Superconducting Cuprates : Properties, Preparation and Applications [1 ed.] 9781613240878, 9781604569193

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Copyright © 2008. Nova Science Publishers, Incorporated. All rights reserved. Superconducting Cuprates : Properties, Preparation and Applications, Nova Science Publishers, Incorporated, 2008. ProQuest Ebook Central,

Copyright © 2008. Nova Science Publishers, Incorporated. All rights reserved. Superconducting Cuprates : Properties, Preparation and Applications, Nova Science Publishers, Incorporated, 2008. ProQuest Ebook Central,

SUPERCONDUCTING CUPRATES: PROPERTIES, PREPARATION AND APPLICATIONS

Copyright © 2008. Nova Science Publishers, Incorporated. All rights reserved.

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Superconducting Cuprates : Properties, Preparation and Applications, Nova Science Publishers, Incorporated, 2008. ProQuest Ebook Central,

Copyright © 2008. Nova Science Publishers, Incorporated. All rights reserved. Superconducting Cuprates : Properties, Preparation and Applications, Nova Science Publishers, Incorporated, 2008. ProQuest Ebook Central,

SUPERCONDUCTING CUPRATES: PROPERTIES, PREPARATION AND APPLICATIONS

KOENRAAD N. COURTLANDT Copyright © 2008. Nova Science Publishers, Incorporated. All rights reserved.

EDITOR

Nova Science Publishers, Inc. New York

Superconducting Cuprates : Properties, Preparation and Applications, Nova Science Publishers, Incorporated, 2008. ProQuest Ebook Central,

Copyright © 2009 by Nova Science Publishers, Inc. All rights reserved. No part of this book may be reproduced, stored in a retrieval system or transmitted in any form or by any means: electronic, electrostatic, magnetic, tape, mechanical photocopying, recording or otherwise without the written permission of the Publisher. For permission to use material from this book please contact us: Telephone 631-231-7269; Fax 631-231-8175 Web Site: http://www.novapublishers.com NOTICE TO THE READER The Publisher has taken reasonable care in the preparation of this book, but makes no expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No liability is assumed for incidental or consequential damages in connection with or arising out of information contained in this book. The Publisher shall not be liable for any special, consequential, or exemplary damages resulting, in whole or in part, from the readers’ use of, or reliance upon, this material. Any parts of this book based on government reports are so indicated and copyright is claimed for those parts to the extent applicable to compilations of such works.

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Independent verification should be sought for any data, advice or recommendations contained in this book. In addition, no responsibility is assumed by the publisher for any injury and/or damage to persons or property arising from any methods, products, instructions, ideas or otherwise contained in this publication. This publication is designed to provide accurate and authoritative information with regard to the subject matter covered herein. It is sold with the clear understanding that the Publisher is not engaged in rendering legal or any other professional services. If legal or any other expert assistance is required, the services of a competent person should be sought. FROM A DECLARATION OF PARTICIPANTS JOINTLY ADOPTED BY A COMMITTEE OF THE AMERICAN BAR ASSOCIATION AND A COMMITTEE OF PUBLISHERS. LIBRARY OF CONGRESS CATALOGING-IN-PUBLICATION DATA

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CONTENTS

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Preface

vii

Chapter 1

Antiadiabatic Theory of the Electronic Ground State of Superconductors Pavol Baňacký

Chapter 2

Dipolon Theory of Kink Structure of Quasi-Particle Energy Dispersion Observed in Photoemission Spectra of High Temperature Superconducters Bi2Sr2CaCu2O8+δ R. R. Sharma

81

Chapter 3

Persistent Photoconductivity in YBa2Cu3Ox and Other HighTemperature Superconductors Wilhelm Markowitsch

101

Chapter 4

Andreev Reflections and Transport Phenomena In Cuprate Superconductors at the Interface with Ferromagnets and Normal Metals Samanta Piano, Fabrizio Bobba, Filippo Giubileo and Anna Maria Cucolo

135

Chapter 5

Influence of Pair Breaking Effects on the Long-Range Odd Triplet Superconductivity in a Ferromagnet/Superconductor Bilayer T. Rachataruangsit

161

Chapter 6

Theoretical Study of YBa2Cu3O7 Beyond the Born-Oppenheimer Approximation and Migdal Theorem: Antiadiabatic Ground State Induced by Phonon Modes Coupling Pavol Baňacký

187

Chapter 7

Theories of Peak Effect and Anomalous Hall Effect for Cuprate Superconductors Wei Yeu Chen and Ming Ju Chou

213

Chapter 8

Mercury-Based Superconducting Cuprates: High Tc and Pseudo Spin-Gap Y. Itoh and T. Machi

235

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vi

Contents

Chapter 9

New Perspectives on Longstanding Issues of the High-Tc Cuprates B. J. Taylor and M. B. Maple

269

Chapter 10

Predicting Superconducting Tc: The Role of Plane Isolation and Bond Ordering as Exemplified by YBa2Cu3Oy and ‘Record’ Hg Based Cuprates H. Oesterreicher

295

Chapter 11

Analysis of Micro- and Nanotexture in Cuprate Superconductors A. Koblischka-Veneva and M. R. Koblischka

311

Chapter 12

Effect and Evolution of the Pseudogap in Y1-xCAxBA2CU3O7-δ: Probed by Charge Transport, Magnetic Susceptibility and Critical Current Density Measurements S. H. Naqib and R. S. Islam

339

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Index

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PREFACE Although cuprate compounds in the normal superconducting state share many characteristics with each other, there is as of 2008 no widely accepted theory to explain their properties. The search for a theoretical understanding of high-temperature superconductivity is widely regarded as one of the most important unsolved problems in physics, and it continues to be a topic of intense experimental and theoretical research. This new book presents the latest research in the field. Chapter 1 - Results of ARPES study of high-Tc cuprates and theoretical results of lowFermi energy band structure fluctuation for different groups of superconductors indicate that electron coupling to pertinent phonon modes drive system from adiabatic into antiadiabatic

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state ( ω > E F ). At these circumstances, not only Migdal-Eliashberg approximation is not valid, but basic adiabatic Born-Oppenheimer approximation (BOA) does not hold. It indicates that current microscopic theories build on effective model Hamiltonians that tacitly assume validity of the BOA, including BCS theory and models of strongly correlated electrons, are inadequate for correct description of the electronic ground state of superconductors. For EP interactions driven transition from adiabatic into antiadiabatic state, nuclear motion dynamics becomes crucial for electronic state in some region of k-space. At these circumstances, electronic structure has to be studied as explicitly dependent on instantaneous nuclear coordinates Q as well as on instantaneous nuclear momenta P. It enables nonadiabatic Q,P-dependent modification of the BOA. Based on this modification, antiadiabatic theory of complex electronic ground state of superconductors is formulated. It is shown, that due to EP interactions that drive system from adiabatic into antiadiabatic state, symmetry breaking is induced and system is stabilized in antiadiabatic state at distorted geometry with respect to adiabatic equilibrium high symmetry structure. Stabilization effect in the antiadiabatic state is due to strong dependence of the electronic motion on the instantaneous nuclear kinetic energy, i.e. on the effect which is neglected on the adiabatic level within the BOA. Antiadiabatic ground state at distorted geometry is geometrically degenerate with fluxional nuclear structure in the phonon modes that drive system into this state. While system remains in antiadiabatic state, nonadiabatic polaron – renormalized phonon interactions are zero in well-defined k region of reciprocal lattice. This, along with geometric degeneracy of the antiadiabatic state, enables formation of mobile bipolarons that can move over lattice as supercarriers without dissipation. More over, it has been shown that due to EP interactions at transition into antiadiabatic state, k-dependent gap in one-electron spectrum has been opened. Gap opening is related to shift of the original adiabatic Hartree-Fock orbital energies and to

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viii

Koenraad N. Courtlandt

the k- dependent change of density of states of particular band(s) at Fermi level. Corrected one-particle spectrum enables to derive thermodynamic properties that are in full agreement with corresponding thermodynamic properties of superconductors. Based on the complex antiadiabatic theory, it has been shown that Fröhlich’s effective attractive electron-electron interaction term represents correction to electron correlation energy at transition from adiabatic into antiadiabatic state due to EP interactions. Increased electron correlation is consequence of stabilization of the system in superconducting electronic ground state, but not the reason of its formation. Chapter 2 - Formerly unexplained experimentally deduced kink in the quasiparticle energy dispersion observed in photoemission spectra of high-temperature superconducters Bi2Sr2CaCu2O8+δ has been explained here naturally via dipolon mediated electronelectron pairing mechanism of high-temperature superconductors. We have made use of the fourmomenta space diagrams in the dressed particle picture to write the selfenergy_(p) by taking the sum of the exchange diagrams involving dipolon propagator, electron Green’s function and screened electron-electron Coulomb repulsion taking into account explicitly the dressed dipolons as mediators of superconductivity and nonrigid electron energy bands considering retardation and damping effects and electron-hole asymmetry to obtain single-quasiparticle energy dispersion in high- TC Bi2Sr2CaCu2O8+δ superconductors. The theory contains Mott renormalization, all important and necessary electron correlations and inherently the interaction on the superconducting layer due to all the planes of atoms and chains of atoms in the system. This constitutes an extension of the strong-coupling dipolon theory which explains also the regular and the pseudo-energy gaps, Tc, T_C and the peak-dip-hump structure of the line shape of the photoemission spectra of high TC superconductors. Our calculations of the single-quasiparticle energy dispersion for Bi2Sr2CaCu2O8+δ obtain a low energy kink at the binding energy near 60 meV which has already been identified in the experiments and also predict two additional high energy kinks at binding energies close to 110 and 170 meV, yet to be identified experimentally, in confirmation with the previous predictions by dipolon theory. The calculations not only explain the observed velocities of the quasiparticles below and above the low energy kink but also predict the velocities below and above the other predicted high energy kinks. The origin of the kinks and the effect of doping, temperature, interaction with phonons and coupling with magnetic modes have been discussed. The Migdal vertex correction does not change our results drastically. Direct relation between the photoemission spectra and the kink structure has been revealed. The knowledge gained from our calculations and explanation of the peak-dip-hump phenomenon and the kink structure has been explicitly written down in the form of five principles for understanding the photoemission results in high- Tc superconductors. The predicted results from our calculations for the high energy kinks, the quasiparticle velocities and the quasiparticle density of states are proposed to be observed by the experiments. Chapter 3 - The persistent photoconductivity (PPC) in high-temperature superconductors has some very special features. The effect is primarily observed in oxygen-deficient 1-2-3 superconductors such as YBa2Cu3Ox, but there are also reports of similar effects in some other compounds. The illumination of these substances leads to significant changes of the electronic properties. The electrical conductivity can be enhanced up to a factor of about 3 and the Hall effect shows that not only the free carrier concentration but also the carrier mobility is modified. Most intriguing is the persistence of the photo-induced effects in YBa2Cu3Ox up to temperatures slightly below room temperature. Below approx. 250 K, PPC

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Preface

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relaxation is practically unobservable. Even at higher temperatures the relaxation is extremely slow, taking hours or even days at room temperature. The photo-excitation, commonly called “photodoping”, is not only able to persistently change the doping level in these substances, but also their crystal structure: the lattice parameters are modified and also the electronic anisotropy is affected by the photo-excitation. Raman and infrared measurements give evidence for re-arrangements of the oxygen ions within the so-called copper-oxide chains in the crystal unit cell. There is a long-standing discussion whether these structural changes are just a side effect of photodoping or are important to stabilise the photo-induced electronic state, or are the actual origin of the persistent photo-induced phenomena in the hightemperature superconductors. Chapter 4 – It has generally been believed that, within the context of the BCS theory of superconductivity, the conduction electrons in a metal cannot be both ferromagnetically ordered and superconducting. In spite of this effect the discovery of ternary rare-earth compounds where magnetism and superconductivity coexist, has opened a new field of research. For this reason “intrinsic” and artificial F/S bilayers have been realized and investigated. This chapter contains our experimental investigation of the symmetry of the superconducting order parameter in cuprate superconductors and of the interaction between superconductivity and magnetism in atomic-size and artificial heterostructures. These objects are interesting not only for the fundamental physics but also for the possible development of useful systems for the realization of quantum electronic devices. The tool of investigation of these systems is the Andreev reflection spectroscopy. The point contact Andreev reflection spectroscopy carried out on RuSr2GdCu2O8 has evidenced the d-wave symmetry of the superconducting order parameter and, due to the presence of a magnetic order in this compound, a reduced amplitude of the superconducting energy gap has been measured. Furthermore, the presence of these two competitive orders has resulted in a peculiar temperature dependence of the superconducting energy gap. In order to investigate the interplay between superconductivity and ferromagnetism, artificial S/F (YBa2Cu3O7−x/La0.7Ca0.3MnO3) heterostructures have been analyzed. Also in this case from the conductance curves a reduced amplitude of the superconducting energy gap has been extrapolated. Chapter 5 - The spin-dependent potential together with the magnetic impurity and the spinorbit scattering are incorporated into the de Gennes-Takahashi-Tachiki theory of a diffusive superconductor-and- ferromagnetic metal to derive a formulation of the odd triplet superconductivity proximity effect. It is found that when the spin exchange interaction is inhomogeneous i.e., pointing in arbitrary directions rather than along the quantization axis, as in the N´eel spiral magnetic order, a new type of triplet condensate is generated, due to the role of the broken time-reversal invariance. The triplet amplitude still contains the s-wave state, similar to the conventional singlet pairing, but the frequency symmetry must be odd to obey the Pauli’s exclusion principle. As a result, the self-consistent order parameter contains only the singlet pair amplitude. The superconducting critical temperature of the bilayer is obtained by solving the secular equation exactly using the multimode method. The necessary condition for the occurrence of the induced long-range triplet component in the ferromagnet layer is characterized by the modulation of the pair amplitudes in the transverse direction. The possibility of the cryptoferromagnetic state is demonstrated in favor of the superconductivity and this may explain the possible origin of the magnetically dead layer. In addition, the influence of the magnetic impurity and the spin-orbit scattering is to decrease the decay

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length and to increase the oscillation period of the pair amplitudes which in turn enhances the critical temperature but in a less pronounced nonmonotonicity manner. Chapter 6 - The phonon modes that at coupling to electronic motion induce transition from adiabatic into antiadiabatic state, in which the Born – Oppenheimer approximation is not valid and electronic motion is dependent not only on nuclear coordinates but also on nuclear momenta, have been identified. It has been shown that for distorted lattice by Ag, B2g, B3g phonon modes, with the O(4), O(2), O(3) atoms displacements, there is a periodic shift of the saddle point of one of the CuO plane (d-pσ) band on the Γ-Y line at Y point of the Brillouin zone across the Fermi level from bonding to antibonding region. At this distortion, the nonadiabatic electron-phonon interactions stabilize the distorted lattice by – 34 meV/unit cell. Distorted lattice is characteristic by a specific - antiadiabatic fermionic ground state that is geometrically degenerate with fluxional arrangements of O(2), O(3) atoms in the CuO planes with the same ground state energy. The nonadiabatic mechanism at the lattice energy stabilization opens two asymmetric gaps in a and b direction in the originally metallic oneparticle spectrum of YBa2Cu3O7. Calculated critical temperature of the transition from adiabatic into antiadiabatic state is 92.8 K that is in good agreement with the experimental data for superconducting state transition. Present study has also revealed that in c direction a new gap that is considerably smaller than gaps in a, b directions should be identified. Appearance of van Hove singularity at Y point has also been calculated. For studied distortions, when saddle point of the d-pσ band in Y point approaches Fermi level, an abrupt change of the dispersion curve – low-Fermi energy electron velocity decrease at about 75 meV below Fermi level is predicted. For YBa2Cu3O7, this effect should be detected close to Y point on the Γ-Y line, i.e. in the off-nodal direction, if the corresponding ARPES experiments are performed.

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Chapter 7 - The high- Tc cuprate superconductors are highly anisotropic type-II superconductors with either tetragonal or orthorhombic crystal structure; however, they share one single common feature that all the cuprate superconductors possess the two-dimensional CuO2 planes. The mechanism of these remarkable cuprate superconductors has now gradually become clear after myriads of studies for more than two decades. The promising potential in application has outlined the unlimited vision in the near future. In this Chapter, the peak effect and anomalous Hall effect for type-II superconductors are investigated in this present study. There exists a peak in the critical current density as the temperature or applied magnetic field of the system increases. This is the peak effect or fishtail effect for the superconductors. The presence of impurities due to quenched disorder, irradiation or doping destroys the long-range order of flux line lattice, after which only shortrange order remains. A first-order phase transition between the short-range order and disorder in the vortex system eventually appears as a result of enormous increase in the dislocations inside the short-range domains when the applied magnetic field or the temperature increases. The origin of peak effect is in this kind of first-order phase transition. The peak value of the critical current density, the exact peak position and its corresponding half-width for a constant temperature as well as for a constant applied magnetic field based on this quasiorder-disorder first-order phase transition of the vortex system are evaluated. The Hall resistivity changes sign from positive to negative as the applied magnetic field (temperature) decreases for constant temperature (applied magnetic field) for many high- Tc and some conventional

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Preface

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superconductors when the temperature of the system is close to the critical temperature. Sometimes the Hall resistivity exhibits the double sign reversal property. This anomalous phenomenon for type-II superconductors is also studied based upon the theory of thermally activated motion of vortex bundles over a directional-dependent energy barrier. It is shown that the directional-dependent potential barrier renormalizes the Hall and longitudinal resistivities, and the Hall anomaly is induced by the competition between the Magus force and random collective pinning force of the vortex bundle. The Hall and longitudinal resistivities as functions of temperature and applied magnetic field for the thermally activated motion of vortex bundles are calculated. The double sign reversal or reentry phenomenon is also investigated. These studies are crucial because they might make available some important information for their future application. Chapter 8 – Mercury-based cuprates HgBa2CuO4+_ with 0 < _ E F ). At these circumstances, not only Migdal-Eliashberg approximation is not valid, but basic adiabatic Born-Oppenheimer approximation (BOA) does not hold. It indicates that current microscopic theories build on effective model Hamiltonians that tacitly assume validity of the BOA, including BCS theory and models of strongly correlated electrons, are inadequate for correct description of the electronic ground state of superconductors. For EP interactions driven transition from adiabatic into antiadiabatic state, nuclear motion dynamics becomes crucial for electronic state in some region of k-space. At these circumstances, electronic structure has to be studied as explicitly dependent on instantaneous nuclear coordinates Q as well as on instantaneous nuclear momenta P. It enables nonadiabatic Q,P-dependent modification of the BOA. Based on this modification, antiadiabatic theory of complex electronic ground state of superconductors is formulated. It is shown, that due to EP interactions that drive system from adiabatic into antiadiabatic state, symmetry breaking is induced and system is stabilized in antiadiabatic state at distorted geometry with respect to adiabatic equilibrium high symmetry structure. Stabilization effect in the antiadiabatic state is due to strong dependence of the electronic motion on the instantaneous nuclear kinetic energy, i.e. on the effect which is neglected on the adiabatic level within the BOA. Antiadiabatic ground state at distorted geometry is geometrically degenerate with fluxional nuclear structure in the phonon modes that drive system into this state. While system remains in antiadiabatic state, nonadiabatic polaron – renormalized phonon interactions are zero in well-defined k region of reciprocal lattice. This, along with geometric degeneracy of the antiadiabatic state, enables formation of mobile bipolarons that can move over lattice as supercarriers without ∗

Reviewed by A.S.Alexandrov, Loughborough University, UK

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Pavol Baňacký dissipation. More over, it has been shown that due to EP interactions at transition into antiadiabatic state, k-dependent gap in one-electron spectrum has been opened. Gap opening is related to shift of the original adiabatic Hartree-Fock orbital energies and to the k- dependent change of density of states of particular band(s) at Fermi level. Corrected one-particle spectrum enables to derive thermodynamic properties that are in full agreement with corresponding thermodynamic properties of superconductors. Based on the complex antiadiabatic theory, it has been shown that Fröhlich’s effective attractive electron-electron interaction term represents correction to electron correlation energy at transition from adiabatic into antiadiabatic state due to EP interactions. Increased electron correlation is consequence of stabilization of the system in superconducting electronic ground state, but not the reason of its formation.

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I. Introduction High value of critical temperature (Tc) in the group of cuprate superconductors [1,2] is a serious challenge for theory even 20 years since discovery of this group of materials. The high value of Tc should be related to corresponding high value of electron-phonon (EP) coupling constant, λ ≥ 1. The standard, generally accepted EP based theories of superconductivity, BCS or BCS-like theories [3-5], assume validity of the Migdal theorem and Eliashberg restriction [4,5] (ME approximation). The first is related to validity of the condition ωλ ⁄ EF > E F . This effect has crucial theoretical impact. At these

circumstances, not only ME approximation is not valid (including impossibility to calculate nonadiabatic vertex corrections [30] that represent off-diagonal corrections to adiabatic ground state), but adiabatic Born-Oppenheimer approximation (BOA) itself does not hold. Low-Fermi energy periodic fluctuation of band structure has been recently reported [31] also for first member of the group of high-Tc cuprates, for YBa2Cu3O7. In this case fluctuation is connected with electron coupling to combination of three phonon modes, Ag, B2g, B3g. It is related to vibration motion of apical O4 oxygen and O2, O3 oxygens in CuO planes. Again, periodic fluctuation of the saddle point of ( d − pσ ) band at Y point of 1st BZ through Fermi level can be identified. Also in this case, it represents reduction of Fermi energy to zero and transition of the system from adiabatic into antiadiabatic ( ω >> E F ). This effect is absent in corresponding nonsuperconducting analogs (e.g. AlB2 and YBa2Cu3O6 [28,31]). Transition from adiabatic ( ω > E F ) state due to EP interactions, seems to be the basic physical effect that is common for superconductors. It

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means, however, breakdown of the adiabatic BOA, i.e. breakdown of the approximation that is the very basic starting point of many-body theory of solids, including BCS theory as well as models of strongly correlated electrons in case of superconductors. On the level of the BOA, the motion of the electrons is a function of the instantaneous nuclear coordinates (usually only parametric dependence is considered), but is not dependent on the instantaneous nuclear momenta (velocities). Nuclear coordinate-dependence modifies nuclear potential energy by so called diagonal BO correction (DBOC) that reflects an influence of small nuclear displacements out-of the equilibrium positions and corrects the electronic energy of clamped nuclear structure. The DBOC enters directly into the potential energy term of nuclear motion (but leaves unchanged the nuclear kinetic energy) and in this way modifies vibration frequencies. The off-diagonal terms of the nuclear part of the equation of motion that mix electronic and nuclear motion over the nuclear kinetic energy operator term are neglected and it enables independent diagonalization of electronic and nuclear motion (adiabatic approximation). Neglecting the off-diagonal terms is justified only if these are very small adiabatic conditions, i.e. if the energy scales of electron and nuclear motion are very different and when relation ω/Ε > E F ) in some region of k-space. In this case, it is necessary to study electronic motion as explicitly dependent on the operators of instantaneous nuclear coordinates as well as on operators of instantaneous nuclear momenta. It is a new aspect for many-body theory of solids. In this work, consistent ab initio theory of complex electronic ground state of superconductors beyond the adiabatic BOA is presented. The theory is based on molecular electron-vibration theory on the Hartree-Fock SCF (HF-SCF) level that we have published in 1992 [32] and that has been recently applied to study superconducting properties of MgB2 [28]. In the present contribution, theoretical background of nonadiabatic P-dependent modification of the adiabatic BOA is presented along with corresponding sequence of canonical transformations of general nonrelativistic form of system Hamiltonian (molecular or solid state system). Solution of the final nonadiabatic form that has been obtained in real space orbital representation is transformed into quasi-momentum k, q space representation of solids. The effect of the transition from adiabatic into antiadiabatic state is analyzed and corresponding equations are presented. It is shown, that due to EP interactions that drive system from adiabatic into antiadiabatic state, symmetry breaking occurs and system in antiadiabatic state is stabilized at distorted geometry with respect to adiabatic equilibrium high symmetry structure. Stabilization effect in the antiadiabatic state is due to strong dependence of the electronic motion on the instantaneous nuclear kinetic energy, i.e. on the effect which is neglected on the adiabatic level within the BOA. Antiadiabatic ground state at distorted geometry is geometrically degenerate with fluxional structure of nuclear positions in the phonon modes that drive the system into this state. It has been shown that while system remains in antiadiabatic state, nonadiabatic polaron – renormalized phonon interactions are zero in well defined k region of reciprocal lattice. Along with geometric degeneracy of the antiadiabatic state it enables formation of mobile bipolarons that can move over lattice as

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Antiadiabatic Theory of the Electronic Ground State of Superconductors

5

supercarriers without dissipation (coherent dynamics). Due to EP interactions at transition into antiadiabatic state, k-dependent gap in one-electron spectrum has been opened. Gap opening is related to shift of the original adiabatic Hartree-Fock orbital energies and to the kdependent change of density of states of particular band(s) at Fermi level. The shift of orbital energies determines in a unique way one-particle spectrum and thermodynamic properties of system. It has been shown that resulting one-particle spectrum yields all thermodynamic properties that are characteristic for system in superconducting state, i.e. temperature dependence of the gap, specific heat, entropy, free energy and critical magnetic field. The kdependent change in the density of states close to Fermi level at transition from adiabatic (nonsuperconducting) into antiadiabatic state (superconducting) can be experimentally verified by ARPES or tunneling spectroscopy as spectral weight transfer at cooling superconductor from temperatures above Tc down to temperatures below Tc. In relation to the ab initio theory of complex electronic ground state of superconductors, the BCS theory and the Fröhlich effective attractive electron-electron interaction Hamiltonian are discussed on corresponding level. It has been shown that, with respect to present-day experimental and theoretical knowledge, the BCS theory, including concept of Cooper’s pairs formation, is oversimplified model description of superconducting ground state. The main points are specified in the paper. It has also been shown, that BCS Hamiltonian as a generic form with attractive electron-electron interaction term, e.g. for models of strongly correlated electrons, is inadequate for description of superconducting ground state. The reason, as it is shown by the ab initio theory, is related to the fact that the Fröhlich effective attractive electron-electron interaction term represents correction to electron correlation energy that arises as a consequence of EP interactions. In this respect, a strong electron correlation that is advocated mainly for cuprates is very realistic assumption but it is not primary reason of superconducting state stabilization. Results of the ab initio theory of complex electronic ground state of superconductors show that increased electron correlation is a consequence of EP interactions that drive system into antiadiabatic state and stabilize it at distorted nuclear geometry. Expressed in the other way, strong electron correlation is a consequence of stabilization of the system in superconducting electronic ground state, but not the reason of its formation. Basic derivations are performed in real space orbital representation. As there are some differences in details of Hartree-Fock treatment in solid state theory and in molecular physics, in section II. (Preliminaries) the quasi-particle form of general molecular Hamiltonian on crude-adiabatic and adiabatic level within HF-SCF solution is presented. In section III, theoretical background of canonical - base transformations of clumped nuclear (crudeadiabatic) Hamiltonian and electronic wave function to adiabatic Q-dependent level and to nonadiabatic Q,P-dependent level (nonadiabatic modification of the BOA) are presented. The results in the following sections are presented in real space orbital representations as well are in quasi-momentum space representation of solids. Dependence of electronic energy terms, i.e. corrections to zero-particle term (ground state energy), to one-particle term, including gap opening and to two-particle term, on nuclear coordinates Q and momenta P are presented in section IV. Antiadiabatic state, i.e. ground electronic state of superconductors is analyzed in section V, including derivation of corresponding thermodynamic quantities. Correction to electron correlation energy due to dependence of electronic motion on nuclear coordinates and momenta is specified in section VI. Effective attractive electron-electron interaction is analyzed in section VII with respect to the ab initio theory and to the original Fröhlich’s and

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Pavol Baňacký

the BCS treatment. In this connection, formation of mobile bipolarons is discussed based on the Q,P-dependent form of the wave function pertaining to system in antiadiabatic state. There is supplementary part, Appendix A – F, that presents details to the particular parts in the main text.

II. Preliminaries General form of nonrelativistic Hamiltonian of molecular or solid state system can be

(

+

written in second quantization formalism as an explicit function of electron a , a

(

+

)

)

and

nuclear b , b creation and annihilation operators,

H = TN (P ) + E NN (Q ) + ∑ hPQ (Q )a P+ aQ + PQ

1 0 v PQRS a P+ a Q+ a S a R ∑ 2 PQRS

(1)

The nuclear potential energy ENN and one-electron core term hPQ (electron kinetic energy plus electron-nuclear coulomb attraction term) are functions of the nuclear coordinate Qr operators (normal modes nuclear displacements out of fixed nuclear geometry R0) and nuclear kinetic energy TN is a quadratic function of the corresponding nuclear momenta operators Pr .The capital letters Q, P which stand for nuclear coordinate and momentum operators should be distinguished from subscripts in matrix elements where these

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letters indicate index of the orbitals, e.g.

ϕ P A ϕ Q = APQ .

In order to keep open possibility of application for real as well for complex wave ( functions, it is assumed that to vibration mode r there exists also corresponding mode r ( ω r = ω r( ) to which in quasi-momentum space of solids corresponds wave vector q or (-q),

(

respectively. In case of molecular systems with real wave functions it holds r ≡ r . For solids, in electronic quasi-momentum k-space representation, it is assumed that for spinorbital ( ϕ R (k , σ ) there is a complex conjugate spinorbital ϕ R (− k ,±σ ) . Also in this case, for real

(

wave functions of molecular systems ϕ R ≡ ϕ R . For nuclear coordinate and momentum operators hold:

( = (b

) ( ) , P = (b

)

(

)

Qr = br + br(+ , Qr( = br( + br+ , Qr+ = br+ + br( = Qr( Pr

r

+ ( r

−b

( r

( r

+ r

)

+ r

(

+ r

)

− b , P = b − br( = − Pr(

The Q-dependence of the terms E NN (Q ) and hPQ (Q ) in (1) can be expressed as the Taylor’s expansion at fixed nuclear configuration R0, ∞

∞

j =1

i =1

(i ) 0 ( j) 0 (R0 ) + ∑ u PQ (Q ) (R0 ) + ∑ E NN (Q ) hPQ (Q ) = hPQ E NN (Q ) = E NN

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Antiadiabatic Theory of the Electronic Ground State of Superconductors

7

Term E NN (R0 ) represents potential energy at fixed nuclear configuration R0, hPQ (R0 ) 0

0

is one–electron core term at fixed nuclear configuration R0 and { u PQ (Q ) } includes matrix (i )

elements of electron – vibration (phonon) coupling, i.e. r (Q ) = u PQ

∂hPQ (Q ) ∂Qr

r ... s r (Q ) = Qr = u PQ Qr ; u PQ

∂hPQ (Q ) ∂Qr ...∂Qs

r ... s Qr ...Qs = u PQ Qr ...Qs (3)

0

Two-electron terms v PQRS (electron-electron coulomb repulsion and exchange integrals) do not depend explicitly on the nuclear operators. It is obvious that the following symmetry relations have to hold, 0 0 0 0 0 0* 0 0 0* 0* hPQ = hQP , hPQ = hQ( P( , ν PQRS = ν QPSR = ν SRQP = ν RSPQ , ν PQRS = ν R(S(P(Q(

(4)

For coefficients of the Taylor’s expansions up to second order, it is required to hold, (

((

((

(

r r* rs sr rs * sr * r r* r E NN = E NN , E NN = E NN = E NN = E NN , u PQ = u QP = u Q( P( , ((

((

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rs sr rs * sr * u PQ = u PQ = u QP = u QP

(5)

There are several possibilities of approximate solution of this many-body problem. Assumption of validity of the Born-Oppenheimer approximation (BOA) is the basis of these treatments. Since the term BOA and subsequent applications are very often understood in a simplistic way, some aspects of the BOA as used in this work are presented in Appendix A. Without the loss of generality, what follows concerns closed shell system in electronic ground state.

II.1. Crude- adiabatic approximation: Standard many-body picture of molecular physics Providing that phonon and electronic energy spectrum are well separated, and

E

te 0

(R ) − E nte (R ) >> hω r

holds for relevant configuration space R near to Req , crude-

adiabatic (clumped nuclear) treatment is justified. In this case, electronic and nuclear parts of the Hamiltonian (1) are treated as statistically independent sets. Electronic Hamiltonian is only parametrically dependent on nuclear configuration, i.e. nuclear geometry is fixed at nuclear configuration R0, 0 0 (R0 ) + ∑ hPQ (R0 )a P+ aQ + H e = E NN PQ

1 0 a P+ aQ+ a S a R ∑ v PQRS 2 PQRS

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8

Pavol Baňacký

Application of the Wick’s theorem to the product of creation and annihilation operators yields for particular terms the following normal product form with corresponding contractions,

[

]

occ

∑ hPQ0 a P+ aQ = ∑ hPQ0 N a P+ aQ + ∑ hII0 PQ

∑v

0 PQRS

a P+ aQ+ a S a R =

PQRS

occ

∑v

PQRS

(

0 0 + ∑ ν IJIJ − ν IJJI

0 PQRS

PQ

[

(7)

I

]

) [

]

⎛ occ 0 ⎞ 0 0 0 N a P+ aQ+ a S a R + ∑ ⎜ ∑ ν PIQI + ν IPIQ − ν PIIQ − ν IPQI ⎟N a P+ aQ + PQ ⎝ I ⎠

(

)

IJ

By application of the Wick’s theorem, renormalized Fermi vacuum has been introduced and total set of orthonormal base orbitals ϕ´P , ϕ Q has been divided on two distinct groups;

(

the set of occupied

)

(ϕ I , ϕ J ) and set of unoccupied (ϕ A , ϕ B ) spinorbitals.

The electronic Hamiltonian (6) is then written in a quasi-particle form as a sum of zero, one and-two particle terms,

(

)

0 H e = H (0 ) + H (1) + H (2 ) = E NN + H e0 + H (1) + H (2 ) = E0te (R0 ) + H (′cad ) (R0 ) + H (′′cad ) (R0 )

(8)

a/ The scalar quantity in this Hamiltonian, H e = H (cad ) (R0 ) (i.e. zero-particle term 0

0

in H (0 ) ), is the result of the operators contractions and has the form,

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occ

H e0 = Φ 0 H e Φ 0 = ∑ hII0 + I

(

)

1 occ 0 0 0 ( R0 ) = E SCF ∑ ν IJIJ − ν IJJI 2 IJ

(9)

This term represents electronic energy calculated by Hartree-Fock SCF (HF-SCF) procedure at fixed nuclear configuration R0. Electronic ground state is represented by renormalized Fermi vacuum Φ 0 . It is antisymmetric electronic wave function that is expressed in the form of single Slater determinant constituted by lowest laying occupied spinorbitals {ϕ I } of complete orthonormal base {ϕ P } ,

Φ 0 (r , R0 ) = ϕ1 .........ϕ I

(10)

b/ The one-particle term H (1) = H (cad ) (R0 ) of the electronic Hamiltonian (8) has the form, '

[

H (1) = ∑ FPQ N a P+ aQ

]

(11)

PQ

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Antiadiabatic Theory of the Electronic Ground State of Superconductors

9

F (R0 ) = h 0 (R0 ) + ∑ (J Q − K Q )

(11a)

Q

Diagonalization of (11), FPQ = ε P δ PQ , i.e. solution of electronic Hartree-Fock equations 0

by HF-SCF procedure,

F (R )ϕ P (r , R0 ) = ∑ ε PQϕ Q (r , R0 )

(11b)

Q

yields set of eigenvalues, i.e. HF-orbital energies 0 0 0 ) ε P0 = hPP + ∑ (ν PQPQ − ν PQQP

(11c)

Q

and corresponding set of eigenfunctions

{ϕ P }

- the orthonormal set of optimized

spinorbitals. It means that one-particle term (11) can always be written in diagonal form and represents set of eigenenergies of system, i.e.

[

H (1) = ∑ ε P0 N a P+ a P

]

(11d)

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P

In an approximate way, the one-particle Hamiltonian H (1) represents complete electronic spectrum of system expressed over occupied and unoccupied - virtual spinorbitals calculated for electronic ground state by HF-SCF procedure. In particular, n-electron excited state wave function Φ hn pn can be constructed by promotion of n electrons from n occupied spinorbitals to n unoccupied spinorbitals (i.e. the same number n of holes (hn) and particles (pn) are created). Electronic energy of such excited state can be calculated over HF-eigenenergies that correspond to optimized spinorbitals of the ground state. In terms of orbital energies (11c), for total electronic ground state energy E0 (R0 ) holds, te

occ occ ⎞ 0 0 0 0 0 (R0 ) + E SCF (R0 ) = E NN (R0 ) + ⎛⎜ ∑ hII0 + 1 ∑ ν IJIJ − ν IJJI E 0te (R0 ) = E NN ⎟= 2 IJ ⎝ I ⎠ (12) occ 0 (R0 ) + 1 ∑ ε I0 + hII0 = E NN 2 I

(

(

)

)

Total electronic ground state energy of the system reaches the minimum at some equilibrium nuclear configuration R0 = Req . Corresponding Slater determinant (10) represents wave function of the electronic ground state at equilibrium nuclear configuration.

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Pavol Baňacký

For closed-shell electronic systems within restricted HF approximation, spinorbitals are expressed over spatial orbitals whereas each spatial orbital can be occupied by two electrons with opposite spins, i.e. spinorbitals ϕ P and ϕ P +1 are replaced by the same spatial orbital with different spin parts,

ϕ P .α and ϕ P .β . In this case, HF operator (11a), HF-orbital energies

(11c) and electronic energy (9,12) are correspondingly modified,

F (R0 ) = h 0 (R0 ) + ∑ (2 J Q − K Q )

(13)

Q

0 0 0 ) ε P0 = hPP + ∑ (2ν PQPQ − ν PQQP

(13a)

Q

occ

occ

I

IJ

(

)

0 0 0 ( R0 ) H e0 = Φ 0 H e Φ 0 = 2∑ hII0 + ∑ 2ν IJIJ − ν IJJI = E SCF

(13b)

occ occ ⎞ 0 0 0 0 0 (R0 ) + E SCF (R0 ) = E NN (R0 ) + ⎛⎜ 2∑ hII0 + ∑ 2ν IJIJ − ν IJJI E 0te (R0 ) = E NN ⎟= IJ ⎝ I ⎠ (13c)

(

occ

(

0 (R0 ) + ∑ ε I0 + hII0 = E NN

)

)

I

c/ The third term of the electronic Hamiltonian (8), i.e. two-particle term has the form,

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H (2 ) = H (''cad ) (R0 ) =

[

1 0 v PQRS N a P+ aQ+ a S a R ∑ 2 PQRS

]

(14)

It formally looks like standard coulomb electron-electron interaction term in (6). With respect to the fact that after application of the Wick’s theorem (7) the renormalized Fermi vacuum has been introduced and zero-particle (scalar) quantity (9) represents electronic energy of the ground state that accounts also for coulomb electron-electron interactions (see (9)) and one-particle term is diagonal (11d) and represents unperturbed HF-orbital energies { ε P }of the system (one-electron energy spectrum), then two-particle term (14) represents 0

perturbation part of the electronic Hamiltonian (8). Since perturbation (14) contains only electron-electron interaction term, contributions of this term represent electron correlation energy of the system in its ground electronic state. It is related to unbalanced treatment of electrons with parallel and antiparallel spins. Within the Hartree-Fock method correlation of pairs of electrons with antiparallel spins is not properly accounted for. Correlation of pairs of electrons with parallel spins is partly reflected over exchange terms (KQ in 11a, or ν IJJI in 9), 0

due to antisymmetry of the wave function (10). Seemingly better results of simple Hartree method that does not include electron spin explicitly (exchange repulsion is absent), is related to the fact that description of correlation effect for electrons with parallel and antiparallel spins is equally poor, but the errors are of different signs and tend to cancel. Introduction of antisymmetry in Hartree-Fock method improves treatments of electrons with parallel spins but leaves electrons with antiparallel spins without correction and cancellation of the errors no longer occurs. In general, correlation problem is not specific problem of the Hartree-Fock

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Antiadiabatic Theory of the Electronic Ground State of Superconductors

11

method, but it is rather the matter related to one-electron treatment that introduces electron motion in a mean field of all other electrons. In this respect, electron correlation energy is treated as a perturbation. Calculation of the electron correlation energy up to higher order of perturbation theory is usually done by diagrammatic many-body perturbation theory. For correlation energy in second order of perturbation theory, analytic expression for closed-shell system can be derived in a simple form, occ unocc

E corr = H (h1h2 p1 p2 ) = ∑ IJ

)

0 − v IJBA ε I0 + ε − ε A0 − ε B0

∑( AB

(2ν

0 IJAB 0 J

)

(14a)

From the fact that with respect to Fermi level, for energies of unoccupied states hold {ε

0 A}

> 0 and for energies of occupied states hold { ε I } < 0, follows that E corr is negative, 0

i.e. it decreases electronic energy of the ground state (this contribution corrects total electronic energy of the system (12)). It holds for arbitrary nuclear geometry R0 until system remains in a bound state. Each eigenfunction ϕ J in Slater determinant (10) can be expressed as a linear combination of the atomic orbital (AO) basis functions {μ },

ϕ P = ∑ c μ μ , that are fixed μ

P

at the positions of the particular nuclei of frozen nuclear configuration R0. It represents fixed basis set {μ ( x,0 )}. +

In second quantization, with single-bar a being creation operator of the crude-adiabatic

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electron, it can be written with respect to Fermi vacuum 0 as,

and,

μ (x,0) = a μ+ (x,0 ) 0

(15)

ϕ P ( x,0 ) = ∑ c μP μ ( x,0 ) = ∑ c μP a μ+ (x,0) 0 = a P+ ( x,0) 0

(15a)

μ

μ

For solids, in electronic quasi-momentum k-space, the basis functions are Bloch-periodic orbitals {μ (k , x,0 )} ,

μ (k , x,0 ) =

1

N

∑e N

ik ⋅tR

μ ( x − tR )

(15b)

tR

μ ( x − tR ) = μ ( x ) . The set of { μ ( x ) } is fixed basis set {μ ( x,0 )}at frozen nuclear configuration R0 (Q = 0). An eigenfunction ϕ P is then crystal orbital CO- ϕ P (band ϕ P ), which is a linear combination of the Bloch-periodic basis functions {μ (k , x,0 )} . In (15b), tR is translation vector, and

In second quantization it has the form, Superconducting Cuprates : Properties, Preparation and Applications, Nova Science Publishers, Incorporated, 2008. ProQuest Ebook Central,

12

Pavol Baňacký

μ (k , x,0 ) =

1

N

∑e N

ik .tR

a μ+ ( x,0) 0

(15c)

tR

and, μ

=

1

N

∑e N

ik .tR

N

1

ϕ P (k , x,0 ) = ∑ c μ(kP) μ (k , x,0) = ∑ c μ(kP)

∑e N

μ

ik .tR

a μ+ (x,0 ) 0 =

tR

.

a Pk+ (k , x,0 ) 0

(15d)

tR

In this case, occupancy of the band is not distinguished by the index of the band (P) itself but it is determined by the value of k-vector of particular band dispersion

ε P0 (k ) with respect

to the energy of Fermi level ε F . 0

The electronic Hartree-Fock equations (11b) are solved for different displaced but fixed nuclear configurations {Rd} at Req and potential energy (hyper)surface (PES) can be calculated. It can be done by gradient technique where nuclear force constants are calculated in the analytic way by minimization of the total electronic energy (12) as a function of nuclear coordinates R (see e.g.[33]). Knowledge of the PES enables calculation of the force constants of vibration motion and subsequently it enables to solve nuclear Schrödinger equation,

{T with nuclear Hamiltonian,

N

+ E 0te (R ) − E 0TS,ν (R )}χ 0,ν (R ) = 0

(16)

H N (R ) = TN + E 0te (R ) ,

(16a)

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i.e. to solve the problem of nuclear motion quantization, mod 1 ⎞ mod ⎛ H N → H Q = ∑ hω r ⎜ br(+ br + ⎟ = ∑ hω r (nr + 1 2 ) 2⎠ r ⎝ r

(16b)

with nuclear vibration wave function, mod

mod

r

r

1

χ (R ) = ∏ χ r , n (R ) = ∏

nr !

(b )

+ nr r

0 ,

(17)

Creation and annihilation operators of crude-adiabatic phonon modes are denoted also as +

single-bare operators, br , br . The operator of nuclear displacements out of fixed nuclear

(

+

geometry R0 (R ∝ R0 +Q) for the crude-adiabatic normal mode r, is Qr = br + br(

(

+

)

) and

corresponding momentum operator is Pr = br − br( . If the influence of the nuclear displacements out of fixed nuclear geometry R0 on the electronic ground state wave function and electronic energy is assumed to be negligible (crude-adiabatic BOA, i.e. Q-independent), the wave function of the total system is,

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Antiadiabatic Theory of the Electronic Ground State of Superconductors

13

Ψ0 (r , R ) = χ (R )Φ 0 (r , R0 ) ,

(18)

Ψ0 (r , Q ) = χ (Q )Φ 0 (r ,0)

(18a)

or in terms of nuclear motion,

Energy of the total system in the ground electronic state is,

E 0TS (R0 ) = E 0te (R0 ) + ∑ hω r (nr + 1 2)

(18b)

r

For fermion and boson creation and annihilation operators, the standard anticommutation and commutation relations hold,

{a

P

, aQ } = 0 , {a P , aQ+ } = δ PQ

[b , b ] = 0 , [b , b ] = δ r

s

r

+ s

rs

(19) (19a)

“Independence” of fermions and bosons, i.e. possibility of simultaneous diagonalization of electronic and nuclear part of system Hamiltonian, requires also validity of the following commutation relations,

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[a

P

]

[

]

, br = 0 , a P , br+ = 0

(19b)

If it is necessary, the effect of electron-vibration (phonon) coupling on the electronic energy and vibration (phonon) spectra are usually calculated by perturbation theory as the corrections to the crude adiabatic ground electronic state and phonons renormalization. On the crude-BOA level these corrections are neglected, however.

II.2. Adiabatic approximation In case of crude-adiabatic approximation, the electrons “see” the nuclei at theirs instantaneous positions at rest and nuclei do not “feel” internal dynamics of electrons. Within the spirit of the BOA it would be correct if the electrons follow nuclear motion instantaneously, i.e. electronic state has to dependent explicitly on instantaneous nuclear positions. In this case, the wave function of the system, instead of the form (18a) with Qindependent electronic part should be rather with Q-dependent electronic part,

Ψ0 (r , Q ) = χ (Q )Φ 0 (r , Q )

(20)

For molecular systems an analytic derivative method is used. The nuclear force constants are calculated by diagonalization of the Hartree-Fock equations that are now functions of

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14

Pavol Baňacký

nuclear coordinates. It results in solution of the set of Coupled Perturbed Hartree-Fock (CPHF) equations (see e.g. [34,35]). Alternative treatment to this problem, denoted as quasi-particle transformation technique, has been proposed and elaborated by Svrcek [36]. In this treatment, the requirement that electrons follow the nuclear motion adiabatically has been expressed through the fermion and boson creation and annihilation operators. In [37] it has been shown that solution of the adiabatic problem by quasi-particle transformation technique is equivalent to the results of the CPHF method. Seemingly it means that there is no extra profit of this treatment. In the present paper it is shown, however, that physical background behind the quasi-particle transformation technique is substantial. It can be effectively generalized, which justify and allow application to more complicated situation when the BOA is not valid, i.e. to

( )

study an intrinsic nonadiabatic state E 0 (R ) − E n Req < hων . te

te

To keep the present paper compact, the main points of the original formulation of the adiabatic quasi-particle transformation technique [36] are presented in Appendix B. Some aspects of different treatments of electron-vibration (phonon) coupling should also be mentioned. Standard solid-state treatment of electron-phonon interaction is based on perturbation theory. Starting Hamiltonian is,

H = H 0 + H ep .

(21)

The unperturbed part (to be consistent with crude-adiabatic notation, the single-bare operators are used) has the form,

(

H 0 = ∑ ε k0 a k+σ a kσ + ∑ hω q b−+q bq + 1 2 Copyright © 2008. Nova Science Publishers, Incorporated. All rights reserved.

kσ

)

(21a)

q

The perturbation Hamiltonian H ep , instead of Λ-perturbation term (A7, A15), is represented by an electron-phonon (EP) interaction term. The simplest form of this term is,

H ep =

∑σu (b q

q

k ,q ,

)

+ b−+q a k++ q ,σ a k ,σ =

∑σu

q

Qq a k++ q ,σ a k ,σ

(21b)

k ,q.

It can be derived at the assumption of small perturbation of rigid-periodic lattice potential due to vibration displacements of nuclei out of equilibrium positions. Within the notation used in the present paper, term (21b) corresponds basically to the first order contribution of Taylor’s expansion of the core Hamiltonian hPQ (Q ) , (see 1, 2), with respect to nuclear displacement Q on the crude-adiabatic level. This is evident from eq. (B19b), when at equilibrium geometry for potential energy of nuclear motion holds E NN = 0 . The r

unperturbed Hamiltonian (21a) is represented by the terms (11d) and (16b). Due to the form of EP interaction H ep (21b), the first order perturbation correction to the electronic ground state Φ0, i.e. diagonal perturbation term equals zero: 〈 Ψ0 Hˆ ep Ψ0 〉 = 0 .

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Antiadiabatic Theory of the Electronic Ground State of Superconductors

15

All interesting physics is then related to higher-order contributions with participation of excited electronic states, i.e. the first possible non-zero contributions are in the second order

(

)

TS TS of perturbation theory, i.e. terms of the form 〈 Ψ0 Hˆ ep Ψn 〉〈 Ψn Hˆ ep Ψn 〉 / E 0 − E n .

These off-diagonal contributions represent, in this treatment, nonadiabatic corrections to the adiabatic ground state, i.e. nonadiabatic correction to electronic ground state energy ΔE 0

te

(calculated as corrections to dispersion of electronic bands Δε P (k ) in solids) and vibration

(phonon) renormalization Δω r (corrections to phonon dispersion Δω r (q ) in solids).

An exact treatment of nonadiabatic corrections calculation assumes independent

{

}

calculation of electronic excited states energies E n (≠ 0 ) (R ) . It would require new e

optimization of excited state wave functions Φ m (r , R ) . It should be extremely complicated

since excited state wave functions have to be orthogonal to the ground state wave function. At practical calculations, an approach is used (see II.1) which is based on the orthonormal orbitals (bands) {ϕ a } already optimized for the ground state electronic wave function

Φ 0 (r , Req ). By promotion of electron(s) from occupied orbital(s) {I, J,..} to virtual –

unoccupied orbital(s) {A,B,...}, excited state(s) configurations { Φ A } can be constructed as a linear combination of corresponding Slater determinants {Φ I → A }. It can be shown that, e.g.

single-electron excitations yield (for closed-shell system) two excited state electronic configurations – lowest lying excited state that is singly excited triplet state Φ A( I → A) , and 3

singly excited singlet state Φ A( I → A) . Differences in the electronic energies of these excited 1

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state configurations with respect to the electronic energy of the ground state are; 3

E Ae ( I → A) (R ) − E 0e (R ) = ε A0 (R ) − ε I0 (R ) − J IiA

(22)

for singly excited triplet state and, 1

E Ae ( I → A) (R ) − E 0e (R ) = ε A0 (R ) − ε I0 (R ) − J IA + 2 K IA

(23)

for singly excited singlet state. For approximations, which do not consider explicitly for two-electron terms, the differences in energies of singly excited triplet and singlet states with respect to the ground state energy are the same and equal to the difference of the energies of involved orbitals,

E Ae ( I → A) (R ) − E 0e (R ) = ε A0 (R ) − ε I0 (R )

(24)

Multiple electronic excitations can be calculated in a similar way, by generation of Slater determinants of p-particle, h( ≡ p )-hole states in notation of particle-hole formalism.

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Pavol Baňacký

In this way, without an explicit calculation of electronic excited state wave functions, the nonadiabatic corrections to (crude-)adiabatic electronic ground state are calculated over optimized eigenfunctions – occupied ( ϕ I ) and unoccupied - virtual ( ϕ A ) orbitals of single

(

)

Slater determinant of the electronic ground state Φ 0 r , Req . Condition for save application of the BOA, expressed in the terms of the ground state orbital energies, is of the form,

ε I0 (R ) − ε A0 (R ) >> hω r

(25)

It has to be valid for relevant configuration space R at Req and for the couple of frontier orbitals, i.e. highest occupied ϕ I ≡ ϕ HOMO and lowest unoccupied ϕ A ≡ ϕ LUMO orbitals. In case of solids, with quasi-continuum of states, in complex k-space representation this inequality can be rewritten in the form,

ε S0 (k c ) − ε F0

Req

>> hω r

(26)

This relation has to hold over the relevant configuration space R at Req for energies of all bands (S) of multi-band system in analytic critical points kc (absolute or local maxima, minima and saddle points) of 1st BZ, with respect to the energy of Fermi level ε F . 0

In molecular quantum theory, different treatment of electron-vibration interaction has also been elaborated. It concerns calculation of the correction to the ground state total electronic energy ΔE (ád ) that corrects potential energy of nuclear motion on the adiabatic Copyright © 2008. Nova Science Publishers, Incorporated. All rights reserved.

0

level. In 1997, Kutzelnigg [38] has proved in a rigorous way that so called Born-Handy ansatz [39,40], according to which the diagonal Born-Oppenheimer correction (DBOC) to adiabatic electronic state - Bnn, (A10) can be calculated directly in laboratory Cartesian coordinate system, is physically correct. It is very crucial results, since it eliminates complicated problem with separation of center-off-mass (COM) motion, which arise at introduction of relative coordinates in a molecule-fixed frame system at practical calculations. Accordingly, the exact adiabatic correction to the total electronic ground state energy is B00 (A10), i.e.

( )

( )

r r ΔE (0ad ) (R0 ) = Φ 0 r , R TN Φ 0 r , R

( )

( )

r r h 2 ∂Φ 0 r , R ∂Φ 0 r , R =∑ ∂Riα ∂Riα iα 2 M i

R0

= −∑ i

( )

( )

r r h2 Φ 0 r , R ∇ i2 Φ 0 r , R 2M i

R0

= (27)

= Λ 00 = B00 R0

The derivatives of the ground state Slater determinant Φ 0 in the bracket of (27) are performed with respect to the Riα , i.e. with respect to the Cartesian component α of the ith nucleus.

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Antiadiabatic Theory of the Electronic Ground State of Superconductors

17

In 1999 it has been shown [37] that this correction is equal to adiabatic correction to the total electronic energy of the ground state (B30a), calculated over the expansion

( )

coefficients c PQ Q of the quasi-particle transformation (B2), i.e.

( )

( )

r r ΔE (0ad ) (R0 ) = Φ 0 r , R TN Φ 0 r , R

Ŕ0

r = ∑ hω r c AI

2

(28)

rAI

r

In (28), c AI stands for derivative of the expansion coefficients (B2) of quasi-particle transformation over the r-normal mode coordinate, r c AI =

∂c AI ∂Qr

(29)

Within the single Slater determinant representation of the ground electronic state, this relation is exact since the eigenfunctions – spinorbitals ϕ Q (R ) of the Hartree-Fock

{

}

equations (11b) form complete basis, i.e. closure property holds,

∑ ϕ (r , R ) Q

ϕ Q (r , R ) = 1

(30)

Q

It means that both sets, i.e. set of occupied

{ϕ I (r , R )} and set of unoccupied {ϕ A (r , R )}

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orbitals are included at calculation of (28). As it is seen from (28), electronic ground state energy correction is due to virtual transitions between occupied {ϕ I (r , R )} and unoccupied

{ϕ A (r , R )}

at nuclear vibration motion. In this respect, even ΔE (ád ) represents exactly DBOC, it covers basically “nonadiabatic” (off-diagonal) corrections in the sense as these are calculated by perturbation theory when excited electronic states are approximated by virtual orbitals, as it has been discussed above. It can be seen very clearly from the expression for correction to frequency of normal modes (B36) that is identical to the one derived by perturbation theory. Correctness of eq. (27, 28) has been verified by high precision calculation of H2, HD, D2 molecules [37] with respect to the exact results for H2 published by Kolos and Wolniewicz [41,42]. This treatment can be used for complex molecular systems and it should be effective also in case of solids. An important conclusion can be made at this place. Electron-vibration (phonon) interactions on the adiabatic level do not stabilize total electronic energy of the ground state. orbitals

0

The adiabatic correction ΔE (ád ) to the ground state total electronic energy (without 0

correlation energy contribution) is small but always positive, (27, 28). More over, correction to the energy of the total system ΔE

TS

, which is composed of the electronic ΔE (ád ) 0

correction and corrections Δω r to vibration (phonon) modes (B36a), is also positive with dominant contribution of the electronic correction, Superconducting Cuprates : Properties, Preparation and Applications, Nova Science Publishers, Incorporated, 2008. ProQuest Ebook Central,

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Pavol Baňacký

2h ω r 2⎛ ⎜⎜1 − 0 r 0 ΔE TS = ∑ hω r c AI εA −εI rAI ⎝

( )

(

)

⎞ ⎟⎟ ⎠

(31)

The only possible stabilization contribution to the ground state electronic energy on the adiabatic level can arise from the correction to electron correlation energy – see (95d). This conclusion does not contradict the Jahn-Teller effect, or Peierls distortion in solids. In these cases, decrease of the total electronic energy connected to nuclear displacements from high symmetry to lower symmetry nuclear arrangement appears already on the crudeadiabatic level within a clumped nuclear Hamiltonian approximation and it is related to removing of degeneracy of occupied and unoccupied states (asymmetry in population of degenerate states). Degeneracy that is present at high symmetry nuclear geometry is not present at lower symmetry structure, which in fact represents actual equilibrium nuclear configuration Req with lower total electronic energy as it corresponds to the structure with higher symmetry. In what follows, a connection of the adiabatic treatment, as presented above and in the Appendix B, to canonical transformations and to introduction of new dynamical variables of the system Hamiltonian is shown. The established link is then extended toward solution of more general problem, when adiabatic condition (25, or 26) is not valid and system is in the intrinsic nonadiabatic state, i.e. when instead of (25, 26) holds

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ε S0 (k c ) − ε F0

Req

ε I0 (R ) − ε A0 (R ) ≤ hω r , or

≤ hω r in case of solids.

III. Base transformation – introduction of new dynamical variables III.1. Q-dependent adiabatic transformation Adiabatic, nuclear displacement Q-dependent electronic wave function Φ 0 (r, Q ) in (20)

assumes existence of complete orthonormal base {ϕ R ( x, Q )} , i.e. validity of the following relations,

( ) ( )

ϕ R x, Q ϕ S x, Q = δ RS ,

∑ϕ R

R

(x, Q) ϕ (x, Q) = 1

(32)

R

Electron creation and annihilation operators that correspond to the Q-dependent moving +

base are denoted as double-bar operators ( a , a ). Boson operators related to the Q-dependent

⎛ ⎝

+

⎞ ⎠

⎛ ⎝

+

⎞ ⎠

moving base are also denoted as double-bar operators, Q r = ⎜ b r + b r( ⎟ and Pr = ⎜br − br( ⎟ Then,

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( )

( ) ( )

( )

a R x , Q 0 = ϕ R x, Q , a R x, Q ϕ R x , Q = 0

19

(33)

Since adiabatic electrons are also fermions, the operators have to obey standard fermion anticommutation relations,

{

}

+ ⎧ ⎫ ⎨a R , a S ⎬ = δ RS , a R , a S = 0 ⎩ ⎭

(34)

( )

+

+

( )

In (34), shorthand notation has been used, a R ≡ a R x, Q and a R ≡ a R x, Q .

Crude-adiabatic electronic wave function Φ 0 (r ,0 ) that does not depend on the nuclear

displacements Q is expanded over fixed basis set functions

{ϕ R (x,0)} that are eigenfunctions

of clumped nuclear electronic Hartree-Fock equations (11b). This is complete and orthonormal base,

ϕ R ( x,0)ϕ S (x,0 ) = δ RS ,

∑ ϕ (x,0) R

ϕ R (x,0 ) = 1

(35)

R

Crude-adiabatic fermion creation and annihilation operators corresponding to the fixed +

basis set are denoted as single-bar operators ( a , a ), i.e.

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a R+ ( x,0) 0 = ϕ R (x,0 ) , a R ( x,0 ) ϕ R ( x,0 ) = 0

(36)

Also in this case, the standard anticommutation relations (19, 19a,b) hold and also in this

case shorthand notation has been used, a R ≡ a R ( x,0 ) , a R ≡ a R ( x,0 ) . +

+

Due to properties (32, 35), the two bases are interconnected by the base transformation of the following form,

( ) ( )

( ( ) ϕ (x, Q)

ϕ R ( x,0 ) = ∑ ϕ S x, Q ϕ S x, Q ϕ R ( x,0) = ∑ c RS Q S

S

+

S

(37)

Then, for fermion operators of second quantization one can write,

()

( ()a

a R = ∑ c RS Q a S , a R+ = ∑ c RS Q S

S

+

+

S

Elements of the Q-dependent transformation matrix c(Q ) in (37,38) are,

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(38)

20

Pavol Baňacký

()

( )

c RS Q = ϕ R ( x,0) ϕ S x, Q Since,

()

(39)

( )

* c RS Q = ϕ S x, Q ϕ R (x,0) ,

(40)

then due to closure property and orthonormality (32, 35) of the bases, it can be derived that base transformation matrix c(Q ) is an unitary matrix,

∑c T

* * c ST = δ RS = ∑ cTR cTS , C + = (C T ) = C −1 *

RT

(41)

T

It can be shown that the base transformation is identical with canonical transformation of operators that satisfy anti-commutation relations (19, 34). There are two basic possibilities of canonical transformations [43]; The standard, most frequently used canonical transformation works with the same set of dynamical variables (Aν), i.e.

( )

~ ~ T : H (Aμ ) ≡ H (Aμ ) = H Aμ = H (U + ( Aν )Aμ U ( Aν )) ~ Aμ = U + ( Aν )Aμ U ( Aν ) = f ( Aν )

(42) (42a)

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Canonical transformation (42) is usually used in an effort to make original Hamiltonian diagonal or “more” diagonal, i.e. to remove off-diagonal interaction matrix elements in a system Hamiltonian. Then, for system Hamiltonian H = H 0 + H int at the canonical

transformation, the anti-hermitean operator S of unitary matrix U = exp(S ) is “constructed” in the form that eliminates presence of interaction term in transformed Hamiltonian completely or at least up to the first order of commutator expansion, i.e. the condition has to be fulfilled,

[H 0 , S ] + H int

=0

(43)

This transformation change the form of the Hamiltonian but preserve original system variables. There is also other possibility, however. By canonical transformation of operators a set of new dynamical variables ( Aν′ ) can be introduced,

Aμ ≡ Aμ ( Aν′ ) = U + ( Aν′ )Aν′U ( Aν′ )

(44)

Then Hamiltonian is not transformed itself, it remains of the original form, but its variables (Aν) are replaced by new variables ( Aν′ ) ,

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(

)

H (Aμ ) ≡ H U + ( Aν′ )Aν′U ( Aν′ )

21 (45)

The Hamiltonian written in new variables is,

~ H (Aμ ( Aν′ )) ≡ H ( Aν′ )

(46)

Since at this transformation, there is not any requirement for fulfillment of condition like (43), the transformation does not make Hamiltonian “more” diagonal, but very often it discloses physical aspects of the problem that are not obvious from non-transformed form of the original variables of system Hamiltonian. Beside the other aspects, appearance of the attractive effective electron-electron interaction term at this transformation will be shown at the other place. For canonical transformation of fermion operators can be written, +

+

a R = U + a RU , a R+ =U a RU +

−1

The unitary matrix ( U = U ) is of exponential form U = e operator S (S+ = - S) is of the bilinear form,

()

(47) S

and anti-hermitean

+

* S = ∑ γ RS Q a R a S , γ RS = −γ SR RS

(48)

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The γ(Q) matrix is Q-dependent. For canonical transformation (47) then holds,

a R = e −S a R e S = a R +

[ ] [ ] ] [[ ] ] ]

1 1 1 aR,S + aR,S ,S + a R , S , S , S + ........ 1! 2! 3!

(49)

Due to fermion anti-commutation relations (34), the commutation expansion (49) can be summed up in a closed form,

⎛ γ RS ⎛⎜⎝ Q ⎞⎟⎠ ⎞ ⎟a S aR = ∑ ⎜ e ⎜ ⎟ S ⎝ ⎠

(50)

In a same way, for creation operator can be derived, +

⎛ γ RS ⎛⎜⎝ Q ⎞⎟⎠ ⎞ + ⎟ aS a = ∑⎜e ⎜ ⎟ S ⎝ ⎠ + R

(50a)

Comparing (50, 50a) with relations (38), one can see that the base transformation is identical with canonical transformation of fermion operators. More over it is identical also with quasi-particle transformation as it is postulated in Appendix B. The exponential form of Superconducting Cuprates : Properties, Preparation and Applications, Nova Science Publishers, Incorporated, 2008. ProQuest Ebook Central,

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Pavol Baňacký

canonical transformation (49) legitimates also Taylor’s expansion of the matrix elements of quasi-particle transformation coefficients (B4, 38), or what is equivalent, the Taylor’s expansion of base transformation coefficients,

()

∞

1 r1 .. rk c PQ Q r1 ...Q rk ∑ ! k k =0 r1 .. rk

c PQ Q = ∑

(51)

The form of transformation relations for boson operators of system Hamiltonian is fully dictated by the factorized form of the total system wave function (20). It expresses possibility of simultaneous, independent diagonalization of electron and boson subsystems. It means that transformed fermion and transformed boson operators obey not only standard anticommutation and commutation relations within the individual subsystems,

{a , a }= 0 , ⎧⎨⎩a , a P

Q

P

+ Q

[ ]

⎫ ⎬ = δ PQ , b r , b s = 0 , ⎭

+ ⎡ ⎤ b r ,bs ⎥⎦ = δ rs , ⎢⎣

(52)

but transformed operators of both subsystems have to commute mutually like the original operators, i.e. also commutation relations have to hold,

[a , b ] = 0 P

r

⎡ ⎢⎣

+

, aP ,br

⎤ ⎥⎦ = 0

(53)

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With respect to the fermion transformation relations (38), the form of transformation relations for boson operators that fully respects commutation relations (53) is (B3, B10), i.e.

()

( ()aa

br = br + ∑ d rPQ Q a P+ aQ , br+ = br+ + ∑ d rPQ Q PQ

PQ

+

+ P

(54)

Q

()

For matrix elements of transformation matrix d Q , the Taylor’ expansion is defined,

()

∞

1 s1 .. s k d rPQ Q s1 ...Q sk ∑ k = 0 k! s1 .. s k

d rPQ Q = ∑

(54a)

()

The relation that holds between transformation matrix d Q and transformation matrix

()

c Q with respect to (53) is specified in Appendix B – (B6, B7). It can be shown that adiabatic transformation preserves the total number of electrons, and nuclear coordinate operator is invariant under transformation, i.e.

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(

23

)

+ ⎞ N = ∑ a P+ a P = ∑ a P+ a P = N , Q r = ⎛⎜ b r + b r( ⎟ = br+ + br( = Qr ⎠ ⎝ P P

(55)

(

+

Up to first order of Taylor’s expansion, the momentum operator Pr = br − br(

)

transforms as, + + + ( ( ⎞ ⎛ r r Pr = ⎜ b r − b r( ⎟ + 2∑ c PQ a P a Q = P r + 2∑ c PQ a P aQ ⎠ ⎝ PQ PQ

⎛ ⎝

+

⎞ ⎠

(56)

The term Pr = ⎜br − br( ⎟ in (56), is nuclear momentum operator on adiabatic level. For adiabatic Q-dependent spinorbitals

( )

ϕ P x, Q , which are the basis functions of the

adiabatic Q-dependent electronic wave function of the ground state Φ 0 (r , Q ) , expressed over crude-adiabatic orbitals can be derived,

( )

( )

+ ⎛ ⎛ 2 ⎞⎞ r Q r a R+ + O⎜ Q ⎟ ⎟ 0 = ϕ P x, Q = a P x, Q 0 = ⎜ a P+ − ∑ c PR

⎝

⎝

rR

⎠⎠

= ϕ P ( x,0 ) − ∑ c Q r ϕ R (x,0 ) + ..... = ϕ P ( x,0 ) − ∑ c Qr ϕ R ( x,0 ) + .... r PR

rR

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(57)

r PR

rR

As it is seen from (57), adiabatic wave function is modulated by the instantaneous nuclear coordinates {Qr} of particular vibration (phonon) modes {r} with the weight r

proportional to transformation coefficients c PR (coefficients of transformation matrix in the r

first order of Taylor’s expansion). In Appendix D it is shown that c PR covers the strength of electron-vibration (phonon) coupling up to the first order of Taylor’s expansion. From eq. (28) it is seen that these coefficients fully determine also exact adiabatic correction to the electronic energy of the ground state ΔE (ád ) . 0

III.2. P-dependent nonadiabatic transformation Save

E

te 0

application

of

te n

to hold not only at Req but also over relevant configuration space

(R ) − E (R ) >> hω r

the

adiabatic

BOA

requires

the

inequality

R=Req±ΔR near to Req. Relevant configuration space is represented at least by amplitudes of pertinent vibration (phonon) modes of the system. Let as consider situation when this inequality is valid for Req but it does not hold for R=Req±Q. Within single Slater determinant approximation of the ground electronic state it can be written as,

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24

Pavol Baňacký

ε I0 (Req ) − ε A0 (Req ) >> hω r

Q

→ ε I0 (Req ± Q ) − ε A0 (Req ± Q ) ≤ hω r

(58)

Corresponding relation holds for solids,

ε S0 (k c ) − ε F0

Req

>> hω r

Q

→ ε S0 (k c ) − ε F0

Req ± Q

≤ hω r

(59)

The inequalities on the rhs of (58, 59) indicate that system at vibration motion is in intrinsic nonadiabatic state. At these circumstances the BOA in the standard Q-dependent form (20) is not valid. The Λ term (A7) that through nuclear kinetic energy operator couples electronic and nuclear motion can be large and can not be treated as a perturbation. It indicates that at the instantaneous nuclear configuration (Req±Q), instantaneous nuclear

~

kinetic energy (momenta) has been significantly changed, i.e. TN → TN . Electrons at these circumstances are not able to follow nuclear motion adiabatically. It means that electronic wave function, in order to respect this fact, should be dependent not only on instantaneous nuclear coordinates Q but it should also be an explicit function of the instantaneous nuclear momenta P, i.e. Φ 0 ≡ Φ 0 (r , Q, P ) . At this moment we assume that wave function of total system can be found in the following factorized form,

Ψ (r , Q, P ) = ∑ χ m (Q, P )Φ m (r , Q, P )

(60)

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m

The form of the wave function (60) is basically P-dependent modification of the original Q-dependent BOA - (A3). Like in adiabatic case, solution of the problem will be restricted to electronic ground state, i.e.

Ψ0 (r , Q, P ) = χ 0 (Q, P )Φ 0 (r , Q, P )

(61)

It means that effect of nuclear momenta will be covered only in the form of Q,P-

~

dependent diagonal correction Λ 00 (Q, P ) = Φ 0 (r , Q, P ) TN Φ 0 (r , Q, P ) , i.e. in a similar

way as it has been covered the effect of instantaneous nuclear coordinates Q on the adiabatic

()

( )

( )

level (27), i.e. Q-dependent adiabatic DBOC, Λ 00 Q = Φ 0 r , Q TN Φ 0 r , Q . Solution of this problem is similar to the transition from crude-adiabatic to adiabatic level as presented above. Now, however, the transition from adiabatic to nonadiabatic level is established. Nonadiabatic, nuclear displacement and momentum (Q,P)-dependent electronic wave function Φ 0 (r , Q, P ) in (61) assumes existence of complete ortonormal basis

set {ϕ R ( x, Q, P )}, i.e. validity of the following relations,

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ϕ R ( x, Q, P )ϕ S ( x, Q, P ) = δ RS ,

∑ ϕ ( x, Q , P ) R

25

ϕ R ( x, Q, P ) = 1

(62)

R

Electron creation and annihilation operators that correspond to the (Q,P)-dependent +

moving base are denoted as bar-less operators ( a , a ). Boson operators related to the (Q,P)-

(

+

dependent moving base are also denoted as bar-less operators, Qr = br + br(

(

+ ( r

)

)

and

Pr = br − b . Then,

a R+ ( x, Q, P ) 0 = ϕ R ( x, Q, P ) , a R ( x, Q, P ) ϕ R ( x, Q, P ) = 0

(63)

Since nonadiabatic electrons remain fermions, the operators obey standard fermion anticommutation relations,

{a

R

}

, a S+ = δ RS , {a R , a S } = 0

(64)

In (64), shorthand notation is used, a R ≡ a R ( x, Q, P ) and a R ≡ a R ( x, Q, P ) . +

+

Since adiabatic Q-dependent moving base derived by adiabatic transformation is complete and orthonormal (32}, then due to (62), the base transformation to nonadiabatic (Q,P)-dependent moving base can be established over the base transformation relation,

( )

( )

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ϕ R x, Q = ∑ ϕ S ( x, Q, P ) ϕ S ( x, Q, P ) ϕ R x, Q = ∑ (cˆ RS (P ))+ ϕ S ( x, Q, P ) S

(65)

S

For fermion operators of second quantization follow, +

a R = ∑ cˆ RS (P )a S , a R = ∑ (cˆ RS (P )) a S+ S

+

(66)

S

Elements of the P-dependent transformation matrix Cˆ (P ) are,

( )

( )

* ( P ) = ϕ S ( x , Q , P ) ϕ R x, Q cˆ RS (P ) = ϕ R x, Q ϕ S ( x, Q, P ) , c RS

(67)

The P-dependent transformation matrix Cˆ (P ) is also unitary matrix, i.e. the relations hold,

∑ cˆ T

RT

( )

* * cˆ ST = δ RS = ∑ cˆTR cˆTS , Cˆ + = Cˆ T

*

= Cˆ −1

(67a)

T

The form of transformation relations for boson operators of system Hamiltonian is fully dictated now by the factorized form of the total system wave function (61). Also in this case, Superconducting Cuprates : Properties, Preparation and Applications, Nova Science Publishers, Incorporated, 2008. ProQuest Ebook Central,

26

Pavol Baňacký

it expresses possibility of simultaneous, independent diagonalization of electron and boson subsystems. It means that transformed-nonadiabatic fermion and transformed nonadiabatic boson operators obey not only standard anticommutation and commutation relations within the individual subsystems,

{a

P

[

]

, aQ } = 0 , {a P , aQ+ } = δ PQ , [br , bs ] = 0 , br , bs+ = δ rs ,

(68)

but, like the original and adiabatic operators, transformed nonadiabatic operators of both subsystems have to commute mutually, i.e. also commutation relations have to hold,

[a P , br ] = 0 , [a P , br+ ] = 0

(69)

With respect to the fermion transformation relations (66), the form of transformation relations for boson operators that fully respects conditions (69) is,

(

+

)

+ b r = br + ∑ dˆ rPQ (P )a P+ aQ , b r = br+ + ∑ dˆ rPQ (P ) a P+ aQ

PQ

(70)

PQ

At this moment, the canonical transformation can be realized. The new nonadiabatic, barless operators (66,70) replace adiabatic double-bar operators in the adiabatic form of system

Hamiltonian (B20a-20e). More details of this transformation, relation between dˆ (P ) and

cˆ(P ) transformation matrices and treatment of the resulting nonadiabatic system

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Hamiltonian is presented in [32a] and in Appendix C. It can be shown that also this transformation preserve total number of particles, i.e.

N = ∑ a P+ a P = N = ∑ a P+ a P = ∑ a P+ a P = N P

P

(71)

P

Invariant of transformation is now momentum operator,

(

)

(

)

+ ⎛ ⎞ Pr = br − br(+ = P r = ⎜ b r − b r( ⎟ ≠ P = br − br(+ , Pr = P r ≠ Pr ⎝ ⎠

(72)

However, coordinate operator is transformed up to first order of Taylor’s expansion as,

(

)

+ ( ( r r (P )a P+ aQ = Q r − 2∑ cˆ PQ (P )a P+ aQ = Qr = br + br(+ = ⎛⎜ b r + b r( ⎞⎟ − 2∑ cˆ PQ ⎝ ⎠ PQ PQ (73) ( r + = Qr − 2∑ cˆ PQ (P )a P aQ PQ

i.e.

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Qr = Q r ≠ Qr

27 (73a)

ϕ P ( x, Q, P ) which are the basis

For nonadiabatic (Q,P)-dependent spinorbitals

functions of the nonadiabatic (Q,P)-dependent electronic wave function of the ground state Φ 0 (r , Q, P ) , expressed over crude-adiabatic orbitals can be derived,

⎛

⎞

r r ϕ P ( x, Q, P ) = a P+ ( x, Q, P ) 0 = ⎜ a P+ − ∑ c PR Qr a R+ − ∑ c PR Pr( a R+ + O(Q 2 , Q P, P 2 )⎟ 0 =

)

rR ⎝ )r ( = ϕ P ( x,0,0 ) − ∑ c Qr ϕ R ( x,0,0 ) − ∑ c PR Pr ϕ R ( x,0,0 ) + ...... ( rR

r PR

rR

⎠

(74)

( rR

Nonadiabatic wave function (74) in contrast to adiabatic wave function (57) is modulated

{ }

not only by the instantaneous nuclear coordinates Qr of particular vibration (phonon)

modes {r} but modulation is also over corresponding instantaneous nuclear momenta {Pr( } .

The weight of momentum modulation is proportional to the P-dependent transformation

)

r coefficients c PR . It represents first derivative of cˆ PR matrix element with respect to nuclear

momentum Pr (coefficient of transformation matrix in first order of Taylor’s expansion), r = cˆ PR

∂cˆ PR (P ) ∂Pr

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From Appendix D, it is obvious that these coefficients reflect not only strength of electron-vibration (phonon) coupling but mainly extent of nonadiabaticity. For nonadiabatic situation, i.e. intrinsic nonadiabatic state ε I (R ) − ε A (R ) < hω r , the weight of such P0

0

modulated state can be considerably large.

IV. Dependence of electronic energy on nuclear vibration displacements and momenta Second quantization, nuclear Q,P-dependent, form of the transformed electronic Hamiltonian (Appendix C) and approximate solution of the coefficients of transformation matrices (Appendix D) allow straightforward derivation of analytic forms of the electronic energy corrections. These corrections are calculated with respect to electronic energy terms that are obtained on crude-adiabatic level at particular fixed nuclear configuration R0.

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Pavol Baňacký

IV.1. Correction to electronic ground state energy – zero-particle term correction With respect to the solution (D2a,b) of approximate GCPHF equations (D1a,b), correction to the electronic ground state energy, i.e. zero-particle term correction (C10a) is, r ΔE (0na ) = ∑ hω r ⎛⎜ c AI ⎝ rAI

2

r 2⎞ − cˆ AI ⎟= ⎠

unocc occ

∑∑ ∑ u A

I

r

r 2 AI

(ε

hω r 0 A

− ε I0

) − (hω ) 2

2

=

unocc occ

∑∑ Ω A

r

AI

(75)

I

The Ω matrix is a symmetric matrix of the form, 2 2 r r r ⎞⎟ = Ω PQ = ∑ hω r ⎛⎜ c PQ − cˆ PQ u PQ ∑ ⎝ ⎠ r r

2

(ε

hω r 0 P

) − (hω ) 2

− ε Q0

(75a)

2

r

As it seen from (75), for standard adiabatic Q-dependent state, the electronic ground state energy correction ΔE (na ) is reduced to the adiabatic DBOC ΔE (ad ) (B30a, 28) which is é

é

always positive. In the extreme case (D3a) of strong adiabatic limit hω r / ε P − ε Q → 0 , the 0

0

correction is basically zero,

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r 2⎞ r ΔE (0sad ) = ∑ hω r ⎛⎜ c AI ⎟ = ∑ u AI ⎝ ⎠ rAI rAI

2

(ε

hω r 0 A

For an intrinsic nonadiabatic state when inequality

− ε I0

)

2

→0

(75b)

ε I0 (R ) − ε A0 (R ) < hω r holds,

correction to electronic ground state energy (75) is negative and represents stabilization contribution to the electronic ground state energy. This contribution can be considerably large and reach the extreme negative value for left-hand side limit toward singular point in (75). Singular point itself is excluded (Appendix D). The correction is always negative for the extreme case (D3b) of strong nonadiabatic limit, hω r / ε P − ε Q → ∞ . However, the 0

0

contribution in this case does not represent the largest possible negative value and it is equal to, r u AI

2

r ⎞⎟ = − ΔE (0sna ) = −∑ hω r ⎛⎜ cˆ AI ∑ ⎝ ⎠ rAI rAI hω r 2

(75c)

For quasi-momentum k,q-space representation of multi-band solids, the corresponding equation for correction to electronic ground state energy can be derived straightforwardly from (75). It is based on the correspondence relations for boson and fermion quantities in real space and complex quasi-momentum space representations. In particular for;

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Antiadiabatic Theory of the Electronic Ground State of Superconductors

29

(

normal modes: r → q , r → − q occupied spinorbital: I → Rk , σ - with σ (α or β) spin and occupation factor f k that obey Fermi-Dir statistics (for T=0 K, f k =1, i.e. occupied state below Fermi level)

unoccupied spinorbital: A → Sk ' , σ ' - with occupation factor (1 − f k ' ) that obey Fermi-Dirac statistics (for T=0 K, f k ' =0, i.e. unoccupied state above Fermi level) one-electron HF-orbital energies: occupied states below Fermi level, ε I → ε Rk 0

0

unoccupied states above Fermi level, ε A → ε Sk ' 0

matrix

element

of

conservation: u

r AI

0

electron-vibration(phonon)

→u

q k 'k

=u =u q

coupling

with

quasi-momentum

k '− k

Then, with respect to k:-k symmetry, the temperature dependent form of energy correction to electronic ground state (75) in quasi-momentum space representation has the form,

⎛ ⎛ 2 hω k ' − k ΔE (0na ) = 2⎜ ∑ ⎜ ∑ u k ' − k f k (1 − f k ' ) 0 0 2 ⎜ R (k ), S (k ' ) ⎜ k < k ,k '> k ε k ' − ε k − (hω k '− k )2 ⎝ F F ⎝

(

)

⎞⎞ ⎟ ⎟ , ϕ Rk ≠ ϕ Sk ' , ⎟⎟ ⎠⎠

(76)

Summation in (76) runs over all bands { ϕ R , ϕ S } and k points of 1st BZ of multi-band system, including intra-band contributions, i.e. ϕ Rk , ϕ Rk ' , k ≠ k ' , while ε k < ε F ; 0

ε k0' > ε F .

