compensated by holes, which are known to be the dominant charge carrier in SrRuO3 at high temperatures. The number of holes are proportional to the ruthenium vacancies as n ≈ 4 V ''''Ru . At low PO2 oxygen vacancies are created as well as Ru vacancies, which is quite different than SrTiO3 where the oxygen and cation vacancies can be separately controlled. If we include the existence of these extrinsic point defects in SrTi1 − xRuxO3, our ability to easily explain the transport properties in this phase becomes more challenging. As we can see, the nuances of which type of defect, defect compensation, and electrical transport mechanism (hopping or band conduction) is most favorable under a given set of circumstances is dependent on many factors, including the oxide or ceramic’s crystal structure, its constituents, the PO2, and the dopant type (solubility), as well as grain size. Fortunately, chemical intuition can sometimes aid us in making reasonable predictions. For example, in SrTiO3, oxygen interstitials are unlikely in such a nearly close-packed structure and can be dismissed. Furthermore, titanium vacancies have a very high formal charge and are not favorable with low donor concentrations. Cation interstitials are also not very favorable in the perovskite structure, but they are in TiO2 (rutile structure). Even in an isostructural series, variation may be observed for comparable doping levels. For example, in Nb-doped SrTiO3, in which niobium functions as a donor substituting for titanium, electronic compensation occurs at least for greater than 10 at% substitution, while in Nb-doped BaTiO3, ionic charge compensation prevails up to at least 20 at% substitution, as has been found from measurements of the PO2 dependency to the electrical conductivity. In the dielectrics industry, titanate oxide powders are frequently doped in order to impart the desired electrical properties, not only the electrical conductivity but also the dissipation factor (a.k.a. loss tangent) and dielectric constant, all of which are temperature and frequency dependent. For disk capacitors, the parent phase (e.g. SrTiO3, BaTiO3) is milled in a slurry with appropriate quantities of the oxide dopants, dried, pressed into a pellet, and sintered. Screen printing processes are used to make multilayer capacitors (MLCCs). The dopant concentration is typically expressed in one of four ways. These are, from most to least common, the atom-% substitution, the molar (atomic) percentage, also given in molar ppm (parts per million); the percentage by weight or mass, given as ppm wt. (parts per million with respect to weight); and the number density, given as the number of ions or atoms per cubic meter or cubic centimeter. Example 6.2 A powder sample is produced by combining 26.581 g of Nb2O5 and 366.980 g of SrTiO3. The powder is pressed into multiple pellets. One pellet is sintered at 1400 C inside a box furnace under an air atmosphere and is found to be light colored and of low conductivity. The second pellet is fired at 1400 C in a tube furnace under flowing forming gas (5%H2/95%N2) and is found to be dark colored and of high conductivity. 1) Write the chemical formula for the sintered products. 2) Calculate the atom-% substitution of niobium in the sintered products, assuming all the Nb5+ cations substitute for Ti4+ sites in SrTiO3. 3) What might you expect to be the donor compensation species for the pellet fired under air? 4) What might you expect to be donor compensation species for the pellet fired in 5%H2/95%N2? 5) Write the net mass + charge balanced charge-compensation reactions using the Kröger–Vink notation for each case. Solution The formula weight of Nb2O5 is 265.81 g. Thus 26.581 g Nb2O5 = 0.10 mol Nb2O5. The formula weight of SrTiO3 is 183.49 g. Thus 366.980 g SrTiO3 = 2.00 mol SrTiO3. This is a relatively large amount of dopant.

265

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6 Transport Properties

Niobium has a valence state of 5+, while that of titanium is 4+. Hence, Nb serves as a donor in SrTiO3, increasing the electron concentration and decreasing the concentration of oxygen vacancies produced under low PO2. For the forming gas to function as a chemical reductant toward SrTiO3, the free energy of formation for H2O must be more negative in sign than the free energy of formation for SrO or TiO2 at 1400 C. Reference to the Ellingham diagram shows that it is not. However, as explained in the text, the forming gas does function to lower the oxygen partial pressure so we might suspect electrons as the defect-compensating species in the product phase fired under low PO2 with high dopant levels and cation vacancies as the defect-compensating species in the product phase fired under high PO2. A mass balanced chemical reaction between 0.10 mol Nb2O5 and 2.00 mol SrTiO3 (assuming all the Nb5+ ions substitute for the Ti4+ ions in SrTiO3) for both PO2 cases might at first guess be written simply as 2SrTiO3 + 0 1Nb2 O5

2SrTi1−x Nbx O3 + 0 20TiO2 + 0 05O2

Since x = 0.10, this yields SrTi0.90Nb0.10O3 for the formula of the main product phase, in which strontium is divalent, titanium is tetravalent, and niobium is pentavalent, with an atom-% substitution of niobium in the sintered products of 10%. There are problems with this, unfortunately. This formula tells us nothing about the charge compensation, and there is an excess positive charge on the product species that must be balanced by either negatively charged electrons (donated to the conduction band) or titanium vacancies. As a workaround, we know that the charge on niobium is 5/4 greater than the charge on titanium. So we should write the chemical reaction as 2SrTiO3 + 0 1Nb2 O5

2SrTi1 − 5x Nb4x O3 + 0 25TiO2 + 0 05O2

Using SrTi1 − 5xNb4xO3 or, equivalently, SrTi1 − 5x/4NbxO3 in which the valence states for each species are the nominal Sr2+, Ti4+, Nb5+, and O2− ensures we are mass and charge balanced. However, we still know nothing about the charge compensation. Niobium ion in Nb2O5 is pentavalent (Nb5+), while the titanium ion in SrTiO3 is tetravalent (Ti4+). Niobium is also not reduced by the forming gas, so it functions as a donor in SrTiO3. With the relatively high concentration of donor dopant in SrTiO3 under oxidizing conditions (air), the compensation should be by cation vacancies, while under reducing conditions (H2/N2), high concentrations of donor dopants in SrTiO3 are compensated by electrons. Using the Kröger–Vink notation, the net mass + charge balanced charge-compensation reactions are the following: 4NbTi• + 10Oo + V Ti (Niobium ions and oxygen atoms are incorporated into the lat2Nb2 O5 tice, and a finite number of titanium vacancies are produced in highly doped samples in order to maintain charge balance when the sample is fired under air.) Nb2O5 2NbTi• + 4Oo + ½O2 + 2e (Niobium ions and oxygen atoms are incorporated into the lattice and electrons are liberated (donated to conduction band) when highly doped samples are fired under low PO2.)

6.2.4

Nitrogen Vacancies in Nitrides

Electrical transport in other ceramics, in addition to oxides, is also affected by the presence of anion vacancy native point defects. Consider nitrogen vacancies in GaN, AlN, and silicon nitride (Si3N4), all of which are prepared from gaseous precursors and show a tendency to develop nitrogen vacancies due to the high growth temperatures and high nitrogen volatility. For all of these compounds, it

6.2 Microscopic Theory of Electrical Transport in Ceramics: The Role of Point Defects

would seem possible for the nitrogen vacancy to be neutral, positively charged (donor states), or negatively charged (acceptor states). The most stable of the possible charge states for the nitrogen vacancy will be determined by their respective defect formation energies, relative to the available thermal energy. For donor and acceptor states, the defect formation energy is dependent on the Fermi level because electrons are transferred across the Fermi level to charge the defect. Thus, it becomes imperative to have some way of estimating the location of the Fermi level, and the reader is instructed to first review Section 5.5 for this purpose. As would be suspected, under N-poor (Ga-rich) conditions, the nitrogen vacancy is more stable than the gallium vacancy for all values of the Fermi energy. Recall that the Fermi level can be considered to be a hypothetical energy level of an electron in an insulator or intrinsic (undoped) semiconductor, such that at thermodynamic equilibrium this energy level would have a 50% probability of being occupied at any given time. The states with the lowest energy are filled first, followed by the next higher ones. At absolute zero temperature (T = 0 K), the energy levels are all filled up to a maximum energy, which is the Fermi level. No states above the Fermi level are filled. At higher temperature, one finds that the transition between completely filled states and completely empty states is gradual rather than abrupt. The Fermi–Dirac distribution function provides the probability of occupancy of energy levels by electrons. In an insulator or an undoped intrinsic semiconductor, at 0 K the Fermi level lies midway between the VBM and the CBM. For the semiconductor case, however, the band gap is narrow enough that at nonzero temperatures a small number of carriers can be thermally excited from the valence band, across the band gap, into the conduction band. When these electrons are injected from the Fermi level into the conduction band, they leave behind holes in the valence band, which can also conduct a current since the holes give the electrons remaining in the valence band a higher degree of freedom. This is further discussed in Section 6.4.1.3. We are now in a position to speculate on what happens when a nitrogen atom is removed from the GaN lattice. A nitrogen vacancy, VN, leaves behind dangling bonds on the gallium neighbors surrounding the vacancy. The question is: what happens to these formerly bonding electrons? The nitrogen vacancy is accompanied by an outward displacement of the surrounding Ga atoms, with the inequivalent Ga atom displacing slightly more than the other three Ga atoms. This large outward relaxation raises the energy levels of the defect states and shifts them into the band gap. If the Fermi level is near, these states become unoccupied as the electrons are excited into the conduction band and that makes the nitrogen vacancy a donor. Indeed, the VN3+ donor state is the most energetically stable state (of lowest formation energy) for the nitrogen vacancy in GaN for Fermi energies just above the VBM all the way to the middle of the band gap, where it transitions to the VN1+ state. However, near the CBM (where the states would need to lie for carrier excitation into a conducting state), the VN+ defect formation energy has been computed as being between 2 and 3 eV for, respectively, Ga- and N-rich crystal growth environments. These are rather high values, indicating that nitrogen vacancies are not likely to be present to an appreciable enough quantity in GaN to explain the observed n-type conductivity, as had widely been believed prior to the 1990s. Unintentional n-type conductivity in GaN, instead, has been attributed to the accidental introduction of extrinsic impurities (silicon and oxygen), which led to a concerted effort within the industry to control these impurities. We will discuss the mechanism of impurity doping in semiconductors like GaN in more detail in Section 6.4.1.3. Even at small concentrations, the positively charged nitrogen vacancies can, nevertheless, still trap electrons to form bound states in the band gap and compensate p-type carriers (Xie et al. 2018). The nitrogen atoms in AlN and β-Si3N4, like those in GaN, are also tetrahedrally coordinated. However, in contrast to GaN (Eg = ~3.4 eV), the band gap is even wider in β-Si3N4 (~5 eV) and

267

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6 Transport Properties

AlN (~6 eV). Because the CBM is so high in energy, we might expect the neutral or acceptor (negatively charged) states for nitrogen vacancies to be more likely than shallow donor states in these compounds. Indeed, some DFT calculations have indicated high formation energies, while some e.s.r. studies have confirmed that the electrons of the dangling silicon bonds surrounding nitrogen vacancies in β-Si3N4 and of the dangling aluminum bonds in AlN tend to stay localized as paramagnetic centers at deep levels in the band gap (the vacancy thus existing as the neutral VN) or to attract electrons from the top of the valence band (the equivalent of donating holes to the valence band), in which case the nitrogen vacancy exits as VN− or VN3−. Hydrogenation of the nitrogen vacancies favors formation of the neutral state, as the hydrogen atom does not form bonds with the surrounding atoms. We have purposely chosen to discuss systems that enabled a simplified treatment. In singlecrystalline and polycrystalline specimens of isotropic phases, the magnitude of the electrical current measured along any axial direction is the same. For anisotropic single crystals exhibiting lower-dimensional transport (e.g. structures in which covalent layers are separated by ionic layers), electron or hole transport may not be possible along certain axes of the crystal. However, if it be recalled from Section 2.4.2, when a polycrystalline sample is free of preferred orientation, that is, if it has a completely random crystallite orientation distribution, its bulk transport properties will be like that of a cubic single crystal, isotropic as a whole regardless of the anisotropy to the single crystallites comprising it. If anisotropic grains are randomly oriented, the macroscopic sample loses any anisotropy in any property.

6.3

Thermal Conductivity

The macroscopic phenomenological equation for heat flow is Fourier’s law, by the mathematician Jean-Baptiste Joseph Fourier (1768–1830). It appeared in his 1811 work, Théorie analytique de la chaleur (The analytic theory of heart). Fourier’s theory of heat conduction entirely abandoned the caloric hypothesis, which had dominated eighteenth-century ideas about heat. In Fourier’s heat flow equation, the flow of heat (heat flux), q, is written as qi = − κij ∇T j

6 18

where κ is a positive quantity called the thermal conductivity (units of W/m/K) and ∇T is the temperature gradient. The thermal conductivity is a second-rank tensor, since the heat flux and temperature gradient are vectors. In general, it may be considered to have contributions from various possible energy-carrying sources, including electrons, phonons, photons, gas molecules, etc. although we shall only discuss the first two of these in this book. Heat flow within the system is in the direction of greatest temperature fall but is not required to be exclusively parallel to the temperature gradient. For example, if a temperature gradient is set up along one axis of a Cartesian coordinate system, transverse heat flow may be measured parallel to the other two axes. Therefore, the equations representing the heat flow along the three axes are qx = κ11 qy = κ12 qz = κ13

∂T ∂x ∂T ∂x ∂T ∂x

+ κ12 + κ22 + κ23

∂T ∂y ∂T ∂y ∂T ∂y

+ κ 13 + κ 23 + κ 33

∂T ∂z ∂T ∂z ∂T ∂z

6 19

6.3 Thermal Conductivity

where each scalar flux is parallel to an axis of a Cartesian coordinate system. In general, because it is the component of the flux parallel to the direction along which the driving force is applied that is measured, the magnitude of any transport property in the direction of the driving force is defined to be the component of the flux parallel to the direction of the driving force divided by the magnitude of the driving force. Thus, the magnitude of a second-rank tensor property in any direction is obtained by applying the driving force in that direction and measuring the component of the response parallel to that direction. As discussed in Section 6.1, the number of independent components needing to be specified is reduced by the crystal symmetry. If, for example, the x-axis is taken as the [1 0 0] direction, the y-axis as the [0 1 0], and the z-axis as the [1 0 0], a cubic crystal or a polycrystalline sample with a random crystallite orientation gives

κ=

κ11 0

0 κ11

0 0

0

0

κ 11

6 20

in which case the thermal conductivity tensor has been reduced to a single independent component, the equivalent of a scalar quantity. Example 6.3 If a temperature gradient ∇T is directed along one of the principal axes of a crystal, say, the x-axis, what will be the angle between the vector q and the vector ∂T/∂x? Solution If ∇T is parallel to the x-axis, ∂T/∂y = ∂T/∂z = 0, and, hence, qy = qz = 0. In this case, q and ∂T/∂x are parallel. It must be stressed that thermal (and electrical) conductivity, like density, is an intensive materials property, meaning its magnitude is independent of the system size (dimensions) or geometry, whereas conductance (units of W/K) is extensive. The ratio of any two extensive properties for the same system is always intensive. Henceforth, for simplicity during our discussions of the underlying physical basis for thermal conductivity, only cubic crystals or nontextured polycrystalline solids will be considered, for which case a single scalar quantity is sufficient. The vast majority of cases in which thermal conductivity is the prerequisite materials property of interest involve polycrystalline phases.

6.3.1 The Free Electron Contribution The thermal conductivity of a metal or alloy consists of two components, a phonon contribution (a phonon is a quantum of acoustic energy, which possesses wave–particle duality), κ ph, and an electronic contribution (free electrons moving through the crystal also carry thermal energy), κel. In pure metals, κ el is the dominant contribution to the total thermal conduction. This free electron contribution to the thermal conductivity is given by the gas kinetic formula as κ el =

π 2 nk 2b Tτ 3me

6 21

269

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6 Transport Properties

where n is the electron density (number of free electrons per cm3), τ is the average time an electron travels between collisions, me is the electron rest mass, T is the absolute temperature, and kb is the Boltzmann constant. The parameter τ can be calculated from τ=

λel VF

6 22

in which λel is the mean free path and VF is the electron velocity at the Fermi surface. The denominator of Eq. (6.22) is related to the Fermi energy, εF, by VF =

2εF me

1 2

6 23

The fact that the thermal conductivity in a pure metal is dominated by the free electron contribution was illustrated in 1853 by Gustav Wiedemann (1826–1899) and Rudolf Franz (1827–1902), who showed that κ el and the electrical conductivity, σ el, are proportionally related (Wiedemann and Franz 1853). A few years later Danish physicist Ludvig Lorenz (1829–1891) realized that this ratio scaled linearly with the absolute temperature (Lorenz 1872). Thus κel = σ el LT

6 24

which is known as the Wiedemann–Franz–Lorenz law. Solving for L and substituting both the expression for κ el (Eq. 6.24) and the Drude expression for σ el (Eq. 6.30) gives L=

χk 2b e2

6 25

in which χ is π 2/3, e is the elementary charge (1.602 × 10−19 coulombs), and kb is the Boltzmann constant (1.380 × 10−23 J/K). The Lorenz number L has the theoretical value 2.45 × 10−8 W Ω/K2. If the experimental value determined for a metal is close to or equal to this, it can be assumed the electronic contribution dominates the thermal conductivity. At room temperature, experimental values for L range from 2.28 × 10−8 W Ω/K2 for silver (the best conductor) to 3.41 × 10−8 W Ω/K2 for bismuth (a poor conductor). The electronic contribution to the thermal conductivity is reduced by electron scattering events, including electron–electron, electron–phonon, and electron-defect scattering. Since they also limit the electrical conductivity, electron scattering mechanisms will be discussed in Section 6.3. In a semiconductor, which has a relatively low electrical conductivity, an appreciable part of the thermal conductivity originates from the phonon contribution. Hence, for semiconductors, Eq. (6.24) must be modified to include a non-electronic (phonon) component, κ , viz. κ total = κ ph + σ elLT. However, in this expression, L is no longer 2.45 × 10−8 W Ω/K2, as χ in Eq. (6.17) is closer to 2.23 than π 2/3 (Price 1957), making L ~ 1.65 × 10−8 W Ω/K2 for extrinsic semiconductors. Example 6.4 At 300 K, the p-type semiconductor (Bi2Te3)1 – x − y(Sb2Te3)x(Sb2Se3)y has the following transport properties: σ = 1290 S/cm, κtot = 15.2 × 10−3 W/cm K, and κ ph = 8.8 × 10−3 W/cm K. 1) Calculate the electronic component of the thermal conductivity. 2) Calculate the experimental value of L for this semiconductor. Solution 1) It is known that κtot = κel + κ ph. Hence, if κtot = 15.2 × 10−3 W/cm K and κ ph = 8.8 × 10−3 W/cm K, then κel = 6.4 × 10−3 W/cm K.

