Theory of SuperconductivityFrom Weak to Strong Coupling

Series in Condensed Matter Physics

Series Editors:

J M D Coey, D R Tilley and D R Vij

Other titles in the series include:

Nonlinear Dynamics and Chaos in SemiconductorsK Aoki

Modern Magnetooptics and Magnetooptical MaterialsA K Zvedin and V A Kotov

Permanent MagnetismR Skomski and J M D Coey

Series in Condensed Matter Physics

Theory of SuperconductivityFrom Weak to Strong Coupling

A S Alexandrov

Loughborough University, UK

Institute of Physics PublishingBristol and Philadelphia

c© IOP Publishing Ltd 2003

All rights reserved. No part of this publication may be reproduced, storedin a retrieval system or transmitted in any form or by any means, electronic,mechanical, photocopying, recording or otherwise, without the prior permissionof the publisher. Multiple copying is permitted in accordance with the termsof licences issued by the Copyright Licensing Agency under the terms of itsagreement with Universities UK (UUK).

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Contents

Preface ix

Introduction xi

PART 1Theory 1

1 Phenomenology 31.1 Normal-state Boltzmann kinetics 31.2 London theory 61.3 Flux quantization 101.4 Ogg’s pairs 111.5 Pippard and London superconductors 121.6 Ginzburg–Landau theory 13

1.6.1 Basic equations 131.6.2 Surface energy and thermodynamic critical field 171.6.3 Single vortex and lower critical field 201.6.4 Upper critical field 231.6.5 Vortex lattice 261.6.6 Critical current 28

1.7 Josephson tunnelling 29

2 Weak coupling theory 332.1 BCS Hamiltonian 332.2 Ground state and excitations 352.3 Meissner–Ochsenfeld effect 402.4 BCS gap, critical temperature and single-electron tunnelling 412.5 Isotope effect 442.6 Heat capacity 452.7 Sound attenuation 462.8 Nuclear spin relaxation rate 482.9 Thermal conductivity 492.10 Unconventional Cooper pairing 50

vi Contents

2.11 Bogoliubov equations 522.12 Landau criterion and gapless superconductivity 552.13 Andreev reflection 572.14 Green’s function formulation of the BCS theory, T = 0 592.15 Green’s functions of the BCS superconductor at finite temperatures 622.16 Microscopic derivation of the Ginzburg–Landau equations 66

3 Intermediate-coupling theory 733.1 Electron–phonon interaction 733.2 Phonons in metal 773.3 Electrons in metal 813.4 Eliashberg equations 853.5 Coulomb pseudopotential 893.6 Cooper pairing of repulsive fermions 91

4 Strong-coupling theory 954.1 Electron–phonon and Coulomb interactions in the Wannier

representation 964.2 Breakdown of Migdal–Eliashberg theory in the strong-coupling

regime 984.3 Polaron dynamics 102

4.3.1 Polaron band 1024.3.2 Damping of the polaron band 1054.3.3 Small Holstein polaron and small Frohlich polaron 1074.3.4 Polaron spectral and Green’s functions 110

4.4 Polaron-polaron interaction and bipolaron 1164.5 Polaronic superconductivity 1194.6 Mobile small bipolarons 122

4.6.1 On-site bipolarons and bipolaronic Hamiltonian 1224.6.2 Inter-site bipolaron in the chain model 1264.6.3 Superlight inter-site bipolarons 129

4.7 Bipolaronic superconductivity 1354.7.1 Bipolarons and a charged Bose gas 1354.7.2 Bogoliubov equations in the strong-coupling regime 1384.7.3 Excitation spectrum and ground-state energy 1404.7.4 Mixture of two Bose condensates 1424.7.5 Critical temperature and isotope effect 1454.7.6 Magnetic field expulsion 1474.7.7 Charged vortex and lower critical field 1484.7.8 Upper critical field in the strong-coupling regime 1524.7.9 Symmetry of the order parameter 156

Contents vii

PART 2Applications to high-Tc superconductors 159

5 Competing interactions in unconventional superconductors 1615.1 High-Tc superconductors: different concepts 1615.2 Band structure and essential interactions in cuprates 1685.3 Low Fermi energy: pairing is individual in many cuprates 1715.4 Bipolaron bands in high-Tc perovskites 174

5.4.1 Apex bipolarons 1745.4.2 In-plane bipolarons 177

5.5 Bipolaron model of cuprates 180

6 Normal state of cuprates 1846.1 In-plane resistivity and Hall ratio 1846.2 Normal-state resistivity below Tc 1886.3 Lorenz number: evidence for bipolarons 1926.4 Spin pseudogap in NMR 1946.5 c-axis transport and charge pseudogap 1956.6 Infrared conductivity 198

7 Superconducting transition 2007.1 Parameter-free description of Tc 2007.2 Isotope effect on Tc and on supercarrier mass 2057.3 Specific heat anomaly 2077.4 Universal upper critical field of unconventional superconductors 209

8 Superconducting state of cuprates 2128.1 Low-temperature penetration depth 2128.2 SIN tunnelling and Andreev reflection 2168.3 SIS tunnelling 2228.4 ARPES 226

8.4.1 Photocurrent 2278.4.2 Self-energy of one-dimensional hole in a non-crossing

approximation 2288.4.3 Exact spectral function of a one-dimensional hole 2308.4.4 ARPES in Y124 and Y123 231

8.5 Sharp increase of the quasi-particle lifetime below Tc 2348.6 Symmetry of the order parameter and stripes 238

9 Conclusion 243

A Bloch states 244A.1 Bloch theorem 244A.2 Effective mass approximation 247A.3 Tight-binding approximation 249

viii Contents

B Quantum statistics and Boltzmann kinetics 252B.1 Grand partition function 252B.2 Fermi–Dirac and Bose–Einstein distribution functions 253B.3 Ideal Fermi gas 254

B.3.1 Fermi energy 254B.3.2 Specific heat 255B.3.3 Pauli paramagnetism, Landau diamagnetism and

de Haas–van Alphen quantum oscillations 257B.4 Ideal Bose gas 261

B.4.1 Bose–Einstein condensation temperature 261B.4.2 Third-order phase transition 262

B.5 Boltzmann equation 264

C Second quantization 265C.1 Slater determinant 265C.2 Annihilation and creation operators 267C.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 layeredcuprates by Bednorz and Muller in 1986 should undoubtedly be rated as oneof the greatest experimental discoveries of the last century, whereas identifyingand understanding the microscopic origin of high-temperature superconductivitystands as one of the greatest theoretical challenges of this century. This book isconceived 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. Thebook starts with the phenomenological ideas by F and H London, Ogg Jr andShafroth-Butler-Blatt, Ginzburg and Landau, Pippard, and proceeds with themicroscopic weak-coupling theory by Bardeen, Cooper and Schrieffer (BCS).The canonical Migdal–Eliashberg extension of the BCS theory to the intermediatecoupling strength of electrons and phonons (or any bosons) is also discussed.Then, proceeding from the dynamic properties of a single polaron, it is shownhow the BCS theory can be extended to the strong-coupling regime, wherethe multi-polaron problem is reduced to a charged Bose gas of bipolarons.Finally, applications of the theory to cuprates are presented in greater detailin part II along with a brief discussion of a number of alternative viewpoints.Superconductivity and, in particular, high-temperature superconductivity is thetopic covered in senior graduate and postgraduate courses virtually in everyphysics department. Therefore the book contains introductions to the Blochstates, quantum statistics and Boltzmann kinetics, second quantization, Green’sfunctions, and canonical transformations in the appendices to make it easy tofollow for senior undergraduate and graduate students with a basic knowledgeof quantum mechanics. The book could also be seen as an attempt to bring thelevel of university training up to the level of modern theoretical condensed matterphysics.