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For T=0 K, relation (76) reduce to,

⎛ ⎛ ΔE (0na ) = 2⎜ ∑ ⎜ ∑ u k '− k ⎜ R (k ),S (k ' ) ⎜ kk ' ⎝ ⎝

2

(ε

hω k '− k 0 k'

− ε k0

) − (hω ) 2

k '− k

2

⎞⎞ ⎟⎟ ⎟⎟ ⎠⎠

(76a)

In (76a), the wave vector k corresponds to states fully occupied below Fermi level ( f k = 1) , and wave vector k’ corresponds to empty – virtual states above Fermi-

level ( f k ' = 0 ) .

IV.2. Corrections to one-particle term Nonadiabatic form of one-particle pure fermion part of the Hamiltonian (boson excitations independent) has the form (C11). There is also boson excitations dependent part that is represented in Appendix C by the expression valid for boson vacuum (C14b). The Q,Pdependent corrections are represented by terms that follow after the first crude-adiabatic term in (C11) and by all terms of (C14b). Restriction to first order of electron-vibration(phonon) coupling allows to neglect first summation in (C14b) and for equilibrium geometry, the second term in (C11) equals zero. The terms that are the products of electronvibration(phonon) coupling and coulomb two-electron interactions can be expected to be

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30

Pavol Baňacký

negligibly small comparing to electron-vibration(phonon) coupling terms and can also be neglected (fifth sum in C11 and third and forth sum in C14b). In solids, due to translation symmetry, the forth sum in (C11) equals zero. Then, one-particle correction has the form,

) [

]

⎛ r r* r r* r r* r r* ⎞ cQA cˆQA cQI cˆQI ΔH ep' = ∑ hω r ⎜ ∑ c PA − cˆ PA − ∑ c PI − cˆ PI ⎟N a P+ aQ + rPQ I ⎠ ⎝ A

(

(

)

(

)

) [

(

]

⎞N a + a ⎟ P P ⎠

r 2 r 2⎞ r r* c PR + ∑ ⎛⎜ ε P0 − ε R0 ⎛⎜ c PR + cˆ PR ⎟ − 2hω r Re cˆ PR ⎝ ⎠ ⎝ rPR

(77)

IV.2.1. Nonadiabatic polarons The diagonal form of the one-particle correction (77) is,

⎛ r 2 r 2⎞ r ΔH ep' (dg ) = ∑ hω r ⎜ ∑ ⎛⎜ c PA − cˆ PA ⎟ − ∑ ⎛⎜ c PI ⎝ ⎠ ⎝ rP I ⎝ A

(

)

2

[

) [

(

]

r 2 ⎞⎞ − cˆ PI ⎟ ⎟N a P+ a P + ⎠⎠

+ r 2 r 2⎞ r r* ⎞ + ∑ ⎛⎜ ε P0 − ε R0 ⎛⎜ c PR + cˆ PR c PR ⎟ − 2hω r Re cˆ PR ⎟N a P a P ⎝ ⎠ ⎝ ⎠ rPR

]

(78)

Substitution for transformation coefficients (D2a,b) yields simple expression for electronvibration(phonon) interaction part of the Hamiltonian,

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r 2 r 2 ⎛ u u PA PI ⎜ ΔH ep' (dg ) = ∑ ⎜ ∑ 0 +∑ 0 0 0 rP ⎜ A ≠ P ε P − ε A − hω r I ≠ P ε P − ε I + hω r ⎝

⎞ ⎟ + ⎟N a P a P ⎟ ⎠

[

]

(78a)

It can be rearranged into the form that is more convenient for solid state interpretation,

ΔH ep' (dg ) = −2

∑u

rPI ( P ≠ I )

u ∑ ( )

rPR P ≠ R

r 2 PI

(ε

r 2 PR

(ε

1 0 P

−ε

0 R

hω r 0 P

− ε I0

[ ) − hω N [a a ]

) − (hω ) 2

]

N a P+ a P −

r

2

+ P

(78b)

P

r

Transcription of (78b) to quasi-momentum k,q-space of multi-band solids is based on the following correspondence: r → q ; P → Pk , σ ; R → Rk − q, σ ; I → Sk − q, σ (occupation factor f k − q );

ε P0 → ε k0 ; ε R0 → ε k0− q ; ε I0 → ε k0− q . The resulting form is,

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Antiadiabatic Theory of the Electronic Ground State of Superconductors

ΔH ep' (dg ) =

⎛

⎞ + ⎟− N a a σ σ k k , , 0 0 ⎟ ε h ω − − k k −q q ⎠ ⎞ hω q + ⎟ N a a k ,σ k ,σ 2 2 0 0 ⎟ ε k − ε k − q − (hω q ) ⎠ 1

∑ ⎜⎜ ∑σ u (ε ⎝

Pk , Rk − q

q 2

qk

⎛ 2 − 2 ∑ ⎜ ∑ u q f k −q ⎜ Pk , Sk − q qkσ ⎝

(

[

)

[

)

31

]

]

(78c)

Expression (78c) represents total one-electron energy correction on the general Q,Pdependent level due to electron-vibration(phonon) interactions. The first term of (78c) is standard adiabatic (Q-dependent) polaron as it can be obtained from Fröhlich Hamiltonian by Lee-Low-Pines transformation [44]. The second term of (78c) is the correction to polaron energy that arises due to dependence of electronic motion not only on nuclear coordinates but also on the nuclear momenta P (nonadiabatic modification of the BOA). This term can be interpreted as a correction to the energy of individual polarons by an effective field created by all other polarons of the system.

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IV.2.2. Correction to orbital energies of occupied and unoccupied states. Gap opening in one-electron spectrum of quasi-degenerate states at Fermi level The expression (78) for correction to one-particle term covers contributions of boson excitation independent part (first sum) as well of boson excitations dependent part (second sum). At finite temperature, due to boson excitations, contribution of the second term will be reproduced as multiples of the contribution of the second sum in the form of (78 ) that represent contribution at boson vacuum (0 K). Since second sum runs over all states, occupied and unoccupied, this contribution does not change character of one-electron spectrum (i.e. position of energy levels with respect to Fermi level), only population of states is changed. More over, for a set of quasi-degenerate occupied and unoccupied states at Fermi level (e.g. intrinsic nonadiabatic state,

ε I0 (R ) − ε A0 (R ) < hω r ) contribution of this term can be

negligibly small since the term is odd function of

(ε

0 P

)

− ε R0 and contributions from occupied

and unoccupied states will tend to cancel mutually. On the other hand, possible temperature dependent change in contribution of the first term that is independent of boson excitations can occur if there is a change in the character of one-electron spectrum of the system (there are separate summations that run over occupied and unoccupied states). In this connection, for investigation of possible changes in the character of one-electron spectrum of the system due to electron-vibration(phonon) interactions on Q,P-dependent level, the first term of (78) is crucial. For correction to orbital energy Δε P of particular state

ε P0 then holds, ⎛ r 2 r 2⎞ r Δε P = ∑ hω r ⎜ ∑ ⎛⎜ c PA − cˆ PA ⎟ − ∑ ⎛⎜ c PI ⎝ ⎠ ⎝ r I ⎝ A

2

occ ⎛ unocc ⎞ r 2 ⎞⎞ − cˆ PI ⎟ ⎟ = ⎜ ∑ Ω PA − ∑ Ω PI ⎟ (79) ⎠⎠ ⎝ A I ⎠

Final, corrected orbital energy is,

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32

Pavol Baňacký

ε P = ε P0 + Δε P

(79a)

Let us consider only the couple of quasi-degenerate states at Fermi level, occupied state ε I and unoccupied state ε A , a situation that can characterize couple of states in intrinsic 0

0

ε A0 (R ) − ε I0 (R ) 0 , Δε I = Ω IA = Ω IA < 0 , Δε I = − Δε A

(80)

It means that orbital energy of unoccupied state has been increased, ε A > ε A , and orbital 0

ε I < ε I0 . The same results follow also from

energy of the occupied state has been decreased,

(78a). This analysis can be generalized for a set of quasi-degenerate occupied

{J } and

unoccupied {B}states (quasi-continuum of states) at Fermi level. Then, with respect to the fact that for intrinsic nonadiabatic state correction to the ground electronic state (75) is negative,

∑Ω

AI

< 0 , the following relations can be derived,

AI

Δε B =

unocc

occ

unocc

occ

A

I

A

I

∑ Ω BA − ∑ Ω BI ≥ 0 , Δε J =

∑ Ω JA − ∑ Ω JI ≤ 0 ,

(80a)

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unocc

r Δε B = ∑ ∑ u BA r

A

unocc

r Δε J = ∑ ∑ u JA r

2

A

2

(ε (ε

hω r

0 B

−ε

occ

) − (hω )

0 2 A

2

−ε

2

− ∑∑ u JIr r

r

2

I

occ

) − (hω )

0 2 A

r

r

hω r

0 J

r − ∑∑ u BI

I

2

hω r

(ε

0 B

(ε

0 J

) − (hω )

−ε

0 2 I

−ε

0 2 I

2

(80b)

r

hω r

) − (hω )

2

(80c)

r

At finite temperature T, for a correction Δε P to an arbitrary state ε P , from the set of 0

quasi-degenerate occupied and unoccupied states at Fermi level can be written,

Δε P (T ) = ∑ Ω PQ (1 − 2 f Q ) , {Q} - set of quasi-degenerate states at Fermi level (81) Q

The occupation factor f Q obeys Fermi-Dirac statistics, −1

⎛ ⎛ εQ − μ ⎞ ⎞ ⎟⎟ + 1⎟ , ε Q = ε Q0 + Δε Q f Q = ⎜⎜ exp⎜⎜ ⎟ ⎝ k BT ⎠ ⎠ ⎝

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(81a)

Antiadiabatic Theory of the Electronic Ground State of Superconductors

33

It is evident that for temperature 0 K, expression (81) reduces to (79). From (81), for temperature dependence of energy gap that is open in one-electron spectrum at Fermi level can be derived [32b],

⎛ Δ(T ) ⎞ ⎟⎟ Δ (T ) = Δ(0 ).tgh⎜⎜ k T 4 ⎝ B ⎠

(82)

The gap is defined as the energy difference of lowest lying corrected unoccupied state ε B ( LU ) and highest lying corrected occupied state ε J ( HO ) . At temperature 0 K holds,

(

Δ(0) = ε B ( LU ) + ε J ( HO )

)

(82a)

Factor 4 in the denominator of the argument of tgh in (82) follows from the assumption that at Fermi level density of quasi-degenerate occupied

{ε }and unoccupied {ε }states of 0 J

0 B

the band with gap opening is the same and consequently ε B ( LU ) =

ε J ( HO ) . This factor can be

larger or smaller than 4, depending on the actual difference in the density of states.

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IV.3. Two-particle term correction. Correction to electron correlation energy Nonadiabatic form of two-particle pure fermion part of the Hamiltonian (boson excitations independent) has the form (C12). Boson excitations dependent part is represented in Appendix C by the expression valid for boson vacuum (C14c). The Q,P-dependent corrections are represented by terms that follow in (C12) after the first crude-adiabatic term (electron correlation energy on crude-adiabatic level for fixed nuclear configuration R0 – see 14, 14a) and by all terms of (C14c). Like in the case of treatment of the one-particle term correction, the terms that are the products of electron-vibration(phonon) coupling and coulomb two-electron interactions can be expected to be negligible comparing to electronvibration(phonon) coupling terms, and can be neglected (i.e. all terms in C14c and fourth, fifth, sixth and seventh sum in C12). In solids, due to translation symmetry, the third sum in (C12) equals zero. Then the correction to electron correlation energy due to dependence of electronic motion on nuclear vibration displacements and momenta reduces to a single term,

∑ hω (c

′′ = ΔH ep

r

r r* PR SQ

c

[

r r* )N a P+ aQ+ a S a R − cˆ PR cˆ SQ

]

(83)

rPQRS

Substitution for transformation coefficients (D2a,b) yields,

ΔH ep'' =

∑ u PRr u SQr*

rPQRS ( P ≠ R ,Q ≠ S )

((ε

((

)(

)

hω r ε P0 − ε R0 ε S0 − ε Q0 − (hω r ) 0 P

−ε

) − (hω ) )((ε

0 2 R

2

r

0 S

−ε

2

)

) − (hω ) )

0 2 Q

2

[

N a P+ aQ+ a S a R

r

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]

(83a)

34

Pavol Baňacký

Transcription of (83a) to quasi-momentum k,q-space representation of solids is based on the following correspondence: r → q ; P → k + q, σ ; Q → k ' , σ ' ; R → k , σ ;

S → k '+ q, σ ' . Resulting final form is, ΔH ep'' =

∑ ( ) (

R k S k ' ) kk 'q

∑u) σσ (

q 2

' q ≠0

((ε

((

)( ) − (hω ) )((ε

)

hω q ε k0+ q − ε k0 ε k0'+ q − ε k0' − (hω q ) 0 k +q

−ε

0 2 k

2

q

0 k '+ q

−ε

2

)

) − (hω ) )

0 2 k'

2

[

N a k++ q ,σ a k+',σ ' a k '+ q ,σ ' a k ,σ

]

(83b)

q

V. Antiadiabatic state - ground electronic state of superconductors Band structure of superconductors, no matter if high-Tc cuprates, MgB2, simple metals or metal alloys, calculated at equilibrium geometry with unit cell atom positions corresponding to particular translation group, are characteristic by at least one band continually crossing Fermi level in some direction of reciprocal lattice. Fermi energy is always greater than corresponding frequencies of pertaining phonon modes; even in case of alkali metal fullerides the ratio ω / E F ≈ 0.3 − 0.8 is relatively large. It indicates that for equilibrium structures the

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adiabatic BOA can be applied, but in case of fullerenes the off-diagonal nonadiabatic corrections can play an important role. As mentioned in the Introduction, it has been recently found that for distorted structures, with atom displacements characteristic for vibration motion in some phonon modes, the band structure of superconductors undergoes significant change. Of particular importance is fluctuation of analytic critical point (maximum, minimum or saddle point) of some band across the Fermi level – see Figure 1 that represents band structure fluctuation in Γ point of 1st BZ at electron coupling to B-B stretching vibration in E2g phonon mode.

a

b Figure 1. Band structure of MgB2 at equilibrium geometry (a) and at distorted geometry (b). At nuclear displacements 0.016 Ao/B-atom in E2g mode, the maximum of σ band in Г point crosses Fermi level and at vibration motion fluctuates between topologies a-b.

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Antiadiabatic Theory of the Electronic Ground State of Superconductors

35

It indicates significant reduction of Fermi energy that is related to formation of intrinsic

nonadiabatic state, ε S (k c ) − ε F 0

< hω r , at vibration motion. It is exactly situation that

0

Req ± Q

is characterized by relation (59). It means that standard adiabatic Q-dependent BOA is not adequate starting point for theoretical study of electronic structure of superconductors. Modified, Q,P-dependent BOA accounts for adiabatic as well for intrinsic nonadiabatic state and can be suitable approximation at theoretical study of superconductors. Let us analyze a superconductor at equilibrium geometry Req when adiabatic approximation is valid. te

( )

On crude-adiabatic level, total ground state electronic energy E 0 Req has minimum at Req,

(dE

te 0

/ dR

)

Req

= 0 , and in HF-SCF approximation it is equal to (12). The only

correction to this energy is electron correlation energy (14a) that is negative and contributes to stabilization of the ground state at geometry Req,

E 0tecorr (Req ) = E 0te (Req ) − E corr

Req

(

te

, dE 0 corr / dR

)

Req

=0

(84)

Nuclear displacements at vibration motion related to any phonon mode increase total electronic energy (potential energy of nuclear motion for particular phonon mode). For displaced geometry Rd on crude-adiabatic level holds,

E 0tecorr (Rd ) = E 0te (Rd ) − E corr

(84a)

Red

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Since two-electron coulomb interactions

{ν

0 PQRS

} do not depend explicitly on nuclear

displacements it can be expected that electron correlation energy has not been changed significantly, i.e. (E corr )R ≈ (E corr )R . On crude-adiabatic level, for an increase of the total eq

d

electronic energy ΔE d due to nuclear displacement Rd then holds

ΔE d (Rd ) = E 0te (Rd ) − E 0te (Req ) > 0

(84b)

In principle, two situations can occur; nuclear displacements of some phonon mode(s) induce formation of intrinsic nonadiabatic state (59), or system remains in adiabatic state for vibration motion in all phonon modes. In the first case, when at displaced geometry Rd related to the phonon mode r an intrinsic nonadiabatic state

ε S0 (k c ) − ε F0

(

Rd Req ± Q

)

< hω r is formed, the crude-adiabatic total

electronic energy is corrected by Q,P-dependent zero-particle term correction (76a),

ΔE(0na ) (Rd ) . For total electronic energy at displaced geometry Rd then holds,

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36

Pavol Baňacký

E 0te,na (Rd ) = E 0te (Req ) + ΔE d (Rd ) + ΔE (0na ) (Rd )

(85)

Since for intrinsic nonadiabatic state this correction is negative, ΔE (na ) (Rd ) < 0 , then if 0

the inequality

ΔE (0na ) (Rd ) > ΔE d (Rd )

(85a)

holds, the electronic state of the system is stabilized at distorted geometry Rd. The reason of it is significant participation of the nuclear kinetic energy terms, i.e. contributions of the r coefficients Cˆ AI of P-dependent transformation - see (75) that stabilize fermionic ground

state energy in the intrinsic nonadiabatic state with distorted nuclear configuration. Stabilization (condensation) energy at transition from adiabatic to intrinsic nonadiabatic state is then, 0 E cond = ΔE d (Rd ) − ΔE (0na ) (Rd )

(85b)

At temperature 0 K, distorted geometry represents true equilibrium geometry Rteq of the system, Rd ≡ Rteq , and corresponding total electronic energy represents electronic ground state energy of the system,

(dE (R ) / dR ) Copyright © 2008. Nova Science Publishers, Incorporated. All rights reserved.

te 0 , na

= 0 , E 0te,na (Rd ) < E 0te (Req )

Red

(85c)

More over, at Rd ≡ Rteq , the one-electron spectrum with quasi-continuum of states at Fermi level has been significantly changed. It concerns not only fluctuating band but also “static” bands that are through phonon mode r in inter-band interaction with the fluctuating band. Quasi-momentum counterpart of the corrections to one-electron spectrum (80b,c) – shift of orbital energies, i.e. change of dispersion in band ϕ P (k ) has the form, Δε (Pk ') =

∑u

k ' − k '1 2

Rk '1 > k F

(1 − f ) k '1

(ε

hω k '− k '1 0 k'

−ε

) − (hω

0 2 k '1

)

2

k ' − k '1

−

∑u

k −k ' 2

Sk < k F

fk

(ε

hω k − k '

0 k'

−ε

) − (hω )

0 2 k

for k ' > k F , and

Δε (Pk ) =

∑u

Rk '1 > k F

k − k '1 2

2

k −k '

(86a)

(1 − f ) k '1

(ε

hω k −k '1 0 k

−ε

) − (hω )

0 2 k '1

2

k − k '1

−

∑u

Sk1 < k F

k − k1 2

fk

(ε

hω k −k1 0 k

−ε

) − (hω )

0 2 k11

for k ≤ k F

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2

k − k1

(86b)

Antiadiabatic Theory of the Electronic Ground State of Superconductors

37

Replacement of discrete summation by integration introduces into derived relations

density of states n(ε k ) ,

∑ ... → ∫ n(ε )... It k

is of crucial importance in relation to the

k

fluctuating band, since at the moment when analytic critical point of the band approaches Fermi level (intrinsic nonadiabatic state), density of states at Fermi level is considerably increased (possibility of van Hove singularity formation at Fermi level as it has been proposed in [45]). As the consequence of the shift of orbital energies, the gap (82) in one-electron spectrum at Fermi level has been opened. The gap is opened close to the k point at the position where the band on crude-adiabatic level has intersected Fermi level. The gap has character of indirect gap as it follows from relations (86a,b). This fact can be observed tunneling spectroscopy or by ARPES (occupied states below Fermi level) and inverse ARPES spectra (unoccupied states above Fermi level) in the form of peaks that appear below and above Fermi level (spectral weight - density transfer) at decreasing temperature from above to below Tc. Since for corrected orbital energy holds,

ε k = ε k0 + Δε k , then for corrected density of

states n(ε k ) due to orbital energy shifts Δε k , the following relation can be derived straightforwardly,

n(ε k ) =

The quantity n

0

1

∂ (Δε k ) 1+ ∂ε k0

n 0 (ε k0 )

(87)

(ε ) in relation (87) stands for uncorrected density of states of particular 0 k

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band (density of states on crude-adiabatic level),

⎛ ∂ε k0 n (ε ) = ⎜⎜ ⎝ ∂k 0

0 k

⎞ ⎟⎟ ⎠

−1

(87a)

From (86a,b, 87a,b), it can be seen that dominant changes in density of states (regions with increased and decreased densities) can be expected close the k point where the band intersect Fermi level on the energy interval that is not larger than ± hω r from Fermi level. If the uncorrected density of occupied and unoccupied states of the band at Fermi level is the same, then position of the peak (with respect to Fermi level) that appears in corrected density of occupied states represents the “half-gap”, Δ (0 ) / 2 . If the uncorrected density of occupied

and unoccupied states is different, the gap Δ (0 ) equals to the distance of the peaks that

appear in corrected density of occupied and unoccupied states. If there is the phonon coupling with two different bands that intersect Fermi level in a given direction of reciprocal lattice at points k1 and k2, then two different gaps in one-electron spectrum close to these points are opened (e.g. in case of MgB2 σ and π gaps are opened along Γ-K direction due to σ − σ and σ − π coupling over E2g phonon mode [28]).

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38

Pavol Baňacký Increase of the total electronic energy ΔE d (Rd ) related to nuclear displacement out of

equilibrium on the crude-adiabatic level (84b), is the only temperature independent correction to the total electronic energy. All other corrections related to nuclear motion are temperature dependent. Due to thermal excitations, at some critical temperature Tc the nonadiabatic correction to electronic ground state energy (76), ΔE (na ) (Rd ) , becomes in absolute value 0

smaller than ΔE d (Rd ) ,

ΔE (0na ) (Rd )

Tc

≤ ΔE d ( R d )

(88)

Consequently, for T > Tc , system is stabilized at equilibrium geometry Req which is identical to the Req that correspond to high-symmetry structure on crude-adiabatic level. System is now in the adiabatic state. It means that over Tc, band structure at Fermi level has to be represented by quasi-continuum of occupied and unoccupied states without the gap in oneelectron spectrum that has been opened at 0 K. Temperature dependence of the gap in oneelectron spectrum is represented by eq. (82). In the limit Δ (T ) → 0 , from (82) for critical temperature follows,

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Tc =

Δ(0) 4k B

(89)

For temperatures T k F , k '< k F

((

hω k '− k ε k0' − ε k0

((ε

0 k'

−ε

) + (hω ) ) N [a 2

2

k '− k 2 2

) − (hω ) )

0 2 k

+ k '↑

]

a −+k '↓ a − k ↓ a k ↑ (95)

k '− k

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Antiadiabatic Theory of the Electronic Ground State of Superconductors

43

Expression (95) shows that due to EP interactions on the Q,P-dependent nonadiabatic level, this correction increases electron correlation energy (14) in the electronic ground state. This correction is due to pairs of electrons with opposite quasi-momentum and antiparallel

(

) ) } (i.e. two-particle (k ↑,− k ↓ )-two-hole (k ' ↑,− k ' ↓ ) excited singlet states) to

spins k ↑,− k ↓ . It has to be noticed that it is contribution of bi-excited configurations

{Φ (

k ', − k ' )→( k , − k

the electronic ground state that is represented by renormalized Fermi vacuum Φ 0 . Expressed explicitly,

first

type Φ 0 ΔH

'' ep

nonzero

contributions

are

from

matrix

elements

of

the

Φ (k ', − k ' )→(k , − k ) , i.e. in second order of perturbation theory. Now,

{ε k } represent particle states that are occupied above Fermi level and {ε k ' }are empty – hole states below Fermi level due to excitations. From the denominator of (95) it is clear that the largest contribution to the correlation energy correction is for pair of electrons at Fermi level when ε k ' − ε k ≤ hω k ' − k . 0

0

For the extreme nonadiabatic regime ε k ' − ε k / hω k ' − k → 0 , from (95) for correction to 0

0

electron correlation energy follows,

u k '− k

ΔH ep'' (red )sna = −2∑

2

hω k '− k

kk '

[

N a k+'↑ a −+k '↓ a − k ↓ a k ↑

]

(95a)

The second extreme case is for strong adiabatic regime hω k ' − k / ε k ' − ε k → 0 . At these 0

0

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circumstances, correction to electron correlation energy goes to zero,

ΔH ep'' (red )sad = −2∑ u k '− k

hω k '− k

2

(ε k ' − ε k )

kk '

2

[

]

N a k+'↑ a −+k '↓ a − k ↓ a k ↑ → 0

(95b)

For an intermediate adiabatic case (i.e. Q-dependent adiabatic level ε k ' − ε k > hω k ' − k ), 0

0

for correction to electron correlation energy from the adiabatic form of this correction (second term in B32), or from (83b) in the limit cˆ = 0 , follows

ΔH ep'' =

∑σσu (ε

hω q

q 2

kk 'q

'

0 k +q

−ε

0 k

)(ε

0 k '+ q

−ε

0 k'

)

[

N a k++ q ,σ a k+',σ ' a k '+ q ,σ ' a k ,σ

]

(95c)

The reduced form has the form,

ΔH ep'' (red )ad = −2∑ u k '− k kk '

2

(ε

hω k ' − k 0 k'

−ε

)

0 2 k

[

N a k+'↑ a −+k '↓ a − k ↓ a k ↑

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]

(95d)

44

Pavol Baňacký

Like for nonadiabatic Q,P-dependent case also for adiabatic Q -dependent level, correction to electron correlation energy is negative. By comparing (95) with (95d) it can be seen that nonadiabatic correction to electron correlation energy is in absolute value larger than corresponding adiabatic correction. More over for antiadiabatic state, when system is close to singular point in (95), nonadiabatic correction can be significant. It means that along with crude-adiabatic correlation energy (14a) also correction to electron correlation energy (95) contributes to stabilization of antiadiabatic state - the ground state of the system at distorted nuclear geometry Rd=Rteq. It should be reminded that even without account for the correction to electron correlation energy (i.e. contributions of second and higher orders of perturbation theory), system is already stabilized in antiadiabatic state at distorted geometry due to the correction to the ground state electronic energy (76) that represents zero-order correction in terms of perturbation theory. In this respect, increased electron correlation is the consequence of EP interactions that have induced transition of the system from adiabatic to intrinsic nonadiabatic state. At finite temperature, the product of Fermi-Dirac occupation factors has to be introduced into derived equations,

∑ ... →∑ f (1 − f )... . With increasing temperature from 0 K, the k'

k

kk '

kk '

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value of the correction (95) that is characteristic for intrinsic noniadiabatic state decreases and at T=Tc when system undergoes sudden transition from intrinsic noniadiabatic state at distorted geometry Rd=Rteq to adiabatic state at undistorted geometry R0=Req and above this temperature, for correction to electron correlation energy holds corresponding T-dependent adiabatic form (95d). Above Tc, this correction along with crude-adiabatic correlation energy (14a) stabilizes adiabatic ground state of system at undistorted geometry R0=Req.

VII. Effective attractive electron-electron interactions, Cooper’s pairs and bipolarons From a formal stand-point, the expressions for correction to electron correlation energy (95) appear to be a kind of effective attractive electron-electron interactions. It can be compared to the Fröhlich effective Hamiltonian of electron – electron interactions [46],

H eff'' (Fr ) =

∑σσu

kk 'q

'

q 2

(ε

hω q 0 k +q

−ε

) − (hω )

0 2 k

2

a k++ q ,σ a k+',σ ' a k '+ q ,σ ' a k ,σ

(96)

q

or, to its reduced form that maximizes attractive contribution of electron – electron interactions, '' (Fr ) = 2∑ u k '− k H red kk '

2

(ε

hω k ' − k

0 k'

−ε

) − (hω )

0 2 k

2

a k+'↑ a −+k '↓ a − k ↓ a k ↑

(96a)

k '− k

This interaction term is either attractive or repulsive depending on sign of the denominator. For nonadiabatic conditions it represents effective attractive electron-electron Superconducting Cuprates : Properties, Preparation and Applications, Nova Science Publishers, Incorporated, 2008. ProQuest Ebook Central,

Antiadiabatic Theory of the Electronic Ground State of Superconductors interactions. In the limit of extreme nonadiabaticity

45

ε P0 − ε Q0 / hω r → 0 , the form of the

Fröhlich two-particle effective Hamiltonian (96) and the correction to electron correlation energy (83b) are identical and equal to,

(

lim ΔH ep''

)

Δε / hω →0

(

)

= lim H eff'' (Fr ) Δε / hω →0 = −

∑σσ

kk 'q

uq '

2

hω q

a k++ q ,σ a k+',σ ' a k ' + q ,σ ' a k ,σ (96b)

Correction to electron correlation energy (83b) has been derived in the explicit assumption of the dependence of electronic motion on nuclear coordinates and momenta, i.e. at the derivation, very general Q,P-dependent form of the electronic Hamiltonian (see Appendix C) has been used. On the other hand, at derivation of (96), starting form has been simple model Hamiltonian (21). However, instead of the simplest form of the EP interaction term (21b), Fröhlich used more general form that can be written as (see [43]),

H ep =

∑σ[g

]

* + q q

b a k+ a k + q + h.c. , g q* = − g q

(97)

k ,q ,

Even it is not immediately seen from (97), this form explicitly introduces the EP interactions to be dependent not only on nuclear coordinates Qq (like term 21b) but also on +

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nuclear momenta Pq ( bq =

1 (Qq − Pq ) ). In this respect, correction to correlation energy 2

(83b) derived from general form of Hamiltonian that describe complex electronic structure of system and term of the effective electron-electron interaction (96) derived by Fröhlich from model Hamiltonian (and corresponding reduced forms 95 and 96a) represents the same physical effect. The difference in the final forms (83b and 96) is due to different kind of canonical transformations used – see Appendix E. Treatment of the Fröhlich effective electron-electron interaction Hamiltonian with respect to development of theory of superconductivity is well known. The first attempt was made by Fröhlich himself [47]. Within the free particle approximation, zero order contribution of the effective interaction Hamiltonian (96) yields correction to ground state electronic energy that after introduction of population factors has the form that is formally identical with ΔE ( na ) as it has been derived in the present work (76), 0

i.e. '' (Fr ) Φ 0 = 2∑ u k '− k f k (1 − f k ' ) ΔE = Φ 0 H red 2

kk '

hω k ' − k

(ε k ' − ε k )2 − (hω k '− k )2

k ' > kF

, k < kF , (98)

This relation was derived by Fröhlich also directly, without canonical transformation, in the form of correction to ground state electronic energy in second order of perturbation theory (E2 term for boson vacuum in [47]). He was looking then for electronic distribution fk,fk’ over the original (metal) one-electron spectrum that would results in maximal stabilization

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Pavol Baňacký

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(extreme negative value of E2 ) of electronic ground state. Corresponding electronic state was identified as superconducting state. Similar results had been obtained also by Bardeen [48] based on a variational approach. Results even correctly describe isotopic effect, have never been accepted as relevant for superconductivity.The reason is related to the fact that on this level of approximation a phase with superconducting properties, including gap formation, can not be identified and energy difference between electronic state that had been supposed to correspond to superconducting state and ground electronic state (condensation energy) seemed to be large. In this respect, Fröhlich noted [46] that free particle approximation cannot be expected to be applicable to study details of the energy spectrum, specific heat and magnetic properties. Different treatment of this problem has been elaborated within the BCS theory [3]. This theory and its modifications are well known and in spite of serious problems with high-Tc superconductors, it remains the basic framework of microscopic mechanism of superconductivity until present. It is not the aim to analyze the BCS theory, but with respect to the results presented in this paper some key points underlying BCS theory should be reminded. The best platform for it offers general derivation based on Bogoljubov -Valatin canonical transformations of the BCS model Hamiltonian - see Appendix F. Let us first point-out relevant results derived within nonadiabatic theory as presented in this paper. Ground state of superconductors and resulting thermodynamics (superconducting properties) are directly related to antiadiabatic state that arises in system with complex electronic structure as a consequence of band structure fluctuation due to EP interactions for this class of solids. Transition to antiadiabatic state is connected to considerable decrease of Fermi energy (chemical potential) with respect to Fermi energy of normal-adiabatic state, μ s (in ) < hω r < μ n (ad ) . EP interactions that drive system into antiadiabatic state decrease electronic energy and stabilize system (76a) at new equilibrium - distorted nuclear configuration (it is related to nuclear motion in respective phonon mode). In order to study antiadiabatic state it is necessary to introduce electronic motion to be dependent on instantaneous nuclear positions as well as on instantaneous nuclear momenta. At these circumstances, in antiadiabatic state not only electronic energy of the ground state is stabilized and chemical potential is reduced but also original adiabatic Hartree-Fock oneparticle (electronic) spectrum H (1)ad =

∑ (ε ) k

0 k R0

[

]

N a k+ a k is changed to

[

H (1)in = ∑ (ε k0 + Δε k )Rd N a k+ a k

]

(99)

k

One-particle spectrum (99) corrected by the correction term Δε k (86b) that incorporates the effect of EP interactions on electronic structure in antiadiabatic state, is crucial quantity that in full extend determines thermodynamic properties (gap in one-particle spectrum, specific heat, entropy, internal energy, free energy and critical magnetic field), i.e. superconducting state of system. As it follows from Bogoljubov-Valatin canonical transformations (Appendix F), the crucial point that is underlying formulation of BCS theory is directly based on possibility to derive one-particle spectrum in the form of (F17). Also in this case, original adiabatic

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Antiadiabatic Theory of the Electronic Ground State of Superconductors

47

Hartree-Fock one-particle (electronic) spectrum is corrected (F16c). The correction Δ k , (F16b), reflects the effect of effective attractive electron-electron interactions (that are consequence of EP interactions) on single-particle excitation spectrum that determines superconducting properties of system. This solution can be derived, however, only if it is correct to assume that ensemble of Cooper’s pairs lying in thick layer (± hω ) from Fermi level can be separated out as a subsystem that is independent of the rest of complex electronic structure of the system and at the same time to assume that chemical potential of the system in normal (adiabatic) metal state is the same as chemical potential of the system in superconducting state, i.e. μ s = μ n , (F22a). Independence of the subsystem of Cooper’s pairs is expressed over condition (F20) that introduces inter-electron interactions to be zero outside the relevant interval and constant attractive (− V ) inside it. However, from the stand-point of complex electronic structure of metals, system is adiabatic and in the limit of strong

(

)

adiabaticity hω / Δε kk ' → 0 the Fröhlich effective inter-electron interactions term (96) that is valid for complex structure goes to zero. On the other hand, correction to electron correlation energy, as presented in this paper, expresses more precisely the influence of nuclear motion on electron correlations than the Fröhlich effective inter-electron interaction term. It has been shown above (and also derived by Wagner in adiabatic Q-dependent representation [49]), that for intermediate adiabatic regime this correction (even small) is always negative (95d), in contrast to the Fröhlich term that is in this regime positive. But, as it follows from (95d), in spite that the main attractive contribution is due to couple of states Δε kk ' ≈ hω , there are still nonzero attractive contributions of couple of states outside this region. Within the BCS solution, condition (F20) with constant attractive value inside and zero outside is crucial, however. It can be seen from derivation of distribution function Copyright © 2008. Nova Science Publishers, Incorporated. All rights reserved.

(F16a), i.e.

uk2 , vk2 coefficients that determine distribution of Cooper’s pairs in

superconducting ground state. In order to derive this distribution that decreases ground state energy, the external constrain (F14) has to be fulfilled. It means that Hamiltonian (F11) of two-particle excited states (pair-excitations) has to be zero. Since pair-excited states, γ k ↑ γ − k ↓ Θ , like single-particle excited states, are “good” excited states (orthogonal +

+

to the ground state Θ ), there is no reason to exclude these states from excitation processes. Then, constrain (F14) means that pair-excitation energies in (F11) are zero over the relevant energy interval hω D
hω k '− k

the results are substantially different. Adiabatic correction to correlation energy has the form (95c) that is always attractive (negative sign, maximum attractive contribution is 95d). On the other hand, adiabatic limit

ε k0' − ε k0 >> hω k '− k of the Fröhlich form (96) results in,

H eff'' (Fr )ad =

∑σσu

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kk 'q

'

q 2

(ε

hω q 0 k +q

−ε

)

0 2 k

a k++ q ,σ a k+',σ ' a k '+ q ,σ ' a k ,σ

(E3)

It is immediately seen that this interaction, in contrast to (95c) is always repulsive. More over, expression (E3) as well as the general form of the Fröhlich effective Hamiltonian (96) has incorrect k, k’ symmetry. It can be shown, however, that strict adiabatic canonical transformation (generator S of the transformation is only Q-dependent – see [43,49]) of the model Hamiltonian with the Fröhlich form of Hep (97), yields effective electron-electron interaction Hamiltonian in the form that is identical to the correction to electron correlation energy on adiabatic level as presented in this paper (95c). The same result is obtained for Hep of the form (21b). This interaction is always attractive and the resulting form of Hamiltonian has correct k,k’ symmetry (see 95c).

Appendix F Superconducting ground state and excitation spectrum of BCS theory In its original form the Fröhlich model Hamiltonian with effective electron-electron interaction when related to normal ground state of metals (i.e. at 0 K all states below Fermi level are occupied and all states above Fermi level are empty) failed at description of

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72

Pavol Baňacký

superconducting phase [46,47]. Instead of this form, within the BCS theory [3] formally similar model Hamiltonian is introduced but with the different philosophy based on the idea of Cooper’s pairs. According to it, electrons situated in a tin layer (up to hω ) above Fermi level can form a stable pairs at the presence of whatever weak but attractive interactions between electrons. Most stable pairs are formed for electrons with opposite quasimomentum

(

)

and antiparalel spins k ↑,− k ↓ . Hamiltonian with only attractive part of effective electronelectron interaction H eff (Fr ) , i.e. valid for electrons fulfilling condition ''

ε k − ε k +q < hω ,

has than the following (reduced) BCS form

H BCS = ∑ ε k a k+ a k + ∑ Wkk ' c k+' c k = 2∑ ε k c k+ c k + ∑ Wkk ' c k+' c k k

In

(F1), +

+

kk '

k

(F1)

kk '

c k+ , c k are creation and annihilation operators of singlet electron +

pairs c k = a k ↑ a − k ↓ , c k = a − k ↓ a k ↑ . These particles obey different anticommutation and commutation relations then those valid for standard electrons and bosons. The matrix element Wkk ' is the sum of standard coulomb electron-electron repulsion and attractive Fröhlich electron-electron interaction H red (Fr ) that must be negative ( Wkk ' 2 is found (Figs. 2 (e),(f)). For comparison, we report the conductance behavior for an anisotropic s-wave superconductor, in which only the amplitude of the order parameter varies in the k-space, while the phase remains constant and Eq. 6 reduces to: ∆+ = ∆− = ∆ cos[2(α − ϕ)].

(7)

Again, in the limit Z → 0, an increase of the conductance for E < ∆ with a triangular profile is found with maximum amplitude GN S (V = 0)/GN N (V  ∆) = 2 at zero bias (Fig. 2 (g)). On the other hand, for higher Z, we obtain tunneling conductance spectra that show the characteristic “V”-shaped profile in comparison to the classical “U”-shaped structure found for an isotropic s-wave order parameter (Figs. 2 (h),(i)). We notice that in this case all the curves are quite insensitive to variation of the α parameter and a zero bias peak is obtained only for low barriers. At the N/S interface the superconducting properties can be inducted in the normal metal, due to the penetration of Cooper pairs: this phenomenon is commonly called the proximity effect. If the electrons’ motion is diffusive, the penetrationp of the Cooper pairs in the metal is proportional to the thermal diffusion length scale L ∼ D/T , where D is the diffusive constant. In the case of a pure metal the characteristic distance is ξ ∼ vF /T , where vF is Fermi velocity. So the order parameter disappears exponentially (see Fig. 3). Simultaneously the leakage of the Cooper pairs weakens the superconductivity near the interface with the normal metal. This effect is called the inverse proximity effect, and results in a decrease of the superconducting transition temperature in a thin layer in contact with a normal metal, with the depression of the superconducting energy gap ∆. If the thickness of a su-

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Andreev Reflections and Transport Phenomena in Cuprate Superconductors...

(a)

141

(b)

TC onset=43K

TCurie=135K

TC =24K

Figure 4. (a) Crystalline structure of the Ru-1212; (b) Resistance vs temperature of the sample reported in this chapter, showing the critical and the Curie temperatures. perconducting layer is smaller than the critical one, the proximity effect totally suppresses the superconducting transition [11].

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2.

Andreev reflections RuSr2 GdCu2 O8

in

an

intrinsic

F/S

system:

In this section we report PCAR studies carried out in the hybrid rutheno-cuprate RuSr2 GdCu2O8 (Ru-1212) system (results reported in Refs.[12, 13, 14, 15]). This compound [16] has recently drawn great attention among theorists and experimentalists in the field of solid state physics due to the coexistence at low temperatures of superconducting and magnetic ordering [17]. The Ru-1212 structure is similar to that of YBa 2 Cu3O7 with magnetic (2D) RuO2 planes substituting the ( 1D) Cu-O chains (see Fig. 4 (a)). The superconducting critical temperature in this compound strongly depends on the preparation conditions with some reports showing transition onset as high as 50K [18]. The Ru-1212 also shows a magnetic phase below 135K. It has been reported that the magnetic order of the Ru moments is predominantly antiferromagnetic along the c axis [19], while a ferromagnetic component has been observed in the RuO 2 planes, that act as charge reservoir [20]. At the moment, due to complexity of this compound, an exhaustive description of the interaction between the magnetic and superconducting layers is still missing as well as an unambiguous evaluation of the symmetry of the energy gap.

2.1.

Experimental conductance curves and theoretical fittings

The Ru-1212 samples used for this study were directionally solidified pellets, grown by means of the Top-Seeded Melt-Textured method starting from Ru-1212 and Ru-1210 (RuSr2 GdO6) powder mixtures with a ratio Ru-1212/Ru-1210 = 0.2. The details of the Superconducting Cuprates : Properties, Preparation and Applications, Nova Science Publishers, Incorporated, 2008. ProQuest Ebook Central,

142

S. Piano, F. Bobba, F. Giubileo and A. M. Cucolo

chamber

thermometer

sample holder

tip holder piston

right-handed theards M1

spring

right-handed theards M2

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Figure 5. Illustrative scheme of our PCAR probe. Close-ups show the sample holder and the micrometric screws that allow to vary the distance between tip and sample. preparation procedure are reported elsewhere [21]. In the X-Ray diffraction patterns, a single Ru-1212 phase was found. In the resistivity measurements versus temperature, the onset of the superconducting transition was observed at T on C ' 43K with T C ' 24K and ∆TC = 12K (∆TC is defined as the difference between the temperatures measured at 90% and 10% of the normal state resistance). We notice that a broadening of the superconducting transition is often observed in polycrystalline samples and it is usually related to the formation of intergrain weak Josephson junctions[22, 17, 23]. We address this point in the next section. To realize our experiments we used a Pt-Ir tip, chemically etched in a 40% solution of HCl, while Ru-1212 samples were cleaned in an ultrasound bath in ethyl alcohol. Sample and tip were introduced in the PCAR probe. The PCAR probe used to investigate the conductance curves is a home-built set up, constituted by three micrometric screws, each driven by its own crank. Two screws allow to vary the distance between tip and sample, with a precision of 1µm and 0.1µm, respectively. The third screw is devoted to change the inclination of the sample holder varying the contact area on the sample surface. The vertical movement of the tip allowed the tuning of the contact resistance from tunneling regime to metallic contact. Our experimental setup resulted to be extremely stable, showing no relevant effects of thermal contraction, so that in many cases it was possible to vary the junction temperature without affecting the contact geometry. A scheme of our PCAR probe is shown in Fig. 5 [15].

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Figure 6. The dI/dV vs V characteristics measured in different Ru-1212/Pt-Ir PC junctions at 4.2K. The experimental data (dots) are shown together with the best theoretical fittings (solid lines) obtained by a modified BTK model for a d-wave symmetry of the superconducting order parameter. To limit surface degradation, after the mounting, the PC inset was placed in liquid He 4 , immediately. The contacts were established by driving the tip into the sample surface at low temperatures. Current and dI/dV vs V characteristics were measured by using a standard four-probe method and a lock-in technique by superimposing a small ac-modulation to a slowly varying bias voltage. Each measurement comprised two successive cycles in order to check for the absence of heating hysteresis effects [24, 25]. A LabVIEW TM software interface was used to display and control the measurements. All measurements were performed in the temperature range between 4.2 K and up to the critical current of the sample. An in-plane solenoid was mounted on the probe to enable magnetic measurements. In Fig. 6, we show three normalized conductance spectra obtained at T = 4.2K by establishing different contacts on different areas of the same Ru-1212 pellet. The junction resistances varied between 10Ω and 100Ω. By using the Sharvin relation [26], it has been possible to achive an estimation of the size of the contact area. Indeed R = ρl/4a2, where ˚ as estimated in Ref.[21]. ρ = 0.4mΩ cm is the low temperatures resistivity and l ≈ 1000 A, ˚ and 1000A. ˚ In our case, we have found that the typical contact size varied between 300A We observe that all the reported spectra are characterized by a Zero Bias Conductance Peak (ZBCP) with a triangular structure; quite often, oscillations are observed on the conductance background, as shown in Fig. 6 (c). We were not able to reproduce the conductance spectra reported in Fig. 6 by using either the conventional s-wave model or the anisotropic one, even by considering small Z values, indicative of low barriers. The solid lines in the figures are the theoretical fittings obtained by considering a d-wave symmetry of the order parameter in the modified BTK model, Eqs. 2-6. A satisfactory agreement is obtained by using as fitting parameters the superconducting energy gap ∆, the barrier strength Z, the angle α and a phenomenological factor ΓDynes [27] to take into account pair breaking effects and finite quasiparticle lifetime [28]. We notice that in the considered spectra, both quasiparticle tunneling and Andreev reflection processes take place, since intermediate Z values have to be used to simulate the barrier strength ( 0.6 ≤ Z ≤ 0.9). Moreover, the angle α varies between 0.43 and 0.51, indicating that the average transport current mainly flows along an intermediate direction between the nodal one ( α = π/4) and that of the maximum amplitude of the energy gap ( α = 0). The modified d-wave BTK model allows to satisfactorily reproduce the variations of slope around ±1mV of the structured ZBCP in Fig. 6(c). We remark that the values of the superconducting energy gap, inferred from

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Figure 7. The dI/dV vs V characteristics measured in different Ru-1212/Pt-Ir PC junctions at 4.2K. The experimental data (dots) are shown together with the best theoretical fittings obtained by a modified BTK model for a d-wave symmetry of the superconducting order parameter by considering a Josephson junction in series (red solid line) and without the junction in series (green dots). tip

N I

VPC grain S Vmeasured

N/I VJ

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grain

S

Figure 8. Intergrain coupling effect in polycrystalline samples. The measured voltage Vmeasured is the sum of two terms: VP C , the voltage drops between tip and sample, the N/S PC junction, and VJ , the voltage drops between two superconducting grains, forming the S/I/S Josephson junction. the theoretical fittings, are all consistent and enable us to estimate an average value of the amplitude of the order parameter ∆ = (2.8 ± 0.2)meV. This value is surprisingly low in comparison with the amplitude of the energy gap in other cuprate superconductors, however the possibility that the presence of the RuO 2 magnetic planes can play an important role in the complex Ru-1212 system has to be taken into account. We notice that the ratio between the smearing factor ΓDynes and the superconducting energy gap results always less than 20%. We consider this fact as an indication of the good quality of our point-contact junctions.

2.2.

Role of the intergrain coupling

To complete our discussion about the spectra measured at low temperatures, we now address the analysis of the conductance curves reported in Figs. 7, that cannot be modeled by the modified BTK model simply. In this respect, we observe that, due to the granularity of the compound, in same cases, an intergrain Josephson junction can be formed in series with the point contact one, as schematically drawn in Fig. 8. This topic has been already addressed

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Figure 9. The dI/dV vs V characteristics measured in different Ru-1212/Pt-Ir PC junctions at 4.2K. These spectra are characteristics of a N–I–S configuration. in the PCAR studies on MgB 2 [29]. To provide a quantitative evaluation of the conductance spectra, we consider a real configuration in which the Pt-Ir tip realizes a PC junction on a single Ru-1212 grain, which, in turn, is weakly coupled to another grain, so forming a Josephson junction. In this case the measured voltage corresponds to the sum of two terms: Vmeasured(I) = VP C (I) + VJ (I) ,

(8)

where VP C and VJ are the voltage drops at the N/S point contact junction and at the S/I/S intergrain Josephson junction, respectively. This last contribution can be calculated by the Lee formula [30] which, in the limit of small capacitance and at low temperatures, reduces to the simplified expression [31]:

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VJ =

(

RJ IJ

p 0

[(I/IJ

)2

− 1]

for I < IJ ; for I ≥ IJ .

(9)

At the same time, for the point contact contribution, we use again the extended BTK model for a d-wave superconductor. The I(V ) characteristic is then calculated by inverting Eq. 8 and the conductance spectrum is given by: dI = σ(V ) = dV



dVJ dVP C + dI dI

−1

.

(10)

We observe that, in this model, two more parameters are needed, namely the resistance RJ and the critical current IJ of the Josephson junction. However, the choice of these two parameters is not completely arbitrary, since the condition RJ + RP C = RN N has to be fulfilled, where RN N is the measured normal resistance and the product RJ IJ necessarily results lower than ∆ [32]. For the spectrum in Fig.7 a) we notice that the modified d-wave BTK model allows to satisfactorily reproduce the variations of slope around 1 mV of the ZBCP but a light discrepancy in modeling the full height of the peak is reported, instead a more satisfactory fitting for this contact can be obtained by taking into account the additional in series intergrain junction. Remarkably, for this spectrum, the best fittings have been obtained by using ∆ = 2.8 meV, consistently with the average value extracted from the other curves in Figs. 6. In some cases, it has been pointed out that dips and a wider ZBCP in the conductance spectra can be related to the presence of intergrain junctions, and for sake of completeness,

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S. Piano, F. Bobba, F. Giubileo and A. M. Cucolo

we have applied our model also to the spectra of Figs. 7 b)–c) and we notice that for different junctions, the effect of the intergrain coupling results more or less evident, depending on ratio RJ /RP C . Remarkably, by these last fittings we have found the same value of the superconducting energy gap previously inferred, ∆ = 2.8 ± 0.2meV. In addition to these spectra that show a ZBCP we have obtained conductance curve that present a “V-shaped” profile characteristic of a d-wave tunnel junctions (Figs.9) as we can see clearly in the theoretical curves of Fig.2 f), also in this case the width of the structures seems to show the presence of junctions in series.

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2.3.

Temperature dependence of the conductance spectra

To achieve information on the temperature dependence of the superconducting energy gap in the Ru-1212 system, we have analyzed the temperature behavior of the conductance spectrum shown in Fig. 7a). Indeed, this PCAR junction resulted to be very stable against temperature variations. In Fig. 10 we show the conductance characteristics measured in the temperature range 4.2K≤ T < 35K. We first notice that the ZBCP decreases for increasing temperature and disappears at about T ' 30K, that we estimate as the local critical temperature TCl of the superconducting Ru-1212 grain in contact with the Pt-Ir tip, consistently with the resistivity measurements[21]. This fact provides further evidence that the ZBCP is a consequence of the superconducting nature of Ru-1212 and is not due to spurious effects like inelastic tunneling via localized magnetic moments in the barrier region [33]. The experimental data for each temperature are then compared to the theoretical fittings calculated by using the d-wave modified BTK model with a small contribution of Josephson junction in series. For all the curves, we fixed the strength of the barrier and the angle α to the values obtained at the lowest temperature. The resulting temperature dependence of the superconducting energy gap ∆(T ) is reported in Fig. 11, where vertical bars indicate the errors in the gap amplitude evaluation, that increase when approaching the critical temperature. Contrarily to what expected for BCS superconductors, we observe that the energy gap, at low temperatures, decreases rapidly for increasing temperatures and goes to zero at TCl in a sublinear way. We notice that the same temperature evolution for the superconducting energy gap is found by using the d-wave BTK model with or without considering any intergrain junction in series; remarkably, in this last case, the superconducting energy gap ∆ remains the only varying parameter. A similar temperature dependence has been reported by G. A. Ummarino et al. [34], however these authors give a larger estimation of the maximum gap amplitude. From the average value of the superconducting energy gap ∆ = 2.8meV and from the measured local critical temperature TCl ' 30K, we obtain a ratio 2∆/KB TCl ' 2 much smaller than the predicted BCS value and also smaller than the values found for high-T C cuprate superconductors[35]. Again we speculate that the simultaneous presence of superconducting and magnetic order is an important key for understanding the behavior of the Ru-1212 system. Recently, it has been found that in conventional F/S structures a dramatic suppression of the amplitude of the order parameter is expected for the high T C superconductors [36] (as we show in the next section) and various examples of anomalous temperature behavior are found in the literature. Gapless superconductivity can be achieved, that

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Figure 10. Temperature evolution of the conductance spectrum of Fig. 7 a) from T = 4.2K to up the critical temperature (dots). The solid lines are the theoretical fittings obtained by a modified d-wave BTK model with the energy gap as only free parameter.

Figure 11. Temperature dependence of the superconducting energy gap as inferred from the theoretical fittings shown in Fig. 10. The solid line is a guide for the eyes. The right hand scale refers to the temperature evolution of the measured height of the ZBCP normalized to the 4.2K value.

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Figure 12. Magnetic field dependence of the normalized dI/dV vs V characteristics at T=4.2K from 0 T to 2 T (dots) for the spectra of Fig. 6b. The full lines are the theoretical fittings obtained as discuss in the text. In the inset the magnetic field dependence of the energy gap is reported. can induce a sublinear temperature dependence of the superconducting energy gap. In the Ru-1212 system, it has been proved that the RuO 2 planes are conducting, however these do not develop superconductivity at any temperature[37]. By means of different experimental techniques, it has been inferred that a large fraction of the charge carriers is not condensed in the superconducting state even at low temperatures [37]. Both findings are consistent with a reduced value of the 2∆/(KB TC ) ratio in this compound. In Fig. 11 we also report (righthand scale) the temperature evolution of the height of the ZBCP normalized to its value at T = 4.2K. It is worth to notice that GN S (V = 0, T ), as directly measured from the experiments, and ∆(T ), as inferred from the theoretical fittings, show a similar scaling with temperature. This correspondence is easily verified for Z = 0 in case of a s-wave superconductor, however here it stands as a quite new interesting result since it has been found for intermediate barriers and unconventional symmetry of the superconducting order parameter.

2.4.

Magnetic field dependence of the conductance spectra

As we already observed, one of the most interesting features of the Ru-1212 is the coexistence of the superconducting phase and magnetic order. Indeed, from Nuclear Magnetic Resonance (NMR) [38, 39] and magnetization [40] measurements, it has been found that in this compound ruthenium occurs in a mixed valence state Ru 4+ , Ru5+ with some higher Ru5+ concentration. The RuO2 planes, from one side, act as charge reservoir for the superconducting CuO 2 planes; on the other hand, as observed in Muon Spin Rotation ( µSR) experiments [20], they show quite homogeneous ferromagnetic order below T C . A weak interaction between the two order parameters, ferromagnetism in the RuO 2 planes and superconductivity in the CuO 2 planes, has been suggested and recently several experiments appear to confirm this hypothesis [37]. In spite of the huge experimental and theoretical

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Figure 13. Normalized conductance curves for the contact of Fig. 7b)–c) measured at T=4.2K in magnetic field up to 3T. When the field is switched off, the original spectra are recovered. efforts focused on the study of the interplay between superconductivity and magnetism, to the best of our knowledge no spectroscopic studies in magnetic field of the superconducting order parameter in Ru-1212 had been reported in literature so far prior to our contribution [14]. In Fig. 12 we show the PCAR spectra measured by applying an external magnetic field, parallel to the tip, with intensity H varying from 0T to 2T. The dI/dV vs V curves refer to the contact reported in Fig. 6b). A reduction of the ZBCP for increasing magnetic fields is observed, that in first approximation can be reproduced by a phenomenological approach. Indeed, addressing the problem of the magnetic field dependence of the conductance characteristics in unconventional superconductors, is in general a quite difficult task, and a complete treatment of PCAR spectroscopy in magnetic field would require the use of an appropriate density of states in calculating the BTK expression for the reflection and transmission coefficients at the N/S interface. In our case, dealing with a polycrystalline unconventional superconductor exhibiting internal magnetic ordering, a way to perform a theoretical fitting is obtained by using an additional pair breaking parameter to simulate the effect due to the magnetic field [41, 42]. In this case, the total broadening effect Γ is considered as the sum of two terms: Γ = ΓDynes + Γext where ΓDynes is the intrinsic broadening due to the quasiparticle lifetime, as used in the previous fittings, while Γext mimics the pair breaking effect due to the external applied magnetic field. The curve at H = 0T (see Fig. 6b) has been fitted by using the d-wave modified BTK model with ∆ = 3.0meV. For increasing magnetic fields we keep constant, in the numerical computation, the strength of the barrier Z= 0.9, the orientation angle α = 0.51 and the intrinsic ΓDynes = 0.7meV, while varying only two parameters: the energy gap ∆ and the magnetic field effect Γext. We observe that the best theoretical fittings (solid lines in Fig. 12) satisfactorily reproduce for any field both the height and the amplitude of the measured spectra. In the inset, we report the magnetic field dependence of the superconducting energy gap (dots) as extracted from the theoretical fittings. The amplitude of the energy gap reduces linearly for H up to 2T and by a linear extrapolation of the data, we find that the energy gap disappears at about H ext ' 30 T, consistently with the estimated critical field reported in Ref.[21]. We have also studied the effect of the magnetic field on the conductance characteristics

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of the junctions showing wider ZBCP, that are formed by two junctions in series. In Fig. 13 we report the dI/dV vs V curves measured up to 3 T for the contacts of Fig. 7b)–c). In this case, we observe that the conductance curves dramatically change with the application of the magnetic field. As discussed in the previous section, the Josephson current due to the intergrain coupling is immediately suppressed by the magnetic field, modifying the spectra towards the narrower, non-structured, triangular shape of the ZBCP. In addition to this, the oscillatory behavior of the background, due to the intergrain coupling disappears in magnetic field. Remarkably, for the junctions of both Figs. 12, 13 the peculiar features of the spectra together with the normal junction resistance, are recovered when the magnetic field is switched off, and no hysteresis is found for increasing/decreasing fields.

3.

Study of the Andreev reflections at the interface between ferromagnetic and superconducting oxides

In this section we analyze the effect of spin-polarized electrons on the tunneling current in a heterostructure constituted by a high-T C superconductor, YBa2 Cu3 O7−x (YBCO), and a Colossal magneto-resistance ferromagnetic oxide La 0.7Ca0.3MnO3 (LCMO), as reported in Refs. [43, 44]. We observe the presence of both Andreev bound states in the YBCO layer, and spin polarization in the LCMO layer. The zero bias conductance peak, appearing in the conductance spectra due to Andreev bound states at the Fermi level of the superconductor, results to be depressed by a proximity effect induced by the magnetic layer. Our results are well interpreted in the framework of the spin-polarized transport theory.

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3.1.

A rapid overview to YBCO and LCMO

In this subsection we explain the main properties of YBCO and LCMO. A schematic structure of YBCO is presented in Fig. 14: the unit cell is developed from that of a tetragonal perovskite tripled along the c-axis and it consists of a sequence of copper-oxygen layers. The dimensions of the unit cell are approximately 1.2 nm and 0.4 nm in the c and in a or b-axis directions respectively. The compound YBa 2 Cu3O6 is an insulator, so it has to

Cu 2+, Cu3+ O 2Y3+

11.6802 A

Ba 2+ c

b

3.8827 A a

3.8227 A

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be doped to gradually become a hole-doped metallic conductor and a superconductor below some critical temperature. The doping is achieved by adding additional oxygen which forms CuO chains. These oxygen ions attract electrons from the CuO 2-planes which therefore become metallic. Note, that the correct formula for YBCO material is YBa 2 Cu3 O7−x , where x corresponds to partial oxygen content. Measurements of resistance vs temperature on YBCO films show a T C ' 91K [45]. The ideal structure of the LCMO oxide is a cubic perovskite. The large sized La trivalent ions and Ca divalent ions occupy the center of the cube with 12-fold oxygen coordination. The smaller Mn ions in the mixed-valence state Mn 3+ –Mn4+ are located at the vertices of the cube and are coordinated 6-fold with the oxygen ions to form octahedrons MnO 6 . In Fig. 15 (a) a schematic view of the cubic perovskite structure is shown. Usually, the crystalline lattice will be distorted by the Jahn-Teller effect (see Fig. 15 (b)). The phase diagram of LCMO was measured by Cheong and Hwang [46], and is given in Fig. 15 (c). The undoped parent compound LaMnO 3 has an antiferromagnetic insulating ground state. LaMnO3 and CaMnO3 are both antiferromagnetic insulators. At first sight a mixture of LaMnO3 and CaMnO3 is expected to show no spectacular effect. But, in their phase (a)

(b)

La, Ca O

(c)

La Ca MnO 1-x

350

x

3

5/8

3/8

300

Temperature (K)

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Mn

4/8

250 200

X = 1/8

CO

7/8

150 FI

100 50

FM AFM

CAF

CAF

CO

0 0.0 0.1 0.2

0.3 0.4 0.5 0.6

0.7 0.8 0.9

1.0

Ca x

Figure 15. (a) Schematic view of the cubic cell of the LCMO. (b) Distortion of the structure due to the Jahn-Teller effect. (c) Phase diagram showing the different phases of LCMO as a function of Ca concentration. FM : Ferromagnetic Metal, FI: Ferromagnetic Insulator, AF: Antiferromagnetism, CAF: Canted Ferromagnetism, and CO: Charge/orbital Ordering

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Figure 16. Scheme of our point contact junction realized between a Pt-Ir tip and a YBCO/LCMO bilayer. diagram a ferromagnetism is found, metallicity and several regions with spin and charge orderings. An important aspect in the unexpected rich phase diagram is the small but relevant distinction in crystal structure, although both compounds are perovskites. LaMnO 3 consists of deformed MnO 6 octahedra, whereas the octahedra are perfect in CaMnO3 . The origin of the deformation is the crystal field splitting of the 3d orbitals. Above 10% Ca doping the ferromagnetic interactions suppress the antiferromagnetic coupling and a ferromagnetic ground state is obtained. In the region of 20% to 50% Ca substitution the ground state is a ferromagnetic metal, dominated by double exchange. According to Cheong’s phase diagram, Fig. 15 (c), the ferromagnetic metallic phase emerges instantly above a critical concentration, at all temperatures below T Curie. Resistance vs temperature measurements without applied magnetic field showed that the metal-insulator transition temperature is in the range of 220-260K [47].

3.2.

Sample preparation and experimental setup

Highly epitaxial, c-axis oriented YBCO/LCMO heterostructures were grown in a pure oxygen atmosphere (p=3.0 mBar) on SrTiO 3 (0 0 1) substrates (STO) by DC sputtering tech˚ was first nique at T=900 ◦C (for further details see Refs. [45, 48]). A YBCO film of 500A deposited; then, by defining the geometry through a shadow mask, we sequentially realized ˚ and dLCMO = 75A, ˚ respeca YBCO/LCMO bilayer with thicknesses dYBCO = 1000A tively. The conductance spectra were measured in the PCAR setup from liquid-helium temperature to room temperature. To realize the point contact experiments, we used mechanically cut fine tips of Pt-Ir, chemically etched in a 40% solution of HCl in an ultrasound bath. Samples and tips were placed in the PCAR probe and the electrical contacts (two on the tip, and two on the first YBCO basis) were realized by indium drops. In Fig. 16 we show the geometry of the junction and the voltage-current terminals.

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Figure 17. Experimental YBCO/LCMO conductance spectra (black line) with the background best fitting (red line) at low temperature. In the inset the YBCO conductance curve is shown with the best fit with d-wave symmetry of the order parameter

Figure 18. Temperature dependence of the spectrum in Fig.17 a). We notice that a ZBCP appears at low temperatures and disappears at about 30K.

3.3.

Experimental data and theoretical interpretation

We present now the conductance spectra obtained between a YBCO/LCMO bilayer and a Pt-Ir tip. We remark that our transport measurements include two different interfaces, YBCO/LCMO and LCMO/Pt-Ir, both responsible for the profile of the conductance curves. The different contributions can be evidenced for instance in the lowest-temperature spectra (see Figs. 17 a) and b)), characterized by an asymmetric, “V”-shaped background. The “V”-shaped background, similar to that reported for other metallic oxide junctions [49, 50], is a signature of the LCMO/Pt-Ir junction, while the YBCO layer is responsible for the asymmetry [51]. On the other hand, the ZBCP, present in the spectrum of Fig. 17 a) is a consequence of the d-wave symmetry of the superconducting order parameter of YBCO, indicating the formation of the Andreev bound states at the YBCO Fermi level [51, 35, 52, 10]. Moreover, the presence of a ZBCP suggests that our tunnel junction is not completely c-axis oriented, but a component in the a-b plane is present as well.

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The nature of the ZBCP can be experimentally investigated by following the temperature evolution of the conductance spectra. In the case of PCAR on pure YBCO, the literature reports a ZBCP decreasing with increasing temperature, and vanishing at the critical temperature of YBCO (∼ 90 K) [51]. In our YBCO/LCMO junction (see Fig. 18), we observe instead a depressed ZBCP. According to the theory of spin transport between a ferromagnetic material and a d-wave superconductor, the depression of the ZBCP follows from the suppression of Andreev reflections at the interface, due to the spin polarization of the ferromagnetic layer [53, 54]. In the measured conductance spectra, the ZBCP disappears at a temperature of about Tc ∼ 30 K, which is in agreement with the resistivity measurements on YBCO/LCMO bilayers [48]. From our conductance spectra, the amplitude of the superconducting order parameter of YBCO can be inferred as well. Namely, we can fit the background according to the model [55, 56]: Z dI d G(V ) = ∝ NF M (E)NSC (E + eV )[f (E) − f (E + eV )]dE, (11) dV dV where NSC is the density of states of the YBCO layer, and NF M is the density of states of the LCMO layer. The latter can be expressed as NF M (E) = NF M (0)[1+(|E|/Λ)n], where Λ is a constant associated with the electron correlated energy of LCMO at the interface and the exponent 0.5 < n < 1 reflects the degree of disorder in LCMO near the YBCO/LCMO interface. Concerning the YBCO, we can assume that, for bias voltages larger than the superconducting energy gap, NSC is approximately constant, except for a linear correction taking into account for the asymmetry of the normal state of YBCO. In a window of bias voltages V ∈ [−V¯ , V¯ ], the density of states of YBCO can thus be written as NSC = 1 − κ(V + V¯ ), where κ is the asymmetry factor and the total conductance is normalized such that G(−V¯ ) = 1. In Fig. 17, we show the lowest-temperature conductance spectra, together with the best fitting curve for the background, following Eq. 11. The superconducting energy gap of YBCO corresponds to the bias voltage at which the theoretical curve for the background deviates from the experimental conductance spectrum. From Fig. 17a), we can estimate an energy gap ∆ ∼ 8 meV, while for Fig.17b), we can estimate an energy gap ∆ ∼ 13 meV, smaller than the reported gap value of 20 meV for YBCO. We speculate that the presence of ferromagnetic order within the superconducting phase results in an effective reduction of the energy gap due to the injection of spin-polarized electrons from LCMO to YBCO. In the case of Fig.17 a) we have extrapolate the contribution of the YBCO layer to the conductance characteristic simply dividing the measured differential conductance (at the lowest temperature) by the modeled background curve: GYBCO (V ) = G(V )exp /G(V )back . We can satisfactorily fit the spectrum GYBCO (V ) with a d-wave BTK model, Eqs. 2–6, regarding as fitting parameters the superconducting gap ∆, the barrier strength Z, the angle α of the order parameter and the smearing factor Γ. Remarkably (see the inset of Fig. 17 a)), the fitting provides a gap ∆ = 8 meV, consistent with our previous findings. This value, smaller than the reported gap value of 20meV for YBCO [51], can be explained by the proximity effect of the Cooper pairs from YBCO to LCMO, or by the injection of spin-polarized electrons from LCMO to YBCO [56]. It has been theoretically predicted that, in a F/S interface, the amplitude of the spin polarization of the ferromagnetic layer can be estimated from the temperature dependence of

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Figure 19. Temperature dependence of ZBCP (dots) with the best theoretical fit (solid line).

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Figure 20. Temperature evolution of the ZBCP for Ru-1212 and YBCO/LCMO conductance spectra.

the ZBCP. The latter is expected to be proportional to the inverse of the temperature for intermediate temperatures [57]. In Fig. 19 we show the temperature dependence of the ZBCP as obtained by the experimental conductance spectra of Fig. 18, together with the best fitting curve. With an extrapolation of the best fitting curve at low temperatures ( T → 0), we can directly compare our ZBCP evolution to the theoretical model of Ref. [57]. This allows us to estimate a spin polarization of the LCMO layer of about 67%, meaning that the electron spins are not fully polarized. This is consistent with the observation of a (depressed) ZBCP: in fact, it would have been completely suppressed if the LCMO polarization had approached unity. In the context of the fundamental interest on the interplay between superconductivity and ferromagnetism, we compare in Fig. 20 the behavior of the ZBCP extrapolated from the temperature dependence of conductance spectra in the Ru-1212 (Fig. 11) and YBCO/LCMO (Fig. 19) tunneling junctions. We notice a similar, sublinear temperature dependence which is found to be strikingly different from the behavior of standard BCS superconductors. So the depressed ZBCP (both in height and in the temperature evolution) together with a reduced superconducting gap, as observed in both intrinsic (Ru-1212) and artificial (YBCO/LCMO) hybrid structures, may be identified as signatures of an injection of spin polarized electrons from the ferromagnetic layer, inducing peculiar modifications to the density of states of the superconducting layer.

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4.

S. Piano, F. Bobba, F. Giubileo and A. M. Cucolo

Conclusion

In conclusion we have presented our experiments of Point Contact Spectroscopy performed on superconducting RuSr 2 GdCu2O8 (Ru-1212) polycrystalline pellets and YBCO/LCMO bilayers. Satisfactory theoretical interpretation is provided for the various illustrated results, and the characteristic traits of the transport phenomena arising in presence of cuprate superconductors coexisting, or in contact with ferromagnetic layers, have been singled out. For the Ru-1212 all the conductance curves at low temperatures have showed a Zero Bias Conductance Peak that decreases for increasing temperatures and disappears at the local critical temperature TCl ' 30K of the superconducting grain in contact with the Pt-Ir tip. The triangular shape of all the measured spectra has been modeled by using a modified BTK model for a d-wave symmetry of the superconducting order parameter. This finding suggests a closer similarity of the Ru-1212 system to the high-T C cuprate superconductors rather than to the magnetic ruthenate Sr 2 RuO4 compound. However, the remarkably low values of the energy gap ∆ = (2.8 ± 0.2)meV and of the ratio 2∆/KB TC ' 2 indicate major differences between the Ru-1212 and the high- TC cuprates. We speculate that the presence of ferromagnetic order within the superconducting phase results in an effective reduction of the energy gap. We have also demonstrated that, when dealing with granular samples, intergrain coupling effects can play a predominant role. In some cases, an intergrain Josephson junction in series with the point contact junction is formed. Taking into account this feature as well, all conductance spectra have been properly modeled by considering a d-wave symmetry of the order parameter, with consistent values of the amplitude of the energy gap. By fixing all the fitting parameters to their values at the lowest measured temperature, and by varying ∆, the temperature dependence of the energy gap has been extracted from the conductance characteristics of a very stable junction. We have found that the energy gap exhibits a sub-linear dependence in temperature. The magnetic field behavior of the spectra has been also studied, showing a linear reduction of the energy gap for fields up to 2 T, from which a critical field H C2 ∼ 30 T is inferred. We have found that both the superconducting features and the normal background in the conductance spectra do not show any hysteresis in magnetic field. These observations seem to suggest a weak coupling between the superconducting and magnetic order parameter. We have also presented the conductance spectra obtained on artificial S/F junctions realized by using a high-T C superconductor, YBCO, and a colossal magnetoresistive material, LCMO. The low-temperature spectra have shown a Zero Bias Conductance Peak, characteristic of a d-wave symmetry of the YBCO, and a “V”-shaped background that is related to the presence of the ferromagnetic layer. From the temperature evolution of the conductance data, we have reported a depressed critical temperature; furthermore, by proper theoretical modeling we have estimated an energy gap ∆ reduced by a factor of about one half compared to the reported gap value of 20 meV for YBCO alone. Also in this case we observe that the presence of ferromagnetic order in proximity of the superconducting phase results in an effective reduction of the energy gap due to the injection of spin-polarized electrons from LCMO to YBCO. The conductance curves are interpreted as well by a d-wave BTK model. Our analysis may be helpful for a deeper understanding of the mechanisms enabling high temperature superconductivity, and its interplay with magnetic order in unconventional

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superconductors like rutheno-cuprates. The comprehension of the coexistence between superconductivity and ferromagnetism in artificial S/F systems can be in turn an important stepping stone towards the realization of magnetic devices such as MRAM and spin valves.

Acknowledgements We acknowledge financial support from CNR INFM.

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[9] Blonder, G. E.; Tinkham, M.; Klapwijk, T. M. Phys. Rev. B 1982, 25, 4515. [10] Kashiwaya, S.; Tanaka, Y. Rep. Prog. Phys. 2000, 63, 1641. [11] Deutscher, G.; Gennes, P. G. D. in Superconductivity, Ed. R. D. Parks (Marcel Dekker, INC.,New York, 1969). [12] Piano, S.; Bobba, F.; Giubileo, F.; Vecchione, A.; Cucolo, A. M. J. of Phys. and Chem. of Solids 2006, 67, 384. [13] Piano, S.; Bobba, F.; Giubileo, F.; Vecchione, A.; Cucolo, A. M. Int. J. Mod. Phys. B 2005, 19, 323. [14] Piano, S.; Bobba, F.; Giubileo, F.; Gombos, M.; Vecchione, A.; Cucolo, A. M. Phys. Rev. B 2006, 73, 064514. [15] Piano, S. Ph.D. thesis, University of Salerno, 2007. [16] Bauernfeind, L.; Widder, W.; Braun, H. F. Physica C 1995, 254, 151. [17] Nachtrab, T.; Bernhard, C.; Lin, C. T.; Koelle, D.; Kleiner, R. Journal Comptes Rendus de l’Academie des Sciences (Comptes Rendus Physique). Special Issue on ”Magnetism and Superconductivity Coexistence” 2005. ArXiv:cond-mat/0508044. Superconducting Cuprates : Properties, Preparation and Applications, Nova Science Publishers, Incorporated, 2008. ProQuest Ebook Central,

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[18] Bernhard, C.; Tallon, J. L.; Br¨ucher, E.; Kremer, R. K. Phys. Rev. B 2000, 61, R14 960. [19] Lynn, J. W.; Keimer, B.; Ulrich, C.; Bernhard, C.; Tallon, J. L. Phys. Rev. B 2000, 61, 14964(R). [20] Bernhard, C.; Tallon, J. L.; Niedermayer, C.; Blasius, T.; Golnik, A.; Br¨ucher, E.; Kremer, R. K.; Noakes, D. R.; Stronach, C. E.; Ansaldo, E. J. Phys. Rev. B 1999, 59, 14099. [21] Attanasio, C.; Salvato, M.; Ciancio, R.; Gombos, M.; Pace, S.; Uthayakumar, S.; Vecchione, A. Physica C 2004, 411, 126. [22] Prester, M.; E. Babi´c, M.; Nozar, P. Phys. Rev. B 1994, 49, 6967. [23] Cimberle, M. R.; Tropeano, M.; M. Feretti, A. M.; Artini, C.; Costa, G. A. Supercond. Sci. Technol. 2005, 18, 454. [24] Naidyuk, Y. G.; Yanson, I. K. J. Phys. : Condens. Matter 1998, 10, 8905. [25] Gloos, K. Phys. Rev. Lett. 2000, 85, 5257. [26] Sharvin, Y. Zh. Ekperim. i. Teor. Fiz. 1965, 48, 984. (Soviet Physics JETP, 21, 655). [27] Dynes, R. C.; Narayanamurti, V.; Garno, J. P. Phys. Rev. Lett. 1978, 41, 1509. [28] Grajcar, M.; Plecenik, A.; Seidel, P.; Pfuch, A. Phys. Rev. B 1995, 51, 16185.

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[29] Giubileo, F.; Bobba, F.; Aprili, M.; Piano, S.; Scarfato, A.; Cucolo, A. M. Phys. Rev. B 2005, 72, 174518. [30] Lee, S. Y.; Lee, J. H.; Ryu, J. S.; Lim, J.; S. H. Moon, H. N. L.; Kim, H. G.; Oh, B. Appl. Phys. Lett. 2001, 79, 3299. [31] Van Duzer, T.; Turner, C. W. Principles of Superconductive Devices and Circuits ; Edward Arnold: London, UK, 1981. [32] Barone, A.; Patern`o, G. Physics and Applications of the Josephson effect ; John Wiley & Sons: New York, US, 1982. [33] Cucolo, A. M. Physica C 1998, 305, 85. [34] Gonnelli, R. S.; Calzolari, A.; Daghero, D.; Ummarino, G. A.; Stepanov, V. A.; Giunchi, G.; Ceresara, S.; Ripamonti, G. Phys. Rev. Lett. 2001, 87, 097001. [35] Dagan, Y.; Krupke, R.; Deutscher, G. Phys. Rev. B 2000, 62, 146. [36] Luo, P. S.; Wu, H.; Zhang, F. C.; Cai, C.; Qi, X. Y.; Dong, X. L.; Liu, W.; Duan, X. F.; Xu, B.; Cao, L. X.; Qiu, X. G.; Zhao, B. R. Phys. Rev. B 2005, 71, 094502. [37] Poˇzek, M.; Dulˇci´c, A.; Paar, D.; Hamzi´c, A.; Basleti´c, M.; E. Tafra, G. V. M. W.; Kr¨amer, S. Phys. Rev. B 2002, 65, 174514. Superconducting Cuprates : Properties, Preparation and Applications, Nova Science Publishers, Incorporated, 2008. ProQuest Ebook Central,

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[38] Kumagai, K. I.; Takada, S.; Furukawa, Y. Phys. Rev. B 2001, 63, 180509(R). [39] Tokunaga, Y.; Kotegawa, H.; Ishida, K.; Kitaoka, Y.; Takagiwa, H.; Akimitsu, J. Phys. Rev. Lett. 2001, 86, 5767. [40] Butera, A.; Fainstein, A.; Winkler, E.; Tallon, J. Phys. Rev. B 2001, 63, 054442. [41] Naidyuk, Y. G.; H¨ausslera, R.; v. Lo¨ hneysenet, H. Physica B 1996, 218, 122. [42] Gonnelli, R. S.; Daghero, D.; Calzolari, A.; Ummarino, G. A.; Dellarocca, V.; Stepanov, V. A.; Jun, J.; Kazakov, S. M.; Karpinski, J. Phys. Rev. B 2004, 69, 100504(R). [43] Piano, S.; Bobba, F.; Santis, A. D.; Giubileo, F.; Scarfato, A.; Cucolo, A. M. Journal of Physics: Conference Series 2006, 43, 1123. [44] Piano, S.; Bobba, F.; Santis, A. D.; Giubileo, F.; Cucolo, A. M. Physica C 2006, 460-462. [45] De Santis, A.; Bobba, F.; Boffa, M. A.; Caciuffo, R.; Mengucci, P.; Salvato, M.; Vecchione, A.; Cucolo, A. M. Physica C 2004, 408, 48. [46] Tokura, Y. (ed.) Colossal Magnetoresistance Oxides ; Gordon and Breach, Monographs in Condensed Matter Science: London, UK, 2000. [47] De Santis, A.; Bobba, F.; Cristiani, G.; Cucolo, A. M.; Frohlich, K.; Habermeier, H.-U.; Salvato, M.; Vecchione, A. J. Magn. Magn. Mater. 2003, 262, 150.

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[48] De Santis, A. Ph.D. thesis, University of Salerno, 2003. [49] Raychaudhuri, A. K.; Rajeev, K. P.; Srikanth, H.; Gayathri, N. Phys. Rev. B 1995, 51, 7421. [50] Cao, D.; Bridges, F.; Worledge, D. C.; Booth, C. H.; Geballe, T. Phys. Rev. B 2000, 61, 11373. [51] Deutscher, G. Rev. Mod. Phys. 2005, 77, 109. [52] Wei, J. Y. T.; Yeh, N.-C.; Garrigus, D. F.; Strasik, M. Phys. Rev. Lett. 1998, 81, 2542. [53] Zhu, J.-X.; Friedman, B.; Ting, C. S. Phys. Rev. B 1999, 59, 9558. [54] Kashiwaya, S.; Tanaka, Y.; Yoshida, N.; Beasley, M. R. Phys. Rev. B 1999, 60, 3572. [55] Altshuler, B.; Aronov, A. in Efros, A. L.; Pollak, M. (eds.) Electron-Electron Interactions in Disordered Systems ; North-Holland: Amsterdam, 1985. [56] Luo, P. S.; Wu, H.; Zhang, F. C.; Cai, C.; Qi, X. Y.; Dong, X. L.; Liu, W.; Duan, X. F.; Xu, B.; Cao, L. X.; Qiu, X. G.; Zhao1, B. R. Phys. Rev. B 2005, 71, 094502. [57] Hirai, T.; Tanaka, Y.; Yoshida, N.; Asano, Y.; Inoue, J.; Kashiwaya, S. Phys. Rev. B 2003, 67, 174501. Superconducting Cuprates : Properties, Preparation and Applications, Nova Science Publishers, Incorporated, 2008. ProQuest Ebook Central,

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In: Superconducting Cuprates Editor: Koenraad N. Courtlandt

ISBN: 978-1-60456-919-3 c 2009 Nova Science Publishers, Inc.