6.3 Thermal Conductivity

2) Next, using the values given for κ tot, κph, and σ in the relation κ total = κ ph + σ elLT, L can be readily calculated as L=

κtot − κ ph 0 0064 = = 1 65 × 10 − 8 WΩ K2 σT 387 000

6.3.2 The Phonon Contribution The thermal conductivity of a pure metal is lowered by alloying, whether the alloy formed is a single-phase (solid solution) or multiphase mixture. There are several reasons for this. First, electrons are scattered by crystal imperfections and solute atoms (electron-defect scattering). Second, a substantial portion of the thermal conductivity in alloys, in contrast to that of pure metals, is by phonons, κ ph (phonons are the sole contribution in electrically insulating solids), and phonons are also scattered by defects. Finally, electron–phonon interactions limit both κel and κ ph. So, what exactly are phonons? Phonons are quantized vibrational excitations (waves) moving through a solid, owing to coupled atomic displacements. The idea of a lattice vibration evolved from work on the theory of the specific heat of solids by Einstein (1906, 1911), Debye (1912), and Born and von Kármán (1912, 1913). Examine a one-dimensional (1D) chain of atoms, of finite length L, such as the one illustrated in Figure 6.2. In this figure, the chemical bonds (of equilibrium length a) are represented by springs. It can be assumed that the interatomic potential is harmonic (i.e. the springs obey Hooke’s law). As the atoms are all connected, the displacement of just one atom gives rise to a vibrational wave, involving all the other atoms, which propagates down the chain. The speed of propagation of the vibrational excitation is the speed of sound down the chain. For atoms constrained to move along one dimension, only longitudinal waves composed of compressional and rarefactional atomic displacements are possible. In general, a chain of N atoms interacting in accordance with Hooke’s law has N degrees of freedom and acts like N independent harmonic oscillators with N independent normal modes of vibration. By analogy, a 3D harmonic solid has 3N normal modes of vibration (with both longitudinal waves and transverse waves possible). For simplicity in the present discussion, fixed boundary conditions have been chosen in which the amplitudes of the atomic displacements are zero at the surface of the crystal, that is, the atoms at the surface are at fixed positions. Note that this does not properly account for the number of degrees of freedom. Nonetheless, with these boundary conditions, an integer number of L

a

Figure 6.2 A one-dimensional chain of atoms, of length L and lattice spacing a, connected by springs that obey Hooke’s law. With the end atoms fixed, only standing waves of atomic displacement are possible.

271

272

6 Transport Properties

phonon half wavelengths must fit along the length of the chain in Figure 6.2 (or along each dimension of a 3D crystal). For the monatomic chain, the allowed frequencies are given by the relation ωk = 2

K M

1 2

sin

ka 2

6 26

where K is the force constant, M is the atomic mass, and k is the wave vector related to the wavelength, λ, of the wave by |k| = 2π/λ = nπ/a (n = ±1, ±2, …). There are N allowed k state in the first BZ (−π/a to π/a) corresponding to the independent normal modes of the system. The mode frequency and energy vary in a periodic way with the wave vector, which gives rise to a dispersion curve and a density of states (DOS), or number of possible modes (vibrational states) per frequency range. The wave function corresponding to one of these normal modes for the monatomic chain of Figure 6.2 is shown with the necessary atomic displacements indicated by arrows. A phonon is now defined as a particle-like entity (representing the wave) that can exist in one of these allowed states. The phonon occupation number at thermal equilibrium, or number of phonons in a given state (i.e. with a particular frequency, ωk, or wave vector, k), follows Bose–Einstein statistics. If a temperature gradient is now set up across the sample, a nonequilibrium distribution of phonons, called a wave packet (or pulse), is produced. The system reacts by attempting to restore the equilibrium distribution. That is, a thermal current occurs only if the phonon occupation number has a nonequilibrium value. The heat flux is, in fact, equal to the sum of the energies of the phonons in the distribution, multiplied by the phonon group velocity, divided by the crystal volume. Heat conduction is reduced by any event that reduces the phonon mean free path or that causes a net change in momentum of the phonons. Phonons can propagate through a defect-free array of harmonic oscillators very easily, for, when phonons collide with one another in such an array, the total energy and momentum of the wave packet is conserved. An infinite defect-free harmonic crystal would possess an infinite thermal conductivity because there would be no limit to the mean free path of the phonons. Real crystals are not defect-free and real interatomic forces are anharmonic, that is, the interatomic potential is nonparabolic for all but small displacements. Anharmonicity gives rise to phonon–phonon scattering. First, however, phonon-defect scattering will be considered. Point defects (including impurities and even different isotopes), dislocations, and grain boundaries in polycrystalline samples all cause elastic phonon scattering at low temperatures where only long-wavelength phonons are excited. The scattering shortens the phonon mean free path and limits thermal conductivity. The easiest way to think of phonon-defect scattering is to compare it to the way a long-wavelength light is Rayleigh scattered by small particles (where the particle radius is much less than the phonon wavelength). The scattering cross section is proportional to the radius of the crystal imperfection. For impurity atoms with masses lighter than those of the other atoms in the lattice, a spatially localized vibrational mode called an impurity exciton is generated, which is of a frequency above that of the maximum allowed for propagating waves. The amplitude of the vibration decays to zero, not far from the impurity. For heavier atoms, the amplitude of the mode is enhanced at a particular frequency. In noncrystalline solids, which lack translational periodicity (e.g. glasses), only those thermal excitations in a certain frequency range become localized, namely, in the band tails of the DOS (the number of vibrational states over a given frequency range). These impurity modes and excitations are actually what cause the phonons to scatter. That is, a phonon is scattered by the alteration in the lattice caused by the exciton. Thus, phonon-defect scattering is really due to exciton–phonon interactions.

6.3 Thermal Conductivity

With regard to the second feature of real crystals mentioned earlier, there are different types of anharmonicity-induced phonon–phonon scattering events that may occur. However, only those events that result in a total momentum change can produce resistance to the flow of heat. A special type, in which there is a net phonon momentum change (reversal), is the three-phonon scattering event called the Umklapp process. In this process, two phonons combine to give a third phonon propagating in the reverse direction. Equation (6.18) gave the macroscopic expression for the heat flux through a solid under a finite temperature gradient. The microscopic expression for the thermal conductivity owing to a single phonon mode in a dielectric monocrystal, in which the electron contribution is negligible, is given from the kinetic theory of gases as κ ph =

1 CΛv 3

6 27

where C is the specific heat, Λ is the phonon mean free path, and v is the phonon group velocity (the velocity at which heat is transported) (Elliot 1998). In reality, one must take account of the contribution from all the phonon modes. Since the level of analysis required is somewhat above that assumed appropriate for this book, it will suffice to just quote the microscopic expression for the lattice thermal conductivity using the single-mode relaxation time method, where the relaxation rate of phonons in a mode is qs, on the assumption that all other phonon modes have their equilibrium distribution (Srivastava 1990): κ ph =

ℏ2 3Vkb T 2

c2s q ω2 qs τqs nqs nqs + 1

6 28

qs

In this equation, ħ is Planck’s constant divided by 2π, V is the crystal volume, T is temperature, kb is the Boltzmann constant, ω is the phonon frequency, cs is the wave packet or phonon group velocity, τ is the effective relaxation time, n is the Bose–Einstein distribution function, and q and s are the phonon wave vector and polarization index, respectively. Equation (6.28) is a rather formidable expression. From the experimentalist’s standpoint, it will be beneficial to point out some simple, yet useful, criteria when toiling with thermal conductivity. First, it is noted that thermal conductivity is not an additive property. It is generally not possible to predict the thermal conductivity of an alloy or compound from the known thermal conductivities of the substituent pure elements. For example, the thermal conductivity of polycrystalline silver and bismuth are, respectively, 429 and 8 W/m K. Yet, the two-phase alloy 50 Bi–50 Ag has a thermal conductivity of only 13.5 W/m K. Often, alloys have thermal conductivities about equal to those of the metal oxides, indicating that a large portion of the total conductivity in alloys is due to phonons. Hence, our limited ability to predict total thermal conductivities (κ tot = κph + κ el) for alloys and compounds is mostly owing to the difficulty in determining the κ ph contribution. One of the best-suited methods of obtaining estimates for the phonon contribution is molecular dynamics. However, it is computationally intensive and most effective for modeling systems of limited size. Depending on the application, one may wish to use either a material with a low or a high thermal conductivity. High thermal conductivity materials are required for thermal management in a broad range of applications. Based on approximate solutions to Eq. (6.28), four simple criteria for choosing high-conductivity single-crystal materials have been established (Srivastava 2001): 1) 2) 3) 4)

Low atomic mass. Strong, highly covalent, interatomic bonding. Simple crystal structure. Low anharmonicity.

273

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6 Transport Properties

A systematic evaluation (Slack et al. 1987) has revealed that most of the high thermal conductivity ceramics (>100 W m/K) are compounds of the light elements that possess a diamond-like crystal structure (e.g. BN, SiC, BeO, BP, AlN, BeS, GaN, Si, AlP, GaP). Conversely, it should be expected that low thermal conductivity materials generally possess complex crystal structures, especially those exhibiting low dimensionality, or have constituents with high atomic masses and stiff chemical bonding, which lack rattling modes. Materials possessing a low thermal conductivity and a high electrical conductivity are desirable for thermoelectric devices (Section 6.4.2). Tailoring the thermal conductivity of polycrystals, on the other hand, is an application where composite manufacturing has borne fruit in the development of new materials. For example, high thermal conductivity can be achieved in some cases by forming a metal matrix composite (MMC) with the alloy as matrix and nano-sized particles of a highly thermally conducting phase (e.g. carbon nanotubes) as the reinforcement material. Reinforcements in MMCs can be continuous or discontinuous. The latter can be isotropic, while the former are typically wire or fibers embedded within the matrix in a certain direction. This results in an anisotropic structure in which the alignment affects the thermal conductivity along those directions. Reinforcements are also used to change the friction coefficient, wear resistance, and strength of materials. Of course, it should be realized that fabricating a composite is more of a continuum-level fix, rather than an atomistic-level solution to a particular materials challenge. Nevertheless, it is an effective one that illustrates how a little engineering ingenuity can be used to overcome a seemingly insurmountable scientific obstacle in order to achieve an objective.

6.4

Electrical Conductivity

The phenomenological equation for electrical conduction is Ohm’s law, which first appeared in Die galvanische kette mathematisch bearbeitet (the galvanic circuit investigated mathematically), the 1827 treatise on the theory of electricity by the Bavarian mathematician Georg Simon Ohm (1789–1854). Ohm discovered that the current through most materials is directly proportional to the potential difference applied across the material. The continuum form of Ohm’s law is ji = σ ij Ej

6 29

where j is the electrical current density vector (the current per unit cross section perpendicular to the current), σ is the intensive materials property called the electrical conductivity tensor (units of Siemens per meter = A/m/V = Ω−1 m−1), and E is the electric field vector. Example 6.5 Assume the electrical conductivity tensor σ for a particular orthorhombic monocrystal (a = 5.10, b = 6.25, c = 2.40) is σ ij =

3 0

0

0 2 0 0

0 5

106 Ω − 1 m − 1

1 Ω−1 m−1 = 1 S m

What current density vector j (units A/m2) is produced by the application of an electric field of 102 V/m along the [1 1 1] direction of the crystal? What is the angle between j and E? What is the magnitude of σ along the [1 1 1]?

6.4 Electrical Conductivity

Solution From the form of Eq. (6.29), it is seen that the components of j are given by j1 = σ 11 E1 + σ 12 E2 + σ 13 E3 j2 = σ 21 E1 + σ 22 E2 + σ 23 E3 j3 = σ 31 E1 + σ 32 E2 + σ 33 E3 The components of the electric field vector E now need to be found along the three mutually perpendicular axes (x, y, z), which are simply the projections of the vector on the axes; for this, direction cosines are used. The vector E is directed along [1 1 1], which is the body diagonal. The body diagonal runs from the origin with Cartesian coordinates (x1, y1, z1) = (0, 0, 0) to the opposite corner of the unit cell, in this case with Cartesian coordinates (x2, y2, z2) = (a, b, c) = (5.10, 6.25, 2.40). So, the direction cosines are given by cos α = cos β = cos γ =

x2 − x1 x2 − x1

2

+ y2 − y1 y2 − y1

2

+ z2 − z1

2

x2 − x1

2

+ y2 − y1 z2 − z1

2

+ z2 − z1

2

x2 − x1

2

+ y2 − y1

2

+ z2 − z1

2

The orthogonal lattice translations are a = 5.10, b = 6.25, and c = 2.40. Hence, 5 10

cos α = 5 10

2

5 10

2

+ 6 25

= 0 6059 2

+ 2 40

2

2

+ 2 40

2

2

+ 2 40

2

6 25

cos β =

+ 6 25

= 0 7426

2 40

cos γ = 5 10

2

+ 6 25

= 0 2851

With the construction of a unit vector along E using the direction cosines as its components along the x, y, and z directions, the vector E can be expressed as E = E cos αi + E cos βj + E cos γk Hence, the application of 100 V/m in the direction of E gives E = E cos α E cos β

E cos γ = 60 59

74 26 28 51

And, from Eq. (6.29), the current density vector is j = σ 11 E cos α σ 22 E cos β = 18 177

14 852

σ 33 E cos γ 106 A m2

14 255 107 A m2

The magnitude of the conductivity in the direction [1 1 1] is σ=

0 6059

2

3 + 0 7426

2

2 + 0 2851

= 1 101 + 1 103 + 0 406 = 2 61 × 106 S m

2

5 10 − 6

275

276

6 Transport Properties

The angle between j and E is given by the dot product 60 59 18 177 + 74 26 14 852 + 28 51 14 255 107

cos θ = 60 59

2

+

74 26

2

+ 28 51

2

18 177 × 107

2

+ 14 852 × 107

2

+ 14 255 × 107

2

Hence, θ = 18 . For simplicity, in this discussion of the physical basis for electrical conductivity, it shall henceforth be presumed that σ is a scalar; that is, consideration will only be given to isotropic media such as cubic crystals or nontextured polycrystalline samples. Likewise, the theory of current flow in bipolar devices, such as through silicon p–n junctions, belongs in the realm of electrical engineering or semiconductor physics and is only briefly discussed in this text. In the classical Drude model (based on the Boltzmann transport formalism) of an electron gas (Drude 1900), electrons are assumed to behave as classical particles colliding with ions. It is predicted that the electronic conductivity, σ (in units of Ω−1 m−1), is proportional to a quantity known as the electron relaxation time, which characterizes the decay of the drift velocity upon removal of the electric field. The DC conductivity, σ 0, and AC conductivity, σ(ω), are expressed as σ0 =

nq2 τ σ0 σ ω = 1 − iωτ me

6 30

where n is the electron density (m−3), q is the elementary electron charge (coulombs), me is the electron mass (kg), ω is the frequency (Hz), and τ is the relaxation time (seconds). Grouping some of the terms into a single parameter for the electron mobility, μn, gives a well-known alternative expression for the DC conductivity: σ = nqμn

6 31

in which μn = qτ/me. The electron mobility may also be defined as μn = v/E where v is the steadystate drift speed (the magnitude of the drift velocity) of the electron in the direction of an electric field of magnitude E. Its units, in this case, are almost always cm2/(V s) or m2/(V s). While only electrons exist in metals, with semiconductors, there may be a contribution from both mobile electrons in the conduction band and mobile holes in the valence band. Thus, the expression for the electrical conductivity in a semiconductor with these two types of charge carriers becomes σ = nqμn + pqμp

6 32

where p is the hole density (m−3) and μp is the hole mobility. In order for the Drude model to yield Ohm’s law (Eq. 6.29), one must neglect the influence of the applied electric field on the drift velocity of the charge carriers. That is, the relaxation time has to be considered independent of the applied field strength. Example 6.6 The mean atomic mass of potassium is 39.10 amu, its density is 0.86 × 103 kg/m3, and its electron configuration is [Ar]4s1. What is n, the number of valence electrons per unit volume? If the electrical conductivity is 0.143 × 108 Ω−1 m−1, what is τ, the relaxation time between collisions? Solution The density of K = 0.86 × 103 kg/m3 = 0.86 g/cm3. Dividing this by the mass of one mole, 39.10 g, gives the number of moles of K atoms per cubic centimeter: 0 86 39 10 = 0 022 mol cm3

6.4 Electrical Conductivity

Multiplying this by Avogadro’s number yields the number of K atoms per cubic centimeter: 6 0223 × 1023 × 0 022 = 1 3 × 1022 atoms cm3 Since there is one 4s valence electron per K atom, the electron density, n, is also equal to 1.3 × 1022 valence electrons per cm3, or 1.3 × 1028 per m3. Drude’s formula can now be used to calculate the relaxation time between collisions: τ = me σ e nq2 τ = 9 11 × 10 − 31 kg 0 143 × 108 Ω − 1 m − 1

1 3 × 1028 m − 3

− 1 6 × 10 − 19 C

2

τ = 3 9 × 10 − 14 s It is not immediately obvious that the units of τ should be seconds. However, this can be confirmed by making the following substitutions: Ω=V A=V C s ,

V = J C,

J = kg m2 s2

Equation (6.30) predicts that electronic conductivity is dependent on the electron relaxation time. However, it suggests no physical mechanisms responsible for controlling this parameter. Since electrons exhibit wave–particle duality, scattering events could be suspected to play a part. In a perfect crystal, the atoms of the lattice scatter electrons coherently so that the mean free path of an electron is infinite. However, in real crystals, there exist different types of electron scattering processes that can limit the electron mean free path and, hence, conductivity. These include the collision of an electron with other electrons (electron–electron scattering), lattice vibrations, or phonons (electron–phonon scattering), and impurities (electron-impurity scattering). Electron–electron scattering is generally negligible owing to the Pauli exclusion principle. Nonetheless, it is important in the transition metals, where s-conduction electrons are scattered by d-conduction electrons. Electron–phonon scattering is the dominant contribution to the electrical resistivity in most materials at temperatures well above absolute zero. Note that both these scattering processes must be inelastic in order to cause a reversal in the electron momentum and increase the electrical resistance. At low temperatures, by contrast, elastic electron-impurity scattering is the dominant mechanism responsible for reducing electrical conductivity. Anderson showed that multiple elastic scattering in highly disordered media stops electron propagation by destructive interference effects (Anderson 1958). It has also been shown that, even in a weakly disordered medium, multiple elastic scattering can be sufficient to reduce the conductivity (Bergmann 1983, 1984). Elastic electron-impurity scattering is further discussed in Section 7.2. In addition to the classical Drude model, there is a semiclassical (Boltzmann) equation and a quantum (Kubo–Greenwood) equation for the conductivity tensor. These equations are not presented for they are quite abstruse and well beyond our scope. Nevertheless, it can be stated that in the Boltzmann model, the electrons behave as particles between collisions, while the collisions themselves are treated with statistical mechanics. This model is more applicable to the case of a small concentration of impurity atoms (rare scattering) within a host. The Kubo–Greenwood formalism is fully quantum mechanical and able to treat even disordered alloys. It is the most common method used to calculate transport properties for materials under extreme conditions. Today, there is research being conducted that attempts to use DFT-derived equilibrium electron density as a generator for electron dynamics and therefore the calculation of transport properties. This is thought possible because the Coulomb interactions in the Kubo–Greenwood model occurs via the Kohn–Sham potential (Duffy et al. 2018).