A S Alexandrov1 January 2003

ix

Introduction

In 1986 Bednorz and Muller [1] discovered the onset of possible superconductiv-ity at exceptionally high temperatures in a black ceramic material comprising fourelements: lanthanum, barium, copper and oxygen. Within the next decade manymore complex copper oxides (cuprates) were synthesized including the mercury-cuprate compounds which, to date, have the highest confirmed critical temper-ature for a superconducting transition, some Tc = 135 K at room pressure andapproximately 160 K under high (applied) pressure. The new phenomenon initi-ated by Bednorz and Muller broke all constraints on the maximum Tc predictedby the conventional theory of low-temperature superconducting metals and theiralloys. These discoveries could undoubtedly result in large-scale commercial ap-plications for cheap and efficient electricity production, provided long lengths ofsuperconducting wires operating above the liquid nitrogen temperature (ca. 80 K)can be routinely manufactured.

Following Kamerlingh–Onnes’ discovery of superconductivity in elementalmercury in 1911, subsequent work revealed that many metals and alloys displayedsimilar superconducting properties, the transition temperature of the alloy Nb3Geat 23 K being the highest recorded prior to the discoveries in the high-Tccuprates. Despite intense efforts worldwide, no adequate explanation of thesuperconductivity phenomenon appeared until the work by Bardeen, Cooper andSchrieffer in 1957 [2]. By that time the frictionless flow (i.e. superfluidity) ofliquid 4He had been discovered below a temperature of some 2.17 K. It had beenknown that the helium atom with its two protons, two neutrons and two electrons,was a Bose particle (boson), while its isotope 3He, with two protons and only oneneutron, is a fermion. An assembly of bosons obey the Bose–Einstein statistics,which allows all of them to occupy a single quantum state. Fermions obey thePauli exclusion principle and the Fermi–Dirac statistics, which dictate that twoidentical particles must not occupy the same quantum state.

F London suggested in 1938 that the remarkable superfluid properties of4He were intimately linked to the Bose–Einstein ‘condensation’ of the entireassembly of Bose particles [3]. Nine years later, Bogoliubov [4] and Landau [5]explained how Bose statistics can lead to the frictionless flow of a liquid. Thebosons in the lowest energy state within the Bose–Einstein condensate thus forma coherent macromolecule. As soon as one Bose particle in the Bose liquid meets

xi

xii Introduction

an obstacle to its flow, for example in the form of an impurity, all the others do notallow their condensate partner to be scattered to leave the condensate. A crucialdemonstration that superfluidity was linked to the Bose particles and the Bose–Einstein condensation came after experiments on liquid 3He, whose atoms werefermions, which failed to show the characteristic superfluid transition within areasonable wide temperature interval around the critical temperature for the onsetof superfluidity in 4He. In sharp contrast, 3He becomes a superfluid only below avery low temperature of some 0.0026 K. Here we have a superfluid formed frompairs of two 3He fermions below this temperature.

The three orders-of-magnitude difference between the critical superfluiditytemperatures of 4He and 3He kindles the view that Bose–Einstein condensationmight represent the ‘smoking gun’ of high-temperature superconductivity.‘Unfortunately’ electrons are fermions. Therefore, it is not at all surprisingthat the first proposal for high-temperature superconductivity, made by Ogg Jrin 1946 [6], involved the pairing of individual electrons. If two electrons arechemically coupled together, the resulting combination is a boson with total spinS = 0 or 1. Thus, an ensemble of such two-electron entities can, in principle,be condensed into the Bose–Einstein superconducting condensate. This idea wasfurther developed as a natural explanation of superconductivity by Schafroth [7]and Butler and Blatt in 1955 [8].

However, with one or two exceptions, the Ogg–Schafroth picture wascondemned and practically forgotten because it neither accounted quantitativelyfor the critical parameters of the ‘old’ (i.e. low Tc) superconductors nor did itexplain the microscopic nature of the attractive force which could overcome thenatural Coulomb repulsion between two electrons which constitute a Bose pair.The same model which yields a rather precise estimate of the critical temperatureof 4He leads to an utterly unrealistic result for superconductors, namely Tc =104 K with the atomic density of electron pairs of about 1022 cm−3, and with theeffective mass of each boson twice that of the electron, m∗∗ = 2me � 2×10−27 g.

The failure of this ‘bosonic’ picture of individual electron pairs became fullytransparent when Bardeen, Cooper and Schrieffer (BCS) [2] proposed that twoelectrons in a superconductor indeed formed a pair but of a very large (practicallymacroscopic) dimension—about 104 times the average inter-electron spacing.The BCS theory was derived from an early demonstration by Frohlich [9] thatconduction electrons in states near the Fermi energy could attract each other onaccount of their weak interaction with vibrating ions of a crystal lattice. Cooperthen showed that electron pairs were stable only due to their quantum interactionwith the other pairs. The ultimate BCS theory showed that in a small intervalround the Fermi energy, the electrons are correlated into pairs in the momentumspace. These Cooper pairs would strongly overlap in real space, in sharp contrastwith the model of non-overlapping (local) pairs discussed earlier by Ogg andSchafroth. Highly successful for metals and alloys with a low Tc, the BCStheory led the vast majority of theorists to the conclusion that there could be nosuperconductivity above 30 K, which implied that Nb3Ge already had the highest

Introduction xiii

Tc. While the Ogg–Schafroth phenomenology led to unrealistically high valuesof Tc, the BCS theory left perhaps only a limited hope for the discovery of newmaterials which could be superconducting at room temperatures or, at least, atliquid nitrogen temperatures.

It is now clear that the Ogg–Schafroth and BCS descriptions are actuallytwo opposite extremes of the same problem of electron–phonon interaction. Byextending the BCS theory towards the strong interaction between electrons andion vibrations, a Bose liquid of tightly bound electron pairs surrounded by thelattice deformation (i.e. of so-called small bipolarons) was naturally predicted[10]. Further prediction was that high temperature superconductivity could existin the crossover region of the electron–lattice interaction strength from the BCS-like to bipolaronic superconductivity [11, 12]. Compared with the early Ogg–Schafroth view, two fermions (now polarons) are bound into a bipolaron bylattice deformation. Such bipolaronic states are ‘dressed’ by the same latticedeformation [13] and, at first sight, they have a mass too large to be mobile.In fact, earlier studies [14, 15] considered small bipolarons as entirely localizedobjects. However, it has been shown later that small bipolarons are itinerantquasi-particles existing in the Bloch states at temperatures below the characteristicphonon frequency (chapter 4). As a result, the superconducting criticaltemperature, being proportional to the inverse mass of a bipolaron, was reducedin comparison with an ‘ultra-hot’ local-pair Ogg–Schafroth superconductivity butturned out to be much higher than the BCS theory prediction. Quite remarkablyBednorz and Muller noted in their original publication, and subsequently intheir Nobel Prize lecture [16], that in their ground-breaking search for high-Tcsuperconductivity, they were stimulated and guided by the polaron model. Theirexpectation [16] was that if ‘an electron and a surrounding lattice distortionwith a high effective mass can travel through the lattice as a whole, and astrong electron–lattice coupling exists an insulator could be turned into a hightemperature superconductor’.

The book naturally divides into two parts. Part 1 describes thephenomenology of superconductivity, the microscopic BCS theory and itsextension to the intermediate-coupling regime at a fairly basic level. Chapters 1–3 of this part are generally accepted themes in the conventional theory ofsuperconductivity. Chapter 4 describes what happens to the conventional theorywhen the electron–phonon coupling becomes strong. Part 2 describes keyphysical properties of high-temperature superconductors. Chapters 5–8 alsopresent the author’s particular view of cuprates, which is not yet generallyaccepted.