Chapter 5

I NFLUENCE OF PAIR B REAKING E FFECTS ON THE L ONG - RANGE O DD T RIPLET S UPERCONDUCTIVITY IN A F ERROMAGNET/S UPERCONDUCTOR B ILAYER T. Rachataruangsit Physics Department, Faculty of Science, Burapha University, Chonburi 20131, Thailand

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Abstract The spin-dependent potential together with the magnetic impurity and the spinorbit scattering are incorporated into the de Gennes-Takahashi-Tachiki theory of a diffusive superconductor-and- ferromagnetic metal to derive a formulation of the odd triplet superconductivity proximity effect. It is found that when the spin exchange interaction is inhomogeneous, i.e. pointing in arbitrary directions rather than along the quantization axis, as in the N´eel spiral magnetic order, a new type of triplet condensate is generated, due to the role of the broken time-reversal invariance. The triplet amplitude still contains the s-wave state, similar to the conventional singlet pairing, but the frequency symmetry must be odd to obey the Pauli’s exclusion principle. As a result, the self-consistent order parameter contains only the singlet pair amplitude. The superconducting critical temperature of the bilayer is obtained by solving the secular equation exactly using the multimode method. The necessary condition for the occurrence of the induced long-range triplet component in the ferromagnet layer is characterized by the modulation of the pair amplitudes in the transverse direction. The possibility of the cryptoferromagnetic state is demonstrated in favor of the superconductivity and this may explain the possible origin of the magnetically dead layer. In addition, the influence of the magnetic impurity and the spin-orbit scattering is to decrease the decay length and to increase the oscillation period of the pair amplitudes which in turn enhances the critical temperature but in a less pronounced nonmonotonicity manner.

PACS 74.25.Dw, 74.45.+c, 74.78.Fk. Keywords: Odd triplet superconductivity; Ferromagnetism; Proximity effect; Transition temperature; Impurity scattering; Spin-orbit scattering. Superconducting Cuprates : Properties, Preparation and Applications, Nova Science Publishers, Incorporated, 2008. ProQuest Ebook Central,

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1.

T. Rachataruangsit

Introduction

The great progress in nano-fabricated technology in the past decade has attracted a lot of interest, in particular much attention has been paid to the study of the hybrid heterostructures of different matters. This active field was pioneered a long time ago by Deutscher and de Gennes [1] who studied the bulk-like sandwiched structures composed of a superconductor (SC) and a normal metal (N). In such system, SC is in contact with N, and the Cooper pairs can physically penetrate the N side in a monotonic decaying manner at a certain length. The induced superconductivity can therefore occur in N, and since then this phenomenon was named the proximity effect. If the N metal is substituted by a ferromagnetic metal (FM), the Cooper pairs do not just spatially decay but also oscillate as affected by the influence of the magnetic ordering. Thus the FM/SC system has two orderings which compete with each other [2]. The two antagonistic phenomena lead to an unusual behavior of the superconducting condensate penetrates the FM region. The induced superconductivity in FM causes the thickness-dependent oscillations of critical temperatures as well as Josephson currents [3–7]. Recently the Buzdin-Tagirov spin switch effect [8,9] and the Radovi´c pi-phase state [22] are discovered to be the most striking phenomena of the FM/SC system. The first one exists in a FM/SC/FM trilayer where theories predicted that an antiparallel magnetization in the FM layers could provide the higher critical temperature than a parallel configuration [10–15]. Many experimental reports showed good agreement with the theory [16–19], but some works yielded the opposite results [20, 21]. While the latter phenomenon is found to occur in SC/FM multilayers. There is a consensus between theories and experiments, that the phase shift of the pair amplitudes between the adjacent SC layers can take either the zero-phase or the pi-phase [23–30]. To explain the above-mentioned effects, a model of a homogeneous exchange interaction in FM is used and the uniform magnetic moments are assumed to exist throughout the sample and orient in the specific direction. In the diffusive regime where the electron mean free path is much smaller than the superconducting correlation length, this type of magnetization causes the pair function in theq SC to penetrate into the FM for a short distance of the order of the exchange length ξI = Df /I, where Df is the diffusion constant and I is the exchange field strength. The pair amplitude oscillates spatially as a result of the broken time-reversal symmetry of the spin singlet pair. However, if the magnetization is inhomogeneous and its direction does not point along the spin quantization axis, the situation changes drastically. In this case a new type of the triplet condensate arises [6]. Apart from the triplet component with the zero spin projection of Cooper pairs Sz = 0 which has the short coherence length ξI like the singlet condensate, there are also other triplet components with exchange field and can nonzero spin projections Sz = ±1 which are independent of theq penetrate into the FM over a long range scale of the order of ξf = Df /2πTc0, where Tc0 is the isolated superconducting critical temperature. This type of the triplet condensate can survive in the diffusive metals when the following conditions are met: (i) the SC layer is a conventional s-wave spin singlet superconductor, (ii) the phonon mediated electron-electron pairing interaction is in the s-wave channel, (iii) the induced triplet pairing is independent of the momentum direction in order to be consistent with the diffusion process but odd in the Matsubara frequency, (iv) the short-range triplet component is generated when the

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exchange field is uniform in the FM, and (v) the long-range triplet components require an inhomogeneous magnetization to induce them. The idea of the odd frequency triplet condensate was first suggested a long time ago by Berezinskii [31] and later it has been applied to study the SC/FM systems [32, 33]. The interplay between superconductivity and inhomogeneous magnetization was first considered by Anderson and Suhl [34]. Their hypothesis suggested a possible mechanism for the existence of superconductivity in a domain-like magnetic structure called the cryptoferromagnetic state. A possibility of this state was investigated theoretically by Buzdin and Bulaevskii [35] and experimentally by M u ¨hge et al. [36] in an attempt to explain the observed magnetically dead layer by magnetization measurements. Because the suppression of superconductivity is affected by the destructive influence of the exchange field, the magnetic moments may be weakened in some regions inside the FM layer, so the effective exchange field is a minimum and remains finite, hence the superconductivity can coexist with the cryptoferromagnetic state in the system. Another aspect of the cryptoferromagnetic state has been developed in the context of domain wall superconductivity where the magnetic structure consists of a domain with alternate regions of fixed magnetization and each domain is separated by a wall. Inside the wall the region is characterized by the presence of the inhomogeneous magnetization, and the long-range triplet components can arise inevitably. The widely used model of the domain wall structure can be catalogued at least into two types, depending on the geometry of the magnetization vector lying in the wall. A Bloch-type wall is typically modeled by the magnetization vector lying in the plane of the FM layer and rotating with increasing length in the perpendicular direction. This model was considered in Refs. [32, 33, 37]. Another type is a N´eel wall, where the magnetization rotates in the plane and is constant in the direction normal to the film. This structure was analyzed theoretically in Refs. [38–42]. Physically, this latter model is more relevant for the thin film system and might be the basic mechanism responsible for the enhancement of the superconductivity in the presence of domain walls during the switching on of the magnetization [43, 44]. Note that all the previous theoretical studies were investigated within the framework of the quasiclassical Green function and the system is in the dirty limit [45–48]. Our particular interest here is paid to the N´eel-type domain wall. In Refs. [40, 41], the simple N´eel-type domain wall structure was considered in the limit of an infinite wall width i.e., there is only the spiral magnetic order rotating continuously in the FM plane. Surprisingly, their analysis showed the absence of the long-range triplet components. Subsequently a more realistic N´eel-type magnetic structure has been investigated in Ref. [39], and the study agreed with Refs. [40, 41]. It is worth mentioning that the inhomogeneity of the magnetic moments unavoidably induces the long-range triplet components [6] and causes the modulation of the pair amplitude in the transverse direction in a form of the FFLO (Fulde-Ferrel-Larkin-Ovchinnikov) phase [49, 50]. Yet, to the best of knowledge, there is no theoretical study of the influence of the pair breaking effects and the inhomogeneous magnetization on the long-range triplet superconductivity. The motivation of this chapter is to reinvestigate the physics of the SC/FM bilayer with emphasis on the possible formation of the long-range triplet pair. This chapter is organized as follows. In Section 2, we present a formulation of the generalized triplet proximity effect by extending the Takahashi-Tachiki theory [51] within the de Gennes’s correlation

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164

T. Rachataruangsit

function approach [52, 53], so the spin-orbit scattering and the magnetic impurity scattering can be included in a phenomenological way [54]. The obtained Usadel transport-like equations are found to agree perfectly with those of Refs. [40, 41] in the absence of the pair breaking scatterers. In Section 3, we study the model of the inhomogeneous N´eel type spiral exchange field in the FM/SC bilayer and calculate exactly the secular equation to determine Tc as a function of a rotating spiral wave vector and an in-plane momentum which represents the modulation of the pair amplitude in the transverse direction. In Section 4, we perform the numerical calculations of the Tc variations to show the behavior of the induced long-range triplet components. Finally, conclusions are presented in Section 5.

2.

General Formalization

In this section we use the generalized de Gennes-Takahashi-Tachiki theory which was developed by Auvil et al. [54] to derive the FM/SC odd triplet proximity effect. According to the de Gennes’ correlation function formalism, the motion of the normal electrons at temperature T is controlled by the diffusion process. Suppose the system is characterized by the position-dependent material parameters such as the density of states N (r), the diffusion coefficient D(r), and the phonon mediated electron-electron pairing interaction V (r) in the s-wave channel. Near the second order phase transition the superconducting order parameter ∆(r) in the s-wave state takes the linearized form [51] ∆(r) = πT N (r)V (r)

XZ

d3sQω (r, s)∆(s),

(1)

ω

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where ω = (2n + 1)πT, with n = 0, ±1, ±2, . . ., and the scalar kernel Qω (r, s) = −

X X φ∗ (s,γ)ργλφ∗ (s,λ)φN (r, α)ραβ φM (r,β) 1 M N , 2πN (r) N,M αβγλ (ξN − iω)(ξM + iω)

(2)

is expressed in terms of the bilinear products of the wave functions. The coordinates (r, α) refer to the position and spin, respectively, while the index N denotes the quantum state. In general, N = (n, σ) specifies the degree of freedom which comes from the translational part and, especially, in an action of the spin-dependent potential, the spin part. ρ = iσy is the spin singlet matrix with σy is the second Pauli matrix. The energy ξN , measured from odinger equation the Fermi level EF , is the eigenvalue of the Schr¨ ˆ 0 φN (r,α) + H

X

ˆαβ φN (r,β) = ξN φN (r,α). U

(3)

β

ˆ =H ˆ 0 + Uˆ is the total Hamiltonian which is composed of the unperturbed part Therefore H ˆ 0 = 1 (p − e A)2 − EF , H 2m c

(4)

where the electromagnetic coupling is included through the vector potential A(r), and the interaction Hamiltonian ˆm + U ˆso , ˆ =U ˆex + U (5) U Superconducting Cuprates : Properties, Preparation and Applications, Nova Science Publishers, Incorporated, 2008. ProQuest Ebook Central,

Influence of Pair Breaking Effects...

165

is the combination of the spin-dependent terms. The first term, the spin exchange inˆex = −I(r).ˆ teraction U σ represents the electron spins coupling to the vector exchange field I(r). The last two terms are the contributions coming from the magnetic impurity ˆm = P Γ(r, ri)Si .ˆ U σ, Γ(r, ri) represents the potential strength of the impurity atom at i ˆso = ∇uso (r).(ˆ site ri due to the localized spin Si , and the spin-orbit scattering U σ × p), where uso (r) is the spin-orbit potential. Note that the ordinary potential scattering needs not be considered because it has no influence on the depairing effect. Instead of solving (3) to find the eigenfunctions and the corresponding eigenvalues, let us express the wave function in the Dirac’s bra-ket notation φN (r,α) = hr, α|N i, then the scalar kernel Qω (r,s) (3) can be written as Qω (r,s) = +

ˆ †δ(R − r)|M ihM |δ(R − s)K|N ˆ i X hN |K 1 , 2πN (r) N,M (ξN − iω)(ξM + iω)

(6)

ˆ = iσy C is the time-reversal operator, C being the complex conjugation operator, where K and R is the position operator. Introducing the energy-dependent ξ correlation function in a matrix form gˆξ (r, s,Ω) =

X

ˆ †δ(R − r)|M i δ(ξ − ξN )δ(Ω + ξN − ξM )hn|K

n,M

ˆ ×hM |δ(R − s)K|ni,

(7)

with Ω = ξM − ξN is the correlated energy, then (6) can be written as

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Qω (r, s) =

1 ˆ TrQω (r, s), 2

(8)

ˆ ω (r, s) is given by where Tr means the diagonal summation, and the matrix kernel Q ˆ ω (r, s) = Q

1 πN (r)

Z

dξdΩ

gˆξ (r, s,Ω) . (ξ − iω)(ξ + Ω + iω)

(9)

In (9) the energy integrals cannot be evaluated immediately, one therefore performs the Fourier transform of gˆξ (r, s,Ω) Z ∞ dt iΩt e gˆξ (r, s,t). gˆξ (r, s,Ω) = −∞

2π

(10)

Substituting (10) into (9) and treating the relevant particle correlation near the Fermi surface ξ ≈ 0, then the energy integrations over ξ and Ω have two nonzero terms; t > 0, ω < 0, and t < 0, ω > 0, and (9) becomes ˆ ω (r, s) = Q

2 πN (r)

Z ∞

dte−2|ω|t gˆξ=0 (r, s,t).

(11)

0

While the inverse Fourier transform of (10) is, when using (7) gˆξ=0(r, s,t) =

X

ˆ †(t)K(0), ˆ δ(ξN )hn|δ(R(t) − r)δ(R(0) − s)|niK

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(12)

166

T. Rachataruangsit

ˆ where K(t) and R(t) are the time-dependent operators in the Heisenberg picture. Taking the t = 0 limit, (12) yields the initial condition gˆξ=0 (r, s,t = 0) = σ0 δ(r − s)N (r),

(13)

here σ0 is the unit Pauli matrix. Having obtained (12) the next task is to determine the diffusion equation of gˆξ=0 (r, s,t). ˆ † (t). First, δ(R(t)−r) The time derivative of (12) deals with the operators δ(R(t)−r) and K may be interpreted as a particle density located at a point r and obeys the continuity equation in the random-walk process ∂ δ(R(t) − r) = D(r)∇2δ(R(t) − r). ∂t

(14)

ˆ †(t) satisfies the Heisenberg’s equation of motion Second, K ∂ ˆ† ˆ K ˆ †(t)], K (t) = i[H, ∂t

(15)

ˆex + U ˆm + U ˆso as stated already. ˆ =H ˆ0 + U with the total Hamiltonian H Since it is well known that a Cooper pair is formed in a zero-momentum spin singlet state as a result of the electrons pairing between the state (p, σ) and the time reversed state ˆK ˆ † = σ0 . In the presence (−p, −σ), so the time-reversal operator satisfies the relation K of an external perturbation the destructive effect on the superconducting phase arises when the time-reversal invariance is broken. To compute the commutator in (15), we consider the ˆ For H ˆ 0 , only the electromagantisymmetric perturbation terms of the total Hamiltonian H. netic coupling term satisfies the requirement Copyright © 2008. Nova Science Publishers, Incorporated. All rights reserved.

ˆ = −(p.A + A.p). ˆ †(p.A + A.p)K K

(16)

ˆm , we have ˆex and U For U ˆex K ˆ = −U ˆex , ˆ †U K

(17)

ˆm K ˆ = −U ˆm . ˆ †U K

(18)

and ˆso , the time-reversal invariance is not broken For U ˆ †U ˆso K ˆ = +U ˆso . K

(19)

ˆex , the spin-orbit scattering may act as the pair breaker by However in the presence of U ˆ and its transpose K ˆ tr equals zero using the Auvil et al.’s trick [54] i.e., a combination of K ˆso can be taken into account as an antisymmetry perturbation. at the time t = 0. Thus U ˆ † , one has Combining all of the above statements to obtain the equation of motion of K ∂φ(t) ˆ † ∂ ˆ† K (t) = −i K (t) ∂t ∂t ˆ †(t)K(0) ˆ U ˆex K ˆ † (t) + K ˆ †(0)) ˆex K +i(U −

2 ˆ† 2 ˆ† ˆ †tr(t)), (K (t) + K K (t) − τm τso

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(20)

Influence of Pair Breaking Effects... where 2e φ(t) = − c

Z r(t)

A(l).dl,

167

(21)

r(0)

is the time-dependent phase factor as obtained in the semiclassical treatment. The scattering rates, the magnetic impurity τm and the spin-orbit τso , are treated in a phenomenological manner and may be related to an averaging of the Matsubara Green’s functions over the impurity sites in the Born Approximation as follows [55] 1 τm 1 τso

= =

Z

2π dΩ nimp N (0)S(S + 1) |um |2 , 3 4π Z 2π dΩ nimp N (0) |uso |2 sin2 θ, 3 4π

(22) (23)

with nimp is the impurity concentration in the dilute limit, S denotes the isotropic spin of the localized magnetic moment, and N (0) is the density of states at the Fermi level. Combining (14) and (20) to obtain the gauge-invariant form of the diffusion equation for the matrix correlation function gˆξ=0 (r, s,t), we have ∂ 2 gˆξ=0(r, s,t) = D(r)Π2gˆξ=0 (r, s,t) − gˆξ=0 (r, s,t) ∂t τm σ) −i(I(r).ˆ σgˆξ=0 (r, s,t) + gˆξ=0 (r, s,t)I(r).ˆ 2 tr − (ˆ gξ=0 (r, s,t) − σy gˆξ=0 (r, s,t)σy ), τso

(24)

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with Π = ∇ − (2ie/c)A(r) is the gauge-invariant differential operator. Utilizing (11), (13), ˆ ω (r, s) in the diffusive and (24) to yield the differential equation for the matrix kernel Q regime, one has 

|ω| +

+



  1 1 ˆ ω (r, s) + i I(r).ˆ ˆ ω (r, s) + Q ˆ ω (r, s)I(r).ˆ − D(r)Π2 Q σQ σ τm 2 2

 1 ˆ ˆ tr(r, s)σy = σ0 δ(r − s). Qω (r, s) − σy Q ω τso

(25)

The Usadel transport-like equations can be obtained quite simply from (25) by defining the matrix pair amplitude Fˆ (r,ω) =

Z

ˆ ω (r, s)iσy ∆(s), d3 sQ

(26)

and the result is given by 



 1 1 i |ω| + − D(r)Π2 Fˆ (r,ω) + I(r) · σ ˆ Fˆ (r,ω) + Fˆ (r,ω)σy I(r) · σ ˆ σy τm 2 2  1 ˆ + F (r,ω) + Fˆtr (r,ω) = iσy ∆(r). (27) τso

Usually, the matrix function Fˆ (r,ω) has the following components Fˆ =

F↑↑ F↑↓ F↓↑ F↓↓

!

.

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(28)

168

T. Rachataruangsit

However, it is useful to expand Fˆ (r,ω) along the superfluid pairing states [56] Fˆ (r,ω) = [Fs (r,ω) + Ft (r,ω).ˆ σ] iσy ,

(29)

where Fs is the scalar spin singlet pair amplitude and Ft = (Ftx , Fty , Ftz ) is the vector spin triplet pair amplitude. In the (S 2, Sz ) spin representation of the composite electrons, the state S = 0 corresponds to the spin singlet state with the zero spin projection on the quantization axis Sz = 0, while the S = 1 case defines the spin triplet state with three values of the spin projections Sz = 0, ±1. Thus Fs and Ftz are the most probable probability amplitudes of finding a paired electron on the spin quantization axis. The other ∓Ftx +iFty , which corresponds to the spin projections Sz = ±1, is possible if the exchange field points in arbitrary directions. Inserting (29) into (27), we have the scalar and the vector equations 



1 1 − D(r)Π2 Fs (r, ω) + iI(r) · Ft (r, ω) = ∆(r), τm 2   1 1 2 |ω| + − D(r)Π2 + Ft (r, ω) + iI(r)Fs(r, ω) = 0. τm 2 τso

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|ω| +

(30) (31)

The obtained formulas are the generalized Usadel transport-like equations which include the perturbation effects on the superconducting phase, namely, the orbital diamagnetism, the vector exchange field, the magnetic impurity, and the spin-orbit scattering. Among the four interactions, the spin exchange interaction is most important because it breaks the singlet pair and generates the triplet pair in compensation. Also, the spin-orbit interaction ˆso preserves the time-reversal can have the destructive effect on the singlet pair, though U symmetry, by coupling together via the triplet components. As a result, the strength of the spin exchange potential is reduced considerably. Note that this mechanism was explained by Demler et al. [57] and the effect of the spin-orbit interaction on the triplet superconductivity has already been investigated by Bergeret et al. [6] within the context of the quasiclassical Green’s functions method in the Keldysh representation. Recently Champel and Eschrig [40, 41] derived the reduction formulas of (30) and (31) by considering only the spin exchange interaction. It is worth noting that in their approach the quasiclassical Green’s functions in the Matsubara representation require an extra condition of the particle-hole symmetry in equilibrium to produce the proper form of the Usadel transport-like equations, apart from the normalization condition of the energy integrated 4 × 4 matrix Green’s function. Another triplet pairing basis was presented by Fominov et σσz instead al. [66], which when transformed to the formal basis, one gets the term −Fˆ σz I.ˆ of the correct Fˆ σy I.ˆ σσy in (27). Thus their formulations cannot be correct when applied to a general case. In the limiting case, where the exchange field points in a particular direction, we choose the z axis as the spin quantization axis so that I(r).ˆ σ = I(r)σz ,

(32)

and introduce the auxiliary kernel Rω (r, s) through the relation 1 ˆ ω (r, s). Rω (r, s) = Trσz Q 2 Superconducting Cuprates : Properties, Preparation and Applications, Nova Science Publishers, Incorporated, 2008. ProQuest Ebook Central,

(33)

Influence of Pair Breaking Effects...

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Applying (8), (32), and (33) to (25), we obtain the coupled differential equations 



1 1 |ω| + − D(r)Π2 Qω (r, s) + iI(r)Rω(r, s) = δ(r − s), τm 2   1 1 2 2 |ω| + − D(r)Π + Rω (r, s) + iI(r)Qω(r, s) = 0, τm 2 τso

(34) (35)

which are the generalized de Gennes-Takahashi-Tachiki formulations as have been derived by Auvil et al. [54]. It can be seen that the application of (32) to (30) and (31) implies the equivalence between the de Gennes-Takahashi-Tachiki formalism and the Usadel equations through the relations Fs (r,ω) ∝ Qω (r, s) and Ftz (r,ω) ∝ Rω (r, s). Therefore the auxiliary kernel Rω (r, s) is indeed the triplet component with zero spin projection. Note that Fs and Ftz are incorporated into the Usadel equations and have been widely used to study the non-monotonic Tc behavior of FM/SC layered structures [22, 58–60]. To discuss the possibility of the triplet pair formation and its implications, we write down the self-consistent s-wave order parameter (1) in the form ∆(r)iσy = πT N (r)V (r)

X

Fˆ (r, ω),

(36)

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ω

ˆ iσy where the matrix pair amplitude (29) contains the antisymmetric iσy , and symmetric σ matrices. Then, according to the Pauli’s exclusion principle, the spatial functions ∆(r) and Fs (r) will have an even symmetry whereas Ft(r) possesses an odd symmetry. But Fˆ (r) is a linear combination of the singlet part and the triplet components, so it does not satisfy the spatial parity condition Fˆ (−r) = Fˆ (r). Nevertheless, by imposing the even spatial parity of Fˆ (r, ω) and taking the even frequency symmetry of Fs (r,ω) and the odd frequency symmetry of Ft(r, ω) into account, a new type of the spin triplet pair amplitude arises, and is called the Berezinskii’s odd frequency triplet condensate [31]. Thus, within this consideration, the mixed pairing states are possible and (36) becomes ∆(r) = 2πT N (r)V (r)

X

Fs (r, ω).

(37)

ω>0

The linearized self-consistent order parameter still possesses a conventional form and tells us that why there is only the spin singlet pairing responsible for the superconducting phase. However, the odd triplet condensate may also exist throughout the hybrid FM/SC proximity system. Hence the equations of the generalized odd triplet superconductivity proximity effect are (30), (31), and (37). In application to the layered structures, the material parameters in each layer are treated separately, and the pairing functions are connected by the KupriyanovLukichev boundary conditions at the interfaces [61, 62] ξs ∇Fˆ s = γξf ∇Fˆ f , ˆf · ∇Fˆ f , Fˆ s = Fˆ f − γb ξf n

(38)

∇Fˆ s(f ) = 0.

(39)

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The superscript s(f ) on the matrix pair amplitude Fˆ belongs to the SC(FM) layer, q ξs(f ) = Ds(f )/2πTc0 is the corresponding coherence length, or may be called the diffusion length, where Tc0 is the isolated superconducting critical temperature. The parameter γ = ρs ξs /ρf ξf is the resistivity mismatch between the two layers in contact with ρs(f ) is the individual resistivity. Usually, ρf > ρs or γ < 1, then the pairing induced in FM is always weaker than in SC, as a result of the pair leakage. γb is the boundary resistance which characterizes the pair jumping from SC into FM in the direction outward normal to the interface n ˆ f . The limit γb = 0 refers to the perfectly transparent interface. The existence of γb has been confirmed experimentally by Aarts et al. [62].

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3.

´ Inhomogeneous NEEL Type Exchange Field

In this section we demonstrate the manifestation of the odd frequency triplet pairing, in particular the long-range triplet component, by calculating the critical temperature Tc of a FM/SC bilayer exactly using the multimode method, and also investigate the influence of the magnetic impurity and the spin-orbit scattering on the induced superconductivity in the FM layer. Assuming that the dirty-limit conditions are held and there is no external magnetic field applying to the system, so the FM/SC proximity effect problems are well described by the generalized Usadel equations (30) and (31) with taking A = 0, and the self-consistent gap function (37). The bilayer structures are composed of two infinite slabs located in the y-z plane and choosing the x-axis as the direction of the propagation of pair amplitudes from SC (0 < x < ds ) to FM (−df < x < 0). Because of the two competing orderings, the interplay between superconductivity and ferromagnetism is feasible when the two orderings are spatially separated, that is in FM, there exists only the exchange potential whereas the pairing interaction and the order parameter are set equal to zero. However, the pair amplitude is permitted to induce superconductivity via the proximity effect. The model of the inhomogeneous exchange field is the spiral magnetic order of the N´eel type i.e., it rotates in an in-plane of the FM layer with a spiral wave vector Q, I(y) = I(0, sinQy, cosQy).

(40)

Within this configuration the Q = 0 limit corresponds to the homogeneous fixed exchange field which has the spin magnetization vector pointing constantly along the z-direction. This limiting case has been thoroughly investigated by many authors [5]. Here the cryptoferromagnetic model, or the domain-like magnetic structure, corresponds to the finite spiral wave vector Q. We are interested to determine the condition for the occurrence of the induced long-range triplet condensate in the FM layer. Since we have assumed the x-axis to be the propagation direction of the pair amplitudes, then the effective equation is needed. Treating the problem in one dimension by introducing the transformations Fs (r,ω) = eipy Fs (x, ω) i(p±Q)y

Ftz (r,ω)±iFty (r,ω) = 2e

Ft± (x, ω),

(41) (42)

where an input parameter p varies in the range 0 < p < Q and represents the modulation of the pair amplitudes in an in-plane direction. In the same manner, the separable form (41) Superconducting Cuprates : Properties, Preparation and Applications, Nova Science Publishers, Incorporated, 2008. ProQuest Ebook Central,

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also holds true for the pair potential ∆(r). Inserting (40) into (30) and (31), and using (41) and (42), the effective one dimensional equations are 



1 1 − D(∂x2 − p2) Fs (x) + iI (Ft+ (x) + Ft− (x)) = ∆(x), τm 2   1 2 1 i ω+ + − D(∂x2 − (p ± Q)2 ) Ft± (x) + IFs (x) = 0, τm τso 2 2 X ∆(x) = 2πT N (x)V (x) Fs (x, ω). ω+

(43) (44) (45)

ω>0

Here the Ftx component has been disregarded because it is not coupled to Fs . The boundary conditions still possess the canonical forms as follows ξs ∂x Fss (x = 0) = Wf p(Q, ω)Fss (x = 0), ∂x Fss (x

(46)

= ds ) = 0,

(47)

where Wf p(Q, ω) =

γξf ∂x Fsf (x = 0) Fsf (x = 0) + γb ξf ∂x Fsf (x = 0)

.

(48)

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The interface boundary function Wf p(Q, ω) contains the information of the singlet pair in the FM. Nevertheless, as can be seen from (43) and (44), the singlet and triplet components are unavoidably coupled together. Thus the other sets of the boundary conditions are

and f (0) = γξf ∂x Ft±

s (ds ) = 0, ∂x Ft±

(49)

f ∂x Fs,t± (−df ) = 0,

(50)

s  ξs ∂x Ft± (0)  f f F (0) + γ ξ ∂ F (0) . b f x t± t± s (0) Ft±

(51)

Next we consider the two layers separately. In SC (0 < x < ds ) we have  



2 ∂x2 − ksp Fss (x, ω) = −



1 ∆(x), πTc0ξs2

2 s ∂x2 − ktp± Ft± (x, ω) = 0,

∆(x) = 2πT λs

(52) (53)

X

Fss (x, ω),

(54)

ω>0

where λs = Ns Vs is the dimensionless BCS coupling constant, and the frequencydependent wave vectors are ksp = ktp± =

s s

ω + p2 , πTc0ξs2

(55)

ω + (p ± Q)2 . πTc0ξs2

(56)

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In writing (52) and (53), we have put I = 0 in (43) and (44) since there is no exchange potential in the SC, but the pair amplitudes can be modulated in the transverse direction. Using the outer boundary condition (49), the solution of (53) is s Ft± (x, ω) = (Const.)× cosh[ktp± (x − ds )].

(57)

This shows that the triplet components spatially decay in the SC layer. In order to determine the critical temperature Tc we have to solve (52) and (54) selfconsistently subject to the boundary conditions (46) and (47). Let us consider the diffusive kernel Kω (x, x0) instead of Fss (x, ω), whose differential equation has the delta function source   1 2 ∂x2 − ksp Kω (x, x0) = − δ(x − x0 ), (58) πTc0ξs2 and the boundary conditions have the same forms as (46) and (47). Therefore Fss (x, ω) is given by an integral equation of kernel Kω (x, x0), Fss (x, ω)

=

Z ds 0

dx0Kω (x, x0)∆(x0).

(59)

Employing the eigenfunction expansion method by expanding Kω (x, x0), and ∆(x) in the Fourier cosine series, due to the even spatial parity of the singlet function, one has ∞ X

Kω (x, x0) = ∆(x) =

m=−∞ ∞ X

Kω (qm , x0) cos(qm x),

(60)

∆(qm ) cos(qm x),

(61)

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m=−∞

with the eigenmode qm = mπ/ds. Transformation of (58) to obtain Kω (qm , x0) through 1 Kω (qm , x ) = ds 0

Z ds

dxKω (x, x0) cos(qm x),

(62)

0

one finds that the self-consistency condition (54) in Fourier space is given by ∞ X

∆(qm ) = 2πT λs

m0 =−∞

∆(qm0 )

X Z ds ω>0 0

dx0 Qω (qm , x0) cos(qm0 x0 ).

(63)

Having obtained Kω (qm , x0), the nontrivial solution of (63) is found to obey the secular equation T X Lmm0 (ω)k = 0, (64) det kδmm0 − 2λs Tc0 ω>0 where Lmm0 (ω) = Ω(ω) =

δmm0 Ω(ω) − 4 2 2 , 2 + q2 ) 2 2 ξs2(ksp ξ (k + q m s sp m )(ksp + qm0 )

(65)

Wf p(Q, ω) , (ds/ξs )(1 + Wf p(Q, ω)/Asp)

(66)

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Asp = ksp ξs tanh(kspds ),

(67)

Atp± = ktp± ξs tanh(ktp±ds ).

(68)

for the singlet case and as for the triplet case The remaining task is to find the explicit form of the interface boundary function Wf p(Q, ω). In FM (−df < x < 0) we have " "

#

−∂x2

 1 1 2i  f f 2 f + (ω + ) + p F (x) + F (x) + F (x) = 0, s t− τm πTc0ξf2 ξI2 t+

(69)

−∂x2

1 1 2 i f + (ω + + ) + (p ± Q)2 Ft± (x) + 2 Fsf (x) = 0. πTc0ξf2 τm τso ξI

(70)

#

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Therefore the two length scales characterizing the induced superconductivity in FM are q q the diffusion length ξf = Df /2πTc0, and the exchange length ξI = Df /I which is originated from the broken time-reversal symmetry. Naturally, I > Tc0 even for a weak ferromagnet, so the exchange length is always shorter than the diffusive coherence length or ξI < ξf . The parameters 1/τm and 2/τso describe the magnetic impurity and the spinorbit scattering in the ferromagnetic alloys. The non-zero exchange field strength (I6=0) makes Fs and Ft± are coupled together. So, the pair amplitudes do not just decay spatially but also oscillate perpendicularly to the layer plane. It is not hard to see that the long-range triplet pair is independent of ξI and has a long penetration length inside the FM. Relating the triplet to the singlet pairs through the following relations f (x, ω) = αFsf (x, ω), Ft+

(71)

f Ft− (x, ω) = βFsf (x, ω),

(72)

where the unknown parameters α and β can be determined by substituting (71) and (72) into the set of coupled equations (69) and (70), one obtains kf2 p =

1 1 2i (ω + ) + p2 + 2 (α + β), τm πTc0 ξf2 ξI

(73)

Equating the spectrum kf2 p of each equation, as a result, three roots kf2 p = (kf2 p0, kf2 p±) are obtained, one of them corresponds to the long-range mode and the others are the short-range ones. For the long-range solution kf2 p0 =

1

(ω πTc0ξf2

αp0 = −βp0 =

+

1 2 + ) + p2 + Q2 , τm τso

1 , 2iζ

(74) (75)

with ζ = pQξI2. The spectrum kf2 p0 does not depend on the exchange field strength I and provides that the pair amplitudes have a long penetration length in a decay manner of the Superconducting Cuprates : Properties, Preparation and Applications, Nova Science Publishers, Incorporated, 2008. ProQuest Ebook Central,

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order of the characteristic length 1/kf p0. The pair-breaking scattering rates τm and τso give an identical feature as well as the in-plane momentum p and the spiral wave vector Q do, in reducing the decay length 1/kf p0. Therefore the long-range triplet amplitudes αp0 and βp0 require the inhomogeneous exchange field and the in-plane momentum p to produce them. For the short-range solution kf2 p = αp = βp =

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ηp =

1

(ω πTc0ξf2

+

1 2i ) + p2 + 2 ηp,− , τm ξI

1 , 2(ηp + iζ) 1 , 2(ηp − iζ)

q

2 + iη , 1 − ζ 2 − ηso so

(76) (77) (78) (79)

with ηso = η + 1/Iτso, η = (QξI )2/4, and  = ±. Now kf2 p± is a complex quantity, so the corresponding characteristic length 1/kf p± contains the penetration length and the oscillation period, which are given by the real part Re(1/kf p±) and the imaginary part Im(1/kf p±), respectively. At this stage we may summarize the obtained results as follows; (i) the inhomogeneous exchange field and the in-plane momentum are the important ingredients for the appearance of the induced long-range triplet, (ii) the magnetic impurity scattering flips the spins of paired electrons, both the singlet and the triplet states, so the penetration lengths are reduced, and (iii) the spin-orbit scattering displays two distinct features simultaneously, it gives the extra decay of long-range mode, while for the short-range modes the exchange field strength is dissipated by coupling with the spin-orbit interaction and as a result the oscillation period of the pair amplitudes is increased. In the case of the modulationless in-plane momentum p = 0, or ζ = 0, and without the pair-breaking effects 1/τm = 1/τso = 0, the eigenvalues (74) and (76) recover the results of Champel and Eschrig [40, 41], but the difference between their work and ours is that in our treatment the long-range mode does not exist because the triplet pair amplitude has been eliminated from the superconductivity nucleation region. Furthermore, in the case of the homogeneous fixed exchange field Q = 0 with a finite spin-orbit scattering rate, but 1/τm = 0, it is necessary to take p = 0 because the rotating magnetization modulates the pair amplitudes in the direction parallel to the layer plane. We find that our eigenvalues are identical with those of Bergeret et al. [6, 63] but the long-range mode in our formalism is excluded. The short-range eigenvalue reproduces our previously obtained formula [64] which covers more ground than the treatment in Ref. [65]. When applying the outer boundary condition (50), the pair amplitude solutions can be written as Fsf (x, ω) =

X

Cj cosh[kf pj (x + df )],

(80)

αpj Cj cosh[kf pj (x + df )],

(81)

βpj Cj cosh[kf pj (x + df )].

(82)

j=0,± f

Ft+ (x, ω) =

X

j=0,± f (x, ω) = Ft+

X

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Therefore the singlet-and-triplet pair functions are expressed as a linear combination of the long-range and short-range modes. The coupled character seems quite difficult to be realized separately. However, the Tc dependence on the FM layer thickness is the most interesting prediction as the signature of the induced long-range triplet superconductivity. We are in a position to find the mechanism responsible for the occurrence of the long-range triplet components. In what follows, we introduce Bf pj =

γ γb + coth(kf pj df )/kf pj ξf

,

(83)

and making use of (81), (82), and the boundary condition (51), one obtains X

αpj (Bf pj + Atp+ )(cosh(kf pj df ) + γb kf pj ξf sinh(kf pj df ))Cj

= 0,

(84)

βpj (Bf pj + Atp− )(cosh(kf pj df ) + γb kf pj ξf sinh(kf pj df ))Cj

= 0.

(85)

j=0,±

X

j=0,±

The function Wf p(Q, ω) can now be evaluated by using (80) and (48), we have P

Cj γkf pj ξf sinh(kf pj df ) . j Cj [cosh(kf pj df ) + γb kf pj ξf sinh(kf pj df )]

Wf p (Q, ω) = P

j

(86)

Calculating C± as a function of C0 and then substituting them back into (86), a tedious algebra leads to the desired result Wf p(Q, ω) =

Mf p0Bf p0 + Mf p+ Bf p+ + Mf p− Bf p− , Mf p0 + Mf p+ + Mf p−

(87)

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where Mf p0 = −ζ 2 (ηp+ + ηp− )(S+ + S− ) − iζ(ηp+ ηp− − ζ 2 )(S+ − S− ), Mf p± =

2 (ηp±

2

+ ζ ) [(ηp∓ − iζ)P± + (ηp∓ + iζ)R±] ,

(88) (89)

P± = (Bf p0 + Atp± )(Bf p∓ + Atp∓ ),

(90)

R± = (Bf p0 + Atp∓ )(Bf p∓ + Atp± ),

(91)

S± = (Bf p+ + Atp± )(Bf p− + Atp∓ ).

(92)

Note that the interface boundary function Wf p(Q, ω) is real. It is interesting to examine (87) when p = 0. In this limit the function Wf p (Q, ω) reduces to Wf,p=0(Q, ω) =

(η+ + η− )Bf + Bf − + (η+ Bf + + η− Bf − )At , η+ Bf − + η− Bf + + (η+ + η− )At

(93)

where η± , Bf ± , and At are, respectively, the short-hand notations of ηp=0,±, Bf,p=0,± , and At,p=0,± . The obtained formula (93) clearly shows the absence of the long-range mode, even when the exchange field is inhomogeneous i.e., it rotates spatially in an in-plane of the FM layer with a spiral wave vector Q. This means that the induced superconductivity is caused by the singlet pair and the short-range triplet components, in the same manner as in the homogeneous exchange field case. Hence the in-plane momentum p is an important

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parameter to indicate the occurrence of the long-range triplet condensate. At this point we are aware of the theoretical results presented in Refs. [66, 67] that they might be incorrect. In those papers the authors studied the Tc of FM/SC/FM triple layers as a function of the relative orientations of the homogeneous exchange fields and reached the conclusions that the long-range triplet arises from intermediate orientations. The secular equation (64) in a simple form may be derived by treating in the Cooper-de Gennes thin films limit, or the so-called single-mode approximation. Under the assumptions that (i) the thickness of the SC layer is very thin compare to the corresponding diffusion length ds  ξs , and (ii) the frequency terms ω in the expressions for the eigenvalues kf p = (kf p0, kf p±) in FM are omitted, so kf2 p0 ≈

1 1 2 ( + ) + p2 + Q2 , 2 πTc0ξf τm τso

kf2 p± ≈

1 1 2i + p2 ± 2 ηp,∓ . 2 πTc0ξf τm ξI

A trivial analytical approach leads to the Abrikosov-Gorkov like-formula [68] as follows 1 1 Tc0 Wf p(Q) Tc = ψ( ) − ψ( + [(pξs )2 + ]), (94) ln Tc0 2 2 2Tc ds /ξs where ψ(x) is the digamma function, and Wf p(Q) is an approximation of (87), by neglecting all terms containing ω in Wf p(Q, ω), which reads explicitly as

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Wf p(Q) = Nf p =

(ηp+ + ηp− )Bf p+ Bf p− , ηp+ Bf p− + ηp− Bf p+ + ζ 2 Nf p ηp+ Bf p+ + ηp− Bf p− (ηp+ + ηp− )Bf p+ Bf p− − . ηp+ ηp− ηp+ ηp− Bf p0

(95) (96)

It is obvious that the long-range mode remains present in this approximation. Consequently, the oscillatory Tc behavior is less pronounced and tends towards a monotonic decay. In a more realistic case, the numerical analysis is needed to be performed in detail.

4.

Numerical Result

The numerical calculations of the critical temperatures Tc versus the spiral wave vector Q of a FM/SC bilayer are shown in Figs. 1-3, to investigate the possibility of the existence of the long-range triplet superconductivity by using the secular equation (64) and associated functions as given by (65), (66), and (87). For simplicity we assume that the metals have the same diffusion length ξs = ξf , in which the SC layer is quite thick (ds/ξs = 3) while the FM layer with the weak exchange field strength (I/πTc0 = 10) is supposed to be thin (df /ξf = 0.5). It is found that the thinner SC layer (ds /ξs ≤ 1) makes the superconductivity vanishes completely. The low value of the resistivity mismatch between the two layers (γ = 0.2) represents the weak proximity effect and the zero interface boundary resistance (γb = 0) characterizes the smoothness of the pair amplitudes penetration from the SC into the FM.

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Figure 1. Reduced critical temperature tC = Tc /Tc0 as a function of dimensionless rotating wave vector Qξs for several values of p/Q. I/πTc0 = 10, ds/ξs = 3, df /ξf = 0.5, γ = 0.2, γb = 0, ξs = ξf , and 1/Iτm = 1/Iτso = 0. In Fig. 1 the variations of the reduced critical temperature tC = Tc /Tc0 versus the dimensionless rotating wave vector Qξs show many significant features depending on the in-plane momentum p/Q. In our treatment the momentum p represents the transverse modulation of the pair amplitudes in the whole structure so it describes the inhomogeneous superconducting state corresponding to the Fulde-Ferrel-Larkin-Ovchinnikov phase. As can be seen clearly for the p = 0 curve, tC rises significantly as Qξs increases and it reaches the highest critical temperature, here the long-range triplet is absent from the nucleation of superconductivity as already stated in (93). Meanwhile, other curves (p6=0) describe the inhomogeneous superconducting state and it is connected to the onset of the long-range triplet superconductivity when the exchange field in the FM layer starts to rotate. We find that tC is very sensitive to the ratio p/Q which results in its rapid reduction with more pronounced non-monotonicity as p/Q increases. This indicates that the long-range triplet superconductivity is less energetically favorable than the localized pairs which are composed of the spin singlet and the short-range triplet components. However, the lower tC curves may be a possible candidate to explain an existence of the magnetically dead layer which corresponds to the cryptoferromagnetic state. Since in a real situation the superconductivity has always been destroyed by the exchange interaction. Fig. 2 illustrates the influence of the spin-orbit interaction on tC . Here we take 1/Iτso = 0.2. It is well known that when the spin-orbit interaction is introduced in the ferromagnetic material, the exchange field strength is reduced considerably which results in the weakening of the influence of the pair breaker. Regarding the complex wave vector of the short-range mode kf p±, which can be written as a combination of the decay length and the oscillation period. The spin-orbit interaction decreases the decay length and increases the oscillation period. So, the short-range components of the pair amplitudes decay in the FM layer faster with less oscillations. An enhancement of tC reveals clearly the important role of the spin-

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Figure 2. tC (Qξs ) curves with varying p/Q, 1/Iτso = 0.2, and other parameters are the same as in Fig. 1. orbit scattering. As the exchange field starts to rotate (Qξs  1) the in-plane momentum p/Q is very small so there exists only the induced short-range triplet components and as the result all curves merge together. Upon the onset of the long-range triplet component, the wave vector kf p0 implies an additional suppression of superconductivity, and this leads to the tC evolutions separate from each other as Qξs increases with the finite p/Q. In Fig. 3, the magnetic impurity scattering does not reduce the exchange field strength, unlike the spin-orbit scattering, but it decreases the decay length of both the long-range and the short-range components, and also increases the oscillation period. However, the uncoupling of the magnetic scattering with the exchange field strength makes it difficult to observe its influence. The evolutions of tC on the FM layer thickness df /ξf are shown in Figs. 4-7. Here the choice Qξs = 0.5 is selected to ensure the contribution of the long-range triplet component. Various characterizations of tC can be observed within a single FM layer thickness depending on the ratio p/Q, as shown in Fig. 4. The non-monotonicity is found for a certain range of p/Q, and the monotonic tC decay is obtained when p/Q = 0.5. This means that the greater long-range contribution provides the non-monotonic tC behavior becomes the monotonic decay. Fig. 5 shows the tc behavior under the influence of the spin-orbit scattering. The significant effect can be displayed when the FM layer thickness is not so thin. As expected, the presence of the spin-orbit interaction is antagonistic to the non-monotonic behavior, and tC is higher for the entire ranges of df /ξf . Note also that the non-monotonic character can be restored from the monotonic decay in the p/Q = 0.5 case. Figs. 6 and 7 demonstrate the influence of the magnetic impurity scattering on

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Figure 3. tC (Qξs ) curves with varying p/Q, 1/Iτm = 0.2, and other parameters are the same as in Fig. 1.

Figure 4. Reduced critical temperature tC as a function of the reduced ferromagnetic layer thickness df /ξf with varying p/Q, I/πTc0 = 10, ds/ξs = 2, γ = 0.2, γb = 0, Qξs = 0.5, ξs = ξf , and 1/Iτm = 1/Iτso = 0.

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Figure 5. tc (df /ξf ) curves for two values of 1/Iτso = 0, and 0.2. The values of other parameters are the same as in Fig. 4. tC (df /ξf ). In the perfect transparency interface case γb = 0, Fig. 6, the Cooper pairs can easily penetrate the FM layer since there is no barrier at the interface, and can be reflected back to the SC layer if the FM layer is thin enough. So, the rapid drop of tC as df /ξf increases is a direct consequence. This situation occurs until the tC drop is lowest at a certain thickness. As df /ξf increases away from this point, the conduction electrons, which result from the destruction of the Cooper pairs due to the exchange interaction, can recombine to form pairs again. Thus tC has tendency to increase effectively. When the dilute magnetic impurity ions were introduced in the FM layer, the additional effect takes place to destroy pairs. Then the decay length becomes shorter, both short-range and longrange modes, in the same manner as the spin-orbit interaction, and so results the more rapid decrease of tC for the very thin FM layer. The intersected curves show that the magnetic impurity scattering can lift tC higher in a narrow interval with a less difference. Although, the magnetic impurity scattering seems not be a promising mechanism to increase tC , yet when one allows the finite interface boundary resistance γb = 0.1, as shown in Fig. 7, the Cooper pairs are more confined in the FM layer. As df /ξf reaches a particular thickness, the tC reduction becomes saturated more quickly, compared between Figs. 6 and 7. Though, the tC difference remains the same in the p/Q = 0 case, but the long-range contribution provides the higher tC in the presence of the magnetic impurity scattering. Thus its influence is most visible in this case.

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Figure 6. tc (df /ξf ) curves for two values of 1/Iτm = 0, and 0.2. The values of other parameters are the same as in Fig. 4.

Figure 7. tc (df /ξf ) curves with γb = 0.1, and other parameters are the same as in Fig. 6. Superconducting Cuprates : Properties, Preparation and Applications, Nova Science Publishers, Incorporated, 2008. ProQuest Ebook Central,

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Conclusion

We have presented a formulation of the odd triplet superconductivity proximity effect within the de Gennes correlation function approach, and applied it to study the behavior of the diffusive FM/SC bilayered structure. The spin-orbit interaction and the magnetic impurity are also incorporated into the Usadel transport-like equation. We found that the spin-orbit interaction tends to break the singlet pair indirectly through the exchange interaction by mixing with it the triplet pairs. While the magnetic impurity flips all spin states. The longrange triplet pair correlation which is defined as the non-zero spin projection on the spin quantization axis is found to be generated in the presence of the inhomogeneous exchange field. We use the model of the N´eel type spiral magnetic order that rotates in the plane and has no contribution in the perpendicular direction to the interface. The linearized selfconsistent order parameter equation has been solved exactly in the multimode method, and the secular equation is derived to determine the dependence of the critical temperature Tc on the spiral wave vector Q and the ferromagnetic layer thickness df . The inhomogeneous superconducting state is characterized by the in-plane modulating wave vector p. We have shown that the induced long-range triplet arises when the pair amplitudes are modulated spatially in the transverse direction. Numerical results demonstrate the possibility of the cryptoferromagnetic state in favor of superconductivity and may be used to explain a possible origin of the magnetically dead layer. The graphs clearly show the evolution of Tc with respect to Q that it is enhanced greatly in the presence of the spin-orbit interaction with less oscillatory characters. The single ferromagnetic layer thickness is a necessary condition for observing various characterizations of Tc behavior. The magnetic impurity scattering with a non-zero boundary resistance can be realized within some ranges of df /ξf , when the long-range triplet pairs are attributed in the process, though Tc is lower than the localized pairs. We conclude that the induced long-range triplet pair condensate is feasible and may be observed through a FM/SC bilayer system.

Acknowlegments This work is supported by the Thailand Research Fund and the Commission on Higher Education. The author would like to acknowledge Prof. S. Yoksan for his encouragement throughout the work.

References [1] Deutscher, G.; de Gennes, P.G. In Superconductivity; Parks, R.D.; Ed.; Dekker: New York, 1969; Vol. 2, pp 1005-1034. [2] Bulaevskii, L.N.; Buzdin, A.I.; Kulic, M.L.; Panjukov, S.V. Adv. Phys. 1985, 34, 175. [3] Izyumov, Yu.A.; Proshin, Yu.N.; Khusainov, M.G. Phys. Usp. 2002, 45, 109. [4] Golubov, A.A.; Kupriyanov, M.Yu.; Il’ichev, E. Rev. Mod. Phys. 2004, 76, 411.

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[5] Buzdin, A.I. Rev. Mod. Phys. 2005, 77, 935. [6] Bergeret, F.S.; Volkov, A.F.; Efetov, K.B. Rev. Mod. Phys. 2005, 77, 1321. [7] Lyuksyutov, I.F.; Pokrovsky, V.L. Adv. Phys. 2005, 54, 67. [8] Buzdin, A.I.; Vedyayev, A.V.; Ryzhanova, N.V. Europhys. Lett. 1999, 48, 686. [9] Tagirov, L.R. Phys. Rev. Lett. 1999, 83, 2058. [10] Baladi´e, I.; Buzdin, A.I.; Ryzhanova, N.; Vedyayev, A.V. Phys. Rev. B 2001, 63, 054518. [11] Baladi´e, I.; Buzdin, A.I. Phys. Rev. B 2003, 67, 014523. [12] You, C.-Y.; Bazaliy, Ya.B.; Gu, J.Y.; Oh, S.-J.; Litvak, L.M.; Bader, S.D. Phys. Rev. B 2004, 70, 014505. [13] Krunavakarn, B.; Sritrakool, W.; Yoksan, S. Physica C 2004, 406, 46. [14] Halterman, K.; Valls, O.T. Phys. Rev. B 2005, 72, 060514(R). [15] Faur´e, M.; Buzdin, A.I.; Gusakova, D. Physica C 2007, 454, 61. [16] Gu, J.Y.; You, C.-Y.; Jiang, J.S.; Pearson, J.; Bazaliy, Ya.B.; Bader, S.D. Phys. Rev. Lett. 2002, 89, 267001. [17] Potenza, A.; Marrows, C.H. Phys. Rev. B 2005, 71, 180503(R).

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[18] Moraru, I.C.; Pratt, Jr. W.P.; Birge, N.O. Phys. Rev. Lett. 2006, 96, 037004. [19] Moraru, I.C.; Pratt, Jr. W.P.; Birge, N.O. Phys. Rev. B 2006, 74, 220507(R). [20] Rusanov, A.Yu.; Habraken, S.; Aarts, J. Phys. Rev. B 2006, 73, 060505(R). [21] Steiner, R.; Ziemann, P. Phys. Rev. B 2006, 74, 094504. [22] Radovi´c, Z.; Ledvij, M.; Dobrosavljevi´c-Gruji´c, L.; Buzdin, A.I.; Clem,J.R. Phys. Rev. B 1991, 44, 759. [23] Kuboya, K.; Takanaka, K. Phys. Rev. B 1998, 57, 6022. [24] Krunavakarn, B.; Yoksan, S. Physica C 2006, 440, 25. [25] Proshin, Y.N.; Zimin, A.; Fazleev, N.G.; Khusainov, M.G. Phys. Rev. B 2006, 73, 184514. [26] Barsic, P.H.; Valls, O.T.; Halterman, K. Phys. Rev. B 2007, 75, 104502. [27] Jiang, J.S.; Davidovi´c, D.; Reich, D.H.; Chein, C.L. Phys. Rev. Lett. 1995, 74, 314. [28] Jiang, J.S.; Davidovi´c, D.; Reich, D.H.; Chein, C.L. Phys. Rev. B 1996, 54, 6119. [29] Obi, Y.; Ikebe, M.; Fujishiro, H. Phys. Rev. Lett. 2005, 94, 057008. Superconducting Cuprates : Properties, Preparation and Applications, Nova Science Publishers, Incorporated, 2008. ProQuest Ebook Central,

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[30] Shelukhin, V.; Tsukernik, A.; Karpovski, M.; Blum, Y.; Efetov, K.B.; Volkov, A.F.; Champel, T.; Eschrig, M.; L¨ ofwander, T.; Sch¨ on, G.; Palevski, A. Phys. Rev. B 2006, 73, 174506. [31] Berezinskii, V.L. JETP Lett. 1974 , 20, 287. [32] Bergeret, F.S.; Volkov, A.F.; Efetov, K.B. Phys. Rev. Lett. 2001, 86, 4096. [33] Kadigrobov, A.; Shekhter, R.;Jonson, M. Europhys. Lett. 2001, 54, 394. [34] Anderson, P.W.; Suhl, H. Phys. Rev. 1959 , 116, 898. [35] Buzdin, A.I.; Bulaevskii, L.N. Sov. Phys. JETP 1988, 67, 576. [36] M¨ uhge, Th.; Garifyanov, N.N.; Goryunov, Yu.V.; Theis-Br¨ ohl, K.; Westerholt, K.; Garifullin, I.A.; Zabel, H. Physica C 1998, 296, 325. [37] Volkov, A.F.; Anischanka, A.; Efetov, K.B. Phys. Rev. B 2006, 73, 104412. [38] Bergeret, F.S.; Efetov, K.B.; Larkin, A.I. Phys. Rev. B 2000, 62, 11872. [39] Volkov, A.F.; Fominov, Y.V.; Efetov, K.B. Phys. Rev. B 2005, 72, 184504. [40] Champel, T.; Eschrig, M. Phys. Rev. B 2005, 71, 220506(R). [41] Champel, T.; Eschrig, M. Phys. Rev. B 2005, 72, 054523. [42] Houzet, M.; Buzdin, A.I. Phys. Rev. B 2006, 74, 214507.

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[43] Rusanov, A.Y.; Hesselberth, M.; Aarts, J. Phys. Rev. Lett. 2004, 93, 057002. [44] Buntinx, D.; Brems, S.; Volodin, A.; Temst, K.; Van Haesendonck, C. Phys. Rev. Lett. 2005, 94, 017204. [45] Eilenberger, G. Z. Phys. 1968, 214, 195. [46] Usadel, K.D. Phys. Rev. Lett. 1970, 25, 507. [47] Serene, J.W.; Rainer, D. Phys. Rep. 1983, 101, 221. [48] Alexander, J.A.X.; Orlando, T.P.; Rainer, D.; Tedrow, P.M. Phys. Rev. B 1985, 31, 5811. [49] Fulde, P. ; Ferrel, R.A. Phys. Rev. 1964, 135, A550. [50] Larkin, A.I.; Ovchinnikov, Y.N. Sov. Phys. JETP 1965, 20, 762. [51] Takahashi, S.; Tachiki, M. Phys. Rev. B 1986, 33, 4620. [52] de Gennes, P.G. Superconductivity of Metals and Alloys , Benjamin, New York, 1966. [53] Ketterson, J.B.; Song, S.N. Superconductivity; Cambridge University Press: Cambridge, 1999. Superconducting Cuprates : Properties, Preparation and Applications, Nova Science Publishers, Incorporated, 2008. ProQuest Ebook Central,

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[54] Auvil, P.R.; Ketterson, J.B.; Song, S.N. J. Low. Temp. Phys. 1989, 74, 103. [55] Maki, K. In Superconductivity; Parks, R.;Ed.; Dekker: New York, 1969; Vol. 2, pp 1035-1105. [56] Mineev, V.P.; Samokhin, K.V. Introduction to Unconventional Superconductivity ; Gordon and Breach: New York, 1999. [57] Demler, E.A.; Arnold, G.B.; Beasley, M.R. Phys. Rev. B 1997, 55, 15174. [58] Tagirov, L.R. Physica C 1998, 307, 145. [59] Fominov, Ya.V.; Chtchelkatchev, N.M.; Golubov, A.A. Phys. Rev. B 2002, 66, 014507. [60] Lee, N.; Choi, H.-Y.; Doh, H.; Char, K.; Lee, H.-W. Phys. Rev. B 2007, 75, 054521. [61] Kupriyanov, M.Yu.; Lukichev, V.F. Sov. Phys. JETP 1988, 67, 1163. [62] Aarts, J.; Geers, J.M.E.; Br¨ uck, E.; Golubov, A.A.; Coehoorn, R. Phys. Rev. B 1997, 56, 2779. [63] Bergeret, F.S.; Volkov, A.F.; Efetov, K.B. Phys. Rev. B 2003, 68, 064513. [64] Rachataruangsit, T.; Yoksan, S. Physica C 2007, 455, 39. [65] Oh, S.; Kim, Y.-H.; Youm, D.; Beasley, M.R. Phys. Rev. B 2000,63, 052501. [66] Fominov, Y.V.; Golubov, A.A.; Kupriyanov, M.Yu. JETP Lett. 2003, 77, 510.

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[67] L¨ ofwander, T.; Champel, T.;Eschrig, M. Phys. Rev. B 2007,75, 014512. [68] Abrikosov, A.A.; Gorkov, L.P. Sov. Phys. JETP 1961,12, 1243.

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In: Superconducting Cuprates Editor: Koenraad N. Courtlandt

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Chapter 6

THEORETICAL STUDY OF YBA2CU3O7 BEYOND THE BORN-OPPENHEIMER APPROXIMATION AND MIGDAL THEOREM: ANTIADIABATIC GROUND STATE INDUCED BY PHONON MODES COUPLING∗ Pavol Baňacký Faculty of Natural Science, Institute of Chemistry, Chemical Physics division, Comenius University, Mlynská dolina CH2, 84215 Bratislava, Slovakia and S-Tech a.s., Dubravská cesta 9, 84105 Bratislava, Slovakia

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Abstract The phonon modes that at coupling to electronic motion induce transition from adiabatic into antiadiabatic state, in which the Born – Oppenheimer approximation is not valid and electronic motion is dependent not only on nuclear coordinates but also on nuclear momenta, have been identified. It has been shown that for distorted lattice by Ag, B2g, B3g phonon modes, with the O(4), O(2), O(3) atoms displacements, there is a periodic shift of the saddle point of one of the CuO plane (d-pσ) band on the Γ-Y line at Y point of the Brillouin zone across the Fermi level from bonding to antibonding region. At this distortion, the nonadiabatic electron-phonon interactions stabilize the distorted lattice by – 34 meV/unit cell. Distorted lattice is characteristic by a specific - antiadiabatic fermionic ground state that is geometrically degenerate with fluxional arrangements of O(2), O(3) atoms in the CuO planes with the same ground state energy. The nonadiabatic mechanism at the lattice energy stabilization opens two asymmetric gaps in a and b direction in the originally metallic oneparticle spectrum of YBa2Cu3O7. Calculated critical temperature of the transition from adiabatic into antiadiabatic state is 92.8 K that is in good agreement with the experimental data for superconducting state transition. Present study has also revealed that in c direction a new gap that is considerably smaller than gaps in a, b directions should be identified. Appearance of van Hove singularity at Y point has also been calculated. For studied distortions, when saddle point of the d-pσ band in Y point approaches Fermi level, an abrupt change of the dispersion curve – low-Fermi energy electron velocity decrease at about 75 ∗

Reviewed by A.S.Alexandrov, Loughborough University, UK

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Pavol Baňacký meV below Fermi level is predicted. For YBa2Cu3O7, this effect should be detected close to Y point on the Γ-Y line, i.e. in the off-nodal direction, if the corresponding ARPES experiments are performed.

I. Introduction

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The ARPES study of high-Tc cuprates [1-3] and theoretical results of low-Fermi energy band structure fluctuation [4-7] in MgB2 indicate that electron coupling to pertinent phonon modes drive superconductors from adiabatic ( ω < E F ) into antiadiabatic state ( ω > E F ). At these circumstances, not only Migdal-Eliashberg approximation is not valid, but basic adiabatic Born-Oppenheimer approximation (BOA) does not hold. Antiadiabatic theory of the complex electronic ground state of superconductors that respect this fact has been discussed in [8]. From the theory follows that due to electron-phonon (EP) interactions that drive system from adiabatic into antiadiabatic state, symmetry breaking is induced and system is stabilized in antiadiabatic state at distorted geometry with respect to adiabatic equilibrium high symmetry structure. Stabilization effect in the antiadiabatic state is due to strong dependence of the electronic motion on the instantaneous nuclear kinetic energy, i.e. on the effect that is neglected on the adiabatic level within the BOA. Antiadiabatic ground state at distorted geometry is geometrically degenerate with fluxional nuclear configuration in the phonon modes that drive system into this state. It has been shown that while system remains in antiadiabatic state, nonadiabatic polaron – renormalized phonon interactions are zero in well defined k region of reciprocal lattice. It enables, along with geometric degeneracy of the antiadiabatic state, formation of mobile bipolarons that can move on the lattice as supercarriers without dissipation in a form of polarized inter-site charge density distribution. More over, it has been shown that due to EP interactions at transition to antiadiabatic state, kdependent gap in one-electron spectrum is opened. Gap opening is related to shift of the original adiabatic Hartree-Fock orbital energies and to the k- dependent change of density of states of particular band(s) at Fermi level. The shift of orbital energies determines in a unique way one-particle spectrum and thermodynamic properties of system. It has been shown that resulting one-particle spectrum yields all thermodynamic properties that are characteristic for system in superconducting state, i.e. temperature dependence of the gap, specific heat, entropy, free energy and critical magnetic field. The k-dependent change in the density of states close to Fermi level at transition from adiabatic (nonsuperconducting) into antiadiabatic (superconducting) state can be experimentally verified by ARPES or tunneling spectroscopy as spectral weight transfer at cooling superconductor from temperatures above Tc down to temperatures below Tc. The theory has been applied recently to the study of MgB2 superconducting state transition [6] and obtained results have been in a good agreement with the experimental data. In the present work, the results of superconducting state transition for the high-Tc cuprate YBCO – YBa2Cu3O7, are presented. The phonon modes, which at the coupling to electronic motion yield significant change in topology of the electronic band structure at Fermi level and induce transition from adiabatic into antiadiabatic state, have been identified. It has been shown that for the distorted lattice by Ag, B2g, B3g phonon modes, with the O(4), O(2), O(3) atoms displacements, there is periodic shift, “back and forth” fluctuation, of the saddle point

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(SP) of one of the Cu(2)-O(2)/O(3) plane (d-pσ) band on the Γ-Y line at Y point of 1st Brillouin zone from below to above-Fermi level position. For displacements 0.0315 Ao of the apical O(4) and 0.0217 Ao / 0.0221 Ao of O(2) / O(3) plane oxygens when the SP of this band in Y point is in the range of ± 72 meV at the Fermi level (value of the vibration energy of the CuO plane B2g, B3g phonon stretching modes), it has been calculated that the ground state energy of the distorted lattice on the adiabatic BOA level has been increased by + 170 meV/unit cell. However, nonadiabatic EP interactions result in net stabilization of such a distorted lattice by about - 34 meV/unit cell. It has also been shown that for the distortion when the SP of this band in Y point approaches Fermi level, van Hove singularity appears close to Y point and an abrupt change of the dispersion curve at about 75 meV below the Fermi level can be identified (electron velocity decrease). It is an effect as it has been reported by the ARPES study [1,2] for a group of high-Tc cuprates (YBCO has not been included) in the nodal direction (0,0,0)-(π,π,0). Our results show that for YBCO, this effect should be identified not in the nodal direction but close to Y point on the Γ-Y line, i.e. in the off-nodal direction (0,0,0)–(0,π,0). It should be the effect related to appearance of Tdependent giant kink as it has been reported [3] for Bi2223. The distorted lattice is characteristic by a specific - antiadiabatic fermionic ground state that is geometrically degenerate with an infinite number of different nuclear configurations, i.e. fluxional geometry arrangements of O(2), O(3) atoms in the CuO planes, with the same ground state energy. It has also been shown that nonadiabatic mechanism at the lattice energy stabilization opens two asymmetric gaps in a and b direction in the originally metallic oneparticle spectrum of YBa2Cu3O7. Calculated critical temperature is 92.8 K. More over, probability of the existence of a small gap in c direction on the Γ-T(U) line has been predicted.

II. Nonadiabatic effects in YBa2Cu3O7 Preliminaries The first member of the family of 90-K superconductors [9], YBa2Cu3O7 (YBCO), is suitable compound for theoretical study of nonadiabatic effects in high-Tc cuprates. It is superconducting at the formula unit composition that is important from the stand point of the HF-SCF theory based on LCAO since it is not necessary to introduce additional approximations related to the effect of doping that is important for other high-Tc cuprates. According to the antiadiabatic theory [8], the study starts with the LCAO-based HF-SCF calculation of the electronic band structure of YBa2Cu3O7 for clumped nuclear configuration at the high-symmetry experimental equilibrium geometry [10]. The band structure calculations have been performed by the computer code [11] SOLID 2000. The code is based on the method of cyclic cluster [12] with the quasi-relativistic INDO [13] Hamiltonian. Based on the results of atomic Dirac-Fock calculations [14], the INDO version used in the SOLID package is parameterized for nearly all elements of the Periodic table. Incorporating INDO Hamiltonian within the cyclic cluster method (with Born-Karman boundary conditions) for electronic band structure calculations has many advantages and some drawbacks as well. The method is not very convenient for strong ionic crystals but it yields good results for

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intermediate ionic and covalent systems. The main disadvantage is an overestimation of the total width of bands. On the other hand, it yields satisfactory results for properties related to electrons at Fermi level (frontier orbital properties) and for calculations of equilibrium geometries [15-17]. The same method is used for study the effects of lattice distortion on the electronic band structure. Study of the nonadiabatic effects is performed as the post-SCF calculations based on the results derived from the band structures. The experimental [10] lattice parameters of YBa2Cu3O7 (orthorhombic structure, space group Pmmm, oP14), with the fractional coordinates of the unit cell atoms: Cu(1) = (0,0,0); Cu(2) = (0,0, ±0.355); Y = (1/2,1/2,1/2), Ba = (1/2,1/2, ±0.186), O(1) = (0,1/2,0), O(2) = (1/2,0, ±0.380), O(3) = (0,1/2, ±0.376), O(4) = (0,0, ±0.156), and lattice constants a = 3.817 Ao, b = 3.882 Ao, c = 11.671 Ao have been used. The unit cell has 13 atoms as it corresponds to the formula unit with the chain oxygen O(1) in b-direction and vacancy in a direction. The basic cluster of the dimension 5x5x5, has been generated by the corresponding translations of the unit cell in the directions of crystallographic axes, a (5), b (5), c (5). All the band structure calculations presented here, have then been performed for the basic cluster 5x5x5 with the value 1.2 of the scaling parameter that is used at the calculations of the one-electron offdiagonal two-center matrix elements of the Hamiltonian (β -“hopping” integrals). The cluster of this size (1625 atoms) generates a grid of 125 points in k-space. The HF-SCF procedure is performed for each k-point of this grid with the INDO Hamiltonian matrix elements that obey the boundary conditions of the cyclic cluster [12]. The Pyykko-Lohr quazirelativistic basis set of the valence electron atomic orbitals (s,p-AO for Ba, O, and s,p,d-AO for Cu and Y) has been used, i.e., 72 AO/unit cell and the total number of STO type functions in the basic cluster has been 9,000.

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A: Band structures of YBa2Cu3O7 Experimental – undistorted geometry Calculated, LCAO-based HF-SCF band structure of YBa2Cu3O7 is presented in Fig.1. So far published band structures of YBa2Cu3O7 have been obtained mainly by application of the DFT- based LAPW or LMTO methods (for review see Ref. [18] and papers referenced therein). In spite of the substantial differences in HF-SCF and DFT-based methods, and fact that the INDO parameterization overestimates the total width of bands, the qualitativetopological agreement of the present calculation and the DFT-based band structures is rather good. However, there is one substantial difference between the present and DFT-based band structures. It concerns mainly the topology of the chain oxygen O(1)–derived pσ band in antibonding region. In the DFT-based calculations [18-21], this band starts above the Fermi level to fall-down in the Y point, crosses Fermi level and enters bonding region on the S-X line. From the present HF-SCF calculation – Fig.1, one can see that this band is going up from the Y point to S point where it reaches maximum, then continues with a little dispersion to X point, from this point starts to fall-down, intersects Fermi level and enters bonding region on the line X-Γ. It is a nontrivial disagreement of the chain oxygen O(1)- derived band topology and it will be discussed on the other place of this paper in relation to opening the gaps in one-electron spectrum on the Γ-Y and Γ-X lines with respect to the recent ARPES results [22,23] on the untwined YBa2Cu3O7.

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Theoretical Study of YBa2Cu3O7 Beyond the Born-Oppenheimer Approximation… 191 Since the specific character of the bands has been discussed in details – see Refs. [18,20], it is not necessary to be analyzed here. From the standpoint of the present study, the important is the topology of the couple of Cu(2)-O(2)/O(3) planes-derived ( d x 2 − y 2 − pσ ) bands with the maximum in antibonding region at the S point.

Figure 1. Part of the band structure of YBa2Cu3O7 calculated at the experimental – undistorted geometry. The fractional coordinates of the selected high symmetry points of the Brillouin zone are: G(≡Γ) (0,0,0), X(1/2,0,0), Y(0,1/2,0), S(1/2,1/2,0), Z(0,0,1/2)

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Both bands cross Fermi level, enter the bonding region on the S-X, respectively S-Y lines and intersect X, respectively Y point, well below the Fermi level. This topology feature is common to the present and DFT – based band structures. At this place the band structure of nonsuperconducting – deoxygenated YBCO, YBa2Cu3O6, which is without the chain oxygen O(1), should be presented – Fig.2.

Figure 2. Part of the band structure of non-superconducting YBa2Cu3O6 calculated at the experimental – undistorted geometry of YBa2Cu3O7 but without the chain oxygen O(1)

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As it can be expected, comparing to the band structure of YBa2Cu3O7 – Fig.1, the Cu(2)O(2)/O(3) planes–derived ( d x 2 − y 2 − pσ ) bands are with the same topology but the chain oxygen O(1)–derived pσ band is absent. As it has been mentioned in the Introduction, recent ARPES experiments [1,2] on the wide group of high-Tc cuprates have shown an abrupt change of the electron velocity in lowFermi energy region, 50-80 meV below the Fermi level, to be universal common feature of the high-Tc cuprate superconductors. In the investigated group of cuprates [1,2], YBa2Cu3O7 has not been involved. Providing that mechanism of superconducting state transition is the same, at least for the family of high-Tc cuprates, one can expect the same or similar abrupt change of the electron velocity in low-Fermi energy region, i.e. 50-80 meV below the Fermi level, also for YBa2Cu3O7. From the character of the calculated band structure of YBa2Cu3O7, shown in Fig.1, as well from the other published DFT-based band structures, it is clear that in the low-Fermi energy region below the Fermi level, there is not any abrupt change in the slope ( ∂ε

∂k

~

velocity (quasi-momentum) of electrons) of bands that cross Fermi level at the basal plane (Γ,X,Y,S). It should be reminded that the feature common to the band structures of all highTc cuprates is the Cu(2)-O(2)/O(3) planes – derived ( d x 2 − y 2 − pσ ) band(s) that cross Fermi

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level in respective basal plane. By an inspection of the published band structures (Ref.[18]) of the investigated group of cuprates [1,2] by ARPES , one can realize that like for YBa2Cu3O7 the same is also true for this group of cuprates. Expressed explicitly, electronic band structures of the high-Tc cuprates calculated for respective high-symmetry experimental structures do not indicate any low-Fermi energy velocity decrease of electrons or importance of nonadiabatic EP coupling at superconducting state transition.

Distorted geometries Early theoretical studies of EP coupling in YBa2Cu3O7, based on rigid-muffin-tin or related rigid-atom approximations [24,25] have indicated EP coupling strength to be very small. Results [26] obtained by LAPW method, using the frozen-phonon technique, are different. For some Ag phonon modes, strong EP coupling with coupling constant λ ≅ 1.7 – 1.9 has been obtained. Much smaller EP coupling has been calculated for B2g and B3g phonon modes, λ < 0.6. In spite of strong EP coupling, mainly of Ag phonon modes, dramatic changes in the topology of bands, e.g. shift of critical point of some band above or below Fermi level, like [4-7] in MgB2, has not be reported for YBa2Cu3O7. Present HF-SCF band structure calculations confirm these results as far as a single – isolated mode distortion is concerned. Displacements of the apical O4 oxygen along the c axes direction (Ag LO-mode of apical O4 vibration) out of the experimental inner fractional coordinate of O4 (0.156) down to 0.151 that represents elongation of Cu2-based pyramid (Cu2 – O4 distance) by 0.0583 Ao exhibit strong EP coupling that results in ground state energy increase (destabilization by + 241 meV/unit cell) but topology of the band structure remains without significant change. The same is true for isolated, a, b plane O2 and O3 atoms displacements (Cu2-O2/O3 bond stretching vibrations, LO-B2g,B3g modes) out of experimental inner coordinates 0.5, by the fraction up to +/- 0.01 that corresponds to change

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Theoretical Study of YBa2Cu3O7 Beyond the Born-Oppenheimer Approximation… 193 of the Cu2-O2/O3 bonds by 0.038 Ao. Again, the ground state energy is destabilized, in this case by +92 meV/unit cell, but without significant change in topology of the band structure. Situation is changed substantially when combination of Ag, B2g and B3g modes (for notation see e.g. [27-29]) is studied. For the c-direction displacement Δfc = - 0.0027 (0.03151 A0) of the apical O4, i.e. change of the inner coordinate 0.156 → 0.1533, and a, b direction displacements of O2 and O3 atoms from 0.5 by Δfa = 0.0057 (0.02175 Ao) for O2, and Δfb = - 0.0057 (- 0.02213 Ao) for O3, the band structure significantly changes the topology – Fig.3.

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Figure 3. Band structure at the coupling to Ag, B2g and B3g modes with displacements of apical O4 (Δfc = - 0.0027), planar O2 (Δfa = 0.0057) and O3 (Δfb = - 0.0057) atoms out of the equilibrium – experimental positions. At this distortion, there is a shift of the SP of one of the Cu(2)-O(2)/O(3) plane – derived d-pσ band on the Γ-Y line at Y point, from below to above-Fermi level position.

For this combination of displacements, the ground state energy with respect to undistorted structure has been increased - destabilized by + 170 meV/unit cell. For single modes displacements destabilization is smaller, but without the shift of the SP across the Fermi level. For apical O4 displacement Δfc = - 0.0027, with undistorted O2/O3 positions 0.5/0.5, ground state energy destabilization is + 41.1 meV/unit cell, and for displacements of planar oxygens O2, O3 by Δfa = 0.0057 and Δfb = - 0.0057 at undistorted apical O4 position 0.156, the ground state energy destabilization is even smaller, + 23.1 meV/unit cell. At the distortion with the displacements of planar O2 by Δfa = 0.0057 and O3 by Δfb = 0.0057 and apical O4 by Δfc = - 0.0027 (inner coordinate 0.1533), when the SP has crossed the Fermi level from below to above-Fermi level position, continuation in displacement of the apical O4 yields again the topology change. For Δfc = - 0.0025 (inner coordinate 0.1529), the SP in Y point is shifted back across the Fermi level to below-Fermi level position and the band structure of the topology of Fig.1 is recovered. At the second Fermi level crossing, calculated change in the ground state energy destabilization, with respect to the first crossing, is very small – 5 meV/unit cell. Expressed explicitly, present results of HF-SCF calculations, show that combination of O4, O2, O3 atoms vibration (Ag, B2g and B3g modes) shifts periodically up and down the SP of one of the Cu(2)-O(2)/O(3) plane – derived d-pσ band on the Γ-Y line at Y point across the Fermi level. The same effect is reached if the displaced position of O4 is fixed and displacements of O3 (O2) atoms are realized.

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Finally, the results of “forced” displacements should be presented. The distortions that result in Fermi level crossing, as reported above, correspond to regular oxygen atoms displacements with respect to nuclear motion in corresponding modes. It can be shown that fictitious – forced cooperative displacements of O2 and O3 atoms that preserve orthogonal character of the displacement vectors with the same |Δf| =0.0057 (O4 remains in the displaced position with Δfc = - 0.0027), generate an infinite number of distorted structures with nearly the same ground state energy destabilization and with the band structure characterized by the SP shift above the Fermi level in the Y point (Fig.3). For illustration, coordinates and ground state energy destabilization with respect to the undistorted - experimental structure of some distorted structures that are characteristic by the first Fermi level crossing on the BOA level are presented in Table 1.

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Table1. Fractional coordinates of the apical O4, and planar O2,O3 oxygen atoms for different distorted structures with nearly the same value of the ground state destabilization energy on the BOA level. Distorted structures, with shift of the SP of the Cu(2)-O(2)/O(3) plane – derived d-pσ band across the Fermi level are characterized by the O2, O3 atoms placed on perimeters of circles centered at the undistorted planar oxygen coordinates with the same radii, Δf = 0.0057. Destabilization Type of distortion and Fractional coordinates energy character of band Atom Atom Atom [eV/unit cell] structure O4 O2 O3 0.0 a - undistorted / Fig.1 0/0/±0.1560 0.5/0/±0.380 0/0.5/±0.376 + 0.169 b - Ag,B2g,B3g / Fig.3 0/0/±0.1533 0.5057/0/±0.380 0/0.4943/±0.376 + 0.161 c -”forced” / Fig.3 0/0/±0.1533 0.5/0.0057/±0.380 0.0057/0.5/±0.376 + 0.187 d -”forced” / Fig.3 0/0/±0.1533 0.4943/0/±0.380 0/0.5057/±0.376 0.5/+ 0.155 e -”forced” / Fig. 3 0/0/±0.1533 0.0057/0.5/±0.376 0.0057/±0.380 + 0.163 b’- Ag,B2g,B3g / Fig.1 0/0/±0.1529 0.5057/0/±0.380 0/0.4943/±0.376 From the Tab.1, it is clear that in the distorted structures, the O2, O3 atoms are placed on perimeters of circles with the same radii Δf = 0.0057. The circles are centered at the undistorted coordinates of O atoms, i.e. O2: 0.5/0/0.380 and O3: 0/0.5/0.376. Basically, keeping out-of phase position of the displacement vectors, an infinite number of different distorted structures can be generated by the cooperative motion of O2, O3 atoms along the perimeters of the specified circles. For the distorted structures, the second Fermi level crossing (shift of the SP from antibonding, back to bonding region below Fermi level) is reached at the increased displacement of O4 in c axis direction at the fractional coordinate 0.1529 keeping unchanged displaced positions of O2 and O3 atoms. The destabilization energy for all distorted structures is again nearly the same. It is to be mentioned that for the distorted structures characteristic by the SP shift across the Fermi level, there is a significant change in the one-particle spectrum with respect to undistorted, high-symmetry experimental structure. It is related also to formation of the new

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Theoretical Study of YBa2Cu3O7 Beyond the Born-Oppenheimer Approximation… 195 Fermi level with down-energy shift by – 0.35 eV with respect to the original Fermi level of the undistorted structure. Results of the distortions for non-superconducting YBa2Cu3O6 are different. The studied distortions do not yield the SP shift of d-pσ band on the Γ-Y line at Y point across the Fermi level over the investigated wide range of Ag, B2g and B3g modes displacements. The distortion that in the case of YBa2Cu3O7 results in Fermi level crossing leaves the band structure of YBa2Cu3O6 – Fig.2, without significant change of the bands topology and without significant shift of the Fermi level. Also the destabilization energy due to distortion, on the BOA level, is in this case smaller: +129 meV/unit cell.