277

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6 Transport Properties

6.4.1

Band Structure Considerations

How both the density and mobility of charge carriers in metals and band semiconductors (i.e. those in which electrons are not localized by disorder or correlation) are influenced by particular features of the electronic structure, namely, band dispersion and band filling, will now be examined. Taking mobility first, this book will briefly revisit the topic of band dispersion. Charge carriers in narrow bands have a lower mobility because they are almost localized in atomic orbitals that overlap very poorly. Although this may seem intuitive enough, a more convincing case can now be built simply by looking at some fundamental physical relationships. Consider the meaning of the wave vector k, which is equal to π/a. Since λ = 2a, k must also be equal to 2π/λ. The de Broglie relationship states that λ = h/p where p is the electron momentum. Therefore, p = hk/2π = ħk. The velocity, v, is related to the momentum through v = ħk/m. However, it is also known that the electron energy E(k) is equal to ħ2k2/2m. Hence, dE(k)/dk = ħ2k/m = vħ. Since dE(k)/dk is the slope of the tangent line to E(k), a conduction electron in a wide band (greater slope) has a higher velocity and higher mobility than one in a narrow band. Consequently, it follows from Eq. (6.31) that a wide band metal, that is, a metal with wide bands in its band structure diagram, will have a higher electrical conductivity than a narrow-band metal. A practical application involves the piezoresistive effect, where mechanical strain results in changes to the interatomic spacings and, hence, bandwidth, which is utilized in strain gauges. Turning now to the other factor, charge carrier density, requires an examination of band filling. In the absence of strong correlation or disorder, band filling determines the type of electrical behavior observed in a solid (insulating, semiconducting, semimetallic, or metallic). When spin degeneracy is included, each CO can hold two electrons per bonding site, for a total of 2N electrons, where N is equal to the number of atoms in the crystal. The lowest COs are filled first, followed by the next lowest, and so on, in a completely analogous manner as done in a MO energy-level diagram. Completely filled low-lying COs make up the valence band. The conduction band consists of the higher energy COs that are either vacant or partially filled. 6.4.1.1 Conductors

Considering only the ground-state band in the tight-binding approximation, Bloch’s picture of the electron wave function in a periodic lattice was successful at explaining how electrons are free to move in a metal and conduct an electrical current. Unfortunately, immediately after Bloch’s work, the existence of insulators became puzzling! The currently accepted explanation for the difference between insulators and metals (we prefer the term “conductors,” as some materials that can pass an electrical current do not exhibit other attributes normally attributed to “metals,” but we may nevertheless use the terms interchangeably in this book) was first proposed by the British physicist Sir Alan Herries Wilson (1906–1995) while working with Bloch and Heisenberg at Leipzig. Recall how the sign of k corresponds to the direction of motion of the electron. Application of an electric field causes a shift, Δk, of the entire electron distribution in an energy band, which accelerates the electrons, increasing their momentum in the direction of the field. In a half-filled band (all electrons have the same spin), there are available k states immediately above the highest filled ground state that can become occupied. For a system of N atoms, the energy separation between states becomes infinitesimal as N approaches infinity. Hence, the field can cause a change in the distribution of the electrons and produce a net current flow in one direction. However, when all the k states in an energy band are filled (two electrons, paired with opposite spin, per bonding site) and a sizable energy gap separates this band from the next highest band, the application of an electric field cannot bring about a shift in the k distribution of the electrons (Wilson 1931).

6.4 Electrical Conductivity

The highest filled ground state is termed the Fermi level, EF. The energy of this level at 0 K is the Fermi energy. The Fermi energy is only defined at absolute zero, while the Fermi level is defined for any temperature. All states above the Fermi level are empty. Thus, the criterion for metallic conduction in the Bloch–Wilson band picture can be stated as follows: If the Fermi level lies in a partially filled band of delocalized states (i.e. there is a nonzero density of states at EF), band theory predicts metallic behavior. It should be noted that in both the high-lying conduction band states and the low-lying valence band states, the Bloch wave functions comprising the band are delocalized throughout the crystal. This may seem counterintuitive at first, since it might appear as though the presence of delocalized electrons is a necessary, but not a sufficient, condition for electronic conduction. Perhaps, the confusion can be cleared up with a more chemical picture. If the MO description of benzene is recalled, a similar situation is found. Even though the π-bonding and antibonding MOs in benzene are delocalized throughout the entire molecule, the benzene molecule is not metallic-like. Benzene, rather, can be considered a molecular resonant tunneling transistor. Single molecules of benzene sandwiched between two gold electrodes can only transport a current upon application of a potential difference via an excited state in which an electron is excited from a filled π-bonding MO to a vacant π∗-antibonding MO, where it can be accelerated by the electric field. At low voltages, the π∗ state is the sole contributor, while at higher voltages, the π states also participate (Pantelides et al. 2002). Molecular electronics, which is based on the goal of using single molecules as active devices, is a concept that has been around for three decades (Aviram and Ratner 1974). Unfortunately, it is not within the scope of this book. The absence of a band gap also means that there are accessible states available for excited electrons to populate. Such solids, in addition to being good conductors, will attenuate/absorb specific wavelengths of visible light (those matching the energies required for the transition, 1.7–3.0 eV) at their surfaces and are thus dark colored. We will discuss optical properties of materials in Chapter 9. The color of a particular solid depends upon the range of wavelengths that are not attenuated/absorbed, i.e. it will depend on those complementary wavelengths that are reflected (in this case, the electrons of atoms on the material’s surface vibrate for short periods of time and then reemit the energy as a reflected light wave). The sum of the absorbance, the reflection, and the transmission (that portion of incident radiant flux passing through a sample and out the opposite side, which is null for an opaque solid) of the incident flux is equal to unity. For most metals, the reflected photons have a wide range of wavelengths, which makes the metallic surface appear silvery. A few metals, such as copper and gold, absorb light in the blue region and reflect light with wavelengths toward the red end region of the spectrum (400–700 nm), and therefore they appear yellowish, as can be deduced by the ROYGBIV color wheel. On the other hand, the intensity of the lustre (a.k.a. luster) of a metal depends upon the amount of light reflected from the surface, which is generally related to the refractive index of the surface of the mineral. Metallic lustre is produced by minerals with a surface having a refractive index of greater than 3. Color does not usually affect lustre. 6.4.1.2

Insulators

Band theory predicts insulating behavior if there is a wide energy gap (say, Eg > 3 eV) separating the top of a completely filled valence band and the bottom of an empty conduction band. For example, the band gaps of diamond and aluminum nitride are, respectively, 5.5 and 6.3 eV. No net current flow can occur in one direction or the other within the filled valence band of an insulator, and the WBG prevents electrons from being thermally excited into singularly occupied states in the conduction band. However, if these materials are doped, they may be considered WBG extrinsic semiconductors (Section 6.4.1.3). Since energy varies with k, in a material with a band gap, the bottom of the conduction band can occur at the same point (termed a direct gap) or at a different point

279

280

6 Transport Properties

EF EF

EF

Insulator wide band gap

Semiconductor narrow band gap

EF

Metal zero band gap

Semimetal zero band gap

Figure 6.3 Schematic diagrams illustrating the different classes of electrical materials at absolute zero based on the density of states (DOS) and the width of the band gap around the Fermi level. At higher temperatures, the DOS is smeared across the Fermi level due to thermal excitations, resulting in a continuous occupation function over the distribution of states. The diagram shown for semiconductors is for the intrinsic type. For extrinsic (doped) semiconductors, the Fermi level still lies within a band gap, but is closer to one of the two band edges.

than the top of the valence band (termed an indirect gap). Band structure diagrams show only k values of one sign, positive or negative. In insulators, the chemical potential of the highest energy electron (Fermi energy) falls midway in the band gap at 0 K where there is a zero DOS, as illustrated in Figure 6.3. One may immediately ask the question of how the Fermi energy can exist inside the band gap where there are no allowed states that can be filled with electrons. Firstly, this situation is a consequence of the Fermi–Dirac distribution, which does not imply that states actually exist between the VBM and the CBM. Rather, it only gives information about whether, if such states were to exist, they would be filled or empty. The Fermi–Dirac distribution, 1/{exp[(ε − μ)/kT] + 1}, must fall within the range 0 < n < 1. For a state with energy εi, if the distribution is close to 1, there is a greater chance the state is occupied. If the distribution is close to 0, there is a greater chance the state is not occupied. The Fermi level εF is defined as that location where the probability of finding an occupied state is, should such a state exist, equal to 50%. A closely related quantity is the Fermi function, 1/{exp[(ει − εF)/kT] + 1}. Hence, the Fermi function is essentially equal to unity up to εF (ει < εF) and zero above εF (ει > εF). The product of the Fermi function (the probability of occupation of the state) and the electron DOS (the number of available states) gives the actual population of electrons at some value of ει. However, in the band gap, there are no electrons because the DOS is zero. We will discuss the topic of dielectrics in Chapter 8. However, now is an appropriate time to briefly distinguish electrical insulators from dielectrics. All dielectrics are electrical insulators, but an electrical insulator need not be a dielectric. As just discussed, an electrical insulator is a material that does not conduct an electric current under the influence of an electric field. The defining feature of an insulator is the large band gap separating a completely filled valence band from the completely empty conduction band above it in energy. However, there is always some voltage (called the breakdown voltage) that gives electrons enough energy to be excited into the conduction band. Once this voltage is exceeded, the material ceases being an insulator and conducts current. On the other hand, a dielectric is an insulator with a high polarizability, expressed by a number called the relative permittivity (also known in older texts as dielectric constant). Dielectrics are useful in energy storage devices. There are a number of properties in addition to polarizability a dielectric must possess to function properly under a set of specified operating conditions (e.g. temperature, electric field, frequency). These include a high breakdown voltage, very low electrical conductivity, and dissipation factor, among others. Most of these are sensitive to the presence of

6.4 Electrical Conductivity

defects, as described in Section 6.2, and oftentimes the desired properties are incompatible or contradictory. Because of this, practitioners in the dielectric formulation industry are fond of claiming that it is as hard to make the perfect dielectric as it is to make the perfect conductor. In a special class of recently discovered band insulators known as topological insulators (e.g. Bi1 − xSbx at x ~ 0.04), spin–orbit effects (Section 8.3.4) are large, and this results in spin ordering in which spins of the opposite sign (inversion of parity) counter-propagate along the edge. As a result, these surfaces have been found to be two-dimensional (2D) conductors. In other words, in a topological insulator, the surface is metallic on all faces while the interior is insulating. To understand how this can be, recall that both the surface and the bulk states of electrons in crystalline solids are described by wave functions obtained from solving Schrödinger’s equation. There will be gaps in the electronic energy spectrum where no wave solutions are possible inside the bulk crystal. If the Fermi level lies inside the band gap, the solid is insulating. However, dangling bonds, or a reorganization of atoms at the surface, can introduce states that have energies that lie within the band gap, but are restricted to move around the 2D surface. In most cases, these conducting surface states are very fragile, and their existence depends on the details of the surface geometry and chemistry. In contrast, in a topological insulator, these surface states are protected, that is, their existence does not depend on how the surface is cut or distorted. The Hamiltonian permits conducting states that circulate along the edge (in a 2D insulator) or the surface (in the 3D case), and no simple deformation to the edge or surface can destroy these conducting states. What appears to be a prerequisite for this topological metallicity is that the surface states are linear in momentum and that they meet at an odd number of points in k-space. Kramers’ theorem states that the degeneracy of states with an even number of electrons that obeys time-reversal symmetry will always be lifted (as are the surface states in graphene). By contrast, the surface states of topological insulators are said to be topologically protected (Zhang 2008). Optical properties are the subject of Chapter 9, but it warrants mentioning here that the presence of a WBG (> ~3 eV) usually means that there are no accessible band states available for electrons to be excited into by incident radiant flux of visible wavelengths (1.7–3 eV). However, color may still arise from excitations involving localized or discrete levels like d–d electron transitions or from transitions involving electrons spatially localized (trapped) at point defects. Perfect, structurally intrinsic defect-free and impurity-free WBG opaque solids are thus colorless, as they do not attenuate incident radiant flux composed of visible wavelengths. If white light falls on them, they appear white because all the incident wavelengths will get reflected, the term “transmittance” being reserved for describing the process by which incident radiant flux leaves a medium from a side other than the incident side, usually the opposite side. However, an appearance of color may arise, depending on the impurity level and the intrinsic defect level, provided these optical processes that may occur obey numerous selection rules. The color of a WBG material can vary all the way from colorless to black depending on the impurities. For example, pure silicon carbide (see Section 6.4.1.3.1) has a band gap around 3.3 eV. The very-high-purity material is colorless, as expected. However, technical-grade SiC appears black due to iron impurities. SiC has a refractive index of 2.65, and although submetallic lustre is normally formed by minerals with refractive indices between 2.6 and 3.0, SiC is lustrous due to a passivation layer of SiO2 that forms on its surface. Optical properties are discussed in Chapter 9. 6.4.1.3

Semiconductors

In addition to metals and insulators, another electrical category is the intrinsic semiconductor. As mentioned briefly at the beginning of Section 6.2, one type of semiconductor transport is known as hopping, or impurity conduction. Our coverage of this topic is deferred to Chapter 7. The ensuing

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discussion will be concerned only with monocrystalline semiconductors for which the band model is valid. While scattering processes in both semiconductors and metals increase with rising temperature, thereby decreasing the mobility of the carriers, the scattering is more than offset in a semiconductor by an increase in the charge carrier concentration. Thus, the electrical resistivity of a semiconductor decreases with increasing temperature, dρ/dT < 0. In a metal, the resistivity has the opposite temperature dependency: dρ/dT > 0. 6.4.1.3.1 The Band Gap

Relative to insulators, in semiconductor materials, the Fermi level resides within a narrow band gap (Eg < 3 eV), allowing electrons to be thermally excited from the valence band to the conduction band. For example, the band gaps for the common semiconductors indium arsenide, germanium, silicon, gallium arsenide, and gallium phosphide are, respectively, 0.36, 0.67, 1.12, 1.43, and 2.26 eV. The excitation of an electron from the valence band produces a hole, or positive charge carrier (it behaves like an electron with a positive charge), in the valence band so that current flow in a semiconductor is composed of both electron flow and hole flow. In such intrinsic semiconductors, under thermal equilibrium, the concentration of electrons and holes is equivalent. In pure silicon, the concentration of electrons (or holes) at 300 K is approximately 1.6 × 1010 cm−3. The concept of a hole was first developed in 1931 by Heisenberg (concurrent with Wilson’s work), based on the work of Peierls (Heisenberg 1931). However, the experimental proof of hole flow is made by measuring the so-called Hall effect (Section 6.3.2.3), which was discovered in 1879 by the American physicist Edwin Herbert Hall (1855–1938) while he was working on his doctoral thesis at John Hopkins University. Above absolute zero, there is always some thermal energy available for the electrons to occupy the energy states that are unoccupied at absolute zero. There are different statistics to describe the occupation of energy levels. The simplest is Boltzmann statistics, which would work for a system of non-interacting particles, but since electrons have a strong repulsive interaction, they obey Fermi–Dirac statistics. Thus, at finite temperatures, the DOS is smeared across the Fermi level due to thermal excitations, resulting in a continuous occupation function over the distribution of states, rather than the states being totally unoccupied as depicted. Most semiconductors with band gaps less than 1.7 eV, as well as metals and semimetals, are dark colored because characteristic wavelengths of visible light (matching the energies required for electronic excitations from the valence band to the conduction band) are absorbed/attenuated while the complementary colors get reflected, provided any applicable selection rules are obeyed (see Chapter 9). Wavelengths in the visible region have energies corresponding from 1.7 to 3.0 eV. In addition to these inter-band electronic transitions, color may also arise from other mechanisms such as transitions involving electrons localized at point defect sites, and even band-to-impurity excitations (we will see this with AlN in just a few paragraphs), so long as the wavelengths correspond to the energy required to promote the electron to an accessible excited state. For most narrow-band-gap semiconductors and semimetals (see Section 6.4.1.4), the transmitted photons have a wide range of wavelengths, which makes the metallic surface gray or silver, in a similar fashion to metals. The so-called WBG semiconductors have band gaps typically between 2 and 4 eV, but sometimes even higher. Examples include 4H-SiC (Eg ~ 3.3 eV), GaN (Eg ~ 3.4 eV), and AlN (Eg ~ 6 eV). These materials allow power electronic components (e.g. insulated-gate bipolar transistors [IGBTs], metal–oxide–semiconductor field-effect transistors [MOSFETs], Schottky diodes) to be smaller, faster, more reliable, and more efficient than their silicon (Si)-based counterparts. This is because, owing to their larger band gaps, high critical field strength, higher electron mobilities, and higher electron saturation velocities, the WBG semiconductors can operate at much higher temperatures,

6.4 Electrical Conductivity

voltages, and frequencies than conventional narrow-gap semiconductors. Applications include electric grid-connected solar photovoltaic (PV) DC-to-AC inverters (which feed the PV energy from solar panels to the power grid with the maximum possible efficiency), power switches, inverters, chargers, and controllers in electric and hybrid vehicles, power conversion circuits, and sensors and military applications. Both SiC and GaN melt incongruently (decomposing into separate phases of different compositions) at atmospheric pressure. This makes it impractical to grow high-purity monocrystalline boules by a melt technique (e.g. Czochralski process, Bridgman–Stockbarger method), as is done with silicon, germanium, or GaAs (although GaAs vaporizes incongruently, it melts congruently). Rather, SiC boules up to about 30 mm in diameter and 20 mm in length are grown by sublimation methods and then sliced into wafers onto which n-type or p-type epitaxial layers are grown. Semiconductor-grade GaN wafers are fabricated by direct deposition via vapor phase epitaxy onto a variety of different substrates, including native GaN, diamond, silicon carbide, sapphire, AlN, and silicon. Both SiC and GaN wafers are thus much more expensive than Si wafers. Some of the WBG semiconductors emit light (not to be confused with absorption and transmittance) in the visible color range of the electromagnetic spectrum. Until the 1990s, SiC was used for commercial yellow and blue light-emitting diodes (LEDs), but it was later replaced by GaN, which is the enabling material behind today’s modern ultra-high-efficiency LEDs used in solid-state lighting. At the present time, the two most widely used WBG semiconductors in commercial applications, including optoelectronics (LEDs, laser diodes), power electronics devices, and radiofrequency (RF) applications are, by far, 4H-SiC and GaN. While GaN has a slightly larger band gap, a slightly higher critical field, and a much higher electron mobility than SiC, it has a much lower thermal conductivity, which makes thermal management a challenge for power system designers. Additionally, at the present time, GaN substrates are only widely commercially available in smaller diameters, which increases the overall cost of producing individual die. Besides their higher cost, there are technical challenges that currently limit the more widespread use of WBG materials in other applications like VLSI and ULSI microprocessor chips with densely packed internal circuitry. For example, the total dislocation density of GaN substrates is still high (~108–1010 defects/cm2) as is the interfacial trap density of SiC with its native SiO2 oxide (two times that of Si/SiO2). Chemical vapor deposited (CVD) diamond is sometimes actually considered the “ultimate” WBG semiconductor material in some contexts, rather than an insulator. The dielectric strength of diamond (10 × 106 V/cm) is much higher than that of SiC (3.3 × 106 V/cm) or GaN (3.5 × 106 V/cm). Therefore, only a few micrometers is required to insulate very high voltages, compared to these other semiconductors. In fact, it is believed that diamond semiconductors could theoretically replace the last few remaining niche applications still using vacuum tubes, such as magnetrons, klystrons, and traveling-wave tubes. However, while it does have higher electron and hole mobilities than either SiC or GaN, as well as a high thermal conductivity, a very high resistivity, and temperature stability, it is particularly difficult to find suitable n-type dopants for diamond (see extrinsic semiconductors below). Its extreme hardness and chemical inertness also require the development of novel fabrication processing. Another material with an even wider band gap than diamond is aluminum nitride, AlN. This compound has Eg ~ 6.1 eV and could thus be referred to as an ultra-WBG semiconductor. Because of its very large direct band gap, AlN appears to be a promising candidate, especially when alloyed with GaN (AlN forms a complete solid solution with GaN), for energy-efficient short-wavelength (blue, near to mid ultraviolet [UV]) optoelectronic devices such as LEDs and lasers. One such potential application is mid-UV LEDs for water and air disinfection. Deep-UV devices using