In the course of writing the book I have profited from valuable andstimulating discussions with P W Anderson, A F Andreev, A R Bishop,J T Devreese, P P Edwards, L P Gor’kov, Yu A Firsov, J E Hirsch, V V Kabanov,P E Kornilovitch, A P Levanyuk, W Y Liang, D Mihailovic, K A Muller,J R Schrieffer, S A Trugman, G M Zhao and V N Zavaritsky. Part of thewriting was done while I was on leave, from Loughborough University, as visiting

xiv Introduction

Professor at the Hewlett-Packard Laboratories in Palo Alto, California. I wish tothank R S Williams and A M Bratkovsky for arranging my visiting professorshipand enlightening discussions. I am thankful to my wife Elena and to my sonMaxim for their help and understanding.

PART 1

THEORY

Chapter 1

Phenomenology

At temperatures below some critical temperature Tc, many metals, alloys anddoped semiconducting inorganic and organic compounds carry the electric currentfor infinite time without any applied electric field. Their thermodynamicand electromagnetic properties in the superconducting state (below Tc) aredramatically different from the normal state properties above Tc. Quite a fewof them might be understood without any insight into the microscopic origin ofsuperconductivity by exploiting the analogy between superconducting electronsand superfluid neutral helium atoms (liquid 4He flows without any friction belowTc � 2.17 K). Realizing this analogy, F and H London developed the successfulphenomenological approach in 1935 describing the behaviour of superconductorsin the external magnetic field. Ogg Jr proposed a root to high-temperaturesuperconductivity introducing electron pairs in 1946 and Ginzburg and Landauproposed the phenomenological theory of the superconducting phase transition in1950 providing a comprehensive understanding of the electromagnetic propertiesbelow Tc.

1.1 Normal-state Boltzmann kinetics

If ions in a metal were to be perfectly ordered and were not to vibrate aroundtheir equilibrium positions, the electric resistivity would be zero. This is due tothe familiar interference of waves scattered off periodically arranged scatteringcentres. The electron wavefunctions in an ideal periodic potential are the Blochwaves, which obey the Schrodinger equation,[

− ∇2

2me+ V (r)

]ψnk(r) = Enkψnk(r) (1.1)

where V (r + l) = V (r) for any lattice vector l connecting two atoms. Here andlater, we use the ‘theoretical’ system of units, where the Planck and Boltzmannconstants and light velocity are unity (� = kB = c = 1). The momentum and

3

4 Phenomenology

the wavevector are the same in these units ( p = k). Single-particle states areclassified with the wavevector k in the Brillouin zone and with the band index n,so that

ψnk(r) = unk(r) exp(ik · r)

where unk(r) is a periodic function of r and the energy is quantized into bands(appendix A). When a weak electric field E is applied, the Bloch states behavelike free particles accelerated by the field in accordance with the classical Newtonlaw

dkdt

= −eE. (1.2)

Here e is the magnitude on the elementary charge. Impurities, lattice defectsand ion vibrations break down the translation symmetry, therefore equation (1.2)should be modified. When the interaction of electrons with these imperfectionsis weak, the electric current is found from the kinetic Boltzmann equation forthe distribution function f (r, k, t) in the real r and momentum k spaces. Thisfunction can be introduced if the characteristic length of field variations is fairlylong. The number of electrons in an elementary volume of this space at time t isdetermined by 2 f (r, k, t) dk dr/(2π)3. In the equilibrium state, the distributiondepends only on the energy E ≡ Enk (appendix B):

f (r, k, t) = nk ≡[

1 + expE − µ

T

]−1

. (1.3)

The Boltzmann equation for electrons in external electric, E, and magnetic, B,fields has the following form (appendix B):

∂ f

∂ t+ v · ∂ f

∂ r− e (E + v × B) · ∂ f

∂k=

(∂ f

∂ t

)c

(1.4)

where v = ∂ Enk/∂k is the group velocity. Here we consider a single band, so theband quantum number n can be dropped. By applying the Fermi–Dirac goldenrule, the collision integral in the right-hand side for any elastic scattering is givenby(

∂ f

∂ t

)c= 2πnim

∑q

V 2sc(q)δ(Ek − Ek+q)[ f (r, k + q, t) − f (r, k, t)] (1.5)

where Vsc(q) is the Fourier component of the scattering potential and nim isthe density of scattering centres. The form of this integral does not depend onthe particles’ statistics. For weak homogeneous stationary electric and magneticfields, the Boltzmann equation is solved by the substitution

f (r, k, t) = nk − ∂nk

∂ EF · v. (1.6)

Normal-state Boltzmann kinetics 5

We consider the case when the transport relaxation time defined as

1

τ (E)≡ 2πnim

∑k′

(1 − cos �)V 2sc(k′ − k)δ(Ek′ − Ek) (1.7)

depends only on energy E . Here � is the angle between v′ = ∂ Ek′/∂k′ and v.Then keeping the terms linear in the electric field E, the Boltzmann equation forthe function F becomes

F · v = −eτ (E)[E · v − F · (v × B · ∇k)v]. (1.8)

If the magnetic field is sufficiently weak (ωτ � 1; ω � B is the Larmourfrequency), one can keep only the terms linear in B with the following resultfor the non-equilibrium part of the distribution function:

F · v = −eτ (E)[E · v + eτ (E)E · (v × B · ∇k)v]. (1.9)

Using the current density

j = −2e∑

k

v f (r, k, t) = 2e∑

k

v∂nk

∂ Ev · F (1.10)

one obtains the longitudinal conductivity σx x ,

σx x = −2e2∑

k

∂nk

∂ Eτ (E)v2

x (1.11)

and the Hall conductivity σyx ,

σyx = −2e3B∑

k

∂nk

∂ Eτ 2(E)(v2

ym−1x x − vyvx m−1

yx ). (1.12)

Here m−1αβ = ∂2 Ek/∂kα∂kβ is the inverse mass tensor (α, β are x, y, z).

To calculate the longitudinal conductivities (σx x and σyy) and the Hallcoefficient RH = −σxy/(Bσx xσyy), we apply the effective mass approximation(appendix A), assuming for simplicity that the inverse mass tensor is diagonal(m−1

αβ = δαβ/mα). If the transport relaxation time is independent of energy(τ (E) = τ ), we obtain, integrating by parts,

σx x = ne2τ

mx(1.13)

and

σyx = ne3 Bτ 2

mymx. (1.14)

6 Phenomenology

Then the Hall coefficient is RH = −1/en, which allows for an experimentaldetermination of the carrier density n = 2

∑k nk . The Hall coefficient is negative

if the carriers are electrons and positive if they are holes.These results also follow from the classical Drude model. For example, in

the case of an electric field alone, the velocity of an electron accelerated by thefield at time t after a collision is v = v0 − eEt/m, where v0 is the velocityimmediately after that collision. Since the electron emerges from a collision ina random direction, there will be no contribution from v0 to the average electronvelocity, which must, therefore, be given entirely by the average of −eEt/m. Theaverage of t is the relaxation time τ and

j = −nevaverage = ne2τ

mE (1.15)

To calculate the current induced by the time-dependent electric field, E(t) =Eeiωt , one can solve the Boltzmann equation using the substitution

f (r, k, t) = nk − ∂nk

∂ EF · veiωt (1.16)

where F does not depend on time. As a result, if we replace 1/τ by −iω + 1/τ ,ac conductivity is obtained from dc conductivity. For example, the real part of aclongitudinal conductivity, which determines the absorption of the electromagneticradiation, is given by

Re σ(ν) = ne2τ

m(1 + ω2τ 2)(1.17)

for the energy spectrum Ek = k2/2m. It satisfies the sum rule∫ ∞

0dω Re σ(ω) = ω2

p/8 (1.18)

where ωp = (4πne2/m)1/2 is the plasma frequency. In our ‘one-band’approximation, there is a band mass m, which comes into the expression for ωp ,rather than the free-electron mass me. In fact, using the approximation we haveto cut the upper limit in the sum rule by a value which is well above the transportrelaxation rate 1/τ but still below the interband gap. One can prove that the bandmass in ωp of the sum rule should be replaced by the free-electron mass, if theupper limit is really infinite.

