Alexandrov A.S.,Theory of Superconductivity - From Weak to Strong Coupling
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Theory of Superconductivity
From Weak to Strong Coupling
Series in Condensed Matter Physics
Series Editors:
JMDCoey, D R Tilley and DRVij
Other titles in the series include:
Nonlinear Dynamics and Chaos in Semiconductors
K Aoki
Modern Magnetooptics and Magnetooptical Materials
A K Zvedin and V A Kotov
Permanent Magnetism
RSkomskiandJMDCoey
Series in Condensed Matter Physics
Theory of Superconductivity
From Weak to Strong Coupling
A S Alexandrov
Loughborough University, UK
Institute of Physics Publishing
Bristol and Philadelphia
c
IOP Publishing Ltd 2003
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agreement with Universities UK (UUK).
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
ISBN 0 7503 0836 2
Library of Congress Cataloging-in-Publication Data are available
Commissioning Editor: Tom Spicer
Production Editor: Simon Laurenson
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Marketing: Nicola Newey and Verity Cooke
Published by Institute of Physics Publishing, wholly owned by The Institute of
Physics, London
Institute of Physics Publishing, Dirac House, Temple Back, Bristol BS1 6BE, UK
US Office: Institute of Ph ysics Publishin g, The Public Ledger Building, Suite
929, 150 South Independence Mall West, Philadelphia, PA 19106, USA
Typeset in L
A
T
E
X2
ε
by Text 2 Text, Torquay, Devon
Printed in the UK by MPG Books Ltd, Bodmin, Cornwall
Contents
Preface ix
Introduction xi
PART 1
Theory 1
1 Phenomenology 3
1.1 Normal-state Boltzmann kinetics 3
1.2 London theory 6
1.3 Flux quantization 10
1.4 Ogg’s pairs 11
1.5 Pippard and London superconductors 12
1.6 Ginzburg–Landau theory 13
1.6.1 Basic equations 13
1.6.2 Surface energy and thermodynamic critical field 17
1.6.3 Single vortex and lower critical field 20
1.6.4 Upper critical field 23
1.6.5 Vortex lattice 26
1.6.6 Critical current 28
1.7 Josephson tunnelling 29
2 Weak coupling theory 33
2.1 BCS Hamiltonian 33
2.2 Ground state and excitations 35
2.3 Meissner–Ochsenfeld effect 40
2.4 BCS gap, critical temperature and single-electron tunnelling 41
2.5 Isotope effect 44
2.6 Heat capacity 45
2.7 Sound attenuation 46
2.8 Nuclear spin relaxation rate 48
2.9 Thermal conductivity 49
2.10 Unconventional Cooper pairing 50
vi
Contents
2.11 Bogoliubov equations 52
2.12 Landau criterion and gapless superconductivity 55
2.13 Andreev reflection 57
2.14 Green’s function formulation of the BCS theory, T = 059
2.15 Green’s functions of the BCS superconductor at finite temperatures 62
2.16 Microscopic derivation of the Ginzburg–Landau equations 66
3 Intermediate-coupling theory 73
3.1 Electron–phonon interaction 73
3.2 Phonons in metal 77
3.3 Electrons in metal 81
3.4 Eliashberg equations 85
3.5 Coulomb pseudopotential 89
3.6 Cooper pairing of repulsive fermions 91
4 Strong-coupling theory 95
4.1 Electron–phonon and Coulomb interactions in the Wannier
representation 96
4.2 Breakdown of Migdal–Eliashberg theory in the strong-coupling
regime 98
4.3 Polaron dynamics 102
4.3.1 Polaron band 102
4.3.2 Damping of the polaron band 105
4.3.3 Small Holstein polaron and small Fr¨ohlich polaron 107
4.3.4 Polaron spectral and Green’s functions 110
4.4 Polaron-polaron interaction and bipolaron 116
4.5 Polaronic superconductivity 119
4.6 Mobile small bipolarons 122
4.6.1 On-site bipolarons and bipolaronic Hamiltonian 122
4.6.2 Inter-site bipolaron in the chain model 126
4.6.3 Superlight inter-site bipolarons 129
4.7 Bipolaronic superconductivity 135
4.7.1 Bipolarons and a charged Bose gas 135
4.7.2 Bogoliubov equations in the strong-coupling regime 138
4.7.3 Excitation spectrum and ground-state energy 140
4.7.4 Mixture of two Bose condensates 142
4.7.5 Critical temperature and isotope effect 145
4.7.6 Magnetic field expulsion 147
4.7.7 Charged vortex and lower critical field 148
4.7.8 Upper critical field in the strong-coupling regime 152
4.7.9 Symmetry of the order parameter 156
Contents
vii
PART 2
Applications to high-T
c
superconductors 159
5 Competing interactions in unconventional superconductors 161
5.1 High-T
c
superconductors: different concepts 161