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AlN are still in the research stage. The yellow impurity in AlN (due to a band-to-impurity absorption involving the excitation of electrons from the valence band to the doubly negative charged 2− 3−/2− 2− 3− , of isolated aluminum vacancies, VAl , described by VAl + hν = VAl + h+) limits state, VAl UV transparency and hence may restrict the applications of AlN substrates for deep-UV optoelectronic devices. AlN is also being studied for use as high-power, high-frequency switching devices with efficient power conversion and as RF components for satellite communications. Particularly, due to its very WBG, high breakdown voltage, and high electrical and thermal conductivities, AlN is in the very early stages of investigation as a potential high-power device, which one day might be able to replace SiC under very high voltage operating conditions, such as in electric power grid, rail traction, and wind turbines (e.g. 10–20 kV), as well as its use in GaN/AlN high electron mobility transistors (HEMTs). First-generation devices of the latter are already commercially available, but they are costly and of low output power and efficiency. As with GaN, AlN is grown by vapor phase techniques. The AlN powder sublimation method is generally recognized as the most promising route currently for the production of the highest-quality bulk AlN polycrystals. As of 2006, AlN crystals grown by this route contained large grains (crystallites) up to 10 mm in diameter (Karas 2006), which can be accessed by post-growth processing. By 2008, 2 in. (50 mm) AlN substrates with 40 mm diameter single-crystal cores and 5 mm thick polycrystalline peripheries had been demonstrated, as well as 1 in. totally single-crystal AlN substrates (Scheel et al. 2010). Even wider, 3 in. diameter totally single-crystal AlN epilayers have been prepared by HVPE, but they have been of lower crystallographic quality, although adequate for many device applications (Segal et al. 2009). The most challenging problem with epilayer deposition appears to be related to the differences in the surface mobilities of Al and Ga adatoms. While the concentration of residual oxygen is high for sublimation-grown AlN, most of the oxygen is incorporated as impurity-forming complexes deep in the band gap, rather than as shallow donors where they would have a much greater impact on conductivity (Bhattacharya et al. 2011). As was discussed in Section 6.2.1, shallow donor and acceptor levels allow the electronic properties of the semiconductor to be treated as though it is unaffected by the impurity atoms, which only increase the electron (or hole) concentration, and thus they mainly control the conductivity. Deep-level traps, by contrast, usually have a negligible effect on electrical conductivity. However, they are generally still undesirable because they offer intermediate states within the band gap and can shorten carrier nonradiative lifetimes, compensate the dominant charge carrier, and enhance radiative recombination and thus need to be controlled for many device applications. The most common acceptor (p-type dopant) in AlN is Mg, while the donor (n-type dopant) of choice has been Si (both residing on Al sites). Both usually result in a low conductivity at room temperature. Doping in AlN is still problematic and controversial. 6.4.1.3.2 Extrinsic Semiconductors: Impurity Doping

Extrinsic semiconductors are those in which the carrier concentration, either holes or electrons, are controlled by intentionally added impurities called dopants. The impurities are termed shallow dopants because their energy levels lie within the band gap close to either the top of the valence band or bottom of the conduction band so that thermal ionization of the impurity with injection of free charge carriers is likely. The gap between these energy states and the nearest energy band is referred to as dopant-site bonding energy, and it is much smaller than the intrinsic semiconductor’s band gap. We have already introduced the subject of dopants in Section 6.2 during our discussion of defect chemistry and its role specifically in controlling electrical conduction in ceramics. There, we defined donors as aliovalent ions with a higher valance than the cation they replace. Likewise, acceptors were defined as aliovalent ions with a lower valance than the cation they

6.4 Electrical Conductivity

replace. For semiconducting elements and compounds, somewhat simpler and perhaps more intuitive definitions are possible. In semiconducting elements and compounds, acceptors (p-type dopants) are impurities that can accept electrons from the valence band, which is the equivalent of introducing holes into the valence band. Acceptors have fewer valence electrons than the host atoms they replace. Donors (n-type dopants) are impurities that can donate electrons to the conduction band. Donors have more valence electrons than the host atoms they replace. There are certain exceptions to the rule concerning the number of valence electrons discussed below. With thermal excitation, dopant ionization occurs, and n-type dopants (donors) are able to donate electrons to the conduction band, while p-type dopants (acceptors) can accept electrons from the valence band, the result of which is equivalent to the introduction of holes in the valence band. Dopants also shift the energy bands relative to the Fermi level. The energy band that corresponds to the dopant with the greatest concentration ends up closer to the Fermi level. Band-gap widening/narrowing may occur if the doping changes the band dispersion. Well-known examples of acceptors in silicon are group III elements (B, Al, Ga, In), while wellknown donors in silicon are group V elements (P, As, Sb). In compound semiconductors, the role of donor or acceptor depends on which atom type the dopant substitutes for in the lattice. The electrical conductivity in III–V nitrides was briefly discussed in Section 6.2.4, where we mentioned that n-type conductivity in GaN for a long time was believed to be due to the presence of native defects like nitrogen vacancies, but is now attributed to the accidental introduction of extrinsic silicon and oxygen impurities. In both GaN and AlN, Si and Mg atoms can substitute for the Al or Ga atoms, which have three valence electrons. Hence, an Mg atom, with its two valence electrons, serves as acceptor, or p-type dopant, while a Si atom, with four valence electrons, serves as donor, or n-type dopant when substituting for aluminum in AlN or gallium in GaN. However, in general, it is possible that a group IVA atom (C, Si, Ge) can become a donor when incorporated on a trivalent atom site (e.g. Al, Ga, etc.) or an acceptor on the pentavalent atom site (e.g. N, As, etc.) of a III–V semiconductor. Likewise, group VIA atoms (O, S, Se) function as donors on the trivalent atom site or as acceptors on pentavalent atom site. When a dopant does simultaneously substitute for atoms on both sublattices, it self-compensates. However, where there are large differences between the atomic radii of the constituent atoms, such as in AlN or GaN, one might expect self-compensation to be blocked by strain effects. For example, a carbon atom in GaN should be able to substitute only for nitrogen, since C and N have similar atomic radii, while the substitution for the much larger gallium might be expected to induce a large lattice strain energy. However, in competition with the strain effects are processes of electron transfer from donors to acceptors. Due to the WBG of nitrides, they lead to large energy gains, thereby enhancing self-compensation. The tendency toward self-compensation is further increased by the formation of donor–acceptor pairs (Bogusławski 1997). The selection of a particular dopant of a specific type (e.g. n-type) is therefore dictated by the dopant’s solubility in the host, its mode of incorporation into the structure (purely substitutional is preferred over partial interstitial in order to eliminate anomalous effects), and its diffusivity. In addition to the already mentioned SiC and GaN, there are a large number of compound semiconductors with a broad range of band gaps that have been under consideration, at one time or another, for various applications. Examples of common III–V phases include GaSb (laser diodes with low threshold voltage, photodetectors with high quantum efficiency, high-frequency devices, booster cells in tandem solar cell arrangements for improved efficiency of PV cells and

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high-efficiency thermophotovoltaic cells and microwave devices), AlSb (microelectronic devices [HEMTs and heterojunction bipolar transistors {HBTs}] that can operate at ultrahigh switching speeds [digital systems] with ultralow power consumption, gamma radiation detector), GaP (LEDs), GaAs (lasers, LEDs, PV cells), AlAs (LEDs, laser diodes, PV devices), AlP (LEDs), and InAs (infrared [IR] detectors, PV devices). There are, of course, numerous others such as the II–VI selenides, tellurides, and sulfides. Each of the compound semiconductors has their own unique challenges or limitations associated with crystal growth, phase stability, doping, carrier mobility, etc. Many can be alloyed with additional elements to afford the possibility of bandgap engineering through altering the ternary phase (sometimes quaternary) composition, which can lead to the creation of highly tailored electrical properties. In the fabrication of integrated circuits on semiconductor wafer substrates, dopants have been added since about the 1980s mostly via room temperature ion implantation techniques where a beam of positive ions (e.g. B+, P+, As+, Sb+) is scanned over the wafer surface, masked by some substance (e.g. photoresist, SiO2, metal) over those areas where doping is undesired. When ions are implanted in a semiconductor wafer, their charge has to be neutralized. In some cases, this can be accomplished by supplying an electric current from the backside of the wafer. The total current used for this can be measured, which allows a precise calculation of the amount of dopant that is deposited. If an insulator is on the top of the wafer, a current cannot pass through to neutralize the positively charged ions and the insulator may get damaged if there is an excessive amount of surface charge buildup. In order to avoid this, a low-energy secondary electron or plasma electron flood is used concurrent with the ion beam to neutralize the ions and reduce surface charging. After ion implantation, the wafers are then thermally annealed in order to move them from interstitial sites to substitutional sites and to repair amorphization damage to the crystal caused by the ion implantation. In the past, before the advent of ion implantation technology, dopants were added by exposing the wafer to a gaseous dopant source (e.g. PH3, AsH3), which decomposes into the elements at high temperatures diffusing into the bulk. This method lacks the control required of modern processes and devices. Impurities with radii much smaller than the host atoms will preferentially occupy interstitial position in the lattice. For example, the alkali- and alkaline-earth ions may cause deep impurity levels in the band gap and act as recombination centers, or, due to their very low electron affinities and ionization, they may function as shallow interstitial donors even though they have fewer valence electrons than the silicon atom. In fact, alkali metals have been studied as silicon dopants for decades. Extensively investigated cases include interstitial site Na, Li, and the double-donor Mg (but it has been suggested that Mg serves as an acceptor when it is a substitutional impurity in silicon). However, there are many problems associated with their use. Among them are their high diffusivities through the interstitial channels of silicon, their tendency to cluster at intermediate doping concentrations yielding an impurity band, and the merging of this impurity band with the conduction band, resulting in metallic behavior, at still higher dopant concentrations. The suitability of an impurity as a dopant is thus seen to be more complex than simply being a function of its valence electron count. In general, there are four factors that can limit the doping ability of a substitutional impurity in a semiconductor. The first is insufficient solubility of the dopant in the host lattice, for example, with large positive formation enthalpies of dopants, ΔHf(q) (of charge q at the Fermi energy) or with a large discrepancy in the atomic radius of the impurity relative to the host. The second is when the ionizability of the dopant does not allow for a sufficiently shallow level within the band gap. In this case, there is an energetically too deep a distance between the dopant electrical transition and the host crystal’s VBM or CBM. In such a deep trap, the wave function of the trapped electron (or hole)

6.4 Electrical Conductivity

is much more localized. When the ionization energy of the impurity is small, on the order of kT, the impurity generally produces shallow states and acts as a dopant. Deep impurities require energies larger than the thermal energy to ionize so that only a fraction of the impurities present in the semiconductor contribute to free carriers. Deep impurities that are more than five times the thermal energy away from either band edge are very unlikely to ionize. Such impurities are effective Shockley–Read–Hall recombination centers for electrons and holes. The first two factors just discussed are primarily what are responsible for the difficulty in finding suitable n-type dopants for diamond. A third factor limiting dopability of an impurity in a semiconductor is when the host crystal responds to the existence of free carriers, introduced by the dopant, with the spontaneous creation of charge-compensating point defects, for example, electron-compensating cation vacancies (recall Section 6.2). This is the predominant difficulty for many III–V compound semiconductors. Finally, practical consideration must also be given to a fourth factor, the diffusion coefficients of potential dopants, as well (see Section 6.5.1). For example, in SiC, the diffusion of large atoms like arsenic is much more hindered than they are in silicon, due to the small interatomic distances in SiC. Only smaller atoms (e.g. B, Al, N, P) have a fast diffusivity and are able to migrate in SiC to a large extent. Even so, thermal diffusion of these elements typically requires much higher temperatures than the analogous processes in silicon. For example, the diffusivity of Al in silicon is about 1 × 10−13 to 6 × 10−10 cm2/s in the temperature range 1100–1400 C, but in SiC temperatures in the range 1800–2300 C are required to achieve diffusivities in the range of about 3 × 10−14 to 6 × 10−12 cm2 s. These elements (B, Al, N, P) also have a relatively high SiC solubility (~1018 – 1021 cm−3 at T > 2500 C), and they produce shallow levels in the SiC band gap, allowing them to be used for the fabrication of low-resistance SiC layers. Other impurities are less often used for SiC doping, except in high-resistance and semi-insulating SiC layers and special types of LEDs. Because of the low solid-state diffusivities at temperatures below about 1800 C, doping of SiC by what are traditionally called “thermal diffusion” processes (with a gas, solid, or liquid source supplying the dopant) is not at all practical, necessitating the use of ion implantation. Nevertheless, in the post-ion implantation annealing (>1500 C), dopant redistribution occurs by solid-state diffusion and therefore must be investigated. With compound semiconductors, there are also mechanisms by which carriers can become localized, or transition from a shallow to a deep center. One famous example is the DX state in AlxGa1 − x. As alloys in which there is the formation of a negatively charged donor–anion complex (DX−) and an ionized donor state (D+), represented as 2D0 D+ + DX−. A DX center can be considered a donor state that is subjected to strong lattice relaxation, so the donor and its nearest neighbors are reversibly displaced from their symmetric lattice site positions driven by the energy of occupied antibonding orbitals. The DX center is a metastable defect because it has more than one structural configuration for a single charge state and can thus cycle reversibly among these different atomic configurations with accompanying changes to the electrical and/or optical properties of the host semiconductor. Another metastable defect is the EL2 center in GaAs. The EL2 defect is a native lattice defect (not related to an impurity), specifically an antisite double donor that introduces two deep localized levels into the band gap of GaAs. The name EL2 derives from the observed characteristic transitions in the photocapacitance spectroscopy of this material. The controlled introduction of dopants, which are a type of zero-dimensional or point defect, is obviously crucial for manufacturing useful semiconductor devices. However, most other kinds of crystalline defects, particularly extended defects (1D, 2D, and 3D), are normally deleterious to transport properties. For example, dislocations (1D defects) of all kinds (edge, screw, mixed) serve as sinks for metallic impurities and intrinsic point defects. Point defects attached to a dislocation line introduce electronic levels within the band gap near the band edges and are electrically active,

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which can cause catastrophic device failure. A single micropipe (a.k.a. micropore, microtube, or pinhole), which is an empty- or open-core screw dislocation with a large Burgers vector, in an electrically active area of the substrate can destroy a high-voltage SiC power device. The reduction in micropipe density in SiC has thus long been an active area of research. In detectors, isolated point defects and defects associated with dislocations manifest themselves as excess dark current, noise, and reduced responsivity. In light-emitting devices, they reduce radiative efficiency and operation lifetime. Defects are generally the culprits for parasitic current paths, decreased gain, and increased threshold current and are source of instability in electronic devices, particularly field-effect transistors (FETs) (Reshchikov and Morkoç 2005). Dislocations can also disrupt dopant diffusion (concentration–distance) profiles (see Section 6.5). Precipitates and inclusions (3D defects) act as sites for generating dislocations. Likewise, stacking faults (2D defects) can cause device degradation due to higher reverse bias currents through unipolar devices or higher forward voltage drift in bipolar devices (described below). In SiC, stacking faults may be caused by basal plane dislocations (BPDs) in the substrate that propagate into the epilayer. The development of stacking faults can also result in the generation of carrier traps, reducing the carrier lifetime. Methods of production for single-crystalline semiconductors with extremely high purity and low defect density was covered in our companion textbook, Inorganic Materials Synthesis and Fabrication (Wiley & Sons 2008). 6.4.1.3.3 Types of Semiconductor Components for Electronic Circuits

Extrinsic (doped) semiconductors are used to fabricate electronic components such as discrete diodes (two terminals) and transistors (three terminals), as well as integrated circuits (a.k.a. “chips”) that contain hundreds, thousands, millions, and even billions of circuit elements with nanometer linewidths in today’s modern devices. The conventional semiconductor diode is a device containing one p–n junction that serves to rectify AC into DC since an electric current can pass through the device under forward bias (i.e. with the negative terminal of the voltage source connected to the n-type side of the device and the positive terminal of the voltage source connected to the p-side of the device) but not under reverse bias. The physics occurring at a semiconductor p–n junction was discovered in 1939 by Russell Shoemaker Ohl (1898–1987) at Bell Labs. In the earliest days of solid-state devices, the combination of a galena (lead sulfide, PbS) crystal and a “cat whisker” was used to detect AM radio signals, with the galena serving as one half of a diode (the cathode) and the tip of the fine metal wire as the other (the anode), thereby forming a device capable of detecting/rectifying an incoming radio signal into a pulsating DC signal that could drive a pair of earphones. Carborundum (silicon carbide, SiC) was also used for this purpose, in place of the galena crystal. Selenium (consisting of selenium plates sandwiched against other metals) and low-power point-contact germanium diodes were used to rectify relatively low voltages and currents. In power supply circuits, germanium was eventually replaced by silicon junction diodes because of the former’s much higher leakage currents in the reverse direction, particularly at high voltages. There are actually a number of other kinds of solid-state diodes in existence, including the Schottky diode (high-power devices), the Zener diode (voltage regulation), and the avalanche diode (surge protection), but we are not concerned with descriptions of these here. The transistor, on the other hand, can be considered the fundamental building block – functioning as a logic gate – in today’s computer microprocessors and memory chips. Bipolar junction transistors (BJTs) contain three regions termed the emitter, base, and collector separated by two junctions (either p–n–p or n–p–n) and utilize both electron and hole carriers. The point-contact transistor, the forerunner of the BJT, was invented in 1947 by Walter Houser Brattain (1902–1987), John Bardeen (1908–1991), and William Bradford Shockley Jr. (1910–1989), all of Bell