1.2 London theory

The first customary macroscopic interpretation of superconductivity as a kindof limiting case of ordinary Drude conductivity met with unresolved difficulties.Following this school of thought one was bound to search for a model of a metalwhich, in its most stable state, contained a permanent current and this without the

London theory 7

assistance of any external field. The analogy with a ferromagnet which in its moststable state contains a permanent magnetization was recalled to support the model.The thermodynamic stability of the superconducting state, and particularly thestability of persistent currents, did not seem to allow for any other conclusion.But disfavouring this concept, a general theorem of quantum mechanics wasused by Bloch, according to which the most stable state of electrons for generalreasons should be without a macroscopic current. So Bloch concluded that theonly theorem about superconductivity which can be proved is that any theory ofsuperconductivity considering the phenomenon as an extreme case of ordinaryconductivity is refutable.

Indeed an experiment by Meissner and Ochsenfeld [17] revealed that thephenomenon must not be interpreted as an extreme case of high metallicconductivity, in spite of the fact that it shows an electric current without anelectric field. From infinite conductivity it only follows that the magnetic fluxin a superconductor must be constant and, therefore, dependent upon the wayin which the superconductor passes the threshold curve between the normaland superconducting states. Meissner’s experiment, however, showed that themagnetic flux in a superconductor was always zero if the magnetic field was lowenough. A superconductor behaves as an enormous ideal diamagnetic atom ofmacroscopic dimensions expelling the entire magnetic flux from its volume. Thescreening of an applied magnetic field is affected by volume currents instead ofan atomic magnetization with a diamagnetic susceptibility

χ = − 1

4π. (1.19)

F London [3] noticed that the degenerate Bose–Einstein gas provides a good basisfor a phenomenological model, such as seemed needed for the superfluid state ofliquid 4He and for the Meissner superconducting state. For superconductivity, inparticular, F and H London [18] showed that the Meissner phenomenon couldbe interpreted very simply by the assumption of a peculiar coupling in themomentum space, as if there were something like a condensed phase of the Bosegas (appendix B). The idea to replace Fermi statistics by Bose statistics in thetheory of metals led F and H London to an equation for the current, which turnedout to be microscopically exact.

The density of the electric current is known to be given in quantummechanics by

j(r) = ie

2m(ψ∗∇ψ − ψ∇ψ∗) − e2

mAψ∗ψ (1.20)

where A ≡ A(r) is the vector potential such that ∇ × A = B. One should sumthis expression over all electron states below the Fermi surface of the normalmetal. The wavefunctions ψ(r) of the electrons with a finite wavevector areconsiderably disturbed by the magnetic field and, therefore, the term in brackets

8 Phenomenology

in equation (1.20) does not vanish. However, it turns out to be of the same order ofmagnitude and of the opposite sign as the last term containing the vector potential.The result is only a very weak Landau diamagnetism (appendix B).

But suppose the electrons are replaced by charged bosons with mass m∗∗and charge e∗. Below the Bose–Einstein condensation temperature the numberof bosons Ns , which occupy the same quantum state, is macroscopic. Theirwavefunction φ(r) obeys the Schrodinger equation,

− 1

2m∗∗ (∇ + ie∗ A)2φ(r) = Eφ(r). (1.21)

The vector potential A can be replaced by another A without any change inphysical observables, if

A = A + ∇ f (1.22)

where f (r) is an arbitrary single-valued function of coordinates. In particular, themagnetic field is the same for both potentials because ∇ × ∇ f = 0. As a result,we can choose A, which satisfies Maxwell’s gauge ∇ · A = 0 by imposing thecondition � f = ∇ · A. In a simply connected superconductor (as a bulk samplewith no holes), f (r) can be always determined for any original choice of vectorpotential A. Then, expanding equation (1.21) and the wavefunction in powers ofA we obtain, in the leading zero and first order,

− 1

2m∗∗ �φ0(r) − 1

2m∗∗ �φ1(r) − ie∗

m∗∗ A∇φ0(r) = E0φ1(r) + E1φ0(r) (1.23)

where φ0(r) is the condensate wavefunction and E0 is the energy in the absenceof the magnetic field, while φ1(r), E1 are the first-order corrections to thewavefunction and to the energy, respectively. In the absence of the external field,free bosons condense into a state with zero momentum, k = 0 and E0 = 0(appendix B). Hence, the unperturbed normalized condensate wavefunction is aconstant (φ0(r) = 1/V 1/2). The first and third terms on the left-hand side andthe first term on the right-hand side of equation (1.23) vanish. The first-ordercorrection to the energy E1 must be proportional to ∇ · A as a scalar, so that itvanishes in the Maxwell gauge. We conclude that φ1(r) = 0 and the perturbationcorrection to the condensate wavefunction is proportional to the square or higherpowers of the magnetic field. Thus, the bracket in equation (1.20) (paramagneticcontribution) vanishes in the first order of A and only the last (diamagnetic) termremains. All condensed bosons have the same wavefunction, φ0(r), so the currentdensity is obtained by multiplying equation (1.20) by Ns as

j(r) = −e∗2ns

m∗∗ A (1.24)

where ns = Ns/V is the condensate density. The non-trivial assumption thatcarriers in the superconductor are charged bosons leads to the simple London

London theory 9

relation between the magnetic field and the current. We can combine the Londonequation with the Maxwell equation ∇ × B = 4π j to obtain an equation for themagnetic field in the superconductor:

B + λ2L∇ × ∇ × B = 0 (1.25)

where λ2L = m∗∗/(4πe∗2ns) is known as the London penetration depth. On a

more phenomenological level, this equation is also derived by minimizing thefree energy of a charged superfluid in the magnetic field,

F = Fs +∫

drm∗∗v2

s (r)2

ns +∫

drB2(r)

8π. (1.26)

Here Fs is the free energy of the unperturbed superfluid, vs(r) is the velocityof condensed carriers, so that the second term is their kinetic energy and thelast term is the energy associated with the magnetic field. Replacing vs(r) byvs(r) = j(r)/(e∗ns) and the current density by j = ∇ × B/4π , we can rewritethe free energy:

F = Fs + 1

8π

∫dr(B2 + λ2

L|∇ × B|2). (1.27)

If B(r) changes by a small vector δB(r), the free energy changes by

δF = 1

4π

∫dr(B + λ2

L∇ × ∇ × B) · δB(r). (1.28)

The field, which minimizes the free energy, must, therefore, satisfyequation (1.25).