B: Nonadiabatic correction to zero–particle term of the fermionic Hamiltonian: B1: Correction to the fermionic ground state energy Results of band structure calculations have shown that at vibration motion the EP coupling induces transition from adiabatic into antiadiabatic state. At these circumstances, standard adiabatic BOA is not valid and electronic motion has to be studied as a function of nuclear coordinates as well of nuclear momenta. It enables antiadiabatic theory [8] that has been discussed in the Introduction. Fermionic ground state energy correction due to EP interactions on the non-adiabatic level is determined by two terms [8],

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r ΔE (0na ) = ∑ =ω r ⎛⎜ c AI ⎝ rAI

2

r 2⎞ − cˆ AI ⎟= ⎠

The first term,

c

r PQ

=u

r PQ

(ε

0 P

unocc occ

∑∑∑ u A

I

− ε Q0

r

)

(=ω r )2 − (ε P0 − ε Q0 )2

r 2 AI

(ε

=ω r 0 A

−ε

) − (=ω )

0 2 I

; P≠Q

2

(1)

r

(1a)

i.e. coordinate Q-dependent coefficients of the quasiparticle canonical transformation, represents the adiabatic diagonal Born-Oppenheimer correction (DBOC).

r

The second term, related to { c PR } - momentum P-dependent coefficients of the quaziparticle canonical transformation, r r cˆ PQ = u PQ

=ω r

(=ω r )2 − (ε P0 − ε Q0 )2

; P≠Q

(1b)

is pure nonadiabatic contribution that expresses influence of the electronic motion on the nuclear kinetic energy and vice-versa. It has to be stressed that on the adiabatic – BOA level, the second term is absent and henceforth, ground state energy correction, even small, is always positive. Negative value of the ground state energy correction (stabilization

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Pavol Baňacký

contribution to the ground state energy) is possible only for antiadiabatic state due to participation of the nuclear kinetic effect. For k-space representation, discrete summation is replaced by integration with corresponding density of states. For interacting pairs of states mediated by phonon mode r it can be written as,

ΔE

0 ( na )

ε k . max ⎞ ⎛ ε k ', max 2 =ω r ⎟ ⎜ = VP 2 ∫ nε k ' dε k ' ∫ u kr − k ' nε k d ε k 2 2 ⎟ ⎜ 0 (ε k − ε k ' ) − (=ω r ) ε k , min ⎠ ⎝

(2)

In (2), VP stands for integration by means of principal value. The expression above is for 0 K and extension for finite temperature is straightforward by introduction of the Fermi-Dirac occupation factors - fk for occupied ε k states below Fermi level and (1-fk’) for unoccupied states

ε k ' above Fermi level.

Inspection of Figs.1,3 indicates that for basal-a,b plain, the main correction to the ground state energy can be expected from EP interactions of occupied states ε k of fluctuating d-pσ

ε k ' of O1-derived pσ band in Γ-Y and Γ-X directions.

Interaction of two d-pσ bands that correspond to different Cu-O layers which are separated nearly by 8.3 Ao, can be expected to be negligibly small. Stabilization contribution (negative value) of EP interactions starts as soon as the SP of the fluctuating d-pσ band approaches Fermi level from the bonding region on the energy distance − =ω and it continues until the SP is not more than + =ω above Fermi level. In principle, Eq. (2) can be solved exactly providing that functional dependences of corresponding density of states n and matrix elements uk-k’ of EP interactions on orbital energies are known. For density of states these functional dependences can be derived from the band structure. Calculated density of states for fluctuating d-pσ band as a function of energy for situation when the SP touches Fermi level at Y point is shown in Fig.4. As it can be seen from this figure, shift of the d-pσ band considerably increases density of states close to Y point at Fermi level as soon as the SP approaches Fermi level or crosses it at fluctuation. 8

6

4

2

nsp - density of states of d-p band at Y point

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band with unoccupied states

0 -0,08

-0,06

-0,04

-0,02

0,00

Energy distance of saddle point from Fermi level [eV]

Figure 4. Density of states of fluctuating d-pσ band as a function of the energy for situation when the SP touches Fermi level at Y point. Superconducting Cuprates : Properties, Preparation and Applications, Nova Science Publishers, Incorporated, 2008. ProQuest Ebook Central,

Theoretical Study of YBa2Cu3O7 Beyond the Born-Oppenheimer Approximation… 197 Density of states of the chain oxygen O1-derived pσ band at k-point(s) where the band intersects Fermi level calculated from the band structure is constant over energy interval ± =ω at Fermi level. For Γ-Y direction it is nΓY = 0.04, and for direction Γ-X corresponding value is even smaller, nΓX = 0.03 states/eV.

Functional dependence u k − k ' ≡ f (ε k ' − ε k ) of the EP interaction matrix elements is not

available on the corresponding level. However, from the SCF-HF calculation the overall value of EP interactions at displacement of vibrating atoms that results to Fermi level crossing

(

)

can be extracted, i.e. change of one-electron core term Δh Qcros sin g = u

(1)

(Q

cros sin g

)≡ u .

In what follows, this value represents maximal value u of EP interaction at transition from adiabatic into antiadiabatic state. For properly displaced O4 (Ag), displacement of O2(O3) within B2g(B3g) results to Fermi level crossing as it has been shown in previous section. At these circumstances, the SCF-HF calculations yield u = 2.5 eV. In what follows, dependence of EP coupling on orbital energy distance is approximated by functional form,

(u z )2 = (u )2 .(0,99 + 0,12.z − 0,31.z 2 )

(3)

for Δε kk ' ≤ 2=ω , and u z = 0 for Δε kk ' > 2=ω . In (3),

z = Δε kk ' =ω

(4)

Then, boundaries of integration in Eq.2 are,

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ε k 'max = =ω , ε k min = −=ω , ε k max = − =ω q ,

(5)

Parameter q ( q ≥ 1 ) in (5) “regulates” energy distance of the SP from Fermi level. For q→ ∞, the SP just touches Fermi level. The functional dependence u ( z ) is an important factor that determines not only the ground state energy correction but also the form of the final density of states. In this respect, reported absolute values of calculated physical parameters should be understood as theoretical simulation rather than the exact numerals. At the calculation of nonadiabatic ground state energy correction, density of states of fluctuating d-pσ band has been approximated by mean value n SP = 2 states/eV (value for nearly degenerate states, Δε kk ' ≈ 0.007 eV – see Fig. 4), and phonon energy of B2g(B3g) phonon mode corresponds to the experimental value =ω = 0.072 eV. Calculated dependence of nonadiabatic ground state energy correction as function of the parameter q is shown in Fig.5.

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198

Pavol Baňacký qparameter of energy distance (−ω/q) of critical point approaching Fermi level from bonding region

Ground state energy correction [eV/u.c.]

-0,075

5

10

15

20

25

30

35

40

45

50

55

60

-0,100

-0,125

-0,150

-0,175

-0,200

-0,225

. Figure 5. The nonadiabatic correction to the fermionic ground state energy,

ΔE (0na ) , of the YBa2Cu3O7

for the distorted structure due to EP coupling to Ag, B2g, B3g modes as function of the parameter q – see text.

As it can be seen from Fig.5, the maximal stabilization effect is reached when the SP of Cu2-O2/O3-derived d-pσ band in Y point just approaches Fermi level from the bonding side, i.e. for q→ ∞. The nonadiabatic correction reaches the value -0.204 eV/unit cell. An equivalent result can be obtained in case of back-crossing. In such a case, the SP would approach Fermi level from the antibonding side and the maximal stabilization effect is reached when the SP is just + =ω above Fermi level

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( ε k 'max = =ω / q, ε k 'min = 0, q = 1).

(6)

The ground state energy loss (destabilization on the BOA level) due to symmetry distortion when the SP of d-pσ band in Y point approaches Fermi level and crosses it, has been calculated to be +170 meV/unit cell (see previous section). At these circumstances, the mean value of the ground state energy correction due to nonadiabatic EP coupling is about 204 meV/unit cell (Fig.5). The net effect of the distortion – symmetry lowering is the fermionic ground state energy stabilization. It means that due to effective nonadiabatic EP coupling, the distorted structure for the specified displacements is by (-204+170) = -34 meV/unit cell more stable than undistorted – equilibrium structure on the BOA level. More over, this new – antiadiabatic ground state is geometrically (quasi-)degenerate. There are an infinite number of the O2, O3 atoms in-plane displacements (as described previously), i.e., different nuclear configurations with the same ground state energy. On the lattice scale, geometric degeneracy of the fermionic ground state energy for distorted structure, i.e. existence of an infinite number of O2, O3 atoms displacements (fluxional structure of Cu2O2/O3 plane), enables cooperative and dissipation-less motion of displaced oxygen atoms along the perimeters of circles centered at theirs equilibrium positions with the radii equal to the fractional displacement ⏐Δfa,b⏐≈ 0.0056 - 0.0057 (≈ 0.022 Ao). This is a new, the coherent - macroscopic quantum state.

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Theoretical Study of YBa2Cu3O7 Beyond the Born-Oppenheimer Approximation… 199 It has to be stressed that the nonadiabatic ground state energy stabilization at broken translation symmetry has to be distinguished from a Peierls distortion (or Jahn-Teller effect on a molecular system). The Jahn-Teller effect/Peierls distortion is a pure adiabatic effect of removal an asymmetry in occupation of degenerate states at Fermi level by geometry distortion that results into lower electronic ground state energy (stabilization) already on the level of the BOA. As it has been shown, in the case of YBa2Cu3O7, the geometry distortion on the level of BOA results into the ground state energy increase – see Table 1. At this place, the effect detectable by the ARPES experiments should be mentioned. For the lattice distortion when the SP of d-pσ band approaches Fermi level at Y point and the ground state energy is stabilized by nonadiabatic EP interactions, the dispersion of the d-pσ band at Fermi level down to – 250 meV along the Γ-Y direction calculated from the band structure is shown in Fig.6. As it can be seen, the deviation of the dispersion curve from the straight-line direction starts at 75 – 80 meV below the Fermi level. This Figure resembles the results obtained by the ARPES experiments [1,2] for high-Tc cuprates. The authors [1,2] have

(

described it as the low-Fermi energy “electron velocity” ∂ ε k

0

)

∂k decrease.

Binding energy - (E-EF), [eV]

0,00

-0,05

-0,10

-0,15

-0,20

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-0,25 0,36

0,38

0,40

0,42

0,44

0,46

0,48

0,50

k - along the line Γ-Y (0,0,0 - 0,1/2,0), kF=1/2

Figure 6. Calculated dispersion of the fluctuating d-pσ band in Γ-Y direction at the lattice distortion due to electron coupling with Ag, B2g, B3g phonon modes for situation when the SP of the band touches Fermi level (0.0 eV) at Y point. As a guide for eyes, the straight line and arrow are drawn to indicate the sudden change of low-Fermi energy “electron velocity” that start at about 75-80 meV below Fermi level.

In case of YBa2Cu3O7, as it is evident from the present study, this effect should be Tdependent and it appears not in the nodal direction but in the off-nodal Γ-Y direction close to Y point at about 75-80 meV below the Fermi level. Recently, beside the presence of Tindependent small kink in the nodal direction as reported in [1,2], formation of the Tdependent giant kink has been measured in off-nodal direction [3] at the Fermi level for Bi2223. This kink is present below Tc and disappears above Tc at about 70 meV from the Fermi level, like the result presented in Fig. 6 that has been calculated for YBa2Cu3O7. It should be stressed that no matter how this effect is interpreted, primarily it represents an effective increase of the density of states of the d-pσ band at the Fermi level,

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Pavol Baňacký

( ) (

n ε k0 = ∂ε k0 ∂k

)

−1

. This fact can be seen in Fig.4, for distorted lattice when the SP of d-

pσ band approaches Fermi level due to electron coupling with Ag, B2g, B3g phonon modes. Increased density of states, which effectively participate at nonadiabatic EP interactions, is the crucial factor enabling ground state energy stabilization of the distorted lattice, see Eq.(2). As it has been mentioned, studied distortions do not change the band structure of nonsuperconducting YBa2Cu3O6. It means that the density of states of d-pσ band remains equal to ≈ 0.04 over the relevant energy interval ± =ω Bľg , šg , and due to the adiabatic character of the bands topology (Fig.2), the EP interactions do not stabilize distorted structure.

C: Nonadiabatic correction to the one–particle term of fermionic Hamiltonian C1: Corrections to the orbital energies and gap opening in a metallic one-particle spectrum According to nonadiabatic theory of EP interactions [8], for nonadiabatic correction

Δε k to an occupied state ε k0 for system in antiadiabatic state follows, ⎞ ⎛ ε k ' max 2 =ω k − k ' 0 ⎟ − ε Δε k = VP⎜ wk ∫ n k ' u k − k ' (1 − f k ' ) d ' k 2 2 0 0 ⎟ ⎜ ( ) = ε ε ω − − 0 k k' k −k ' ⎠ ⎝ 0 ⎛ ⎞ =ω k − k1 2 0 ⎟ − VP⎜⎜ wk ∫ n k1 u k − k1 f k1 d ε k1 ⎟ 2 0 0 2 ⎜ ε ⎟ − − = ε ε ω − k k k k k min 1 1 1 1 ⎝ ⎠ 0

(

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(

)

) (

(7)

)

In (7), k , k1 ≤ k F ≤ k and wk represents the dimensionless weight factor of the state '

ε k0 for particular density of states nk over considered energy interval. The original HF state ε k (orbital energy of the occupied state for calculated distorted 0

clumped nuclear structure) is by this correction shifted on the energy scale to a new position,

ε k = ε k0 + Δε k For nonadiabatic correction Δε k ' to an unoccupied state

(8)

ε k0' holds,

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Theoretical Study of YBa2Cu3O7 Beyond the Born-Oppenheimer Approximation… 201 ε k '1 max ⎛ =ω k '− k '1 2 ⎜ Δε k ' = VP wk ' . ∫ n k '1 u k '1 − k ' 1 − f k '1 2 ⎜ ε k0' − ε k0'1 − =ω k '− k '1 0 ⎝ 0 ⎛ ⎞ 2 =ω k − k ' 0⎟ d ε − VP⎜ wk ' ∫ n k u k − k ' f k k 2 ⎜ ⎟ ε k0' − ε k0 − (=ω k − k ' )2 ⎝ ε k0min ⎠ 0

(

(

)

(

) (

)

2

⎞ dε k0'1 ⎟ − ⎟ ⎠

(9)

)

In (9), k ≤ k F ≤ k ' , k '1 and wk ' represents the dimensionless weight factor of the state

ε k0' for particular density of states nk ' over the considered energy interval. The original HF state ε k ' (orbital energy of the unoccupied state for calculated distorted 0

clumped nuclear structure) is shifted to a new position,

ε k ' = ε k0' + Δε k '

(10)

Nonadiabatic shift of the HF orbital energies induces change of the original density of states at Fermi level for the band where the gap in one-particle spectrum has been opened. For resulting final – corrected density of states can be derived in the form,

(

n(ε k ) = 1 + ∂ (Δε k ) ∂ε k0

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Term n

0

(ε ) = (∂ε 0 k

0 k

∂k

)

−1

)

−1

( )

n 0 ε k0

(11)

in (11) stands for the original – uncorrected density of

states (original density of states of calculated distorted HF clumped nuclear structure). Equation (11) holds for k F < k < k F . Calculations of the nonadiabatic corrections to one-particle HF states for system in antiadiabatic state when the SP of d-pσ band approaches Fermi level have been done with the same parameters as it has been used for calculation of the nonadiabatic correction to the ground state energy. Calculated corrections to the HF states (at 0 K) of the O1- derived pσ band in the Γ-Y direction at k point where the band intersects Fermi level (uncorrected density of states at Fermi level is n pσ = 0.04) are shown in Fig.7 a, b. In a same way corrections to the HF states of the O1- derived pσ band in the Γ-X direction are calculated at k point where the band intersects Fermi level. In this case, however, uncorrected density of states at Fermi level is n pσ = 0.03. Final – corrected density of states for bonding and antibonding region of the chain oxygen O1- derived pσ band are calculated according to Eq.11. The results are presented in Fig.8 and 9. From these figures, it can be seen that asymmetric gaps in one particle spectrum are opened in the b (Γ-Y) and a (Γ-X) directions. While the gap (energy distance of the peaks at

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Fermi level) in b direction is Δ b (0 ) = 35.7 meV, the gap in a direction is smaller,

Δ a (0) = 24.2 meV.

-0,05

-0,04

-0,03

-0,02

-0,01

0,00 0,00 -0,01 -0,02 -0,03 -0,04 -0,05 -0,06 -0,07

Correction to orbital energy [eV]

Uncorrected orbital energy [eV]

-0,08 -0,09

a

Correction to orbital energy [eV]

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0,07

0,06

0,05

0,04

0,03

0,02

0,01 0,00

0,01

0,02

0,03

0,04

0,05

Uncorrected orbital energy [eV]

b Figure 7. The nonadiabatic corrections Δεk to the orbital energies and nonadiabatic corrections Δεk’ to the orbital energies

ε

0 k'

ε k0

of the occupied states {k}pσ (a)

of the unoccupied states {k’}pσ (b) of the

chain oxygen O1-derived pσ band for the distorted lattice in Γ-Y direction

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Theoretical Study of YBa2Cu3O7 Beyond the Born-Oppenheimer Approximation… 203

10

8

Density of states

12

6

4

2

0 -0,05

-0,04

-0,03

-0,02

-0,01

0,00

0,01

0,02

0,03

0,04

0,05

Energy distance from Fermi level [eV], Y-Γ direction

Figure 8. Corrected density of states of the O1- derived pσ band in Γ-Y direction at k point where the band intersects Fermi level.

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16 14 12

Density of states

18

10 8 6 4 2 0 -0,04

-0,02

0,00

0,02

0,04

Energy distance from Fermi level [eV], X-Γ direction

Figure 9. Corrected density of states of the O1- derived pσ band in Γ-X direction at k point where the band intersects Fermi level.

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Temperature dependence of energy gap that is open in one-electron spectrum at Fermi level has be derived [8] in the form,

Δ (T ) = Δ(0 ).tgh(Δ (T ) 4k B T ) The gap extinction is at critical temperature Tc. Then, in the limit Δ (T ) → 0 , for critical

temperature Tc follows: Tc = Δ(0 ) 4k B .Calculated value of the critical temperature that

corresponds to the larger gap is Tc = 92.8 K. The critical temperature corresponds to temperature at which the system undergoes transition from antiadiabatic state (superconducting ground state) into adiabatic state that does not exhibit superconducting properties. The experimental value [9] of Tc for SC state transition in YBCO is in the range 90 – 94 K. Calculated density of states simulates the a, b asymmetry of the ARPES spectra as it has been recorded [22] for untwined YBa2Cu3O7. Experimental ratio of the gaps in the a and b direction is (Δ a (0 ) Δ b (0 ))exp ≈ 0.66. The present calculations yield for this ratio

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(Δ a (0)

Δ b (0))theor ≈ 0.68, that is in good agreement with the experimental results.

In this connection, existence of the gap in the a direction (X-Γ line) should be mentioned. According to the present results, this gap in one-electron spectrum is opened in the chain oxygen O1- derived pσ band at the k point where the band intersects X-Γ line (Fig. 1, 3). Existence of this gap can hardly be expected from the band structures based on DFT calculations since according to the corresponding published results, pσ band intersects Fermi level on X-S line and there is no band at all that intersects Fermi level on X-Γ line. Related to the mechanism of gap formation in one-electron spectrum of YBa2Cu3O7, unexpected new result should be mentioned. Calculated band structure has no bands crossing Fermi level in vertical directions (Γ-Z, X-U, Y-T, S-R lines). In this direction, there is a little dispersion of bands restricted separately to bonding and antibonding regions that indicates strong 2D character of this material. Closer inspection of the band structure reveals, however, that 2D character is disturbed by strong dispersions of two bands; the Cu-O plane d-pσ band with fluctuating SP and chain oxygen O1-derived pσ band. These bands interests Fermi level in c-direction. In particular, the d-pσ band intersects Fermi level in b-c plane on Γ-T line and O1-derived pσ band intersects Fermi level in a-c plane on Γ-U line – see Fig.10. Also in this case it can be expected that for antiadiabatic state when the SP at Y point approaches Fermi level, nonadiabatic EP interactions open gaps in one-electron spectrum. For d-pσ intraband interactions, gap should appear in Γ-T direction and interbands interactions should give rise to a gap on pσ band in Γ-U direction. By simple rescaling of u with respect to Γ-Y and Γ-T (or Γ-U) distances, mean value of EP interaction in Γ-T direction is approximated by the value u ΓT ≈ 0.6 eV. Calculated uncorrected density of states of d-pσ band at k-point(s) where the band intersects Fermi level in Γ-T direction is for energy interval ± =ω at Fermi level constant and equal to nΓT = 0.06 states/eV.

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Figure 10. Dispersion of the bands; Cu-O plane ( d x 2 − y 2

− pσ

) band in Γ-T direction and chain

oxygen O1-derived pσ band in Γ-U direction.

Density of states

Final – corrected density of states for bonding and antibonding region of the CuO- plane derived d-pσ band in Γ-T direction at Fermi level calculated according to Eq.11 is presented in Fig.11. 0,50

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0,25

-0,020

-0,015

-0,010

-0,005

0,000

0,005

0,010

0,015

0,020

Energy distance from Fermi level [eV], direction Γ-T

Figure 11. Corrected density of states of the CuO-plane derived d-pσ band in Γ-T direction at k point where the band intersects Fermi level.

As it can be seen from this figure, opened gap Δ ΓT (0 ) ≈ 5 meV is considerably smaller comparing to the gaps that are opened in the basal plane. The gap in Γ-U direction that should be opened on chain oxygen O1 –derived pσ band is even smaller.

C2: Formation of Van Hove singularity in DOS of the Cu-O plane d-pσ band Possibility of the existence of Van Hove singularities in the density of states (DOS) at Fermi level was proposed [30,31] at the very beginning to be responsible for high Tc in Superconducting Cuprates : Properties, Preparation and Applications, Nova Science Publishers, Incorporated, 2008. ProQuest Ebook Central,

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cuprates. For a long time there were only indirect evidences for existence of such peak(s) in the DOS at Fermi level, e.g. discontinuity in the specific heat at Tc, and studies of thermopower and quasiparticle lifetime broadening [32-34]. For the first time, direct experimental evidence of the Van Hove singularity in DOS of high-Tc cuprates has been reported by Gofron et al [35]. The existence of the Van Hove singularity in DOS of CuO d-pσ band at 19 meV below the Fermi level has been measured at the study of superconducting YBa2Cu4O8. While, the peak is present below Tc at Y point of the Brillouin zone along Y-Γ direction, it is absent above Tc. This effect has been found to be a common feature of the DOS of CuO plane d-pσ band of all high-Tc cuprate superconductors. Formation of the Van Hove singularity follows straightforwardly from the nonadiabatic EP interaction mechanism. From the band structure at high-symmetry equilibrium nuclear configuration (Fig.1) one can see that analytic critical points of the d-pσ bands of Cu-O plane at Y point that are related to singularities in DOS, are far below the Fermi level. This situation corresponds to adiabatic regime, and Van Hove singularities at the Fermi level do not exist, as it is experimentally detected above Tc. However, electron coupling to the phonon modes generates formation of the antiadiabatic state which is stabilized at distorted nuclear configuration. At these circumstances, the SP of fluctuating d-pσ band of Cu-O plane at Y point has approached Fermi level – Fig.6. It has been shown above that as a consequence of the band shift, the ground state is stabilized at distorted nuclear configuration and the gap in the one-particle spectrum of the chain oxygen O1-derived pσ band has been opened due to effective nonadiabatic EP interactions. The nonadiabatic EP interactions which result in opening of the gaps in the one-particle spectrum (Fig.8,9) of the chain oxygen O1-derived pσ band influences also DOS of the d-pσ band of Cu-O plane at Y point in Y-Γ direction. Without the account for nonadiabatic EP interactions the DOS of the fluctuating d-pσ band, at the moment when the SP of this band touches Fermi level, has the maximum at 0 eV (see Fig.4 with rescaled energy value of the Fermi level to 0 eV). The analytic expression of the uncorrected DOS corresponding to this situation is,

(

) (

n 0 ε k0,d − pσ = ∂ε k0,d = pσ ∂k

)

−1

(11a)

The nonadiabatic interactions shift not only the orbital energies of the chain oxygen O1derived pσ band (gap opening as described above), but the EP interactions shift also orbital energies of the fluctuating d-pσ band in the downward direction away from Fermi level to the new positions,

ε k ,d − pσ = ε k0,d − pσ + Δε k ,d − pσ

(12)

For corrected DOS that reflects the shift due to nonadiabatic EP interactions in the antiadiabatic state holds Eq.11. Calculation of final corrected DOS of the d-pσ band of Cu-O plane at the Y point according to Eq.11 yields result presented in Fig.12.

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DOS of Cu-O2 plane d-p band

10

8

6

4

2

0 -0,06

-0,05

-0,04

-0,03

-0,02

-0,01

0,00

B inding energy at Y point [eV ]

Figure 12. Resulting density of states of the Cu-O plane derived d-pσ band at Y point with the account for nonadiabatic EP interactions. The energy of the Fermi level is rescaled to 0 eV.

As it can be seen from this figure, the peak in the DOS is at ≈ 17 meV below the Fermi level, which is in a good agreement with the experimental results [35] measured below Tc.

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III. Discussion and Conclusion Application of the nonadiabatic theory of EP interactions at theoretical study of YBa2Cu3O7 offers substantially different scenario of SC state transition than BCS or BCS-like pairing theories. For the antiadiabatic conditions with broken translation symmetry, the Bose–Einstein condensation can be related to inter-site bipolarons formation, instead of Cooper’s pairs formation at equilibrium geometry of clumped nuclear BOA structure. In this respect, “condensate” is represented by “charged bosons”- real space singlet pairs, rather than by BCS superfluid. This picture is consistent with the experimental results on cuprates [36,37], where it has been concluded that charged bosons are bipolarons. The condensation energy is related to the fermionic ground state energy stabilization due to nonadiabatic EP interactions in the antiadiabatic state at broken translation symmetry. In order to stabilize distorted structure, the energy gain due to nonadiabatic EP interactions has to be greater than energy loss due to symmetry lowering. Meaning of the gap is also different here. Within the antiadiabatic picture, the gap has its usual meaning, i.e. quasiparticle (“non-adiabatic polaron”) excitation energy in the one-particle spectrum. At finite temperatures, with increasing temperature from 0 K, due to the Fermi statistics of the one-particle state populations, the nonadiabatic EP corrections become smaller (T– dependent contribution of the antiadiabatic state at distorted nuclear geometry), and at crossing critical temperature Tc, the electronic energy loss due to symmetry lowering (T– independent contribution) becomes greater than the energy gain due to nonadiabatic EP interactions. The system, in order to minimize fermionic ground state energy, goes to normal

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– adiabatic metal state with the equilibrium geometry of the higher symmetry. For cooling, the situation is opposite. At crossing Tc in downward direction, the distorted structure becomes more stable than the undistorted one. The reason is that at lowered symmetry, there is the proper structure of the one-particle spectrum at Fermi level. This structure, the antiadiabatic state, effectively enables to switch-on and maximizes the nonadiabatic EP interactions. The nonadiabatic energy gain starts to prevail over the energy loss at the distortion. With respect to presented results, the SC state transition in YBa2Cu3O7 can be characterized, like in the case [6] of MgB2, as a nonadiabatic sudden increase of the cooperative kinetic effect at lattice energy stabilization. It is exactly participation of the nuclear kinetic energy term on the antiadiabatic level that stabilizes (negative contribution) fermionic ground state energy at a distorted structure. At the adiabatic conditions, nuclear kinetic effect is absent, and adiabatic correction (DBOC - effect of the nuclear positions) to the fermionic ground state energy is small but always positive for equilibrium as well as for distorted clumped nuclei structures. It should be pointed out that for undistorted – experimental equilibrium geometry, the one-particle spectrum is of adiabatic metal – like character and nonadiabatic EP interactions do not contribute to the ground state energy stabilization. Only at decreased symmetry, i.e., distorted structure, when the orbital energies are shifted into suitable positions at Fermi level (formation of the new one-particle spectrum, i.e. the antiadiabatic state), the non-adiabatic EP interactions can become operative and effective. Geometrical degeneracy of the fermionic ground state energy for distorted structure (i.e. existence of an infinite number of in-plane displacements of O2, O3 atoms in Cu-O planes, due to electron coupling to the Ag and, B2g, B3g phonon modes) enables cooperative and dissipation-less motion of displaced O2, O3 atoms along the perimeters of circles centered at the undistorted O2, O3 atoms positions. The radii of the circles are equal to the fractional displacements ⏐Δfa,b⏐cr. In the case of YBa2Cu3O7, it is⏐Δfa,b⏐cr = 0.0057 (expressed in fractional unit), that is 0.022 Ao in the absolute value. The cooperative nuclear motion, i.e. “fluxional microcirculations” of O2, O3 nuclei in the Cu-O layers, induces dynamic - cooperative formation of shortened and elongated O2 – O3 “bond” distances in CuO layers on the lattice scale. This microcirculation is connected with a dynamic formation of increased and decreased interatomic charge densities in the a-b plane of Cu-O layers (see Fig.13 a-c). Incorporation of the effect of nonadiabatic nuclear momenta results in further increase of the inter-atomic charge density polarization – see eq. (100) in [8]. It is related to formation of mobile nonadiabatic bipolarons [8]. Possibility of bipolaron superconductivity in high-Tc cuprates has also been proposed and discussed in great details by Alexandrov, see e.g. [38, 39] and references therein. Nonadiabatic bipolarons are charge supercarriers and theirs motion on the lattice in the a,b plane is coherent and dissipation-less. It has been shown [8] that in strong nonadiabatic

(

)

limit, lim =ω / ε Y , d − pσ − ε F → ∞ , which is characteristic for antiadiabatic state at broken translation symmetry, interaction energy of new quasi-particles (nonadiabatic polarons) and nonadiabatic phonons is zero, ΔH nFB → 0 . More over, due to geometrical degeneracy of '

the fermionic ground state energy (fluxional structure of Cu-O layers), there are not energy barriers for the motion of bipolarons on the lattice and tunneling mechanism is not necessary to be considered.

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a/

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b/

c/

d/ Figure 13. Valence electron isodensity lines in the Cu2 – O2,O3 plane (a-b plane) for undistorted experimental (a) and distorted structures (b,c,d) - see Tab.1. The highest electron density (dark-blue) is localized at O2, O3 atoms positions for equilibrium structure (a). The lowest electron density (white) is at Cu2 atom positions with characteristic angular character of

d x2 − y2

orbital. The displacement of the

O2, O3, (⏐Δf⏐a,b = 0.0057) and O4 (⏐Δf⏐c = 0.0027) atoms at electron coupling to the Ag and, B2g, B3g phonon modes induces alternating O2-O3 inter-site charge density delocalization, different for the particular types of the distortion (see b, c).

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210

Pavol Baňacký For adiabatic systems, i.e. in the limit, lim

(ε

Y , d − pσ

)

− ε F / =ω → ∞ , which is

characteristic for normal metal state of superconductors at equilibrium, high-symmetry nuclear geometry, interaction energy of the adiabatic quasi-particles (polarons and

( ) / (ε

renormalized phonons) is ΔH nFB ≈ u k 'k '

2

k

− ε k ' ). It is basically well known polaron

energy. As it has been shown, the nonadiabatic effects, which are present in YBa2Cu3O7, are absent in YBa2Cu3O6. This is the reason, from the standpoint of the antiadiabatic theory, why the deoxygenated YBCO is not superconductor. Present study suggests possibility of the experimental verification of the described antiadiabatic mechanism at transition to SC state. The effect of sudden decrease of low-Fermi energy (50-80 meV) electron velocity by ARPES experiments for a wide group of high–Tc cuprates has been measured [1,2] in the nodal (0,0,0)- (π,π,0) direction. The results concerning YBa2Cu3O7 had not been reported. According to present calculations (see Fig.6) this effect should be registered also for YBa2Cu3O7, but in the off-nodal (0,0,0) – (0,π,0) direction close to Y point on the Γ - Y line at about 75-80 meV below the Fermi level. Like in the case [3] of Bi2223, this effect should be T-dependent, i.e. it should appear only at T≤Tc. The high precision ARPES or tunneling spectroscopy should also detect the existence of small gap in c-direction ( Γ − T (U ) line – Fig.12), the existence of which has not been reported so far.

Acknowledgements

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The author is acknowledged to S-Tech a.s. for financial support of the project, as well for partial support due to grant VEGA 1/2465/05, 1/0013/08

References [1] Lanzara, A.; Bogdanov, P.V.; Zhou, X.J.; Kellar, S.A.; Feng, D.L.; Lu, E.D.; Yoshida, T.; Eisaki, H.; Fujimori, A.; Kishio, K.; Shimoyama, J.I.; Noda, T.; Uchida, S.; Hussain, Z. and Shen, Z.X. Nature 2001, 412, 510 [2] Zhou, X.J.; Yoshida, T.; Lanzara, A.; Bogdanov, P.V.; Kellar, S.A.; Shen, K.M.; Yang, W.L.; Ronning, F.; Sasagawa, T.; Kakeshita, T.; Noda, T.; Eisaki, H.; Uchida, S.; Lin, C.T.; Zhou, F.; Xiong, J.W.; Ti, W.X.; Zhao, Z.X.; Fujimori, A.; Hassain, Z. and Shen, Z.X. Nature 2003, 423, 398 [3] Takahashi, T.; Sato, T.; Matsui, H. and Terashima, K. New J. Phys. 2005, 7, 105 [4] An, J.M. and Picket, W.E. Phys.Rev.Lett. 2001, 86, 4366 [5] Yilderim, T; Gulseren, O.; Lynn, J.W.; Brown, C.M.; Udovic, T.J.; Huang, Q.; Rogado, N.; Regan, K.A.; Hayward, M.A.; Slusky, J.S.; He, T.; Hass, M.K.; Khalifah, P.; Inumaru, K. and Cava, R.J. Phys.Rev.Lett. 2001, 87, 037001 [6] Banacky, P. Int.J.Quant.Chem. 2005, 101, 131 [7] Boeri, L.; Cappelluti, E. and Pietronero, L. Phys.Rev.B 2005, 71, 012501

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Theoretical Study of YBa2Cu3O7 Beyond the Born-Oppenheimer Approximation… 211 [8] Banacky, P. Antiadiabatic theory of electronic ground state of superconductors, Nova Science Publishers. [9] Hor, P.H.; Meng, R.L.; Wang, Y.Q.; Gao, L.; Hung, Z.J.; Bechtold, J.; Forster, K. and Chu, C,W. Phys. Rev. Lett. 1987, 58, 1191 [10] Muto, Y.; Kobayashi, N. and Syono, Y. in Novel Superconductivity, eds. S. Wolf and V Kresin, Plenum Press, New York, 1987 [11] SOLID 2000, Computer code for electronic structure calculation of periodic systems. STech a.s., Bratislava, Slovakia (www.stech.sk) [12] Noga, J.; Baňacký, P.; Biskupič, S.; Boca, R.; Pelikan, P. and Zajac, A. J. Comp. Chem. 1999, 20, 253 [13] Pople, J.A. and Beveridge, D.L. in Approximate Molecular Orbital Theory, McGrawHill Inc, New York, 1970 [14] Boca, R. Int. J. Quant. Chem. 1987, 31, 941; 1988, 34, 385 [15] Zajac, A.; Pelikán, P.; Noga, J.; Banacky, P.; Biskupic, S. and Svrcek, M. J. Phys. Chem. B 2000, 104, 1708 [16] Zajac, A.; Pelikán, P.; Minar, J.; Noga, J.; Straka, M.; Banacky, P. and Biskupic, S. J. Solid. State Chem. 2000, 150, 286 [17] Pelikán, P., Kosuth, M.; Biskupič, S.; Noga, J.; Straka, M.; Zajac, A. and Banacky, P. Int. J. Quant. Chem. 2001, 84, 157 [18] Picket, W.E. Rev. Mod. Phys. 1989, 61, 433 [19] Picket, W.E.; Cohen, R.E. and Krakauer, H.A. Phys. Rev. B 1990, 42, 8764 [20] Andersen, O.K.; Jepsen, O.; Liechtenstein, A.I. and Mazin, I.I. Phys. Rev. B 1994, 49, 4145 [21] Kouba, R.; Ambrosch-Draxl, C. and Zangger, B. Phys. Rev. B 1999, 60, 9321 [22] Lu, D.H.; Feng, D.L.; Armitage, N.P.; Shen, K.M.; Damascelli, A.; Kim, C.; Ronning, F.; Shen, Z.X.; Bonn, D.A.; Liang, R.; Hardy, W.N.; Rykov, A.I. and Tajima, S. Phys. Rev. Lett. 2001, 86, 4370 [23] Damascelli, A.; Hussain, Z. and Shen, Z.X. Rev. Mod. Phys. 2003, 75, 473 [24] Weber, W. and Mattheiss, L.F. Phys. Rev. B 1988, 37, 599 [25] Allen, P.B.; Picket, W.E. and Krakauer, H. Phys. Rev. B 1988, 37, 7482 [26] Cohen, R.E.; Picket, W.E. and Krakauer, H. Phys. Rev. Lett. 1990, 64, 2575 [27] Chung, J.H.;Egami, T.; McQuinney, R.J.; Yethiraj, M.; Arai, M.;Yokoo, T.; Petrov, Y.; Mook, H.A.; Endoh, Y.; Tajima, S.; Frost, C. and Dogan,, F. Phys. Rev. B 2003, 67, 014517 [28] Lin, R.; Thomsen, C.; Kress, W.; Cardona, M.; Gegenheimer, B.; deWette, F.W.; Arade, J.; Kulkarni, A.D. and Schroder, U. Phys. Rev.B 1988, 37, 7971 [29] McCarty, K.F.; Lin, J.Z.; Shelton, R.N. and Radomsky, M.B. Phys. Rev. B 1990, 41, 8792 [30] Hirsh, J.E. and Scalapino, D.J. Phys. Rev. Lett. 1986, 56, 2735 [31] Labbe, J. and Bok, J. Europhys.Lett. 1987, 3, 1225 [32] Tsuei, C.C.; Chi, C.C.; Newns, D.M. Phys. Rev. Lett. 1992, 69, 2134 [33] Newns, D.M.; Krishnamurthy, H.R.and Pattnaik, P.C. Phys. Rev. Lett. 1992, 69, 1264 [34] Pattnaik, P.C.; Kane, C.L. and Newns, D.M. Phys Rev.B 1992, 145, 5714 [35] Gofron, K.; Campuzano, J.C.and Abrikosov, A.A. Phys. Rev. Lett. 1994, 73, 3302 [36] Zhao, G.; Hunt, M.B.; Kellerand, H. and Muller, K.A. Nature (London) 1997, 385, 236

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[37] Franck, J.P. in Physical Properties of High Temperature superconductors IV, ed. D. Ginsberg, World Scientific, Singapore,1994 [38] Alexandrov, A.S. Phys. Rev. B 2000, 61, 12315 [39] Alexandrov, A.S. in Studies in high temperature superconductors, Vol.50, p.1-69, ed. A.V.Narlikar, Nova Science Publishers, 2006

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In: Superconducting Cuprates Editor: Koenraad N. Courtlandt

ISBN: 978-1-60456-919-3 © 2009 Nova Science Publishers, Inc.

Chapter 7

THEORIES OF PEAK EFFECT AND ANOMALOUS HALL EFFECT FOR CUPRATE SUPERCONDUCTORS Wei Yeu Chen and Ming Ju Chou Department of Physics, Tamkang University, Tamsui 25137, Taiwan

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Abstract The high- Tc cuprate superconductors are highly anisotropic type-II superconductors with either tetragonal or orthorhombic crystal structure; however, they share one single common feature that all the cuprate superconductors possess the two-dimensional CuO2 planes. The mechanism of these remarkable cuprate superconductors has now gradually become clear after myriads of studies for more than two decades. The promising potential in application has outlined the unlimited vision in the near future. In this Chapter, the peak effect and anomalous Hall effect for type-II superconductors are investigated in this present study. There exists a peak in the critical current density as the temperature or applied magnetic field of the system increases. This is the peak effect or fishtail effect for the superconductors. The presence of impurities due to quenched disorder, irradiation or doping destroys the long-range order of flux line lattice, after which only short-range order remains. A first-order phase transition between the short-range order and disorder in the vortex system eventually appears as a result of enormous increase in the dislocations inside the short-range domains when the applied magnetic field or the temperature increases. The origin of peak effect is in this kind of firstorder phase transition. The peak value of the critical current density, the exact peak position and its corresponding half-width for a constant temperature as well as for a constant applied magnetic field based on this quasiorder-disorder first-order phase transition of the vortex system are evaluated. The Hall resistivity changes sign from positive to negative as the applied magnetic field (temperature) decreases for constant temperature (applied magnetic field) for many high- Tc and some conventional superconductors when the temperature of the system is close to the critical temperature. Sometimes the Hall resistivity exhibits the double sign reversal property. This anomalous phenomenon for type-II superconductors is also studied based upon the theory of thermally activated motion of vortex bundles over a directional-dependent energy barrier. It is shown that the directional-dependent potential barrier renormalizes the Hall and longitudinal resistivities, and the Hall anomaly is induced by the competition between the Magus force and random collective pinning force of the vortex bundle. The Hall and longitudinal resistivities as functions of temperature and applied magnetic field for the thermally activated motion of vortex bundles are calculated. The double

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214

Wei Yeu Chen and Ming Ju Chou sign reversal or reentry phenomenon is also investigated. These studies are crucial because they might make available some important information for their future application.

Keywords: Peak effect; Quasiorder-disorder phase transition; Anomalous Hall effect; Collective pinning; Random walk theorem.

1. Introduction The discovery of high temperature superconductivity, especially the high- Tc cuprate

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superconductors has encouraged enormously scientific research. The properties of these compounds are outlined as: highly anisotropic type-II superconductors [1-42] with either tetragonal or orthorhombic crystal structure, and they share one common feature that all the cuprate superconductors possess the two-dimensional CuO2 planes. The normal state above superconducting transition temperature is the charge-spin separation non-Fermi liquid normal state. The mechanism of these remarkable cuprate superconductors has now gradually become clear after myriads of studies for more than two decades [2] which is the forming of cooper pairs by charge spin recombination. The attractive force between the dressed holons are generated by the exchanging of dressed spin excitations via the interaction between dressed holons occurring directly through the kinetic energy. Recently, we [6] have developed a theory for the quasiorder-disorder first-order phase transition in the vortex system and the theory of thermally activated motion of vortex bundles over a directional-dependent energy barrier [7]. Based on the framework of our theories, the peak effect and anomalous Hall effect for the fascinating cuprate superconductors are investigated. The presence of impurities due to quenched disorder, doping or irradiation destroy the long-range order of flux line lattice, after which only short-range order, the vortex bundles, retains [6, 7, 8]. For low temperature and applied magnetic field, the transverse size of short-range order, the vortex bundle, is approximately 10 −6 m. This is the region of large vortex bundles. If the applied magnetic field increases for a fixed temperature or the temperature increases for a fixed magnetic field, a quasiorder-disorder first-order phase transition between the short-range order and disorder in the vortex system, or the peak effect eventually occurs [6] due to the enormous increase in the dislocations inside the short-range domains. In this case, the vortex lines become a disordered amorphous vortex system. However, they are not individual single quantized vortex lines, the vortex lines still bounded close together to form small vortex bundles of the dimension R ≅ 10 −8 m . This is the region of small vortex bundles of the vortex system. In this Chapter, we shall determine the peak value of the critical current density J c , the exact peak position and its corresponding halfwidth for a constant temperature as well as for a constant applied magnetic field by calculating the critical current density explicitly. On the other hand, based on the present theory of thermally activated motion of vortex bundles jumping over the directionaldependent potential barrier, the anomalous Hall effect for cuprate superconductors is investigated. It is shown that the directional-dependent potential barrier renormalizes the Hall and longitudinal resisitivities strongly and the Hall anomaly is induced by the competition between the Magnus force and the random collective pinning force of the vortex bundles. We

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also find that the domains of anomalous Hall effect includes two regions [7], namely, the region of thermally activated motion of small vortex bundles and that of the large vortex bundles, separated by the contour of the quasiorder-disorder first-order phase transition or the peak effect [6] of the vortex system. The Hall and longitudinal resistivities as functions of temperature and applied magnetic field are also calculated. This Chapter is organized as follows. The theory of peak effect is presented in Sec.2. It is shown that the peak effect is induced by the quasiorder-disorder first-order phase transition of the vortex system. The volume of the short-range order of the vortex system are obtained by evaluating the random, thermal, and quantum averages of the displacement of the vortex lattice inside the short-range order. The peak effect is then investigated by evaluating the critical current density of the vortex bundle explicitly. The peak value of the critical current density J c , the exact peak position and the corresponding half-width for constant temperature and constant applied magnetic field are also obtained. In Sec.3, the theory of thermally activated motion of vortex bundles jumping over a directional-dependent potential barrier is developed. Based on this theory, the anomalous Hall effect for type-II superconductors and the condition for the appearance of this fascinating phenomenon are studied. The Hall and longitudinal resistivities are also calculated. Sec. 4, a conclusive remark is given.

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2. Theory of Peak Effect for Type-II Superconductors The critical current density for type-II superconductors usually decreases with increasing applied magnetic field for a fixed temperature or as the temperature is increased for a fixed applied magnetic field. However, there are many reports that the peak effect in the critical current density is observed as a function of the applied magnetic field for a constant temperature or as a function of temperature for a constant applied magnetic field in quenched disorder type-II superconducting cuprate. The presence of impurities due to doping or quenched disorder destroys the long-range order of flux line lattice, only short-range order, the vortex bundles, remains [6, 7, 8]. Especially, if the applied magnetic field increases for a fixed temperature or the temperature increases for a fixed magnetic field, a quasiorderdisorder first-order phase transition between the short-range order and disorder in the vortex system, or the peak effect eventually arises [6] due to the vast increase in the dislocations inside the short-range domains. In this section we would like to investigate this quasiorderdisorder first-order phase transition in great details.

2.1 Mathematical model The Hamiltonian of the fluctuation for the flux line lattice (FLL) in the z − direction for type-II conventional and high- Tc superconductor is given by [9, 10, 11],

H = H f + HR ,

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Wei Yeu Chen and Ming Ju Chou

where H f = H kin + H e presents the Hamiltonian for the free modes, with H kin the kinetic energy part [9],

H kin =

1 2ρ

∑ Pμ ( K ) Pμ (− K ) ,

(2)

Kμ

H e the elastic energy part [9, 12], He =

1 1 C L K μ K ν S μ ( K ) Sν ( − K ) + 2 Kμν 2

∑

∑μ (C

2 66 K ⊥

+ C 44 K z2 ) S μ ( K ) S μ (− K ) ,

(3)

K

and H R represents the random Hamiltonian, given as [9, 10, 11],

H R = ∑ f Rμ ( K ) S μ (− K ) ,

(4)

Kμ

with ( μ ,ν ) = ( x, y ) , ρ stands for the effective mass density of the flux line [13],

K ⊥2 = K x2 + K y2 , Pμ ( K ), S μ ( K ) are the Fourier transformations of the momentum and displacement operators and C L , C11 , C 44 and C 66 are temperature- and K -dependent bulk modulus, compression modulus, tilt modulus and shear modulus, respectively. f R (K ) denotes

the

Fourier

transformation

of

the

total

random

pinning

force

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f R (r ) = −∇ (VCR (r ) + VSR (r )) , with VCR (r ) the random potential energy of the collective pinning, which is the sum of the contributions of defects within a distance ξ of the vortex core position r , where ξ is the temperature-dependent coherent length and VSR (r ) the random potential energy of strong pinning. The correlation functions of the total random pinning force are assumed to be the short-range correlation,

th are the thermal, quantum, and random averages, and

β (T , B) is the

temperature- and magnetic field-dependent correlation strength. It is understood that the free Hamiltonian can be diagonalized [9]

S fμ ( K ) =

2 ρω Kμ

(α −+Kμ + α Kμ ) ,

with the eigenmodes spectrum

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1

1

ω K 1 = [ (C11 K ⊥2 + C 44 K z2 )] 2 , ρ 1

1

ω K 2 = [ (C 66 K ⊥2 + C 44 K z2 )] 2 , ρ

(7)

where μ = 1 presents the component that parallel to K ⊥ direction, while μ = 2 perpendicular to the K ⊥ direction, with α K+μ , α Kμ stand for the creation and the annihilation operators for the corresponding eigenmodes, respectively.

2.2

Critical current density and transverse size of vortex bundle and peak effect for cuprate superconducting films

The presence of impurities as a result of doping or quenched disorder demolishes the long-range order of flux line lattice, only short-range order, the vortex bundles, resides [6, 7, 8]. By taking into account the balance of the Lorentz force and collective pinning force, the critical current density J c can be obtained as follows: 1

1 β (T , B) 2 Jc = [ ] , B π R2 d

(8)

where B is the applied magnet field, d is the thickness of the superconducting films, with Copyright © 2008. Nova Science Publishers, Incorporated. All rights reserved.

d > λ0 , where λ0 is the penetration depth at zero temperature, R represents the transverse size of the short-range order which can be determined by the following condition,

>th = η 2 ,

(9)

where η is the corresponding collective pinning force range. In consideration of the fact that C11 >> C 66 , after some algebra, we obtain

d 2k nπ z 2 [ 1 − cos ( + K ⊥ R cosθ )] ∑ 2 ∫ d n = −∞ (2π ) d ω2 f B ( ω2 ) β (T , B) ], (10) ×[ + π nπ 2 2 n 2 2 2 C66 K ⊥ + C44 ( ) (C66 K ⊥ + C44 ( ) ) d d

>th =

π

∞

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Wei Yeu Chen and Ming Ju Chou

where f B is the Bose-Einstein distribution function, R = | R | , θ signifies the angle between

2π , … . Taking the d d classical limit, and pondering the non-dispersive regime of the above equation, we arrive at

K ⊥ and R , and the possible values of k z are given by 0, ±

π

,±

2 π k B T 1 β (T , B) R 2 ⋅ + , 2 2 d C66 C44C66 ξ 0

>th =

(11)

where k B is the Boltzmann constant and ξ 0 is the coherent length at zero temperature. In deriving the above equation, the summation has been carried out by contour integration, and 1 the cutoff value for small k has been used. The transverse size of the vortex bundle is R therefore given as 1

R =[

2 2 d C66 2π k B T (η 2 − ) ]2 . β (T , B ) ξ 0 C44C66

(12)

The conditions for the quasiorder-disorder first-order phase transition is given by

>th = γ a02 ,

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where

γ is a dimensionless constance, a 0 = (

(13)

1

2Φ 0

) 2 is the lattice constant and Φ 0 stands

3B for the unit flux. Inserting Eqs (11), and (13) into Eq. (8), we obtain,

β (T , B)

Jc =

( 2 π ) d B C66 [ γ

where C 66 =

B Φ0

16 π μ 0 λ2

,

C 44 =

B2

μ0

a02

1

2 π kB T 1 2 − ⋅ ] C44C66 ξ 0

, λ = λ0

1 − (T / Tc ) , ξ = ξ 0

.

(14)

1 − (T / Tc ) , μ 0

is the permeability and Tc is the critical temperature of the superconductor, λ is the temperature dependent penetration depth. For a constant applied magnetic field the numerator in Eq. (14) is a decreasing function of temperature due to the reduction of condensation energy, while the denominator is also a decreasing function of temperature; therefore, there will exist a peak of the critical current density J c for constant applied magnetic field. This peak critical current density J c appears at temperature TP , when the conditions are satisfied,

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∂ J c (TP , B ) ∂ 2 J c (TP , B ) | B = 0 and |B < 0 , ∂T ∂T2

(15)

and the half-width of the peak is obtained as,

ΔThalf − width = [

1

J c (TP , B ) ]2 . 2 ∂ J c (TP , B ) |( |B ) | ∂T 2

(16)

However, in the case of constant temperature, if the applied magnetic field is not too close to the upper critical field BC 2 , the numerator of Eq. (14) is an increasing function of the applied magnetic field due to the increasing vortex density, while the denominator is also an increasing function of the applied magnetic field B. In this case, there will be a peak critical current density J c at some value B P of the applied field. The peak of the critical current density appears when the following criteria are satisfied:

∂ J c (T , BP ) ∂ 2 J c (T , BP ) |T = 0 and |T < 0 , ∂B ∂ B2

(17)

The corresponding half-width of the peak is given as

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ΔBhalf − width = (

1

J c (T , BP ) )2 . 2 ∂ J c (T , BP ) |( |T ) | ∂B 2

(18)

All the above derivations are independent of the mechanism for the superconductivity. Therefore the present theory is applicable for both conventional and high- Tc superconductors.

2.3

Transverse and longitudinal sizes of vortex bundle, critical current density and peak effect for superconducting bulk materials

In this subsection the critical current density for three-dimensional superconducting bulk materials is calculated. By considering the balance of Lorentz force and collective pinning force, the critical current density J c is given as,

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Wei Yeu Chen and Ming Ju Chou 1

Jc =

[ β (T , B)]2 B [π R L] 2

1 2

,

(19)

where R and L are the transverse and longitudinal sizes of the vortex bundle, and can be determined by the following condition

> th = η 2 .

(20)

Once again, > th are the random, thermal and quantum averages and η is the corresponding collective pinning force range. After some algebra, we obtain

>th =∫

ω2 f B ( ω2 ) β (T , B) d 3k + ] , (21) 2 [ 1 − cos ( K ⊥ ⋅ R + k z L)] ⋅ [ 3 2 2 C66 K ⊥ + C44 k z (C66 K ⊥2 + C44 k z2 ) 2 (2π )

where f B stands for the Bose-Einstein distribution function. In deriving the above equation, we have taken into account the fact that C11 >> C 66 . Taking the classical limit, considering only the non-dispersive regime of the vortex bundle, the above equation now becomes

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>th =

1 kB T ⋅ + 2 π C44C66 ξ0

β (T , B ) ( R 2 +

a02 L2

λ2

1

)2

2π 2C66 C44C66

, (22)

In deriving the above equation we have performed the average over ϕ , which is the angle between the

k S = 2 (R 2 +

K ⊥ and R , and exploited the cutoff values for small k as a 02 L2

λ

2

1

) 2 and for large k as k L =

1

ξ0

, respectively. The transverse and

longitudinal sizes of the vortex bundle are given as

R =[ L =[

2 π 2 C66 C44C66

β (T , B )

2 π 2 C66 C44C66

β (T , B )

(

λ a0

(η 2 −

kB T

1

1

π 2 C44C66 ξ 0

) (η 2 −

kB T

) ]2

1

π 2 C44C66 ξ 0

1

) ]2 .

The conditions for the quasiorder-disorder first-order phase transition is given by

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Theories of Peak Effect and Anomalous Hall Effect for Cuprate Superconductors

221

>th = γ a02 ,

(24)

where

γ is a dimensionless constance. Inserting Eqs. (22) and (24) into Eq. (19), we obtain,

finally,

[ β (T , B)]2

Jc = B[

πλ a0

1 ]2

2

[ 2 π C66 C44C66 ( γ

a02

3

1 k T − 2 B ⋅ ) ]2 π C44C66 ξ 0

.

(25)

It is worthwhile to point out that, from the above equation, for the constant applied magnetic field, B , the numerator is a decreasing function of temperature, T, due to the reduction of condensation energy; however, the denominator is also a decreasing function of temperature, therefore there will exist a peak in the critical current density J c for a constant applied magnetic field B. On the other hand, considering the situation of constant temperature, if the applied magnetic field is not too close to the upper critical field BC 2 the numerator is an increasing function of the applied magnetic field owing to the increasing of vortex density. Moreover, the denominator is also an increasing function of the applied magnetic field B, hence there will exist a peak in the critical current density J c at some value BP of the applied magnetic field. For a constant applied magnetic field, as we have pointed out above, there will be a peak in the critical current density J c at some value of temperature TP for a constant applied

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field B , the conditions for the peak of the critical current density J c appearing at temperature

TP are ∂ 2 J c (TP , B ) ∂ J c (TP , B ) | B = 0 and |B < 0 , ∂T ∂T2

(26)

and the half-width of the peak is given by

ΔThalf − width = [

1

J c (TP , B ) ]2 . 2 ∂ J c (TP , B ) |( |B ) | ∂T 2

(27)

It is interesting to make the numerical estimates of the above results: for an applied TP = 90.6 K , the magnetic field B = 0.5 T we have obtained the peak temperature transverse size of the short-range order R = 2.77 × 10 short-range

order L = 3.35 × 10

−5

−5

m , the longitudinal range of the

m , the critical current density of the peak

J c (TP , B) = 9 ×105 Am −2 and the half-width of the peak for a constant applied magnetic Superconducting Cuprates : Properties, Preparation and Applications, Nova Science Publishers, Incorporated, 2008. ProQuest Ebook Central,

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Wei Yeu Chen and Ming Ju Chou

field ΔThalf − width = 0.3K . From the above results, we find that the vortex bundle is indeed in the non-dispersion regime, which is consistent with our original assumption. These values are in good agreement with the experimental results for YBCO superconducting bulk materials [14]. In obtaining the above results, the following approximate data have been used:

TC = 97 K , β C (TP , B) = 1.637 ×10 − 2 ∂ 2 β C (TP , B) ∂T

2

= −7.34 × 10

N2

−3

3

m K

2

,

N2 m3

,

N2 ∂β c (TP , B ) = −1.81× 10 − 3 3 , ∂T m K

γ = 0.09 , ξ 0 = 1×10 − 9 m , λ0 = 2 ×10 − 8 m .

Considering the situation when the temperature of the system is kept at a constant value T ; if the applied magnetic field is not too close to BC 2 there will be a peak of the critical current density J c at some value B P of the applied field. The conditions for the peak of the critical current density J c are

∂ J c (T , B P ) ∂ 2 J c (T , BP ) |T = 0 and |T < 0 . ∂B ∂ B2

(28)

The corresponding half-width of the peak is given by

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ΔBhalf − width

1

J (T , BP ) =( 2 c )2 . ∂ J c (T , BP ) |( |T ) | ∂B 2

(29)

The numerical estimation of the above results shows that when T = 90.2 K , we obtained −5

the B P = 0.68 T , the transverse size of the short-range order R = 3.486 × 10 m , the −5

longitudinal range of the short-range order L = 4.769 × 10 m , the peak of the critical current density for the constant temperature J c (T , B P ) = 5.2 × 10

5

A , and the m2

corresponding half-width of the peak ΔBhalf − width = 0.02 T . From the above results, we show that the vortex bundle is indeed in the non-dispersion regime in consistence with our original assumption. These values are exactly in consistence with the experimental results for YBCO superconducting crystal [14], where the following approximate data have been exploited:

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TC = 97 K , β C (T , BP ) = 2.264 × 10− 2

223

2 N 2 ∂β C (T , BP ) −2 N , = 5 . 826 × 10 , m3 m3 T ∂B

N2 ∂ 2 β C (T , BP ) −9 −8 = − 28 . 823 , γ = 0.09 , ξ 0 = 1 × 10 m , and λ0 = 2 × 10 m . 2 3 2 m T ∂B

3. Theory of Anomalous Hall Effect for Type-II Superconductors

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In this section the theory of anomalous Hall effect for cuprate superconductors is developed and the conditions of appearance of this Hall anomaly are derived based upon the theory of thermally activated motion of vortex bundles jumping over the directionaldependent energy barrier. This anomalous sign reversal of the Hall resisitivity below the superconducting transition temperature Tc is the most confusing and controversial phenomenon for the last forty years since it was first observed by van Beelen et al. [15] in 1967. Experiments indicated that when the temperature is slightly below the critical temperature of the superconductors, the Hall resistivity in many type-II superconductors changes its sign as the temperature or applied magnetic field decreases. In some occasions, the observed Hall resistivity displays the double sign reversal, or the reentry phenomenon. Numerous theories have been proposed trying to clarify this anomaly, such as the flux flow model [16], the large thermomagnetic model [17], vortex and effective antivortex model [18], opposing drift of quasiparticles model [19], and many others [20]; however, up to the present time, there is no satisfactory elucidation for this sign reversal of Hall resistivity. Still the origination of Hall anomaly remains unsolved. Recently, we [7] have developed a self-consistent theory for the thermally activated motion of the vortex bundles, under the steady-state condition, jumping over the directionaldependent potential barrier generated by the Magnus force, the random collective pinning force and the strong pinning force inside the vortex bundle. The directional-dependent energy barrier denotes that the energy barrier is different when the direction of thermally activated motion is different. Based on the present theory, it is shown that the Hall anomaly is universal for type-II conventional and high- TC superconductors as well as for superconducting bulk materials and thin films, provided certain conditions are satisfied. It is shown that the directional-dependent potential barrier of the vortex bundles renormalizes the Hall and longitudinal resistivities, and the anomalous Hall effect is induced by the competition between the Magnus force and the random collective pinning force of the vortex bundle. Our results demonstrate that the domain of Hall anomaly includes two regions: the region of thermally activated motion of the small vortex bundles and that of the large vortex bundles separated by the contour of the quasiorder-disorder first-order phase transition or the peak effect [6] of the vortex system. For applied magnetic field (temperature) below the quasiorder-disorder first-order phase transition magnetic field B p (temperature Tp ), the vortex system belongs to the region of thermally activated motion of large vortex bundles with the transverse size of the short-range order is approximately 10 −6 m. However, if the applied magnetic field (temperature) increases beyond B p ( T p ), a quasiorder-disorder firstSuperconducting Cuprates : Properties, Preparation and Applications, Nova Science Publishers, Incorporated, 2008. ProQuest Ebook Central,

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order phase transition between the short-range order and disorder in the vortex system, or the peak effect eventually occurs [6] due to the enormous increase in the dislocations inside the short-range domains. In this case, the vortex lines become a disordered amorphous vortex system. However, they are not individual single quantized vortex lines, the vortex lines still bounded close together to generate small vortex bundles of the dimension R ≅ 10 −8 m . In this case, the region belongs to that of the thermally activated motion of small vortex bundles. Under the framework of present theory, the Hall and longitudinal resistivities as functions of temperature as well as applied magnetic field are evaluated for type-II superconducting bulk materials and thin films.

3.1 Mathematical description of the model To proceed, let us first calculate the coherent frequency of the vortex bundle by random walk theorem. From the theory of forced oscillations, it is understood that the response function of the vortex line oscillates inside the potential barrier owing to thermal agitation. By identifying the oscillation energy of the vortex line inside the potential barrier with the thermal energy, the thermal oscillation frequency of the individual vortex inside the potential barrier can be expressed as

ν =ν T ,

(30)

kB 1 , A stands for the average π A 2m amplitude of the oscillation, k B is the Boltzmann constant and m is the mass of the vortex Copyright © 2008. Nova Science Publishers, Incorporated. All rights reserved.

where the proportional constant ν is given by ν =

line [13]. It is worthwhile to point out that the viscous damping of the vortex line is included implicitly in the average amplitude A . However, the oscillations of vortex lines inside the vortex bundle are not coherent, namely, their oscillations are at random. To obtain the coherent oscillation frequency ν C of the vortex bundle as a whole, by applying the random

walk’s theorem, the frequency ν in equation (30) must be divided by the square root of N, the number of vortices inside the vortex bundle

νC =

ν N

=

ν T Φ0 R πB

,

(31)

where Φ 0 is the unit flux, R is the transverse size of the vortex bundle and B is the value of the applied magnetic field. Next we would like to evaluate the root-mean-square of the angle between the random collective pinning force and positive y-direction. Let us consider the case for p-type superconductors with current flowing in the positive x-direction and the applied magnetic field in the positive z-direction. If we assume that the mean angle between the random collective pinning force of a vortex line inside the vortex bundle and the positive y-direction

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measured in counterclockwise sense is θ , this temperature- and field-dependent θ can be obtained as follows: Since θ is small, we can approximately write θ ≅

| f el | | fL |

, | f el | and

| f L | are the magnitudes of the elastic force and the Lorentz force of the vortex line. Taking into account the fact that the compression modulus C11 is much larger than the shear modulus C66 [6], owing to the thermal fluctuations, the magnitude of the displacement vector

| S f ( r ) | of the vortex line inside the vortex bundle as well as its corresponding magnitude of the elastic force | f el | is proportional to

k B C66 or ( 1

B)

T TC − T [6, 9, 20], TC is

the critical temperature of the superconductors. The temperature- and field-dependent θ can now be expressed as

θ (T , B ) = α '

1 B

T , TC − T

(32)

α ′ is a proportional constant. The mean angle Θ (T , B ) between the random collective pinning force of vortex bundle and the positive y-direction measured in counterclockwise sense, by the theory of random walk, can be written as

Θ(T , B ) = N θ (T , B ) = α

T , TC − T

(33)

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with α = α ′ R π / Φ 0 . The root-mean-square value of the angle between the random collective pinning force for the vortex bundle and the positive y-direction in counterclockwise sense can now be obtained as π

Ψ

2 = [ ∫φ2 −π 2

exp(

−φ2 Θ 2 (T , B )

π

) dφ /

2 ∫ −π 2

exp (

−φ2 Θ 2 (T , B )

1

) dφ ] 2 .

(34)

Keeping in mind the fact that Θ(T , B ) is usually very small in our theory, we obtain

Ψ ≅ Θ(T , B ) = α

T . TC − T

(35)

Finally, we evaluate the Hall and longitudinal resistivities through the calculation of directional-dependent energy barrier of the vortex bundles formed by the Magnus force, the random collective pinning force together with the strong pinning force inside the vortex bundle for the magnetic field in the z-direction B = B ez and the transport current in the x-

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Wei Yeu Chen and Ming Ju Chou

direction J = J e x . Considering the case where the magnitude of Lorentz force JB is slightly greater than the magnitude of the random collective pinning force, after some algebra, the directional-dependent energy barrier of the vortex bundles both in the positive and negative x-direction as well as the positive and negative y-direction are obtained respectively as

U + V R ( JB

vby

U − V R ( JB

vby

vT vT

− < Fp x > R ) ,

(36)

− < Fp x > R ) ,

(37)

U + V R ( JB − JB

vbx − < Fp y > R ) , vT

(38)

U − V R ( JB − JB

vbx − < Fp y > R ) , vT

(39)

where the potential barrier U is generated by the strong pinning force due to the randomly distributed strong pinning sites inside the vortex bundle, vb ( vT ) is the velocity of the vortex bundle (super current), V is the volume of the vortex bundle, once again, R represents the transverse size of the vortex bundle, the range of U is assumed to be the order of R and

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< F p > R stands for the random average of the random collective pinning force per unit volume. The self-consistent equations for the velocity of the thermally activated motion of the vortex bundles jumping over the directional-dependent energy barrier are therefore obtained in components as

vbx = ν C R {exp[ − exp[

v −1 ( U − V R ( JB by − < Fp x > R )) ]} , k BT vT

and

vby = ν C R {exp[ − exp [

v −1 ( U + V R ( JB by − < Fp x > R ))] k BT vT (40)

v −1 ( U + V R ( JB − JB bx − < Fp y > R ))] k BT vT

v −1 ( U − V R ( JB − JB bx − < Fp y > R ))]} , k BT vT

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227

where ν C is the coherent oscillation frequency of the vortex bundle jumping over the directional-dependent potential barrier from one equilibrium position to another. Taking into v account the fact that bx R | sin Ψ )] R πB k BT k BT vT

ν

vbx = (

− exp[

+ V R − JB | vby | ( + |< Fp > R | sin Ψ )]} , k BT vT

(42)

ν

vby = (

T Φ0 −U −V R ) R exp( ) {exp[ ( JB − |< Fp > R | cos Ψ )] R πB k BT k BT − exp[

+V R ( JB − |< Fp > R | cos Ψ )]} , k BT

(43)

where Ψ is the root-mean-square value of the angle between the random collective pinning force of the vortex bundles and the positive y -direction measured in the counterclockwise

Ey Ex , ρ xy = together with Eq. J J (35) and keeping in mind that Ψ is usually very small, the longitudinal and Hall resistivities, can now be written respectively as follows:

sense. By considering the identities E = −vb × B , ρ xx =

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ρ xx =

ν BT Φ0 J π

ρ xy =

−ν

1

exp(

B T Φ0 J π − exp [

and

with (

exp(

1 | vby | V R β C (T , B ) 2 T −U ){exp [ (( ) α )] − JB vT k BT k BT TC − T V

1 | vby | − V R β C (T , B ) 2 T − JB (( ) α )]} , vT k BT TC − T V

| vby |= Jρ xx / B ,

β C (T , B )

1

−U −VR β C (T , B ) 2 β C (T , B ) 2 , (44) VR ){exp [ ( JB − ( ) )] − exp[ ( JB − ( ) )]} k BT k BT k BT V V

1 )2

(45) (46)

is the magnitude of the random collective pinning force per unit volume, V α is a proportional constant. It is clear that, from the above calculations, the directionaldependent potential barrier of the vortex bundle renormalized the Hall and longitudinal resistivities. Taking into account the fact that the arguments in the exponential functions inside the curly bracket of Eqs. (44) and (45) are very small when the Lorentz force is close to the random collective pinning force, we finally obtain the temperature- and field-dependent longitudinal and Hall resistivities as

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Wei Yeu Chen and Ming Ju Chou

ρ xx = ρ xy =

−ν

ν B Φ0

J πT

(47)

1 | vby | β C (T , B ) 2 T − U 2V R )( ) [( ) α ] , − JB k BT kB TC − T vT V

(48)

J πT

B Φ0

exp(

1

β C (T , B ) 2 −U 2V R ) ( ) [ JB − ( ) )] , k BT kB V

exp (

with | vby |= Jρ xx / B . Let us assume that the vortex system is initially in the region of the thermally activated motion of small vortex bundles. From our previous study [6], the value of 1

1 β C (T , B) 2 ( ) increases with decreasing applied magnetic field when the temperature of B V the system is kept at a constant value T ; if this value passes the value of | vby | T ( ) / (α ) , then ρ xy changes sign from positive to negative with decreasing vT TC − T 1

1 β C (T , B) 2 ) increases and that of applied magnetic field. On the other hand, the value of ( B V

α

T decreases as temperature decreasing when the applied magnetic field is kept at a TC − T 1

T 1 β C (T , B) 2 constant B . Therefore, the term ( ) α exists a maximum at some B V TC − T

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| vby |

, then ρ xy possesses the sign vT reversal property. From the above analysis, the sign reversal of the Hall resistivity appears when the following condition is satisfied: temperature T , if this maximum is greater than J

1

| vby | T 1 β C (T , B ) 2 >J , ( ) α vT B V TC − T

(49)

provided T p ( B p ) is not too close to Tc ( Bc2 ). The sign reversal phenomenon could therefore be observed for cuprate superconductors. The above results show the fact that the anomalous Hall effect is induced by the competition between the Magnus force and the random collective pinning force. In certain occasion, ρ xy might appear the double sign 1

1 β C (T , B) 2 reversal property, this will be discussed in Sec. 3.4. Since the value of ( ) B V increases with decreasing temperature (applied magnetic field), therefore, ρ xx decreases monotonically as temperature (applied magnetic field) decreases. Finally when temperature (applied magnetic field) decreases below T p ( B p ) , the quasiorder-disorder first-order phase transition temperature (magnetic field) of the vortex system, the region crosses over to that of

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thermally activated motion of large vortex bundles. In this case, both the Hall and longitudinal resistivities decay to zero quickly with decreasing temperature (magnetic field). From the above analysis, the Hall anomaly is independent of the mechanism for the superconductors and the expression of V , the volume for the vortex bundle. Hence the Hall anomaly is universal for type-II conventional and high- Tc superconductors as well as for superconducting bulk materials and thin films, provided the conditions given above are satisfied.

3.2

Anomalous Hall effect for type-II superconducting bulk materials Let us focus on superconducting bulk materials, the volume V for the vortex bundle in

Eqs. (47) and (48) is given as V = π R L , where R (L) is the transverse (longitudinal) size 2

of the vortex bundle. The longitudinal and Hall resistivities for type-II superconducting bulk materials now obtain as

ρ xx = ρ xy =

(50)

|v | − U 2 π R 3 L β C (T , B) 2 T − JB by ] , )[ ] [( ) α vT k BT kB V TC − T

(51)

J π T

−ν Φ0 B J π T

exp(

1

− U 2π R3 L β C (T , B) 2 )[ ][ JB − ( ) ] , k BT kB V

ν Φ0 B

exp(

1

with | vby |= Jρ xx / B . As we have indicated in Sec. 3.1, the above equations give rise to the

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phenomenon of anomalous Hall effect, namely, the value of ρ xy changes it sign from positive to negative with decreasing applied magnetic field (temperature). The cases for constant temperature and constant applied magnetic field are investigated separately as follows. Under the framework of present theory, suppose the system is in the region of thermally activated motion of small vortex bundles, ρ xy possesses the desired anomaly properties as applied magnetic field decreases when the temperature of the system is kept at a constant value T . Moreover, from our previous study [6], it is shown that the quasiorder-disorder first-order phase transition of the vortex system occurs if the applied field decreases below B p , then the region crosses over to that of the thermally activated motion of large vortex bundles. In this regime, the potential barrier U generated by the randomly distributed strong pinning sites inside the vortex bundle becomes large; hence, both the Hall and longitudinal resistivities approach to zero quickly with decreasing applied magnetic field. In the case of constant applied magnetic field, within the present theoretical framework, the Hall resistivity changes its sign from positive to negative as the temperature decreases. From our [6] previous study, it has shown that the quasiorder-disorder first-order phase transition of the vortex system occurs when the temperature decreases below Tp , then the region crosses over to that of the thermally activated motion of large vortex bundles. The potential barrier U generated by the randomly distributed strong pinning sites within the

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Wei Yeu Chen and Ming Ju Chou

vortex bundle becomes large, therefore, both the Hall and longitudinal resistivities reduce to zero quickly with decreasing temperature. Under the framework of present theory, ρ xy possesses the desired anomaly properties as applied magnetic field decreases when the temperature of the system is maintained at a constant value T . The numerical calculations of ρ xy and ρ xx of Eqs (50) and (51) when the temperature is kept at T = 91K , the Hall as well as longitudinal resistivity as functions of applied magnetic field in Tesla is given as follows:

ρ xy ( B = 3.5) = 1.2914 × 10 −9 Ω − m , ρ xy (3.03) = 1.7658 ×10 −15 Ω − m , ρ xy (2.5) = −1.4627 × 10 −9 Ω − m , ρ xy (2.0) = −2.7907 × 10−9 Ω − m , ρ xy (1.5) = −3.9996 × 10−9 Ω − m , ρ xy (1.0) = −4.6405 × 10−9 Ω − m ρ xy (0.75) = −3.9801× 10 −9 Ω − m , ρ xy (0.5) = −2.0098 × 10 −9 Ω − m ; ρ xx (3.5) = 1.8739 × 10 −6 Ω − m ,

ρ xx (3.03) = 1.5916 × 10 −6 Ω − m , ρ xx (2.5) = 1.2663 × 10 −6 Ω − m , ρ xx (2.0) = 9.5142 × 10 −7 Ω − m , ρ xx (1.5) = 6.2539 × 10 −7 Ω − m , ρ xx (1.0) = 2.9669 × 10 −7 Ω − m , ρ xx (0.75) = 1.6825 × 10 −7 Ω − m

ρ xx (0.5) = 1.226 × 10 −7 Ω − m . In obtaining the above results, the following approximate data have been employed: R = 2 × 10 −8 m , L = 10 −6 m , J = 106 −5

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α = 5.59 × 10 T (

β C (3.03) V

1

−U β C ( B = 3.5) 2 exp( ) = 2.07 × 10 − 2 m , ( ) = 3.4506 × 10 6 N / m 3 , kB T V

1

) 2 = 2.9849 × 106 N / m3 , (

1 β C (1.5) 2 ( )

V

−1 2 ,

A , TC = 92 K , vT = 10 3 m / sec , ν = 1011 sec −1 , 2 m

= 1.4748 × 106 N / m3 ,

β C (2.5) V

1 β C (1) 2 ( )

and (

V

β

C

1

) 2 = 2.4605 × 106 N / m3 , ( = 9.854 × 105 N / m3 , (

1 (0.5) 2 )

V

β C (2) V

β C (0.75) V

1

) 2 = 1.9668 × 106 N / m3 , 1

) 2 = 7.4042 × 105 N / m3 ,

= 4.915 × 10 5 N / m 3 .

The quasiorder-disorder first-order phase transition of the vortex system occurs [6] when the applied field decreases below 0.5 Tesla, the region crosses over to that of the thermally activated motion of large vortex bundles. In this case, the potential barrier U generated by the randomly distributed strong pinning sites inside the vortex bundle becomes large; hence, both of the Hall and longitudinal resistivities approach to zero quickly with decreasing applied magnetic field. These results are in agreement with the experimental data on YBa2Cu 3O7 −δ high- TC bulk materials [21].

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3.3 Anomalous Hall effect for type-II superconducting films Now let us investigate the anomalous Hall effect for type-II superconducting films. In this case, the volume V for the vortex bundles in Eqs. (47) and (48) is given by V = π R d , 2

where R is the transverse size of the vortex bundle and d is the thickness of the film. The Hall and longitudinal resistivities become as

ρ xx = ρ xy =

ν Φ0 B

(52)

|v | − U 2 π R 3d T β C (T , B) 2 − JB by ] , )[ ] [( ) α k BT kB V TC − T vT

(53)

J π T

−ν Φ 0 B J π T

exp (

1

β C (T , B) 2 − U 2π R 3d )[ ][ JB − ( ) ] , k BT kB V

exp(

1

with | vby |= Jρ xx / B . The above equations, as discussed in Sec.3.1, bring about the phenomenon of anomalous Hall effect for both the constant applied magnetic field and constant temperature. Within the framework of present theory, it is shown that when the system is initially in the region of thermally activated motion of small vortex bundles, the Hall resistivity ρ xy changes its sign from positive to negative as the applied magnetic field decreases when the temperature of the system is kept at a constant value T . The quasiorderdisorder first-order phase transition of the vortex system takes place [6] when the applied magnetic field decreases below B p . In this case, the region belongs to the thermally activated motion of large vortex bundles. The potential barrier U generated by the randomly distributed strong pinning sites within the vortex bundle becomes large; hence both ρ xy and

ρ xx reduce to zero rapidly. Copyright © 2008. Nova Science Publishers, Incorporated. All rights reserved.

For constant applied magnetic field when T > T p , based on the present theory, the Hall resistivity ρ xy changes its sign from positive to negative as the temperature of the system decreases when the applied magnetic field is kept at a constant value B . The quasiorderdisorder first-order phase transition of the vortex system takes place [6] when the temperature decreases below Tp , then the region crosses over to that of thermally activated motion of large vortex bundles. In this region, the potential barrier U generated by the randomly distributed strong pinning sites inside the vortex bundle becomes large. Therefore, both ρ xy and

3.4

ρ xx decrease quickly to zero with deceasing temperature. Reentry phenomenon for anomalous Hall effect

The double sign reversal or the reentry phenomenon for the anomalous Hall effect is studied in this subsection. The crucial conditions for occurring this fascinating reentry phenomenon are

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Wei Yeu Chen and Ming Ju Chou 1

| vby | T 1 β C (T , B ) 2 >J ( ) α vT B V TC − T and

TP < TR ,

(54)

where T p is the quasiorder-disorder first-order phase transition [6] temperature, TR is the temperature for ρ xy crossing over back from negative to positive value. It is understood that

Tp decreases with increasing (

β (T , B)

1

) 2 , total random pinning force, namely, the random

V collective pinning force plus the strong pinning force of the vortex system [22]. Therefore, materials with large (

β (T , B)

1

) 2 , such as YBa2Cu3O7 −δ [23], Tl2 Ba2Cu 2O8 [24], the

V reentry phenomenon could be observed. The quasiorder-disorder first-order phase transition of the vortex system takes place [6] when the temperature decreases below Tp , the system crosses over to the region of thermally activated motion of large vortex bundles. In this case, the potential barrier U generated by the randomly distributed strong pinning sites inside the vortex bundle becomes large, therefore, both ρ xy and ρ xx decrease rapidly to zero with decreasing temperature.

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4. Conclusion Based on our recently developed theories, for the quasiorder-disorder first-order phase transition in the vortex system and the thermally activated motion of vortex bundles jumping over a directional-dependent potential barrier, the peak effect and anomalous Hall effect for type-II superconductors are investigated. It is found that the half-widths of the peaks of critical current density are very small in conventional or high- Tc superconductors, bulk materials or thin films, these phenomena are universal for processes induced by first-order phase transition, that the directional-dependent potential barrier renormalized the Hall and longitudinal resistivities strongly, and that the Hall anomaly is induced by the competition between the Magnus force and random collective pinning force. We also find that the domain of Hall anomaly includes two regions: the regime of thermally activated motion of small vortex bundles and that of large vortex bundles separated by the contour of quasiorderdisorder first-order phase transition of the vortex system. The peak values of the critical current density, the exact peak positions and its corresponding half-widths, the Hall and longitudinal resistivities are calculated for constant applied magnetic field and constant temperature. The conditions for the appearance of sign reversal and double sign reversal are also discussed. All the results are in good agreement with the experiments.

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[28] LeBlanc, M. A. R.; Little, W. A. Proc. of the Seven Int. Conf. on Low Temperature Physics (University of Toronto Press, Toronto, 1960), p. 198. [29] Niessen, A.K.; Staas, F. A. Phys. Lett. 1965, 15, 26-28. [30] Bardeen, J.; Stephen, M.J. Phys. Rev. 1965, 140, A1197-A1207. [31] Van Vijfeijken, A. G.; Niessen, A. K. Phys. Lett. 1965, 61, 23-24; Nozie’res, P.; Vinen, W.F. Philos. Mag. 1966, 14, 667-688. [32] Fiory, A. T. Phys. Rev. B 1973, 8, 5039-5044. [33] Kes P. H. and Tsuei C. C. Phys. Rev. B 1983, 28, 5126-5139; Wördenweber, R.; Kes, P. H.; Tsuei, C. C. Phys. Rev. B 1986, 33, 3172-3180. [34] Niessen A.K.; Staas F.A. and Weijsenfeld C.H. Phys. Lett. 1967, 25A, 33-35; Weijsenfeld, C. H. Phys. Lett. 1968, 28A, 362-363. [35] Koshelev, A. E.; Vinokur, V. M. Phys. Rev. B 1998, 57, 8026-8033; Mikitik G. P.; Brandt, E. H. Phys. Rev. B 2001, 64, 184514-1-184514-14. [36] Smith, A.W.; Clinton, T. W.; Tsuei, C. C.; Lobb, C. J. Phys. Rev. B 1994, 49, 1292712930; Hagen, S. J.; Smith, A.W.; Rajeswari, M.; Peng, J. L.; Li, Z. Y.; Greene, R. L.; Mao, S. N.; Xi, X. X.; Bhattacharya, S.; Li, Q.; Lobb, C. J. Phys. Rev. B 1993, 47, 10641068. [37] Essmann, U.; Träuble, H. Phys. Letters 1967, 24A, 526-527; Gammel, P.L.; Bishop, D. J.; Dolan, G. J.; Kwo, J. R.; Murray, C. A.; Schneemeyer, L. F.; Waszczak, J. V. Phys. Rev. Lett. 1987, 59, 2592-2595; Kleiner W.H.; Roth, L. M.; Autler, S. H. Phys. Rev. 1964, 133, A1226-A1227; Lasher, G. Phys. Rev. 1965, 140, A523-A528; Kramer, E. J. J. Appl. Phys. 1978, 49, 742-748. [38] Beasley, M.R.; Labusch, R.; Weber, W. W. Phys. Rev. 1969, 181, 682-700; DewHughes D. Cryogenics 1988, 28, 674-677; Yeshurun, Y.; Malozemoff, A. P. Phys. Rev. Lett. 1988, 60, 2202-2205. [39] Moore M.A. Phys. Rev. B 1989, 39, 136-139; Brandt E.H. Phys. Rev. Lett. 1989, 63, 1106-1109; Gazman L.I. and Koshelev A.E. Phys. Rev. B 1991, 43, 2835-2843. [40] Brandt, E. H. J. Low Temp. Phys. 1977, 26, 709-733, ibid. 735-753; 28, 263-289, ibid. 291-315; Brandt E.H. J. Low Temp. Phys. 1986, 64, 375-393; Fisher M.P.A. Phys. Rev. Lett. 1989, 62, 1415-1418; Nelson D.R. and Seung H.S. Phys. Rev. B 1989, 39, 91539174. [41] Clem J.R. Phys. Rev. B 1991, 43, 7837-7846; Artemenko S.N. and Kruglo A.V. Phys. Lett. A 1990, 143, 485-488; Fisher K.H. Physica C 1991, 178, 161-170 (1991); Lawrence W.E. and Doniach S. Proc. 12th Internatl. Conf. of Low Temperature Physics LT 12 (E. Kanda,. Academic Press of Japan, Kyoto, 1971) p.361. [42] Reed, W. A.; Fawcett, E.; Kim, Y. B. Phys. Rev. Lett. 1965, 14, 790-792; Zavaritsky N.V.; Samoilov, A.V.; Yurgens, A.A. Physica C 1991, 180, 417-425; Chien, T. R.; Jing, T. W.; Ong, N. P.; Wang, Z. Z. Phys. Rev. Lett. 1991, 66, 3075-3078.