6.4 Electrical Conductivity

Labs. These men shared the 1956 Nobel Prize in Physics for their work. The BJT was conceived in 1948 by Shockley, and a series of critical experiments conducted by many individuals, including John Northrup Shive (1913–1984), Joseph Adam Becker (1897–1961), Gordon Teal (1907–2003), and Morgan Sparks (1916–2008), paved the way for the realization of a working prototype by 1951. While bipolar transistors rely on both majority and minority carriers, n-channel or p-channel unipolar transistors (e.g. junction field-effect transistors [JFETs]) use only a single type of charge carrier. The JFET was also proposed by Shockley, in 1952, and first demonstrated by George Clement Dacy (1921–2010) and Ian Munro Ross (1927–2013) of Bell Labs in 1953. However, a similar device was envisioned and patented as far back as 1925 by Julius Edgar Lilienfeld (1882–1963). An important variant of the basic JFET is the MOSFET developed at Bell Labs in 1959. MOSFETs, like the Schottky diode, are used in high-power devices. Silicon and SiC MOSFETs both use thermally grown native SiO2 as the gate oxide, which is required for a MOSFET. For GaN MOSFETs, both the native oxide on GaN (Ga2O3) and silicon nitride (Si3N4), which readily forms an interface to GaN, have been tried. Unfortunately, these materials have narrow band gaps (1 mS/cm) remains a challenge in solid-state electrolytes. One famous example is NASICON, the acronym for sodium (Na) super ionic conductor, a family of solids with the chemical formula Na1 + x Zr2(PO4)3 − x(SiO4)x (see Section 3.4.1.11), which has been proposed as solid electrolyte for sodium-ion batteries. In these phases, a rigid framework of vertex-sharing ZrO6 octahedra and ZrO4/SiO4 tetrahedra contains a network of connected pores through which sodium ions migrate. Other solid electrolytes exist in the glassy state. For example, certain fast lithium-ion conducting glasses made from lithium thiophosphate, lithium sulfide, and barium-doped lithium hydroxide and lithium chloride have recently been proposed as an electrolyte to replace the flammable liquid electrolyte in conventional LIBs. A lithium-ion conducting glass electrolyte would not be as prone as batteries with liquid electrolytes to the safety hazards arising from lithium finger or dendrite formation during continual cycling, which can result in short circuits between the anode and cathode (Braga et al. 2016). Although the dendrites would still grow from the anode, it would not be as easy for them to pass through the electrolyte material, pierce the separator, and reach the cathode. In many ceramics the ionic conduction occurs predominantly via the movement of one type of ion. For example, in the alkali halides, tcation ~ 1, whereas in CeO2, BaF2, and PbCl2, tanion ~ 1. For any particular solid, the relative activation barriers for the available mechanisms determine whether the anions or cations are responsible for the ionic conduction. For example, in yttriastabilized ZrO2 (YSZ), with the formula Zr1 − xYxO2 − (x/2), aliovalent substitution of Zr4+ by Y3+ generates a large number of oxygen vacancies, giving rise to a mechanism for fast oxide ion conduction. Indeed, it is found that the O2− anions diffuse about six orders of magnitude faster than the cations. It is for this reason that YSZ has been used as the electrolyte (allowing conduction of oxygen ions) in

Practice Problems

SOFCs. Many families of compounds have been classified as superionic conductors, due to an open structure or a high degree of dynamic disorder associated with the crystalline defects present, giving rise to high ionic conductivity. Included among these are silver and copper halides (particularly the iodides), silver and copper chalcogenides, fluorite-structured halides, pyrochlores, spinels, and perovskites. The microscopic mechanisms for ionic conduction are the same as those for chemical diffusion, namely, the vacancy and interstitial models discussed in the previous section. Thus, ionic conduction is facilitated by the same sort of geometric features as chemical diffusion is, such as open channels that result in easier ion movement, since this lowers the magnitude of the ΔHm terms in Eqs. (6.46) and (6.47). Another example is β-alumina solid electrolyte (BASE), wherein Al2O3 is complexed with a mobile ion (e.g. sodium beta alumina) in which the mobile ions are located in sparsely populated layers pillared between spinel blocks. Accordingly, they move through these channels easily due to the large number of vacancies present. In fact, due to its rapid transport of sodium ions, sodium beta alumina has been used as the solid electrolyte in sodium–sulfur batteries, which was pioneered by the Ford Motor Company in the 1960s to power early model electric vehicles.

Practice Problems 1

Give the definition of a second-rank tensor.

2

Write the expression for the ellipsoid of a symmetrical second-rank tensor referred to its principal axes in a monoclinic crystal.

3

What is a phonon? In what types of materials do they contribute substantially to thermal transport?

*4

Many high-temperature superconductors contain copper oxide layers in the ab plane of a tetragonal unit cell. What would be the differences in the magnitude of the electrical conductivity along the a axis, b axis, and ab diagonal in the ab plane?

5

As the grain size of a polycrystalline material decreases, what happens to the thermal and electrical conductivity? Why?

6

What is a wide-band metal and why does it have a higher electrical conductivity than a narrowband metal?

*7

Use the modified Wiedemann–Franz–Lorenz law to estimate the phonon contribution to the thermal conductivity of a semiconductor at 298 K whose total thermal conductivity is 2.2 W/ m/K if the electrical conductivity is 0.4 × 105 S/m.

8

HgCdTe is the traditional semiconductor used for long-wavelength infrared radiation photodiodes operating in the wavelength range from 8 to 12 μm. What must be the band-gap range in these materials?

9

Why are low band-gap materials not well suited for photovoltaic solar power conversion?

321

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10

Compare/contrast the vacancy, direct interstitial, and interstitialcy mechanisms of mass transport.

11

Assuming a nearest-neighbor tetrahedral interstitial distance of √3a/4, use Eq. (6.43) to derive a simple expression for nearest-neighbor interstitial diffusion through silicon (diamond cubic lattice). Hint: Recall this structure can be considered two primitive interpenetrating FCC lattices offset by a displacement of a/4, a/4, a/4, with one half the tetrahedral interstitial sites occupied by silicon self-interstitials.



For solutions, see Appendix D.

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7 Hopping Conduction and Metal–Insulator Transitions It has been seen that, in the Bloch/Wilson band picture, insulators are materials with a completely filled valence band and an empty conduction band, with the two separated by a sizable energy gap. Semiconductors are similar but have a smaller energy gap between the valence and conduction bands. Thermal excitation of charge carriers across the band gap ensures that the electrical conductivity of a band insulator or semiconductor will increase with increasing temperature. In metals (conductors), where the Fermi energy lies in the conduction band, electron scattering mechanisms decrease the electrical conductivity as the temperature is raised. Commonly, the reciprocal quantity, or resistivity, ρ, is used to describe this behavior. An experimental criterion for metallic behavior (and semimetals) is thus that dρ/dT > 0, while for insulators and semiconductors dρ/dT < 0. A material need not possess a low electrical resistivity (e.g. copper has an electrical resistivity of about 10−6 Ω cm) to be regarded as metallic-like; rather its resistivity must merely have a positive temperature coefficient. A material that meets the dρ/dT > 0 criterion for metallicity, but which, nevertheless, exhibits high resistivity, is termed a marginal metal. In addition to band conduction, which was discussed amply in earlier chapters, another type of electrical transport termed hopping conduction (a type of semiconduction) is also possible. This is a thermally assisted motion of the charge carriers between localized sites. It may occur between nearest neighbors or be of variable range. Hopping is important both for materials with disorder and for those where the internuclear distances are above some critical value. Additionally, hopping is also important for materials in which the same transition metal ion is present in two oxidation states since this allows the transfer of charge from one ion to the other without any overall change in energy of the system. In some cases, it is possible to dope many of these materials and drive them to a metal–insulator (M–I) transition, in which the hopping semiconduction converts to band conduction, in which the charge carriers become itinerant (delocalized) so that metallic transport behavior is observed in the presence of an electric field. This is appropriately termed a dopinginduced or filling control metal–insulator transition, since one is filling a formerly vacant conduction band with charge carriers. We will expand on that topic later in this chapter. First, however, we need to more fully develop the concept of the charge carrier localization. About one decade after the development of band theory, two Dutch industrial scientists at the NV Philips Corporation, Jan Hendrik de Boer (1899–1971) (de Boer was later associated with Technological University, Delft) and Evert Johannes Willem Verwey (1905–1981), reported that many transition metal oxides with partially filled bands that band theory predicted to be metallic were poor conductors and some were even insulating (de Boer and Verwey 1937). Rudolph Peierls (1907–1995) first pointed out the possible importance of electron correlation in controlling the electrical behavior of these oxides (Peierls 1937). Dynamic correlation is the term applied to the Principles of Inorganic Materials Design, Third Edition. John N. Lalena, David A. Cleary, and Olivier B. M. Hardouin Duparc. © 2020 John Wiley & Sons, Inc. Published 2020 by John Wiley & Sons, Inc.

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interaction between electrons via Coulombs law, while static correlation is the deviation from the exact energy caused by attempting to represent the wave function by just one determinant. Fermi correlation is due to the Paul antisymmetry of the wave function, but this is accounted for at the single-determinant level. Somewhat later, Nevill Francis Mott (1905–1996), who was at the University of Bristol at the time (Mott became Cavendish professor of physics at Cambridge in 1954), intuitively predicted a metal–insulator transition resulting from a strong Coulomb attraction between the conduction electrons and the positively charged ion cores, which are screened from any one conduction electron by all the other conduction electrons (Mott 1949). A conduction electron, diffusing under an electric field, encounters other electrons as it moves among singularly occupied sites. John Hubbard (1931–1980), who was at the Atomic Energy Research Establishment in Harwell, England, later introduced a model in which a band gap opens in the conduction band because of the Coulomb repulsion between two electrons at the same site (Hubbard 1963, 1964a, b). Thus, by this definition the Mott–Hubbard insulator is only realizable when the conduction band is at integer filling but only partially filled (i.e. an integer number of conduction electrons per atomic site). That is only possible with multicomponent systems, namely, compounds. The opening of a band gap within the conduction band obviously leads to nonmetallic behavior. However, the difference between the origin of gaps in correlated systems and systems with noninteracting electrons must be emphasized. In a Bloch/Wilson insulator, electrons in the valence band are delocalized in the sense that the COs are extended throughout the volume of the crystal. As a particle, the electron has an equal probability of existing in any cell of the crystal. However, the Pauli principle prohibits an electron from moving into an already completely filled CO. There is also a sizeable energy gap (>3 eV) separating the top of the filled valence band and the bottom of the empty conduction band. Hence, no energetically near-degenerate states (immediately above the Fermi level) exist in which an electron can be accelerated by an electric field. By contrast, in a Mott–Hubbard insulator, the Coulomb repulsion between two conduction electrons (with antiparallel spin) at the same bonding site is strong enough to keep these electrons away from one another and spatially localized in individual atomic orbitals, rather than delocalized Bloch functions. A narrow bandwidth and concomitant band gap results. The localized conduction electrons in such a solid are rather like the electrons in the very narrow valence band of an ionic crystal (e.g. KCl). Because of this picture of localized wave functions, such insulators are also sometimes called Heitler–London-type insulators. An electron or hole can jump between its resident localized site and its nearest-neighbor localized site in a completely randomly manner if it receives energy from a photon. If the excitation energy is supplied by an electric field, such small-polaron hopping occurs on average along the field. This motion is referred to as nearest-neighbor hopping, and it is the type of carrier motion in Mott–Hubbard insulators. Like with band insulators (semiconductors), nearest-neighbor hopping exhibits an Arrhenius-type temperature dependency to the conductivity (σ(T) = σ a exp–Ea/kBT), with the activation energy Ea corresponding to the band-gap energy) at high temperatures, which becomes non-Arrhenius as the temperature is reduced. More importantly, at some doping level, such a system may transition from hopping (semi)conduction to band conduction once the magnitude of Ea required for conduction decreases and the conductivity becomes high. The intrasite Coulomb gap is completely unaccounted for in the Hartree–Fock (HF) theory, since both dynamic correlation and static correlation are neglected. In fact, the formal definition of correlation energy is Ec = Eexact − EHF, where EHF represents the HF approximation to the exact quantity. The precise description of this behavior requires the full account of the multi-determinant nature of the N-electron wave function and of the many-body terms of the electronic interactions.

7.1 Correlated Systems

Electron correlation is underestimated in the local density approximation to the density functional theory (LDA–DFT). Simple LDA–DFT calculations are also generally poor at reproducing the metallic state near the Mott transition. There are several reasons for this. First, it is difficult to treat the nonuniform electron densities of correlated systems with localized wave functions. Second, in density functional formalisms there is the inclusion of a self-interaction correction term in the exchange-correlation potential for the Coulomb interaction energy of an electron in a given eigenstate with itself and which is fitted to the results for a uniform electron density. However, in the HF method, the non-local exchange exactly cancels the self-interaction and introduces a potential that splits the manifold of states in a way expected from the Hubbard model. The Hubbard potential is repulsive for less-than-half-filled orbitals and attractive for others. This is the mechanism by which the correction discourages fractional occupations and favors Mott localization. Modern DFT methods have been developed, such as LDA + U, which emulate some of the important features of the HF Hamiltonian and yield better results for the on-site interactions. Essentially, the approach is one of applying the Hubbard Hamiltonian (Eq. 7.2 below) to the strongly correlated states (typically, localized d and f orbitals) and the standard LDA approximation to the remaining valence electrons. The main difference between LDA + U and HF is that the former applies a corrective functional (which formally resembles the HF interaction energy) to a subgroup of single-particle Kohn–Sham wave functions that do not have any physical meaning (other than producing the ground-state density). It is assumed this correction, through its effects on the exchange-correlation functional, will capture the effects of static correlation that result from the multi-determinant nature of the wave function and give a better description of the properties of the correlated system. Of course, in HF calculations, by contrast, no XC energy exists, and the optimized single-particle wave functions are associated with a physical meaning. Electrons in periodic solids can also be localized by disorder. This is a quite different mechanism involving multiple elastic scattering of the conduction electrons by the impurities. When the disorder and, hence, scattering are strong enough, the electrons can no longer propagate through the solid. This is known as Anderson localization, after Philip Warren Anderson (b. 1923), the physicist who discovered it while he was at Bell Laboratory (Anderson was later at Princeton University). However, unlike Bloch/Wilson band insulators and Mott insulators, no band gap is opened in Anderson insulators. There remains a finite DOS at the Fermi level, but the electrons are localized on individual sites, as in Mott insulators, rather than existing as itinerant, or delocalized, particles. Anderson argued that in such systems, electrical transport is semiconduction by non-diffusive, phonon-assisted hopping between localized centers (Anderson 1958). The most frequent hopping is not, however, between nearest-neighbor sites as in Mott–Hubbard systems. The hopping electron or hole will try to find the shortest hopping distance and the lowest activation energy, and because these two conditions are not usually satisfied simultaneously in disordered systems, the electrical transport occurs by variable range hopping (VRH) rather than Arrhenius-type behavior in Anderson insulators. Anderson insulators can also switch from hopping semiconduction to band conduction by a doping-induced M–I transition.

7.1

Correlated Systems

Mott originally considered an array of monovalent metal ions on a lattice, in which the interatomic distance, d, may be varied. Very small interatomic separations correspond to the condensed crystalline phase. Because the free-electron-like bands are half-filled in the case of ions with a single valence electron, one-electron band theory predicts metallic behavior. However, it predicts that the

327

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array will be metallic, regardless of the interatomic separation. Clearly, this can’t be true given that, in the opposite extreme, isolated atoms are electrically insulating. For an isolated atom, the single valence electron feels a strong Coulomb attraction to the ion core, V = −e2/r, where e is the proton charge and r is the distance separating the electron and ion core. In the solid state, the Coulomb attraction of any one valence electron to its ion core is reduced because the electron is screened from the core by all the other electrons of the free electron gas. The potential now becomes V = (−e2/r) exp(−ξr), where ξ is the screening parameter. Mott argued that when the potential is just strong enough to trap an electron, a transition from the metallic to the nonmetallic state occurs. The Mott criterion for the occurrence of a metallic–insulator (M–I) transition was found to be n1c 3 a∗0 = 0 25

71

where nc is the critical electron density and a∗0 is the effective Bohr radius. The screening parameter, ξ, increases with increasing magnitude of the tight-binding hopping integral, which, in turn, increases with decreasing internuclear separation between neighboring atoms. In the context of the tight-binding or LCAO method, rather than considering the screened potential, one usually speaks in terms of the bandwidth, which also increases with decreasing internuclear separation (owing to a stronger overlap between adjacent orbitals). Hence, metallic behavior would be expected for internuclear distances smaller than some critical value. A large internuclear distance results in more localized atomic-like wave functions for the electrons (i.e. the Heitler–London model), as opposed to extended Bloch functions, since the hopping integral and the overlap integral are very small in this case. Thus, Mott recognized that there should be a sharp transition to the metallic state at some critical bandwidth: W = 2zb (Eq. 5.38), where z is the coordination number and b is the hopping integral (Mott 1956, 1958). These arguments are particularly important for the d electrons, since they do not range as far from the nucleus as s or p electrons. The d electrons in solids are best described with localized atomic orbitals if the magnitudes of the hopping and overlap integrals between neighboring atoms are very small (i.e. narrow-band systems). Moving to the right in a period, and for smaller principal quantum numbers, an increase in the extent of d electron localization is expected because of a contraction in the spatial extension of the d atomic orbitals. Valence s and p orbitals are always best described by Bloch functions, while 4f electrons are localized and 5f are intermediate. For heavy elements, however, one may also need to consider s and p orbital contraction owing to both increased electrostatic attraction toward the nucleus (the lanthanide and actinide contraction) and direct relativistic effects. The latter, in fact, causes a d and f orbital expansion, termed the indirect relativistic effect, owing to the increased shielding of those orbitals from the nucleus. A semiempirical approach for characterizing the transition region separating the localized (Heitler–London) and itinerant (Bloch) regimes in terms of the critical internuclear distance was developed by Goodenough (1963, 1965, 1966, 1967, 1971). This method employs the tightbinding hopping integral (transfer integral), b, which is proportional to the overlap integral between neighboring orbitals. The approach is useful for isostructural series of transition metal compounds, since b can be related to the interatomic separation. In the ANiO3 perovskites (where A = Pr, Nd, Sm, Eu), for example, the M─I transition is accompanied by a very slight structural change involving coupled tilts of the NiO6 octahedra. In the ABO3 perovskites, BO6 octahedral tilting is in response to nonoptimal values of the geometrical tolerance factor, t, which places the B–O bonds under compression and the A–O bonds under tension or vice versa (Goodenough and Zhou 1998). Tilting relieves the stresses by changing the B–O–B angles, which govern the transfer

7.1 Correlated Systems

interaction. The smaller the tolerance factor (for t < 1), the greater the deviation of the B–O–B angle from 180 . This reduces the one-electron d bandwidth and localizes the conduction electrons. For ANiO3, only when A is lanthanum, is the oxide metallic. The t’s are smaller (t < 1) for the other oxides in the series, which are insulators. The maximum Ni–O–Ni angle deviation is ~ −0.5 . The structural transition is thus a subtle one. Nonetheless, it may be considered the driving force for the M─I transition, because the changes to the Ni–O bond length affect the d bandwidth.