The London equation explains the Meissner–Ochsenfeld effect. In simplegeometry, when a superconductor occupies half of the space with x ≥ 0, thefield B(x) is parallel to its surface and depends only on x . The London equationbecomes a simple equation for the magnitude B(x),

λ2L

d2 B

dx2− B = 0 (1.29)

with the boundary condition B(0) = H , where H is the external field. Thesolution is

B(x) = H e−x/λL. (1.30)

Therefore, a weak magnetic field penetrates only to a very shallow microscopicdepth λL. Indeed, with the atomic density of carriers ns = 1022 cm−3,m∗∗ = me and e∗ = e, one obtains the London penetration depth as small asλL = [mec2/(4πnse2)]1/2 ≈ 600 A (in ordinary units).

10 Phenomenology

�

Figure 1.1. Flux trapped in the hole is quantized: �B = p�0, p = 1, 2, 3, . . . .

1.3 Flux quantization

The condensate wavefunction φ(r) should be a single-valued function. Thisconstraint leads to a quantization of the magnetic flux. Let us consider a holein a bulk superconductor (figure 1.1) with the trapped magnetic flux

�B =∫

ds · B (1.31)

where the surface integral is taken over the cross section, which includes the hole.The magnetic field does not penetrate into the bulk deeper than λL. Hence,

we can find a contour C surrounding the hole, along which the field andthe current are zero. Normalizing the condensate wavefunction as φ(r) =n1/2

s exp(i�) and taking j = 0 in equation (1.20), we can express the vectorpotential along the contour as

A(r) = −∇�

e∗ . (1.32)

Then the magnetic flux becomes

�B =∮

Cdl · A(r) = δ�

e∗ . (1.33)

Here δ� is a change of the phase in the round trip along the contour. Thewavefunction is single-valued if δ� = 2πp where p = 0, 1, 2, . . . . Hence, theflux is quantized (�B = p�0) and the flux quantum (in ordinary units)

�0 = π�c

e= 2.07 × 10−7 G cm2 (1.34)

for e∗ = 2e as observed experimentally [19].

Ogg’s pairs 11

1.4 Ogg’s pairs

The London equation successfully explained the Meissner–Ochsenfeld effect butcould not, of course, explain superconductivity, as electrons do not obey Bosestatistics. In 1946 Richard Ogg Jr [6] proposed that the London ‘bosonization’ ofelectrons could be realized due to their pairing. If two electrons are chemicallycoupled together, the resulting combination is a boson with the total spin S = 0or 1. Ogg suggested that an ensemble of such two-electron entities could, inprinciple, form a superconducting Bose–Einstein condensate. The idea wasmotivated by his demonstration that electron pairs were a stable constituent offairly dilute solutions of alkali metals in liquid ammonia. Sufficiently rapidcooling of the solutions to temperatures in the range from −90 to −180 ◦Cresulted in the production of homogeneous deep-blue solids. All of the solidsamples proved to be good electrical conductors. No abnormal resistancechange accompanying solidification was observed, except for solutions in theconcentration range characterized by the phase separation into two dilute liquidphases at sufficiently low temperatures. Extremely rapid freezing of suchsolutions caused an enormous decrease in measured resistance. The resistanceof the liquid sample at −33 ◦C was some 104 �, while that of the solid at −95 ◦Cwas only 16 �. Ogg argued that even such a small residual resistance wasdue to faulty contact with platinum electrodes, and the solution in the specialconcentration range was actually a high-temperature superconductor up to itsmelting point of the order of 190 K. Other experimental studies showed the soluteto be diamagnetic in the concentration range characterized by liquid–liquid phaseseparation. This suggested the electron constituent to be almost exclusively inthe electron pair configuration. In a more dilute phase, the electrons were stillpredominantly paired according to Ogg but their Bose–Einstein condensationtemperature was low enough due to a low concentration. In a more concentratedphase, the electron pairs became unstable, and one had essentially a liquidmetal with a small temperature-independent Pauli paramagnetism (appendix B).By extremely rapid cooling, it appeared that the liquid–liquid phase separationwas prevented, and that the system became frozen into the superconductingBose–Einstein condensate. Ogg proposed that his model could also explain thepreviously observed superconductivity of quasi-metallic alloys and compounds.

While independent experiments in metal-ammonia solutions did not confirmOgg’s claim, his idea of real-space electron pairing was further developed as anatural explanation of superconductivity by Schafroth, Butler and Blatt [7, 8].However, with one or two exceptions, the Ogg–Schafroth picture was condemnedand practically forgotten because it neither accounted quantitatively for the criticalparameters of the ‘old’ (i.e. low-Tc) superconductors nor did it explain themicroscopic nature of the attractive force which could overcome the naturalCoulomb repulsion between two electrons, which constitute the Bose pair. Themicroscopic BCS theory showed that in a small interval round the Fermi energy,electrons are paired in the momentum space rather than in the real space. The

12 Phenomenology

Cooper pairs strongly overlap in real space, in sharp contrast with the model ofnon-overlapping (local) pairs, proposed by Ogg and discussed in more detail bySchafroth, Butler and Blatt.

1.5 Pippard and London superconductors

There is only one characteristic length λL in the London theory. A substantialstep towards the microscopic theory was made by Pippard [20] and by Ginzburgand Landau [21], who introduced the second characteristic length of thesuperconductor, the so-called coherence length ξ . Pippard noticed that theLondon equation (1.24), which couples the current density at some point rwith the vector potential in the same point A(r), should be modified to betterfit experimentally observed penetration depths in superconducting metals andalloys. Pippard proposed a phenomenological non-local relation, which couplesthe current density at one point with vector potential at all neighbouring points:

j(r) = − 3

16π2λ2Lξ

∫dr ′ (r − r ′) A(r ′) · (r − r ′)

|r − r ′|4 e−|r−r ′|/ξ . (1.35)

If the London penetration depth is large, λL � ξ , the vector potential variesslowly enough to take it out of the integral at point r. In this case Pippard’sequation yields the London equation. However, in the opposite case λL � ξ ,the vector potential is essentially non-zero only in a thickness λH smaller than ξ ,and the integral in equation (1.35) is reduced by the factor λH/ξ . We can roughlyestimate the result of integration to be

j(r) ≈ −λHe∗2ns

ξm∗∗ A(r). (1.36)

Applying the Maxwell equation, we obtain the exponential penetration law for themagnetic field as in the London case but with the penetration depth λH differingfrom the London penetration depth λL:

1

λ2H

≈ 4πλHe∗2ns

ξm∗∗ . (1.37)

In this Pippard limit, the true penetration depth becomes larger than the Londonvalue:

λH ≈ λL(ξ/λL)1/3 > λL. (1.38)

There is another characteristic length in ‘dirty’ superconductors, which is theelectron mean free path l limited by the presence of impurities. Here it is naturalto expect that the non-local relation between the current density and the vectorpotential holds within a distance of the order of l rather than ξ, if l � ξ . Thus,we need to replace ξ in the exponent of equation (1.35) by l. By doing so, Pippard

Ginzburg–Landau theory 13

assumed that the contribution to j(r) coming from distances less than the meanfree path is not modified by the impurities. As a result, we obtain from Pippardand Maxwell equations in the dirty limit λH, ξ � l:

λH ≈ λL(ξ/ l)1/2. (1.39)

In this limit the penetration depth is proportional to the square root of theimpurity concentration in agreement with experimental observations. Actuallythe difference between the London superconductors with λL � ξ (type II) andthe Pippard superconductors, where λL � ξ (type I), is much deeper as followsfrom the Ginzburg–Landau theory.