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ISBN: 978-1-60456-919-3 c 2009 Nova Science Publishers, Inc.

Chapter 8

M ERCURY-B ASED S UPERCONDUCTING C UPRATES: H IGH Tc AND P SEUDO S PIN -G AP Y. Itoh1 and T. Machi2 1 Department of Chemistry, Graduate School of Science, Kyoto University, Kyoto 606-8502, Japan 2 Superconductivity Research Laboratory, International Superconductivity Technology Center, 1-10-13 Shinonome, Koto-ku, Tokyo 135-0062, Japan

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Abstract Mercury-based cuprates HgBa2CuO4+δ with 0 < δ < 0.2 (Hg1201) are the superconductors with a single CuO 2 layer in unit cell and the optimally oxidized one has the highest Tc = 98 K among the ever reported single-CuO 2-layer superconductors. Double CuO 2 layered cuprates HgBa2 CaCu2O6+δ with 0.05 < δ < 0.35 (Hg1212) have the highest Tc = 127 K at the optimal oxygen concentration. This is the highest Tc among the ever reported double-CuO 2-layer superconductors. The Hg1201 has the nearly perfect fat CuO 2 plane. The Hg1212 has the flattest CuO 2 plane among the other lower Tc double-layer cuprates, which is associated with the mystery of the highest Tc . Both systems have a pseudo spin-gap in the magnetic excitation spectrum of the normal states. In this article, we present the microscopic studies of magnetic and electric properties of the Hg-based superconducting cuprates using nuclear magnetic resonance (NMR) and nuclear quadrupole resonance (NQR) techniques. NMR and NQR are powerful to detect local information through the nuclear sites in materials and have supplied us with information on low frequency magnetic response of electronic systems. Although the structure analysis indicates the flat CuO 2 planes, zero field 63,65Cu NQR spectra, which are sensitive to local electric charge distribution, show inhomogeneous broadening. The local electrostatic states are rather inhomogeneous. Although the d-wave superconductivity must be fragile to imperfection and non-magnetic impurities, the pure Hg-based superconducting cuprates show impure 63,65 Cu NQR spectra but rather robust pseudo spin-gap in the 63 Cu NMR Knight shift and nuclear spin-lattice relaxation rate over the wide doping regions. There had been an issue whether the pseudo spin-gap results from a double-layer coupling or a single layer anomaly. The NMR results for Hg1201 served as the evidence for the existence of the single-layer pseudo spin-gap. The pseudo spin-gap is explained by a precursory phenomena of superconducting pairing fluctuations or spin singlet correlation. The

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Y. Itoh and T. Machi different doping dependence of the pseudo spin-gap of Hg1201 and Hg1212 is associated with the different Fermi surface contour. The similar temperature dependence of the 199Hg and the 63Cu nuclear spin-lattice relaxation times indicates uniform interlayer coupling. The double-layer coupling effect is revisited through the comparison of Hg1201 and Hg1212.

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1.

Introduction

The BCS theory is one of the most successful theories in physics. It has not only revealed the mechanism of phonoic superconductivity but also developed our understanding of spontaneous breaking of gauge symmetry. The BCS theory as a field theory has been applied to various stages of physics, elementary particles, condensed matter and cosmology. The discovery of high-Tc cuprate superconductors has renewed our interests of superconductivity. Low dimensionality on layered compounds, carrier doping effect, electron correlation effect, antiferromagnetic ordering and Mott transition play the key roles to understand the high-Tc physics. Since the high-Tc superconductivity emerges in close to antiferromagnetic instability, the significant role of magnetism is associated with the highTc mechanism. Since Coulomb repulsion between electrons can produce spin and charge fluctuations, a close relation between magnetism and superconductivity is suggested. The superconducting properties except the high Tc and vortex matter physics are conventional. The electronic and magnetic properties in the normal conducting state are unconventional. The high-Tc cuprate superconductors have still attracted great interests. We may anticipate new development in understanding the solid state physics. Mercury-based cuprates HgBa2CuO4+δ with 0 < δ < 0.2 (Hg1201) are the superconductors with a single CuO 2 layer in unit cell and the optimally oxidized one has the highest Tc = 98 K among the ever reported single-CuO 2-layer superconductors [1, 2]. Mercurybased cuprates HgBa2 CaCu2 O6+δ with 0.05 < δ < 0.35 (Hg1212) are the superconductors with double CuO 2 layers in unit cell and the optimally oxidized one has the highest Tc = 127 K among the ever reported double-CuO 2-layer superconductors [3]. Hg1201 and Hg1212 are the layered compounds and contain one and two CuO 2 planes in unit cell, respectively [4]. Triple-CuO 2-layer HgBa2 Ca2 Cu3 O8+δ (Hg1223) is the highest Tc = 134 K at ambient pressure [3] and Tc = 164 K at high pressure [5]. Hg1234 is also known to be synthesized [6]. But it has lower Tc than Hg1223 [7]. In this article, we present the Hg-based superconducting cuprates and the microscopic magnetic properties studied through nuclear magnetic resonance (NMR) and nuclear quadrupole resonance (NQR) techniques. Especially, we present the Cu NMR evidence of the existence of a pseudo spin-gap in the norma-state magnetic excitation spectrum of the single-layer cuprate superconductor. The discovery of a big pseudo spin-gap of the underdoped single-layer cuprate superconductor has turn our attention to the unconventional electronic state above Tc.

2.

Pseudo Spin-Gap: Secondary or Inherent?

Conventional metallic states are well described by the Landau-Fermi liquid theory. Uniform magnetic susceptibility χs exhibits Pauli paramagnetism. Since the density of electron Superconducting Cuprates : Properties, Preparation and Applications, Nova Science Publishers, Incorporated, 2008. ProQuest Ebook Central,

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237

states is finite at the Fermi level, χs is finite at T = 0 K and nearly independent of temperature. Nuclear spin-lattice relaxation time T1 satisfies Korringa relation with the spin Knight shift. Since the quasi-particle scattering through the finite density of states induces the nuclear spin relaxation, 1/T1T is finite at T = 0 K. Magnetic itinerant compounds involving the transition metal elements often exhibit Curie-Weiss magnetism, in spite of the absence of localized moments. Itinerant magnetism breaks down the Korringa relation. In antiferromagnetic compounds, the nuclear spin-lattice relaxation rate divided by temperature 1/ T1T is enhanced in a Curie-Weiss law, more than the uniform spin susceptibility χs . The normal-state pseudo spin-gap was first confirmed by NMR experiments [8–13]. The decrease of a static uniform spin susceptibility χs with cooling down was observed in the NMR Knight shift measurements for the widely-studied double-layer YBa 2 Cu3 O7−δ . The decrease of the nuclear spin-lattice relaxation rates divided by temperature 1/ T1T was also observed for the underdoped cuprates in the normal state. Thus, the existence of a pseudogap in the magnetic excitation spectrum was observed in the underdoped doublelayer superconductors in the normal states. However, the existence of such a pseudogap in the magnetic excitation spectrum was not obvious for the single-layer superconductor La2−x Srx CuO4 . Then, there had been an issue whether the pseudogap in the magnetic excitation spectrum is intrinsic in a CuO 2 plane or it is secondary due to spin singlet formation between adjacent CuO2 planes in the double layer. The NMR studies of Hg1201 gave us an obvious evidence of the pseudo spin-gap in the single layer. The review articles of the intensive NMR studies of the high- Tc YBa2 Cu3 O7−δ can be seen in [14–17]. Hence, we focus on the NMR studies of Hg1201 and Hg1212.

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3. 3.1.

Crystal Structure and Phase Diagram Record High Tc

Figure 1 shows the crystal structures of the Hg-based superconductors Hg1201 and Hg1212 [4]. Strong covalent O-Hg-O bonds along the c-axis, which look like “dumbbell” shape, and dilute oxygen concentration in the HgO δ layers are characteristics of the crystal structures. The change in oxygen concentration of the HgO δ layers can yield a wide carrier doping region. Figure 2 (a) shows the oxygen concentration dependences of Tc of Hg1201, Hg1212, and Hg1223 [18]. The typical “bell”-shaped dependences of Tc are seen for Hg1201 and Hg1212. With changing only the oxygen concentration, the electronic states of Hg1201 and Hg1212 develop from the underdoped to the overdoped regions Although no magnetic ordering states nor insulating states were confirmed for the deeply underdoped samples of Hg1201 and Hg1212, the existence of Hg-based cuprate insulators was observed for Hg2Ba2 YCu2 O8−δ (Hg2212) which contains the double Hg layers [19–21]. Figure 2 (b) shows Tc plotted against the in-plane a-axis lattice parameter for Hg1201, Hg1212, and Hg1223 [18, 22]. The in-plane Cu-Cu distances of Hg1212 and Hg1223 are shorter than that of Hg1201. The shrunk CuO 2 planes characterize the higher Tc cuprates. Figure 2 (c) shows Tc plotted against the hole concentration Psh defined by the ionic formal valence of Cu 2+Psh [18]. We estimated Psh = 2δ for Hg1201 and Psh = δ for Hg1212.

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Y. Itoh and T. Machi

HgBa2CaCu2O6+

HgBa2CuO4+δ BaO CuO2 BaO HgOδ BaO CuO2 BaO

δ

BaO CuO2 Ca CuO2 BaO HgOδ BaO CuO2 Ca CuO2 BaO

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HgOδ

Figure 1. Crystal structures of single-layer HgBa 2 CuO4+δ and double-layer HgBa2 CaCu2 O6+δ . Strong O-Hg-O bonds like “dumbbell” along the c-axis and dilute oxygen concentration in the HgO layers are characteristics of these crystal structures.

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150 (a)

239

Hg1223 Hg1212

Tc (K)

Hg1201 100 50 0 0.0

0.1 0.2 0.3 Excess oxygen content δ

0.4

150 (b)

Tc (K)

underdoped 100

underdoped overdoped

50

150

3.86 3.88 3.90 In-plane lattice constant a (A) (c) Hg1212

Tc (K)

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0 3.84

100 50 0 0.0

Hg1201

0.1 0.2 0.3 0.4 2+Psh Hole concentration Psh of Cu

Figure 2. Tc phase diagrams of Hg1201 and Hg1212 reproduced from [18, 22]. Tc is plotted against excess oxygen concentration δ [18](a), the in-plane a-axis lattice constant [18,22](b), and hole concentration Psh defined by the ionic formal valence of Cu 2+Psh (Psh = 2δ for Hg1201 and Psh = δ for Hg1212) [18](c).

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Y. Itoh and T. Machi

O-Hg-O dumbbell apical oxygen bond angle

Cu Cu O O CuO2 plane

HgB2CaCu2O6+δ

Figure 3. A CuO2 plane (left) and two key structure parameters (right) are illustrated. The two key structure parameters (right) are the distance from the plane-site Cu to the apical oxygen and the bond angle between the Cu ions via the oxygen ion in the CuO 2 plane. The flatness of the CuO 2 plane is believed to be a significant factor to the higher Tc .

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The optimal hole concentration is nearly the same as the typical values of 0.18-0.20. Then, the mechanism of the record high Tc is not only due to the substantial hole concentration.

3.2.

Flat CuO2 Plane

Figure 3 illustrates a CuO 2 plane and two key structure parameters. The two key parameters are the distance from the plane-site Cu ion to the apical oxygen and the bond angle between the plane-site Cu ions via the plane-site oxygen. Both parameters are associated with the flatness of the CuO 2 planes. The large separation between Cu and the apical oxygen and the nearly 180 degree of the in-plane Cu-O-Cu bond angle are observed in Hg1201 and Hg1212 [23–28]. The optimal Tc = 38 K La2−x Srx CuO4 with the single CuO 2 layers and the optimal Tc = 93 K YBa2 Cu3 O6.9 with the double CuO 2 layers have the buckling structures in the CuO 2 planes. The CuO2 planes of Hg1201 and Hg1212 are the flattest among the reported superconducting cuprates [23–28]. Many researchers believe that the nearly perfect flat CuO 2 plane is a significant factor to realize the higher Tc. Actually, the double-Hg-layer (Hg, Tl)2212 has the relatively lower Tc and involves the bucking in the CuO 2 planes [21, 29]. In the (Hg, Tl)2212, the in-plane Cu-O-Cu bond is bended and the bond angle is about 170 degree. The flatness of the CuO 2 planes is associated with a higher Tc mechanism or at least an inevitable background of crystal structure.

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4.

241

Two Dimensional Conductor

Anisotropy of electrical resistivity has been measured for Hg1201 single crystals [30] and Hg1212 epitaxial thin films on vicinal substrates [31, 32]. Figure 4 shows the in-plane resistivity ρab and the out-of-plane resistivity ρcc for Hg1201 single crystal of Tc = 97 K reproduced from [30] (a) and those of (Hg, Re)1212 of Tc = 117 K reproduced from [31,32] (b). Dash lines are T linear functions for the in-plane resistivity ρab . The T -linear resistivity is associated with the conduction electron scattering due to two dimensional antiferromagnetic spin fluctuations [33, 34]. The deviation from the T -linear behavior is associated with the scattering suppression due to the opening of a pseudo spin-gap in the magnetic excitation spectrum. The onset temperatures of deviation from the T linear functions are denoted by T ∗. At T ∗ , the out-of-plane resistivity ρcc also takes the minimum value. The out-of-plane resistivity ρcc is two or three order higher than the in-plane resistivity ρab and the temperature dependence is different from each other. The metallic in–plane resistivity ρcc and the semi-conducting out-of-resisitivity ρcc above T ∗ is an evidence of two dimensional electrical conduction in Hg1201 and Hg1212. This has been known for the other superconducting cuprates. The in-plane and the out-of-plane electrical conduction is correlated with each other with respect to the T ∗. The two dimensional electrical resistivity is understood in terms of a hot spot and a cold spot on the two dimensional Fermi surface [34].

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5.

Cu NQR, NMR and Magnetic Fluctuations

Nuclear magnetic resonance (NMR) and nuclear quadrupole resonance (NQR) are powerful techniques to characterize microscopically magnetic insulators, metals, superconductors, alloys and compounds [35]. Microscopic studies using the NMR and NQR techniques have provided us with rich information inside unit cell through the nuclear sites in a site-selective way. Using the NMR and NQR techniques, one can obtain static and dynamic information of the electronic systems.

5.1.

Inhomogeneous Cu NQR Spectrum of Pure Hg-Based Cuprate

63,65

Cu nuclei have spin I = 3/2 and then quadrupole moments Qm . In a non-cubic crystalline symmetry, they interact with the electric field gradient Vαβ (α, β = x, y, z) of electric static crystalline potential V [35]. Nuclear quadrupole Hamiltonian is given by HQ =

e2 qQm η 2 2 [(3Iz2 − I 2 ) + (I+ + I− )] 4I(2I − 1) 2

(1)

where q is the maximum component of the electric field gradient eq ≡

∂ 2V ∂z 2

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(2)

242

Y. Itoh and T. Machi

0.8

400

0.4

→ ρcc (mΩcm)

↑ T* ↓

200

←

0.0 0.8

0 600

ρab (mΩcm)

(Hg, Re)1212 (Tc ~ 117 K) ↑ T* ↓

0.4

←

0.0 0

100

→ 400

ρcc (mΩcm)

ρab (mΩcm)

Hg1201 (Tc ~ 94 K)

200

200

0 300

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T (K)

Figure 4. Anisotropic electrical resistivity of ρab and ρcc of Hg1201 (upper panel) [30] and Hg1212 (lower panel) [31,32]. Dash lines for the in-plane resistivity are T linear functions. The onset temperatures of deviation from the T linear functions are denoted by T ∗ , which are also the minimum temperatures of the out-of-plane resistivity.

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HgBa2CuO4+δ

Tc ~ 98 K

Tl2Ba2CuO6+δ

Tc ~ 85 K

Tc ~ 38 K

HgBa2CaCu2O6+δ

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243

Tc ~ 127 K

Bi2.1Sr1.9Ca0.9Cu2O8+δ Tc ~ 93 K

La1.85Sr0.15CuO4

Tc ~ 93 K

YBa2Cu3O6.93 63

Cu(2)

65

Cu(2)

63, 65

Cu nuclear spin-echoes (arb. units)

Mercury-Based Superconducting Cuprates: High Tc and Pseudo Spin-Gap

Tc ~ 28 K

(Ca, Na)2CuO2Cl2

Tc ~ 82 K YBa2Cu4O8

63

Cu(2)

65

Cu(2)

20

30 Frequency (MHz)

40 10

20 30 Frequency (MHz)

Figure 5. Zero-field plane-site 63,65Cu NQR frequency spectra for single-CuO 2-layer superconducting cuprates (left panels) (a) and double-CuO2-layer superconducting cuprates (right panels) (b) at T = 4.2 K.

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and η is called the asymmetry parameter η≡ 63,65

Vxx − Vyy . Vzz

Cu nuclear quadrupole resonance frequency νNQR is given by s

νNQR = νQ 1 +

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(3)

η2 , 3

(4)

where νQ = e2 qQm /2. The ratio of the natural abundance of 63Cu and 65Cu atoms is about 69.1 to 30.9. The ratio of the nuclear quadrupole moments Qm is about 0.211 to 0.195. For one crysallographic Cu site, a pair of 63Cu and 65 Cu NQR lines is observed. Figure 5 shows actual zero-field plane-site 63,65Cu NQR frequency spectra for various optimally doped single-CuO 2-layer superconducting cuprates (left panels) (a) and doubleCuO2-layer superconducting cuprates (right panels) (b) at T = 4.2 K. The Cu NQR spectra were measured for Hg1201 in [36,37], Tl 2Ba2 CuO6+δ [38], La1.85Sr0.15CuO4 in [39], (Ca, Na)2 CuO2Cl2 in [40], Hg1212 in [41], Bi 2.1Sr1.9 Ca0.9 Cu2O8+δ in [42], YBa2 Cu3O6.93 in [43] and YBa2 Cu4O8 in [43]. Solid curves are simulations using multiple Gaussian functions. More than two pairs of 63Cu and 65 Cu NQR lines were needed to reproduce the broad NQR spectra except YBa2 Cu3O7 and YBa2 Cu4 O8 . Non-stoichiometry in compounds gives rise to inhomogeneous broadening in NMR and NQR spectra, because magnetic shift and quadrupole frequency are distributed. YBa2 Cu4 O8 with double CuO 2 planes and double CuO chains in unit cell has Tc = 82 K. In Fig. 5, this stoichiometric and naturally underdoped compound shows sharp Cu NQR spectra. YBa2 Cu3 O6.93 with double CuO 2 planes and a single CuO chain in unit cell has Tc = 93 K. In Fig. 5, this stoichiometric and slightly overdoped compound shows sharp Cu NQR spectra. Although the CuO 2 planes of these compounds have the buckling structures, the Cu NQR spectra are rather sharp. The superconducting Hg1201 and Hg1212 at any doping level, however, show the inhomogeneously broad Cu NQR spectra [44–46]. In general, crystalline imperfection is the origin of the inhomogeneous broadening of an NQR spectrum. In Fig. 5, Bi2.1Sr1.9 Ca0.9Cu2 O8+δ (Bi2212) also shows a broad Cu NQR spectrum. This is attributed to a wide range structural modulation of the BiO layers. In spite of the dilute oxygen concentration in the HgO layers of Hg1201 and Hg1212, most of the Cu nuclei feel inhomogeneous electric field gradients. Thus, the effect of dilute oxygen ions on the crystalline potential may be long ranged. Long-range Friedel oscillations from the excess oxygen ions may yield such a broad NQR spectrum. The inhomogeneously broad Cu NQR spectrum is observed in the nearly perfect flat CuO 2 plane. This is instructive for us to understand the relation between NQR and the crystal structure. The broad Cu NQR spectrum results from the nonstoichiometry but not from the buckling of the CuO 2 planes. In Fig. 5, one should note that the value of Cu νNQR of Hg1201 is similar to that of Tl2Ba2 CuO6+δ and about a half of those of YBa 2 Cu3 O7 and YBa2 Cu4 O8. The local density approximation calculations of the NQR frequencies account for these experimental similarity and difference [47].

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Mercury-Based Superconducting Cuprates: High Tc and Pseudo Spin-Gap -4

χ (emu/mole)

2.0x10

245

HgBa2CuO4+δ (Tc = 98 K)

1.0 χspin + χvv 0.0 χdia = — 1.44 × 10

— 4

-1.0 0

200 T (K)

400

Figure 6. Uniform magnetic susceptibility χ of the optimally doped powdered sample of Hg1201 (Tc = 98 K) in [22] at a magnetic field of 1 Tesla [48].

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5.2.

Bulk Magnetic Susceptibility

A static magnetic field H is applied to a material along the z-axis and then a magnetization Mz is measured by a magnetometer. The bulk magnetization Mz is the sum of respective electron spins hSiz i X Mz = hSiz i, (5) i

where hSiαi is a thermal average of the electron spin along the α-axis. Then the bulk magnetic susceptibility χ is defined by χ=

P

i hSiz i

Hext

.

(6)

The paramagnetism of 3d transition metal oxides results from the unpaired d electron spins and the orbital momentums. The bulk magnetic susceptibility is expressed by the sum of the spin susceptibility χspin, the Van Vleck orbital susceptibility χvv , and the diamagnetic susceptibility χdia of inner core electrons χ = χspin(T ) + χvv + χdia .

(7)

Fgiure 6 shows the uniform magnetic susceptibility χ of the optimally doped powdered sample of Hg1201 (Tc = 98 K) in [22] at a magnetic field of 1 Tesla [48]. The pseudo Superconducting Cuprates : Properties, Preparation and Applications, Nova Science Publishers, Incorporated, 2008. ProQuest Ebook Central,

246

Y. Itoh and T. Machi

spin-gap behavior is seen in the bulk magnetic susceptibility. The magnitude of χ above 150 K is the same order of the other high- Tc superconducting cuprates .

5.3.

Knight Shift

Nuclear spins of constituent ions in a crystal interact with electron spins through an electron-nuclear hyperfine coupling. When a static uniform magnetic field Hext is applied to the system along the z-axis, the electron medium is polarized along the z-axis. The magnetic polarization produces an additional magnetic field at the nuclear sites through the hyperfine coupling and then leads to a shift of the resonance field (frequency) of the nuclear spins. This is Knight shift K. The static nuclear spin Hamiltonian of a Zeeman coupling and the electron-nuclear hyperfine coupling is given by Hint = −γn I ·

X

(Hext + Aiz hSiz i)

(8)

i

where γn is the nuclear gyromagnetic ratio and Aiα is a hyperfine coupling constant along the α-axis with the i-site electron spin. The experimental Knight shift is defined by a shift of the observed resonance frequency ωres from the reference frequency ωref ωres = γn Hext(1 + K)

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= ωref (1 + K).

(9)

In the 3d transition metal oxides, the Knight shift K is decomposed into the spin shift Ks due to unpaired electron spins and the orbital shift Korb due to the Van Vleck orbital susceptibility (10) K = Ks(T ) + Korb. The spin Knight shift Kα is given by 0

χ (q = 0, ω = 0) . Ks = A(q = 0) s NA µB

(11)

where A(q = 0) is the uniform Fourier component of the hyperfine coupling constant and χ0(q = 0, ω = 0) is the static (ω = 0) uniform (q = 0) electron spin susceptibility. µB is the Bohr magneton. NA is the Avogadro number. χα 0 is the measured bulk susceptibility in emu/mole-atom. The hyperfine coupling constant reflects the characters of the wave functions of the electron orbitals. Measurement of the Knight shift at each nuclear enables us to obtain the site-specific information. The temperature dependence of the Knight shift reveals that of the intrinsic uniform spin susceptibility χspin at each site. In covalent bonded compounds, the transferred and supertransferred hyperfine coupling constants play important roles. In the high- Tc cuprate superconductors, the Fourier transformed hyperfine coupling constants at the plane-site Cu and the oxygen are expressed by ACu α (q) = Aα + 2B{cos(qx ) + cos(qy )}

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247

AO α (q) = 2Ccos(qx /2),

(13)

and where Aα (α = ab and cc) is an on-site anisotropic hyperfine coupling constant, B(> 0) is a supertransferred isotropic hyperfine coupling constant from a Cu to a Cu through an oxygen, and C(> 0) is a transferred hyperfine coupling constant from a Cu to an oxygen [16, 49]. For the optimally doped Hg1201, we estimate ACu ab (q = 0) = 145 kOe/mole-Cu-µB from the K − χ plot, where the Knight shift Kab [37] is plotted against the bulk magnetic susceptibility χ in Fig. 6 with temperature as an implicit parameter.

5.4.

Nuclear Spin-Lattice Relaxation

Nuclear spins are coupled by the fluctuating hyperfine fields of electron spins through a time-dependent hyperfine coupling Hamiltonian Hint(t) =

1X (I− Ai δSi+ (t) + I+ Ai δSi− (t)). 2 i

(14)

In a spin-echo recovery technique, the nuclear moments excited by an inversion rf -pulse are in a thermal non-equilibrium state. The energy dissipation from the nuclear moments to a lattice takes place through the fluctuating hyperfine fields. The recovery time of the nuclear moments to a thermal equilibrium state is the nuclear spin-lattice relaxation time T1 . Moriya derived a general expression of T1 [50],

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1 T1

=

Z γn2 X 2 ∞ Ai h{δSi+ (t)δSi− (0)}ie−iωn tdt 2 i −∞

=

2γn2 kB T g 2µ2B ωn

Z

00

dD qAab (q)2χab (q, ωn ).

(15)

D is the dimension of space on which the electronic system lies. 1/ T1T is the square of the hyperfine coupling constant A(q)2 times the wave vector averaged low frequency 00 dynamical spin susceptibility χ (q, ω) [50]. Thus, from measurements of the Knight shift and the nuclear spin-lattice relaxation time, one can infer the q dependence of the dynamical spin susceptibility χ”(q, ω). The modified Korringa ratio can give us a criterion which the 00 electronic system is, ferromagnetic or antiferromagnetic and where χ (q, ω) is enhanced in the q space. In general, the q dependence of the hyperfine coupling constant A(q) is slower than 2 that of χ(q). But, the form factor of the plane-site oxygen AO α (q) of Eq. (13) acts as a filter for a finite q correlation. The antiferromagnetic correlation of Q = [π, π] and Q∗ = [π(1±δ), π(1±δ)] is cancelled at the plane-site oxygen through the form factor of Eq. (13). The plane-site Cu nuclei can probe the antiferromagnetic correlation through Eq. (12) .

5.5.

Gaussian Cu Nuclear Spin-Spin Relaxation

Strong indirect Cu nuclear spin-spin interaction was first found in YBa 2 Cu3 O7 [51, 52]. Gaussian decay in the transverse relaxation of the plane-site Cu nuclear moments is preSuperconducting Cuprates : Properties, Preparation and Applications, Nova Science Publishers, Incorporated, 2008. ProQuest Ebook Central,

248

Y. Itoh and T. Machi

dominantly induced by the indirect nuclear spin-spin interaction through the in-plane antiferromagnetic electron spin susceptibility. Strong antiferromagnetic fluctuations persist in all the high-Tc cuprate superconductors. Thus, the Gaussian decay rate of the plane-site Cu nuclear spin-spin relaxation provides us fruitful information on the antiferromagnetic electron spin susceptibility [52–57]. The indirect nuclear spin-spin interaction is given by HII =

X

Φ(rij )Ij · Ij

(16)

i,j

where a range function Φ(rij ) is given by Φ(rij ) =

Z

0

dD qA(q)2χ (q).

(17)

The Gaussian decay rate 1/T2g of the nuclear spin-spin relaxation is given by 1 T2g

!2

∝

X

≈

Z

Φ(rij )2

j 0

dD qA(q)4χ (q)2.

(18)

The Kramers-Kronig relation is 0

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χ (q) =

2 π

Z ∞ 0

00

dω

χ (q, ω) . ω

(19)

The Gaussian decay rate 1/T2g reflects the full range frequency integration of the dynamical 00 spin susceptibility χ (q, ω). The cross section of inelastic neutron scattering is expressed by 1 00 ∂ 2σ ∝ χ (q, ω), ∂ω∂Ω 1 − e−βω

(20)

where q is a momentum transfer of the neutron, ω the energy transfer, and β = 1/kB T [58]. 00 In the high-T c cuprate superconductors, the enhancement of χ (q, ω) was observed over a 00 finite range of ω and at around q = Q or Q∗ [59]. In general, it is hard to see χ (q, ω) over the full range. From measurements of the nuclear spin-lattice relaxation and the Gaussian decay rate of nuclear spin-spin relaxation, one can infer the dynamical spin susceptibility at low and high frequency regions.

5.6.

Two Dimensional Nearly Antiferromagnetic Spin Fluctuation Model

The electronic state of a CuO 2 plane of the high-Tc cuprate superconductor is described by a single band picture. The anisotropy of Knight shift and nuclear spin-lattice relaxation and the site differentiations are explained by anisotropic hyperfine coupling and different q dependence of the coupling constant due to transferred and supertransferred hyperfine couplings. That is Mila-Rice-Shastry hyperfine coupling Hamiltonian [60, 61]. Two

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249

dimensional antiferromagnetic spin fluctuations through the Mila-Rice-Shastry hyperfine coupling yield the anisotropy and the site difference [13, 62, 63]. The two dimensional spin fluctuation models were successfully applied to account for NMR, neutron scattering and conductivity results [33, 64, 65]. We employ the two dimensional nearly antiferromagnetic spin fluctuation model in [33]. The dynamical spin susceptibility is expressed by a relaxation mode χ(q, ω) =

χ(q) . 1 − iω/Γ(q)

(21)

The leading terms of χ(q) and Γ(q) in a random phase approximation (RPA) are expressed using a long wave length expansion around a specific mode q = Q by χ(q) ≈

χ0 (Q) κ2 + (q − Q)2

(22)

and Γ(q) ≈ Γ0 (Q)(κ2 + (q − Q)2 ).

(23)

κ is the inverse of a magnetic correlation length ξ defined around q = Q, ξ2 = −

1 ∂ 2 χ(q) . χ(Q) ∂q 2 q=Q

(24)

χ0 (Q) is the spin fluctuation amplitude at q = Q and Γ0 (Q) is the characteristic spin fluctuation energy. The staggered spin susceptibility χ(Q) is then

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χ(Q) = χ0 (Q)ξ 2 .

(25)

We obtain the plane-site Cu nuclear spin-lattice relaxation rate expressed by the leading term of ξ 1 T1T

Z

χ(q) Γ(q) Z dD q χ0 (Q) ∝ Γ0 (Q) {κ2 + (q − Q)2 }2 Z χ0 (Q) ξqB dD (ξ q˜) ∝ ξ 4−D Γ0 (Q) 0 {1 + (ξ q˜)2}2 2

≈ A(Q)

dD q

∝ ξ 4−D ∝ χ(Q)2−D/2,

(26)

where qB is the spherical radius of the same volume as the first Brillouin zone and ξqB  1. We also obtain the Gaussian Cu nuclear spin-spin relaxation rate expressed by the leading term of ξ 1 T2g

!2

≈ A(Q)4

Z

dD qχ(q)2

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Y. Itoh and T. Machi Z

dD q {κ2 + (q − Q)2}2 Z dD (ξ q˜) 4−D 2 ∝ ξ χ0 (Q) {1 + (ξ q˜)2}2 2

∝ χ0 (Q)

∝ ξ 4−D ∝ χ(Q)2−D/2.

(27)

Thus, for D = 2 we have

and

T1 T ∝ Γ0 (Q)ξ −1 , T2g

(28)

T1 T ∝ Γ0 (Q)χ0 (Q). (T2g )2

(29)

In the self-consistent renormalization (SCR) theory for two dimensional antiferromagnetic spin fluctuations, the Curie-Weiss behavior of the sqaure of the antiferromagnetic correlation length ξ 2 (∝ the staggered spin susceptibility χ(Q)) is reproduced as a function of the distance from the quantum critical point and the spin fluctuation energy Γ0 (Q) [33]. Using χ0 (Q) = 1/2αsTA (α = U χ0 (Q)), Γ0 (Q) = 2πT0 and t = T /T0, we obtain [41] T1 T ∝ T0ξ(t)−1 , T2g

(30)

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which represents the inverse of the antiferromagnetic correlation length with a unique parameter T0, and T0 T1 T ∝ , (31) 2 (T2g ) TA which represents the integrated spin fluctuation weight. Using Eqs. (30) and (31), the NMR relaxation data for Tl 2Ba2 CuO6+δ , YBa2 Cu3O7 and YBa2 Cu4 O8 were analyzed [54–57].

6. 6.1.

Normal-State Pseudo Spin-Gap Single-Layer Pseudo Spin-Gap: Hg1201

The left panels of Fig. 7 show 63Cu NMR results of the underdoped Hg1201 of Tc = 50 K [36,37]. The NMR experiments were performed for the powdered polycrystalline sample. All the powder samples were magnetically aligned along the c axis. In general the mercury compounds cannot be easily aligned by a magnetic field. Then, the NMR experiments were performed for partially oriented powder samples. This does not indicate that the NMR data are the partially powder-averaged ones. The sharply aligned NMR lines can be separated in the powder pattern, so that the selected signals surely come from the aligned grains. The plane-site 63 Cu Knight shift Kab, nuclear spin-lattice relaxation rate (1/T1T )cc, and Gaussian nuclear spin-echo decay rate (1/T2g)cc are shown as functions of temperature. The subscripts of cc and ab denote the data in the external magnetic field Hext ∼ 8 Tesla along the c and ab axis, respectively. No appreciable field dependence was observed within Hext = 4 ∼ 8 Tesla.

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Tc = 96 K

Tc = 50 K

63

Kab (%)

1.0

251

Tc = 30 K

0.5

↔ ↔

Kspinab

Korbab

0 ↓Ts

)

15 —1 —1

(1/T1T)cc (s K

↑ Ts

Ts ↓

63

10

5

0

10

63

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(1/T2g)cc (ms)

—1

20

0 0

100 200 T (K)

300 0

100 200 T (K)

300 0

100 200 T (K)

300

Figure 7. 63Cu NMR results of single-layer superconductors Hg1201. Temperature dependences of the plane-site 63Cu Knight shifts Kab (top panels), nuclear spin-lattice relaxation rates (1/T1T )cc (middle panels), and Gaussian decay rates (1/T2g)cc of nuclear spin-spin relaxation (bottom panels) are shown for the underdoped (left), the optimally doped (center) and the overdoped samples (right). Dash lines denote the respective Tc ’s. The data are reproduced from [36, 37, 73].

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The drastic decreases of the Cu Knight shift K and nuclear spin-lattice relaxation rate 1/T1T with cooling down are clearly seen from room temperature. This is an obvious evidence of the existence of the pseudo spin-gap in the single layer cuprate. The decrease of the Cu Knight shift is independently found in [66]. The pseudo spin-gap behavior of the uniform spin susceptibility is also found by the in-plnae 17 O NMR experiments [67]. In passing, for the underdoped triple-layer Hg1223, the pseudo spin-gap behavior has been observed by Cu NMR experiments [68]. The absence of the Hebel-Slichter peak of 1/ T1T just below Tc excludes the weak coupling s-wave pairing symmetry. The slow decrease of the Cu 1/T2g is also seen below 200 K. It suggests that the large pseudo spin-gap leads to the loss of the total weight of the frequency integrated χ” (q, ω) 0 and then that the static staggered spin susceptibility χ (Q) decreases. The drastic decrease of 1/T1T but the moderate decrease of 1/T2g are theoretically reproduced by the numerical calculations involving the self-energy correction due to the enhanced dx2 −y2 -wave superconducting fluctuations [69]. In this theory, the pseudogap is a consequence from the resonance scattering in the two dimensional strong coupling superconductivity. The finite 1/ T2g below Tc is an evidence for the dx2 −y2 wave pairing symmetry [70–72]. Figure 7 shows the hole doping dependence of 63 Cu NMR results of Hg1201 [36,37,73]. Temperature dependences of the plane-site 63Cu Knight shifts Kab (top panels), nuclear spin-lattice relaxation rates (1/T1T )cc (middle panels), and Gaussian decay rates (1/T2g)cc of nuclear spin-spin relaxation (bottom panels) are shown for the underdoped (left), the optimally doped (center) and the overdoped samples (right). The pseudo spin-gap temperature Ts is defined by the maximum temperature or the onset of the decrease of (1/ T1T )cc. Ts decreases from the underdoped to the overdoped samples. The temperature region of the Curie-Weiss behavior of (1/T1T )cc above Ts is broadened with the hole doping.

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6.2.

Double-Layer Pseudo Spin-Gap: Hg1212

The middle panels of Fig. 8 show 63 Cu NMR results of the optimally doped Hg1212 of Tc = 127 K [41], which is the maximum Tc among the ever reported single and double layers. The plane-site 63 Cu Knight shift Kab, nuclear spin-lattice relaxation rate (1/T1T )cc, and Gaussian nuclear spin-echo decay rate 1/T2g are shown as functions of temperature. The decrease of the 63 Cu Knight shift Kab with cooling down is obvious even at the optimal Hg1212. Above Ts = 200 K, 1/T1T shows a Curie-Weiss behavior, because of the development of the antiferromagnetic correlation length. The decrease of 1/ T1T starts below about Ts = 200 K. This Ts = 200 K of Hg1212 is larger than Ts = 140 K of the optimally doped Bi2212 with Tc = 86 K in [74]. The record optimal Tc of Bi2212 is about 96 K [75]. For the double layer systems of Hg1212 and Bi2212, the ratio ( ∼ 1.4) of the optimal Ts ’s is nearly the same as that of Tc. The slight decrease of 1/T2g is also seen, because of the large pseudo spin-gap effect on the frequency integrated χ” (q, ω). The finite 1/T2g below Tc indicates the dx2 −y2 wave pairing symmetry [70–72]. Figure 8 shows the hole doping dependence of 63Cu NMR results of Hg1212 [41]. Temperature dependences of the plane-site 63Cu Knight shifts Kab (top panels), nuclear spin-lattice relaxation rates (1/T1T )cc (middle panels), and Gaussian decay rates (1/T2g)cc of nuclear spin-spin relaxation (bottom panels) are shown for the underdoped (left), the optimally doped (center) and the overdoped samples (right). The pseudo spin-gap temperature

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Tc = 127 K

Tc = 103 K

Tc = 93 K

63

Kab (%)

1.0

253

↔ ↔

0.5 Kspinab Korbab

0.0 ↓Ts

10

5

63

—1

Ts ↓

Ts ↓

—1

(1/T1T)cc (s K )

15

0

—1

(1/T2g)cc (ms)

15 10

63

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20

5 0 0

100

200 T (K)

300 0

100

200 T (K)

300 0

100

200 T (K)

300

Figure 8. 63Cu NMR results of double-layer superconductors Hg1212. Temperature dependences of the plane-site 63Cu Knight shifts Kab (top panels), nuclear spin-lattice relaxation rates (1/T1T )cc (middle panels), and Gaussian decay rates (1/T2g)cc of nuclear spin-spin relaxation (bottom panels) are shown for the underdoped (left), the optimally doped (center) and the overdoped samples (right). Dash lines denote the respective Tc ’s. The data are reproduced from [41].

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Ts decreases from the underdoped to the overdoped samples. The values of Ts of Hg1212 are higher than those of Bi2212 [74]. The temperature region of the Curie-Weiss behavior in (1/T1T )cc above Ts is broadened with the hole doping. It should be noted that the overdoped sample shows the Curie-Weiss behavior in (1/ T1T )cc but not Korringa behavior above Ts [46].

6.3.

Interlayer Coupling Via Hg NMR

The Hg site is located just halfway between the CuO 2 planes and between the respective Cu ions. The Hg nuclei can serve as a probe of the interplane coupling. If the interplane coupling is antiferromagnetic, the in-plane antiferromagnetic correlation is also cancelled out at the Hg site. If the inteplane coupling is uniform and ferromagnetic, the in-plane antiferromagnetic correlation is also seen at the Hg site. Figure 9 shows 63 Cu and 199Hg nuclear spin-lattice relaxation rates 1/ T1T for the optimally doped Hg1201 (a) and for the overdoped Hg1201 (b) [37]. The temperature dependence of 199(1/T1T ) is nearly the same as that of 63 (1/T1T ) for both samples. Thus, the inteplane coupling is not antiferromagnetic but uniform. This uniform interplane coupling is also observed by 63 Cu and 199Hg NMR 1/T1T for Hg1212 [46, 76]. In contrast to the reports [77, 78] that 63Cu and 199Hg nuclear spin-lattice relaxation rates 1/ T1T show the different behaviors with each other in Hg1201, these results in Fig. 9 indicate that the interplane coupling is uniform in Hg1201 irrespective of the doping level.

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6.4.

Pseudo Spin-Gap Phase Diagram

Figure 10 shows the magnetic phase diagrams of Hg1201(a) and Hg1212 (b) [37, 41, 73]. Pseudo spin-gap temperatures Ts and superconducting transition temperatures Tc are plotted against the hole concentration Psh . The hatched region of the overdoped Hg1201 indicates a pseudo Korringa behavior below Ts . It should be noted that the pseudo spin-gap persists at the optimally doped regions for Hg1201 and H1212. The doping dependence of Ts of Hg1212 is different from that of Hg1201 even in the underdoped region. The Ts as a function of Psh does not seem to be universal. We present two theoretical explanations for the doping dependence of Ts . In the two dimensional t-J model with spinon-holon decomposition technique, the pseudo spin-gap temperature Ts is regarded as the onset temperature TRVB of a spinon singlet RVB (resonating valence bond) state [79–81]. The real transition Tc is given by a Bose-Einstein condensation temperature TBEC of holons, leading to the underdoped regime. With an existing approximation, TRVB is a second order phase transition temperature. Figure 11 (a) shows a numerical TRVB as a function of doped hole concentration [81]. The doping dependence of TRVB depends on the contour of a basal Fermi surface. For each high- Tc family with different Fermi surface, TRVB exhibits the different doping dependence. In the two dimensional superconducting fluctuation theory with the strong coupling, the pseudo spin-gap temperature Ts is regarded as the onset of enhancement of dx2 −y2 -wave superconducting fluctuations and a mean-field Tc. The actual Tc is reduced by the strong superconducting fluctuations so that the underdoped regime appears. Thus, the mean-field Tc is a crossover temperature. Figure 11 (b) shows the mean-field TcMF and the suppressed Tc [69]. The doping dependence of TcMF depends on the shape of the Fermi surface.

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—1 63

5

0.00

—1 63

5 overdoped 0 0

Cu (Tc = 30 K ) H // c-axis 199 Hg (Tc = 30 K) H // ab-axis

0.05

— 1

0.10 10

199

63

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— 1

—1

(1/T1T)cc (s K )

0 15

0.05

199

Cu (Tc = 96 K ) H // c-axis 199 Hg (Tc = 96 K) H // ab-axis H // c-axis

— 1

0.10 10

(1/T1T)ab (s K )

63

— 1

—1

(1/T1T)cc (s K )

15 optimally doped

255

(1/T1T)ab, cc (s K )

Mercury-Based Superconducting Cuprates: High Tc and Pseudo Spin-Gap

0.00 100

200

300

T (K)

Figure 9. 63 Cu and 199Hg nuclear spin-lattice relaxation rates 1/ T1T for optimally doped Hg1201 (a) and for overdoped Hg1201 (b) [37]. Temperature dependence of 199(1/T1T ) is nearly the same as that of 63(1/T1T )cc for both samples.

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Y. Itoh and T. Machi

300

HgBa2CuO4+δ

Tc, Ts (K)

Ts

Curie-Weiss χ(Q)

200

100

Pseudo Spin-Gap State Superconducting State

Tc 0 300

HgBa2CaCu2O6+δ

Tc, Ts (K)

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Ts

Curie-Weiss χ(Q)

200 Pseudo Spin-Gap State 100

0 0.0

Tc

0.1

Superconducting State 0.2

0.3

2+Psh

Psh of Cu

Figure 10. Magnetic phase diagrams of Hg1201 and Hg1212 [37, 41, 73]. Pseudo spingap temperatures Ts and superconducting transition temperatures Tc are plotted against the hole concentration Psh per plane-site Cu. The value Psh of Cu2+Psh is estimated from the excess oxygen concentration δ and the charge neutrality condition. Solid and dash curves Superconducting Cuprates : Properties, Preparation and Applications, Nova Science Publishers, Incorporated, 2008. ProQuest Ebook Central,

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257

0.2

TRVB / t

(a)

0.1 singlet RVB formation

0.0 0.0

0.2

0.4

Tc / t

Tc

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0.05

MF

(= Ts ) (b)

superconducting fluctuations

Tc

0.00 0.0

0.1 0.2 hole concentration δ

Figure 11. (a) Theoretical hole doping dependences of pseudo spin-gap temperatures Ts as siglet-RVB formation temperatures TRVB in a two dimensional t-J model with two type Fermi surfaces reproduced from [81]. t is a Cu-to-Cu transfer integral and J is a superexchange interaction in charge-transfer type compounds. The inset figures are the two type Fermi surfaces. (b) Theoretical hole doping dependences of the mean-field TcMF and the true Tc suppressed by dx2 −y2 -wave superconducting fluctuations reproduced from [69].

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From the above two theories, one may conclude that the difference in the hole doping dependence of Ts of Hg1201 from Hg1212 can be attributed to the different shape in the basal Fermi surface of Hg1201 and Hg1212. The band structure calculations have been performed by the full potential linear muffintin orbital method for Hg1201, Hg1212 and Hg1223 [82]. The band calculations indicate the different electronic structures and Fermi surfaces between Hg1201 and the others [82]. The Fermi surface of a single crystal Hg1201 was observed by angle-resolved photoemission measurement [83].

7.

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7.1.

Spin Fluctuation Spectrum Scaling

The ratios of T1T and T2g tell us the spin fluctuation parameters through Eqs. (28), (29), (30) and (31). In Figs. 12 and 13, T1T /T2g (a) and T1 T /(T2g)2 (b) for Hg1201 and Hg1212 are plotted against temperatures from the underdoped to the overdoped samples [36,37,41,73]. The pseudo scaling of T1T /T2g = constant is observed in the limited temperature region for Hg1212. The pseudo scaling temperature region decreases and shifts at lower temperatures by the hole doping. At high temperatures, T1T /(T2g )2 = constant holds. The value of T1T /(T2g )2 indicates the product of the spin fluctuation amplitude and the spin fluctuation energy. Thus, the product of χ(Q)Γ(Q) decreases from the underdoped to the overdoped regimes of Hg1201 and Hg1212. Equation (28) indicates that T1 T /T2g is proportional to the inverse of the antiferromagnetic correlation length ξ and has a unique scale parameter T0. In Figs. 12 and 13, T1T /T2g decreases from the underdoped to the overdoped samples of Hg1201 and Hg1212. This is inconsistent with the theoretical doping dependence of the magnetic correlation length near the two dimensional quantum critical point. The ξ −1 in the SCR theory increases from the weakly antiferromagnetic to the nearly antiferromagnetic regimes and away from the quantum critical point [33]. The decrease of T1 T /T2g suggests the decrease of the spin fluctuation energy T0 from the underdoped to the overdoped samples. In the overdoped regime, the decrease of Tc can be associated with the decrease of the spin fluctuation energy T0 . In the RPA for the two dimensional t-J model, both T1T /T2g (a) and T1T /(T2g )2 (b) depend on temperature more or less [84]. The value of T1T /T2g increases with the hole doping [84], that is inconsistent with the experimental doping dependence. But the decreases of T1T /(T2g)2 with doping is reproduced within the RPA calculations. In the numerical calculations for the small size t-J model using the Lanczos diagonalization method, the doping dependence of T1 T /T2g below T = 1390 K agrees with the experimental tendency. Above T = 2320 K, the doping dependence of T1T /T2g above T = 2320 K reproduces those of the SCR and the RPA calculations [85].

7.2.

Spin Fluctuation Parameters

Within the framework of Eliashberg-Nambu strong coupling superconductivity theory, the actual Tc is determined by competition between the pairing effect and the depairing efSuperconducting Cuprates : Properties, Preparation and Applications, Nova Science Publishers, Incorporated, 2008. ProQuest Ebook Central,

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3

T1T / T2g (10 K)

Tc = 50 K 72 K 96 K 30 K

(a)

2

259

1

0 underdoped underdoped optimally doped overdoped

40

2

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6 — 1

T1T / (T2g) (10 s K)

(b)

20

0 0

100

200 T (K)

300

Figure 12. Hole doping dependence of T1 T /T2g (a) and T1T /(T2g)2 (b) for Hg1201. The data are reproduced from [36, 37, 73]. Superconducting Cuprates : Properties, Preparation and Applications, Nova Science Publishers, Incorporated, 2008. ProQuest Ebook Central,

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(a)

Tc = 103 K Tc = 127 K Tc = 93 K

3

T1T/T2g (10 K)

3

2

1

0 underdoped optimally doped overdoped

(b)

2

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6 —1

T1T/(T2g) (10 s K)

80

40

0 0

100

200 T (K)

300

Figure 13. Hole doping dependence of T1 T /T2g (a) and T1T /(T2g)2 (b) for Hg1212. The data are reproduced from [41]. Solid lines are eye guides for T1T /T2g = constant (a) and T1T /(T2g )2 = constant (b).

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300

Hg1212 Hg1201 Tl 1212

Curie-Weiss law

200 Pseudo Spin-Gap

T c,T

s

(K)

261

overdoped

100 Superconducting State

0 0

10 20 30 2 6 —1 (T1T)/(T2G) [ ∝T0/TA ] (10 s K)

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Figure 14. Superconducting transition temperatures Tc are plotted against the spin fluctuation parameters of Hg1201, Hg1212 and Tl1212 (TlSr 2 CaCu2 O7−δ ). Pseudo spin-gap temperatures Ts are also plotted. The Cu NMR results of Hg1201 and Hg1212 are reproduced from [36, 37, 41, 73]. The Cu NMR results of Tl1212 are reproduced from [94]. fect [86–88]. In the spin-fluctuation-mediated superconductors, the depairing effect due to low frequency spin fluctuations competes the paring effect due to high frequency ones. For the dx2 −y2 superconductivity on a square lattice, the numerical calculation and theoretical consideration tell us that Tc is proportional to the characteristic energy scale of antiferromagnetic spin fluctuations [89], (32) Tc ∝ T0. The spin-fluctuation-induced superconductivity theories are studied for the two and three dimensional Hubbard models [90, 91]. As a general tendency, two dimensional systems have higher Tc than three dimensional ones [89,92,93]. Then, the mystery of the high Tc of the layered compounds is traced back to the two dimensionality and the large scale of the antiferromagnetic spin fluctuation energy. Figure 14 shows Tc plotted against T1T /(T2g)2 of Hg1201, Hg1212, and Tl2212 reproduced from [94]. If one assumes T1T /(T2g )2 ∝ T0/TA of Eq. (29), the linear relation between Tc and the spin fluctuation energy does not seem to hold. The spin fluctuation product χ(Q)Γ(Q) decreases monotonically with the hole doping as in Figs. 12 and 13. This has been recognized in the other systems [55]. Thus, T1T /(T2g )2 is a monotonic function of the doped hole concentration and then the indicator. At some threshold value of T1T /(T2g )2 in the overdoped regime, Tc starts to increase toward the optimally doping level. Beyond the optimally doping level, Tc decreases and Ts increases as the hole concentration is reduced. The antiferromagnetic spin fluctuation spectrum is different between the underdoped and the overdoped samples. Thus, the overdoped part of the Tc -vs-T1T /(T2g)2 curves indicates

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Y. Itoh and T. Machi 1.0

Tc = 30 K

Tc = 127 K

Tc = 103 K

Kab (%)

63

Tc = 96 K

Tc = 50 K

Tc = 93 K

0.5

0

—1

(1/T1T)cc (s K

—1

)

15

63

10

5

0

10

63

(1/T2g)cc (ms)

—1

20

0 0

100 200 T (K)

300 0

100 200 T (K)

300 0

100 200 T (K)

300

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Figure 15. Hg1201 versus Hg1212. 63Kab, 63 (1/T1T )cc, and 63(1/T2g)cc for Hg1201 (black symbols) and Hg1212 (red symbols) from Figs. 7 and 8 [36, 37, 41, 73]. some correlation between Tc and the spin fluctuation parameters, but the underdoped part indicates the suppression of Tc due to the grown of the large pseudo spin-gap.

7.3.

Double-Layer Coupling: Revisited

From the fact that the uniform spin susceptibility χs is suppressed at low temperatures in YBa2 Cu3 O6.6 and YBa2 Cu4 O8 more than La2−x Srx CuO4, the bilayer (double-layer) coupling effect had been proposed to be the primary origin of the pseudo spin-gap [95–98]. The double-layer exchange scattering effect was propsed to account for the different behavior of 1/T 1T and 1/T2g below Ts [99, 100]. Now the single CuO 2 layer Hg1201 is found to possess the pseudo spin-gap in the low-lying excitation spectrum. However, the existence of the double-layer coupling was actually confirmed by the neutron scattering [59] and NMR [101] experiments for low doped YBa 2 Cu3 O6+δ . Spin-echo double resonance techniques have been applied to estimate the double-layer coupling constant in Y2 Ba4 Cu7 O15−δ [102, 103] and in Bi 2 Sr2 Ca2 Cu3 O10 [104]. The angle dependence of the Gaussian decay rate 1/T2g was measured to estimate the like-spin interlayer coupling for Hg1223 [105]. Figure 15 shows 63 Kab, 63 (1/T1T )cc, and 63 (1/T2g)cc for Hg1201 (black symbols) and Hg1212 (red symbols) from Figs. 7 and 8 to compare two systems [36, 37, 41, 73]. The

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263

NMR results of Hg1212 is quantitatively different from those of Hg1201. As to the underdoped Hg1201 and Hg1212 samples, 63(1/T1T )cc of Hg1212 is nearly the same as that of Hg1201 but the Tc is about twice higher than Hg1201. The different point is the underdoped 63 (1/T2g)cc of Hg1212 higher than that of Hg1201. Thus, not the low frequency spin fluctutions but the high frequency ones contribute the higher Tc. As to the optimally doped Hg1201 and Hg1212 samples, the spin part of the Cu Knight shift of Hg1212 and 63(1/T1T )cc are smaller than those of Hg1201, whereas the 63 (1/T2g)cc is higher than that of Hg1201. The electron spin-spin correlation function χinter due to the double-layer coupling is competitive in the in-plane correlation function χintra of 63Kab and 63 (1/T1T )cc but is additive in 63 (1/T2g)cc [100]. Although the double-layer coupling is not a primary origin of the pseudo spin-gap, it surely affects the microscopic magnetic properties.

8.

Conclusion

The flat CuO2 plane and the large pseudo spin-gap are the characteristics of the mercurybased high-T c superconducting cuprates Hg1201 and Hg1212. The role of the pseudo spingap in the higher Tc is still unclear. Is the pseudo spin-gap a consequence of the enhanced superconducting fluctuations to suppress the mean field TcMF ? Then, we should explore the layered compounds with higher energy spin fluctuations and any method to suppress the superconducting fluctuations to get higher Tc .

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Acknowledgments We would like to thank S. Adachi, A. Yamamoto, A. Fukuoka, K. Tanabe, N. Koshizuka, K. Yoshimura, and H. Yasuoka for the fruitful collaboration, and Y. Ohashi, J. Kishine, and Y. Yanase for the valuable discussions on theoretical studies.

References [1] Putilin S. N.; Antipov E. V.; Chmaissem O.; Marezio M. Nature 1993, 362, 226-228. [2] Fukuoka A.; Tokiwa-Yamamoto A.; Itoh M.; Usami R.; Adachi S.; Yamauchi H.; Tanabe K. Physica 1996, B 265, 13-18. [3] Schilling A.; Cantoni M.; Guo J. D.; Ott H. R. Nature 1993, 363, 56-58. [4] Antipov E. V.; Loureiro S. M.; Chaillout C.; Capponi J.; Bordet P.; Tholence J. L.; Putilin S. N.; Marezio M. Physica 1993, C 215, 21-24. [5] Gao L.; Xue Y. Y.; Chen F.; Xiong Q.; Meng R. L.; Ramirez D.; Chu C. W.; Eggert J. H.; Mao H. K. Phys. Rev. 1994, B 50, 4260-4263. [6] Antipov E. V.; Loureiro S. M.; Chaillout C.; Capponi J. J.; Bordet P.; Tholence J. L.; Putilin S. N.; Marezio M. Physica 1993, C 215, 1-10.

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[7] Usami R.; Adachi S.; Itoh M.; Tatsuki T.; Tokiwa-Yamamoto A.; Tanabe K. Physica 1996, C 262, 21-26. [8] Imai T.; Yasuoka H.; Shimizu T.; Ueda Y.; Yoshimura K.; Kosuge K. Physica 1989, C162-164, 169-170. [9] Yasuoka H.; Imai T.; Shimizu T. in Strong Correlation and Superconductivity ; Springer Series in Solid-Sate Sciences 89; Springer-Verlag: Berlin, 1989, pp 254261. [10] Alloul H.; Ohno T.; Mendels P. Phy. Rev. Lett. 1989, 63, 1700-1703. [11] Warren, Jr. W. W.; Walstedt R. E.; Brennert G. F.; Cava R. J.; Tycko R.; Bell R. F.; Dabbagh G.; Phys. Rev. Lett. 1989, 62, 1193-1196. [12] Horvati´c M.; S´egransan P.; Berthier C.; Berthier Y.; Butaud P.; Henry J. Y.; Couach M.; Chaminade J. P. Phys. Rev. 1989, B 39 7332-7335. [13] Takigawa M.; Reyes A. P.; Hammel P. C.; Thompson J. D.; Heffner R. H.; Fisk Z.; Ott K. C. Phys. Rev. 1991, B43, 247-257. [14] Pennington C. H.; Slichter C. P. in Physical Properties of High Temperature Superconductors; Ginsberg, D. M.; Ed.; World Scientific Publishing Co.: New Jersey, 1998; Vol. 2, pp 269-367.

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[15] Berthier C.; Julien M. H.; Horvatic M.; Berthier Y.; J. Phys. I France 1996, Vol. 6, 2205-2236. [16] Walstedt R. E. The NMR Probe of High-T c Materials; Springer Tracts in Modern Physics; Springer-Verlag: New York, 2008. [17] Itoh Y. cond-mat/0711.1688. [18] Fukuoka A.; Tokiwa-Yamamoto A.; Itoh M.; Usami R.; Adachi S.; Tanabe K. Phys. Rev. 1997, B 55, 6612-6620. [19] Radaelli P. G.; Marezio M.; Perroux M.; de Brion S.; Tholence J. L.; Huang Q.; Santoro A. Science 1994, 265, 380-383. [20] Radaelli P. G.; Marezio M.; Tholence J. L.; de Brion S.; Santoro A.; Huang Q.; Capponi J. J.; Chaillout C.; Krekels T.; van Tendeloo G. J. Phys. Chem. Solids 1995, 56, 1471-1478.  268, 191[21] Tokiwa-Yamamoto A.; Tatsuki T.; Adachi S.; Tanabe K. Physica 1996, C 196. [22] Yamamoto A.; Hu W.-Z.; Tajima S. Phys. Rev. 2000, 63, 024504-1-6. [23] Wagner J. L.; Radaelli P. G.; Hinks D. G.; Jorgensen J. D.; Mitchell J. F.; Dabrowski B.; Knapp G.; Beno M. Physica 1993, C 210, 447-454. Superconducting Cuprates : Properties, Preparation and Applications, Nova Science Publishers, Incorporated, 2008. ProQuest Ebook Central,

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[34] Yanase Y.; Yamada K. J. Phys. Soc. Jpn. 2000, 69, 2209-2220. [35] Slichter C. P. Principles of Magnetic Resonance 3rd; Springer Series in Solid-Sate Sciences 1; Springer-Verlag: Tokyo, 1990. [36] Itoh Y.; Machi T.; Fukuoka A.; Tanabe K.; Yasuoka H. J. Phys. Soc. Jpn. 1996, 65, 3751-3753. [37] Itoh Y.; Machi T.; Adachi S.; Fukuoka A.; Tanabe K.; Yasuoka H. J. Phys. Soc. Jpn. 1998, 67, 312-317. [38] Itoh Y.; Michioka C.; Yoshimura K.; Hayashi A.; Ueda Y. J. Phys. Chem. Solids 2007, 68, 2031-2034. [39] Yamagata H.; Miyamoto H.; Nakamura K.; Matsumura M.; Itoh Y. J. Phys. Soc. Jpn. 2003, 72, 1768-1773. [40] Itoh Y.; Yoshimura K. unpublished works. [41] Itoh Y.; Tokiwa-Yamamoto A.; Machi T.; Tanabe K. J. Phys. Soc. Jpn. 1998, 67, 2212-2214. [42] Nishiyama M.; Kinoda G.; Zhao Y.; Hasegawa T.; Itoh Y.; Koshizuka N.; Murakami M. Supercond. Sci. Technol. 2004, 17, 1406-1410. Superconducting Cuprates : Properties, Preparation and Applications, Nova Science Publishers, Incorporated, 2008. ProQuest Ebook Central,

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[55] Itoh Y.; Yasuoka H. J. Phys. Soc. Jpn. 1994, 63, 2518-2521. [56] Itoh Y. J. Phys. Soc. Jpn. 1994, 63, 3522-3527. [57] Itoh Y. Physica 1996, C 263, 378-380. [58] Marshall W.; Lowde R. D. Rep. Prog. Phys. 1968, Vol. 31, 705-775. [59] Rossat-Mignod J.; Regnault L. P.; Vettier C.; Bourgers P.; Burlet P.; Bossy J.; Henry J. Y.; Lapertot G. Physica 1991, C185-189, 86-92. [60] Mila F.; Rice T. M. Physica 1989, 157, 561-570. [61] Shastry B. S. Phys. Rev. Lett. 1989, 63, 1288-1291. [62] Imai T. J. Phys. Soc. Jpn. 1990, 59, 2508-2521. [63] Walstedt R. E.; Warren Jr. W. W.; Bell R. F.; Cava R. J.; Espinosa G. P.; Schneemeyer L. F.; Waszczak J. V. Phys. Rev. 1990, B 41, 9574-9577. [64] Millis A. J.; Monien H.; Pines D. Phys. Rev. 1990, B42, 167-178. [65] Bulut N.; Hone D. W.; Scalapino D. J.; Bickers N. E. Phys. Rev. 1990, B41, 17971811. Superconducting Cuprates : Properties, Preparation and Applications, Nova Science Publishers, Incorporated, 2008. ProQuest Ebook Central,

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[76] Horvati´c M.; Berthier C.; Carretta P.; Gillet J. A.; S´egransan P.; Berthier Y.; Capponi J. J. Physica 1994, 235-240, 1669-1670. [77] Suh B. J.; Borsa F.; Sok J.; Torgeson D. R.; Xu M.; Xiong Q.; Chu C. W. Phys. Rev. 1996, B 54, 545-548. [78] Suh B. J.; Borsa F.; Xu M.; Torgeson D. R.; Zhu W. J.; Huang Y. Z.; Zhao Z. X. Phys. Rev. 1994, B 50, 651-654. [79] Kotliar G. Phys. Rev. 1988, B37, 3664-3666. [80] Suzumura Y.; Hasegawa Y.; Fukuyama H. J. Phys. Soc. Jpn. 1988, 57, 2768-2778. [81] Tanamoto T.; Kohno H.; Fukuyama H. J. Phys. Soc. Jpn. 1992, 61, 1886-1890. [82] Novikov D. L.; Freeman A. J. Physics 1993, C216, 273-283. [83] Lee W. S.; Yoshida T.; Meevasana W.; Shen K. M.; Lu D.H.; Yang W. L.; Zhou X. J.; Zhao X.; Yu G.; Cho Y.; Greven M.; Hussain Z.; Shen Z.-X. cond-mat/0606347. [84] Tanamoto T.; Kohno H.; Fukuyama H. J. Phys. Soc. Jpn. 1994, 63, 2739-2759. [85] Jakliˇc J.; Prelovˇsek P. Phys. Rev. Lett. 1995, 74, 3411-3414. [86] Ohashi Y.; Shiba H. J. Phys. Soc. Jpn. 1993, 62, 2783-2802. Superconducting Cuprates : Properties, Preparation and Applications, Nova Science Publishers, Incorporated, 2008. ProQuest Ebook Central,

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[100] Kishine J. preprint entitled by Spin Fluctuations in Magnetically Coupled Bi-layer Cuprates, Ph. D. thesis, University of Tokyo, 1996. [101] Matsumura M.; Nishiyama S.; Iwamoto Y.; Yamagata H. J. Phys. Soc. Jpn. 1993, 62, 4081-4092. [102] Stern R.; Mali M.; Roos J.; Brinkmann D. Phys. Rev. 1995, B 52, R15734-15737. [103] Suter A.; Mali M.; Roos J.; Brinkmann D. Phys. Rev. Lett. 1999, 82, 1309-1312. [104] Statt B. W.; Song L. M.; Bird C. E. Phys. Rev. 1997, B 55, 11122- 11125. [105] Goto A.; Clark W. G.; Vonlanthen P.; Tanaka K. B.; Shimizu T.; Hashi K.; Sastry P. V. P. S. S.; Schwartz J. Phys. Rev. Lett. 2002, 89, 127002-1-4.

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In: Superconducting Cuprates Editor: Koenraad N. Courtlandt

ISBN: 978-1-60456-919-3 c 2009 Nova Science Publishers, Inc.

Chapter 9

N EW P ERSPECTIVES ON L ONGSTANDING I SSUES OF THE H IGH -Tc C UPRATES B. J. Taylor and M. B. Maple University of California, San Diego, CA, USA

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Abstract An overview of the history of longstanding issues in the study of high temperature superconductivity is given in light of recent developments. We review experimental approaches to and analyses of the boundary between the electrically dissipation-less vortex solid state and the dissipative vortex liquid state, commonly referred to as the irreversibility line, Hirr (T), the vortex solid melting line, Hm (T), or the resistive upper critical field, Hc2(T ), highlighting the conjoined development of various theoretical perspectives as to the nature of this transition. From the perspective of recent work, the nature of the irreversibility line is inherently linked to the properties that determine the temperature at which superconductivity is established in zero field. Measurements of the vortex solid melting line, Hm (T), of certain high-Tc cuprates have recently been performed to higher H and lower T (T/Tc ≈ 0.03) than heretofore explored. A phenomenological melting line model, based on an extension of the Lindemann melting theory of Blatter and Ivlev, has been shown to describe the Hm (T) curve over the entire range of the measurements. Examination of the model yields an expression for Tc that is equivalent to the universal relation proposed by Homes et al., for the high Tc cuprates. A key quantity in this model is the ratio of the superconducting condensation energy within the coherence volume of a Cooper pair to the energy scale kB Tc . It was demonstrated that in hole-doped high- Tc cuprates this ratio is near to unity. Future directions of experimental exploration of the H − T phase diagram of cuprate superconductors in the context of the phenomenological melting line model are considered. Implications of the latter relationship with respect to the critical temperature-doping phase diagram, Tc − x, are also examined.

1.

Introduction

Shortly after the discovery of superconductivity by Bednorz and M¨uller in the Ba-La-Cu-O system [1], investigation of the magnetic susceptibility revealed the existence of an intermediate region, characterized by a reversible magnetic response in M (H), between the Superconducting Cuprates : Properties, Preparation and Applications, Nova Science Publishers, Incorporated, 2008. ProQuest Ebook Central,

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diamagnetic onset of superconductivity and a magnetically irreversible state [2]. This boundary between these two regions in the H − T phase space has come to be known as the irreversibility line Hirr (T ). We provide below a selected review of the history of the “irreversibility line,” Hirr (T), and the ensuing 22 years of experimental investigation and theoretical interpretations by various groups as to the nature of this boundary in the H − T phase diagram. Ensuing investigations of the “irreversibility line” in many other cuprate-based high- Tc superconducting compounds have involved a variety of techniques, leading to sometimes confusing results, various types of notation, and contrasting perspectives as to its true nature and thus to implications concerning the mechanism of superconductivity in these materials. From electrical transport measurements, a field-temperature line, marking a boundary between zero (or exponentially low) resistance and normal resistive states is found, representing a resistive upper critical field, denoted as HR (T ) (or sometimes Hc2 (T )). In a picture where the zero resistance state is comprised of a solid ensemble of vortices, the transition to the resistive state is known as a vortex-lattice or a vortex-glass melting transition, with the phase boundary denoted as Hm (T ) or Hg (T ), respectively. Depending on the type of disorder, various vortex-glass states can exist, further adding to the growing collection of names. While a final consensus concerning the nature of the boundary in question has not yet been arrived at, generally speaking there is agreement that all of the above “lines” represent the same phenomenon, and thus have come to be interchangable in meaning. Throughout the remainder of this chapter, when referring to the work of different authors, we attempt to retain their original language and notation keeping in mind the evolving measurements and theoretical perspectives. We have attempted to highlight key aspects of various experiments and theoretical developments relevant to the latter sections of this chapter, and to provide an overview of a range of theoretical viewpoints. Finally, we review and expand upon recent and ongoing research of our own into the properties of this phase boundary and on its relationship to fundamental properties of superconductors in general.

2.

Thermal vortex de-pinning and thermal vortex lattice melting

A common feature of the irreversibility line, found in the early investigations of different high-Tc compounds, was a temperature dependence where Hirr ∼ (1 − T /Tc)3/2. M¨uller et al., interpreted this as evidence for a superconductive-glass state, analogous to a spin glass [2]. Alternatively, Yeshurun and Malozemoff [3], and Malozemoff et al. [4], provide an explanation based upon the flux-creep model of Anderson and Kim [5, 6], referred to as giant flux creep. In the giant flux creep scenario a flux line, pinned to a structural defect, is thermally activated over this barrier with an energy U0 = Hc2ξ 3 /8π, where Hc2/8π is the condensation energy density and ξ is the Cooper pair coherence length. Crucial to this picture is the assumption of independent motion of finite bundles of vortex-line segments. Subsequently it was shown that, within this picture, the region characterized by irreversible magnetic behavior never becomes truly dissipationless with the consequence that, if achieved, any room temperature superconductor would be useless [3,4,7–9]. The extension of the model of Yeshurun and Malozemoff [3] by Tinkham [7] proved to be quite successful

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in accounting for the observed broadening of the resistive transition with increasing applied magnetic field. Inui et al. [10], however, make use of a single flux line-depinning model to consistently explain temperature dependence of both resistivity and ac susceptibility in samples of Bi2.2Sr2 Ca0.8 Cu2 O8+δ . Evidence for single flux line-depinning via frequency dependent ac susceptibility, penetration depth, and oscillating magnetization measurements were inferred in both single crystals and films of YBa 2 Cu3 O7−δ and Bi2.2 Sr2 Ca0.8Cu2 O8+δ by Gammel [11]. In contrast to the above flux creep picture, Brandt [12] considers the process of threedimensional melting of a bulk flux line lattice, wherein the irreversible region should be viewed as a solid vortex lattice (in analogy to an atomic lattice) which, due to thermal vortex displacements, melts at a field dependent temperature. Taking into account the effects of the large values of the ratio of the penetration depth λ to the coherence length ξ (given by the Ginzburg-Landau parameter κ = λ/ξ), on both local and nonlocal elastic behavior, an expression is found for the vortex lattice melting line where B(Tm ) = Bc2 (0)(1 − t2 )(1 − t4 )(t∗/t)2 , where t ≡ T /Tc, t∗ ≡ T ∗ /Tc, and T ∗ ≈ 40 K for YBa2 Cu3 O7−δ [13,14]. Evidence for the first order melting of the vortex lattice was observed via calorimetric measurements of the specific heat of untwinned YBa 2 Cu3 O7−δ single crystals by Schilling et al. [15]. Further measurements by Junod et al. [16] show that, depending upon the extent of disorder, the transition can be either a first-order melting of a vortex-lattice or a second-order melting of a vortex-glass.

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3.

The vortex-glass and Bose-glass states

In contrast to the flux creep model of thermally activated vortex motion, wherein independent motion of finite bundles of vortex segments are assumed, Fisher, Fisher, and Huse [17–19] propose a scenario in a superconductor with sufficient disorder involving the collective dynamic behavior of vortices which, at given temperatures, undergo a ‘freezing’ transition into an immobile state at which the superconductor become truly superconducting. This leads to a very different picture of the H − T phase diagram of strongly type-II M F (T ) is broadened by superconductors, where now the mean field upper critical field, Hc2 fluctuations into a crossover region between the fully normal state and the vortex-liquid state. The vortex-liquid phase consists of moble, fluctuating vortices. As the system is cooled further, it orders into a vortex-lattice or vortex-glass phase. The FFH picture of the melting line is based upon an extension of the critical properties of the superfluid density, ρs of the zero field superconducting transition to the case of a penetrating magnetic field. The result is that the melting line is to be considered a line of critical points belonging to the same (3D-XY) universality class, i.e. Tc → Tg . Hence, various scaling relations are predicted to exist in the region of the vortex-glass to vortex-liquid for the case of isotropic quenched disorder. Amongst these are scaling forms for the vanishing of the resistivity, ρ(T ), and the power law behavior of E(J) at the critical temperature, Tg , given respectively as, ρ ∼ |T − Tg |ν(z−d+2) , and E ∼ J (z+1)/d−1 , where ν and z are static and dynamic critical exponents and d is the dimensionality of the system. The first experimental evidence for the vortex-glass state was carried out by Koch et al., on epitaxial films of YBa2 Cu3 O7−δ [20]. Numerous studies have since followed providing additional evidence of the existence of the vortex glass phase in various superconducting materials [21–28].

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If correlated columnar disorder is dominant, then the critical dynamics of the vortex solid-liquid transition is described by the Bose glass model [29] where the relevant length scale is the wandering length of a localized vortex line transverse to the field direction, 0 0 `(T ) ∼ (TBG − T )ν , and the relaxation time of a fluctuation diverges as τ ∼ `⊥z , where ν 0 and z 0 are new critical exponents. However, Lidmar and Wallin [30] have shown that the scaling properties of a Bose glass also arise from the scaling properties of the superfluid density; thus, the two vortex glass melting theories are quite related. The vortex dynamics in the melting region described by the Bose glass model also leads to a power law scaling of the dc resistivity, ρ(T ), with the same form as that found by the FFH vortex glass model with the exponent ν(z − d + 2) → s0 , where s0 is defined via the scaling relations for || d−2 ) and ρ = `d−ζ−1−z ρ the resistivity, ρ⊥ = `d+ζ−3−z ρ˜⊥ ˜± (H⊥ `d−2 ), where ζ is || ± (H⊥ ` an anisotropy exponent, with ζ = 1 for unscreened long-range interactions and ζ = 2 for correlated disorder [30]. In the case when H is parallel to the columnar defects, the relevant resistivity is ρ⊥ , so that s0 ≡ ν 0 (z 0 + 3 − ζ − d), with ζ = 2. The correct form for the critical exponent can be determined by the angular dependence of the scaling of the resistivity at the melting transition, where, if the vortex ensemble 0 0 0 0 0 0 is a Bose glass [31], ρ⊥ (t, θ) = |t|ν⊥ (z −2) f± (θ/|t|ν⊥ ) and ρ||(t, θ) = |t|ν⊥ z g± (θ/|t|ν⊥ ). The resulting behavior can be observed via a cusp in the phase boundary TBG(H⊥ ) c (T ) at the Bose glass to vortex liquid transition varies where the perpendicular field H⊥ 0 c ν as H⊥ ∼ ±(TBG (0) − T ) [30, 31]. However, the Bose glass phase eventually gives way to the vortex glass phase as the field increases sufficiently past a characteristic field at which the number of vortex lines is equal to the number of columnar defects [32].

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4.

Universality of the irreversibility line of electron- and hole-doped high-Tc superconductors

The cuprate-based superconductors (both electron- and hole-doped) share a common structural feature; the perovskite unit cell with Cu-O layers within the a−b plane separated in the c-direction by the remaining chemical constituents by varying distances, depending on the compound. This quasi-2D structure was anticipated to have consequences on the properties of vortices and vortex structures, particularly in the highly anisotropic bismuth and mercury based compounds where the anisotropy parameter γ ≡ (ρc /ρab)1/2 ∼ 100 − 1000 [33]. Evidence for a universal temperature dependence of Hirr (T ) for both electron- and holedoped cuprate superconductors, was reported by Almasan et al. [34]. A universal expression for Hirr (T ) argues strongly for common underlying physics in spite of the many physical disparities of the high-Tc cuprate superconductors. Numerous experimental investigations of the boundary between the electrically dissipationless, magnetically irreversible region and the electrically dissipative, magnetically reversible region; i.e. the irreversibility / melting line, Hirr (T ) / Hm (T), of both electronand hole-doped cuprate superconductors indicated anomalous behavior arising as the magnetic field increased and the temperature decreased [35–38]. In particular, from the universal form of Hirr (T ) observed by Almasan et al. [34], it is seen that at temperatures T ≥ 0.6 Tc , Hirr (T ) follows a (1 − T /Tc)m form with m ≈ 3/2, giving way to a more rapid temperature dependence at lower temperatures. For instance, de Andrade et al. [35]

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found, for the electron-doped compound Sm 1.85 Ce0.15CuO4−y , the low temperature region follows the same analytic form with a larger value of the exponent m. Schilling et al. [37], however, observed that the low temperature region of Hirr (T ) of a highly anisotropic (γ ∼ 370) Bi2 Sr2 CaCu2 O8 single crystal was best described by an exponential expression, Hirr (T ) ∼ exp(1/T ), which was taken as evidence for a crossover from a low field 3D to high field 2D vortex regime. Numerous additional studies have supported this picture [39–42], although this conclusion is not universally accepted [43]. While generally considered to be a much less anisotropic system, the irreversibility line of YBa2 Cu3 O7−δ wth various oxygen concentrations, determined by magnetization or magnetic susceptibility measurements, was found to be consistent with a 3D-2D crossover model by Gray et al. [44]. Additionally, following the analysis of Schilling et al. [37], Almasan et al. [45] also found evidence for a crossover in dimensionality of the vortex ensemble in the Y 1−x Prx Ba2 Cu3 O6.97 system.

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5.

Quantum melting model of Blatter and Ivlev

Soon after the discovery of the high-Tc cuprate superconductors it was recognized that, due to the typically large values of their Ginzburg numbers Gi , the effects of thermal fluctuations on the order parameter were relevant to the fluctuation regime about the (zero field) critical temperature. The Ginzburg number defines a temperature range, Tf , about the transition temperature, Tc , over which (thermal) critical fluctuations of the order parameter become significant, given by, (1 − Tf /Tc ) = Gi . Additionally, as seen below, the large values of Gi result in thermal fluctuations also being significant with respect to the displacement of vortices in a pinned solid lattice state. Blatter and Ivlev (BI) showed that quantum fluctuations are also a relevant source of vortex flux line displacements, particularly in the high-Tc superconductors [46, 47]. The vortex lattice melting theory of BI takes into account the contribution of both quantum and thermal fluctuations to the melting of the vortex lattice, resulting in a universal form of the vortex lattice melting line based upon a Lindemann criterion approach, wherein a melting transition occurs when the mean squared amplitude of fluctuations of a lattice approaches a sizable fraction of the lattice constant a0 ,

2  u (Tm ) ≈ c2L a20 , where cL ∼ 0.1 - 0.3 and a0 ≈ (Φ0/B)1/2 for a vortex lattice. From their model it is seen that quantum fluctuations are present for all temperatures, but are most relevant above a characteristic magnetic field, which they find is ∼ 2-3 tesla in the high-Tc cuprates, so that quantum fluctuations must be accounted for over most of the melting line. In refs. [46] and [47], BI arrive at two slightly different expressions describing the temperature-field dependence of the melting line. In their initial work, [46] when calculat 2 ing the mean squared displacement amplitude u , a term involving compressional modes is dropped. In the latter work, [47] this term is retained. For the first case, BI obtain Hm (t) =

4Hc2(0)θ2 √ (1 + 1 + 4Qθ)2

(1)

√ where θ √ is a reduced temperature given by θ = (πc2L/ Gi )(1 − t), ˜ u /(π 2 Gi )]Ωτr is a parameter measuring the relative strength of quantum Q = [Q ˜ u = e2 ρN is the dimensionless quantum of resistance, to thermal fluctuations, t ≡ T /Tc , Q h d ¯

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cL is the Lindemann number, Gi is the Ginzburg number, Ω is a cutoff frequency, τr is the scattering relaxation time of the quasiparticles in the vortex core given by the Drude 2 formula ρ−1 N = σN = e nτr /m, (σN is the normal state conductivity, n is the free charge-carrier density, and m the electron mass), and here, d is the distance p between the superconducting planes. Ω is given by Ω = min[Ωρ , Ωi], where Ωρ ≈ η` /µ` τr is the kinetic cutoff frequency associated with the electromagnetic contribution to the vortex mass, (with η` and µ` and denoting respectively the flux line viscous drag coefficient and the electromagnetic contribution of the vortex line mass [48, 49]), and Ωi ≈ ¯h2 ∆ is the intrinsic cutoff given by the energy gap due to the creation of quasiparticles by vortex motion. For the latter case, they find,

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Hm =

4Hc2(0)θ2 p (1 + 1 + 4Sθ/t)2

(2)

√ q q 2 β where now θ = c2L βGthi TTc (1 − t), S = q + c2L βGthi , and q = π3 th √QGu Ωτr . Either exi pression can be approximated by the power-law form H m ∼ (1 − t)α over temperatures T ranging from Tc down to 0.6 Tc . As BI note, the value of α depends on the quantum parameter Q and the reduced temperature θ such that, as lower temperatures are reached along the melting line, the value of α increases, effectively dividing the melting line into segments which can each be described by the power law form above with distinct values of α. This provides an explanation for the experimentally observed increase of α as the temperature drops below T ≈ 0.6 Tc . [34] Blatter et al. provide a comprehensive review of issues surrounding the properties of vortices in high-Tc superconductors in ref. [50]. From their perspective, there are two things which govern the statistical mechanics and the dynamics of vortices in superconductors: ( i) (dynamic) thermal and quantum fluctuations and (ii) (static) quenched disorder. The importance of each source of disorder are quantified by the Ginzburg number Gi , the quantum of resistance Qu = (e2/¯ h)(ρn /ξ), and the critical current-density ratio jc /jo , where jc and jo are the depinning and depairing current densities, respectively, ρn is the normal-state resistivity, and  is the anisotropy parameter.