Sir Nevill Francis Mott (1905–1996) received his bachelor’s degree (1927) and master’s degree (1930) from the University of Cambridge. He became a professor of theoretical physics at the University of Bristol in 1933 and returned to Cambridge in 1954 as Cavendish professor of physics, a post from which he retired in 1971. In his early career, Mott worked on nuclear physics, atomic collision theory, the hardness of metals, the electronic structures of metals and alloys, and latent image formation in photographic emulsions. In the 1940s, he postulated how a crystalline array of hydrogen-like electron donors could be made insulating by increasing the interatomic separation. Mott was to devote the rest of his career to the study of the metal–insulator transition. He was a corecipient of the 1977 Nobel Prize in physics for his work on the magnetic and electronic properties of noncrystalline substances. Mott was elected a Fellow of the Royal Society in 1936. He was knighted in 1962. (Source: Nobel Lectures, Physics 1971–1980, World Scientific Publishing Company, Singapore.) (Photo courtesy of AIP Emilie Segrè Visual Archives. Reproduced with permission.

The crucial step in our present-day understanding of the M–I transition in systems with interacting electrons was Hubbard’s introduction of a model that included dynamic correlation effects – the short-range Coulomb repulsion between electrons with antiparallel spin at the same bonding site (Hubbard 1963, 1964a, b). Long-range (intersite) electron–electron interactions were neglected. The inclusion of dynamic correlation effects in narrow-band systems, such as transition metal compounds, form the basis for what is now known as the Mott–Hubbard M–I transition.

7.1.1 The Mott–Hubbard Insulating State The Hubbard picture is the most celebrated and simplest model of the Mott insulator. It is composed of a tight-binding Hamiltonian, written in the second quantization formalism. Second quantization is the name given to the quantum field theory procedure by which one moves from dealing with a set of particles to a field. Quantum field theory is the study of the quantum mechanical interaction of elementary particles with fields. Quantum field theory is such a notoriously difficult subject that this textbook will not attempt to go beyond the level of merely quoting equations. The Hubbard Hamiltonian is Η = −t i,j

C iσ Ajσ + U i, j, σ

ni ni

72

i

Equation (7.2) contains just two parameters: t, the transfer integral (a.k.a. hopping coefficient), and a local on-site interaction, U. The entire first term is the electron kinetic energy. It describes the hopping, or the destruction of an electron with a certain spin on one site and its creation on an adjacent site. The entire second term is the interaction energy. The local interaction may be dominated by the Coulombic repulsion (U > 0) of two antiparallel spin electrons at the same site, or alternatively, it may be dominated by an attraction (U < 0) that is not a priori known. Many theorists assume the

329

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attraction interaction originates from the coupling of electrons to polarons or local phonons. Finally, some texts include a third term in Eq. (7.2) accounting for the chemical potential that controls the band filling (in statistical mechanics, one can work with either an ensemble of fixed particle number, or one in which the particle number is allowed to vary, controlled by the chemical potential). If U = 0, the Hamiltonian is quadratic. Finding the single-particle states and occupying them accordingly give four eigenstates whose associated operators and energies are easily found. They are just the tight-binding results for two electrons in which the ground state has one electron in each orbital. For small U/t, the wave function is for free electrons in which every electron is shared equally between all sites. However, if we move the ions farther apart, t becomes smaller without changing the wave function. By definition, U does not depend on the interatomic distance, so that in the limit as U/t ∞ one has the up spin entirely at one site and the down spin at the other site. HF actually gets this trend correct. It predicts metallic behavior for small U/t and insulating behavior for large U/t. HF even mimics the antiferromagnetic (opposite spin) ordering of electrons in the Mott insulating phase with one electron per ion. With ferromagnetic (same spin) insulators, the Pauli principle forbids any electron from hopping to its neighbor. Thus, for large U/t, we can replace the Hubbard Hamiltonian with one describing the coupling between the spins localized at two sites, which at least works well if we are interested in the low-energy properties. This is called the Heisenberg Hamiltonian and is described in Chapter 8. The repulsive Hubbard model on a 2D square lattice away from half-filling (average lattice filling of less than one fermion per site) has been proposed as the minimum model necessary to explain d wave high-temperature superconductivity in the doped cuprates. It is interesting, however, that with repulsive (U > 0) interactions (the kind we are concerned with in this chapter), the Mott insulating state, metal–insulator transitions, and d wave superconductivity are predicted, while with small (weakly) attractive U interactions (U < 0), BCS-type electron pairing (Section 6.4.3) and superfluidity is predicted. For sufficiently negative U, the only allowed states for every lattice site would be zero or double occupancy (all electrons paired). The attractive and repulsive regimes of U are related by a transformation as the doped lattice at U > 0 away from half-filling maps onto a U < 0 system at finite magnetization. The situation is very complex and well beyond our scope. It is possible to include interactions between electrons on adjacent sites, and, in fact, 30 years earlier Semen Petrovitch Shubin (1908–1938) and Sergeı ̌ Vasil’evich Vonsovskiı ̌ (1910–1999) (respectively known as Semyon Schubin and variously spelled as “Wonsowsky” or “Vonsovsky” in the Western literature. Shubin was one of the victims of Stalin’s terror, one year after Vadim Gorskiı,̌ Lev Shubnikov, and others (see Hardouin Duparc and Krajnikov 2017) of the Russian school put forth such a model (Schubin and Wonsowsky 1934; Izyumov 1995), in which a third term, containing the matrix element of Coulomb interaction between different sites, is added to Eq. (7.2). Nowadays, the inclusion of next-nearest-neighbor interactions is called the extended Hubbard model. The Hubbard Hamiltonian can be solved exactly only in the one-dimensional case and requires numerical techniques for higher dimensions. Nonetheless, it often adequately describes tendencies and qualitative behavior. The first term in Eq. (7.2), representing the electron kinetic energy, contains the Fermi creation operator (Ciσ ) and annihilation operator (Ajσ) for an electron or hole, at sites i and j with spin σ = or . In quantum field theory, Hamiltonians are written in terms of Fermi creation and annihilation operators, which add and subtract particles from multiparticle states just as the ladder operators (raising and lowering operators) for the quantum harmonic oscillator add and subtract energy quanta. Multiparticle wave functions are specified in terms of single-particle occupation numbers. The second term in Eq. (7.2) representing the interaction energy is a potential energy term where the number of electrons at the site is given by ni. This term runs through all

7.1 Correlated Systems

the sites and adds an energy U if it finds that the site is double occupied. In the presence of a twobody Coulomb interaction, rather than a many-body electron system, one can substitute the field operators with Wannier operators (maximally localized orbitals that are Fourier transforms of Bloch functions; see Eq. 5.27) to express the generalized interaction parameter between two fermionic particles (fermions have half-integer spins) with opposite spin at the same site, U, as U = g dr w r

4

73

Equation (7.3) is called the on-site interaction strength. In an experimentally realizable atomic version of the Hubbard model, such as might be produced by ultralow temperatures for an optical lattice, interacting (e.g. colliding) fermionic atoms have a low kinetic energy so that the coupling constant g is equal to 4πasħ2/m (in which as is the atomic scattering length and m is the atomic mass) and w(r) is the Wannier function of an atom on a single lattice site. Ultracold Fermi gases loaded onto an optical lattice provide a means of independently controlling both the hopping amplitude t and on-site interaction U. In fact, the scattering length as and, hence, U can be tuned to negative or positive values by exploiting Feshbach resonances. In many cases, it is possible to tune the interaction between the ultracold atoms from strongly attractive to strongly repulsive by simply changing the magnetic field. However, in this chapter, we are concerned only with the repulsive (U > 0) interaction between electron charge densities of opposite spin (σ, σ ) and spatial coordinates (r, r ) as given by the pair products. Specifically, for two electrons on the same crystalline lattice site, the electrostatic energy, may be expressed as (note that different conventions exist in the literature for the definition of U) U = e2

φRσ r

2

φRσ r r−r

2

drdr

74

which is the same on all lattice sites. In Eq. (7.4), φ is a localized orbital and R is the lattice vector. Equation (7.4) is called the Hubbard intra-atomic or on-site interaction, and it is not accounted for in conventional band theory, in which the independent electron approximation is invoked. It should also be noted that this on-site Coulomb repulsion interaction had been implicitly introduced earlier in the Anderson model describing a magnetic impurity coupled to a conduction band (Anderson 1961). In fact, it has been shown that the Hubbard Hamiltonian reduces to the Anderson model in the limit of infinite-dimensional (Hilbert) space (Izyumov 1995). Hence, Eq. (7.4) is sometimes referred to as the Anderson–Hubbard repulsion term. The Hubbard model was actually independently conceived by Martin Charles Gutzwiller (1925–2014) and Junjiro Kanamori (b. 1930) in 1963 (the same year Hubbard published his original paper on electron correlation in narrow-band energy bands) to describe, respectively, the metal–insulator transition and itinerant ferromagnetism. Under the name Pariser–Parr–Pople model, it had been used to describe extended π-electron systems in 1953 (Pariser and Parr 1953; Pople 1953). The Hubbard intra-atomic energy is one of three critical quantities, which determine whether a solid will be an insulator or a conductor, the other two being band filling and bandwidth. The Hubbard energy is the energy cost of placing two electrons on the same site. In order to see this, consider two atoms separated by a distance a. In the limit a ∞ (i.e. two isolated atoms), the excitation energy required for electron transfer from an atomic orbital on one of the atoms to an atomic orbital on the other atom, already containing an electron, is defined as U = I el − χ

75

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7 Hopping Conduction and Metal–Insulator Transitions

where Iel is the ionization energy and χ is the electron affinity (the minus sign signifies that a small portion of Iel is recovered when the electron is added to a neutral atom). In a solid, the situation is analogous, but U is given by Eq. (7.4), instead of Eq. (7.5). The Coulomb repulsion owing to Eq. (7.4) is typically on the order of 1 eV or less. This is much smaller in magnitude than the energy gaps between the valence and conduction bands in covalent or ionic insulators (>3 eV). For wide-band metals, such a small Coulomb repulsion term is insufficient to drive the system to the insulating state. However, in narrow-band systems, it may. As alluded to in Section 5.5.4, the transition metal monoxides with the rock-salt structure are the archetypical examples of correlated systems. Of these oxides, only TiO is metallic. The others, NiO, CoO, MnO, FeO, and VO, are all insulating, despite the fact that the Fermi level falls within a partially filled band (in the independent electron picture). Direct electron transfer between two of the transition metal cations (in the rock-salt structure, d–d interactions are important owing to the proximity of the cations), say, manganese, is equivalent to the disproportionation reaction: Mn + + Mn3 + Mn2 + + Mn2 + d5 + d5 d6 + d4

76

It costs energy, U, to transfer an electron from one Mn2+ into the already occupied d orbital of another Mn2+ ion (d5 d6), thereby opening a Coulomb gap. The d band is split into two equivalent sub-bands, shifted symmetrically relative to the Fermi level – a half-filled, singly occupied lower Hubbard band (LHB) in which the electrons at each ion are aligned antiferromagnetically (with opposite spin) at low temperatures (below the Néel temperature) and an empty upper Hubbard band (UHB), separated by a band gap of the order of U. A schematic representation of this type of electronic structure is shown in Figure 7.1a. The LHB represents low-energy configurations with no doubly occupied sites, that is, Mn2+ Mn3+. The UHB is the high-energy configuration corresponding to doubly occupied sites, that is, Mn2+ Mn+. In general, for a Mott–Hubbard insulator, charge excitations across the gap correspond to dn + dn dn + 1 + dn − 1. In Hubbard’s original approximation, the system is an insulator for any value of U, however, small. In what is known as the Hubbard-3 approximation (a)

(b)

U

Δ

EF

U EF

p band

Δ

p band

Figure 7.1 The band gap is determined by the d–d electron correlation in the Mott–Hubbard insulator (a), where Δ > U. By contrast, the band gap is determined by the charge-transfer excitation energy in the charge-transfer insulator (b), where U > Δ.

7.1 Correlated Systems

(the models are named in accord with the order of Hubbard’s three papers), band splitting occurs only when U is sufficiently large. In fact, the Hubbard-3 approximation describes the occurrence of a transition from insulating to metallic behavior, as first predicted by Mott (1961), when U ~ W, where W represents the one-electron bandwidth. Hence, the bandwidth is another quantity that influences the type of electronic behavior exhibited by a solid. The critical value of the ratio (W/U)c corresponding to the Mott M–I transition is W U

≈1

77

c

with numerical calculations showing the actual value to be 1.15 (Mott 1990). Although there is some disagreement in the literature over the exact value of the ratio at which a M–I transition occurs, the gist of the matter is that a correlated system is expected to become metallic (the Hubbard band gap closes and hopping semiconduction transitions to band conduction) when U becomes smaller than the bandwidth. In terms of the simple picture described earlier, this may be attributed to a large portion of the energy penalty for transferring an electron to an occupied atomic orbital being recovered. The amount of energy regained corresponds to the amount that the COs are stabilized, relative to the free atomic orbitals, as reflected in the bandwidth. A dynamical mean-field theory (DMFT) has been developed for the Hubbard model, which enables one to describe both the insulating state and the metallic state, at least for weak correlation. However, solution of the mean-field equations is not trivial and need not be discussed here (see Georges et al. 1996). Suffice it to say that the basic idea of DMFT is that a solid be viewed as an array of atoms exchanging their electrons, rather than a gas of interacting electrons moving in a periodic potential. The approximation is then to focus on a single atom exchanging electrons with a self-consistent effective medium, considering local many-body correlations. Spatial fluctuations are frozen, the lattice is rigid, but local temporal fluctuations due to electron–electron correlations are fully considered, hence the name “dynamical” associated to “mean field.” The effective medium is a representation of the whole system (minus the atom under focus), and it is self-consistently determined in order to reflect the fact that all atoms are equivalent. The formalism implies the second quantification formalism, with P.W. Anderson’s Hamiltonian of the Hubbard problem (with its U term) and the use of local Green’s functions: this local Green function is to DMFT’s basic concept what the electronic local density is to DFT. Recent extensions are, for instance, extended DMFT, cluster DMFT, diagrammatic expansions. John Hubbard (1931–1980) received his BS and PhD degrees from Imperial College, University of London, in 1955 and 1958, respectively. Hubbard spent most of his 25-year career at the Atomic Energy Research Establishment in Harwell, England, as head of the solid-state theory group. He is best known for a series of papers, published in 1963–1964, on dynamic correlation effects in narrow-band solids. The terms “Hubbard Hamiltonian” and “Mott–Hubbard metal–insulator transition” became part of the jargon of condensed matter physics. Walter Kohn once described Hubbard’s contributions as the basis of much of our present thinking about the electronic structure of large classes of magnetic metals and insulators. Hubbard also conducted research on the nature of gaseous plasmas in nuclear fusion reactors and the efficiency of centrifuges in isotope separation and developed the functional integral method manybody technique, as well as a first-principles theory of the magnetism of iron. In 1976, Hubbard joined the IBM Research Laboratory in San Jose, where he remained until his premature death at the age of 49. (Source: Physics Today Vol. 34, No. 4, 1981. Reprinted with permission from Physics Today. Copyright 1981, American Institute of Physics.) (Photo courtesy of AIP Emilie Segrè Visual Archives. Copyright owned by IBM. Reproduced with permission.

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7.1.2

Charge-Transfer Insulators

Unlike the Mott–Hubbard insulator, MnO described above the band gap in the isostructural oxide NiO is much smaller than expected from intrasite Coulomb repulsion. Fujimori and Minami showed that this is owing to the location of the NiO oxygen 2p band – between the lower and upper Hubbard sub-bands (Fujimori and Minami 1984). This occurrence can be rationalized by considering the energy level of the d band while moving from Sc to Zn in the first transition series. Moving to the right in a period, the d orbital energy drops owing to an increasing effective nuclear charge. Consequently, the energy of, say, the nickel 3d (conduction) band in NiO or in LaFexNi1−xO3 decreases to a level close to the oxygen 2p (valence) band, which allows Ni 3d–O 2p hybridization. Although the d band still splits from on-site Coulomb repulsion, the O 2p band now lies closer to the UHB than the LHB does. The magnitude of the band gap is thus determined by the charge-transfer energy, Δ, associated with the process: dn dn+1L1, where L1 represents a hole in the ligand O 2p band. In other words, the band gap is now Δ not Udd, as illustrated in Figure 7.1b. Jan Zaanen (b. 1957), George Albert Sawatzky, and James W. Allen (b. 1945) named these type systems charge-transfer insulators (Zaanen et al. 1985). Thus, in a charge-transfer insulator, hopping semiconduction results from the transfer of charge carriers between the metal and anion; in contrast to a Mott–Hubbard insulator, hopping semiconduction results from transfer of carriers between adjacent metal atoms. Whether the band gap is of the charge-transfer type or Mott–Hubbard type is determined by the relative magnitudes of Δ and U. If U < Δ, the band gap is determined by the d–d Coulomb repulsion energy. When Δ < U, the charge-transfer excitation energy determines the band gap. It is important to remember that for either type of insulator, charge carriers must be thermally excited or otherwise promoted into the UHB to impart conduction in the presence of an electric field. Crude estimates for U can be obtained using Eq. (7.4). However, neglecting hybridization effects in this manner gives an atomic value, resulting in the values for the solid state being overestimated. The Coulomb repulsion energy is primarily a function of the spatial extension of the orbital. The charge-transfer energy, by contrast, depends on the effective nuclear charge. Both of these parameters, in turn, are dependent upon the transition metal’s atomic number and oxidation state, as well as hybridization effects and the ligand electronegativity. A systematic semiempirical study of the core-level photoemission spectra of a wide range of 3d transition metal compounds has been carried out (Bocquet et al. 1992, 1996). The values for U and Δ obtained from a simplified CI cluster model analysis are demonstrated in Figure 7.2. As can be inferred from the graphs, the heavier 3d transition metal compounds shown in the figure are expected to be charge-transfer insulators, whereas the compounds of the lighter metals are generally expected to be of the Mott–Hubbard type.