1.6 Ginzburg–Landau theory

1.6.1 Basic equations

The condensate density ns is homogeneous in the London theory due to a stiffnessin the condensate wavefunction to small magnetic perturbations. With increasingmagnetic field, the stiffness no longer holds, and the superfluid density becomesinhomogeneous, ns = ns(r). If we normalize the condensate wavefunction bythe condition |φ(r)|2 = ns(r), the supercurrent density is

j(r) = ie∗

2m∗∗ (φ∗∇φ − φ∇φ∗) − e∗2

m∗∗ A(r)ns(r). (1.40)

To describe the superconductor in a finite magnetic field, we need an equationfor the condensate density and the phase �(r) of the condensate wavefunctionφ(r) ≡ ns(r)1/2 exp[i�(r)]. This equation might be rather different fromthe ideal Bose gas (equation (1.21)) because pairs are strongly correlated inreal superconductors. Remarkably, Ginzburg and Landau (GL) [21] formulatedthe equation for φ(r), which turned out to be general enough to satisfy themicroscopic theory of superconductivity. They applied the Landau theory of thesecond-order phase transitions assuming that φ(r) is an order parameter whichdistinguishes the superconducting phase, where φ(r) �=0, and the normal phase,where φ(r) = 0. In the vicinity of the transition they expanded the superfluid freeenergy Fs (equation (1.26)), in powers of φ(r) keeping the terms up to the fourthpower:

F − Fn =∫

dr{α|φ(r)|2 + β

2|φ(r)|4

+ 1

2m∗∗ |[∇ + ie∗ A(r)]φ(r)|2 + B2(r)8π

}. (1.41)

Here Fn is the free energy of the normal phase in the absence of a magneticfield. There are no odd terms in this expansion. Such terms (for example, linear

14 Phenomenology

and cubic in φ(r)) would contain an arbitrary phase � of the order parameterand should be excluded because F is a physical observable. In the frameworkof the Landau theory of phase transitions, the phenomenological coefficient α isproportional to the temperature difference α � (T − Tc) and β is temperatureindependent. Varying φ∗(r) by δφ∗(r) and integrating by parts, one obtains

δF =∫

dr δφ∗(r){αφ(r) + β|φ(r)|2φ(r) − 1

2m∗∗ [∇ + ie∗ A(r)]2φ(r)}

+ 1

2m∗∗

∮δφ∗(r) ds · [∇ + ie∗ A(r)]φ(r) (1.42)

where the second integral is over the surface of the superconductor. Inequilibrium, δF = 0 and both integrals should vanish for any δφ∗(r). As a result,we obtain the master equation of the theory:

− 1

2m∗∗ [∇ + ie∗ A(r)]2φ(r) + β|φ(r)|2φ(r) = −αφ(r). (1.43)

It looks like the Schrodinger equation for the condensate wavefunction,equation (1.21), but with a nonlinear ‘potential energy’ proportional to |φ(r)|2and a fixed total energy −α. For homogeneous superconductors without amagnetic field, there are two solutions for the order parameter, φ(r) = 0, whichcorresponds to the normal state, and |φ(r)|2 = ns = −α/β, which describes thehomogeneous superconducting state. The superfluid density should vanish at thetransition, T = Tc, which is the case if α(T ) � (T − Tc) and β is a constant.The coherence length is a fundamental feature of the GL theory. To determinethis new length, let us consider a situation when the real order parameter changesonly in one direction, φ(r) = n1/2

s f (x), and there is no magnetic field. The GLequation for a dimensionless function f (x) becomes

−ξ(T )2 d2 f (x)

dx2− f + f 3 = 0 (1.44)

where

ξ(T ) = 1

[2m∗∗|α(T )|]1/2

is the natural unit of length for the variation of φ(r).Varying A(r) by δ A(r) in the free energy, equation (1.41), and setting

δF = 0, we obtain the Maxwell equation ∇×∇× A = 4π j with the supercurrentdensity as in equation (1.40). The master equation and the supercurrentequation (1.40) provide a complete description of the magnetic properties ofinhomogeneous superconductors for any magnetic field. To make sure that thesecond integral in the variation of F, equation (1.42), vanishes, the equationsshould be supplemented by the boundary condition at the surface:

n · [∇ + ie∗ A(r)]φ(r) = 0 (1.45)

Ginzburg–Landau theory 15

where n is a unit vector normal to the surface. Using this condition and the currentequation (1.40), we obtain n · j = 0, i.e. the normal component of the currentshould be zero on the surface. Hence, this boundary condition is applied not onlyto the interface with vacuum but also to a superconductor–insulator interface.

The London equation follows from the GL theory for a weak magneticfield because there is no contribution to φ(r) in equation (1.43) linear withrespect to the vector potential in Maxwell’s gauge. However, it differs from theLondon theory as there is another characteristic length ξ(T ), which is importantin inhomogeneous superconductors. The ratio of the London penetration depthand the coherence length, equation (1.44), plays a crucial role in superconductingelectrodynamics:

κ = λL

ξ. (1.46)

It is known as the Ginzburg–Landau parameter. Substituting λL =[m∗∗/(4πnse∗2)]1/2, ξ = 1/(2m∗∗|α|)1/2 and ns = − α/β we obtain thetemperature independent

κ = m∗∗

e∗

[β

2π

]1/2

. (1.47)

In 1950 the pairing hypothesis was not yet accepted, therefore Ginzburg andLandau assumed e∗ = e in their pioneering paper [21]. Meanwhile Ogg’sphenomenology and BCS theory predict e∗ = 2e as a result of real or momentumspace pairing, respectively.

The GL theory has its own limitations. Since the phenomenologicalparameters α and β were expanded near Tc in powers of (Tc − T ), the theoryis confined to the transition region

|Tc − T |Tc

� 1. (1.48)

It predicts the local London relation between the current density and the vectorpotential which requires λL(T ) � ξ(0), as we know from the Pippard theory.This inequality leads to another constraint:

|Tc − T |Tc

� κ2 (1.49)

because λL(T ) � (Tc − T )−1/2. This constraint is more stringent thanequation (1.48) in superconductors with κ � 1 like Al, Sn, Hg and Pb.

There is also a general constraint on the applicability of the Landau theoryof phase transitions. The expansion of F in powers of the order parameter makessense if a statistically averaged amplitude of fluctuations |δφ| in the coherencevolume Vc remains small compared with the order parameter itself, |δφ| � n1/2

s .The probability of such fluctuations is

w � exp(−δF/T ) (1.50)

16 Phenomenology

which gives an estimate for the fluctuation part of the free energy as δF ≈Tc. However, the free-energy fluctuation in the coherence volume Vc(T ) ≈4πξ(T )3/3 is some δF ≈ |α||δφ|2Vc, which yields

|δφ|2 ≈ Tc

|α|Vc. (1.51)

As a result, the fluctuations are small, if [22]

|Tc − T |Tc

� Gi (1.52)

where

Gi =[

m∗∗Tc

ns(0)Vc(0)1/3

]2

. (1.53)

In low-temperature superconductors, Vc(0) is about (EF/Tc)3n−1 (chapter 2),

where EF is the Fermi energy and n ≈ ns(0) is the electron density. Hencethe number Gi is extremely small, Gi ≈ (Tc/EF)4 < 10−8 and the fluctuationregion is practically absent. In novel high-temperature superconductors, a fewexperiments measured an extremely small coherence volume of some 100 A3,reduced superfluid density, ns(0) ≈ 1021 cm−3, and an enhanced effectivemass of supercarriers, m∗∗ ≈ 10me (part 2). With these parameters Gi turnsout to be larger than unity when Tc ≥ 30 K. Here the GL theory does notapply in its canonical form because equations (1.52) and (1.48) are incompatible.There might be other reasons which make the expansion in powers of the orderparameter impossible or make the temperature dependence of the coefficientsdifferent from that in GL theory. GL considered the superconducting transitionas the second-order phase transition by taking α(T ) � (T − Tc) and β as aconstant near the transition. Indeed, in a homogeneous superconductor with nomagnetic field, the difference of free energy densities of two phases (the so-calledcondensation energy, fcond = (Fn − Fs)/V ) is

fcond ≡ −(αns + βn2s /2) = α2(0)