6.

The melting line in YBa 2 Cu3 O6+x and the modified vortex glass model of Rydh, Rapp, and Andersson

The evolution of the melting line as a function of oxygen content in the YBa 2 Cu3 O6+x system was investigated by Lundqvist et al. [51], to magnetic fields up to 12 tesla, corresponding to a temperature range as large as 0.25 Tc ≤ T ≤ Tc . They found empirically that the entire melting line data of all samples could be well described by the expression, Bm ≈

1.85Φ0 [(1 − t)/t]α , 2 (γd)

where γ = 1/ is the anisotropy parameter, d the interlayer spacing, and α ≈ 1. The ability to smoothly fit a single expression to the data, i.e., beyond the previously observed Superconducting Cuprates : Properties, Preparation and Applications, Nova Science Publishers, Incorporated, 2008. ProQuest Ebook Central,

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275

characteristic universal crossover temperature T ∗ ≈ 0.6 Tc , was therefore taken as evidence that the vortex ensemble remains 3-dimensional across the entire field-temperature range examined. Subsequently, Rydh, Rapp, and Andersson (RRA) [52, 53] developed a modified vortex glass model, based in part upon the model of FFH. Where FFH only explicitly accounted for the temperature dependence of the critical dynamic behavior of vortices as the melting transition is approached, RRA add to this the assumption that the pinning energy scale changes with both temperature and magnetic field so that energy difference kB T - U0(B, T ), where U0 is the current independent mean pinning energy, fully characterizes the vortex glass transition. This leads to a modification of the scaling form of the resistivity as the melting transition is approached, ρ = ρ0 |

T (Tc − Tg ) − 1 |ν(z−1) , Tg (Tc − T )

where ρ0 is taken as the normal state resistivity just above T c . Furthermore, by assuming a scaling form for the mean pinning energy, U 0, such that, U0 (B, T ) = kB Tc

f (t) (B/B0 )β

,

where f (t) is a function of t = T /Tc , and B0 and β are temperature and field independent constants, the above empirically determined form of the melting line can be recovered.

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7.

Vortex-liquid or fluctuation suppressed upper critical field?

Cooper et al. [54, 55] and Babi´c et al. [56] put forth a different position on the nature of the resistive upper critical field, i.e., the melting line. Cooper [55] raises the point that, while neutron scattering studies show that the flux line lattice disappears at Hirr (≡ Hg ) [57, 58], they provide no indications that line vortices actually survive to form a flux-line liquid. This opens the possibility of a picture wherein the solid vortex state gives way to a state consisting of large, thermally driven, critical superconducting fluctuations, so that the vortex melting line actually corresponds to an upper critical field line, Hc2 (T ), that has been suppressed by thermodynamic fluctuations. In refs. [55, 56], irreversibility lines determined by magnetization M(H) measurements of samples of YBa 2 Cu3 O6+x across a wide range of oxygen doping were examined in the context of the 3D-XY scaling model. It is demonstrated that above Hirr , the magnetic response attributable to fluctuations Mfluc scales as expected from the 3D-XY model. From this it is concluded that ( i) Hirr is a fixed fraction of the upper critical field in the critical region, ( ii) along the irreversibility line, the free energy in a correlation volume is a fixed multiple of kB T , and (iii) thermodynamic fluctuations determine the location of Hirr (T ). These results are consistent with both a flux pinning scenario
and a critical fluctuation scenario. From the latter it is found that, at 0 2 ξ 0 = Dk T , where D is a constant of order 4. Finally, it is also H = 0, Hc (0)2 ξab B c c observed that the low field portion of the irreversibility fields, Hirr scale in accordance with what is expected from the 3D-XY model, i.e., Hirr = H ∗(1 − T /Tc)2ν where ν ≈ 2/3. It is interesting that, while the position concerning the (non) existence of vortex flux lines above the irreversibility (melting) lines of Cooper et al. is in stark contrast to the

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B. J. Taylor and M. B. Maple

vortex-lattice (glass) melting scenario of FFH, both models invoke the existence of critical behavior along the entire boundary, Hg (T ) or Hc2 (T ), resulting in the scaling of various properties whose temperature dependencies in the critical region are dependent upon the values of exponents expected from the 3D-XY model. In light of this common basis concerning thermal fluctuations, an intriguing result from an early investigation of the resistively determined upper critical field, HR(T ), (which we would identify here as Hirr (T )), of Sm1.85Ce0.15 CuO4−y , is recalled, wherein, via a scaling analysis of the fluctuation conductivity, σf l, Han et al. [59] determined an H −T phase diagram consisting of both HR (T ) and Hc2(T ). They find that the scaling behavior of σf l in the region between HR (T ) and Hc2 (T ) crosses over from a 3D behavior near Hc2 (T ) to a 2D behavior midway between HR (T ) and Hc2 (T ).

8.

Connection between fundamental properties of high-Tc superconductors: The critical temperature, Tc , upper critical field, Hc2 (0), critical behavior of the superfluid density, ρs , and the vortex-solid melting line, Hg (T )

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In this section, we review selected recent work of our own on the issues surrounding the nature of the vortex solid-liquid transition and a connection to the relationship between the value of the superconducting critical temperature, Tc , and the condensation energy density, Hc0/2µ0 . We also present new results concerning the evolution of charge carrier density in the YBa2 Cu3 O6+x system, deduced within this context from the analysis of various data available in the literature, and find behavior consistent with the dynamic charge stripe scenario [60, 61].

8.1.

Fuctuation conductivity and the quantum-thermal vortex solid melting transition scenario

As seen from the preceding sections, there are many processes that must be considered when trying to accurately describe the transition from an electrically dissipation-less vortex-solid state to the dissipative vortex-liqiud state, including: thermal and quantum fluctuations of vortex flux lines, strength and kind of pinning structures, anisotropy of superconducting properties, coupling of the vortex flux lines to the underlying electronic structure, and the critical behavior of vortices as the melting transition is approached. For over two decades, a consistent theory describing the vortex-solid melting scenario over the entire vortex lattice melting line with a single expression has not been forthcoming. Recently, however, via an empirical modification of the quantum-thermal vortex-lattice melting model of Blatter and Ivlev [46], (discussed above in section 5.), an expression was arrived at that was found to be capable of accurately describing the entire vortex glass melting line H g (T ) of Y1−xPrx Ba2 Cu3 O6.97 thin film samples (x = 0 - 0.4) and an ultra high purity oxygen deficient YBa 2 Cu3 O6.5 single crystal in magnetic fields up to 45 tesla, corresponding to a temperature range as large as 0.03 ≤ T /Tc ≤ 1 [62, 63]. We use the terms vortex lattice and vortex glass and the notation Hm (T ) and Hg (T ) interchangeably, mainly due to the different terminology used in refs. [18, 46, 47], however, the expression

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277

we arrive at describing the H − T dependence of the vortex solid-liquid boundary is for a second order vortex glass melting transition. 8.1.1. Modification of the Blatter and Ivlev model In section 5., two related expressions for the temperature dependence of the vortex-lattice melting line Hm from the quantum vortex-lattice melting model of Blatter and Ivlev [46, 47] were examined. The key parameters of the model in the two melting line expressions are the quantum parameter Q or q, respectively, and the Lindemann number c L , which characterizes the stability of the vortex lattice to fluctuations. The value of the quantum parameter consists of four further characteristic parameters, given by Q=

˜ Q √u Ωτrv , π Gi

(3)

2

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˜ u = e ρN is the quantum of resistance, Gi is the Ginzburg number, where Q h ξ ¯ Ω = min[Ωµ , Ω∆ ] is a kinetic cutoff (µ) or gap limiting (∆) frequency, and τrv is the relaxation time of a single vortex displaced by a quantum/thermal fluctuation. In ref. [62], the results of an experimental investigation of the vortex glass melting lines Hg (T ) of Y1−x Prx Ba2 Cu3 O6.97 thin film samples (x = 0 - 0.4) and an ultra high purity oxygen deficient YBa2 Cu3O6.5 single crystal in magnetic fields up to 45 tesla were initially analyzed in the context of the theory of Blatter and Ivlev. However, starting from the (first) universal form of the vortex lattice melting line it was found that the entire vortex glass melting line H g (T ) of each sample investigated can be described by the expression of BI when the temperature/field dependences of parameters held constant in their expression are accounted for, and if it is assumed that the relaxation time of a single vortex line in the region of the melting transition takes the form τrv

= τ0



T Tc

s 

T 1− Tc

−s

,

(4)

where Tc is the superconducting critical temperature in zero field. This modification was also extended to the second melting line expression of Blatter and Ivlev in ref. [63]. The resulting two expressions are given by, (πc2 )2

Hg (t) = 1+

r

L 4Hc2(0) Gi (H (1 − t)2 g)

1+

2 ˜ 0Ω0τ0 ) cL ts˜(1 − 4(Q πGi (Hg )

!2 ,

(5)

t)2−˜s

and β c4

Hg (t) = 1+

4Hc2(0) Gith(HLg ) (t−1 − 1)2

!2 .  ˜  2 cL βth −1 −1 2Q0 Ω0 τ0 s˜ 2 1−˜ s 1+4 t (1 − t) + cL G (Hg ) t (t − 1) π3

r

i

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(6)

278

B. J. Taylor and M. B. Maple

Q(Hg , Tg ) =

q[µ,∆]

˜ 0 Ω0τ0 Q q

(7)

√ ˜ 0Ω0 τ0 2 βth Q q = ts˜(1 − t)1−˜s , π3 Gi (Hg ) 

e2 ρN h ξ0 , ¯

s

π 2 Gi (Hg )

Ω0[µ, ∆] = Ωµ0 ≡ ˜0 ≡ Q

s

t 2 (1 − t)1− 2 ,

4cλd ξ0

r



2∆0 πµ0σN , Ω∆0 ≡ , τ0 ¯ h

s˜[µ, ∆] = [s/2, s], t ≡ T /Tc = Tg /Tc , η` ≈ 1 ξ2 Hc2 4µ0 c2



(8)

2

Φ20 σN 2πξ 2

(9)

is the viscous drag

λ coefficient [48], µem = is the electromagnetic contribution of the vortex ` λd mass [49], and Gi (H) is the field dependent expression of the Ginzburg number, [33]

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Gi (Hg ) ≈ (Gi )

1 2α



Hg Hc2 (0)

1

α

,

(10)

where H is evaluated along the melting line at H = Hg , and α ≈ 3/2. The Hg (T ) data from the above study are shown in Fig. (1) in linear and semi-log plots, to emphasize the quality of the fits to the data of the phenomenological melting line equation given in Eq. (5). Equally good fits to the data are also obtained using Eq. (6) as shown in ref. [63], with consistent values of the parameters. The result that both of the above expressions can be fit over the entire melting line of each sample implies that the physical process of the melting of the vortex solid can be described continuously over the entire temperature − field range, with an underlying common physical mechanism. Additionally, examination of the ratio of the quasiparticle and London current densities (inside and surrounding the vortex core, respectively), and the power dissipated by normal currents outside the vortex core as T → Tc within the context of the modified BI model above indicates that quantum fluctuations of the vortex core are relevant to the melting process for all T < Tc . 8.1.2. Fluctuation conductivity and the single vortex relaxation time The above expression for τrv finds a natural explanation in the context of the scaling behavior of the fluctuation conductivity [18, 64]. That is, if the dynamical behavior of the vortex flux lines is determined by the properties of the conductivity within the vortex core, and if the fluctuation conductivity is significant, then the above scaling form of τrv is expected from which it follows that the exponent s is a critical exponent associated with the scaling form of the superfluid density, ρs . The critical scaling behavior of the superfluid density comprises the foundation of the vortex-glass model of Fisher, Fisher, and Huse (FFH) [17,18] wherein the melting line, Hg (T ), actually consists of a line of critical points, i.e., the critical behavior of the superconducting transition at (H = 0, T = Tc ) is dragged along the boundary between the dissipative and dissipation-less states. From the form of ρs in the critical region FFH show that the resistivity vanishes as, ρ ∼ (T − Tg )ν(z+2−d) as the critical temperature Tg is approached, where ν and z are the static and dynamic critical exponents, and d is the dimensionality of the vortex system. It is argued in [62] and [63]

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New Perspectives on Longstanding Issues of the High- Tc Cuprates Y Pr Ba Cu O 1-x

2

3

6.96

& YBa Cu O 2

3

(a)

6.5

(b)

300 200

10

100 0

100 g

g

x

H (kOe)

H (kOe)

400

279

0

20

40

T (K)

60

80

0

x=0 x = 0.1 x = 0.2

20

40

T (K)

x = 0.3 x = 0.4 x = 0.0 ; y = 6.5

60

80

1

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Figure 1. Vortex melting line H g vs T data from ref. [62] for Y1−x Prx Ba2 Cu3 O6.97 films (1kOe < H < 450 kOe) and a YBa2 Cu3 O6.5 single crystal (100 Oe < H < 450 kOe) with fits of the modified melting line Eq. (5) shown in linear (a) and semi-log (b) plots, emphasizing the quality of the fits. that the expression for relaxation time τrv arises from the scaling form of the conductivity σ = 1/ρ and hence the exponent s is the same critical exponent characterizing the critical behavior of ρ at the vortex solid-liquid boundary, i.e., s = ν(z + 1 − d). If instead the dynamics of the vortex ensemble are described by the Bose glass model, the exponent s is then identified with the Bose glass exponent that describes the vanishing of the resistivity [29, 30]. This argument is based upon the fact that both scaling forms of the resistivity from the vortex-glass and Bose-glass model are arrived at from the critical scaling behavior of the superfluid density. The value of the exponent s, found by fitting Eq. (5) to the data, was seen to be in close agreement with that found by scaling the resistivity data for each sample examined in Fig. (1), suggesting that the two different approaches of BI and FFH to the problem of the melting of the solid vortex ensemble are both valid and their perspectives on the physics involved are compatible. Subsequently, any changes in the critical dynamic behavior of the vortices along the melting transition will necessarily correspond to a change of the value of the critical exponent over these various regions, so that each section will be described via the above expressions for the melting line with with the different exponents. The result that the melting line can be fit over the entire melting line of each sample with a single exponent implies that no crossover in dynamics from 3D to 2D takes place. Finally, discussed in section 6., above RRA [52, 53] developed a modified vortex-glass model based upon the original model of FFH, [18] and on the empirical observation of Lundqvist et al., [51] where the vortex glass melting line (over a temperature range 0.25Tc ≤ T ≤ Tc ) was found to be described by, Bm ≈

1.85Φ0 [(1 − t)/t]α . 2 (γd)

(11)

It is easily seen that Eq. (11) can be recovered as a high Q or q limiting case of Eqns. (5) or (6) where α = s˜, lending further support to the notion that many of the various approaches to the problem of the melting transition are in fact compatible.

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8.2.

B. J. Taylor and M. B. Maple

First universal Tc equation

In addition to being able predict what materials can support superconductivity, and to provide an understanding of the physical mechanism responsible for superconductivity, one of the primary goals of any theory of superconductivity is to be able to accurately predict the critical temperature Tc below which the material enters the superconducting state. In this section and the following, we review our recent development two equivalent expressions for the critical temperature Tc in terms of the condensation energy contained within the coherence volume of a single Cooper pair (Hc (0)2/2µ0)Vcoh . In both expressions, these two energy scales are related by a characteristic parameter, denoted as CL , where kB Tc = CL (Hc(0)2/2µ0 )Vcoh . In this section a definition of CL is arrived at in terms of parameters involved in the characterization of the vortex glass melting line Hg (T ) as described above. An independently defined expression for CL in terms of the density of states at the Fermi level N(0)EF is given in the following section. In ref. [65] we make use of the result that both Eqns. (5) and (6) describe the vortexglass melting line data shown in Fig. (1) equally well, and subsequently extrapolate to the same field values at low temperatures. From the empirically determined expression for the single vortex relaxation time τrv τrv

= τ0



T Tc

s 

T 1− Tc

−s

,

(12)

it follows that as T → 0, τrv , Q and q → 0. From this we then have at T = 0, for Eqs. (5) and (6), respectively; π 2c4L Hg (0) = 1/2α , (13) Hc2 (0) G Copyright © 2008. Nova Science Publishers, Incorporated. All rights reserved.

i

and Hg (0) = 1. Hc2(0)

(14)

where, as discussed above, the exponent α characterizes the approximate form of the melting line at high temperatures, via the field dependent expression for the Ginzburg number. Combining Eqs. (13) and (14), gives, 

Gi = π 2 c4L

2α

,

(15)

The Ginsburg number defines a temperature range, Tf , about the transition temperature, Tc , over which (thermal) critical fluctuations of the order parameter become significant, given by, (1 − Tf /Tc) = Gi. For a 3-D system, Gi is defined as, Gi = 12 [kB Tc /((8πHc2/2µ0 )Vcoh )]2. It is a measure of the relative size of the energy density Hc2 /2µ0 associated with the condensation energy of a single Cooper pair coherence volume, with respect to the critical superconducting temperature, Tc . The coherence volume is conventionally defined as Vcoh = ξaξb ξc [= ξ 3 in most cases]. We propose that a more accurate determination of the unit volume, Vcoh , for a type-II superconductor, can be found by the geometry over which the order parameter is suppressed within a unit √ length of 2 a vortex flux line [aligned along the c-axis], VΦ` ≈ πrrmsξc , where rrms = ξa ξb . This

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New Perspectives on Longstanding Issues of the High- Tc Cuprates 100

281

YBa Cu O 2

3

x

60 40

c

T (K) [calculated]

80

underdoped overdoped 1/8 doping optimal critical doping

20 0

0

20

40

60

T (K) [experimental]

80

100

c

Figure 2. Calculated value of Tc vs. the experimentally determined value of Tc . The value of Tc is calculated from Eq. (18) as described in the text. The experimentally determined value of Tc is taken from ref. [67]

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reasoning follows from the expression for the thermodynamic critical field, Hc , Φ0 Φ0 Hc = √ = √ , 2 2πλξ 2 2A

(16)

2 where, in this case, A = πr√ rms is the area of a circle given by the length scales λ and ξ with an rms radius rrms = λξ. Furthermore since we are concerned with the expulsion of the flux Φ0 from within a cross sectional area determined by √ the energy scale of the , the same geometric factor, 2 2, ought to apply so that thermodynamic critical field, H c √ 3 VΦ` = 2 2πξ . Combining Eq. (15) and the above definition of the Ginzburg number, a straight forward relationship between the critical temperature, Tc , and the condensation energy is found, wherein, α H 2 (0) √  c VΦ` . (17) kB Tc = 8π 2 π 2 c4L 2µ0

The above expression for Tc can actually be traced back, implicitly, to the pioneering work of Ginzburg, where the expression for Gi comes from the Ginzburg-Landau free energy expression for a second order phase transition. As stated in a footnote, “. . . there is also a dimensionless parameter that characterizes the ratio of the volume and correlation energies in the Hamiltonian itself.” . . . ‘and, to a considerable extent, determines the value of the transition temperature’. -V. L. Ginzburg, Sov. Phys. Solid State, 1960.

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B. J. Taylor and M. B. Maple √
α For convenience we define CL ≡ 8π 2 π 2c4L . CL is then the fraction of the condensation energy contained within the coherence volume VΦ` that is used in the formation of a single Cooper pair. Using the unit volume VΦ` and the definition of CL , we arrive at the universal expression, Tc =

"

#

C Φ20 ξ0 √L 4 2π µ0 kB λ20

≈ (1.39 × 10−2 Km)CL

(18) ξ0 . λ20

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In Fig. (2), calculated values of the critical temperature Tc are plotted vs the experimentally determined values of Tc for samples of YBa 2 Cu3Ox presented in the work of Tallon et al., [67] using Eq.(17) and the data Tallon et al. give for λ0, values of  from Chien et al. [68], and data for the coherence length ξ0 from Ando et al., Grey et al., (see Fig. (5)), and calculated values of ξ0 in the overdoped region from Konsin and Sorkin [69]. The values α = 3/2 and cL = 0.30 are used for all samples. The value of c L = 0.30 is in close agreement with the values of 0.31 and 0.28 found for the pure YBa 2 Cu3 O7−δ and YBa2 Cu3 O6.5 samples, respectively, in [62] by fitting the melting line data with the modified vortex glass melting line expression of Blatter and Ivlev. We find that the variation is only ∆cL = 0.015; that is, by allowing cL to vary for each sample by less 0.015, we can achieve Tc (K)[calc] = Tc (K)[expt] for all of the data. We could also achieve this by allowing for a less than 10% variation in λ0, ξ0 , or 0 . It was found that CL ≈ 0.9 − 1.0 in all regions, under-, over-, and optimally doped indicating that the coherence volumes of Cooper pairs at low temperatures in YBa 2 Cu3Ox overlap to a very small extent throughout the entire doping range.

8.3.

Second universal Tc expression

As mentioned above, the value of CL can be viewed as representing the fraction of the condensation energy within a given volume that is used in forming a single Cooper pair. Equivalently, it is the ratio of two energy densities, that of a single Cooper pair, defined by the ratio of the correlation energy kB Tc to its coherence volume VΦ` to the condensation energy density, Hc2(0)/2µ0. From another viewpoint, CL measures the extent to which the coherence volume VΦ` of each Cooper pair overlaps with the volume occupied by other P pairs. Following this perspective, a definition of the form CL ≡ aV / VΦ` , where V is the volume of the system and a is a geometrical factor, is natural. Similar to this geometrical argument, we have instead chosen [70] to count the number of one-electron states at the Fermi energy (for one direction of spin) contained in the coherence volume, VΦ` , with respect to the number used in the formation of the Cooper pair (2 electrons). In the simplest (flat band) approximation, this definition is then, CL ≡

2 8 = N (0)EF VΦ` 3nn VΦ`

(19)

h3 is the density of states at the Fermi level for a free electron where N (0) = m∗2vF /2π 2¯ Superconducting Cuprates : Properties, Preparation and Applications, Nova Science Publishers, Incorporated, 2008. ProQuest Ebook Central,

New Perspectives on Longstanding Issues of the High- Tc Cuprates HgBa Ca Cu O 2

2

3

8

HgBa2CaCu2O6 HgBa2CuO4

YBa2Cu3O6.96 MgB2

10

Pr1.85Ce0.15CuO4

Nd1.85Ce0.15CuO4 YNi2B2C

In CeCoIn5 UPd2Al3 PrOs4Sb12 URu2Si2 Al PrRu4Sb12 UBe13 CeCu2Si2 UPt3

Pb

CeRu2

c

T (K) [calculated]

100

283

1

0.1 0.1

1

10 T (K) [expt]

100

c

Figure 3. Calculated values of the superconducting critical temperature, Tc , using the expression given in Eq. (21) vs. the experimentally determined values. The compounds shown here represent many different types and classes of superconductors. With the exception of Pb, In, and Al, see ref [70] for a table of values of experimental parameters used to calculate Tc and the references from which they are taken. For experimental parameter values of the elemental superconductors see refs. [71–81] here. 

k3

∗

3

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is the normal state gas, EF = m∗ vF2 /2 is the Fermi energy, nn = 3πF2 = 3π12 m ¯hvF ∗ carrier density (m is the electron effective mass, and vF is the Fermi velocity). This gives, 8 Hc2(0) . (20) 3nn 2µ0 which is independent of the geometrical definition of the Cooper pair coherence volume, Vcoh . √ We next define the effective penetration depth as, λef f (0) ≡ λL (0)/ 4π, where λL (0) is the London penetration depth, given by λ2L (0) = m∗s /(µ0 ns e2 ) (in mks units) with ns the superconducting carrier density and m∗s the charge carrier effective mass in the supercon√ ducting state. Our definition of λef f (0) is a factor of π larger than that defined in ref. [82], where the (effective) London penetration depth p), is √ p (as a function of doping concentration, defined as λL (p) ≡ 1/2πωps , with ωps = µ0 ns e2 /m∗. The factor of 1/ 4π was chosen as it was found to bring the values of Tc expected from Eq. (21) below into close agreement with the measured values of Tc for all data in Fig. (3). By inserting the above expression for λef f (0) and Eq. (19) into Eq. (17) we arrive at, kB Tc =

Tc =



2π ns 3kB nn



¯ h ξ0

2

1 . m∗s

(21)

Calculated values of the critical temperature Tc vs the experimentally determined values of Tc of a large number of compounds belonging to many different classes of superconducSuperconducting Cuprates : Properties, Preparation and Applications, Nova Science Publishers, Incorporated, 2008. ProQuest Ebook Central,

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B. J. Taylor and M. B. Maple

tors, including optimally doped electron- and hole-doped cuprates, heavy fermion systems, and multi-band superconductors, are plotted in Fig. (3). It can be seen that the relationship given by Eq.(21) holds for over two decades of Tc values, with the notable (and curious) exception of the elemental superconductors examined, for which we consistently calculate values of Tc less than the experimental value. At this time we do not know if the discrepancy is of a real physical origin, or, if the data, which is up to 35 years old for these elements, are unreliable. For instance, if we use values of the coherence length ξ0 = 920 nm for aluminum films with Tc = 1.36 K, reported recently by Bruyndoncx et al., [83] and the same effective mass m∗c = 0.11 me used in the calulation for the bulk material, we find Tc [calc] = 1.99 K. Or, using the experimental value of Tc = 1.36 K, we find a calculated effective mass, m∗ = 0.19 me , well within the range m ∗c = 0.11 - 0.4 me reported in ref. [78]. It is important to resolve this issue. With the exception of the hole-doped cuprate systems, the value (ns /nn = 1) is assumed. Tanner et. al. [84] observed (ns /nn ) ≈ 1/4 for optimally doped hole-doped cuprate systems. Additionally, the values used for the effective electron mass are those obtained by either de Haas-van Alphen, m ∗c , or optical conductivity measurements, m∗opt , instead of the effective mass m ∗ estimated from measurements of the electronic specific heat coefficient. For materials where multiple branch masses are reported, the mass value that gives, via Eq. (21), the calculated value of Tc in closest agreement with the experimental value was used [70]. In general, it was found that use of the thermodynamic effective mass from specific heat measurements in Eq. (21) gives calculated values of Tc a factor of ∼ 2 - 20 less than the experimental value. It can be seen that the relationship given by Eq. (21) holds for nearly three decades of Tc values.

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8.4.

Evolution of charge carrier density in YBa2 Cu3 O6+x

Because the expression for Tc given in Eq. (21) is equivalent to Eq. (18), it should also accurately provide values of Tc for the YBa2 Cu3 O6+x system, examined above in section 8.2.. We make use of this here to examine properties of the normal state and superfluid charge carrier densities, nn and ns , as a function of oxygen content from the under-doped to over-doped regimes. First we make use of Eq. (20) to find nn (x). We use the of data of Loram et al. [85] (as shown in Cooper et al. [55]) of the condensation energy density Hc2 /2µ0 as a function of oxygen content in YBa 2 Cu3O6+x . This is done so as to provide independent results from those found in section 8.2. to which a comparison can then be made. The values of Tc (x) are obtained from Liang et al. [86]. The calculated values of nn (x) are shown in Fig. (4), which we comment on further below. Next, using the same Hc2/2µ0 data, the penetration depth data λ0(Tc ), reported by Zuev et al. [87], and the same values of Tc (x), (x) via the Ginzburg-Landau expression for the thermodynamic critical field we calculate ξ0√ Hc (0) = Φ0/2 2πλ0ξ0 . We also calculate values of ξ0 (x) using the following method. Since CL is by (one) definition the ratio of the binding energy of a Cooper pair to the condensation energy contained within the volume of the Cooper pair, VΦ` , then we must have, CL ≡

8 ≤ 1. 3nn VΦ`

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(22)

New Perspectives on Longstanding Issues of the High- Tc Cuprates 0.2

YBa Cu O 2

8

p

n

0

6.0

6.2

6.4

6.6

6+x

6.8

6 4

n

2 7.0

-1

0.05

width

, 30*HWHM [Å ]

q

0.1

-27

p (holes/Cu)

10

6+x

n *10

0.15

3

285

0

Figure 4. Normal state charge carrier density nn number of holes per copper atom p and the width of an off-resonance neutron diffraction peak ∆q vs oxygen content 6 + x. Values of nn were obtained using Eq. (20) as described in the text. Values of p and ∆q were taken from refs. [86] and [90] respectively. √ using the definition of VΦ` = 2 2ξ03 , gives,

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"

√ #1/3 2 2 , ξ0 ≥ 3πnn

(23)

from which we obtain values of ξ0 (x) using the values of nn (x) found above. Both sets of calculated values of ξ0 (x) are shown in Fig. (5), along with experimentally determined values from Ando and Segawa [88], and Grey et al. [89] for comparison, and are found to be in good agreement with the experimental values, with perhaps the exception of the two lowest doping values found using Eq. (23). Due to the steepness of the reported values of Hc (0)2 in the region of 6 + x ≈ 6.5 and some uncertainty in matching the values of Hc (0)2 from Cooper et al. [55] with the values of Tc from Liang et al. [86] as a function of 6 + x, it is possible that the first two data points have a larger error associated with the corresponding error in nn (see Fig. (4)); however, this feature can also be seen to match well with observed values of the width of an off-resonance peak in neutron scattering data [90], which we examine further below. Nevertheless, these results lend credibility to the obtained values of nn (x) shown in Fig. (4), and the subsequent analysis below. Also shown in Fig. (4) are the values of p(x), the number of holes per copper atom in the CuO2 planes, as determined from measurements of the c-axis lattice parameter by Liang et al. [86]. Very different behavior in nn (x) and p(x) is readily discernible. A possible explanation for the apparent discrepancy between the charge p, associated with the superconducting Cu-O planes, and the normal state carrier density, nn , is the formation of charge stripes [61], a point we discuss further below.

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286

B. J. Taylor and M. B. Maple 100

10

6+x

c

T (K)

6

c

60

8

n

4

20

-27

40

40

2

c

0

3

80

60 ξ [Å]

2

n ~(1/T )*(H )*10

80

YBa Cu O

2

6.4

6.6

6.6

6.7 6+x

6+x

6.8

7.0

0

20

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0

6.4

6.5

6.8

6.9

7.0

Figure 5. Calculated (circles) and experimentally determined (squares) values of the superconducting coherence length ξ0 as a function of oxygen content. Filled squares are from Grey et al. [89] and open squares are from Ando et al. [88]. Open circles are calculated using penetration depth data from Zuev et al. [87] and critical field data from Cooper et al. [55], filled circles are calculated using Eq. (23) as explained in the text. Note that the “dip” at x = 6.6 may be better viewed as a “peak” at x = 6.66, wherein the “reduction of Tc ” in the region 6.5 ≤√x ≤ 6.8 is, in part, due to a reduction of the condensation energy Hc2(0)/2µ0 = (Φ0/2 2πλ0ξ0)2 ; i.e., a corresponding increase of ξ0 . Inset: Critical temperature Tc and normal state carrier density n n vs oxygen content.

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287

5 4

0

2

s

n

2

n

s

4 π( µ e / m * ) n λ = (n /n )

6

3 2 1

6.4

6.5

6.6

6.7

6+x

6.8

6.9

7.0

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Figure 6. Ratio of the normal state charge carrier density to the superfluid charge density nn /ns vs oxygen content in YBa 2 Cu3 O6+x as determined by the values of nn calculated here, and penetration depth data from Zuev et al. [87]. Note that n n /ns ≈ 4 at optimal doping as has been observed by Tanner et al. [84].

Next we examine the ratio of the normal state carrier density to the superfluid carrier density nn /ns across the YBa2 Cu3 O6+x system.
Using the definition of λef f (0) given in section 8.3., in Fig. (6) we plot 4π µ0 e2 /m∗s nn λ2ef f = (nn /ns ) vs oxygen content. The results shown here are in good agreement with those of Tanner et al. [84] where they found ns /nn ≈ 1/4 for the optimally doped case. The apparent suppression towards the conventional value nn /ns = 1 from the larger values of nn /ns ≈ 4 at both ends of the under-doped to optimally doped region coincides with the 1/8 doping value. This result, combined with the order of magnitude lower values of nn in the doping range x ≤ 6.9, suggests that, although charge is being introduced via oxygen doping, a portion of the charge is not available for conduction; i.e., some of the (normal state) charge carriers are being localized. Finally, in Fig. (4) we show inelastic neutron scattering peak width data ∆q (HWHM) from χ(q, ω), for YBa2 Cu3 O6+x from Balatsky and Bourges [90]. It is suggested in ref. [90] that the behavior of ∆q in this system, i.e., that the critical temperature is observed to be linearly proportional to ∆q as a function of oxygen content, is possible evidence for a dynamic charge stripe phase. As noted above, values of nn (x) calculated for the two lowest oxygen doping values may be attributable to an increased error associated with accurately determining values of Tc and Hc (0)2 as a function of oxygen content. However, the qualitatively similar behavior in ∆q(x) and nn (x), i.e., the shoulder at x ≈ 0.5, a plateau over the range 0.5 < x < 0.7, and a subsequent increase at higher x values, suggests a correlation between the two. Recently Homes et al. [91] demonstrated that a universal scaling relationship exists

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B. J. Taylor and M. B. Maple

for a large number of high-Tc superconducting materials extending over the underdoped to overdoped regimes, wherein ρs (0) = (120 ± 25)σn(Tc )Tc ,

(24)

holds for both the a-b plane and c-axis conductivities ( ρs (0) is the superfluid density). It was shown in ref. [65], that the empirical observation of Homes et al. [91], combined with the expression for Tc given in Eq. (18) gave the result that for the high- Tc cuprate superconductors examined, the value of the conductivity at Tc is related to the quantum of conductivity by σn (Tc) ≈ (25 CL−1 ) σQ . With CL ≈ 1, this ratio is in agreement with the value σn (Tc ) ≈ 10 σQ estimated by Emery and Kivelson [92] within the context of their theory of superconductivity in materials with a low superfluid carrier density; i.e., “bad metals”. Subsequently Emery, Kivelson, Carlson, Fradkin, and coauthors [60, 61, 93] have developed this theoretical perspective further from which the idea of the formation of dynamic charge stripes as a mechanism of superconductivity has been proposed. While results from neutron scattering studies are considered as strong evidence for the formation of charge stripes [60, 61], the notion that charge stripes are ultimately responsible for establishing superconductivity remains controversial. However, the above results; i.e., ( i) the contrasting behavior of nn and p as a function of oxygen content, (ii) a decrease in the ratio of normal state to superfluid carrier densities to nn /ns ≈ 1 with a minimum at the x = 1/8 doping level, and (iii) the similar dependence of ∆q, which is interpreted as a measure of the width of charge stripes [61, 90], and nn on oxygen content, in conjunction with the observation in ref. [65] can be added to the growing list of experimental results that are consistent with this perspective on the mechanism of superconductivity in the high- Tc cuprate superconductors.

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9.

Concluding remarks

We have presented a selection of works spanning the past two decades comprising many of the landmark experimental and theoretical developments concerning the investigation of the irreversibility (melting) line in high- Tc cuprate superconductors. Given that the topic is vast and complicated, inevitably, certain studies, issues, and points of view of various authors have not been included. We have attempted to maintain a balanced presentation of some of the more contrasting experimental results and theoretical points of view, as well as to point out where we believe that some of these come together. Being outside the scope of this work, we have also not provided an extensive review of various competing theories of superconductivity though we do find, via the above analysis, evidence for behavior consistent with the charge stripe model [61]. We have not examined our results in the context of these various competing theories, so this remains to be pursued further. As mentioned in the introduction, full agreement has not yet been achieved as to the true nature of the melting line in high- Tc superconductors. However we favor a picture, built upon the work of Blatter and Ivlev and Fisher, Fisher, and Huse, where critical fluctuations of vortices, both quantum and thermal, are present in the region along the entire vortex-solid melting line (vanishing as T → 0). The nature of the criticality is that of the superfluid in a critical regime. The energy scales which determine the relative importance of quantum to thermal fluctuations, kB Tc and Hc (0)2, also determine the temperature at which these

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289

materials (and superconductors, in general) become superconducting. Finally, from this picture and independent considerations, we have developed two equivalent expressions for Tc from which it can be seen how certain basic properties of superconductors are balanced against each other. These expressions can be used as potentially powerful tools by which to investigate the general properties of superconductivity in a compound or system as the critical temperature evolves upon variation of various parameters.

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[10] Inui, M.; Littlewood, P. B.; Coppersmith, S. N. Phys. Rev. Lett. 63, 2421 (1989). [11] Gammel, P. L. J. Appl. Phys. 67, 4676 (1990). [12] Brandt, E. H. Phys. Rev. Lett. 63, 1106 (1989). [13] Brandt, E. H. Physica B 169, 91 (1991). [14] Brandt, E. H. J. Supercond. 6, 201 (1993). [15] Schilling, A.; Fisher, R. A.; Phillips, N. E.; Welp, U.; Dasgupta, D.; Kwok, W. K.; Crabtree, G. W. Nature 382, 791 (1996). [16] Junod, A.; Roulin, M.; Genoud, J. -Y.; Revaz, B.; Walker, E.; Erb, A.; Marcenat, C.; Calemczuk, R.; Bouquet, F. Physica C 282, 1425 (1997). [17] Fisher, M. P. A. Phys. Rev. Lett. 62, 1415 (1989). [18] Fisher, D. S.; Fisher, M. P. A.; Huse, D. A. Phys. Rev. B 43, 130 (1991). [19] Huse, D. A.; Fisher, D. S.; Fisher, M. P. A. Nature 358, 553 (1992). [20] Koch, R. H.; Foglietti, V.; Gallagher, W. J.; Koren, G.; Gupta, A.; Fisher, M. P. A. Phys. Rev. Lett. 63, 1511 (1989). Superconducting Cuprates : Properties, Preparation and Applications, Nova Science Publishers, Incorporated, 2008. ProQuest Ebook Central,

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[32] Oussena, M.; de Groot, P. A. J.; Deligiannis, K.; Volkozub, A. V.; Gagnon, R.; Taillefer, L. Phys. Rev. Lett 76, 2559 (1996). [33] Bennemann K. H.; Ketterson J. B. (Eds.), The Physics of Superconductors, Vol I., Conventional and High-Tc Superconductors (Springer-Verlag, Berlin Heidelberg, New York, 2003). [34] Almasan, C. C.; de Andrade, M. C.; Dalichaouch, Y.; Neumeier, J. J.; Seaman, C. L.; Maple, M. B.; Guertin, R. P.; Kuric, M. V.; Garland, J. C. Phys. Rev. Lett. 69, 3812 (1992). [35] de Andrade, M. C.; Almasan, C. C.; Dalichaouch, Y.; Maple, M. B. Physica C 184, 378 (1991). [36] Garland, J. C.; Almasan, C. C.; Maple, M. B. Physica C 181, 381 (1991). [37] Schilling, A.; Jin, R.; Guo, J. D.; Ott, H. R. Phys. Rev. Lett. 71, 1899 (1993). [38] Dalichaouch, Y.; Lee, B. W.;Seaman, C. L. ; Markert, J. T.; Maple, M. B. Phys. Rev. Lett. 64, 599 (1990). [39] Gaifullin, M. B.; Matsuda, Y.; Chikumoto, N.; Shimoyama, J.; Kishio, K. Phys. Rev. Lett. 84, 2945 (2000). Superconducting Cuprates : Properties, Preparation and Applications, Nova Science Publishers, Incorporated, 2008. ProQuest Ebook Central,

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[71] ξ0 of Pb estimated from the value of Hc (0) and λ0 reported in refs. [72] and [73] below. [72] Aliev, A. E.; Lee, S. B.; Zakhidov A. A.; Baughman, R. H. Physica C, 453, 15 (2007). [73] Poole, Jr., C. P.; Farach, H. A.; Creswich, R. J. Superconductivity (Academic Press, New York, 1995). [74] Anderson, J. R.; O’Sullivan, W. J.; Schirber, J. E. Phys. Rev. B 5, 4683 (1972). [75] ξ0 of In estimated from the value of Hc (0) and λ0 reported in refs. [76] and [77] below. [76] Shaw, R. W.; Mapother, D. E.; Hopkins, D. C. Phys. Rev. 120, 88 (1960). [77] Lynton, E. A. Superconductivity (Metheun and Company, Ltd., London, 1962). [78] Fawcett, E. Phys. Rev. Lett. 3, 139 (1959). [79] ξ0 of Aluminum estimated from the value of Hc (0) and λ0 reported in refs. [80] and [81] below. [80] Harris E. P.; Mapother, D. E. Phys. Rev. 165, 522 (1968). [81] Strunk, C.; Bruyndoncx, V.; Moshchalkov, V. V.; Van Haesendonck, C.; Bruynseraede, Y.; Jonckheere, R. Phys. Rev. B 54, R12701 (1996). Superconducting Cuprates : Properties, Preparation and Applications, Nova Science Publishers, Incorporated, 2008. ProQuest Ebook Central,

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[82] Hwang, J.; Timusk, T.; Gu, G. D. J. Phys.: Condens. Matter 19, 125208 (2007). [83] Bruyndoncx, V.; Rodrigo, J. G.; Puig, T.; Van Look, L.; Moshchalkov, V. V.; Jonckheere R. Phys. Rev. B 60, 4285 (1999). [84] Tanner, D. B.; Liu, H. L.; Quijada, M. A.; Zibold, A. M.; Berger, H.; Kelley, R. J.; Onellion, M.; Chou, F. C.; Johnston, D. C.; Rice, J. P.; Ginsberg D. M.; Markert, J. T. Physica B 244, 1 (1998). [85] Loram, J. W.; Mirza, K. A.; Cooper, J. R. ; Liang, W. Y. Phys. Rev. Lett. 71, 1740 (1993). [86] Liang, R.; Bonn, D. A.; Hardy, W. N. Phys. Rev. B 73, 180505(R) (2006). [87] Zuev, Y.; Kim, M. S.; Lemberger, T. R. Phys. Rev. Lett. 95, 137002 (2005). [88] Ando Y.; Segawa, K. Phys. Rev. Lett 88, 167005 (2002). [89] Gray, K. E.; Kim, D. H.; Veal, B. W.; Seidler, G. T.; Rosenbaum, T. F.; Farrell, D. E. Phys. Rev. B 45, 10071 (1992). [90] Balatsky A. V.; Bourges, P. Phys. Rev. Lett. 82, 5337 (1999). [91] Homes, C. C.; Dordevic, S. V.; Strongin, M.; Bonn, D. A.; Liang, R.; Hardy, W. N.; Komiya, S.; Ando, Y.; Yu, G.; Kaneko, N.; Zhao, X.; Greven, M.; Basov, D. N.; Timusk, T. Nature 430, 539 (2004).

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[92] Emery V. J.; Kivelson, S. A. Nature 374, 434 (1995). [93] Schrieffer J. R.; Brooks, J. S. (Eds.), High-Temperature Superconductivity (Springer Science + Business Media, LLC, New York, 2007).

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In: Superconducting Cuprates Editor: Koenraad N. Courtlandt

ISBN: 978-1-60456-919-3 © 2009 Nova Science Publishers, Inc.

Chapter 10

PREDICTING SUPERCONDUCTING TC: THE ROLE OF PLANE ISOLATION AND BOND ORDERING AS EXEMPLIFIED BY YBA2CU3OY AND ‘RECORD’ HG BASED CUPRATES H. Oesterreicher Department of Chemistry, UCSD, La Jolla, CA 92093-0506

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Abstract The isolation of the plane is the most important parameter for the magnitude of Tc as it determines both the magnitude of coupling and of the optimal hole concentration hop. The highest isolation of the planes occurs with cuprates based on Hg. Not surprisingly those are the record holders for Tc and hop. Previously an empirical parameter for the isolation of the plane from the apical O has been given as f=1-1.6sc, with Tc=hpf600 , where hp are the total holes on the planes as pairs and sc is the bond valence in the c-direction. Here we further define the origin of the apical factor f and discuss the general high Tc-phenomenology connecting empirical Tc -Rules and aspects of a bond order model with an electronic crystal on the planes. On BO principles we expect ‘characteristic features’ in the doping curves [Tc vs. holes, h] such as sharp Onsets, Kinks, and regions with near Linear sections up to sharp Optima in Tc or Tc -plateaus occurring at characteristic hole concentrations as observed on non-compromised system such as YBa2Cu3Oy. For the ideal Linear section in the doping curves, where all holes are considered to be converted to pairs, an empirical Tc prediction holds generally when reliable data for h, such as from Knight shift, are employed. Accordingly Tc=[hp-hc]600 where hc represents an ineffective hole concentration in the cdirection. It is related to hybridization between Cu, plane and apical O, as outlined earlier. Here we identify hc=sc-scp by analyzing structural literature on YBa2CuCu2Oy as the difference in bond valence in the c-direction between superconductor [sc] and un-doped parent at y=6 [scp]. This difference is physically meaningful as it corresponds to ‘extra’ charge drawn out of the planes proper beyond the equilibrium charge distribution of the non-superconducting parent. Where scp or sc[h] are not directly accessible, they can be determined from empirical Tc rules such as sc=scp/[1-1.6h] or hc=h1.6sc, indicating low Tc for high sc. As examples, materials with low [Hg analogs] and high sc [Bi analogs] are successfully dealt with and the formalism is extendible to others such as oxypnictides. We also elaborate on the concept of special charge-lattice lock-ins for ‘characteristic features’ in the doping curves. Accordingly a

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H. Oesterreicher fundamentally significant, comprehensive and useful scheme is derived for modeling high Tc phenomenology for planar cuprates based purely on structural data. These empirical rules provide an encompassing phenomenological frame for further theory that was so far missing.

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1. Introduction Charge ordering effects are ubiquitous in transition metal oxides [1-3]. Not surprisingly, mounting experimental [4-7] evidence exists for the operation of such effects also in high Tc superconductivity. Microscopic experiment [STM, ARPES] [4-7] further emphasizes charge ordering. Such a view was previously explored in terms of the t-j [band-exchange] model or the direct involvement of doped charge in stripes [8-10] of single holes. It was also anticipated within an alternative phenomenological bond order [BO] model [11-18]. In the latter, local super-exchange pairs form electronic crystal arrangements in superconducting pair plaids [11, 14-18], that can be created out of non-superconducting stripes [19] by rearrangement. They represent unique selected charge–lattice lock-in patterns, explaining trends to characteristic optimal hole numbers. Pairs can coexist with or outperform stripes due to a successful competition of super-exchange with repulsion when the dimension becomes comparable of the pair to an intervening formal insulator. Different regions in the doping curves [Tc-holes] mark events in the pair-single competition such as optima or kinks. Here we explore implications of local models on the example of the BO variety. We explore implications of local models on the example of the BO variety. We show that a quantitative phenomenological algorithm for doping curves [Tc vs. holes] can be based on the periods of electronic crystals and degree of isolation of the planes. In this respect we will introduce a physically meaningful connection with the empirical rule in the difference of bond valence in the c –direction between the undoped parent and the optimal superconductor. In fact the isolation of the planes from the apical O influences also the magnitude of optimal holes or the magnitude of the pseudogap or the Neel temperature of the undoped parent phase. Contrary to earlier assumptions, experiment indicates that superconductivity is accordingly predominantly structurally dictated. It is here further discussed how the many new concepts inherent in this different perception promise to form the basis of an encompassing theoretical understanding for ‘all’ high Tc superconductivity.

1.1

Charge ordering:

The microscopic basis for high Tc superconductivity in the bond order model [BOM][810 for theoretical models, 11-18 for semi-empirical ones] is related to a uniform system of local pairs in homo-conjugate position [also referred to as trijugate or 3J at distance ~3a0/2] that is correlated into orthogonal plaid structures. The organizing principles are based mainly on crystal chemical arguments of minimizing stress or repulsion [e.g. through alternating pair orientation]. Pair structures are in competition with arrangements of singles [stripes] that are advantageous at low doping and can coexist with pairs in certain regions of the doping curve. In this real space [rather than momentum space] picture, doped charge condenses into a bond order of strands of local pairs that can be seen as assembled of stripes of singles. A local pair accordingly represents 2 uncompensated radical half bonds that are formally derived from drawing apart a ‘compensated bond’, composed of spin up and down. The

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pairing energy of this ‘split’ bond is transmitted by a super-exchange type interaction within the pair. Experiment gives now pictorial representations for this situation in the related Pseudo-gap State. Characteristic optimal hole-values of hop~0.16, 0.22 or 0.25 are observed in Knight shift [20]. These characteristic optimal hole concentrations are here identified as representing unique selected charge–lattice lock-in patterns. Questions are discussed concerning the actual location of the doped charge in the resulting electronic crystals. In the search for a common denominator for other high Tc materials [e.g. cuprates or C60] the concept of ‘local’ super-exchange pairs can be extended to a large variety of materials. Selection of one of the characteristic hole values depends on the magnitude of exchange. The latter is given by the degree of isolation of the plane, that is, its interaction with the third dimension due to electronic hybridization in the c-direction [21]. Periods of pair plaid, indicating magnitude of elastic distortions, represent a central aspect for an algorithm of Tc or doping curve predictions on the Plane Isolation Model. Pair plaids can be in competition or coexist with stripes of nonsuperconducting singles at low doping, depicting a phase stability aspect of superconductivity. They can also form templates for more complex overfilled structures at higher doping [over-doping]. Conditions for the disordering of pair plaids, leading to conventional metallicity, can have to do with charge placement, more or less exclusively, in the third dimension.

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1.2. Doping curve predictions If charge ordering is indeed a central issue, then one can expect the phenomenology to depict characteristic events in this competition and it can be shown that indications to this effect abound. Charge lattice commensurability explains the extra stability of characteristic BO that occur at characteristic hole concentrations in the doping curves such as Onsets, Kinks or Optima. Examples are the doping curves of YBa2Cu3Oy that either show a linear section in the doping curves with a characteristic Kink and ‘sharp’ Optimum, or Tc plateaus indicating special BO stability. This choice depends on preparation. As charge-lattice lock-ins favor selected commensurable patterns for optimal hole concentrations, a ‘harmonic’ Tc level structure can be expected. It is not widely recognized that this is indeed the case when one takes into consideration reliable hole counts such as from Knight shift. In addition one has to allow for the concept that the degree of isolation of the plane influences Tc, as it is plausible that the c-axis coordination individualizes the rather uniform plane. For the relevant doping curve phenomenology, a quantitative algebraic representation utilizes exchange and elasticity [15]. The former is related to the strength of coupling between the uncompensated moments of the radical bonds. This parameter is reflected and made visible by TN of the coupled Cu moments of the undoped parent compound. The second parameter corresponds to the bond stiffness or elasticity. The degree of isolation of the planes as indicated by the apical factor [fa] accordingly dictates the range where hop are to be expected, while charge –lattice lock-in conditions fine-tune these values. Linear or parabolic sections in the doping curves obtain, depending on the magnitude of exchange. This model can serve for a large number of materials, e.g. systems with or without magnetic moments [see cuprates vs. C60]. It also can represent hole or electron doping.

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1.3 Empirical rules: The period of the electronic crystal is here related to hole-concentration and to an empirical algorithm for doping curves. This period Pab, where a and b are multiples of lattice parameter [e.g.P4x4 for 4a0x4b0 also depicted as h=0.25=2/42], gives the number of pairs np=1/Pab. The latter also represents the elastic energy [for the linear ‘source’ region, where all holes are transformed into pairs 2np=h]. In the Plane Isolation Model Tc=2fa600np=2fa600/Pab Here, fa represents the diminished correlation of the plane configurations, when the apical system interacts with the plane. It depends on the bond valence [18] in the c-direction. However, to first approximation fa =1, 2/3, ½ for apical layer coordination of 0,1 and 2 [CuO2, CuO3, CuO4] respectively. By depicting Pab, experiment therefore connects with quantitative phenomenological rules that support a charge order view for high Tc superconductivity. These rules deal with characteristic feature of doping curves with chargelattice lock-ins, Tc dependence on period of charge modulations etc. The trends for fa and Pab to characteristic values organize cuprates into ’musical’ Tc families. Empirical rules allow for compound specific theoretical Tc and doping curve predictions purely on structural parameters. The phenomenological rules discussed previously are rich in indications for the operation of BO effects in high Tc superconductivity. As the BO model corresponds to an alternative self-contained way of looking at high Tc superconductivity as a charge order effect, it should have a large number of further ramifications. Currently, there are a number of clues that BO effects play a central role in high Tc superconductivity of cuprates:

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• • • • •

BO filling mechanisms are often quantitatively connected with characteristic features [onsets, kinks, ‘sharp’ optima, off conditions] in selected doping curves. Prescriptions for BO constructions as orthogonal 3J strands of the Pair Plaid Model are born out in experiment [STM, ARPES] for the related Pseudo-gap State. Selected charge-lattice lock-in patterns are identifiable for characteristic oftenobserved optimal hole-values of 0.16, 0.20, 0.22 or 0.25 The elastic energy 1/Pab of a pair BO is empirically related to the magnitude of Tc [modified by interaction in the third dimension [fa] in the Plane Isolation Model] BO of stripes of nonsuperconducting singles can coexist with superconducting pairs and are therefore assumed to be able to generate a corresponding pair BO by ‘dimerization’.

2. Results The BO concepts are used in the following to explain selected aspects of high Tc superconductivity in cuprates. A sampling of Tc on the fa approximation is given in Table 1 in order to compare it with calculations of this study.

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Table 1 Charge order periodicity and resulting characteristic hole numbers for the main types of bond patterns. They are the basis of a ‘musical’ Tc level scheme. Tcmax would be obtainable if fa=1. hopK are Knight shift data [bracketed are calculated]. OP and IP stand for outer and inner plane respectively. o stands for observed [3], c for calculated on Tc= fa Tcmax. For HgBa2CaCu3O8+x OP, fa =0.88>2/3 with Tc=132cK is due to its unusually large distance to the apical O. Materials with anomalous low c/a can be nonsuperconducting such as PrBa2Cu3O7. Pab with a=b are considered to represent one BO family and are assumed to be preferred by tetragonal symmetry depicted Tet [orthorhombics are depicted by Ort]. The undoped YBa2Cu3O6 and La2CuO4 have TN~350 and 150K respectively

h=n/Pab 0.444=4/3x3 0.333=4/3x4 0.250=4/4x4 0.222=4/3x6 0.222=2/3x3 0.200=4/4x5 0.167=2/3x4 0.160=4/5x5 0.125=2/4x4 0.125=1/2x4

Tcmax[K]=600h Examples 267 200 150 Bi2Sr2CaCu2O8.25 ‘Tet’ HgBa2Ca2Cu3O8+x [OP] Tet 133 YBa2Cu3O6.95 Ort 133 HgBa2Ca2Cu3O8+x [IP] Tet 120 YBa2Cu4O8 Ort 100 YBa2Cu3O6.7 Ort 96 La1.84Sr0.16CuO4 75 GdBa2RuCu2O8 Tet ‘Tranquada stripes’

hopK

fa

Tc[K]

0.25 0.25 0.22 0.21 0.19 0.16 [0.16] [0.125]

2/3 0.88 2/3 1 2/3 2/3 ½ 2/3

100c, 92o 132c, 133o 89c, 95o 133c, 133o 80c, 80o 66c, 64o 48c, 38o 50c, 48o

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2.1. Superconductors display a ‘musical-type’ Tc level structure because of their electronic crystal nature: It is plausible that the electronic crystal structure on the planes, deduced from phenomenology and corroborated by experiment, will lead to trends for a harmonic Tc level structure for optimal values. This Tc level structure is related to stepwise variations in the period of charge ordering. The latter is due to charge–lattice commensurability. A select number of charge–lattice lock-in patterns is identified as the origin of trends to characteristic numbers in the observed optimal hole concentrations hop. As an example charge periods of 5a0x5b0 leads to a situation with 16 out of 100 Cu atoms formally assigned a hole, or hop=0.16. Other examples are shown in Table 2. ARPES experiments show fragments of several of these predicted charge orders. A second major influence pertains to the selected modes of coordination of the planes. Since this is also taking place in selected types, albeit with some variation in strength, trends to a level structure can be expected. These points shall be further detailed. We present first an orbital interpretation of BO as visualized in the experimental ARPES charge blots. Generally, the doped bond corresponds to an abnormality in a uniform sea of bonds. For the undoped insulator, the presence of antibonding Cu-O bonds has to be assumed as hole doping decreases the lattice constant. Hole and electron doping can then be understood in the following way.

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Table 2. Selected high Tc-families analyzed on ‘parent calibration’ model. Calculated and observed parameters are given [Tcs=he600] for examples of various degrees of plane isolation. Superscripts o correspond to observed, cp to calculated on procedure-p [he =hhc with hc=sc-scp=sc1.6h] and a to assumed. h values are from Knight shift [K], except for other super-scripts. Where values are close to optimal, the assumed ideal optimal charge-lattice lock-ins are given in brackets. One notices the pronounced decrease in the detrimental hc with increasing dc. In this respect Hg and Bi analogs represent high and low plane isolation [dc] families respectively with the former having the higher optimal Tc. For double O coordination hc=2sc1.6h. For La2-xSrxCuO4 we start with dcp=0.240nm and calculate dc=0.228nm comparable to literature [21]

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n=2 HgBa2CaCu2O6+z HgBa2CaCu2O6 -

h=2np

Tcs [K]=he600 he=h-hc hc=sc1.6h

sc

0.22 c [0.222]

125 o, 123ca

dc [nm]

0.204

0.0178

0.0501 0.2787 0.0323

YBa2CuCu2O6.95 YBa2CuCu2O6.64 YBa2CuCu2O6.0

0.22 K [0.222] 95o, 94 cp 0.16 K [0.167] ~64o, 69 cp -

0.156 0.115

0.066 0.045

0.186 0.174 0.120

0.2301 0.2341 0.2471

Bi2Sr2CaCu2O8.25 Bi2Sr2CaCu2O8 n=1 La1.84Sr0.16CuO4 La2CuO4

0.25K [0.250]

92o, 85 cp

0.141

0.108

0.271 0.163

0.216

0.16 [0.160] -

38o, 37 cp

0.0622 0.0489x2

0.142

0.191 0.240

Hole doping corresponds to the removal of antibonding electrons, resulting in the extension of antibonding orbitals from adjacent bonds into the region of the doped bond. It is assumed that this represents the charge density blots seen in microscopy. We assume that the antibonding orbitals involve components of Cu orbitals such as x2-y2. This charge density emanates from a node between Cu and the super-exchange O and extends its lobe predominantly ‘into’ the Cu region. These lobes contain one uncompensated spin and can fluctuate also into the perpendicular directions. Another charge is displayed in the 3J positions. These 2 charges represent the compensated pair in Cu2O7 units with distance between the 2 centers of charge of d[P]>a0~1.3-1.5 a0. Charge satellites can occur in adjacent CuO4 diamonds. Correlation effects will create a charge kernel Cu4O12, as a recognizable unit and force charge somewhat into the diagonal of the kernel. Kernels arrange into plaid or tile patterns, fragments of which are observed in microscopy. Electron doping can be understood in a similar way. The addition of an antibonding electron to a Cu-O bond results in the extension of its antibonding orbitals into the region of adjacent bonds. Charge accordingly extends ‘out’ of the doped bond. It again envelopes the Cu nucleus but d[P] may now actually be somewhat smaller than in the corresponding situation with hole doping. This difference is predicted to be observable in microscopy.

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2.2. Correlation effects lead to plaid [tile] structures:

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In Fig 1 we depict isotropic and anisotropic plaid series Pab. A systematic alteration of the pair direction is shown, which equalizes lattice pressure. This orthogonal orientation of pairs leads to plaid patterns with a set of primary and secondary channels of charge. The former collect 2 charges from the parallel pairs together with one charge of the perpendicular pair. The primary channels are the reference for the lesser- charged secondary channels, which contain only the perpendicular end of the pair. Their distance reflects the pair kernel at 1.3-1.5a0.

Figure 1. Schematic of assumed superconducting electronic crystal with regions of hole doped Pab a] h=0.222=2/32 and b] h=0.167=2/3x4. Circles represent O, while line crossings stand for Cu. Shaded ovals represent correlated hole density of pairs. Alternate pair orientation creates primary and secondary charge channels.

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An example for the anisotropic series is YBa2Cu3Oy~6.6, which due to its anisotropic blocking layer is orthorhombic over some y. Preparations involving annealing result in 2 Tc plateaus, indicating special stability for characteristic Pab. The Tc~60K plateau region has h=0.16 as per Knight shift. This is identified as h=0.167=2/3x4. For the ideal BO along the a– axis this indicates a charge density of 3/6 and 1/6 for primary and secondary channels respectively. For the b-axis this is 3/8 and 1/8 respectively, indicating the required anisotropy. For y=~7, h=0.22, as given by Knight shift. This is identified as derived from h=0.222=2/32. This is an isotropic Pab. Partial lifting of the orthogonal pair orientation could induce different charge density along a and b. Different degrees of anisotropy can quite generally be achieved by complex superstructures concerning pair orientation. In any case the period 3a0 appears maintained for the respective BO in both plateau regions as shown in the figure. This facilitates transitions between these Pab, depicting a BO family.

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2.3. Proposals for BO transformations and superconducting charge propagation We can now investigate the mode of transformation of one lock-in charge order into another. We investigate the transition from h=0.167=2/3x4 to h=0.222=2/3x3. Pair density is increased in a manner that minimizes elastic or electrostatic energy. This can occur through the gradual insertion of the new BO motifs. Within one channel along the b-axis [4b0 period] a local region with period 3b0 is inserted. It takes 4 such 3b0 motifs to equal 3 of the original 4b0 periods. We term this the adjustment strip. These strips will be placed in an elastically advantageous pattern that will become increasingly denser on doping. Conversely, on diluting pairs from h=0.222=2/3x3, adjustment strips of 4b0 periods are placed into the 3b0 matrix. In the intermediary situation mixes of the full tile patterns of the respective patterns are expected to result from detailed energy calculations. The question of BO transformations is related to the mechanism of superconducting charge propagation as well as an observed asymmetry in super-fluid density. The asymmetry of Pab is here related to the observed asymmetry in super-fluid density. For YBa2Cu3O6.7 the super-fluid density was observed along the b-axis to be 1.7x larger compared to the a-axis [22]. Taking h=0.16 from Knight shift and h=0.167=2/3x4 with Pa=3 and Pb=4 we get a ratio of 4/3=1.33, in qualitative agreement with experiment. This procedure can serve as a rough guide for determining ratios of super-fluid density in anisotropic materials. This is in line with the assertion that large charged objects propagate, as was proposed [23]. This makes high Tc superconductivity relatively immune [24] to defect density. More fundamental treatments of elastic textures [25] and their propagation appear relevant. BOM provides plausible mechanism of charge transport in block movements within BO. The principle of charge propagation can involve local BO disproportion or shifts. An example is given in Fig 3 for h=0.222=2/32. It is shown how a combination of shifts between primary and secondary charge channels and further dislocations can lead to charge propagation of one block vs. another. A shift of part of the BO by e.g. one tile distance can be accomplished involving a region of partly disordered tiles. One can also see the generating principles of this movement as local shifts of basic units such as kernels or super-exchange O at a minimum of local stress. The proposed charge propagation corresponds to shifts of blocks within the total BO. Alternative proposals involve propagation confined along selected charge channels. This

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is a mechanism that is less stable towards local defects. As there exists little effect of local disorder on charge propagation, the former proposal is the more likely one.

2.4. Relations between empirical and semi-empirical Tcs algorithms: It has been empirically shown [14-18] that Tcs = [ h -hc]Te with Te = 600K, as a best-fit type parameter for cuprates and he=h-hc representing an effective hole number [within the Tcs concept it is understood that h=hp]. This has been approximated by Tcs = h f 600, with f=11.6sc and sc the bond valence in the c-direction. sc has been routinely obtained near the Tc optimum. This procedure defines the apical factor f as an empirical parameter with a more specific accounting for structural detail compared with the related more generally defined fa. We will here identify hc as corresponding to the difference between bond valence sc and the one of the undoped parent scp according to

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Tcs=[h-sc+scp]600. This physically meaningful expression ties in with the successful empirical rules and provides an encompassing Tc modeling for cuprates, based on structure. Within the plane isolation or effective hole [he] model we search for an empirical relationship of hc and sc-scp where sc=exp[(d0-dc)/0.37], with d0=1.679A and dc the distance to the apical O. scp refers to the respective value for the undoped insulator. In Fig 2a we analyze literature data [26-30] of YBa2Cu3Oy=6+z. Data for dc [26], h [e.g. Knight shift, hK] and Tc are not from identical preparations but the consistency amongst the results bears out this procedure. We select hole-concentrations that are close to the ones corresponding to ideal charge-lattice lock-ins so that we do not encounter deviations from Tcs as found in the plateaus. For hop=hK=0.22=2/32 we use dc=0.2301nm and Tc=95K at z~0.90 [27]. From this we obtain z=h0.90/0.22 for extrapolation. For the ‘ideal’ center of the Tc~60K plateau at an assumed h=0.167=2/3x4 [z~0.68, hK=0.16] we use dc=0.2347nm [z~0.64] and Tc=64K [28]. For h~0.080=2/52 expected at z=0.33 [corresponding to the onset of Tcs with the location of a Kink in the doping curve] we use Tc=31K [29] and extrapolate dc=0.241nm. For the undoped YBa2Cu3O6, we calculate scp=0.118 from dcp=0.2471nm. sc as a function of holes then displays a near linear relationship [omitting Tc plateau and Tc rise region]. In the search for an empirical relation of hc and bond valence one can define hc=c[sc-scp] with c a constant. It appears to be physically significant that we can reproduce the observed Tcs values for YBa2Cu3Oy with c~1.0. This indicates that hc is identified as sc-scp, which is physically meaningful as it corresponds to ‘extra’ charge drawn out of the planes proper beyond the equilibrium charge distribution of the insulator. In this insulator calibration, hc therefore corresponds to an inert hole part, that is not involved in establishing the potentials that lead to pair coupling in the planes.

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Figure 2a. Semi-empirical relation sc=scp/1-1.6h as dashed line obtained from hop. An approximating straight line is also given. Focus is on hc=sc-scp in YBa2Cu3Oy=6+z. Materials near special chargelattice lock-ins are selected for the bond valence sc vs. holes representation. Filled circles correspond to ‘calculated from experiment’ open circles are extrapolated. The parameter hc=sc-scp=0.066 is indicated for Tc optimum. For the parent [at y=6] one notes satisfying agreement of calculated parent scp represented as diamonds with experiment. An example for related materials with higher dc [e.g. Hg analogs] is included. One notices the decrease of detrimental hc with higher dc. This increases Tc for a given h, as indicated in Fig 2b. Figure 2b. Calculation of Tcs=600he with he=h-hc and hc=sc-scp on parent calibration model. Also presented is Tcmax, the theoretical value without c-axis interaction. From this value the respective hc equivalents have to be subtracted. Examples are chosen for high and relatively low dc. Faint vertical lines give positions of theoretical charge-lattice lock-ins. Open circles indicate calculated optimal Tc. From optimal Tc, Tcs are extrapolated as solid lines. Crosses and filled circles give selected experimental points. For YBa2Cu3Oy a Kink in the doping curves is given at ‘ideal’ h=0.080=2/5x5, close to observation. A universal Tc onset near h=0.050 is assumed [corresponding to a stripe like feature of single holes at a theoretical h=0.050=1/2x10]. The unusually large dc for Hg analogs is the origin of record Tc.

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However, scp are often not known. One can indirectly obtain scp from superconducting data by working back from Tcs or he [scp=sc+he-h]. One can more directly obtain it from the relationships of the empirical rules as scp=sc-sc1.6h where sc1.6h =hc. It allows also an empirical relation of sc and h according to sc=scp/[1-1.6h]. This indicates that sc[h] is nearly but not exactly linear and shows curvature for higher h. using this relationship one can calculate scp=0.120 from sc=0.186 at h=0.22. This shows consistency with scp=0.118, obtained directly from dc. The most expedient approach then utilizes sc determined from dc near optimum and calculate the full curve from that value. This is here done routinely. One can now directly employ the concept of a reduction of Tc through hc. This reduction for YBa2Cu3Oy is Tc=hc600=40K for a calculated hc=0.66 at hop=0.22 so that Tcop=133-40=93K. When we expand this procedure to other representative high Tc compound families [TAB 1] we find similar satisfactory agreement with experiment. This is done in Fig 2b per example of the difference to the high dc analog HgBa2CaCu2Oy using literature data [31-32]. Satisfactory agreement with experiment obtains under the assumption of hop=0.22. This represents also the observed ideal charge-lattice lock-in for YBa2Cu3Oy~6.9 and is close to the value for the higher member of the series as per Knight shift. The record optimal Tc of this family is understandable as a result of their unusually high dc. Other materials were satisfactorily analyzed on the empirical scheme elsewhere [14-18] and no obvious exceptions appear to obtain. For La1.84Sr0.16CuO4 calculated Tc [taking 2sc for double coordination] are also within the trend. We assume that strongly c-axis contracted materials such as PrBa2Cu3Oy~7 are non-superconductors due to unusually low dc. It should finally be mentioned that the formalism can also be successfully expanded to predict Tc of oxypnictides under the reasonable assumption of fa=1/2 [33].

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2.5. Summary of success of BOM We summarize that BOM • • • • • • •

Correlates details of doping curves [e.g. ‘sharp’ Optima or Kinks] with BO filling events [see YBa2Cu3Oy and contrast to parabolic model]. Organizes Tc families with respect to mode of O coordination [in contrast to secondary nature of structural sensitivity of conventional theory] Correlates characteristic BO with often-observed hop and fragments of patterns in the electronic crystal of ARPES graphs and relates their period with Tc. Gives detailed empirical dependence of degree of plane isolation and TcS , hop or TN [e.g. ExEl model in contrast to the lack of guidance from other theory]. Explains absence of Tc at low apical distance [e.g. YBa2Cu3O6.7, PrBa2Cu3Oy] Predicts independent properties for differently coordinated planes such as found in Knight shift [in contrast to inter-planar tunneling model]. Gives prescriptions for BO constructions on correlated pair Kernels of 3J bonds with distance between primary and secondary channels of 1.3-1.5a0 as corroborated by ARPES.

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H. Oesterreicher •

• •

Supplies explanation for BO transitions between singles and pairs at constant doping [e.g. La1.875Sr0.125CuO4 type] and isotropic [Bi2Sr2CaCu2O8.25] or an-isotropic [YBa2Cu3Oy] BO filling families. Explains under- and over-doping as single vs. pair equilibriums [in contrast to tj model] corresponding to specific regions in the doping curves. Provides plausible mechanism of charge transport in block movements within BO, which is largely insensitive to defects.

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3. Conclusion It is plausible that microscopic experiments for the Pseudo-gap State reflect essential aspects of high Tc superconductivity, strengthening the relevance of local models. The predicted orthogonal plaid patterns of strands of super-exchange pairs of the BO model are corroborated in local regions of experiment. They can be identified as corresponding to the limited number of postulated charge-lattice lock-ins for optimal doping, giving a rational understanding for the corresponding characteristic hole numbers. Tc are directly related to the period of these BO. Also the presence of primary and secondary charge channels is a natural consequence of the crystallographic stress relief principles of orthogonal pair placement in the BO model. Their period of ~1.3-1.5 a0 is roughly in line with experiment. The progressive switching of primary and secondary charge channels can provide a pictorial mechanism of superconducting charge transport. In addition a tie exists with experiment concerning quantitative doping curve prediction on the plane isolation model. Here it is plausible that the blocking layer individualizes the rather uniform plane, influencing ‘all’ important parameters for the phenomenological description. Not surprisingly, but contrary to earlier assumptions, experiment in ‘all documented’ cases indicate the paramount importance of structure for superconductivity. The highest degree of isolation of the planes accordingly leads to the highest Tc and hop as in the Hg analogs. At the other extreme anomalous c-axis contraction can lead to absence of superconductivity due to strong hybridization with the c-axis [e.g. YBa2Cu3O6.7 or PrBa2Cu3O7 in special preparations, creating special c-axis contractions]. A high degree of isolation of the CuO2 planes is also the major factor in increasing the magnitude of pseudogaps or the Neel temperature of the boundary phase. Where reliable values for hop are available [e.g. by Knight shift], no exceptions to the Plane Isolation model have been found to date in predicting Tcop. In addition, plane distortions through the blocking layer can result in preference for anisotropic characteristic Pab. Different BO filling schemes will be expected for iso- and anisotropic planes. Accordingly one understands that systematic changes in symmetry on preparation can lead to the well-known degradation of Tc, such as observed in La1.875Sr0.125CuO4. In BOM this is explained as due to gradual transformation of pairs into singles. The concept of a parent calibration supplies a deeper meaning to the empirical Tc rules. It further supports the notion that the degree of the isolation of the CuO2 planes represents the major factor in determining aspects of the doping curves. This dependence can be related to other exchange-related parameters such as the optimal number of doped holes [hop] and the corresponding pseudo-gap.

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The effective hole model is related to band calculations. It simplifies and generalizes the problem of accounting for an ineffective hole part in the c-direction. The present model offers a practical solution of employing bond valences and allowing the construction of complete doping curves for ‘all’ materials. This construction can be accomplished directly where structural detail is known. Conversely, structural detail can be obtained from Tc and h. The effective hole model also depicts Tc family relationships and contains a direct relation to a microscopic picture for the BO and allows for a pictorial understanding of the mechanism of charge transport. If charge ordering is important, then one expects that a phenomenology obtain that deeply reflects this. Indeed general trends exist to the canonical predicted hop=n/P2, with P running as integers from 3 to 5 and n=2 for simple cells. Also Tc plateaus or onsets of Tcs are connected with special BO. For an evolving BO with h, continuous adaptations between the BO are expected. An example is the transition between h=0.16=2/3x4 and h=0.22=2/32 [e.g. Tc plateaus in YBa2Cu3Oy]. For continuous adaptations one predicts a reciprocal relationship of electrostatic or elastic energy with averaged BO period, which corresponds to a linear relation of this energy or Tcs with hole concentration. This is modified in terms of a secondary relationship with dc. Accordingly an encompassing set of relations between superconductivity and structural parameters exists. However, this view asks for a rethinking of some conventional assumptions. One of these is the universality of featureless parabolic doping curves with optima near hop=0.16. Neither of these features is observed in non-compromised systems such as YBa2Cu3Oy [hop=0.22 as per Knight shift]. Here one finds typical BO features such as ‘Linear Regions’ leading to ‘sharp’ Optima or Kinks and Plateaus. Other conventional assumptions include the operation of stripes or band abnormality. However, these models are not in line with ARPES evidence for the presence of some type of BO and have not produced predictive schemes. One concludes that structural and BO effects permeate the phenomenology of high Tc materials and make doping curves predictable purely on structural terms. This is rather different from conventional superconductors. Generally, electronic crystal behavior with characteristic periods [P] and the limited number of coordination in the structure [fa] work in concert to produce ‘musical’ superconductors, that is a level-structure in hop and Tcop. This leads to special features in the doping curves, which have their roots in these harmonious relations. For the first time superconducting properties for any class of materials can be predicted quantitatively and purely on structural data. Local models such as BOM answer to an impressive array on the wish list for high Tc superconductivity. This array ranges from quantitative doping curve predictions to pictorial representation of superconductivity. It includes essential requisites for its generation, such as planar multi-center ring structures [8 in cuprate planes] of covalent bonds with high exchange due to high isolation from adjacent layers. BOM as based on elasticity and exchange is qualitatively distinct in assumptions and conclusions, e.g. from the somewhat related t-j model. As an example, a version of the latter considers the initial Tc Rise region to contain superconducting stripes that become 2-dimensional at higher Tc. By comparison the BO model considers only pair strands to superconduct. It contains direct proportionality of np and Tc and straightforward mechanisms for coexistence of single holes and pairs and their mutual transformations. However, a convergence and synergy amongst these approaches appears possible.

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BOM is built on surprisingly straightforward principles. Tc depends primarily on the number of pairs [np]. This number can directly reflect the number of holes as in the Source region. In adjacent regions, separated by Kinks, equilibria exist between single and paired holes [Rise and Drain range]. Phenomenology indicates that this competition reflects the magnitude of Ex. In principle future calculations based on electrostatic energies can be expected to yield the hole-values for non-linearity [Kinks] directly. However, the ExEl model gives a useful prescription for these limits of the Source slope based on the approximation that the pair-only region is terminated where Ex~-El. A connection of np to the electronic crystal nature involves the corresponding charge periods with a modification concerning their effectiveness [fa] predictable from structure. Characteristic features within the doping curves are therefore primarily related to the different modes of BO fillings. The essential aspects of BO construction follow rules concerning minimizing elastic distortions [orthogonal pair placement] in the presence of super-exchange. Many body quantum effects operate by utilizing classical crystal or bonding principles such as local distortions and super-exchange. Pairs can be pictorially represented. Details of doping curve follow directly from filling or overfilling [single insertion] of charge-lattice lock-in patterns. Within BOM, only exchange and elasticity related parameters for a given plane and its environment enter the phenomenological formalism. The BO model therefore stresses the relative independence of planes in setting independent Tc [see experiments in ref 31] given by different plane coordination. In another approach [34], quantum tunneling between the planes has been invoked to explain the high Tc of multi-layer HgBa2Can-1CunO2n+2. Both approaches similarly argue that the limit of Tc is dependent on imbalances in the doping between the planes in question. The BO model sees the decrease of Tc after a maximum around n=3 as a result of a reduction of hop on approach of the coordination corresponding to the infinite layer type. It considers that 2 different BO have to correspond to the different coexisting hop. As an example for n=3 Cu layers it is h=4/4x5=0.20 for the inner plane and h=4/42 =0.25 for the outer planes. The corresponding fa nearly equalize the corresponding Tc. This independent filling is born out in experiment, as are the independent Tc. Also the infinite layer type does generally not exhibit record Tc, weakening an argument for its importance. There exists therefore no need for more complex concepts beyond independent BO filling of the planes in question. Optimization of Tc therefore rests primarily with the question of how to further increase Ex for the CuO2 or related planes. Where do we go from here? The influence of structural detail on parameters such as exchange and pseudo-gap should be related with computations of electrostatic or elastic energy for specific compounds. This approach is taking a further step toward a new era in establishing an encompassing crystal chemistry of high Tc superconductivity or more specifically the energetic of hole placement as a phase stability question. The computational tools and theoretical framework need to be developed for understanding how multiple influences work together to produce overall charge ordering and transport. Quantum chemical definition of the proposed 5-center bond for the pair needs to be elaborated and explored with respect to the former question. A further challenge is to identify the consequences of the success of phenomenological rules for theory. In this respect future research must address the similarities and relevance of various approaches in the light of this encompassing empirical understanding.

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Further investigation within the BO theory demands more detailed normative data and their comparisons, especially in the areas of structural detail and structure-electronic relations. Does variation in planar network architecture and metrics itself lead to discernible influences on the doping curve, and if so, in what ways? Does it lead to new Tc families? What can BO filling reveal about the development in electronic behavior such as resistivity or Hall effect? What is the nature of the assumed ‘devil staircase derivatives’ for regions between the postulated charge-lattice lock-ins. Do materials with high hole numbers display the predicted dense BO in experiment [e.g. 0.25=4/42] and in what detailed ways? Can conditions exist for even higher ones [e.g. 0.44=4/32]. The correlation between high Tc superconductivity and exaggerated structural characteristics can be explored by detailed study of regions that are known to be close to phase instability. Novel preparative methods hold continuing interest in this respect. They can also be studied theoretically by extrapolating from the empirical rules. Do BO aspects apply to planar cuprates, because only in this restricted situation can ordered charge patterning occur? Or might it apply more broadly even to high symmetry 3 dimensional systems such as doped diamond, creating a 3-D analog Plaid Pattern? Lastly, even if the BO theory can explain some core characteristics of high Tc superconductivity, it will be important to establish which other characteristics, if any, may require different or expanded explanations and in what way transitions occur to the more classical types of superconductivity. In summary, a comprehensive understanding is at hand of high Tc superconductivity manifesting itself as a charge order phenomenon. It is plausible that the electronic crystal structure, deduced from phenomenology and corroborated by experiment, is related to the observed harmonic Tc level structure. This Tc level structure reflects the period of charge ordering and the degree of isolation of the planes. Fragments of patterns in ARPES are identified as representing the enigmatic characteristic hop as observed in Knight shift and predicted on first principles. These concepts provide a realistic empirical framework for theory. Evidence reviewed above suggests that electronic crystal aspects reflect themselves in a broad range of phenomenology and the microscopic origin of high Tc superconductivity. Now that the assumptions of local pairs are further born out in experiment, the challenge ahead will be to test local theory in its various guises across the whole spectrum of high Tc materials. The promise and prize is now a unified view of the field. This will include materials based on planar structural elements of diverse electronegative elements such as B, C, N. In addition the mode of transitions to other more or less related mechanisms for superconductivity will be of interest.