7.1.3

Marginal Metals

It was pointed out in the introduction to this chapter that an experimental criterion for metallicity is the observation of a positive temperature coefficient to the electrical resistivity. The so-called bad or marginal metals are those that meet this criterion but in which the value for the resistivity is relatively high (ρ > 10−2 Ω cm). Many transition metal oxides behave in this manner, while others (e.g. ReO3 and RuO2) have very low electrical resistivities, similar in scale to those of conventional metals (ρ < 10−4 Ω cm). Consider the Ruddlesden–Popper ruthenates. Both strong Ru 4d─O 2p hybridization and weaker intrasite correlation effects compared with the 3d transition metals

7.1 Correlated Systems

10

8 7

Ti V Cr Mn Fe Co Ni Cu

6 5 4 3 2 1 0

2+

4+ 3+ Cation charge

5+

Change transfer energy (eV)

Coulomb repulsion energy (eV)

9

8 6 4 2 0

2+

3+

4+

5+

Ti V Cr Mn Fe Co Ni Cu

–2 –4 Cation charge

Figure 7.2 The on-site d–d Coulomb repulsion energy (left) and the charge-transfer excitation energy (right) for the 3d transition metal oxides. Source: Adapted from Bocquet et al. (1992, 1996).

are expected because of the greater spatial extension of the 4d atomic orbitals. Nonetheless, U is sufficiently strong to open a band gap in the ruthenates. Optical conductivity measurements indicate the one-electron bandwidth in the n = 1 Ruddlesden–Popper insulator Ca2RuO4 is ~1 eV, while the on-site Coulomb interaction is estimated to be in the range 1.1–1.5 eV (Puchkov and Shen 2000). Since U > W, correlation prevents metallicity. However, in the isostructural Sr2RuO4, the conductivity in the ab plane of the perovskite layers is metallic with a low resistivity (ρ ~ 10−4 Ω cm), and the out-of-plane transport along the c-axis is marginally metallic at low temperatures (ρ ~ 10−2 Ω cm) and semiconducting at high temperatures. Below 1 K, Sr2RuO4 exhibits superconductivity. For the n = 3 Ruddlesden–Popper phase Ca3Ru2O7, the bandwidth is expected to be slightly larger, while the on-site Coulomb interaction is presumed to be essentially unchanged from that of Ca2RuO4. Likewise, Ca3Ru2O7 and Sr3Ru2O7 not only are marginally metallic in the ab planes of the perovskite blocks (ρ ~ 10−2 Ω cm) but also exhibit a very slight metallic conductivity in the perpendicular out-of-plane direction along the c-axis (Puchkov and Shen 2000). The anisotropy is smaller than in the n = 1 Ruddlesden–Popper phases. It is a general observation that the anisotropy in transport properties of such layered structures decreases with increases in the number of layers, n. It is believed that electron correlation plays an important role with the anomalously high resistivity exhibited in marginal metals. Unfortunately, although the Mott–Hubbard model adequately explains behavior on the insulating side of the M─I transition, on the metallic side, it does so only if the system is far from the transition. Electron dynamics of systems in which U is only slightly less than W (i.e. metallic systems close to the M–I transition) are not well described by a simple itinerant or localized picture. The study of systems with almost localized electrons is still an area under intense investigation within the condensed matter physics community. Other materials that are sometimes classified as marginal metals are quasicrystals, which meet one criterion for semimetallic behavior since they have a nonzero DOS at the Fermi level, but in a contradictory manner these materials display a semiconductor-like temperature dependence to their electrical resistivity, i.e. dρ/dT < 0, rather than the expected dρ/dT > 0 for semimetals and metals. This can be rationalized as being due to the lack of translational invariance in which strictly equivalent sites cannot be found in the fully extended quasilattice, thereby limiting conduction to

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hopping between almost identical sites. In this respect, they are more like amorphous and disordered materials where electrons are localized by disorder, a topic now to be discussed.

7.2

Anderson Localization

It has been seen in the previous section that the ratio of the on-site electron–electron Coulomb repulsion and the one-electron bandwidth is a critical parameter. The Mott–Hubbard insulating state is observed with narrow-band systems such as transition metal compounds in which the charge carriers become localized when U > W. Disorder is yet another condition that localizes charge carriers. In crystalline solids, there are several possible types of disorder. One kind arises from the random placement of impurity atoms in lattice sites or interstitial sites. The term Anderson localization is applied to systems in which the charge carriers are localized by this type of disorder. Anderson localization is important in a wide range of materials, from phosphorusdoped silicon to the perovskite oxide strontium-doped lanthanum vanadate, La1 − xSrxVO3. In a crystalline solid, the presence of a strong disorder results in destructive quantum interference effects from multiple elastic scattering, which is sufficient to stop propagation of the conduction electrons. Inelastic scattering also increases electrical resistivity but does not cause a phase transition. Anderson’s original model is the tight-binding Hamiltonian on a d-dimensional hyper cubic lattice with random site energies, characterized by a probability distribution. On a hyper cubic lattice, each site has 2d nearest neighbors and 2d(d − 1) next nearest neighbors. In the second quantization language, the tight-binding Hamiltonian is written as Η = −t i,j

εiσ niσ

Ciσ Ajσ + i, j, σ

78



where tij is the nearest-neighbor hopping matrix (assumed constant). The Ciσ and Ajσ terms are creation and annihilation operators at the lattice sites i and j, and niσ = Ciσ Aiσ . The site energies, εiσ, are taken to be randomly distributed in the interval [−B/2, B/2], where B is the width characterizing the probability distribution function. Random site energies correspond to varying potential well depths, or diagonal disorder (referring to the diagonal elements of the Hamiltonian matrix), and they arise from the presence of random substitutional impurities in an otherwise periodic solid. Accordingly, diagonal disorder is also sometimes referred to as compositional disorder. When there is variation in the nearest-neighbor distances, this gives rise to fluctuations in the hopping elements tij. That is referred to as off-diagonal disorder, and it is the type of disorder present in amorphous solids like glasses and polymers, in which there is no long-range crystalline structure periodicity. Off-diagonal disorder is sometimes called positional disorder, lateral disorder, and Lifshitz disorder (Lifshitz 1964, 1965) after the Russian physicist Evgeny Mikhailovich Lifshitz (1915–1985). Consider the simplest possible case, a monatomic crystalline solid. The potential at each lattice site is represented by a single square well in the Kronig–Penney model (de Kronig and Penney 1931) by Ralph Kronig (1904–1995) and William G. Penney (1909–1991). For a perfect monatomic crystalline array (Figure 7.3a), all the potential wells are of the same depth and the same distance apart. However, the random introduction of impurity centers in the crystalline lattice produces variation in the well depths (diagonal, or Anderson, disorder), as illustrated in Figure 7.3b. Amorphous substances like glasses exhibit variations in nearest-neighbor distances. This off-diagonal disorder essentially produces unevenly spaced potential wells (Figure 7.3c), that is, the disorder is in the positions of the wells. The multiple elastic scattering from fluctuations in site energies and positions, owing to disorder, is sufficient to reduce the mean free path traveled by the electrons between scattering events to a

7.2 Anderson Localization

Figure 7.3 (a) In the Kronnig–Penney model, the potential at each lattice site in a monatomic crystalline solid is of the same depth. (b) The random introduction of impurity atoms in the crystalline lattice produces variation in the well depths, known as diagonal, or Anderson, disorder. (c) Amorphous (noncrystalline) substances have unevenly spaced potential wells, or off-diagonal disorder.

(a)

(b)

(c)

value comparable with the electron wavelength. Destructive quantum interference between different scattering paths ensues, localizing the electron wave functions. It must be noted that this localized electron wave function does not exist solely on a single isolated atomic center, but rather exponentially decays over several atomic centers, referred to as the localization length. It is found that both diagonal and off-diagonal disorder can produce electron localization and both types of systems can, as well, exhibit M–I transitions. Most metallic glasses, however, do not show a M–I transition, but only moderate changes in resistivity with temperature (Mott 1990). Off-diagonal disorder is obviously assumed to dominate the charge transport at temperatures above the glass transition temperature, that is, in the liquid regime. In all substances, at high temperatures, the electrical resistivity is dominated by inelastic scattering of the electrons by phonons and other electrons. As classical particles, the electrons travel on trajectories that resemble random walks, but their apparent motion is diffusive over large length scales because there is enough constructive interference to allow propagation to continue. Ohm’s law holds, and with increasing numbers of inelastic scattering events, a decrease in the conductance can be directly observed. By contrast, we can think of Anderson localization as occurring at some critical number of elastic scattering events, in which the free electrons of a crystalline solid experience a phase transition from the diffusive (metallic) regime to the non-diffusive (insulating) state. In this state, nondiffusive transport occurs via a thermally activated quantum mechanical hopping of localized states from site to site. It is believed that quantum interference does not occur in liquids (with their offdiagonal disorder), where all collisions are thought to be inelastic, which gives rise to a minimum metallic conductivity. There is one very important aspect of pure Anderson localization that distinguishes it from band-gap insulators. In the latter case, nonmetallic transport is owing to the lack of electronic states at the Fermi level, whereas with Anderson localization, there remains a finite DOS at EF. The critical disorder strength needed to localize all the states via the Anderson model is W B

≈2

79

complete

although a value as low as 1.4 has also been calculated (Mott 1990). To treat cases of weak disorder, Mott introduced the concept of a mobility edge, Ec, which resides in the band tail and separates

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(a)

(b)

Ec E

Pseudo-gap

Ec′

Ec EF

N(E)

Figure 7.4 (a) A schematic DOS curve showing localized states below a critical energy, Ec, in the conduction band. Conduction electrons are localized unless the Fermi energy is above Ec. (b) In weakly disordered metals, a pseudo-gap forms over which states are localized around the Fermi energy, owing to an overlap between the valence band and conduction band tails.

localized from nonlocalized states (Mott 1966). In weakly disordered systems, the states in the tail of the conduction band tend to be localized. The electrons are nonmetallic for all values of the electron energy that fall below Ec into the localized regime, as illustrated in Figure 7.4a. In fact, since the valence band and conduction bands overlap in a metal, the localized states in the tails of weakly disordered systems form a pseudo-gap, in which electrons are localized around the Fermi energy (Figure 7.4b). Percolation theory is helpful for analyzing disorder-induced M–I transitions (recall the classical percolation model that was used to describe grain-boundary transport phenomena in Chapter 2). In this model, the M–I transition corresponds to the percolation threshold. Perhaps the most important result comes from the very influential work by Abrahams et al. (1979), based on scaling arguments from quantum percolation theory. This is the prediction that no percolation occurs in a one-dimensional or two-dimensional system with nonzero disorder concentration at 0 K in the absence of a magnetic field. It has been confirmed in a mathematically rigorous way that all states will be localized in the case of disordered one-dimensional transport systems (i.e. chain structures). In two-dimensional systems (e.g. layered structures with alternating conducting and insulating sheets), the effect of disorder is controversial. Theoretically, two-dimensional systems with any amount of disorder are believed by many theoreticians to behave as insulators at any temperature. Such behavior has been confirmed for several compounds. For example, in the phase Na2−x+yCax/2La2Ti3O10, two-dimensional metallic transport should be possible within the perovskite slabs. This is because electrons have been donated to the formally empty conduction band by a two-step sequence. The first step consists of alloying the alkali metal sites in the NaO rock-salt layers of the parent phase Na2La2Ti3O10 with Ca2+, thereby introducing vacancies (yielding Na2−xCax/2La2Ti3O10 where 0.48 < x < 1.66). This is followed by intercalation of additional sodium atoms into the vacancies (yielding Na2−x+yCax/2La2Ti3O10), which chemically reduces a portion of Ti4+ (d0) cations in the octahedral layers to the Ti3+ (d1) state and donates electrons to the conduction band. However, the random electric fields generated by the distribution of impurities (Ca2+) in the rock-salt layer are primarily responsible for the nonmetallic behavior in this system, possibly with weak correlation effects (Lalena et al. 2000). By contrast,

7.2 Anderson Localization

in the analogous three-dimensional transport system, the mixed-valence perovskite La1−xTiO3 (Ti3+/Ti4+), metallic conduction is observed at x = 0.25. Over the last two decades, the prediction that metallicity is impossible in disordered 2D systems appears to have been contradicted in some semiconductors, such as high-quality, high mobility n-doped silicon MOSFETs. It should be noted that while the MOSFET device is physically a three-dimensional system, the carriers are confined within a potential such that their motion in the third direction is restricted, thus leaving a two-dimensional momentum or k-vector characterizing carrier motion in a plane normal to the confining potential. Electrical conduction occurs at the interface of the semiconductor silicon and the insulating silicon dioxide, which constitutes a conducting sheet, or two-dimensional electron gas. We could therefore refer to this at least as a quasi-two-dimensional transport system. The conductivity can be separated into three distinct regimes in these materials (Sarma and Hwang 2015). At very low carrier densities, the system is a disorder-driven strongly localized insulator with a VRH resistivity, as expected. At high carrier densities, the electrical resistivity is essentially temperature independent. However, at “intermediate” but still low densities, there appears to be a strong apparent metallic temperature dependence to the 2D electrical resistivity with weak localization effects perhaps manifested at very low temperatures. Similarly, Si (100)-MOSFET (n-GaAs) also is thought to exhibit metallicity, and a metallic regime has been observed at zero applied field, down to the lowest accessible temperatures, in GaAs/Ga1−xAlxAs heterostructures with a high density of InAs quantum dots incorporated just 3 nm below the heterointerface (Ribeiro et al. 1999). Some have argued that this is an artifact and that other effects mask the M–I transition. Furthermore, some high-quality 2D semiconducting systems manifest much weaker metallicity than others. A high carrier mobility (low disorder) is clearly necessary, but the manifestation of metallicity is strongly dependent on the semiconducting material system under investigation. More recently, field-effect transistors made from true 2D semiconductors based on transition metal chalcogenides (e.g. MoS2), phosphorene, hexagonal boron nitride, and silicene with atomically thin layers have begun to appear. In such devices all carriers are without a doubt uniformly influenced by the gate voltage. Unfortunately, no research has been performed on the M–I transition or the effects of disorder in these systems. The possibility of metallicity in disordered 2D transport systems is thus still a hotly debated topic within the physics community. Philip Warren Anderson (b. 1923) received his PhD in theoretical physics under John H. Van Vleck in 1949 from Harvard University. From 1949 to 1984, he was associated with Bell Laboratories and was the chairman of the theoretical physics department there from 1959 to 1961. From 1967 to 1975, he was a visiting professor of theoretical physics at the University of Cambridge. In 1975, he joined the physics department at Princeton University, becoming emeritus in 1997. Anderson was the first to estimate the magnitude of the antiferromagnetic super-exchange interaction proposed by Kramers and to point out the importance of the cation–anion–cation bond angle. In addition to basic magnetics, Anderson has made seminal contributions to the understanding of the physics of disordered media, such as the spin-glass state and the localization of noninteracting electrons. For this work, he was a corecipient of the 1977 Nobel Prize in Physics. He has worked almost exclusively in recent years on high-temperature superconductivity. Anderson was on the United States National Academies of Sciences Council from 1976 to 1979, and he was awarded the United States National Medal of Science in 1983. (Source: Nobel Lectures, Physics 1971–1980, World Scientific Publishing Company, Singapore.) (Photo ourtesy of the Department of Physics, Princeton University. Reproduced with permission.

339

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7 Hopping Conduction and Metal–Insulator Transitions

7.3 Experimentally Distinguishing Disorder from Electron Correlation The main effect of both types of electron localization, of course, is hopping semiconduction. Nevertheless, it would be very beneficial to have a method of experimentally distinguishing between the effects of electron–electron Coulomb repulsion and disorder. In cases where only one or the other type of localization is present, this task is relatively simpler. The Anderson transition, for example, is predicted to be continuous. That is, the zero-temperature electrical conductivity should drop to zero continuously as the impurity concentration is increased. By contrast, Mott predicted that electron correlation effects lead to a first-order or discontinuous transition. The conductivity should show a discontinuous drop to zero with increasing impurity concentration. Unfortunately, experimental verification of a true first-order Mott transition remains elusive. Disorder and correlation are often both present in a system. One then has the more difficult task of ascertaining which is the dominant electron localizing mechanism. As might be expected, the most useful experimental approaches to this problem involve magnetic and conductivity measurements. In the paramagnetic regime at high temperatures, the magnetic susceptibility, χ, of a sample with a nonzero density of states at the Fermi level (e.g. metals and Anderson-localized states) is temperature independent. This is known as Pauli paramagnetism (see Example 7.1 and Figure 7.6a later), since it results from the Pauli exclusion principle. The Pauli susceptibility is given by 3Nze μ2B 2EF 0

7 10

where Nze is the number of atoms times the number of electrons per atom in the band at the Fermi energy, EF, and μB is the Bohr magneton, 9.27 × 10−24 J/T (Pauli 1927). For narrow bands and high temperatures, however, there may be a slight temperature dependency to the Pauli susceptibility (Goodenough 1963). Disorder cannot be the sole mechanism of electron localization when there also exists experimental evidence suggesting the presence of a band gap. Direct methods for detecting the presence of a band gap include optical conductivity and photoemission spectroscopy. However, magnetic characterization is also very useful here as well. This is because electron correlation induces the exchange interactions responsible for spontaneous magnetization in insulators. Goodenough has emphasized the use of magnetic criteria as a means of characterizing the transition region between the itinerant and localized regimes. The validity of this approach may be seen by considering the Heisenberg Hamiltonian: H ex = −

J ij Si Sj

7 11

This is the expression for the exchange interaction between localized magnetic moments. In Eq. (7.11), J is the exchange integral, given by J= −

2b2 4S2 U

7 12

where S is the spin quantum number of the system (Anderson 1959). In a mean-field approximation, the exchange integral can be incorporated into a dimensionless constant (the Weiss constant), and the Heisenberg Hamiltonian can be rewritten in terms of this constant and the magnetization

7.3 Experimentally Distinguishing Disorder from Electron Correlation

dρ/dT < 0 χ = χCW

Increasing U

dρ/dT < 0 χ = χspin glass

dρ/dT < 0 χ = χcw + χPauli

dp/dT > 0 0

Increasing B

Figure 7.5 An electronic phase diagram illustrating the location of a metallic regime, at low U (Coulomb repulsion energy) and B (disorder), where dρ/dT > 0. With increasing U and/or B, the system becomes B). nonmetallic (dρ/dT < 0) but may still exhibit Pauli paramagnetism if there is no band gap (U

of the sample. Thus, Eq. (7.10) for the Pauli susceptibility of itinerant electrons and Eq. (7.12) relating J to the on-site Coulomb energy and the tight-binding transfer integral establish how magnetic data can be used to characterize the itinerant-to-localized transition region. Systems exhibiting both strong disorder and electron correlation, so-called disordered Mott– Hubbard insulators, are difficult to evaluate. The description of electronic states in the presence of both disorder and correlation is still an unresolved issue in condensed matter physics. Whether disorder or the correlation is the predominant factor in controlling transport properties in a material depends on a complex temperature/composition-dependent three-way interplay between U, B, and W. Many materials exhibit the behavior illustrated in the electronic phase diagram shown in Figure 7.5. For cases of strong disorder with weak correlation, a temperature-independent (Pauli) paramagnetic susceptibility, χ(T), is usually observed at high temperatures because of the finite density of states at EF. However, this may crossover to a Curie- or Curie–Weiss-type behavior (linear dependence of χ on T) at low temperatures, as illustrated in Figure 7.6a of Example 7.1 (shown later). The crossover from Pauli-type to Curie-type behavior has been shown to be owing to correlation, which gives rise to the occurrence of singularly occupied states below the Fermi energy, EF (Yamaguchi et al. 1979). A Curie law results whenever the probability of finding two electrons in a localized state is less than the probability of finding one electron (because of the Coulomb B), repulsion) (Kaplan et al. 1971). If electron correlation is much stronger than disorder (U the magnetic susceptibility curve typically has the general appearance of an antiferromagnetic insulator. Another characteristic feature of amorphous and disordered systems (as well as most quasicrystals) is the VRH mechanism to the electrical conduction, which is observed on the nonmetallic side of the M–I transition at low temperatures, whereas the temperature dependence to the conductivity is of the Arrhenius-type (σ(T) = σ aexp–Ea/kBT, with the activation energy Ea corresponding to the band-gap energy) for both band insulators/semiconductors and Mott–Hubbard insulators. The phenomenon is also known as impurity conduction. As has been discussed, this arises from the hopping of charge carriers between localized states, or impurity centers, that need not be