2β

(Tc − T

Tc

)2

(1.54)

near the transition. Calculating the second temperature derivative of fcond yieldsthe difference of the specific heat C = −T ∂2 F/∂T 2 of the superconducting andnormal phases which turns out to be finite at T = Tc:

�C = (Cs − Cn)T =Tc = α2(0)

βTc. (1.55)

The finite jump at Tc of the second derivative of the relevant thermodynamicpotential justifies the assumption that the transition is a second-order phasetransition. However, such definition of the transition order might depend on the

Ginzburg–Landau theory 17

approximation made. In particular, there is a critical region close to the transition,|Tc−T | ≤ TcGi , where the specific heat deviates from the GL prediction. Landauproposed a more general definition of the second-order phase transition. Whiletwo phases with zero- and finite-order parameters coexist at Tc of the first-ordertransition, the ordered phase of the second-order transition should have zero-orderparameter at Tc. In that sense, the second-order phase transition is a continuoustransition. This definition does not depend on our approximations (for the theoryof phase transitions see [23]). In general, the transition into a superfluid statemight differ from the second-order phase transition if the order of the transitionis defined by the derivative of the thermodynamic potential. For example, thetransition of an ideal Bose gas into the Bose-condensed state is of the third orderwith a jump in the third derivative of the free energy (appendix B).

1.6.2 Surface energy and thermodynamic critical field

The GL theory allows us to understand the behaviour of superconductors in finitemagnetic fields. In particular, it predicts qualitatively different properties fortype I and II superconductors in sufficiently strong fields. If the external fieldH is fixed by external currents, the relevant thermodynamic potential describingthe equilibrium state is the Gibbs energy

G(T, N, H ) = F(T, N, B) − 1

4π

∫dr H · B(r). (1.56)

Let us consider the equilibrium in the external field between the normal andsuperconducting phases separated by an infinite plane boundary at x = 0,figure 1.2. If both λL and ξ are taken to be zero, the boundary would be sharpwith no magnetic field penetrating into the superconducting phase on the right-hand side, x ≥ 0 and with no order parameter in the normal phase on the left-hand side of the boundary, x ≤ 0. Then the Gibbs energy per unit volume of thesuperconducting phase, where B = 0, is

gs = fs(T, N, 0) (1.57)

and the Gibbs energy density in the normal phase, where B = H , is

gn = fn(T, N, 0) + H 2

8π− H 2

4π= fn(T, N, 0) − H 2

8π. (1.58)

Two phases are in equilibrium if their Gibbs energy densities are equal. Thisis possible only in the so-called thermodynamic critical field H = Hc, where

H 2c

8π= fcond = α2

2β. (1.59)

The thermodynamic critical field is linear as a function of temperature near Tc:

Hc =(

4πα(0)2

β

)1/2 (Tc − T

Tc

). (1.60)

18 Phenomenology

�

�

κ<<1

����

φ���

�

κ>>1

�

Figure 1.2. Magnetic field and order parameter near the boundary between the normal andsuperconducting phases of type I (a) and type II (b) superconductors.

In fact, the boundary is not sharp because both characteristic lengths are finite.To describe the effect of the finite boundary ‘thickness’, we introduce the surfaceenergy of the boundary defined as the difference between the true Gibbs energyand that of a homogeneous sample in the external field H = Hc:

σs = 1

8π

∫ ∞

−∞dx [B2(x) − 2HcB(x) + H 2

c ]

+∫ ∞

−∞dx

[α|φ(x)|2 + β

2|φ(x)|4 + 1

2m∗∗

∣∣∣∣dφ(x)

dx

∣∣∣∣2

+ e∗2 A2

2m∗∗ |φ(x)|2]

.

(1.61)

The term proportional to A · ∇φ(x) vanishes because the vector potential hasno x-component (A = {0, Ay, 0}) when the field is parallel to the boundary.We can always choose the vector potential in this form. Also using the gaugetransformation (1.22), the order parameter can be made real, � = 0, in anysimply connected superconductor. It is convenient to introduce the dimensionlesscoordinate x = x/λL, the dimensionless vector potential a = Ay/(21/2HcλL)

and the dimensionless magnetic field h = B/(21/2Hc). Then the GL equationsdescribing the surface energy take the following form:

f ′′ + κ2[(1 − a2) f − f 3] = 0 (1.62)

anda′′ = a f 2 (1.63)

where the double prime means the second derivative with respect to x . Theboundary conditions in the problem are:

f = 0 a′ = h = 2−1/2 (1.64)

Ginzburg–Landau theory 19

for x ′ = −∞ (in the normal phase) and

f = 1 a′ = 0 (1.65)

for x ′ = ∞ (in the superconducting phase). Multiplying equation (1.62) by f ′and integrating it over x yields the first integral as

f ′2/κ2 + (1 − a2) f 2 − f 4/2 + a′2 = 1/2 (1.66)

where the constant (= 1/2) on the right-hand side is found using the boundaryconditions. Equation (1.66) allows us to simplify the surface energy as follows:

σs = λL H 2c

4π

∫ ∞

−∞dx {(a′ − 2−1/2)2 + (a2 − 1) f 2 + f 4/2 + f ′2/κ2}

= λL H 2c

2π

∫ ∞

−∞dx {a′(a′ − 2−1/2) + f ′2/κ2}. (1.67)

Let us estimate the contributions to the surface energy of the first and secondterms in the last brackets. The first term a′(a′ − 2−1/2) is zero both in the normaland superconducting phases. Its value is about −1 in the boundary region ofthickness |x ′| < 1, because a′ = h < 2−1/2 in any part of the sample. Hence,the contribution of the first term is negative and is about −1. In contrast, thecontribution of the second term is positive. This term is non-zero in the region ofthe order of |x ′| � 1/κ , where its value is about 1. Hence, its contribution to theintegral is about +1/κ . We conclude that the surface energy is positive in extremetype I superconductors where κ � 1 but it is negative in type II superconductorswhere κ � 1.

The exact borderline between the Pippard and London superconductors isdefined by the condition σs = 0. Integrating the last term f ′2/κ2 in thefirst integral of equation (1.67) by parts and substituting f ′′ from the masterequation (1.62) we obtain

σs = λL H 2c

4π

∫ ∞

−∞dx {(a′ − 2−1/2)2 − f 4/2}. (1.68)

Hence, the surface energy is zero if

a′ = 2−1/2(1 − f 2). (1.69)

Now the second equation of GL theory (equation (1.63)) yields a′′ = −21/2 f f ′ =a f 2 or f ′ = −a f/21/2. Substituting this f ′ and a′ (equation (1.69)) into the firstintegral of motion (equation (1.66)), we obtain

a2 f 2(

1 − 1

2κ2

)= 0. (1.70)

20 Phenomenology

�

�ξ

�

Figure 1.3. Single-vortex core screened by supercurrent.

Hence, the borderline between type I and II superconductors is found at

κ = 1

21/2. (1.71)

Due to the negative surface energy, type II superconductors are inhomogeneous insufficiently strong magnetic fields. Their order parameter is modulated in spaceso that the normal and superconducting regions are mixed.