References [1] [2] [3] [4]

A.R. Bishop. Current Opinion in Solid State and Material Science 2, 244 (1997. J.B. Goodenough and J.S. Zhou. In: Structure and Bonding 98, 17 (2001). C.Park, R.L.Snyder J. Am. Cer. Soc.78,3171 (1995) J.E. Hoffman, E.W. Hudson, K.M. Lang, VV. Madharan, H. Eisaki, S. Uchida, J.C. Davis. Science 295, 466 (2002). [5] H.A. Mook, P. Dai, F. Dogan. Phys. Rev. Letters 88, 97004 (2002). [6] T. Hanagurii, Nature, 430, 1001 (2004)

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[7] K. M. Shen, F. Ronning,D. H. Lu, F. Baumberger, N. J. C. Ingle, W. S. Lee, W. Meevasana, Y. Kohsaka, M. Azuma, M. Takano,H. Takagi, Z.-X. Shen, Science, 307, 901-904 [2005] [8] Y. Zhang, E. Demler, S. Sachdov. Phys Rev. B 66, 94501 (2002). [9] J. Zhu, I. Martin, R.A. Bishop Phys Rev. Letters 89, 67003 (2002). [10] M. Voita Phys Rev. B, 66, 104505 (2002). [11] H. Oesterreicher, J. Solid State Chemistry, 158,139 (2001) [12] J.B. Goodenough. Europhys Letters 57, 550 (2002). [13] H. Oesterreicher, J. Superconductivity 16, 507,(2003) [14] H. Oesterreicher, J. of Alloys and Compounds, 366, 1, (2003) [15] H. Oesterreicher, J. Superconductivity 17, 439,(2004) [16] H.Oesterreicher Solid State Communications, 137, 235-240 [2006] [17] H. Oesterreicher, AIP Proc. 24th international conference on low temperature physics, p573 [2006] [18] H.Oesterreicher Solid State Communications, 142, 583-586 [2007] [19] J.M. Tranquada, B.J. Sternlieb, J.D. Axe, Y. Nakamura and S. Uchida. Nature 375, 561-563 (1995). [20] H. Kotegawa, Y. Tokunaga, K. Ishida, G.-Q. Zhang, Y. Kitaoka, H. Kito, A. Iyo, K. Tokiwa, T. Watanabe, and H. Ihara. Phys. Review B, 64, 064515 [2001] [21] L.F. Feiner, M. Grilli, C. DiCastro, Phys Rev. B 45, 10647 (1992). [22] H.A.Mook, P. Dai, F.Dogan, R.D.Hunt, Nature 404, 729 (2000) [23] A.V. Balatsky and Z.X. Shen, Science 284, 1137 [1999] [24] P.W. Anderson, Science 288, 480 [2000] [25] K.H.Ahn, J.Zhu, Z.Nussinov, T.Lookman, A.Saxena, A.V.Balatsky, A.R.Bishop, J. Superconductivity 17, 7 (2004) [26] D.M.de Leeuw, W.A. Groen, L.F.Feiner, E.E.Havinga, Physica 166C133 [1990] [27] R.J. Cava, A.W. Hewat, E.A. Hewat, B. Batlogg, M. Marezio, K.M. Rabe, J.J. Krajewski, W.F. Peck and L.W. Rupp. Physica C 165, 419 (1990). [28] J.M.Tranquada, S.M.Heald, A.R.Moodenbaugh, Y.Xu Phys. Rev.38, 8893 [1988] [29] J.P.Emerson, D.A.Wright, R.A.Fisher, N.E.Phillips, Czech J. Physics, 46, 1209 [1996] [30] Y. Nakazawa, M. Ishikawa, Physica C158, 381 [1989] [31] B.A. Hunter, J.D. Jorgensen, J.L. Wagner, P.G. Radaelli, D.G. Hinks, H. Shaked and R.L. Hitterman. Physica C 221, 1-10 (1994) [32] J.L. Wagner, P.G. Radaelli, D.G. Hinks, J.D. Jorgensen, J.F. Mitchell, B. Dabrowski, G.S. Knapp and M.A. Beno Physica C 210, 447-454 (1993) [33] H. Oesterreicher, arxiv.org/abs/0811.2792 [34] S. Chakravarty, H. Kee and K. Voelker, Nature 428, 53 (2004).

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In: Superconducting Cuprates Editor: Koenraad N. Courtlandt

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Chapter 11

ANALYSIS OF MICRO- AND NANOTEXTURE IN CUPRATE SUPERCONDUCTORS 1

A. Koblischka-Veneva1 and M. R. Koblischka2

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Institute of Functional Materials, Saarland University, P.O.Box 151150, D-66041 Saarbrücken, Germany 2 Institute of Experimental Physics, Saarland University, P.O.Box 151150, D-66041 Saarbrücken, Germany The micro- and nanotexture of high-Tc cuprate superconductor samples of technological interest is studied studied by means of electron backscatter diffraction (EBSD). The samples analyzed comprise Ag-clad (Pb,Bi)2Sr2Ca2Cu3Ox (Bi-2223) tapes, YBa2Cu3Ox (YBCO) coated conductors and melt-textured YBCO bulk samples. A dedicated surface preparation technique is required in order to achieve high image quality of the recorded individual Kikuchi patterns. This further enables multi-phase EBSD scans on e.g., the Bi-2223 tapes including Bi-2223, Bi2Sr2CaCu2Ox, Bi2Sr2CuOx, (Sr,Ca)14Cu24O41 and Ag to be performed. For the EBSD scans a maximum spatial resolution of 30 nm was reached enabling a detailed orientation analysis. Close to the Ag-cladding, a homogeneous distribution of Bi-2223 grains is observed with misorientations of ±10° around the [0 0 1] direction, while in the tape center a larger amount of secondary phases is residing. These misorientations are visualized using various EBSD maps. Furthermore, EBSD enables the spatially resolved mapping of the misorientation angles within each phase separately. The influence of these grain boundaries on the current transport properties is discussed. On YBCO coated conductors and melt-textured samples, it is essential to study the interaction of the YBCO with the formed Y2BaCuO5 (Y-211) particles, which can provide flux pinning sites but also show negative effects on the formation of the superconducting YBCO matrix.

Keywords: EBSD, high-Tc superconductors, tapes, melt-textured superconductors, texture, YBCO, Bi-2223

1. Introduction The analysis of the texture achieved in high-Tc superconductors (HTSC) samples is essential to achieve the rquired high critical current densities. As the coherence length of the

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HTSC cuprates is so small (3 – 5 nm at low temperatures), the presence of any type of grain boundary within the current flow direction poses a hughe problem. Due to this behavior, the texture in cuprate HTSC has to be controlled on the microscale as well as on the nanoscale. The microscale comprises the grain orientations, the large-angle grain boundaries and larger secondary phase particles, whereas the nanoscale comprises small-angle grain boundaries, embedded nanoparticles as flux pinning sites, twin boundaries, nano-stripe patterns, and many more. The texture analysis on the nanoscale is going beyond the capabilities of commonly used texture analysis methods like X-ray analysis, and furthermore, the analysis has to be performed on multi-phase systems. Secondary phase particles are often found within the superconducting matrix, as the secondary phases are often unaviodable during the processing steps, but also, such particles may be added deliberately in order to create the required flux pinning sites. On this background, it is extremely important for the development of real conductors to have a suitable texture analysis tool available. The texture achieved during the processing of silver-sheathed (Pb,Bi)2Sr2Ca2Cu3Ox (Bi2223) tapes is an essential parameter to obtain tapes with high critical currents. The core of monofilamentary Bi-2223 tape consists mainly of Bi-2223, but even the currently best tapes contain about 20% secondary phases [1,2]. Especially intergrowths of Bi2Sr2CaCu2Ox (Bi2212) do occur and (Sr,Ca)14Cu24O41 as non-superconducting particles may be present as well. Smaller contributions of Bi2Sr2CuOx (Bi-2201) and the amorphous phase Pb3(Bi0.5Sr2.5)Ca2CuOy (3321 phase) may also be present [3,4]. However, the determination of texture in multi-phase compounds is not straightforward, as integral methods only deliver averaged information. For a detailed orientation analysis, a spatially highly-resolved analysis technique is required thus enabling the interplay between the different phases to be studied. The electron backscatter diffraction (EBSD) technique [5,6] provides here a unique tool for this task, and with the recent development to the EBSD technique concerning the image recording system, a high spatial resolution down to about 50 nm even on oxidic samples can be achieved [7,8]. In recent works, we have demonstrated the usefulness of the EBSD technique on various bulk high-Tc samples, where the interaction of the YBa2Cu3Ox (YBCO) and the Y2BaCuO5 (Y-211) phase could be analysed directly [9,10]. Furthermore, the EBSD technique was previously applied also to Ag-sheathed Bi-2223 tapes [11-13], but up to now, only a single phase analysis was performed which, however, does not completely describe the situation found in Bi-2223 tapes.

2. Experimental Sample preparation Bi-2223 monofilamentary tapes were produced by the standard powder-in-tube method [14]. The critical current Ic was measured to be 27 A (self-field, 77 K), and the critical current density Jc was about 20 kA/cm2. The critical current density of these tapes is not optimized. However, this does not have an influence on the basic conclusions of the present paper, which focuses on the application of the EBSD technique to the texture analysis of Bi-2223 tapes in general. The YBCO coated conductors investigated here are YBCO thin films prepared by pulsed laser deposition on a CeO2-buffered Al2O3 substrate and by MO-CVD on a cube textured Ni

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substrate employing a three-layered buffer structure composed from CeO2/Y-stabilized ZrO2/CeO2 [15,16] Melt-textured YBCO samples were produced using a standard top seeded melt growth (TSMG) procedure [17,18]. The fully melt processed samples were subsequently oxygenated in a separate process in the usual way to obtain the superconducting YBa2Cu3O7-δ phase (Y123). Additions of Y2BaCuO5 (Y-211) and Y2Ba4CuMOx (with M = Nb, Zr, Ag, etc., abbreviated M-2411) nanoparticles were added to the precursor mixtures. A dedicated polishing procedure is required for most of the cuprate superconductor samples in order to achieve a high image quality enabling a multi-phase EBSD scan to be performed. For this study, the top surface of the silver-sheathed tape was carefully polished away using 2400 mesh grinding paper. For the surface finish, the tape was subsequently polished using diamond paste in four steps down to 1/4 µm grain size, and then a final step using colloidal silica (OP-S) [19] was applied. Only ethanol was allowed for cleaning purposes. This polishing procedure yields a rms roughness of about 5 nm [20], and, the superconducting properties are retained fully as confirmed by magneto-optic imaging performed on such samples [21]. An optimum polishing procedure yielding a high image quality of the resulting Kikuchi patterns is required to enable a multi-phase EBSD scan to be performed.

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Electron backscatter diffraction (EBSD) The EBSD system employed consists of a FEI dual beam workstation (Strata DB 235) equipped with a TSL OIM analysis unit [22]. The Kikuchi patterns are generated at an acceleration voltage of 20 kV, and are recorded by means of a DigiView camera system, allowing a maximum recording speed of the order of about 14 pattern/s. The time employed in the case of a multi-phase scan is somwhat longer, of the order of 10 pattern/s, as a higher image quality/confidence index is required. To produce a crystallographic orientation map, the electron beam is scanned over a selected surface area and the resulting Kikuchi patterns are indexed and analyzed automatically (i.e., the Kikuchi bands are detected by means of the software). An image quality (IQ) parameter and a confidence index (CI) is recorded for each such Kikuchi pattern. The dimensionless IQ parameter is the sum of the detected peaks in the Hough transform employed in the image recording; the CI value yields information about how exact the indexation was carried out. The CI value ranges between 0 and 1 [22]. Based on the analysis of the recorded CI value, a multi-phase analysis is realized. A detailed description of the measurement procedure can be found in Refs. [23,24]. The results of the EBSD measurement are presented in form of maps, the most important thereof are the socalled inverse pole figure (IPF) maps, indicating the crystallographic orientation of each individual point. Automated EBSD scans were performed with a minimum step size of 30 nm; the working distance was set to 10 mm.

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3. Results and Discussion 3.1. Kikuchi patterns of HTSC In the following, we present Kikuchi patterns obtained on a variety of materials.

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(a)

(b)

(d)

(c)

(e)

Figure 1. Kikuchi patterns of (a) orthorhombic YBCO, (b) tetragonal YBCO, (c) orthorhombic 211, (d) Bi-2212 and (e) NdBCO. The first row gives the measured Kikuchi pattern, the second the indexation. Superconducting Cuprates : Properties, Preparation and Applications, Nova Science Publishers, Incorporated, 2008. ProQuest Ebook Central,

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This includes YBCO with different oxygen content [Fig. 1 (a,b)], the green phase, Y2BaCuO5 (Y-211) [Fig. 1 (c)], Bi-2212 [Fig. 1 (d)] and NdBa2Cu3Ox (NdBCO) [Fig. 1 (e)]. Most of the high-Tc superconductors do not have a simple, cubic unit cell, but often are orthorhombic or tetragonal. Accordingly, the indexation of these more complicated Kikuchi patterns is less unique. An additional challenge is the presence of secondary phases within one sample, which − like in the case of Y-211 − may be orthorhombic as well. In Figs. 1 (a) – (e), the first row presents the measured Kikuchi patterns; the second row the indexation as performed by the EBSD software. The third row shows the orientation of the unit cell as determined from the measured Euler angles, and in the fourth row, the corresponding simulations of the Kikuchi patterns. The patterns of orthorhombic 123 and tetragonal 123 are quite similar, so an automated distinction between the two phases is difficult. The green phase pattern, however, is remarkably different from 123 due to the different unit cell. The NdBCO pattern is again very similar to that of pure YBCO. The Bi-2212 pattern shows the characteristic three parallel lines.

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3.2 Texture and phase analysis of Ag-sheathed Bi-2223 tapes Figure 2 presents the obtained EBSD mappings on a monofilamentary Bi-2223 tape (Tape 1). The selected section for the EBSD scan is located close to the silver sheath, where the best Bi-2223 texture is expected to be obtained. In Fig. 2 (a), the image quality (IQ) map is shown, resembling a SEM-BSE image of the sample. The detected IQ values range between 50 and 330. However, it is important to note that the IQ parameter carries the information about the crystallinity of the investigated spot, so that we have an idea about the homogeneity of the grain. In Fig. 2 (b) the corresponding phase map is presented. The Bi2223 phase is depicted in light blue, the Bi-2212 phase in red, the (Sr,Ca)14Cu24O41 phase in yellow, the Bi-2201 phase in blue and Ag in green. This phase map enables a direct comparison to the orientation map of Fig. 2 (c). From this phase map, it is clearly visible that the majority of the grains are the Bi-2223 phase as expected (see Table 1). In Fig. 2 (c) the inverse pole figure (IPF) map in [0 0 1] direction (i.e. perpendicular to the sample surface) is shown, which gives the crystallographic orientation of the grains according to the stereographic triangles for each phase. The rolling direction (RD) is directed along the selected scan area. This implies that the IPF map is illustrating the c-axis spread of the Bibased compounds present within the investigated tape section.Characteristically for Bi-2223 tapes, some large Bi-2223 grains are found to be present, which are, however, in the present case not aligned to the rolling direction, but form even an angle > 30°. On top of the map, there is even a Bi-2223 grain oriented in [1 0 0] direction. This is, of course, the reason for the small achieved critical current density of this tape. Most of the smaller Bi-2223 grains are more or less oriented parallel to the rolling direction (RD). The main orientation obtained here is the [0 0 1] direction, with some deviations leading to small-angle grain boundaries. The EBSD maps clearly reveal that the phase distribution in the Bi-2223 tapes is entirely different from the bulk YBCO superconductors investigated earlier [18,19]. The large Bi2223 grains are not uniform themselves but within the grains some smaller subgrains are located, which can even be oriented in [1 0 0] direction.

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Bi-2223 Bi-2212 (Sr,Ca)14Cu24O41 Bi-2201 Ag

Figure 2. Image quality (IQ) map (a), phase map (b) and inverse pole figure (IPF) orientation map (c) in [0 0 1] direction (perpendicular to the sample surface) of a selected area close the Ag sheath. The scale bar in the maps is 15 µm long. The rolling direction (RD) and transverse direction (TD) are indicated by arrows. The phase map reveals the spatial distribution of the five investigated phases, Bi-2223, Bi2212, Bi-2201, (Sr,Ca)14Cu24O41, and metallic Ag (color code on the right side). The crystallographic orientations of the IPF map are given by the stereographic triangles displayed on the right side.

The presence of such subgrains may be a serious obstacle for the current flow in an otherwise homogeneous grain, especially because the presence of these grains limits the available current paths. Besides these big grains, there are also (much smaller) Bi-2223 grains with more or less random orientation. These small grains act quasi as filler between the big grains. However, such small grains are not favourable for the current transport, as many highangle grain boundaries are involved in the current path. The arrow at position '1' marks another interesting feature: This big grain exhibits one stripe-like area which is misoriented to the rest of the grain by about 25°. Such situations were found repeatedly within the tape, so this type of defect is quite common within such a monofilamentary tape. The presence of this defect also implies a spatial reduction of the available current path within a grain.

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The secondary phases Bi-2212 (red) and the non-superconducting particles (Sr,Ca)14Cu24O41 (yellow) are located mainly along the 'big' Bi-2223 grains, with some separated bigger grains also being present. This arrangement, however, implies that these secondary phase particles are not really contributing to the flux pinning, which is expected to be active mainly inside the grains. Even worse, by the presence of these particles, the boundary to the neighboring grain is strongly affected and rendered useless for the current flow.

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Bi2223

(b)

Bi2212

(c)

(Sr,Ca)14Cu24O41

(d)

Bi2201

(e)

Ag

Figure 3. Orientation (IPF) maps for the same area as shown in Fig. 2, but for each phase separately; the other phases are represented in black. At the bottom of each map, the corresponding pole figure in (0 0 1) direction is given as well.

The phase analysis of the EBSD scan of Tape 1 is summarized in Table 1. The selected section of the tape contains (total fraction) 91.7% Bi-2223, 4.6% Bi-2212, 2.2% Bi-2201, 0.5% (Sr,Ca)14Cu24O41 and 1.1% Ag. The partition fraction represents the amount of a phase recognized by the EBSD software. The two values may differ from each other if a given phase is detected, but the corresponding CI value is zero. In X-ray analysis, the contents were determined to be >90% for Bi-2223, and 5% for Bi-2212. This corresponds well to the EBSD data, which however, yield local information. The amorphous phase, Pb3(Bi0.5Sr2.5)Ca2CuOy (3321 phase), cannot be detected by means of EBSD as there will be no Kikuchi pattern generated.

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Table 1. EBSD-determined phase distribution in Tape 1. The total fraction represents the amount of the phase detected, while the partition fraction describes the amount of the phase recognized by the EBSD software. Not-indexed points can either be voids (pores) or also the amorphous phase Pb3(Bi0.5Sr2.5)Ca2CuOy (3321 phase), which does not show a Kikuchi pattern. Phase

Total fraction

Partition fraction

Bi-2223 Bi-2212 Bi-2201 (Sr,Ca)14Cu24O41 silver unindexed points

0.913 0.046 0.005 0.022 0.011 0.003

0.913 0.046 0.005 0.022 0.011 0.003

Figure 3 presents the phase analysis in detail for each phase separatedly; the other phases are represented in black. Below each map, the corresponding pole figure in [0 0 1] direction for the selected scan area is given. The color code of the IPF maps can be found in the respective stereographic triangle below each map. The Bi-2223 phase represents the majority of the sample, but there is no homogeneous [0 0 1] texture, but a much broader distribution together with some grains being fully misoriented. Most misorientations of the grains are about ±15°, with some grains being off by more than 40° (light green, blue). The Bi-2212 phase shows the same orientations like the Bi2223 phase. The Bi-2212 phase forms mainly own grains, but there are also some smaller units embedded within the Bi-2223 phase. (Sr,Ca)14Cu24O41 forms only small spots scattered over the entire scan area, but practically always only along the grain boundaries of the Bi2223 or the Bi-2212 phase. These small grains, which may act as effective flux pinning sites only if located inside the Bi-2223 grains [25], do not exhibit any texture similar to the Y-211 phase particles in melt-textured YBCO. This implies that it is an essential task for the further development of the Ag-clad Bi-2223 conductors that one finds a type of nanoparticle which can be embedded within the superconducting grains. Then the required increase of the critical current densities, especially at elevated temperatures, can be achieved. The Bi-2201 phase follows the Bi-2212 phase distribution and orientation; mostly, these two phases appear together. The also present Ag particles are again randomly distributed and forms separated grains located at the grain boundaries of the superconducting matrix. Figure 4 presents the determined orientation distribution (ODF) functions in [0 0 1] direction for all 5 phases. This representation of the data gives a better overview of the distribution details as the simple pole figures, which are also shown for comparison. The ODF fuction confirms the good texture of the Bi-2223 phase, together with the large-angle misoriented grains. The Bi-2212 texture is similar to that of Bi-2223, but less homogeneous. (Sr,Ca)14Cu24O41 and Ag are randomly oriented, while Bi-2201 exhibits a pattern with the same basic features as of Bi-2223 but being concentrated on some spots.

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Figure 4. The orientation distribution functions (ODF) for each phase as determined by EBSD; (a) – Bi2223, (b) – Bi-2212, (c) – Sr, (d) – Bi-2201 and (e) metallic silver. Clearly visible is here that the ODF’s for Bi-2223, Bi-2212 and Bi-2201 follow essentially the same pattern.

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Figure 5. IQ-maps with the misorientation boundaries indicated in different colors. The left map gives the misorientations between 1° and 5°, 5° and 10° and 10° and 30°. The right map gives the misorientations between 40° and 60°, 60° and 90° and 90° and 120°.

Figure 5 gives a detailed grain boundary analysis for all phases as determined by EBSD. The represented map is in both cases an IQ map, where the detected grain boundaries are marked in color. Figure 5 (a) shows the misorientations from 1° to 30°, (b) the large angles between 40° and 120°. Together with the map, there is an analysis of the character of the grain boundaries. The majority of the boundaries in (a) is given by boundaries in the range 10° to 30° (yellow), while only a small number of low-angle grain boundaries is observed. Figure 5 (b) shows that even higher misorientation angles can be found, which is due to the misoriented (Sr,Ca)14Cu24O41 and Ag particles. Such grain boundary analysis is useful to understand the growth mechanism as shown for the case of YBCO with embedded nanoparticles in Ref. [26]. Figure 6 shows the determined misorientation angles of each individual phase in a diagram. The curve of Bi-2223 decays from a number fraction of 0.2 smoothly, but shows a plateau-like behavior up to about 45°, before it decays further. In contrast to that, Bi-2212 exhibits a maximum at 10°, followed by some smaller maxima at 25°, 45° and 60°. Bi-2201 is found to behave similar to Bi-2212, but exhibits a maximum at 90°.

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Figure 6. Misorientation diagram for all phases.

Figure 7 gives the misorientation angle analysis within the Bi-2223 phase only. The underlying map is here a phase map where the Bi-2223 phase is indicated in red. Grain boundaries towards other phases are disregarded in this analysis. Boundaries are shown between 1° and 5° (yellow), 5° and 10° (green) and between 10° and 30° (blue). The largest amount of boundaries is obtained in the range between 10° and 30° with an overall length of 6.38 mm, while the sum of both small-angle boundaries is 6.2 mm. This figure confirms the results of Figs. 5,6, indicating that the misorientations within the Bi-2223 phase are not so small as one might expect from e.g., the brick-wall model for the grains within Bi-2223 tapes. The situation corresponds is more similar to the railway-switch model [27], but essentially, much more complex.

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Figure 7. Misorientations within the Bi-2223 phase. The underlying map is a phase map (Bi-2223 drawn in red).

Figure 8 presents crystal direction (CD) maps. Here, the colour code is concentrated on 10° around the (0 0 1) direction in (a) and on 10° around the (1 0 0) direction in (b). This way of presenting the data clearly reveals that only 22% of the Bi-2223 is oriented within this limit, whereas even 10% of the Bi-2223 grains have an orientation close to (1 0 0). Fully oriented grains do practically not exist. This implies that a current flowing through the tape parallel to RD has to face a large amount of obstacles formed by high-angle grain boundaries.

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Figure 8. Crystal direction (CD) maps. The left map gives all grains within 10° of the (0 0 1) direction, and the right map all grains within 10° of the (1 0 0) direction. All other grains are plotted in white.

Of course, in the present multifilamenary tapes, the achieved texture is much better as in the present sample, but the main problem of secondary phases within a filament still remains; the filaments are practically shells of Bi-2223 enclosing the secondary phases [1]. It will especially be a problem to control the intergrowths of Bi2Sr2CaCu2Ox and Bi2Sr2CuOx, which are in principle of the same type, but have one or two CuO2 planes less. If one regards the same section of tape with a tolerance of 20° around the (0 0 1) direction as shown in Fig. 9, then 61.7% of the Bi-2223 grains fall in this limit. Here, we are now closer to obtain at least a good current path through the sample, but still, a fully superconducting current path is not available. This is the reason for the strong field dependence of the transport currents as observed in the Bi-2223 tapes. In current state-of-the-art multifilamentary tapes (see e.g. Ref. [1]), the amount of grains not oriented in the RD direction is considerably reduced, but the intrinsic tendency of Bi-2223 grains to form subgrains and intergrowths cannot easily be controlled during the tape processing.

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Figure 9. CD-map with a tolerance of 20° around the (0 0 1) direction (compare with Fig. 8).

In order to increase the amount of intragranular current density, one should find a way to employ the existing secondary phases as flux pinning sites within the grains, but it is an essential task to keep the grain boundaries free from these obstacles. One possibility to improve the flux pinning could be the addition of specific nanoparticles; another approach could be the addition of a oxide to the precursor powder which influences the formation of the (Sr,Ca)14Cu24O41) particles. Figure 10 presents a small section of the first EBSD analysis in detail. An IQ-map (a), a phase map (b) and a IPF-map (c) is given. Within the IPF map, the phase boundaries are marked in white and the grain boundaries in black. Note the large grain on the left side, where a grain boundary is running through the entire grain. Furthermore, here it is clearly visible that most of the secondary phases are located at the grain boundaries of the Bi-2223 phase, so that the (Sr,Ca)14Cu24O41 particles cannot act as flux pinning centers as would be desired. This is only given in the light green grain (i.e., highly misoriented Bi-2223) where Bi-2201

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granules are embedded within the Bi-2223 matrix, so that we may expect an improved flux pinning at temperatures around and above the critical temperature of Bi-2201. In this higher magnification, it becomes clear that pores and other, non-indexed phases [e.g., the amorphous phase Pb3(Bi0.5Sr2.5)Ca2CuOy (3321 phase)] play also an important role as obstacles for the current flow.

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Bi2223 Bi2212 (Sr,Ca)14Cu24O41 Bi2201 Ag Figure 10. Selected detail of a EBSD scan. (a) is the IQ map, (b) the phase map and (c) the orientation (IPF) map. For the color code in the stereographic triangle, see Fig. 2.

A detail of the IPF map is presented in Fig. 11 for the Bi-2223 phase only; contributions of other phases are plotted in grey. Further, the respective unit cells are indicated as wireframes. This helps to obtain an impression how oriented the individual grains are within the tape matrix. Here, it is visible that along large Bi-2223 grains, several small, but misoriented ones are formed. These small grains are acting quasi as a filler material. The resulting misorientation angles are mostly above 20°. As the big grain in the center is not oriented parallel to the rolling direction (RD), it is itself an obstacle for the current flow parallel to RD. The boundary through this big grain is also remarkable, as the unit cells plotted are indicating a resulting ‘V’-shape with a misorientation of about 30°. This detail section also shows clearly that within the big Bi-2223 grains, no other phases are present which may act as flux pinning sites. The flux pinning of the Bi-2223 is therefore entirely intrinsic.

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Figure 11. IPF map for the Bi-2223 phase only; other phases are plotted in grey. Further, the orientation of the unit cells are indicated. For the color code of the determined orientations (stereographic triangle), see Fig. 2.

Finally, Fig. 12 presents an EBSD analysis (3-phase scan) of Tape 2, which is a section of the same piece of tape located in the central region of the core. Here we find a much larger amount of pores, respective the amorphous phase, which pose a large problem for the current flow. The c-axis spread of the Bi-2223 phase is, however, similar to that of Tape 1, but the resulting Bi-2223 grains are much smaller. The results of the phase analysis are presented in Table 2. The total fraction of the detected phases is here not close to 1, but due to the amount of pores and amorphous phase being present only at 0.86. It is clearly visible that the amount of the (Sr,Ca)14Cu24O41-phase is much larger than in Tape 1, as well as the amount of the Bi2212 phase. Also in this case, we find that the (Sr,Ca)14Cu24O41-phase particles are located only at the grain boundaries of the Bi-2223 phase, and not inside a Bi-2223 grain, thus being useless as flux pinning sites. Of course, the amount of pores, the presence of the amorphous phase and the larger amount of secondary phases is responsible for a much smaller critical current density of the present tape as compared to the section tape 1. Therefore, here is the largest potential for improvement.

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(c)

Bi2223

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Bi2223 Bi2212 (Sr,Ca)14Cu24O41

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Figure 12. EBSD scan of a tape section in the tape center. (a) is the IQ map, (b) the phase map and (c) the orientation (IPF) map.

Table 2. EBSD-determined phase distribution in Tape 2. Phase

Total fraction

Partition fraction

Bi-2223 Bi-2212 Bi-2201 (Sr,Ca)14Cu24O41 silver unindexed points

0.767 0.056 -0.035 -0.135

0.767 0.056 -0.035 -0.135

The increase of the transport currents in Ag-clad Bi-2223 tapes is twofold. One effort has to concern the improvement of the texture in general, the other one the improvement of the critical current density within a Bi-2223 grain. However, in the Bi-2223 tapes there is no intrinsic phase which could, if optimized, provide extra flux pinning sites within a grain (compare the Y-211 grains in bulk YBCO samples). This may be overcome by embedding nanoparticles into the Bi-2223 phase as in the case of bulk, melt-textured YBCO samples with Y2Ba4CuMOx (M = Nb, Zr, Ag) nanoparticles. EBSD analysis revealed that these nanoparticles even follow the texture of the surrounding matrix [26]. Colonies of Bi-2223

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grains as shown in [1,3,4] are thought to share the c-axis direction. However, if such subgrains as marked in Fig. 1 are created, there is a much larger misorientation angle created as the two grains fit together like a ‘V’. Furthermore, the microtexture forming the connections between the grains has to be optimized as the transport currents have to flow through these connections. The formation of big Bi-2223 grains is not really useful with respect to the transport current densities. Therefore, the tailoring of the microstructure of the Bi-2223 tapes remains a challenge. It would be, therefore, desirable to perform an EBSD analysis also on state-of-the-art multifilamentary tapes. However, it was up to now not possible to polish a multifilamentary tape in order to expose several filaments and, at the same time, to avoid the contribution of the silver sheath to the resulting Kikuchi patterns which is much stronger than that of Bi2223, and therefore, obscuring the measurement of Bi-2223 within such a multifilamentary tape. If this problem can be solved, may be using extracted individual Bi-2223 filaments, a detailed EBSD analysis should become possible. In summary, we have presented an EBSD analysis of a monofilamentary Bi-2223 tape. EBSD scans could be performed with 5 phases, and a resolution of ~70 nm. The phase map reveals that the secondary phases are located mainly along the grain boundaries. The detected misorientation angles within the Bi-2223 phase are found to be mainly of the order of 10° to 30°.

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3.3. YBCO melt-textured samples Bulk, melt-textured YBCO samples play an important role as magnetic bearings and superconducting permanent magnets [28]. Also here, a good texture is extremely important for the performance of the samples. The most important problem for the bulk samples is to achieve a high critical current density at T = 77 K (temperature of liquid nitrogen), so therefore, a large number of flux pinning sites has to be created during the processing steps [29]. The most commonly used addition is therefore the addition of the Y-211 phase, which needs to be refined by adding additionally Pt, PtO2 or CeO2. The EBSD analysis [30,31] has shown that the Y-211 particles may have different effects onto the superconducting YBCO matrix: (i) large Y-211 particles or clusters may cause a heavy misorientation of the surrounding YBCO matrix and (ii) small Y-211 particles do not disturb the YBCO matrix – the smaller the particles, the less the negative influence. This is an important finding of the EBSD measurements, as the so-called subgrains, which may disturb the current flow through a melt-textured YBCO sample, are caused by these misorientations and then grow through the sample in form of a misoriented YBCO grain or ‘subgrain’. In turn, this feature implies that one should be able to control the growth and orientation of the Y-211 particles. A fair chance is given by the fact that the small Y-211 particles do also not disturb the growth of the YBCO matrix.

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Figure 13. Interplay of the embedded Y-211 particles with the superconducting YBCO matrix. Shown is a phase map (light grey/dark grey), the IPF maps of YBCO and Y-211 together with the Kikuchi patterns at some selected places and the respective orientations of the unit cells.

This observation is illustrated in the EBSD-mappings shown in Fig. 13. Y-211 particles with specific orientations do not exhibit any negative effect on the YBCO matrix. However, during the regular processing of the melt-textured YBCO samples, it cannot be guaranted that the particles have only one orientation. Nanometer-sized particles as flux pinning centers were the goal for a long time since the coherence length of the high-Tc superconductors is so small [32-35]. Based on the results of a chemical analysis of melt-textured YBCO samples with additions of depleted uranium oxide, a new phase with the composition Y2Ba4CuMOx (M = Nb, Zr, Ag, etc.) could be identified. Nanopowders of this phase were added successfully to the starting powders to prepare melttextured YBCO samples [36,37]. Now, it is an important question how the superconducting matrix is reacting concerning the embedded nanoparticles, and how many such nanoparticles may be embedded within a sample. In Refs. [38-40], the Kikuchi patterns of the M-2411 phase could be identified. With the achieved high spatial resolution of the EBSD technique, it is possible to determine the crystallographic orientation even of these embedded nanoparticles within the YBCO matrix [41,42]. Figure 14 presents the results of a three-phase EBSD analysis on a melt-textured YBCO sample with Nb-2411 additions. Figure 14 gives the EBSD-determined phase maps for each individual phase, the other phases are plotted in black. In this way, the different types of grains can directly be recognized. As in most melttextured YBCO samples, some large Y-211 grains are present. In contrast to that, the Nb2411 grains retain their nanoscale size, and no clustering of these particles is observed.

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However, these particles clearly form stripes, which is of course not fully ideal. This specific distribution of the nanoparticles, which is related to the growth of the YBCO phase, must be better controlled in order to achieve a homogeneous critical current density within the sample. However, it is remarkable how many such nanoparticles can be embedded within the superconducting YBCO matrix.

Figure 14. EBSD-phase map for each phase separatedly; the respective other phases are plotted in black. The YBCO phase is plotted red, Y-211 in green and the Nb-2411 nanoparticles are shown in yellow.

In Fig. 15, the IPF orientation maps of the same three phases are plotted, again for each phase separatedly. The YBCO phase is found to exhibit only minor misorientations, and the map is nearly homogeneously red. The only misorientations found are due to a high density of Y-211 particles as visible from a comparison of Figs. 14 and 15. The important result achieved here is that the embedded Nb-2411 nanoparticles are found to be fully homogeneously oriented analog to the YBCO matrix. Therefore, it becomes possible to add a large number of such, pre-prepared nanoparticles to the YBCO matrix, which is in turn not

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disturbed by the presence of these nanoparticles. In contrast to the nanoparticles, the also existing Y-211 particles do not show such a texture, even if their size is considerably reduced. The result of this measurement, which was confirmed also for other types of M-2411 particles [43-45], allows to draw some important conclusions.

Figure 15. EBSD-orientation maps (IPF) or each phase separatedly; the other phases are plotted in grey. The orientations can be deduced from the stereographic triangles on the right side of the figure. Y-123 and Y-211 are here both orthorhombic, the Nb-2411 particles have a cubic unit cell. The corresponding pole figures for each phase are given below the map; the intensities are normalized to that of the YBCO phase.

Therefore, it is essential to reduce the amount of the Y-211 particles in such melttextured samples to an unavoidable minimum. M-2411 nanoparticles, prepared prior to the melt-texturing, may be added in larger numbers without disturbing the growth of a homogeneous YBCO matrix. This orientation of the nanoparticles is demonstrated in Figs. 16 and 17, showing the complete orientation distribution function (ODF) for YBCO (Fig. 16) and for the Nb-2411 nanoparticles (Fig. 17). The strong texture is indicated by the fully yellow elements at 0° (corresponding to the orientation in [0 0 1] direction).

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Figure 16. ODF function for the YBCO phase, corresponding to Fig. 14.

Figure 17. ODF function for the Nb-2411 phase, corresponding to Fig. 14.

3.4. YBCO coated conductors Coated conductors are the second-generation of HTSC tapes. Mainly, two different approaches to prepare such samples are published in the literature. The common feature is, however, that a buffer layer is needed on the substrate in order to enable an oriented growth of the YBCO phase. Therefore, the texture analysis of such samples is extremely essential in order to achieve high critical current densities.

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Figure 18. EBSD-measurements on „coated conductors“, i.e., a YBCO layer on a Ni substrate with a suitable buffer layer. The YBCO layer is about 400 nm thick. Left: IQ map, center: Phase map (orthorhombic YBCO – pink, 81.8 %, Y-211 particles – green, 3.0 %, and tetragonal YBCO – blue, 17.1%); right: IPF map in [0 0 1]-orientation. The scale bar in the EBSD maps is 25 µm long.

The EBSD analysis as presented in Fig. 17 indicates that the coated conductors exhibit features which are strongly related to the features found in melt-textured YBCO superconductors. Also here, we have a large amount of Y-211 particles being present, and in areas with a high Y-211 particle density, the YBCO phase may show strong misorientations, which otherwise do not appear in the samples. Due to the thickness of the YBCO layer of about several micrometers, the embedding of non-superconducting particles as flux pinning sites is essential as well. Of course, to achieve high critical current densities, the added particles should be of a small size and being distributed homogeneously. Only in this way, an overall high critical current is achieved.

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orth

tetr

Y-211

Figure 19. EBSD-measurements on „coated conductors“, here it is a YBCO layer on a CeO2-buffered Al2O3. The YBCO layer is about 400 nm thick. Left: IQ map, center: Phase map (orthorhombic YBCO – pink, 65.9 %, Y-211 particles – green, 0.02%,, and tetragonal YBCO – blue, 4.1%); right: IPF map in [0 0 1]-orientation. The color code for the crystallographic orientations corresponds to that given in Fig. 18. The scale bar is 5 µm long.

A similar behavior is exhibited by a different type of coated conductor as shown in Fig. 19. The desired [0 0 1] orientation is achieved due to the buffer layer. Nevertheless, the tetragonal YBCO and the Y-211 particles are also present

4. Conclusions In this contribution, we have shown that the EBSD technique is a very useful tool to determine the texture of HTSC cuprate samples on the microscale as well as on the nanoscale. The currently achieved EBSD resolution fits exactly to the requirements posed by the HTSC cuprates, if the necessary surface preparation can be handled well. The possibility of the EBSD technique to perform a multi-phase analysis offers an unique way to study the interplay of the different phases encountered in cuprate HTSC samples.

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These secondary phases may be of intrisic origin (and are therefore, unavoidable during the processing) or can be added deliberately in order to create the required flux pinning sites. In the present contribution, a multi-phase EBSD analysis was performed on different sections of Ag-clad, monofilamentary Bi-2223 tapes. As a high image quality was achieved through a dedicated surface preparation process, an automated EBSD scan with up to five phases could be performed. The EBSD measurements reveal the locations of the various phases within a tape. The typical intrinsic phases, Bi-2212, Bi-2201, and (Sr,Ca)14Cu24O41 are located mainly along the grain boundaries of the Bi-2223 phase, so that they cannot act as suitable flux pinning sites. These particles may even change the character of the grain boundaries to worse. A full insight to the growth mechanism of the Bi-2223 phase may be achieved using detailed EBSD analysis on state-of-the-art multifilamentary tapes, if a special preparation technique can be established. The same conclusions do apply for the analysis of melt-textured YBCO samples and coated conductors, which also employ YBCO as superconductor. In a dedicated experiment, the Kikuchi pattern of the cubic M-2411 phase could be determined, thus enabling the multi-phase EBSD analysis to be extended to the determination of the crystallographic orientations of nanoparticles embedded within the superconducting YBCO matrix. The EBSD analysis demonstrated that M-2411 nanoparticles being prepared prior to the melt-texturing, may be fully oriented with the surrounding YBCO matrix. As a result, a larger number of such particles (up to 20 vol.-%) may be added to a YBCO superconductor without disturbing the growth of a homogeneous superconducting matrix. This result is an important finding for the processing of such bulk samples, and may also have impact on future generations of thin films and coated conductors of the YBCO-type as well.

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Acknowledgements We acknowledge several collaborations within the European Forum for Processors of Bulk Superconductors (EFFORT), which is funded by the Engineering and Physical Sciences Research Council (EPSRC) of the U.K. government.

References [1] T. G. Holesinger, J. A. Kennison, S. Liao, Y. Yuan, J. Jiang, X. Y. Cai, E. E. Hellstrom, D. C. Larbalestier, R. M. Baurceanu, V. A. Maroni, Y. Huang, IEEE Trans. Appl. Supercond. 15, 2514 (2005). [2] P. Yao, R. Zeng, H. K. Liu, S. X. Dou, Physica C 337, 174 (2000). [3] O. Eibl, Supercond. Sci. Technol. 8, 833 (1995) [4] D. P. Grindatto, J. C. Grivel, G. Grasso, H. U. Nissen, R. Flükiger, Physica C 298, 41 (1998); S. X. Dou, R. Zeng, X. K. Fu, Y. C. Guo, J. Horvat, H. K. Liu, T. Beales, M. Apperley, IEEE Trans. Appl. Supercond. 9, 2436 (1999). [5] K. Z. Baba-Kishi, J. Mat. Sci. 37, 1715 (2002). [6] F. J. Humphreys, 2004 Scripta Materialia 51, 771 (2004) [7] M. R. Koblischka and A. Koblischka-Veneva, Physica C 392-396, 545 (2003). [8] A. Koblischka-Veneva, M. R. Koblischka, N. Hari Babu, D. A. Cardwell, L. Shlyk, G. Krabbes, Supercond. Sci. Technol. 19, S562 (2006).

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[9] A. Koblischka-Veneva, M. R. Koblischka, F. Mücklich, K. Ogasawara, M. Murakami, Supercond. Sci. Technol. 18, S158 (2005) [10] A. Koblischka-Veneva, M. R. Koblischka, K. Ogasawara, F. Mücklich, M. Murakami, J. Superconductivity 18, 469 (2005) [11] T. T. Tan, S. Li, J. T. Oh, W. Gao, H. K. Liu, S. X. Dou, Supercond. Sci. Technol. 14, 78 (2001). [12] T. T. Tan, S. Li, H. Cooper, W. Gao, H. K. Liu, S. X. Dou, Supercond. Sci. Technol. 14, 471 (2001). [13] A. Godfrey, W. Liu, D. Lin, Q. Liu, Mat. Sci. Forum 495-497, 1467 (2005). [14] T. M. Qu, Z. Han, R. Flükiger, Physica C 444, 71 (2006); T. M. Qu, C. Gu, M. Y. Li, Z. Han, Physica C 426, 1159(2005). [15] G. Celentano, V. Galluzzi, A. Mancini, A. Rufoloni, A. Vannozzi, A. Augieri, T. Petrisor, L. Ciontea, A. Tuissi, E. Villa, and U. Gambardella, IEEE Trans. Appl. Supercond. 15, 2691 (2005). [16] A. Augieri, G. Celentano, L. Ciontea, V. Galluzzi, U. Gambardella, J. Halbritter, T. Petrisor, A. Rufoloni, and A. Vannozzi, Physica C 460, 829 (2007). [17] N. Hari Babu, K. Iida, Y. Shi, and D. A. Cardwell, Physica C 445, 286 (2006). [18] N. Hari Babu, Y. Shi, K. Iida, and D. A. Cardwell, Nature Materials 4, 476 (2005); N. Hari Babu, M. Kambara, P. J. Smith, D. A. Cardwell, and Y. Shi, J. Mat. Res. 15, 1235 (2000). [19] OP-S, Struers Co., Denmark, product flyer. [20] A. Koblischka-Veneva, M. R. Koblischka, in: Magneto-Optical Imaging, ed. T. H. Johansen, D. V. Shantsev, Kluwer Acad. Press (2004) p. 242 [21] M. R. Koblischka, T. H. Johansen, B. H. Larsen, N. H. Andersen, H. Wu, P, SkovHansen, M. Bentzon, P. Vase, Physica C 341-348, 2583 (2000); M. R. Koblischka, W. G. Wang, B. Seifi, P. Skov-Hansen, P. Vase, N. H. Andersen, IEEE Trans. Appl. Supercond. 11, 3242 (2001). [22] Orientation Imaging Microscopy software version V4.0, user manual, TexSEM Laboratories (TSL), Draper, UT. [23] A. Koblischka-Veneva, M. R. Koblischka, P. Simon, F. Mücklich, and M. Murakami, Supercond. Sci. Technol. 15, 796 (2002); A. Koblischka-Veneva, M. R. Koblischka, K. Ogasawara, M. Murakami, Crystal Engineering 5, 265 (2002). [24] A. Koblischka-Veneva, M. R. Koblischka, P. Simon, K. Ogasawara, M. Murakami, Physica C 392-396, 601 (2003). [25] M. R. Koblischka, S. L. Huang, K. Fossheim, T. H. Johansen, H. Bratsberg, Physica C 300, 207 (1998). [26] A. Koblischka-Veneva, M. R. Koblischka, F. Mücklich, N. Hari Babu, D. A. Cardwell, M. Murakami, Physica C 426-431, 618 (2005). [27] B. Hensel, G. Grasso, R. Flükiger, Phys. Rev. B 51, 15456 (1995). [28] M. Tomita and M. Murakami, Nature 421, 517 (2003). [29] M. Murakami, in: Melt Processed High Temperature Superconductors, ed. M. Murakami (World Scientific, Singapore, 1993). [30] A. Koblischka-Veneva, M. R. Koblischka, N. Hari Babu, D. A. Cardwell, L. Shlyk, and G. Krabbes, presented at 5th PASREG workshop, Oct. 21.-23. 2005, Tokyo, Supercond. Sci. Technol. 19, S562 (2006).

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[31] A. Koblischka-Veneva, M. R. Koblischka, K. Ogasawara, F. Mücklich, and M. Murakami, J. Superconductivity 18, 469 (2005); A. Koblischka-Veneva and M. R. Koblischka, J. Low Temp. Phys. 131, 635 (2003); K. Ogasawara, N. Sakai, M. R. Koblischka, A. Koblischka-Veneva, and M. Murakami, Supercond. Sci. Technol. 17, S61 (2004). [32] N. Vilalta, F. Sandiumenge, S. Pinol, and X. Obradors, J. Mater. Sci. 12, 38 (1997). [33] J. L. Macmanus-Driscoll, S. R. Foltyn, Q. X. Jia, H. Wang, A. Serquis, L. Civale, B. Maiorov, M.E. Hawley, M. P. Maley, and D. E. Peterson, Nature Materials 3, 439 (2004). [34] T. Haugan, P. N. Barnes, R. Wheeler, F. Meisenkothen, and M. Sumption, Nature 430, 867 (2004). [35] N. Takezawa and K. Fukushima, Physica C 290, 31 (1997). [36] N. Hari Babu, E. Sudhakar Reddy, D. A. Cardwell, and A. M. Campbell, Supercond. Sci. Technol. 16, L44 (2003). [37] N. Hari Babu, E. S. Reddy, D. A. Cardwell, A. M. Campbell, C. D. Tarrant, and K. R. Schneider, Appl. Phys. Lett. 83, 4806 (2003). [38] A. Koblischka-Veneva, F. Mücklich, M. R. Koblischka, N. HariBabu, and D. A. Cardwell, presented at EUCAS 2005, Vienna, J. Phys.: Conf. Ser. 43, 438 (2006). [39] JCPDS powder diffraction, Alphabetical Indexes (1997) file no. 480120 [40] A. Koblischka-Veneva, C. Holzapfel, F. Mücklich, M. R. Koblischka, N. Hari Babu, D. A. Cardwell, presented at EUCAS 2005, Vienna, J. Phys.: Conf. Ser. 43, 527 (2006). [41] A. Koblischka-Veneva, M. R. Koblischka, N. Hari Babu, D. A. Cardwell, F. Mücklich, Physica C 445-448, 379 (2006). [42] A. Koblischka-Veneva, M. R. Koblischka, presented at the 6th PASREG meeting, Cambridge, Mat. Sci. Eng. B 151, 65 (2008). [43] M. R. Koblischka, A. Koblischka-Veneva, presented at the 6th PASREG meeting, Cambridge, Mat. Sci. Eng. B 151, 47 (2008). [44] A. Koblischka-Veneva, F. Mücklich, M. R. Koblischka, N. Hari Babu, and D. A. Cardwell, J. Am. Ceram. Soc. 90, 2582 (2007). [45] A. Koblischka-Veneva, M. R. Koblischka, N. Hari Babu, D. A. Cardwell, F. Mücklich, Microscopy and Microanalyis 13 (Suppl. 3), 360 (2007).

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In: Superconducting Cuprates Editor: Koenraad N. Courtlandt

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Chapter 12

EFFECT AND EVOLUTION OF THE PSEUDOGAP IN Y1-XCaXBa2Cu3O7-δ: PROBED BY CHARGE TRANSPORT, MAGNETIC SUSCEPTIBILITY AND CRITICAL CURRENT DENSITY MEASUREMENTS S. H. Naqib∗ and R. S. Islam Department of Physics, University of Rajshahi, Raj-6205, Bangladesh

Abstract

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In recent years Ca-substituted Y123 (Y1-xCaxBa2Cu3O7-δ) has proven to be an extremely useful compound both from the point of view of fundamental research and applications. The major advantage of Y1-xCaxBa2Cu3O7-δ over Y123 is the fact that substitution of Ca2+ for Y3+ makes the deeply overdoped side experimentally accessible. The methods of synthesis, characterization, and annealing of sintered polycrystalline and c-axis oriented crystalline pulsed laser deposited (PLD) thin films of Y1-xCaxBa2Cu3O7-δ are described in the present report. Almost all the normal and superconducting state properties of Y1-xCaxBa2Cu3O7-δ were found to be strongly dependent on the hole concentration in the CuO2 plane, p. This can be primarily attributed due to the strongly p-dependent behavior of the pseudogap in the quasiparticle spectral density for samples with p < 0.19. We have investigated in detail the effect and evolution of the pseudogap from the resistivity, ρ(T), static (q = 0) magnetic susceptibility, χ(T), and zero-field critical current density, Jc0(T), measurements for a series of Y1-xCaxBa2Cu3O7-δ compounds with different values of p. Hole concentration was varied by changing both Ca (x) and oxygen deficiency (δ). The essential characteristics of ρ(T), χ(T), and Jc0(T) were found to be dominated by the evolution of the pseudogap with hole content. From the analysis of the experimental data, we have found clear indications that the pseudogap vanishes quite abruptly at p ~ 0.19. At the same hole content, ρ(T) becomes completely linear, χ(T) becomes T-independent in the normal state, and low-T Jc0 is maximized. We also report a (non-magnetic) Ca induced Curie enhancement in the static magnetic susceptibility in Y1-xCaxBa2Cu3O7-δ in this article.

∗

Corresponding author. E-mail: [email protected]

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1. Introduction The complete understanding of the physics of cuprate superconductivity and the various normal state properties remains one of the fundamental issues in condensed matter physics, still after more than two decades of their discovery. Almost all the normal and superconducting (SC) state properties of high-Tc cuprates are strongly dependent on the level of hole doping. Therefore, investigation of these doping dependences is of paramount importance, both for the understanding of the basic physics and exploring the possibilities for future applications. The high-temperature SC compound YBa2Cu3O7-δ (Y123) remains the most extensively studied among all the cuprates. Nevertheless, there is a material specific problem associated with this system, namely, the overdoped (OD) side of the phase diagram cannot be accessed. Pure Y123 with full oxygenation (δ = 0), is slightly OD (with hole concentration per CuO2 plane, p ~ 0.18, whereas superconductivity is expected up to p ~ 0.27). Further overdoping can be achieved by substituting the trivalent Y3+ by a suitable divalent atom, namely Ca2+, which adds hole carriers in the CuO2 plane irrespective of the state of oxygenation in the CuO1-δ chains [1, 2]. 20%Ca substitution plus full oxygenation raises the hole content to p ~ 0.235, significantly higher than 0.18 that can be achieved in pure fully oxygenated Y123 [1, 2]. Thus the compound Y1-xCaxBa2Cu3O7-δ (Ca-Y123) enables one to study various properties in the normal and the SC states over a much wider doping range. In this article, we report detailed results for three important bulk properties of Ca-Y123: (i) the temperature dependence of the resistivity, ρ(T), (ii) the T-dependent static (q = 0, where q is the wave-vector) magnetic susceptibility, χ(T), and the (iii) T-dependent zero-magnetic field critical current density, Jc0(T), over a wide range of hole concentrations. All the results shown here came from the measurements on high-quality c-axis oriented crystalline thin films and high-quality sintered polycrystalline Y1-xCaxBa2Cu3O7-δ samples. Although charge transport and magnetic studies on single crystals give more quantitative information (e.g., regarding the anisotropy and c-axis charge dynamics), it is extremely difficult to grow good quality Ca-Y123 single crystals and to control their stoichiometry and homogeneity. We have found oxygenation and de-oxygenation of flux-grown single crystals particularly difficult. For the epitaxial thin films the in-plane resistivity can be measured with high degree of accuracy. In the sintered compounds transport properties are mainly dominated by the (comparatively) highly conducting ab-plane. Oxygen content can also be varied over a wide range fairly easily and homogeneously for sintered and thin film samples. From the analysis of ρ(T), χ(T), and Jc0(T) data, it was found that all these fundamental bulk properties are highly dependent on the hole concentration in the normal (in case of Jc0(T), in the SC) state. These strong pdependences can be largely ascribed to the existence and extent of the pseudogap (PG) in the quasi-particle (QP) energy density of states near the chemical potential [3]. The characteristic PG energy scale, Eg, decreases almost linearly with increasing p and goes to zero at p = 0.19 ± 0.01. At this precise doping ρ(T) becomes linear over the entire temperature range (if strong SC fluctuations are disregarded near Tc), χ(T) becomes T-independent (again except in the vicinity of Tc, due to diamagnetic fluctuations), and Jc0(T) is maximized.

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2. Experimental samples and their characterizations

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(a) Polycrystalline Y1-xCaxBa2Cu3O7-δ: Synthesis Sintered samples of Y1-xCaxBa2Cu3O7-δ were prepared following the method described in refs. [1, 2, 4]. These samples were also used as target materials for the thin films grown by the pulsed laser ablation process. Samples were synthesized using the following powdered chemicals supplied by SIGMA-ALDRICH (purity is given in brackets): Y2O3 (99.999%), BaCO3 (99.999%), CaCO3 (99.999%), CuO (99.9999%), and ZnO (99.999%). All these chemicals were dried in alumina (Al2O3) crucibles in a furnace at 500°C in air (CaCO3 in a separate furnace at 450°C) for 12 hours. Weight losses (in brackets) after drying were: Y2O3 (0.52%), BaCO3 (0.365%), CaCO3 (1.01%), CuO (0.21%) and ZnO (0.23%). Before this drying, all the crucibles were cleaned and heated for 6 hours in air at 1000oC to eliminate possible contamination of the powdered chemicals through any reaction with unwanted residue left in them. The dried powders were weighed in the proper stoichiometric ratios and ground together in an agate mortar with cyclohexane (99.94%) to help a homogeneous mixing of the powders. These well-ground powdered mixtures were then put into a box furnace at 150oC for further drying. The loose powdered mixtures were first calcined at 900°C in air for 12 hours (heating/cooling rate = 300°C/hour). This procedure was repeated thrice (with intermediate grindings at each step) but at temperatures 910oC, 920oC, and 935oC respectively. Pellets were formed using a pressure of 7 tons/cm2 and the 935oC (in air) run was repeated once more. After each run we have noted the mass change. Significant weight losses were observed after the first sintering of the pellets at 935oC, all our samples lost ~ 12% of their initial masses. This we believe was due to nearly the complete loss of unwanted CO2. Sintering runs above 935oC were performed in high-purity (5-Grade) O2 at a pressure of 1 bar. The sintering temperatures were increased stepwise (with intermediate grindings) by 10oC from 960oC to 980oC (kept 12 hours at each temperature). Samples were furnace-cooled to room-temperature after each sintering in O2. We have used a gas purification system during all the sintering runs which removed possible presence of moisture and traces of CO2 in the gases used. The evolution of various phases in the sample with different calcination conditions were tracked by X-ray diffraction (XRD) on both powders and pellets. After the 980oC sintering in O2, all of the samples appeared almost completely phase-pure, few of them showed minute traces of impurity phases at levels < 1.5% of XRD count. Typical XRD spectra are shown in Figs. 1. The X-ray spectra are plotted as graphs of diffracted X-ray intensity (in square-root scale, which enhances lines of low intensity) versus 2θ, where θ is the angle of incidence (in degrees). Most of the graphs show results in the 2θ-range from 20o to 55o, as all the prominent diffraction peaks for Y123 (and for Y1-xCaxBa2Cu3O7-δ) and possible impurity phases lie within this range. Figs. 1a and 1b show the XRD profiles of Y0.95Ca0.05Ba2Cu3O7-δ both in the sintered pellet and finely ground powdered forms respectively. Similar representative XRD patterns for other Y1-xCaxBa2Cu3O7-δ sintered compounds are shown in Figs. 1c and 1d. As can be seen from these XRD profiles, Y0.95Ca0.05Ba2Cu3O7-δ is phase-pure. There is no trace of the most commonly occurring impurities, e.g., BaCuO2, Ba4CaCu3O8 (both arise due to unwanted partial substitution of Ca for Ba), and Y211 (the so-called green phase, Y2BaCuO5). Another interesting feature is the

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enhancement of (00l) peaks (compared to the powder) in the sintered pellet, indicative of some degree of c-axis alignment present in this case. 100

(013)/ (103)

Intensity (%)

80

60

40 (003)

(006)/ (020)

(005)/ (104) (113)

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(016)/ (115) (106)

(114)

35

40

45

50

55

2θ Figure 1a. XRD profile of Y0.95Ca0.05Ba2Cu3O7-δ powder, the plane indices (hkl) are shown.

(003)

80 Intensity (%)

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100

60 (005)

40 (006)

20 (004)

0

20

25

30

(007)

35

40

45

50

2θ Figure 1b. XRD profile of Y0.95Ca0.05Ba2Cu3O7-δ pellet, the enhanced (00l) peaks are shown. Superconducting Cuprates : Properties, Preparation and Applications, Nova Science Publishers, Incorporated, 2008. ProQuest Ebook Central,

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Effect and Evolution of the Pseudogap in Y1-xCaxBa2Cu3O7-δ

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Intensity (%)

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25

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2θ Figure 1c. XRD profile of Y0.90Ca0.10Ba2Cu3O7-δ pellet.

80 Intensity (%)

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100

60 40 20 0 20

25

30

35

50

2θ Figure 1d. XRD profile of powdered Y0.80Ca0.20Ba2Cu3O7-δ.

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Figs. 2 show the effect of sintering temperatures on the phase-purity of the samples. The impurity phases present decrease with increasing sintering temperature and finally vanishes (i.e., below the instrumental resolution of the XRD machine) when sintered at 980oC in oxygen (see Fig. 1a). 100

Intensity (%)

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*X 0 20

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2θ Figure 2a. XRD of Y0.95Ca0.05Ba2Cu3O7-δ powder sintered in air at 935oC. Impurity peaks are marked in figure (*: BaCuO2 [7%] and X: Ba4CaCu3O8+x [5%]). Intensities of these impurity peaks are given in the square brackets. 100

Intensity (%)

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80 60 40 20 0 20

*X 25

30

35 40 2θ

45

50

55

Figure 2b. XRD of Y0.95Ca0.05Ba2Cu3O7-δ powder sintered in oxygen at 960oC. Impurity peaks are marked in figure (*: BaCuO2 [3%] and X: Ba4CaCu3O8+x [4%]). Intensities of these impurity peaks are given in the square brackets.

Oxygen stoichiometry The details of the various annealing conditions applied to the sintered samples and their outcomes are described in this section. Oxygen contents were varied by quenching the Y1-xCaxBa2Cu3O7-δ samples from higher temperatures under different oxygen partial pressures into liquid nitrogen (LN2). A standard sample of mass ~ 1gm was used to detect the changes

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in weight after each annealing. Table 1 describes the annealing conditions and the respective changes in the oxygen stoichiometry for Y0.80Ca0.20Ba2Cu3O7-δ sintered pellets. All the annealings described in Table 1 were done following an earlier work by Loram et al. [5]. Weight changes, and hence the oxygen deficiencies, found in this study agree very well with the previous study [5]. Table 1. Annealing and oxygen stoichiometry of Y0.80Ca0.20Ba2Cu3O7-δ. Annealing Identity AR Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10 Q11

Description As received 300oC in O2 for 96 hours 450oC in O2 for 72 hours 520oC in O2 for 48 hours 543oC in O2 for 36 hours 565oC in O2 for 36 hours 590oC in O2 for 24 hours 615oC in O2 for 24 hours 640oC in O2 for 18 hours 670oC in O2 for 18 hours 690oC in O2 for 12 hours 550oC in 1%O2 + 99%N2 for 24 hours

Oxygen stoichiometry (δ) [± 0.02] 0.00 [6] 0.055 0.123 0.196 0.235 0.283 0.321 0.366 0.411 0.444 0.465 0.505

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Table 2 shows the annealing conditions and the oxygen stoichiometry of Y0.90Ca0.10Ba2Cu3O7-δ. For the 5%Ca-Y123 compound the same treatments were employed with an additional initial annealing (A0) after the final sintering at 980oC in oxygen. The condition for this annealing was as follows-

Figure 3. First annealing of Y0.95Ca0.05Ba2Cu3O7-δ pellets.

where the high-temperature (700oC) annealing was for the uniform oxygenation inside the bulk and the low-temperature (175oC) one was done to oxygenate the grain boundaries. Table 3 gives the oxygen deficiencies for the Y0.95Ca0.05Ba2Cu3O7-δ.

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S. H. Naqib and R. S. Islam Table 2. Annealing and oxygen stoichiometry of Y0.90Ca0.10Ba2Cu3O7-δ.

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Table 3. Annealing and oxygen stoichiometry of Y0.95Ca0.05Ba2Cu3O7-δ.

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The pure Y123 (Ca-free) sintered sample used in this study came from Dr. J. R. Cooper†. The details of annealings for this sample can be found elsewhere [7]. It is worth noting in Tables 1-3 that a higher annealing temperature is required for the most Ca deficient sample (5%Ca-Y123) to get the same oxygen deficiency (δ) as that of 10%Ca-Y123. This annealing temperature for 5%Ca-Y123 has to be even higher for identical δ-values of 20%Ca-Y123. Thus, annealing samples with different amount of Ca at the same temperature masks, to some extent, the effect of Ca on the carrier concentration due to unequal oxygen deficiencies. The general rule is, higher the carrier concentration, p, (higher for samples with larger amount of Ca, when δ is constant), oxygen loss takes place at a lower temperature. This important fact, we believe, has not been recognized by some earlier works [8 - 10] where questions have been raised regarding the role played by Ca in raising the overall hole concentration in Y1-xCaxBa2Cu3O7-δ compounds.

Hole concentration, p Number of doped holes per CuO2 plane in Y1-xCaxBa2Cu3O7-δ was calculated from the roomtemperature thermopower, S[290K], following the relations obtained by Obertelli et al. and Tallon et al. [11, 12]. For polycrystalline samples, the measured room-temperature thermopower is representative of the ab-plane property. S[290K] is almost completely independent of the quality of the grain boundaries and other possible disorders in the sample. These makes S[290K] a very reliable estimator of p. We have obtained p directly from S[290K] using the following relations [11, 12]

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S [ 290 K ] = − 139 p + 24 . 2 S [ 290 K ] = 992 exp( − 38 . 1 p )

for p > 0.155 for 0.05 < p < 0.155

(1a) (1b)

Almost identical p-values were also obtained from the parabolic Tc(p) relation valid for a number of HTS cuprates [13], given by Tc(p) = Tcmax[1 - 82.6(p - popt)2]

(2)

Where Tcmax is the maximum superconducting transition temperature and popt = 0.16 is the optimum hole content. Fig. 4 shows the normalized superconducting transition temperature, Tc(p)/Tcmax, versus p. Tc was determined from the resistivity, ρ(T), measurements. Tc was defined at zero resistance (within the instrumental noise level of ± 10-6 Ω). We have also obtained Tc from the low field (Hrms = 0.1 Oe; f = 333.3 Hz) ac susceptibility (ACS) measurements as follows: a line was drawn on the steepest part of the diamagnetic ACS curve and another one as the almost T-independent base line associated with negligibly small normal state (NS) signal. The intercept of the two lines gave Tc. Tc values obtained by these two methods agree quite well within ± 1K for all the experimental samples.

†

Dr. J. R. Cooper, Shoenberg Laboratory for Quantum Matter, Department of Physics, University of Cambridge, J. J. Thomson Avenue, Cambridge CB3 0HE, United Kingdom.

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1

x = 0.05 x = 0.10 x = 0.20

T c/T cmax

0.9

0.8

0.7

0.6 0.05

0.1

0.15

0.2

0.25

p Figure 4. Tc(p)/Tcmax versus p for Y1-xCaxBa2Cu3O7-δ compounds. Circles give p-values obtained from S[290K] and the dashed line gives the parabolic Tc(p) relation [13].

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(b) Crystalline epitaxial Y1-xCaxBa2Cu3O7-δ thin films: Synthesis and characterization c-axis oriented thin films of Y1-xCaxBa2Cu3O7-δ (with x = 0.0, 0.05, 0.10, and 0.20) were fabricated using the method of pulsed LASER deposition (PLD). The basic principle of the laser deposition technique is as following. The high-energy beam from a pulsed laser source strikes a target of the desired composition and evaporates a thin surface layer. Since this is an evaporative process, the target layer comes off perpendicular to the surface and its direction does not depend on the angle of the incident laser beam. The layer is ejected with high kinetic energy and is subsequently re-condensed on a suitable substrate positioned a few centimeters away opposite and parallel to the target surface. Before the discovery of cuprate superconductors, laser ablation was a successful method for the deposition of a wide variety of solid films, e.g., elemental semiconductors, silicon carbides, polymers, and various oxide materials [14]. Laser ablation was first used to deposit Y123 immediately after its discovery in 1987 [15] and initially was an ex situ technique, i.e., the sample required high temperature post-deposition annealing in air to form a crystalline structure. Presently, laser ablation is established as a highly successful in situ deposition technique. The main difficulty that must be overcome to grow good Y123 film is the need to achieve the correct film cation stoichiometry. Laser ablation has the advantage that the target stoichiometry is transferred very accurately to the substrate and it is not necessary to make the target rich in any one element, as is sometimes the case in sputtering technique [16]. In addition, the highly excited

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nature of the laser ablated "plume" enables high background oxygen pressures to be used during the deposition, ensuring a high level of oxygenation of the as-grown films. A schematic diagram for a PLD system is shown in Fig. 5. All the films were grown on 10 x 5 x 1mm3 SrTiO3 substrates, highly polished on one side. The crystallographic orientation of the substrate was (100) (SrTiO3 is a cubic system with a lattice constant of 3.905 Å). Highdensity targets of Y1-xCaxBa2Cu3O7-δ (x = 0.05, 0.10, and 0.20) were used for PLD. To grow pure (Ca-free) Y123, we have used a commercially available polycrystalline Y123 as the target material.

Figure 5. Schematic diagram of the laser ablation chamber.

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The films were fabricated using a Lambda Physik LPX210 KrF laser with a wavelength of 248 nm. The laser intensity inside the chamber was measured before each PLD run and was adjusted to the desired value using focusing/attenuating systems. A pulse rate of 10 Hz was used during thin film deposition. Prior to the deposition, the substrate heater block was fully out-gassed by heating it to the deposition temperature with a turbo-pump running until the chamber pressure reached ~ 10-5 mbar. The deposition temperature, Tds, was measured by a thermocouple fixed to the heater block. The selected target-substrate distance depended on the oxygen partial pressure, PO2, and on the incident laser energy, as these two were the main factors determining the geometry of the plume. Tds was varied within the range from 740oC to 820oC, whereas the oxygen partial pressure lied within the range from 0.70 mbar to 1.20 mbar during deposition. The best thin-film samples are grown at deposition temperatures in the range 800oC to 820oC with oxygen pressures in the range 0.95 to 1.20 mbar. The X-ray (diffracted intensity versus 2θ ) result for a representative Y1-xCaxBa2Cu3O7-δ film is shown in Fig. 6. Result from the rocking-curve (Ω-scan) analysis is shown in the inset. This is often interpreted as a measure of the c-axis orientation of the film. We have used the (007) diffraction peak for rocking-curve analysis, following [14]. The expected XRD pattern from a c-axis oriented film would be a series of (00l) reflections. Very clear and sharp sets of (00l) peaks were observed for all the films used in the present study with no sign of any impurity phase. Most of the films show very narrow (007) peak indicative of a very high degree of c-axis orientation. These peaks have double structures, arising from the CuKα2 component of the incident X-rays. The hole concentrations of the films were obtained from the Tc(p) and S[290K], as before. Information regarding the oxygen deficiency was obtained from the c-axis lattice parameters [17, 18] after different oxygen annealings. Identical annealings yield almost the same value of δ for films and sintered Y1-xCaxBa2Cu3O7-δ. Thickness and surface features of the films were studied using atomic force microscopy (AFM), details of which can be found elsewhere [19]. Thickness of all the films lie within (2800 ± 500) Å. Films grown under optimal conditions exhibited high surface quality. We have found the maximum Tc to be consistently lowered by ~ 2K for the films compared with the Tc for sintered samples at the same hole concentration. A possible reason for this might be the non-uniform epitaxial strain due to the lattice mismatch between the target material and the substrate. Alternatively it could be caused by in-plane atomic disorder in the films. According to a proposed universal relation between Tc and residual resistivity for cuprate superconductors [20], a shift of 2K in Tc corresponds approximately to an intra-grain residual resistivity of only 4 μΩ cm, and in our work this is not detectable relative to the larger grain boundary contribution (see Figs. 10).

3. Resistivity measurements and results (a) Sintered samples Resistivity measurements were carried out in a standard Oxford Instruments continuous flow cryostat (CFC). Cooling runs at 4K/minute were used followed by warming runs at 1K/minute to obtain accurate values of Tc of the samples. A temperature difference, ΔT, of less than 0.5K was observed between these warming and cooling runs for all of the samples.

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Figure 6. XRD profile of c-axis oriented Y0.95Ca0.05Ba2Cu3O7-δ thin film. Inset shows the result of rocking-curve analysis.

High density (5.7 to 6.0 gm/cm3, i.e., 89% to 94% of X-ray density) sintered bars (length 6-9 mm and cross-sectional area 3-6 mm2) were used for ρ(T) measurements. Resistivity was measured using the conventional four-terminal method with an injected ac current of 1 mA (77 Hz) using copper wires (40 micron diameter) and room temperature silver paste to make the contacts. Current contacts were placed over the whole area of the end sides of the bar to ensure homogeneous current flow and the voltage contacts were made narrow to minimize error in measuring the distance between them. Typical contact resistances were below 2Ω. The signal was detected by a lock-in amplifier (LIA) (EG&G PRINCETON APPLIED RESEARCH MODEL 5210) after it had been filtered through a low noise transformer. Representative ρ(T) data for sintered Y0.80Ca0.20Ba2Cu3O7-δ and Y0.90Ca0.10Ba2Cu3O7-δ compounds are shown in Figs. 7 for different hole concentrations.

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Figure 7a. Resistivity of sintered Y0.80Ca0.20Ba2Cu3O7-δ. Hole contents (within ± 0.004) are shown in the plot.

Figure 7b. Resistivity of sintered Y0.90Ca0.10Ba2Cu3O7-δ. Hole contents (within ± 0.004) are shown in the plot.

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Figs. 7 show the systematic changes in ρ(T) with p. ρ(T) for samples with p > 0.19 gradually develops an increasingly positive curvature as T decreases whereas a negative curvature grows with decreasing p for the UD compounds. The slope of ρ(T) increases systematically as p is reduced. The residual resistivity also increases monotonically with underdoping. The evolution of ρ(T) with p provides a way of establishing the T-p phase diagram of high-Tc superconductors and can give measures of T*(p), the characteristic pseudogap temperature. Conventionally T* is defined as the temperature at which ρ(T) starts to decrease at a faster rate than from its high-T linear behavior [21]. One particular problem associated with this method is that near the vicinity of the optimum doping level and above, T*(p) is quite close to Tc(p) and the fall of ρ(T) due to strong SC fluctuation masks any possible gradual reduction in ρ(T) due to the presence of the PG as observed at higher-T in the UD compounds. The proposed scenarios to explain the origin and the p-dependence of the pseudogap could be classified mainly into two categories [3, 22, 23]. The first one is based on the precursor pairing scenario, where the pseudogap is derived from strong fluctuations of SC origin for systems with low dimensionality and low superfluid density. In the second scenario, the pseudogap is attributed to correlations of some other (non-superconducting) type, which coexists (i.e., T*(p) exists below Tc(p)) and often competes with superconductivity. The experimental situation is often thought to be rather inconclusive regarding the origin of the pseudogap [3, 22 - 24]. The close proximity between T* and Tc near and above the optimum hole concentration does not pose any problem for theories belonging to the first group where T*(p) itself is essentially derived from strong superconducting fluctuations and T*(p) merges with the Tc(p) in the OD region (p ≥ 0.21) where the pair formation temperature (T*) is essentially the same as the phase coherence temperature (the temperature at which there is long-range order and the superconducting transition takes place in the thermodynamical sense, i.e., Tc). For the second scenario, the closeness of T* and Tc poses a serious problem as superconducting pairing fluctuations (and superconductivity itself) mask the signatures of the predicted pseudogap in the vicinity of (and below) Tc. In this study two methods have been employed to determine T*. As can be seen from Fig. 8a, representative plots of dρ(T)/dT versus T and [ρ(T) - ρLF] versus T yield very similar T* values (within ± 5K). Here ρLF is a linear fit of the form ρLF = b + cT, in the high-temperature region of the ρ(T). Also using T*/T as a scaling parameter it is possible to normalize ρ(T). The result of such scaling is shown below in Fig. 8b, where, leaving aside the superconducting transitions, all resistivity curves collapse onto one p-independent universal curve over a wide temperature range, signifying the same dominant electronic scattering dynamics at play, characterized by the presence and extent of the PG energy. Also using T*/T as a scaling parameter it is possible to normalize ρ(T). The result of such scaling is shown below in Fig. 8b, where, leaving aside the superconducting transitions, all resistivity curves collapse onto one p-independent universal curve over a wide temperature range, signifying the same dominant electronic scattering dynamics at play, characterized by the presence and extent of the PG energy.

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Figure 8a. Methods employed to determine T* for Y0.80Ca0.20Ba2Cu3O7-δ (p = 0.115) (see text for details).

Figure 8b. Normalized resistivity, with T*(p) as scaling parameter for Y0.80Ca0.20Ba2Cu3O7-δ. p-values (within ± 0.004) are shown in the plot. ρ(0) is the residual resistivity. Superconducting Cuprates : Properties, Preparation and Applications, Nova Science Publishers, Incorporated, 2008. ProQuest Ebook Central,

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It also shows that the extent of the PG (T*) sets the dominant energy scale for low-lying QP excitations in Ca-Y123 for compounds with p < 0.18. It is worth mentioning that the departure form the scaling invariably starts at a temperature ~ 20 ± 5K above Tc. At this temperature strong SC fluctuations are expected due to short-lived Cooper pairs [25]. The validity of the scaling at higher temperature and not below ~ 20K above Tc points towards different origin for the gradual downturn in the resistivity at T* and that in the vicinity of Tc. T*(p) obtained from such scaling agrees quite well with those obtained by using the other two methods (shown in Fig. 8a). It is interesting to note that, since ρ(T) of samples with p > 0.19 develops increasingly positive curvatures with increasing overdoping (Fig. 7), a scaling with any kind of characteristic temperature will deviate from the universal scaling shown in Fig. 8b. This could be an indication of a fundamental change in the electronic ground states of cuprates in the overdoped side (p > 0.19), namely, the absence of a pseudogap in the electronic density of states.

(b) Epitaxial thin films

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The ab-plane resistivity, ρab(T), measurements were made on patterned thin films of c-axis oriented Y1-xCaxBa2Cu3O7-δ. Fig. 9 illustrates the geometry of a patterned film. The pattern consisted of a bar of the thin-film sample with dimensions 7.0 x 0.5 mm2. There were six contact pads: four voltage contacts and two current contacts. The width of the constrictions between the bar and the voltage contact pads was ~ 0.05 mm. The distance between the constrictions was 3 mm.

Figure 9. A schematic diagram showing the geometry of a patterned Y1-xCaxBa2Cu3O7-δ film.

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Gold was evaporated onto the contact pads of the films. This gives low contact resistance. Resistivity was measured employing the four-terminal configuration as used for the sintered compounds described previously. Hole content for Y1-xCaxBa2Cu3O7-δ thin films were varied by oxygen annealing similar to the ones used for the polycrystalline samples.

Figure 10. In-plane resistivity of Y1-xCaxBa2Cu3O7-δ thin films. Ca contents and p-values (within ± 0.004) are given.

Fig. 10 shows some representative ρab(T) data for Y1-xCaxBa2Cu3O7-δ with different values of hole and Ca contents. For a given value of p, T* is independent of the crystalline state of the samples. This is shown in Fig. 11, which also demonstrates that for sintered samples, charge transport is mainly governed by the ab-plane charge scattering mechanisms. At this point it is worth noting that the systematic evolution (super-linear → linear → sublinear) of the ρ(T, p) with decreasing hole concentration has an electronic origin since the phonon spectrum does not vary significantly with p in cuprate superconductors. This may also indicate that phonons are not strongly influencing the charge scattering processes and therefore superconductivity in cuprates originates mainly from non-phonon mechanisms.

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Figure 11. Resistivity and T* of sintered and c-axis oriented Y0.95Ca0.05Ba2Cu3O7-δ compounds for p = 0.155 ± 0.003.

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4. Static magnetic susceptibility measurements and results The possible effects of Ca in Y123 other than changing the hole content in CuO2 plane have not yet been explored in details. χ(T) of Y1-xCaxBa2Cu3O7-δ provides a way to study the effect of Ca, if any, on the magnetization over a wide range of p. The most important information that can be extracted from the analysis of χ(T, p) is about the temperature and p dependences of low-energy electronic density of states (EDOS) for Y1-xCaxBa2Cu3O7-δ. As EDOS as a function of energy is at the centre of any problem associated with PG, χ(T, p) is indeed a powerful experimental probe to study various p-dependent features of this phenomenon. It is in fact the intrinsic spin part (the Pauli spin susceptibility, χspin) of χ(T, p) that represents the QP spectral density near the Fermi- level. Pauli spin susceptibility arises from the coupling of intrinsic spins of the mobile carriers with the applied magnetic field and, for ordinary Fermiliquids, can be expressed as

χ spin = μ B 2 N (ε F )

(3)

where, μB is the Bohr magneton and N(εF) is the EDOS at the Fermi-energy. In cuprates, where strong electronic correlations are present, χspin at a particular temperature T still approximately represents the thermal average value of EDOS near the Fermi-level over an energy width of ~ εF ± 2kBT [26]. χ(T) measurements on sintered Y1-xCaxBa2Cu3O7-δ were performed using Quantum Design SQUID magnetometers. During the measurements the samples were mounted

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between two quartz tubes of similar dimensions. This whole configuration was then attached to a sample probe rod. The tubes were cleaned before each measurement to avoid contamination by any magnetic particles. Data were collected, usually in the range of 5K to 400K, using a dc magnetic field of 5 Tesla. A scan length of 6cm was used. Sometimes plastic straws were used as sample holder. The background signals were subtracted from the data to obtain the magnetic moment of the sample, from which the molar susceptibility would be obtained. Fig. 12 shows the background moment from a piece of non-magnetic plastic straw and the uncorrected moment from one of the samples.

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Figure 12. Background moment from plastic straw compared to that from a Y0.80Ca0.20Ba2Cu3O7-δ compound (p = 0.215 ± 0.004), at 5 Tesla.

The uniform magnetic susceptibility data consists of a number of contributions coming from different origins. It is therefore important that we are able to separate these different contributions from each other so that only the terms directly related to the PG (i.e., to the QP excitation spectrum) and Ca substitution can be studied. Considering all these contributions to the uniform susceptibility, we can express χ(T) as follows [27, 28]

χ ( T ) = χ spin (T ) + χ Curie (T ) + χ core + χ VV + χ imp (T )

(4)

where the Larmor or the core susceptibility and the Van Vleck (VV) susceptibility are expected to be p- and T-independent over the experimental temperature range [27, 28]. χCurie(T) is the paramagnetic Curie susceptibility. We have ignored Landau diamagnetic susceptibility, which has the same T-dependence as χspin(T), but is only from 2% to 5% of the χspin(T) in magnitude [27]. χimp(T) is the contribution from possible impurity phases present in the compounds. Figs. 13 show the experimental χ(T) and χ(T)T data for Y1-xCaxBa2Cu3O7-δ compounds and readily illustrate two important points. The first one, related to the existence of the pseudogap, is the fact that both χ(T) and χ(T)T becomes strongly p-dependent only for p < 0.19 (± 0.005). The second important finding is the fact that Ca modifies the normal state magnetic properties of Y1-xCaxBa2Cu3O7-δ compounds quite drastically. This is seen from the

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Effect and Evolution of the Pseudogap in Y1-xCaxBa2Cu3O7-δ

progressive growth of Curie-like behavior in χ(T) with increasing Ca contents. χ(T) for Casubstituted Y123 shows features similar to Co- or Ni-substituted Y123 [7]. This is quite surprising, because unlike Co and Ni, Ca is non-magnetic. In fact Ca2+ has a full shell electronic configuration (3p6), and no Curie-like contribution to the magnetization is expected. It is also seen from Figs. 12 that χ(T)T deviates from linearity at lower temperatures above Tc even for samples with no pseudogap (p > ~ 0.19). Namely χ(T)T rises above the high-temperature linear behavior for the Ca-substituted overdoped samples, implying the presence of a Curie-Weiss-like contribution in these compounds. This is due to the presence of small quantity of an unwanted p-independent magnetic impurity phase in these compounds, namely BaCuO2+z. A detailed analysis of the contribution from the impurity phases to magnetic susceptibility can be found elsewhere [29]. We have analyzed the effect of Ca on χ(T) using the following procedure. For the high-T region where χ(T)T is linear, one can express,

χ ( T )T = χ h T + C

(5a)

where χh is the susceptibility of the host material (Ca-free Y123 in this case) and C, the molar Curie constant, is given by,

N A n μ B p eff 2

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C =

2

3k B

(5b)

with n is the concentration of Ca, NA is the Avogadro’s constant, and peff denotes the effective Bohr magneton number. Considering the form of eqn. (5a), one can obtain C from the extrapolated values of high-temperature linear fits to χ(T)T at T = 0K. When C-values are plotted against the respective Ca concentrations for these Y1-xCaxBa2Cu3O7-δ samples, we find C(x) increases systematically with increasing Ca content, as shown in Fig. 14. As C(x) is nonlinear, the slope of this trend gives C/n, at a particular Ca content and can be used to determine peff2/Ca (giving the Ca-induced magnetic moment in the units of μB2 at that particular Ca content) from eqn. (5b). Using the data point for x = 0.20 shown in Fig. 14, we obtain peff2/Ca = 1.93 (in the units of μB2). Therefore, a quite significant Curie-like magnetic contribution comes from Ca substitution in Y123. In the above analysis of χ(T)T data we have not taken into account the unequal oxygen contents of these Y1-xCaxBa2Cu3O7-δ compounds for the given value of p. This does not affect the calculated values of peff2/Ca significantly. For example, for almost fully oxygenated 5%Ca, 10%, and 20%Ca-Y123, one get peff2/Ca = 1.85 ± 0.20 (in the units of μB2). It is worth noting that for these nearly fully (δ-values in the range 0.02 to 0.05) oxygenated compounds, p-values are unequal (in the range 0.227 to 0.206, due to the different amount of Ca). But this also does not affect the value of peff2/Ca in any appreciable way since the high-T χ(T)T is almost p-independent for p > 0.19 (see Fig. 13). The values of peff2/Ca in the UD side are also similar to those for the OD samples. For example, peff2/Ca turns out to be 1.98 (in the units of μB2 for the x = 0.20 sample) when p = 0.145 ± 0.005. Hence the Curie-like contribution due to Ca is fairly insensitive to p and the oxygen contents in the CuO1-δ chains.