341

7 Hopping Conduction and Metal–Insulator Transitions

(a)

(b) 100 000

0.10 La1–x TiO3

0.08 0.06

La.70 La.75 La.80 La.88 La.92

0.04 0.02 0.00

La.70 La.75 La.80 La.88 La.92

10 000 Resistivity (mΩ–cm)

CHI × 102 (emu/mcI)

342

1000

La1–x TiO3

100 10 1

0

100 200 Temperature (K)

300

0.1

0

100 200 Temperature (K)

300

Figure 7.6 Temperature variation of magnetic susceptibility (a, left) and electric resistivity (b, right) for several metallic composition in La1-xTiO3. Source: From MacEachern, M.J., et al. (1994). Chem. Mater. 6: 2092. Reproduced by permission of the American Chemical Society.

nearest neighbors in the case of amorphous systems. Experimentally, it is indicated by characteristic temperature dependency to the DC conductivity. For three-dimensional systems with noninteracting electrons, the logarithm of the conductivity and T −1/4 are linearly related, in accordance with the equation given by Mott (1968): σ = A exp −

T0 T

n

n=1 4

7 13

The exponent n, which is known as the hopping index, is actually equal to 1/(d + 1) where d is the dimensionality. Hence, for two-dimensional systems n = 1/3. Strictly speaking, Eq. (7.13) holds only when the material is near the M–I transition and at sufficiently low temperatures. At high temperatures, conduction proceeds by thermal excitation of electrons or donors into the conduction band, or injection of holes or acceptors into the valence band. The Anderson transition is described in terms of noninteracting electrons. For real materials, electron–electron interactions cannot be ignored, even when disorder is the primary localization mechanism. In other words, with Anderson-localized states correlation effects are normally present to some extent. Of course, the reverse is not true – correlated systems may be completely free of disorder. Electrical conduction in insulators with band gaps is also thermally activated. Hence, it can be difficult to determine the value of n in strongly correlated disordered systems unambiguously. The hopping index, however, is often found to be equal to one half at low temperatures, with one quarter still being observed at higher temperatures. For highly correlated systems, a T2 dependence to the resistivity is frequently observed in the metallic regime. Notable exceptions to this rule are the high-temperature superconducting cuprates, in which conduction occurs within CuO2 layers. Electron correlation is believed to be important to the normal state (nonsuperconducting) properties. In fact, all the known high-temperature superconducting cuprates are compositionally located near the Mott insulating phase. Both holedoped superconductors (e.g. La2−xSrxCuO4, YBa2Cu3O7−y) and electron-doped superconductors (e.g. Nd2−xCexCuO4) are known, in which the doping induces a change from insulating to metallic behavior with a superconducting phase being observed at low temperatures. However, a T-linear

7.4 Tuning the M–I Transition

dependence to the resistivity, rather than a T2 dependence, is widely observed in the nonsuperconducting metallic state above the critical temperature, Tc, in these materials. Lengthy review articles on disorder-induced M–I transitions, with minor contributions by electron correlation, are available (Lee and Ramakrishnan 1985; Belitz and Kirkpatrick 1994). An extensive review article on correlated systems has also been published (Imada et al. 1998).

7.4

Tuning the M–I Transition

There are two basic approaches for tuning the composition of a phase to induce metallicity. One is the band filling control approach, in which the chemical composition of an insulator or semiconductor (either a band type or correlated type) is varied in such a way as to increase the charge carrier (hole or electron) concentration. The sodium tungsten bronzes, introduced in Chapter 2, and La-doped perovskite SrTiO3 are examples of filling control. Both WO3 and SrTiO3 are band insulators. However, as sodium is intercalated into the interstitials of WO3, giving NaxWO3, or when La3+ is substituted for Sr2+ to give La1−xSrxTiO3, a portion of the transition metal cations are chemically reduced, and mixed valency is introduced. The extra electrons are donated in each case to a conduction band that was formally empty. Hole doping may also be used to bring about metallicity, as in La1−xSrxMO3 (M = Ti, Mn, Fe, Co) where hole doping of x ~ 0.05 drives the ground state from an antiferromagnetic Mott insulator to a paramagnetic metal. Introducing substitutional donors directly on the cation sublattice can sometimes also induce metallic behavior. For example, the perovskite oxide LaCoO3 is a band insulator at 0 K (i.e. it contains a filled valence band and an empty conduction band separated by a band gap) since Co3+ is in the low-spin d6 t 62g ground state. Upon increasing the temperature to approximately 90 K, the Co3+ ion is excited into a high-spin state. Nonetheless, the oxide remains in the insulating state, owing to correlation effects. Upon further increasing the temperature to about 550 K, the oxide becomes conducting. Metallic behavior can be induced at much lower temperatures, however, by electron doping LaCoO3 with Ni3 + t 62g e1g

to give LaNi1−xCoxO3 (Raychaundhuri

et al. 1994). Despite its success in LaNi1−xCoxO3, the strategy of alloying the transition metal site is not often attempted because of the danger of reducing the already narrow d bandwidth, which promotes electron localization. Transition metal compounds generally tend to have a narrow d band (small W) that becomes pronounced if dopant atoms are not near in energy to those of the host, owing to a different effective nuclear charge or orbital radius. For example, low concentrations of impurities may have energy levels within the band gap and effectively behave as isolated impurity centers. With heavier doping, appreciable overlap between the orbitals on adjacent impurities may occur with the formation of an impurity band. However, nonthermally activated metallic conduction would be predicted in the Bloch/Wilson band picture only if the impurity band and conduction band overlap. This is believed to be the mechanism for the M–I transition in phosphorus-doped silicon. The filling control approach has even been applied to some nanophase materials. For example, the onset of metallicity has been observed in individual alkali metal-doped single-walled zigzag carbon nanotubes. Zigzag nanotubes are semiconductors with a band gap around 0.6 eV. Using tubes that are (presumably) open on each end, it has been observed that upon vapor phase intercalation of potassium into the interior of the nanotube, electrons are donated to the empty conduction band, thereby raising the Fermi level and inducing metallic behavior (Bockrath 1999).

343

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7 Hopping Conduction and Metal–Insulator Transitions

A second way of inducing metallicity is known as bandwidth control. Usually, the lattice parameters of a phase are varied while maintaining the original structure. Thus, U remains essentially the same, while W is made larger. This may be accomplished by the application of pressure or the introduction of substitutional dopants. The former is commonly exhibited in V2O3, while the latter is successful in Ni2S2−xSex, where the Se doping widens the S 2p bandwidth, closing the chargetransfer gap. There has been disagreement in the literature as to whether NiS2 is a charge-transfer or Mott–Hubbard insulator. It has been speculated that at low temperatures, the compound is of the Mott–Hubbard type and, at higher temperatures, of the charge-transfer type (Honig and Spałek 1998). As with filling control, alloying of the transition metal cation sublattice is not usually successful at driving a system metallic. In the ABO3 perovskites, bandwidth control may be accomplished by changing the ionic radius of the A-site (twelve-coordinate) cation. Essentially, this mimics the effect of pressure, which also effectively changes the B–O–B angle (see discussion in Section 7.1). The intrasite correlation energy usually changes somewhat also during bandwidth control, in accordance with the expectations outlined earlier. However, the method primarily relies on increasing the hybridization, in order to obtain a wider bandwidth favorable for metallic behavior. In some cases, structure-type changes may occur, for example, as with the transformation of a cubic perovskite to a rhombohedral or orthorhombic structure. Example 7.1 The M–I transition has been studied in powder samples (prepared by the ceramic method) of the series of perovskite oxides La1−xTiO3 with 0 ≤ x ≤ 0.33 (MacEachern et al. 1994). Below are the curves of the temperature dependencies to the magnetic susceptibility (Figure 7.6a) and electrical resistivity (Figure 7.6b) for different compositions, as well as the cell parameters for the phases: 1) For what value of x does metallic behavior first appear? 2) Is electron localization primarily due to disorder or correlation? 3) Speculate as to the nature of the M–I transition. Does varying the chemical composition (the value of x) primarily result in filling control or bandwidth control? Cell parameters Comp.

a (Å)

b (Å)

c (Å)

vol (Å3)

La0.70TiO3 La0.75TiO3 La0.80TiO3 La0.88TiO3 La0.92TiO3

5.464 5.541 5.557 5.582 5.606

7.777 7.793 7.817 7.882 7.914

5.512 5.528 5.532 5.559 5.584

234.22 238.70 240.30 244.58 247.74

Solution For x = 0.30, dρ/dT is negative over the entire temperature range, indicating insulating behavior. For x = 0.25, dρ/dT is negative at high temperatures but changes sign, becoming positive (i.e. the resistivity begins to increase with increasing temperature) around 150 K and remains positive for all temperatures below this point. Hence, the M–I transition appears at the composition x = 0.25. 1) For these materials, there is evidence for both disorder and electron correlation. The temperature-independent component to the magnetic susceptibility at high temperatures is too great in magnitude for Van Vleck paramagnetism but, rather, is Pauli-like paramagnetism indicating the

7.5 Other Types of Electronic Transitions

presence of a finite density of states at the Fermi level (Anderson localization). Furthermore, plots of ln σ versus T−1/4 were reported as being linear, indicating VRH. However, there is also a Curie–Weiss-like behavior at temperatures below 50 K in the two lower curves of Figure 6.6a and below about 150 K in the two upper curves, which is indicative of electron correlation. Since Ti is to the left of the first transition period, relatively strong electron correlation in the Ti 3d orbitals would be expected. In fact, a linear dependency of the resistivity to T2 (plots of ρ versus T2 were linear) was also reported in the metallic phases, supporting the presence of correlation effects. Hence, for La1−xTiO3, both disorder and correlation are probably important. These phases are best considered disordered Mott–Hubbard insulators. 2) The unit cell volume expands with increasing lanthanum content. However, the metallic behavior appears with the larger unit cells. This is contradictory to bandwidth control, in which greater atomic orbital overlap would be expected to overcome correlation effects with contractions in the unit cell. For La0.66TiO3, titanium is fully oxidized to Ti4+ (d0). Hence, the conduction band is empty. As the Ti3+ (d1) content increases with increased lanthanum deficiency, however, the conduction band becomes occupied with electrons. Thus, the M–I transition appears to be under filling control.

7.5

Other Types of Electronic Transitions

In this chapter, the focus has been on the Mott (Mott–Hubbard) and Anderson transitions. When charge ordering is present, other types of transitions are also possible. A classic example is the mixed-valence spinel Fe2O3. There are two types of cation sites in Fe2O3, denoted as A and B. The A sites are tetrahedrally coordinated Fe3+ ions, and the B sites are a 1:1 mixture of octahedrally coordinated Fe2 + t 42g e2g and Fe3 + t 32g e2g ions. Electrical transport is on the B sublattice and is nonmetallic (dρ/dT < 0) thermally activated hopping below about 300 K. Interestingly, it is found that at 120 K, the resistivity increases by two orders of magnitude, the oxide remaining nonmetallic. This is the Verwey transition in Fe2O3. Since the system remains nonmetallic, it is sometimes called an insulator–insulator transition. At the Verwey transition, the mixed-valence ions on the B sublattice become ordered. There is some controversy about the exact structure of the ordered state. Verwey first proposed that the mixed-valence cations are ordered onto alternate B-site layers (Verwey and Haayman 1941; Verwey et al. 1947). Charge ordering transitions are also observed in Ti4O7, as well as many transition metal perovskite oxides, such as La1−xSrxFeO3, La1−xSr1+xMnO4, and La2−xSrxNiO4 (the latter in which the charge ordering drives a true M–I transition). Another type of charge ordering, called the charge density wave (CDW) state, can open a band gap at the Fermi level in the partially filled band of a metallic conductor. This causes a M–I transition in quasi-one-dimensional systems or a metal–metal transition in quasi-two-dimensional systems, as the temperature is lowered. At high temperatures, the metallic state becomes stable because the electron energy gain is reduced by thermal excitation of electrons across the gap. Most CDW materials contain weakly coupled chains, along which electron conduction takes place, such as the nonmolecular K0.3MoO3 and the molecular compound K2Pt(CN)4Br0.33H2O (KCP). Perpendicular to the chains, electrical transport is much less easy, giving the quasi-onedimensional character necessary for CDW behavior.

345

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7 Hopping Conduction and Metal–Insulator Transitions

ρ(r)

(a) a

ρ(r)

(b)

Atoms

2a

E(k)

Atoms

E(k)

Metal

Insulator EF –π/a

–kF

0

Gap kF

π/a

k

–π/a

–kF

0

kF

π/a

k

(c)

q = 2kF

Figure 7.7 The Peierls distortion of a one-dimensional metallic chain. (a) An undistorted chain with a half-filled band at the Fermi level (filled levels shown in bold) has an unmodulated electron density. (b) The Peierls distortion lowers the symmetry of the chain and modulates the electron density, creating a CDW and opening a band gap at the Fermi level. (c) The Fermi surface nesting responsible for the electronic instability.

A CDW is a periodic modulation of the conduction electron density within a material. It is brought about when an applied electric field induces a symmetry-lowering lattice modulation in which the ions cluster periodically. The modulation mechanism involves the coupling of degenerate electron states to a vibrational normal mode of the atom chain, which causes a concomitant modulation in the electron density that lowers the total electronic energy. In one-dimensional systems, this is the classic Peierls distortion (Peierls 1930, 1955). It is analogous to the JT distortion observed in molecules. Figure 7.7a shows the extended-zone electronic band structure for a one-dimensional crystal – an atom chain with a real-space unit cell parameter a and reciprocal lattice vector π/a – containing a half-filled (metallic) band. In this diagram, both values of the wave vector, ±k, are shown. The wave vector is the reciprocal unit cell dimension. The Fermi surface is a pair of points in the first BZ (Figure 7.7c). When areas on the Fermi surface can be made to coincide by mere translation of a wave vector, q, the Fermi surface is said to be nested. The instability of the material toward the Peierls distortion is due to this nesting. In one dimension, nesting is complete, and a onedimensional metal is converted to an insulator because of a Peierls distortion. This is shown in Figure 7.7b, where the real-space unit cell parameter of the distorted lattice is 2a and a band gap opens at values of the wave vector equal to half the original values, π/2a. The total electronic energy is reduced because filled states are lowered in energy and empty states are raised, relative to the same states in the undistorted lattice. The distortion is favored so long as the decrease in electronic energy outweighs the increase in elastic strain energy (Elliot 1998). Note that the Peierls transition does not involve electron correlation effects. In higher dimensions, only

References

sections of the Fermi surface may be translated and superimposed on other portions. The band-gap opening is thus only partial, and a metal–metal transition, rather than a M–I transition, is observed. A related phenomenon is the spin-density wave (SDW), in which the Coulomb repulsion between electrons in doubly occupied sites results in a spin modulation, rather than a charge density modulation. The SDW state leads to a band-gap-opening M–I transition like the CDW state. However, the lattice is not distorted by a SDW state, so they cannot be detected by X-ray diffraction techniques. SDW instabilities may be induced in low-dimensional metals by magnetic fields as well as by lowering the temperature (Greenblatt 1996). Hence, neutron diffraction can be used for observing the existence of a SDW state, since it results in a spatially varying magnetization of the sample.

Practice Problems 1

Name and describe the four basic types of electron localization mechanisms in solids.

2

What does the term marginal metal signify?

3

Which type of atomic orbital is especially prone to localization effects and why?

4

Describe the two basic approaches for inducing metallicity, or otherwise tuning the electrical behavior of solids.

5

What is diagonal disorder? What is off-diagonal disorder? Which type is responsible for Anderson localization?

6

Name one possible method of experimentally distinguishing between the different localization mechanisms.

7

Is it possible for a system to exhibit both disorder and electron correlation? Which of the following is possible: A system with disorder and no electron correlation or a system with electron correlation and no disorder?

References Abrahams, E., Anderson, P.W., Licciardello, D.C., and Ramakrishnan, T.V. (1979). Phys. Rev. Lett. 42: 673. Anderson, P.W. (1958). Phys. Rev. 109: 1492. Anderson, P.W. (1959). Phys. Rev. 115: 2. Anderson, P.W. (1961). Phys. Rev. 124: 41. Belitz, D. and Kirkpatrick, D.R. (1994). Rev. Mod. Phys. 6: 261. Bockrath, M.W. (1999). Carbon nanotubes: electrons in one dimension. Ph.D. dissertation. Department of Physics, University of California, Berkeley, 63–67. Bocquet, A.E., Fujimori, A., Mizokawa, T., Saitoh, T., Namatame, H., Suga, S., Kimizuka, N., and Takeda, Y. (1992). Phys. Rev. B 46: 3771. Bocquet, A.E., Mizokawa, T., Morikawa, K., Fujimori, A., Barman, S.R., Maiti, K.B., Sarma, D.D., Tokura, Y., and Onoda, M. (1996). Phys. Rev. B 53: 1161. de Boer, J.H. and Verwey, E.J.W. (1937). Proc. Phys. Soc. Lond. A 49: 59. Elliot, S. (1998). The Physics and Chemistry of Solids. Chichester: Wiley. Fujimori, A. and Minami, F. (1984). Phys. Rev. B 30: 957.

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8 Magnetic and Dielectric Properties The phenomenon of magnetism was observed in antiquity. The earliest discovery of the magnetic properties of lodestone (Fe3O4) was either by the Greeks or the Chinese. The relationship between magnetism and electricity was discovered in the nineteenth century. In 1820, the Danish physicist Hans Christian Ørsted (1777–1851) demonstrated that bringing a current-carrying wire close to a magnetic compass caused a deflection of the compass needle. This deflection is caused by the magnetic field generated by the electric current. While the French physicist André-Marie Ampère (1775–1836) immediately started a physicomathematical theory of electrodynamics, the British physicist Michael Faraday (1791–1867) conversely showed that a changing magnetic field produces an electric field. In 1855, still more than 40 years before the discovery of the electron, Scottish physicist James Clerk Maxwell