5.2 Band structure and essential interactions in cuprates 168
5.3 Low Fermi energy: pairing is individual in many cuprates 171
5.4 Bipolaron bands in high-T
c
perovskites 174
5.4.1 Apex bipolarons 174
5.4.2 In-plane bipolarons 177
5.5 Bipolaron model of cuprates 180
6 Normal state of cuprates 184
6.1 In-plane resistivity and Hall ratio 184
6.2 Normal-state resistivity below T
c
188
6.3 Lorenz number: evidence for bipolarons 192
6.4 Spin pseudogap in NMR 194
6.5 c-axis transport and charge pseudogap 195
6.6 Infrared conductivity 198
7 Superconducting transition 200
7.1 Parameter-free description of T
c
200
7.2 Isotope effect on T
c
and on supercarrier mass 205
7.3 Specific heat anomaly 207
7.4 Universal upper critical field of unconventional superconductors 209
8 Superconducting state of cuprates 212
8.1 Low-temperature penetration depth 212
8.2 SIN tunnelling and Andreev reflection 216
8.3 SIS tunnelling 222
8.4 ARPES 226
8.4.1 Photocurrent 227
8.4.2 Self-energy of one-dimensional hole in a non-crossing
approximation 228
8.4.3 Exact spectral function of a one-dimensional hole 230
8.4.4 ARPES in Y124 and Y123 231
8.5 Sharp increase of the quasi-particle lifetime below T
c
234
8.6 Symmetry of the order parameter and stripes 238
9 Conclusion 243
A Bloch states 244
A.1 Bloch theorem 244
A.2 Effective mass approximation 247
A.3 Tight-binding approximation 249
viii
Contents
B Quantum statistics and Boltzmann kinetics 252
B.1 Grand partition function 252
B.2 Fermi–Dirac and Bose–Einstein distribution functions 253
B.3 Ideal Fermi gas 254
B.3.1 Fermi energy 254
B.3.2 Specific heat 255
B.3.3 Pauli paramagnetism, Landau diamagnetism and
de Haas–van Alphen quantum oscillations 257
B.4 Ideal Bose gas 261
B.4.1 Bose–Einstein condensation temperature 261
B.4.2 Third-order phase transition 262
B.5 Boltzmann equation 264
C Second quantization 265
C.1 Slater determinant 265
C.2 Annihilation and creation operators 267
C.3 -operators 271
D Analytical properties of one-particle Green’s functions 274
E Canonical transformation 279
References 283
Index 294
Preface
The observation of high-temperature superconductivity in complex layered
cuprates by Bednorz and M¨uller in 1986 should undoubtedly be rated as one
of the greatest experimental discoveries of the last century, whereas identifying
and understanding the microscopic origin of high-temperature superconductivity
stands as one of the greatest theoretical challenges of this century. This book is
conceived as a fairly basic introduction to the modern theory of superconductivity.
It also sets out an approach to the problem of high-temperature superconductivity,
based on the extension of the BCS theory to the strong-coupling regime. The
book starts with the phenomenological ideas by F and H London, Ogg Jr and
Shafroth-Butler-Blatt, Ginzburg and Landau, Pippard, and proceeds with the
microscopic weak-coupling theory by Bardeen, Cooper and Schrieffer (BCS).
The canonical Migdal–Eliashberg extension of the BCS theory to the intermediate
coupling strength of electrons and phonons (or any bosons) is also discussed.
Then, proceeding from the dynamic properties of a single polaron, it is shown
how the BCS theory can be extended to the strong-coupling regime, where
the multi-polaron problem is reduced to a charged Bose gas of bipolarons.
Finally, applications of the theory to cuprates are presented in greater detail
in part II along with a brief discussion of a number of alternative viewpoints.
Superconductivity and, in particular, high-temperature superconductivity is the
topic covered in senior graduate and postgraduate courses virtually in every
physics department. Therefore the book contains introductions to the Bloch
states, quantum statistics and Boltzmann kinetics, second quantization, Green’s
functions, and canonical transformations in the appendices to make it easy to
follow for senior undergraduate and graduate students with a basic knowledge
of quantum mechanics. The book could also be seen as an attempt to bring the
level of university training up to the level of modern theoretical condensed matter
physics.
A S Alexandrov
1 January 2003
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