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Figure 13. χ(T) and χ(T)T for Y1-xCaxBa2Cu3O7-δ. p-values are given.

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C (10-4 emu-K/mole)

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361

400

200

0 0

0.05 0.1 0.15 Ca content (x)

0.2

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Figure 14. High-T extrapolated values of χ(T)T at T = 0K for different Y1-xCaxBa2Cu3O7-δ compounds with p = 0.180 ± 0.005 (see text for details). The intercept for pure Y123 was taken from ref. [4].

It is important to notice that high-T linear extrapolation of χ(T)T for pure Y123 yields a negative intercept at T = 0K, as seen from Fig. 14. This is due to the presence of an approximately triangular non-states conserving PG [5, 26]. Such a PG introduces a negative Curie-like term in χ(T) and eqn. (5a) still holds. As the width of the PG increases with underdoping, this intercept becomes more and more negative. Positive intercept-values for Ca substituted Y123 with p < 0.19 does not imply the absence of PG for these compounds, rather, it represents a real additional (to that for pure Y123) paramagnetic Curie contribution in χ(T) due to Ca substitution. Ca itself does not affect the magnitude of the pseudogap [4, 21, 22]. To extract the characteristic PG energy scale, Eg, we have used a V-shaped gap in the EDOS centered at the Fermi-level, N(ε) = N0 for |ε - εF| > Eg and N(ε) = N0|ε - εF| for |ε - εF| < Eg, giving [30] T = N0[1 – D-1ln{cosh(D)}]

(6)

where, D = Eg/2kBT. Eqn. (6) captures the essential features of the PG on the electronic entropy [26]. Fig. 15 shows results of the fits of χ(T, p) data for pure (0%Ca) Y123 samples from 400K to ~ Tc + 30K (to avoid significant diamagnetic SC fluctuations near Tc) using eqns. (3) and (6) with a p-independent value of N0.

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-4

χ (10 emu/mole)

3

2

0.176 0.161 0.143 0.111 0.094

1 p-values

0 0

100

200 T(K)

300

400

Figure 15. χ(T, p) of pure Y123. Thick full lines show the fits to eqn. 6 (taken from [30]).

400 T

c

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E g/kB; Tc (K)

300

E /k g

B

200 100 0 0

0.1

0.2

0.3

p Figure 16. Eg(p)/kB and Tc(p) of pure Y123. The dashed line is drawn as a guide to the eyes.

The values of Eg obtained from the analysis of the χ(T, p) data agrees well with the earlier reports [3, 21]. A similar analysis of the χ(T, p) data for Ca-substituted Y123 also yields identical values for Eg(p) once the p-independent Ca induced Curie term is subtracted. Eg(p) values extracted from the fits are shown in Fig. 16 together with the Tc(p) for pure Y123.

5. Zero magnetic field critical current density of Y1-xCaxBa2Cu3O7-δ thin films Maximizing the superconducting critical current density, Jc, is one of the most important goals of applied superconductivity research. Jc is primarily limited by the depairing effect and by the flux depinning processes [31, 32]. The depairing effect is due to the breaking of the

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Effect and Evolution of the Pseudogap in Y1-xCaxBa2Cu3O7-δ

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paired charge carriers by the induced super-current and depends mainly on the superconducting condensation energy. The depinning critical current density, on the other hand, is governed by the interplay between flux motion and various possible flux pinning mechanisms. Large superconducting critical current densities can be achieved in double CuO2-layer YBa2Cu3O7-δ superconductor [33]. The high superconducting critical currents are clearly due to an improvement in the type and distribution of pinning centres as well as stronger c-axis coupling in this system. Also in recent years significant improvement in the grain boundary conductivity due to overdoping has led to an enhancement in the critical current density for Ca substituted Y123 [34, 35]. Some studies have also revealed that the oxygen content, distribution, and hole concentration are important parameters for obtaining the highest critical currents in the high temperature superconducting cuprates [36 - 38]. In a previous study it was shown that Jc and the irreversibility temperature, Tirr, of a c-axis aligned Y0.8Ca0.2Ba2Cu3O7-δ powdered sample reached a maximum in the overdoped (OD) side of the superconducting phase diagram at p ~ 0.19 per planer-Cu [36]. It has also been argued that p ~ 0.19 resembles a quantum critical point [3, 36] where the normal-state pseudogap correlation tends to vanish [3, 5, 21, 22, 24, 26, 29, 36] and the superconducting condensate density is maximized [3, 5, 36]. Previous studies [39, 40] also showed that oxygen content and oxygen ordering play parts in determining the magnitude and the magnetic field and temperature dependences of Jc for Y123 compounds. It is therefore of fundamental physical and technical interest to investigate the effects of hole concentration and oxygen disorder and/or deficiency on Jc in Ybased double-layer cuprate superconductors. As Ca can induce mobile carriers in the CuO2 planes irrespective of the level of oxygenation in the CuO1-δ chains, therefore, using Y1-xCaxBa2Cu3O7-δ compounds with different level of Ca, it is possible to differentiate between the contributions to the critical current density due to oxygen disorder and hole concentration. In this section we report the effects of hole content and oxygen deficiency on the zerofield ab-plane critical current density, Jc0(T), for c-axis oriented Y1-xCaxBa2Cu3O7-δ thin films over a wide range of p, x, and δ values. We have found that the low-temperature Jc0 is primarily determined by the hole concentration, reaching a maximum at p ~ 0.185 ± 0.005, irrespective of the oxygen deficiency. This indicates that the oxygen disorder and the contribution from the CuO1-δ chains to the superconducting condensate (due to possibly a SC proximity effect) play a secondary role and the intrinsic Jc0 is mainly governed by the inplane carrier concentration. Jc0 is maximized near p ~ 0.19 where the SC condensation energy as well as the superfluid density are at their largest [3, 26]. Thickness of the experimental thin films used for critical current measurement lied within the range (2700 ± 300) Å. As mentioned earlier the ab-plane room-temperature thermopower, Sab[290K], was used to calculate the planar hole content following the study by Obertelli et al. [11]. The level of oxygen deficiency was determined from an earlier work where the relation between δ and p were established [4]. Also, as an independent check, systematic changes in the c-axis lattice parameters were noted as δ was varied as a result of oxygen annealings. ρab(T) measurements gave information regarding the impurity content and also about the quality of the grain-boundaries of the films. All our samples used for Jc0(T) measurements had low values of ρab(300K) and the extrapolated zero temperature resistivity, ρab(0K) [19]. As prepared thin film samples were annealed at various temperatures and

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S. H. Naqib and R. S. Islam

oxygen partial pressures to vary the oxygen content and hence the hole concentration in the CuO2 planes, irrespective of the Ca content. Annealing times were kept within the range from 2 to 4 hours depending on the annealing temperatures. Samples were furnace cooled. Details of the annealings and their outcomes can be found in ref. [17]. The c-axis lattice parameters after the annealings for the Y0.80Ca0.20Ba2Cu3O7-δ thin film are plotted versus the oxygen deficiency and the hole content in Figs. 17a and 17b respectively. The systematic changes found here in the c-axis lattice parameter with δ are consistent with earlier studies by Jorgensen et al. [18] and by Hejtmánek et al. [41] on pure Y123 and Ca-substituted Y123 respectively.

m)

1 1 .74

c-axis lattice parameter (10

-10

a 1 1 .73

1 1 .72

1 1 .71

m) c-axis lattice parameter (10

0 .2

0 .3

δ

0 .4

0 .5

0 .1 8

0 .2

1 1 .7 4

b

-10

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1 1 .70 0 .1

1 1 .7 3

1 1 .7 2

1 1 .7 1

1 1 .7 0

0 .1 4

0 .1 6

p Figure 17. The c-axis lattice parameter of Y0.80Ca0.20Ba2Cu3O7-δ thin film versus (a) oxygen deficiency, δ and (b) planar hole content, p. Dashed lines are drawn as guide to the eye.

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Effect and Evolution of the Pseudogap in Y1-xCaxBa2Cu3O7-δ

The superconducting transition temperature, Tc, and Jc were measured using a vibrating sample magnetometer (VSM). An applied magnetic field of 0.5 mT was used for measuring Tc. Jc was measured at various fixed temperatures with magnetic fields ramped up to 1 Tesla, with the applied magnetic field (H) perpendicular to the surface of the film, i.e., HIIc in all cases. Typical M-H hysteresis loops for a representative Y0.95Ca0.05Ba2Cu3O7-δ thin film is shown in Fig. 18.

3

Magnetization (emu)

2 1

5%Ca-Y123 HIIc

p = 0.17

p = 0.12

0 -1

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-2 -3 4 -1 10

-5 10

3

0

0 10 Field (Oe)

3

5 10

1 10

4

Figure 18. Magnetization versus magnetic field data. The hole contents are given.

Jc0(T) was determined from the (zero-field) width of magnetization loops and dimensions of the thin films following the method developed by Brandt and Indenbom given in ref. [42]. We have shown the p-dependence of the low-T (16K) Jc0 in Fig. 19a. Reason for selecting low-T values of Jc0 is that they are more intrinsic in the sense that at low temperatures the superconducting energy gap is almost fully developed and the presence of weak links (e.g., due to grain boundaries) has lesser contributions to the critical current density. It is seen from Fig. 19a that Jc0(16K) is maximized at p ~ 0.185 independent of the oxygen deficiency. Jc0(16K) versus δ plot is shown in Fig. 19b, once again illustrating that Jc0(16K) is maximized at different values of δ depending on the Ca content.

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2

J (16K) (10 A/cm )

20

c0

6

15

0%Ca 5%Ca 10%Ca 20%Ca

10

5 a 0 0.1

0.15

0.2

p

2

J (16K) (10 A/cm )

20

c0

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6

15

10

5

0 0

0%Ca 5%Ca 10%Ca 20%Ca

0.1

0.2

b

δ

0.3

0.4

0.5

Figure 19. (a) Jc0(16K) versus p for Y1-xCaxBa2Cu3O7-δ thin films. The dashed curves show the trends of Jc0(16K) as a function of p for the 10% and 20% Ca-substituted Y123. The thick vertical line shows the position of the optimum doing level where Tc is maximum. (b) Jc0(16K) versus δ for Y1-xCaxBa2Cu3O7-δ thin films. The dashed curves show the trends of Jc0(16K) as a function of δ.

Next, we have plotted the SC condensation energy, U0, obtained from earlier specific heat measurements on Y0.80Ca0.20Ba2Cu3O7-δ [43] and Jc0(16K) of Y0.80Ca0.20Ba2Cu3O7-δ thin film versus p in Fig. 20. A clear and direct correspondence is seen between the p-dependences of U0 and Jc0(16K). A recent study by Wen et al. [44] has found similar correspondence between U0 and Jc0 for LSCO.

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Effect and Evolution of the Pseudogap in Y1-xCaxBa2Cu3O7-δ

J

U

c0

0

30

2

J (16K) (10 A/cm )

12

6

0

20

8 10

c0

U (J/mole)

10

6 0.1

0.15

p

0.2

0 0.25

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Figure 20. Jc0(16K) and the superconducting condensation energy, U0, versus p for Y0.80Ca0.20Ba2Cu3O7-δ [17, 43].

The intrinsic depairing critical current density is directly related to the superconducting condensation energy and therefore, a peak in the condensation energy, U0, should result in a peak in the critical current density, as we have indeed observed in the present study. Also within the collective pinning model for the magnetic flux-lines, Jc is expected to vary as ~ U0ξ0 [43], where ξ0 is the zero-temperature SC coherence length. A sharp peak in U0(p) thus should enhance the flux pinning in the compound as the p-dependence of ξ0 is rather weak [43]. It can be seen from Fig. 19a that, qualitatively the p-dependent Jc0(16K) shows identical features independent of Ca content, namely Jc0(16K) increases with p in the underdoped side, reaches a maximum at p ~ 0.185 and then decreases again with further overdoping. This implies that oxygen deficiency/disorder plays a secondary role in determining the value of low-T critical current density. The magnitude of the maximum Jc0(16K) for each individual film, on the other hand also show little dependence on the Ca content except for the 20%Ca substituted thin film for which the maximum Jc0(16K) is significantly lower than the other films with lesser amounts of Ca. This could be due to the lack of the contribution to the superfluid density from the CuO1-δ chains for the 20%Ca-Y123. Like the superconducting condensation energy, superfluid density is another measure of the strength of the superconducting pairing. Tallon et al. [45] have found from their muon spin relaxation (μSR) study that disorder free (δ → 0) CuO chains enhance the superfluid density significantly. As δ increases, chain contribution to the superfluid density diminishes quite rapidly [45]. As it can be observed from Fig. 19b, the maximum of Jc0(16K) is reached at oxygen deficiencies of ~ 0.11, ~ 0.16, and ~ 0.24 for 5%, 10%, and 20%Ca substituted thin films respectively. Therefore, the lower value of the maximum Jc0(16K) for Y0.80Ca0.20Ba2Cu3O7-δ is probably

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due to larger chain disorder. This indicates that in order to maximize Jc in Y123, large amount of Ca substitution may not be very helpful, because larger the Ca content, larger will be the oxygen deficiency required to reach p ~ 0.185. An optimal system will be the one having 5% to 10%Ca where p ~ 0.185 is attainable with relatively lower values of δ.

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6. Discussion and Conclusion We have reported systematic results of three of the most important bulk properties of high-Tc Y1-xCaxBa2Cu3O7-δ compounds, namely, temperature dependent resistivity, static magnetic susceptibility, and low-T zero-field critical current density in this study. All these bulk properties were found to be strongly p-dependent. From the analysis of the experimental data, we have found clear indications that these p-dependences actually follow the strongly p-dependent characteristic energy scale of the PG. The values of the characteristic energy scale for PG obtained from ρ(T) and χ(T) show the same qualitative features (as far as the p-dependence is concerned), but numerically T*(p) (extracted from ρ(T)) is always lower than Eg/kB (obtained from the analysis of χ(T)). Generally Eg/kB = ηT*(p), where η is a constant in the range 1.3 to 1.4. This is somewhat expected because different experiments probe the quasi-particle spectral density in different ways near the Fermi-energy. It should be noted that the effect of the PG have been investigated in the normal state for ρ(T) and χ(T), whereas for Jc, the effect of PG is manifested in the SC state. The same systematic behavior observed in all three cases (ρ(T) becomes completely linear in the normal state, χ(T) becomes flat in the normal state, and Jc0 is maximized in the SC state at p ~ 0.19) implies that the PG observed in the normal state persists in the SC state as the temperature is lowered. This contradicts the theoretical models [46] where PG vanishes as T → 0K, where only the SC gap exists. Also the analyses of the data presented in this report indicate that the characteristic PG energy scale falls below Tc in the slightly OD region and does not exist beyond p > 0.19. This does not support any of the existing versions of the precursor pairing scenarios [3, 23, 46-49] where the PG is originating from phase incoherent pairing fluctuations present at temperatures above Tc in the UD regime and becomes identical to the SC energy gap in the OD side. The origin of PG remains as one of the key mysteries in the physics of cuprates and the general consensus in the high-Tc community is that once the mechanism for PG is well understood, we will be able to describe the normal and superconducting states of the cuprates in a coherent fashion. In a broader sense the question regarding the origin of the PG has a much deeper conceptual consequence. For example, if PG originates from non-SC correlations then the segmented Fermi-arcs observed in angle-resolved photoemission spectroscopy (ARPES) [50, 51] poses a serious problem for the condensed matter theorists. The Fermi-arc scenario interpolates between small hole pocket and large Fermi surface quite satisfactorily as hole concentration is varied, but this is not permitted in the band theory of Fermi-liquids [52]. In this study, from the analysis of three bulk properties of Y1-xCaxBa2Cu3O7-δ, we have found indications that PG is unrelated with superconducting correlations. Pseudogap in fact competes with superconductivity, at least in the sense that its presence removes quasi-particle spectral weight near the chemical potential which otherwise would have been available for SC condensation. We have also observed that non-magnetic Ca substitution induces a

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significant paramagnetic Curie term in χ(T). The Curie enhancement does not depend on hole content or oxygen deficiency. The precise reasons for these rather interesting effects are not entirely clear to us and further study is required in this regard.

Acknowledgements The authors would like to thank Dr. J. R. Cooper (University of Cambridge, UK), Dr. J. W. Loram (University of Cambridge, UK), and Prof. J. L. Tallon (Victoria University, Wellington, New Zealand) for their help, thoughtful comments, and suggestions over a long period of time. We would also like to thank Commonwealth Scholarship Commission (UK), Trinity College, Darwin College, the Cambridge Philosophical Society, the Lundgren Fund, the Department of Physics and the IRC in Superconductivity, Cambridge, for funding and providing with the facilities for this work at various stages. SHN also thanks the Quantum Matter group, University of Cambridge, for their hospitality.

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References [1] C. Bernhard, Ch. Niedermayer, U. Binninger, A. Hofer, Ch. Wenger, J. L. Tallon, G. V. M. Williams, E. J. Ansaldo, and J. I. Budnick, Phys. Rev. B 52, 10488 (1995). [2] C. Bernhard and J. L. Tallon, Phys. Rev. B 54, 10201 (1996). [3] J. L. Tallon and J. W. Loram, Physica C 349, 53 (2001). [4] S. H. Naqib, Ph.D. thesis, University of Cambridge, UK, 2003 (unpublished). [5] J. W. Loram, K. A. Mirza, J. R. Cooper, and J. L. Tallon, J. Phys. Chem. Solids 59, 2091 (1998). [6] J. L. Tallon, private communications. [7] J. R. Cooper and J. W. Loram, J. Phys. I France 6, 2237 (1996) and A. Carrington, Ph.D. thesis, University of Cambridge, UK, 1993 (unpublished). [8] A. Sedky, Anurag Gupta, V. P. S. Awana, and A. V. Narlikar, Phys. Rev. B 58, 12495 (1998) and V. P. S. Awana and A.V. Narlikar, Phys. Rev. B 49, 6353 (1994). [9] J. T. Kucera and J. C. Bravman, Phys. Rev. B 51, 8582 (1995). [10] B. Fisher, J. Genossar, C. G. Kuper, L. Patlagan, G. M. Reisner, and A. Kinzhink, Phys. Rev. B 47, 6054 (1993). [11] S. D. Obertelli, J. R. Cooper, and J. L. Tallon, Phys. Rev. B 46, 14928 (1992). [12] J. L. Tallon, C. Bernhard, H. Shaked, R. L. Hitterman, and J. D. Jorgensen, Phys. Rev. B 51, 12911 (1995). [13] M. R. Presland, J. L. Tallon, R. G. Buckley, R. S. Liu, and N. E. Flower, Physica C 176, 95 (1991). [14] K. Scott, Ph.D. thesis, IRCS, University of Cambridge, UK, 1992 (unpublished). [15] D. Dijkkamp, T. Venkatesan, X. D. Wu, S. A. Shahen, N. Jisrawi, Y. H. Minlee, W. L. Mclean, and M. Croft, Appl. Phys. Lett. 51, 619 (1987). [16] D. J. C. Walker, Ph.D. thesis, IRCS, University of Cambridge, UK, 1994 (unpublished). [17] S. H. Naqib and A. Semwal, Physica C 425, 14 (2005). [18] J. D. Jorgensen, B. W. Veal, A. P. Paulikas, L. J. Nowicki, G. W. Crabtree, H. Claus, and W. K. Kwok, Phys. Rev. B. 41, 1863 (1990).

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[19] S. H. Naqib, R. A. Chakalov, and J. R. Cooper, Physica C 403, 73 (2004). [20] F. Rullier-Albenque, P. A. Vieillefond, H. Alloul, A. W. Tyler, P. Lejay, and J. Marucco, Europhys. Lett. 50, 81 (2000). [21] S. H. Naqib, J. R. Cooper, J. L. Tallon, and C. Panagopoulos, Physica C 387, 365 (2003). [22] S. H. Naqib, J. R. Cooper, J. L. Tallon, R. S. Islam, and R. A. Chakalov, Phys. Rev. B 71, 054502 (2005). [23] T. Timusk and B. Statt, Rep. Prog. Phys. 62, 61 (1999). [24] S. H. Naqib, J. R. Cooper, R. S. Islam, and J. L. Tallon, Phys. Rev. B 71, 184510 (2005). [25] J. Corson, R. Mallozzi, J. Orenstein, J. N. Eckstein, and I. Bozovic, Nature 398, 221 (1999). [26] J. W. Loram, J. Luo, J. R. Cooper, W. Y. Liang, and J. L. Tallon, J. Phys. Chem. Solids 62, 59 (2001) and J. W. Loram, K. A. Mirza, and J. R. Cooper, Research Review 1998 (IRCS), p.77. [27] R. E. Walstedt, R. F. Bell, L. F. Schneemeyer, J. V. Waszczak, and G. P. Espinosa, Phys. Rev. B 45, 8074 (1992) and C. Allgeier and J. S. Schilling, Phys. Rev. B 48, 9747 (1993). [28] S. Blundell, Magnetism in Condendensed Matter, Oxford University Press, 2001. [29] S. H. Naqib, J. R. Cooper, and J. W. Loram, Phys. Rev. B 2009 (accepted) and condmat0709.4075. [30] S. H. Naqib and J. R. Cooper, Physica C 460-462, 750 (2007) and R. S. Islam, S. H. Naqib, Supercond. Sci. Technol. 21 125020 (2008). [31] M. Tinkham, Introduction to superconductivity (McGraw-Hill International edition, 1996). [32] J. Mannhart, in Earlier and Recent Aspects of Superconductivity, edited by J. G. Bednorz and K. A. Müller, Springer Series in Solid State Sciences Vol. 90 (SpringerVerlag, Berlin 1990). [33] A. Semwal, N. M. Strickland, A. Bubendorfer, S. H. Naqib, S. K. Goh, G. V. M. Williams, Supercond. Sci. Technol. 17, S506 (2004). [34] G. Hammerl, A. Schmehl, R. R. Schulz, B. Goetz, H. Belefeldt, C. W. Schneider, H. Hilgenkamp, J. Mannhart, Nature 407, 162 (2000). [35] A. Schmehl, B. Goetz, R. R. Schulz, C. W. Schneider, H. Belefeldt, H. Hilgenkamp, J. Mannhart, Europhys. Lett. 47, 110 (1999). [36] J. L. Tallon, J. W. Loram, G. V. M. Williams, J. R. Cooper, I. R. Fisher, C. Bernhard, Physica Status Solidi (b) 215, 531 (1999). [37] E. C. Jones, D. K. Christen, J. R. Thompson, R. Feenstra, S. Zhu, D. H. Lowndes, J. M. Phillips, M. P. Siegal, J. D. Budai, Phys. Rev. B 47, 8986 (1993). [38] J. G. Ossandon, J. R. Thompson, D. K. Christen, B. C. Sales, H. R. Kerchner, J. O. Thompson, Y. R. Sun, K. W. Lay, J. E. Tkaczyk, Phys. Rev. B 45, 12534 (1992). [39] Akihiro Oka, Satoshi Koyama, Teruo Izumi, Yuh Shiohara, Physica C 314, 269 (1999). [40] H. Schmid, E. Burkhardt, B. N. Sun, J. -P. Rivera, Physica C 157, 555 (1989). [41] J. Hejtmánek, Z. Jirák, K. Knížek, M. Dlouhá, S. Vratislav, Phys. Rev. B. 54, 16226 (1996). [42] E. H. Brandt, M. Indenbom, Phys. Rev. B 48, 12893 (1993). [43] J. L. Tallon, J. W. Loram, Physica C 338, 9 (2000).

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[44] H. H. Wen, H. P. Yang, S. L. Li, X. H. Zeng, A. A. Soukiassian, W. D. Si, X. X. Xi, Europhys. Lett. 64, 790 (2003). [45] J. L. Tallon, C. Bernhard, U. Binninger, A. Hofer, G. V. M. Williams, E. J. Ansaldo, J. I. Budnick, Ch. Niedermayer, Phys. Rev. Lett. 74, 1008 (1995). [46] J. Stajic, A. Iyengar, K. Levin, B. R. Boyce, T. R. Lemberger, Phys. Rev. B 68, 024520 (2003) and Q. Chen, I. Kosztin, B. Janko, K. Levin, Phys. Rev. Lett. 81, 4708 (1998). [47] V. J. Emery, S. A. Kivelson, Nature 374, 434 (1995). [48] M. R. Norman, D. Pines, C. Kallin, Advances in Physics 54, 715 (2005) and the references therein. [49] V. Z. Kresin, Y. N. Ovchinnikov, S. A. Wolf, Phys. Rep. 431, 231 (2006) and the references therein. [50] J. C. Campuzano, M. Norman, M. Randeria in Physics of Conventional and Unconventional Superconductors, edited by K. H. Bennemann and J. B. Keterson, (Springer, Berlin 2003). [51] A. Damascelli, Z. Hussain, Z. -Y. Shen, Rev. Mod. Phys. 75, 473 (2003) and references therein. [52] P. A. Lee, Rep. Prog. Phys. 71, 012501 (2008).

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INDEX

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A Aβ, 20, 137, 138, 164, 167, 246, 247, 249 Abrikosov, 176, 185, 211, 233 absorption, 84 accounting, 271, 303, 307 accuracy, 340 ACS, 347 activation, 123 activation energy, 123 adiabatic, vii, viii, x, 1, 2, 3, 4, 5, 7, 11, 12, 13, 14, 15, 16, 17, 18, 19, 22, 23, 24, 25, 26, 27, 28, 29, 31, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 46, 48, 50, 51, 52, 53, 54, 55, 56, 57, 58, 60, 61, 62, 63, 66, 67, 68, 69, 70, 71, 187, 188, 195, 197, 199, 200, 204, 206, 207, 208, 210 adjustment, 302 Ag, x, xiii, 3, 187, 188, 192, 193, 194, 195, 197, 198, 199, 200, 208, 209, 311, 312, 313, 315, 316, 317, 318, 320, 327, 329, 335 aging, 105, 111, 114, 116, 123, 124, 125, 126 aging process, 123, 125 AIP, 310 air, 167, 169, 170, 296, 341, 344, 348 algorithm, 296, 297, 298 alkali, 34 alloys, 34, 173, 241 alternative, 127, 296, 298 aluminum, 284, 292 ambient pressure, 236 amorphous, 214, 224, 312, 317, 318, 325, 326 amplitude, ix, 135, 137, 140, 143, 144, 146, 149, 154, 156, 161, 162, 163, 164, 167, 168, 169, 170, 174, 224, 249, 273 analog, 61, 305, 309, 330

anisotropy, ix, 101, 104, 109, 110, 113, 116, 117, 119, 121, 126, 241, 248, 249, 272, 274, 276, 302, 340 annealing, xiii, 103, 302, 339, 344, 345, 346, 347, 348, 356, 364 annihilation, 6, 8, 12, 13, 14, 18, 19, 25, 56, 72, 74, 217 anomalous, x, 146, 213, 214, 215, 223, 228, 229, 231, 232, 272, 299, 306 antagonistic, 162, 178 antibonding, x, 187, 190, 191, 194, 198, 201, 204, 205, 299, 300 antiferromagnetic, 141, 151, 152, 236, 237, 247, 248, 249, 250, 252, 254, 258, 261 APP, 351 application, x, 6, 8, 10, 14, 16, 23, 52, 60, 61, 65, 67, 68, 69, 87, 150, 169, 190, 213, 312 argon, 84 argument, 33, 122, 124, 279, 282, 308 assumptions, 47, 48, 78, 176, 296, 306, 307, 309 asymmetry, viii, 18, 81, 83, 96, 153, 154, 199, 204, 244, 302 asymptotic, 109, 122 atmosphere, 152 atomic force, 350 atomic force microscopy (AFM), 350 atomic orbitals, 190 atoms, viii, x, 3, 48, 49, 50, 81, 83, 88, 96, 97, 102, 103, 105, 111, 120, 121, 126, 129, 130, 187, 188, 189, 190, 192, 193, 194, 197, 198, 208, 209, 244, 299 averaging, 167 Avogadro number, 246

B band gap, 112 Bangladesh, 339

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374

Index

barrier, xi, 117, 118, 119, 122, 137, 138, 143, 146, 149, 154, 180, 213, 214, 215, 223, 224, 225, 226, 227, 229, 230, 231, 232, 270 barriers, 104, 118, 119, 139, 140, 143, 148, 208 basis set, 11, 19, 24, 190 BCS theory, vii, ix, 1, 4, 5, 46, 48, 51, 71, 72, 107, 135, 136, 236 behavior, xiii, 40, 91, 140, 146, 150, 155, 156, 162, 164, 169, 176, 178, 182, 241, 246, 250, 252, 254, 270, 271, 272, 275, 276, 278, 279, 285, 287, 288, 307, 309, 312, 320, 334, 339, 353, 359, 368 behaviours, 104 bell, 237 bias, 138, 140, 143, 150, 154 binding, viii, 81, 82, 83, 88, 89, 90, 92, 93, 94, 95, 97, 98, 284 binding energies, viii, 81, 82, 90, 92, 93, 94, 97 binding energy, viii, 81, 82, 83, 89, 90, 93, 94, 95, 98, 284 bismuth, 272 bleaching, 120 blocks, 302 Bohr, 246, 357, 359 Boltzmann constant, 218, 224 bonding, x, 187, 190, 191, 194, 196, 198, 201, 204, 205, 308 bonds, 193, 237, 238, 296, 297, 299, 300, 305 Bose, 84, 207, 218, 220, 254, 271, 272, 279 Bose-Einstein, 218, 220, 254 boson, 13, 14, 18, 22, 25, 26, 28, 29, 31, 33, 45, 56, 60, 61, 62, 63, 65, 67, 68, 95, 97 bosons, 13, 72, 82, 95, 96, 97, 98, 207 boundary conditions, 169, 171, 172, 189, 190 breakdown, 4 buffer, 313, 332, 333, 334 bulk materials, 219, 222, 223, 229, 230, 232 butterfly, 49, 50

C Ca2+, xiii, 339, 340, 359 calibration, 69, 300, 303, 304, 306 calorimetric measurements, 271 capacitance, 145 CAR, 139 carbides, 348 carrier, viii, 83, 101, 105, 106, 107, 109, 112, 113, 114, 115, 116, 117, 119, 120, 122, 123, 126, 127, 129, 236, 237, 274, 276, 283, 284, 285, 286, 287, 288, 347, 363 cation, 348 cell, ix, x, xi, 34, 101, 102, 103, 109, 128, 129, 150, 151, 187, 189, 190, 192, 193, 194, 195, 198, 235, 236, 241, 244, 272, 315, 331

ceramics, 82 channels, 301, 302, 305, 306 charge density, 39, 48, 49, 50, 51, 188, 208, 209, 287, 300, 302 chemical properties, 54 chemicals, 341 cladding, xiii, 311 classes, 283 classical, 2, 140, 218, 220, 308, 309 cleaning, 313 closure, 17, 20 clustering, 329 clusters, 120, 121, 124, 128, 328 coherence, xii, 88, 117, 170, 173, 269, 270, 271, 280, 282, 283, 284, 286, 311, 329, 353, 367 collaboration, 263 colors, 320 community, 368 compensation, 168 competition, xi, 3, 213, 214, 223, 228, 232, 258, 296, 297, 308 complexity, 141 components, 105, 109, 113, 162, 163, 164, 167, 168, 169, 171, 172, 175, 177, 178, 226, 300 composition, 189, 329, 348 compounds, vii, viii, ix, xiii, 101, 127, 135, 136, 137, 152, 214, 236, 237, 241, 244, 246, 250, 257, 261, 263, 270, 272, 283, 308, 312, 315, 339, 340, 341, 347, 348, 351, 353, 355, 356, 357, 358, 359, 361, 363, 368 comprehension, 157 computation, 149 concentration, viii, xi, xii, xiii, 103, 105, 109, 111, 114, 115, 116, 117, 119, 120, 126, 127, 129, 148, 151, 152, 167, 235, 237, 238, 239, 240, 244, 254, 256, 257, 261, 283, 295, 298, 307, 339, 340, 347, 350, 353, 356, 359, 363, 364, 368 condensation, xii, 36, 46, 77, 207, 218, 221, 254, 269, 270, 276, 280, 281, 282, 284, 286, 363, 366, 367, 368 condensed matter, 236, 340, 368 conductance, ix, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 152, 153, 154, 155, 156 conduction, ix, 39, 103, 104, 105, 112, 117, 135, 136, 180, 241, 287 conductivity, viii, 101, 102, 103, 104, 106, 108, 109, 112, 113, 114, 117, 118, 119, 120, 121, 123, 124, 125, 127, 129, 236, 249, 274, 276, 278, 279, 288, 363 conductor, 140, 151, 334 confidence, 313

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Index configuration, 6, 7, 8, 9, 11, 16, 18, 23, 24, 27, 33, 36, 38, 42, 46, 52, 53, 69, 130, 139, 145, 162, 170, 188, 189, 206, 356, 358, 359 conjugation, 87, 165 consensus, 162, 270, 368 conservation, 29 construction, 307, 308 contamination, 341, 358 continuity, 166 contractions, 8, 60, 65, 66, 69, 74, 306 control, 143, 323, 328, 340 convergence, 77, 307 cooling, 5, 51, 111, 188, 208, 237, 252, 341, 350 Cooper pair, xii, 138, 140, 154, 162, 166, 180, 269, 270, 280, 282, 283, 284, 355 copper, ix, 101, 103, 104, 110, 117, 123, 127, 129, 150, 285, 351 correlation, viii, xi, 2, 5, 10, 17, 18, 33, 35, 42, 43, 44, 45, 47, 51, 60, 67, 71, 96, 98, 162, 163, 164, 165, 167, 182, 216, 235, 236, 247, 249, 250, 252, 254, 258, 262, 263, 275, 281, 282, 287, 298, 309, 363 correlation function, 164, 165, 167, 182, 216, 263 correlations, viii, 2, 47, 81, 83, 84, 97, 353, 357, 368 cosine, 172 Coulomb, viii, 3, 53, 54, 81, 82, 85, 86, 87, 88, 89, 91, 96, 236 Coulomb interaction, 3, 85, 86 couples, 24, 49 coupling, vii, viii, x, xi, xii, 1, 2, 3, 7, 13, 14, 23, 27, 29, 33, 34, 37, 48, 51, 60, 66, 81, 82, 83, 85, 90, 92, 93, 94, 95, 96, 97, 98, 144, 146, 150, 152, 156, 164, 165, 166, 168, 171, 174, 187, 188, 192, 193, 195, 197, 198, 199, 200, 206, 208, 209, 235, 236, 246, 247, 248, 249, 252, 254, 258, 262, 263, 276, 295, 297, 303, 357, 363 coupling constants, 246 covalent, 190, 237, 246, 307 covalent bond, 246, 307 creep, 270, 271 critical behavior, 276, 278 critical current, x, xiii, 143, 145, 213, 214, 215, 217, 218, 219, 221, 222, 232, 274, 311, 312, 315, 318, 326, 327, 328, 330, 332, 333, 339, 340, 362, 363, 365, 367, 368 critical current density, x, xiii, 213, 214, 215, 217, 218, 219, 221, 222, 232, 312, 315, 326, 327, 328, 330, 339, 340, 362, 363, 365, 367, 368 critical points, 16, 206, 271, 278 critical temperature, ix, x, xi, xii, 2, 38, 41, 42, 103, 129, 141, 146, 147, 151, 156, 161, 162, 170, 172, 176, 177, 179, 182, 187, 189, 204, 207, 213, 218,

375

223, 225, 269, 271, 273, 276, 277, 278, 280, 281, 282, 283, 287, 289, 325 critical value, 40 criticism, 126 crossing over, 232 cross-sectional, 351 crystal lattice, 123 crystal structure, ix, x, 88, 102, 103, 105, 109, 128, 129, 130, 152, 213, 214, 237, 238, 240, 244, 299, 309 crystalline, xiii, 106, 151, 241, 244, 339, 340, 348, 356 crystallinity, 315 crystals, 189, 271, 296, 297, 340 cubic system, 349 CVD, 312 cycles, 143 cyclohexane, 341

D damping, viii, 81, 83, 87, 96, 224 dark conductivity, 124 decay, ix, 102, 112, 162, 172, 173, 174, 176, 177, 178, 180, 229, 247, 248, 250, 251, 252, 253, 262 decomposition, 254 defects, 112, 121, 122, 126, 129, 130, 216, 272, 303, 306 deficiency, xiii, 127, 339, 347, 350, 363, 364, 365, 367, 369 definition, 280, 281, 282, 283, 284, 285, 287, 308 deformation, 152 degenerate, vii, x, 1, 4, 18, 31, 32, 33, 38, 51, 187, 188, 189, 197, 198, 199 degradation, 143, 306 degrees of freedom, 82, 96, 98 delocalization, 209 Denmark, 336 dependent variable, 62 deposition, 106, 312, 348, 350 depressed, 150, 154, 155, 156 depression, 140, 154 derivatives, 16, 309 destruction, 48, 76, 180 deviation, 199, 241, 242 DFT, 190, 191, 192, 204 diamagnetism, 40, 168 diamond, 309, 313 diamonds, 300, 304 dielectric function, 121 differential equations, 169 diffraction, xii, 102, 110, 142, 285, 311, 312, 313, 337, 341, 350 diffusion, 115, 140, 162, 164, 166, 167, 173, 176

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Index

diffusion process, 162, 164 diffusivity, 123, 126 dimensionality, 236, 261, 271, 273, 278, 353 dimerization, 298 discontinuity, 206 dislocations, x, 213, 214, 215, 224, 302 disorder, x, 105, 120, 154, 213, 214, 215, 217, 218, 220, 223, 228, 229, 230, 231, 232, 271, 272, 274, 303, 350, 363, 367 dispersion, viii, x, 12, 15, 36, 81, 82, 83, 84, 85, 88, 90, 91, 92, 93, 94, 95, 96, 97, 98, 108, 118, 187, 189, 190, 199, 204, 222 displacement, 3, 14, 18, 24, 35, 38, 52, 55, 193, 194, 197, 198, 209, 215, 216, 225, 273 distortions, x, 187, 194, 195, 200, 297, 306, 308 distribution, xi, xii, xiii, 39, 45, 47, 48, 49, 50, 51, 78, 105, 111, 118, 119, 122, 188, 218, 220, 235, 295, 303, 311, 315, 316, 318, 319, 327, 330, 331, 363 distribution function, 47, 218, 220, 319, 331 division, 1, 187 domain walls, 163 dominance, 4, 49 doped, xii, 83, 92, 120, 121, 129, 151, 239, 244, 245, 247, 251, 252, 253, 254, 255, 258, 259, 260, 261, 262, 263, 269, 272, 273, 282, 284, 287, 295, 296, 297, 299, 300, 301, 306, 309, 347 doping, viii, ix, x, xi, xii, 2, 81, 83, 101, 102, 103, 107, 110, 121, 126, 127, 128, 129, 151, 152, 189, 213, 214, 215, 217, 235, 236, 237, 244, 252, 254, 257, 258, 259, 260, 261, 269, 275, 281, 282, 283, 285, 287, 288, 295, 296, 297, 298, 299, 300, 302, 303, 304, 305, 306, 307, 308, 309, 340, 353 drying, 341

E earth, ix, 135, 136 EBSD, xii, 311, 312, 313, 315, 317, 318, 319, 320, 324, 325, 326, 327, 328, 329, 330, 331, 333, 334 elasticity, 297, 307, 308 electric charge, xi, 235 electric current, 40 electric field, 49, 87, 96, 241, 244 electric potential, 39, 51 electrical conductivity, viii, 101, 102, 103, 106, 108, 109, 112, 114, 120 electrical properties, 107 electrical resistance, 106 electromagnetic, 164, 274, 278 electron beam, 313 electron density, 49, 50, 209 electron pairs, 72 electron state, 282

electronic structure, vii, 1, 35, 45, 46, 47, 48, 49, 211, 258, 276 electronic systems, xi, 10, 241 electron-phonon, x, 2, 14, 39, 83, 95, 96, 98, 187, 188 electrons, vii, ix, 1, 2, 3, 4, 5, 9, 10, 13, 14, 19, 22, 25, 38, 39, 40, 43, 48, 51, 52, 53, 54, 56, 57, 58, 72, 73, 76, 77, 82, 88, 92, 93, 94, 97, 98, 102, 104, 112, 113, 118, 119, 122, 123, 124, 126, 128, 129, 130, 135, 136, 138, 139, 140, 150, 151, 154, 155, 156, 164, 166, 168, 174, 180, 190, 192, 236, 245, 282, 300 elementary particle, 236 elongation, 192 emission, 90, 94, 97 encouragement, 182 energy density, 82, 90, 270, 276, 282, 284, 340 energy of system, 38, 39, 78 energy transfer, 248 ENN, 6 entropy, 5, 39, 40, 46, 48, 51, 78, 188, 361 environment, 308 epitaxial films, 271 equality, 48 equilibrium, vii, xii, 1, 4, 9, 14, 18, 29, 34, 35, 36, 38, 39, 46, 48, 49, 50, 51, 52, 55, 136, 137, 168, 188, 189, 193, 198, 206, 207, 208, 209, 210, 227, 247, 295, 303 equilibrium state, 136, 247 estimator, 347 ethanol, 313 ethyl alcohol, 142 evolution, xiii, 104, 108, 109, 110, 117, 146, 147, 148, 154, 155, 156, 182, 274, 276, 339, 341, 353, 356 excitation, ix, xi, 31, 47, 62, 71, 76, 77, 78, 83, 90, 91, 94, 97, 101, 102, 107, 110, 114, 119, 120, 130, 207, 235, 236, 237, 262, 358 exclusion, ix, 161, 169 expansions, 7, 70 experimental condition, 115, 126 exponential functions, 118, 227 expulsion, 281 extinction, 204 extrapolation, 149, 155, 303, 361 eyes, 147, 199, 256, 362

F family, 2, 189, 192, 254, 299, 302, 305, 307 family relationships, 307 fat, xi, 235 Fermi energy, vii, x, 1, 2, 3, 34, 35, 46, 51, 73, 89, 107, 187, 188, 192, 199, 210, 282, 283

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Index Fermi level, vii, x, 2, 3, 5, 11, 12, 16, 29, 31, 32, 33, 34, 36, 37, 38, 39, 40, 43, 47, 48, 49, 51, 71, 76, 77, 91, 94, 95, 96, 97, 98, 138, 150, 153, 164, 167, 187, 188, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 201, 202, 203, 204, 205, 206, 207, 208, 210, 237, 280, 282 Fermi liquid, 214, 236 Fermi surface, xi, 88, 165, 236, 241, 254, 257, 258, 368 Fermi-Dirac, 29, 32, 40, 44, 48, 77, 78, 196 fermions, 13, 19, 25, 48, 56, 60, 65, 73 ferromagnetic, ix, 103, 136, 137, 148, 150, 151, 152, 154, 155, 156, 161, 162, 163, 170, 173, 177, 179, 182, 247, 248, 254 ferromagnetism, ix, 135, 137, 148, 152, 155, 157, 170 field theory, 236 field-dependent, 216, 225, 227 filament, 323 film, 106, 108, 109, 110, 111, 127, 136, 152, 163, 231, 340, 348, 350, 355, 363, 365, 367 films, 106, 109, 113, 118, 151, 217, 224, 231, 271, 279, 284, 340, 348, 350, 356, 363, 366, 367 financial support, 78, 157, 210 first principles, 309 flatness, 240 flow, 223, 312, 316, 317, 325, 326, 328, 350, 351 fluctuations, xi, 225, 235, 236, 241, 248, 249, 250, 252, 254, 257, 261, 263, 271, 273, 275, 276, 277, 280, 288, 340, 353, 355, 361, 368 fluid, 302 flux pinning, xiii, 275, 311, 312, 317, 318, 324, 325, 326, 327, 328, 329, 333, 335, 363, 367 focusing, 350 force constants, 12, 13 Fourier, 165, 172, 216, 246 Fourier transformation, 216 fragmentation, 105 France, 264, 369 free energy, 5, 46, 48, 51, 78, 188, 275 freedom, 82, 96, 98, 164 freezing, 271 fulfillment, 21 fullerenes, 34 funding, 369

G GaAs, 102 gas, 283, 341 gases, 341 gauge, 167, 236 Gaussian, 244, 247, 248, 249, 250, 251, 252, 253, 262

377

generation, 15, 101, 307, 332 Germany, 311 glass, 270, 271, 272, 274, 275, 276, 277, 278, 279, 280, 282 goals, 280, 362 government, 335 grain, xiii, 144, 145, 146, 156, 311, 312, 313, 315, 316, 317, 318, 320, 322, 324, 325, 326, 327, 328, 335, 345, 347, 350, 363, 365 grain boundaries, xiii, 311, 312, 315, 316, 318, 320, 322, 324, 326, 328, 335, 345, 347, 365 grains, xiii, 144, 250, 311, 315, 316, 317, 318, 321, 322, 323, 324, 325, 326, 327, 329 granules, 325 ground state energy, x, 5, 9, 15, 16, 17, 28, 36, 38, 46, 47, 49, 55, 66, 77, 187, 189, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 207, 208 groups, vii, 1, 8, 51, 60, 65, 270 growth, 111, 120, 313, 320, 328, 330, 331, 332, 335, 359 growth mechanism, 320, 335

H H2, 17 Hamiltonian, 4, 5, 6, 7, 8, 9, 10, 12, 13, 14, 18, 20, 21, 22, 25, 26, 27, 29, 30, 31, 33, 44, 45, 46, 47, 52, 53, 54, 55, 57, 58, 60, 61, 62, 63, 65, 66, 67, 69, 70, 71, 74, 75, 164, 166, 189, 190, 195, 200, 215, 216, 241, 246, 247, 248, 281 Hartree-Fock method, 10 heat, 40 heating, 128, 143, 341, 350 height, 145, 147, 148, 149, 155 Heisenberg, 166 Heisenberg picture, 166 helium, 152 hemisphere, 139 heterostructures, ix, 135, 136, 137, 152, 162 high pressure, 236 high resolution, 82, 94, 97 high temperature, xi, 82, 84, 96, 102, 106, 107, 109, 111, 112, 113, 116, 117, 118, 123, 125, 126, 129, 137, 156, 212, 214, 258, 269, 280, 348, 363 high-Tc, vii, xii, 1, 2, 3, 34, 46, 51, 52, 79, 97, 188, 189, 192, 199, 206, 208, 311, 312, 315, 329, 340, 353, 368 hip, 287 Hm, xi, 269, 270, 272, 273, 274, 276, 277 Holland, 80, 159 homogeneity, 315, 340 hospitality, 369 host, 359 HTS, 347

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Index

Hubbard model, 3, 261 hybrid, 137, 141, 155, 162, 169 hybridization, xii, 295, 297, 306 hypothesis, 148, 163 hysteresis, 143, 150, 156, 365 hysteresis loop, 365

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I identification, 97 Illinois, 81 illumination, viii, 101, 106, 107, 108, 109, 110, 111, 112, 114, 115, 116, 117, 118, 120, 121, 122, 123, 124, 127, 128, 129 imaging, 313 imbalances, 308 impurities, x, xi, 136, 213, 214, 215, 217, 235, 341 in situ, 348 in transition, 296 incidence, 341 independence, 308 indication, 144, 355 indices, 342 indium, 152 induction, 40 industry, 136 inelastic, 146, 248, 287 inequality, 16, 23, 28, 36, 55, 69 inert, 303 infinite, 38, 49, 72, 163, 170, 189, 194, 198, 208, 308 infrared, ix, 101, 118, 119, 120, 128, 129 inhomogeneity, 163 injection, 136, 139, 154, 155, 156 insertion, 302, 308 insight, 335 inspection, 192, 204 Inspection, 196 instability, 236, 309 insulators, 96, 98, 151, 237, 241 integration, 37, 47, 139, 196, 197, 218, 248 intensity, 89, 108, 112, 115, 120, 121, 149, 341, 350 interaction, viii, ix, xiii, 2, 3, 5, 10, 14, 16, 20, 21, 30, 36, 39, 44, 45, 47, 51, 57, 69, 70, 71, 72, 73, 76, 81, 83, 95, 96, 98, 122, 125, 135, 141, 148, 162, 164, 168, 170, 174, 177, 178, 180, 182, 197, 204, 206, 208, 210, 214, 247, 248, 257, 297, 298, 304, 311, 312 interactions, vii, viii, x, 1, 2, 3, 4, 5, 10, 17, 29, 31, 33, 35, 42, 43, 44, 45, 46, 47, 48, 51, 66, 67, 69, 70, 72, 77, 83, 87, 97, 136, 152, 168, 187, 188, 189, 195, 196, 197, 199, 200, 204, 206, 207, 208, 272 interface, 137, 138, 139, 140, 143, 149, 150, 154, 170, 171, 173, 175, 176, 180, 182

interpretation, 30, 112, 114, 127, 153, 156, 299 interstitial, 128 interval, 37, 47, 76, 180, 197, 200, 201, 204 intrinsic, ix, 3, 14, 18, 24, 27, 28, 31, 32, 35, 36, 37, 44, 47, 69, 135, 136, 141, 149, 155, 237, 246, 274, 323, 325, 327, 335, 357, 363, 365, 367 inversion, 83, 247 ionic, 189, 237, 239 ions, ix, 83, 96, 101, 102, 105, 106, 110, 111, 112, 113, 121, 122, 123, 127, 151, 180, 240, 244, 246, 254 IRC, 369 irradiation, x, 127, 130, 213, 214 Islam, 339, 370 isolation, xii, 47, 295, 296, 297, 300, 303, 305, 306, 307, 309 isotope, 84, 96, 98 isotropic, 140, 167, 247, 271, 301, 302, 306 ISS, 267 Italy, 135 iterative solution, 69

J Japan, 234, 235 Jc, 312, 362, 363, 365, 367, 368 Josephson effect, 158 Josephson junction, 102, 142, 144, 145, 146, 156 Jun, 159, 233

K kernel, 164, 165, 167, 168, 169, 172, 300, 301 kinetic energy, vii, 1, 3, 4, 6, 24, 36, 49, 51, 53, 54, 57, 69, 188, 195, 208, 214, 216, 348 kinks, viii, 81, 82, 90, 92, 93, 95, 97, 98, 296, 298

L language, 270 laser, 106, 107, 110, 112, 115, 121, 128, 348, 349, 350 laser ablation, 341, 348, 349 lattice parameters, ix, 101, 103, 109, 111, 122, 123, 190, 350, 363 law, 118, 237, 261, 271, 272, 274 leakage, 140, 170 lending, 279 lifetime, 143, 149, 206 linear, xiii, 11, 15, 52, 91, 105, 112, 118, 121, 125, 127, 149, 154, 156, 169, 175, 241, 242, 258, 261, 278, 279, 297, 298, 303, 305, 307, 339, 340, 353, 356, 359, 361, 368 linear dependence, 156 linear function, 241, 242

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Index links, 365 liquid nitrogen, 111, 328, 344 liquid phase, 271 liquids, 357, 368 localised, 105 localization, 49, 50 location, 275, 297, 303 London, 80, 99, 100, 158, 159, 211, 278, 283, 292 long period, 369 losses, 341 low temperatures, 102, 106, 107, 111, 113, 116, 118, 119, 126, 129, 139, 141, 143, 144, 145, 146, 148, 153, 155, 156, 262, 280, 282, 312, 365 low-temperature, 345, 363 lying, 15, 33, 47, 163, 262, 355

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M M1, 142 magnet, 217 magnetic field, x, 5, 40, 41, 42, 46, 48, 51, 78, 114, 117, 136, 148, 149, 150, 152, 156, 170, 188, 213, 214, 215, 216, 218, 219, 221, 222, 223, 224, 225, 228, 229, 230, 231, 232, 245, 246, 250, 271, 273, 274, 275, 276, 277, 340, 357, 358, 362, 363, 365 magnetic field effect, 149 magnetic moment, 146, 162, 163, 167, 297, 358, 359 magnetic particles, 358 magnetic properties, 46, 236, 263, 358 magnetic resonance, 241 magnetic structure, 163, 170 magnetism, ix, 135, 136, 149, 236, 237 magnetization, 128, 148, 162, 163, 170, 174, 245, 271, 273, 275, 357, 359, 365 magnetoresistance, 114, 136 magnetron, 106 magnets, 328 manganites, 136 many-body theory, 4 mapping, xiii, 311 mask, 152, 353 matrix, xiii, 6, 7, 8, 19, 20, 21, 22, 23, 25, 27, 28, 29, 32, 42, 43, 48, 49, 50, 51, 62, 69, 71, 72, 88, 95, 97, 120, 164, 165, 166, 167, 168, 169, 170, 190, 196, 197, 302, 311, 312, 318, 325, 327, 328, 329, 330, 331, 335 Maxwell equations, 40 measurement, 108, 110, 143, 258, 313, 328, 331, 358, 363 measures, 118, 121, 282, 353 media, 120 mediators, viii, 81, 83, 96 Meissner effect, 40, 42, 78

379

melt, xiii, 270, 280, 311, 313, 318, 327, 328, 329, 331, 333, 335 melting, xi, 269, 270, 271, 272, 273, 274, 275, 276, 277, 278, 279, 280, 282, 288 melts, 271 mercury, xi, 235, 236, 237, 239, 241, 243, 245, 247, 249, 250, 251, 253, 255, 257, 259, 261, 263, 265, 267, 272 metal oxide, 245, 246, 296 metal oxides, 245, 246, 296 metals, 34, 47, 71, 72, 107, 162, 176, 241, 288 MgB2, 2, 3, 4, 34, 37, 48, 49, 50, 52, 188, 192, 208 microcirculation, 208 microscopy, 300, 350 microstructure, 328 misleading, 122 MIT, 103, 104, 105, 106, 109, 112, 124 mixing, 182, 341 mobility, viii, 101, 109, 113, 114, 115, 116, 117, 123, 125, 126, 129 modeling, xii, 137, 145, 156, 296, 303 models, vii, 1, 2, 3, 4, 5, 51, 96, 98, 121, 122, 125, 126, 129, 130, 249, 261, 276, 296, 306, 307, 368 modulation, ix, 27, 48, 50, 143, 161, 163, 164, 170, 244 modulus, 216, 225 moisture, 341 mole, 245, 246, 247 molecules, 17 momentum, 4, 5, 6, 11, 12, 23, 24, 26, 27, 28, 29, 30, 34, 36, 43, 49, 58, 63, 64, 85, 87, 95, 97, 162, 164, 166, 174, 175, 177, 178, 192, 195, 216, 248, 296 Moon, 158 motion, vii, x, xi, 1, 2, 3, 4, 5, 11, 12, 13, 14, 16, 17, 24, 31, 33, 34, 35, 38, 39, 40, 45, 46, 47, 49, 50, 51, 52, 53, 54, 57, 58, 61, 66, 69, 140, 164, 166, 187, 188, 194, 195, 198, 208, 213, 214, 215, 223, 226, 228, 229, 230, 231, 232, 270, 271, 274, 363 motivation, 163 movement, 142, 302 multiples, 31, 298 muon, 367

N nanometer, 136 nanoparticles, 312, 313, 320, 324, 327, 329, 330, 331, 335 nation, 109, 127 NATO, 80 natural, 244, 278, 282, 306 Nb, 313, 327, 329, 330, 331, 332 Nd, 129, 283

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380

Index

neglect, 29, 69, 176 network, 309 New Jersey, 264 New Orleans, 100 New York, 80, 157, 158, 184, 185, 211, 233, 264, 292, 293 New Zealand, 369 Ni, 312, 333, 359 nitrogen, 111, 114, 328, 344 noise, 347, 351 nonlinear, 76, 85 non-linear, 308, 359 non-linearity, 308 non-magnetic, xi, xiii, 339, 358, 359, 368 non-uniform, 350 normal, vii, xi, xiii, 6, 8, 12, 17, 29, 38, 46, 47, 60, 61, 62, 65, 66, 71, 72, 74, 77, 83, 102, 106, 107, 108, 109, 117, 127, 129, 136, 137, 138, 139, 140, 142, 145, 150, 154, 156, 162, 163, 164, 170, 207, 210, 214, 235, 236, 237, 270, 271, 274, 275, 278, 283, 284, 285, 286, 287, 288, 339, 340, 347, 358, 363, 368 normalization, 61, 72, 168 NQR, xi, 235, 236, 241, 243, 244 nuclear, vii, x, xi, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 16, 17, 18, 19, 22, 23, 24, 27, 31, 33, 34, 35, 36, 38, 42, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 60, 61, 63, 66, 69, 187, 188, 189, 194, 195, 198, 200, 201, 206, 207, 208, 210, 235, 236, 237, 241, 243, 244, 246, 247, 248, 249, 250, 251, 252, 253, 254, 255 Nuclear magnetic resonance (NMR), xi, 148, 235, 236, 237, 241, 244, 249, 250, 251, 252, 253, 254, 261, 262, 263, 264 nucleation, 174, 177 nuclei, 2, 11, 13, 14, 38, 39, 48, 49, 52, 53, 54, 57, 208, 241, 244, 247, 254 nucleus, 16, 300 numerical analysis, 176

O observations, 93, 96, 111, 120, 129, 156 one dimension, 170, 171 operator, 4, 8, 10, 11, 12, 20, 21, 22, 23, 24, 26, 56, 58, 63, 64, 165, 166, 167 optical, 2, 84, 106, 107, 112, 120, 121, 122, 123, 124, 128, 284 optimization, 15 orbit, ix, 161, 164, 165, 166, 167, 168, 170, 173, 174, 177, 178, 180, 182 orientation, xiii, 50, 139, 149, 296, 301, 302, 311, 312, 313, 315, 316, 318, 319, 322, 325, 326, 327, 328, 329, 330, 331, 333, 334, 349, 350

orthorhombic, x, 103, 129, 190, 213, 214, 302, 314, 315, 331, 333, 334 oscillation, x, 161, 174, 177, 178, 224, 227 oscillations, 87, 143, 162, 177, 224, 244 overdoped, xiii, 83, 99, 127, 237, 244, 251, 252, 253, 254, 255, 258, 259, 260, 261, 281, 282, 288, 339, 340, 355, 359, 363 oxide, 101, 103, 104, 110, 117, 123, 127, 129, 137, 150, 151, 153, 324, 348 oxides, 150, 245, 246 oxygenation, 340, 345, 349, 363

P PACS, 161 pairing, viii, ix, xi, 2, 72, 81, 83, 84, 85, 87, 95, 96, 97, 98, 136, 161, 162, 164, 166, 168, 169, 170, 207, 235, 252, 258, 297, 353, 367, 368 parabolic, 297, 305, 307, 347, 348 paramagnetic, 358, 361, 369 parameter, ix, xii, 2, 84, 85, 87, 88, 89, 91, 96, 108, 109, 110, 111, 118, 129, 135, 136, 137, 138, 139, 140, 143, 144, 146, 147, 148, 149, 153, 154, 156, 161, 164, 169, 170, 176, 182, 190, 197, 198, 237, 244, 247, 258, 271, 272, 273, 274, 277, 280, 281, 283, 285, 295, 297, 298, 303, 304, 312, 313, 315, 353, 354, 364 particle density, 166, 333 particles, xiii, 9, 26, 48, 56, 63, 72, 73, 76, 77, 208, 210, 236, 311, 312, 317, 318, 320, 324, 326, 328, 329, 330, 331, 333, 334, 335, 358 partition, 317, 318 patterning, 309 Pb, xiii, 283, 292, 302, 311, 312 perception, 296 performance, 328 periodic, x, 3, 11, 14, 50, 117, 187, 188, 211 periodicity, 299 permeability, 218 perovskite, 150, 151, 272 perovskites, 152 perturbation, 4, 10, 13, 14, 17, 24, 43, 44, 45, 54, 55, 62, 75, 166, 168 perturbation theory, 11, 13, 14, 15, 17, 43, 44, 45, 54, 62, 75 phase boundaries, 324 phase diagram, xii, 2, 152, 239, 254, 256, 269, 270, 271, 276, 340, 353, 363 phase space, 270 phenomenology, xii, 295, 297, 299, 307, 309 philosophy, 72 phonon, vii, x, 1, 2, 3, 4, 7, 12, 13, 14, 15, 17, 23, 27, 29, 30, 31, 33, 34, 35, 36, 37, 38, 39, 40, 46, 48, 49, 51, 55, 60, 66, 67, 69, 77, 96, 98, 162, 164,

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Index 187, 188, 192, 196, 197, 199, 200, 206, 208, 209, 356 phonons, viii, 13, 56, 82, 95, 96, 98, 208, 210, 356 photoconductivity, viii, 101, 102, 112, 122 photoemission, viii, 81, 82, 83, 85, 89, 90, 94, 95, 96, 97, 98, 368 photoexcitation, 102, 106, 107, 108, 109, 110, 111, 112, 118, 120, 122, 125, 126, 127, 128, 129, 130 photo-excitation, ix photon, 106, 107, 108, 111, 112, 113, 115, 119, 120, 122, 128 photons, 106 physics, vii, ix, 5, 7, 15, 48, 72, 89, 135, 141, 163, 236, 272, 279, 310, 340, 368 pI, 88 planar, xii, 136, 137, 193, 194, 296, 305, 307, 309, 363, 364 plastic, 358 play, 34, 87, 129, 144, 156, 236, 246, 298, 325, 328, 353, 363 PLD, xiii, 339, 348, 350 polarization, 49, 50, 88, 150, 154, 155, 208, 246 polycrystalline, xiii, 106, 139, 142, 144, 149, 156, 250, 339, 340, 347, 349, 356 polymers, 348 poor, 10, 105, 122, 124, 126 population, 18, 31, 45, 48, 137 pores, 318, 325, 326 powder, 141, 250, 312, 324, 337, 342, 344 powders, 329, 341 power, 52, 128, 271, 272, 274, 278 prediction, xii, 83, 93, 123, 175, 295, 306 preference, 306 pressure, 84, 236, 301, 341, 350 private, 369 probability, 72, 77, 93, 113, 139, 168, 189 probe, 142, 143, 152, 247, 254, 357, 358, 368 propagation, 170, 302 property, xi, 17, 20, 38, 41, 42, 213, 228, 347 proportionality, 307 pseudo, viii, xi, 81, 83, 84, 87, 95, 235, 236, 237, 241, 245, 252, 254, 257, 258, 262, 263, 306, 308 p-type, 224 pulse, 112, 247, 350 pulsed laser, xiii, 312, 339, 341, 348 pulsed laser deposition, 312 pulses, 112 purification, 341

Q quadrupole, xi, 235, 236, 241, 244 quantization, ix, 6, 11, 12, 19, 25, 27, 57, 161, 162, 168, 182

381

quantum, ix, 16, 135, 164, 198, 215, 216, 220, 250, 258, 273, 274, 276, 277, 278, 288, 308, 363 quantum fluctuations, 273, 274, 276, 278 quantum state, 164, 198 quantum theory, 16 quartz, 358 quasiclassical, 163, 168 quasiparticle, viii, 81, 82, 83, 84, 85, 86, 87, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 136, 139, 143, 149, 195, 206, 207, 278 quasiparticles, viii, 81, 89, 90, 92, 93, 94, 95, 97, 98, 138, 223, 274

R radiation, 112, 119, 123, 128 radical, 296, 297 radius, 38, 249, 281 Raman, ix, 101, 102, 106, 120, 123, 129 Raman spectra, 120 Raman spectroscopy, 102, 120, 123 random, xi, 105, 111, 122, 123, 166, 213, 214, 215, 216, 220, 223, 224, 225, 226, 227, 228, 232, 249, 316 random walk, 224, 225 range, ix, x, xii, 3, 82, 89, 109, 112, 126, 143, 146, 152, 161, 162, 163, 164, 170, 173, 174, 175, 176, 177, 178, 180, 182, 189, 195, 204, 213, 214, 215, 216, 217, 220, 221, 222, 223, 226, 244, 248, 269, 270, 272, 273, 274, 275, 276, 278, 279, 280, 282, 284, 287, 297, 308, 309, 315, 320, 321, 340, 341, 350, 353, 357, 358, 359, 363, 368 rare earth, 103 rare earth elements, 103 RAS, 121, 126 reasoning, 281 recall, 90, 92, 95 recombination, 102, 112, 122, 129, 214 recombination processes, 102 recovery, 247 reduction, 3, 4, 35, 109, 110, 112, 120, 121, 125, 127, 149, 154, 156, 168, 177, 180, 218, 221, 286, 305, 308, 316, 353 reflection, ix, 135, 137, 138, 139, 143, 149 regular, viii, 81, 83, 84, 95, 194, 329 relationship, xii, 102, 269, 270, 276, 281, 284, 287, 303, 305, 307 relationships, 305, 307 relative size, 280 relaxation, viii, xi, 101, 102, 108, 110, 111, 114, 118, 119, 123, 125, 128, 129, 235, 236, 237, 247, 248, 249, 250, 251, 252, 253, 254, 255, 272, 274, 277, 278, 279, 280, 367 relaxation process, 111, 114

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382

Index

relaxation processes, 111, 114 relaxation rate, xi, 102, 119, 125, 235, 237, 249, 250, 252, 254, 255 relaxation time, xi, 102, 111, 118, 119, 236, 237, 247, 272, 274, 277, 278, 279, 280 relevance, 306, 308 renormalization, viii, 13, 15, 81, 83, 85, 87, 96, 250 research, vii, ix, xiii, 135, 136, 214, 270, 308, 339, 362 researchers, 240 reservoir, 128, 136, 141, 148 reservoirs, 102, 103 resistance, 40, 42, 103, 106, 107, 108, 109, 110, 112, 117, 119, 120, 125, 127, 128, 142, 145, 150, 151, 170, 176, 180, 182, 270, 273, 274, 277, 347, 356 resistive, xi, 137, 269, 270, 271, 275 resistivity, x, xiii, 104, 105, 108, 109, 110, 112, 114, 120, 136, 142, 143, 146, 154, 170, 176, 213, 223, 228, 229, 230, 231, 241, 242, 271, 272, 274, 275, 278, 279, 309, 339, 340, 347, 350, 353, 354, 355, 356, 363, 368 resolution, xiii, 2, 82, 94, 97, 311, 312, 328, 329, 334, 344 retardation, viii, 81, 83, 87, 96 rolling, 315, 316, 325 room temperature, viii, 101, 102, 106, 107, 110, 111, 115, 116, 117, 123, 125, 129, 152, 252, 270, 341, 347, 351, 363 root-mean-square, 224, 225, 227 Rössler, 233 roughness, 313 ruthenium, 148

S sample, 92, 99, 107, 108, 110, 111, 122, 125, 126, 127, 128, 141, 142, 143, 144, 162, 245, 250, 254, 277, 278, 279, 282, 315, 316, 318, 323, 328, 329, 341, 344, 347, 348, 355, 358, 359, 363, 365 sampling, 298 saturation, 108, 115, 122 scalar, 8, 10, 60, 65, 74, 164, 165, 168 scaling, 148, 190, 258, 271, 272, 275, 276, 278, 279, 287, 353, 354, 355 scaling relations, 272, 287 scatter, 164, 262 scattering, ix, 3, 39, 114, 120, 123, 126, 127, 137, 139, 161, 164, 165, 166, 167, 168, 170, 173, 174, 178, 180, 182, 237, 241, 248, 249, 252, 262, 274, 275, 285, 287, 288, 353, 356 Schmid, 370 Schrödinger equation, 12, 52, 53, 54 scripts, 300 search, vii, 136, 297, 303

secular, ix, 61, 69, 161, 164, 172, 176, 182 selecting, 365 SEM, 315 semiconductor, 104, 129 semiconductors, 112, 122, 348 sensitivity, 305 separation, 2, 16, 109, 214, 240 series, xiii, 56, 58, 63, 64, 70, 144, 145, 146, 150, 156, 172, 301, 302, 305, 339, 350 services, iv shape, viii, 49, 50, 81, 83, 120, 150, 156, 237, 254, 258, 325 shear, 216, 225 shock, 114 short period, 125 short-range, x, 3, 213, 214, 215, 216, 217, 221, 222, 223 shoulder, 287 SIGMA, 341 sign, x, 44, 71, 73, 213, 223, 228, 229, 231, 232, 350 signals, 250, 358 signs, 10, 138 silica, 313 silicon, 348 silver, 312, 313, 315, 318, 319, 327, 328, 351 similarity, 107, 111, 123, 156, 244 simulation, 52, 197 simulations, 244, 315 Singapore, 212, 336 single crystals, 106, 121, 241, 271, 340 singular, 28, 44 singularities, 70, 205, 206 sintering, 341, 344, 345 sites, xi, xiii, 50, 83, 105, 106, 118, 167, 226, 229, 230, 231, 232, 235, 241, 246, 311, 312, 318, 324, 325, 326, 327, 328, 333, 335 Slater determinants, 15 Slovakia, 1, 187, 211 Sm, 273, 276 smoothness, 176 software, 143, 313, 315, 317, 318, 336 solid state, xi, 4, 5, 6, 30, 141, 236, 269 solid-state, 14, 69, 89 spatial, xiii, 10, 169, 172, 311, 312, 316, 329 specific heat, 5, 39, 40, 46, 48, 51, 78, 88, 188, 206, 271, 284, 366 spectroscopy, ix, 5, 37, 51, 82, 94, 97, 121, 123, 126, 135, 136, 137, 149, 188, 210, 368 spectrum, vii, x, xi, 2, 5, 7, 9, 10, 31, 33, 36, 37, 38, 39, 42, 45, 46, 51, 55, 66, 71, 76, 77, 78, 145, 146, 147, 153, 154, 173, 187, 188, 189, 190, 194, 200, 201, 204, 206, 207, 208, 216, 235, 236, 237, 241, 244, 261, 262, 309, 356, 358

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Index speed, 313 spin, ix, xi, 10, 29, 96, 136, 150, 152, 154, 155, 156, 157, 161, 162, 164, 165, 166, 167, 168, 169, 170, 173, 174, 177, 178, 180, 182, 214, 235, 236, 237, 241, 243, 245, 246, 247, 248, 249, 250, 251, 252, 253, 254, 255, 256, 257, 258, 261, 262, 263, 270, 282, 296, 300, 357, 358, 367 spin polarized electrons, 155 sputtering, 106, 152, 348 square lattice, 261 SQUID, 357 stability, 128, 130, 277, 297, 302, 308 stabilization, viii, x, 2, 5, 18, 28, 35, 42, 44, 45, 51, 187, 189, 195, 198, 199, 200, 207, 208 stabilize, x, 5, 17, 36, 42, 46, 187, 200, 207 stages, 236, 369 statistical mechanics, 274 statistics, 29, 32, 48, 84, 207 stiffness, 297 STM, 296, 298 STO, 152, 190 stoichiometry, 244, 340, 344, 345, 346, 348 strain, 350 strength, 23, 27, 92, 93, 94, 95, 97, 121, 137, 143, 146, 149, 154, 162, 165, 168, 173, 174, 176, 177, 178, 192, 216, 273, 276, 297, 299, 367 stress, 2, 83, 296, 302, 306 stretching, 34, 38, 48, 50, 189, 192 strikes, 348 stripe patterns, 312 structural changes, ix, 101, 102, 111, 118, 119, 122 structural characteristics, 309 structural defect, 128, 130 structural defects, 128, 130 structural sensitivity, 305 STRUCTURE, 81 substances, viii, 101, 105 substitution, xiii, 41, 127, 128, 130, 152, 339, 340, 341, 358, 359, 361, 368 substrates, 106, 109, 152, 241, 349 successive approximations, 88 Sun, 370 superconducting gap, 154, 155 superconducting materials, 271, 288 superconductivity, vii, viii, ix, xi, 2, 45, 46, 48, 77, 79, 81, 82, 83, 87, 96, 97, 98, 106, 131, 135, 136, 137, 140, 146, 148, 149, 156, 161, 162, 163, 168, 169, 170, 173, 174, 175, 176, 177, 178, 182, 208, 214, 219, 235, 236, 252, 258, 261, 269, 270, 280, 288, 289, 296, 297, 298, 302, 306, 307, 308, 309, 340, 353, 356, 362, 368, 370 superconductor, ix, xii, 2, 5, 35, 51, 96, 129, 136, 137, 138, 139, 145, 148, 149, 150, 151, 154, 156,

383

161, 162, 188, 210, 215, 218, 236, 237, 248, 270, 271, 280, 295, 296, 311, 313, 335, 363 superfluid, 168, 207, 271, 272, 276, 278, 279, 284, 287, 288, 353, 363, 367 superposition, 125 suppression, 120, 146, 154, 163, 178, 241, 262, 287 surface area, 313 surface layer, 348 surface properties, 106 susceptibility, xiii, 236, 237, 245, 246, 247, 248, 249, 250, 252, 262, 269, 271, 273, 339, 340, 347, 357, 358, 359, 368 s-wave, ix switching, 163, 306 symbols, 66, 85, 262 symmetry, vii, ix, 1, 4, 7, 18, 29, 30, 33, 38, 49, 51, 53, 71, 83, 84, 87, 135, 136, 137, 139, 141, 143, 144, 148, 153, 156, 161, 162, 168, 169, 173, 188, 189, 191, 192, 194, 198, 199, 206, 207, 208, 210, 236, 241, 252, 299, 306, 309 synchronous, 110 synthesis, xiii, 339 systems, ix, xi, 2, 6, 13, 17, 80, 83, 84, 88, 94, 96, 102, 127, 130, 135, 136, 137, 157, 163, 190, 210, 211, 235, 252, 261, 262, 284, 297, 307, 309, 312, 350, 353

T Taiwan, 213 targets, 349 TBG, 272 technology, 162 temperature dependence, ix, xi, 5, 33, 40, 51, 102, 103, 104, 109, 112, 113, 114, 116, 118, 123, 126, 135, 137, 146, 148, 154, 155, 156, 188, 236, 246, 270, 271, 272, 275, 277, 340, 363 terminals, 152 Tesla, 230, 245, 250, 358, 365 Thailand, 161, 182 thermal activation, 118 thermal energy, 102, 118, 224 thermal equilibrium, 247 thermodynamic, viii, 2, 5, 39, 46, 48, 51, 73, 74, 75, 78, 188, 275, 281, 284 thermodynamic properties, viii, 2, 5, 39, 46, 48, 51, 78, 188 thermodynamics, 41, 46, 48, 78 thin film, xiii, 104, 106, 109, 110, 114, 125, 128, 137, 163, 176, 223, 229, 232, 241, 276, 277, 312, 335, 339, 340, 341, 348, 350, 351, 355, 356, 362, 363, 364, 365, 366, 367

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Index

thin films, xiii, 104, 106, 109, 114, 125, 128, 137, 176, 223, 229, 232, 241, 312, 335, 339, 340, 341, 348, 355, 356, 362, 363, 365, 366, 367 Thomson, 347 three-dimensional, 219 thresholds, 129, 261 tics, 105 time, ix, 40, 47, 76, 102, 103, 108, 109, 110, 111, 112, 114, 115, 116, 118, 120, 121, 122, 123, 125, 145, 161, 162, 163, 165, 166, 167, 168, 173, 206, 223, 237, 247, 272, 274, 277, 278, 279, 280, 284, 307, 313, 328, 329, 369 tin, 48, 72, 192, 258 Togo, 290 Tokyo, 235, 265, 267, 268, 336 tolerance, 323, 324 topological, 190 topology, 188, 190, 191, 192, 193, 195, 200 total energy, 39, 67 traits, 137, 156 transfer, 5, 37, 51, 106, 112, 120, 121, 122, 123, 126, 129, 188, 248, 257 transformation, 14, 17, 18, 19, 20, 21, 22, 23, 25, 26, 27, 30, 31, 33, 36, 45, 48, 49, 50, 51, 55, 56, 57, 58, 61, 62, 63, 68, 69, 70, 71, 72, 75, 195, 216, 302, 306 transformation coefficients, 22, 23, 27, 30, 33, 61, 68, 70 transformation matrix, 19, 20, 22, 23, 25, 27, 48, 49, 50, 51, 62 transformation operators, 56 transformations, 4, 5, 18, 20, 45, 46, 56, 57, 58, 60, 63, 65, 71, 170, 216, 302, 307 transition, vii, viii, x, xi, 1, 2, 3, 4, 24, 36, 38, 39, 40, 44, 48, 49, 50, 51, 84, 103, 105, 106, 108, 112, 113, 114, 129, 140, 141, 142, 152, 164, 187, 188, 192, 195, 197, 204, 207, 208, 210, 213, 214, 215, 218, 220, 223, 228, 229, 230, 231, 232, 236, 237, 245, 246, 254, 256, 261, 269, 270, 271, 272, 273, 275, 276, 277, 278, 279, 280, 281, 302, 307, 347, 353, 365 transition metal, 237, 245, 246 transition temperature, 84, 105, 140, 152, 214, 223, 228, 254, 256, 261, 273, 281, 347, 365 transitions, 17, 53, 62, 112, 113, 121, 302, 306, 309, 353 translation, 11, 30, 33, 34, 38, 49, 199, 207, 208 translational, 164 transmission, 149 transparency, 180 transparent, 138, 170

transport, xiii, 102, 120, 121, 124, 138, 139, 143, 150, 153, 154, 156, 164, 167, 168, 182, 225, 270, 302, 306, 307, 308, 311, 316, 323, 327, 340, 356 transport phenomena, 156 transpose, 166 traps, 102, 118, 119, 122, 129 trend, 305, 359 triggers, 111 tunneling, 5, 37, 51, 104, 117, 119, 136, 140, 142, 143, 146, 150, 155, 188, 208, 210, 305, 308 two-dimensional, x, 87, 213, 214

U ubiquitous, 296 ultrasound, 142, 152 unconventional superconductors, 149 uniform, xi, 162, 163, 236, 237, 245, 246, 252, 254, 262, 296, 297, 299, 306, 315, 345, 350, 358 universality, 271, 307 uranium, 329 uranium oxide, 329 UV, 113, 129

V vacancies, 105, 113, 120, 121, 122, 124, 125, 127, 128, 129 vacuum, 8, 10, 11, 29, 31, 33, 43, 45, 67, 72, 73, 76 valence, xii, 39, 48, 49, 88, 112, 148, 151, 190, 237, 239, 254, 295, 296, 298, 303, 304 validity, vii, 1, 2, 7, 13, 18, 24, 48, 51, 52, 70, 83, 94, 115, 118, 355 values, xiii, 3, 49, 88, 89, 90, 92, 94, 96, 97, 104, 105, 106, 118, 124, 143, 146, 156, 168, 177, 180, 181, 197, 218, 220, 222, 232, 240, 254, 271, 273, 274, 276, 278, 280, 282, 283, 284, 285, 286, 287, 297, 298, 299, 300, 303, 306, 308, 315, 317, 339, 347, 348, 350, 353, 354, 356, 359, 360, 361, 362, 363, 365, 368 variables, 18, 20, 21, 62, 63, 71 variation, 2, 84, 123, 127, 140, 282, 289, 299, 309 vector, 6, 11, 12, 29, 92, 163, 164, 165, 168, 170, 174, 175, 176, 177, 178, 182, 225, 247, 340 velocity, x, 2, 90, 92, 94, 95, 97, 140, 187, 189, 192, 199, 210, 226, 283 vibration, 3, 4, 6, 7, 12, 13, 14, 15, 16, 17, 23, 24, 27, 29, 30, 31, 33, 34, 35, 38, 48, 50, 55, 56, 57, 60, 61, 65, 66, 67, 69, 189, 192, 193, 195 vibrational modes, 120 Victoria, 369 visible, 94, 101, 106, 119, 120, 127, 129, 180, 297, 315, 319, 324, 325, 326, 330 voids, 318

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Index vortex, x, xi, 213, 214, 215, 216, 217, 218, 219, 220, 221, 222, 223, 224, 225, 226, 227, 228, 229, 230, 231, 232, 236, 269, 270, 271, 272, 273, 274, 275, 276, 277, 278, 279, 280, 282, 288 vortices, 224, 270, 271, 272, 273, 274, 275, 276, 279, 288

W wave vector, 6, 29, 92, 164, 170, 171, 174, 175, 176, 177, 178, 182, 247 weight loss, 341 wires, 351 workstation, 313

X X-ray analysis, 312, 317 X-ray diffraction (XRD), 341, 342, 343, 344, 350, 351

Y

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YBCO, xiii, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 150, 151, 152, 153, 154, 155, 156, 188, 189, 191, 204, 210, 311, 312, 313, 314, 315, 318, 320, 327, 328, 329, 330, 331, 332, 333, 334, 335 yield, 3, 15, 47, 82, 125, 167, 188, 195, 197, 204, 237, 244, 249, 308, 317, 350, 353

Z ZnO, 341

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385