1.6.3 Single vortex and lower critical field

With increasing external field, the normal region in the bulk type IIsuperconductor appears in the form of a single vortex line with the normal core ofthe radius about ξ surrounded by a supercurrent. Similar vortex lines were foundin superfluid rotating 4He and discussed theoretically by Onsager and Feynman.A generalization to superconductors is due to Abrikosov [24]. The propertiesof a single vortex are well described by the GL equations with proper boundaryconditions. Let us assume that the vortex appears at the origin of the coordinatesystem and has an axial symmetry, i.e. the order parameter depends only onr of the cylindrical coordinates {r,�, z} with z parallel to the magnetic field,figure 1.3.

Then the GL equations become

1

κ2ρ

d

dρρ

d f

dρ− 1

f 3

(dh

dρ

)2

− f 3 = 0 (1.72)

and1

ρ

d

dρ

ρ

f 2

dh

dρ= h. (1.73)

Ginzburg–Landau theory 21

These equations are written in a form which introduces the dimensionlessquantities f (ρ) = n−1/2

s φ(r), ρ = rλL and h(ρ) for the order parameter, lengthand magnetic field, respectively. The second equation is readily derived replacingthe current in equation (1.40) by j = ∇ × B/4π and taking the curl of both partsof the equation. We choose the gauge where the order parameter is real, � = 0and the paramagnetic term of the current in equation (1.40) is zero.

There are four boundary conditions in the single-vortex problem. Three ofthem are found at ρ = ∞. Here the superconductor is not perturbed by themagnetic field, so that h = dh/ρ = 0 and the dimensionless order parameteris unity ( f = 1). The fourth boundary condition is derived using the fluxquantization. The total flux carried by the vortex is

�B = 23/2πλ2L Hc

∫ ∞

0dρ ρh(ρ). (1.74)

Applying equation (1.73), we obtain

�B = −κ�0

(ρ

f 2

dh

dρ

)ρ=0

(1.75)

which should be equal to p�0. Hence,

dh

dρ= −p

f 2

κρ(1.76)

for ρ = 0, where p is a positive integer.Let us first consider the region outside the vortex core, where ρ � 1/κ . The

order parameter should be about one in this region. Then, the second GL equationis reduced to its London form in the cylindrical coordinates:

d2h

dρ2+ 1

ρ

dh

dρ− h = 0. (1.77)

The solution which satisfies the boundary conditions, h = dh/dρ = 0 for ρ = ∞,is

h = constant × K0(ρ). (1.78)

Here K0(ρ) is the Hankel function of imaginary argument of zero order. Itbehaves as the logarithm for small ρ (�1):

K0(ρ) ≈ ln

(2

ργ

)(1.79)

with γ = 1.78, and as the exponent for large ρ (�1):

K0(ρ) ≈(

π

2ρ

)1/2

exp(−ρ). (1.80)

22 Phenomenology

Figure 1.4. Vortex core in the BCS superconductor.

The magnetic field and current decrease exponentially in the exterior of the vortex,where ρ > 1. However, the magnetic field is almost constant in the interior ofthe vortex, ρ � 1. Because the magnetic field and its derivative change over thecharacteristic length ρ � 1, the boundary condition (1.76) is applied for the wholeinterior (0 ≤ ρ � 1), not only at zero. Hence, we can use the flux quantizationcondition to determine the constant in equation (1.78), which is applied for 1/κ ≤ρ < ∞. If κ � 1, the two regions overlap. Therefore, dh/dρ calculated usingequation (1.76) with f = 1 and equation (1.78) should be the same, which is thecase if the constant = p/κ . Hence, the magnetic field around the vortex core(r � ξ in ordinary units) is

B(r) = �0 p

2πλ2L

K0(r/λL). (1.81)

The flux quantization boundary condition allows us to ‘integrate out’ the magneticfield in the master equation for the interior of the vortex with the following result(ρ � 1):

1

ρ

d

dρρ

d f

dρ− p2

ρ2 f − κ2 f 3 = 0. (1.82)

This equation is satisfied by a regular solution of the form f = cpρp for ρ → 0.

The constant cp has to be found by numerical integration of equation (1.82). Thenumerical result for p = 1 is shown in figure 1.4 where c1 � 1.166.

The order parameter is significantly reduced inside the core (ρ � 1/κ) butit becomes almost one in the region ρ � 1/κ ,

f ≈ 1 − p2

κ2ρ2 . (1.83)

The vortex free energy�p is defined as the difference in the free energies of a bulksuperconductor with and without a single vortex. The GL equations are reducedto the single London equation outside the vortex core where the order parameter is

Ginzburg–Landau theory 23

almost a constant, f ≈ 1. When κ � 1, this region yields the main contributionto the vortex energy, while the contribution of the core is negligible. Then �p isgiven by the London expression (1.27). It comprises the kinetic energy due to thecurrent and the magnetic energy:

�p ≈ 1

8π

∫dr (B2 + λ2

L|∇ × B|2)

= Lλ2L H 2

c

2

∫ ∞

κ−1dρ ρ[h2 + (dh/dρ)2]. (1.84)

Here L is the vortex length along the field lines. Integrating the second term underthe integral by parts, we obtain

�p = −λ2L L H 2

c

2

[h(ρ)ρ

dh

dρ

]ρ=κ−1

= Lλ2L H 2

c

2

p2

κ2ln κ. (1.85)

The single-vortex Gibbs energy is obtained as

G p = �p − 1

4π

∫dr B · H. (1.86)

Here the integral is proportional to the flux carried by the vortex, so that

G p = �p − pL

4π�0 H. (1.87)

G p > 0 and the vortex state is unfavourable, if the external field is weak.However, if the field is strong enough so that G p < 0, the vortex state becomesthermodynamically stable. The first (lower) critical field Hc1, where the vortexappears in bulk type II superconductors, is defined by the condition G p = 0,

Hc1 = 4π�p

p�0. (1.88)

We see that the lowest field corresponds to a vortex carrying one flux quantum(p = 1),

Hc1 ≈ Hcln κ

21/2κ. (1.89)

The first (lower) critical field appears to be much smaller than the thermodynamiccritical field in type II superconductors with a large value of κ � 1. If κ isnot very large, the vortex penetration and the determination of Hc1become morecomplicated.

1.6.4 Upper critical field

Bulk type I superconductors remain in the Meissner state with no field inside thesample if the external field is below Hc. They suddenly become normal metals

24 Phenomenology

when the external field is above Hc. A bulk type II superconductor also exhibitscomplete flux expulsion in the external field H < Hc1 < Hc. However, whenH > Hc1, vortices penetrate inside the sample but the flux passing through thesample remains less than its normal state value. Only for a larger magnetic field,H � Hc2 does the sample become entirely normal with no expulsion of the flux,B = H . The vortex state for Hc1 < H < Hc2 is still superconducting withpermanent currents in the sample. The upper (second) critical field Hc2 is oneof the fundamental characteristics of type II superconductors. One can measureHc2 by continuously decreasing the field from a high value. At a certain field,H = Hc2, superconducting regions begin to nucleate spontaneously, so that theresistivity and magnetization start to deviate from their normal state values. Inthe regions, where the nucleation occurs, superconductivity is just beginning toappear and the density of supercarriers ns = |φ(r)|2 is small. Then approachingfrom the normal phase, the GL equation for the order parameter can be linearizedby neglecting the cubic term:

1

2m∗∗ [∇ + ie∗ A(r)]2φ(r) = αφ(r). (1.90)

With the same accuracy the magnetic field inside the sample does not differfrom the external field, so that A(r) = {0, x Hc2, 0}. Then the master equationbecomes formally identical to the Schrodinger equation for a particle of charge2e in a uniform magnetic field, whose eigenvalues and eigenfunctions are wellknown [25]. The Hamiltonian (1.90) does not depend on coordinates y andz and the corresponding momentum components, ky, kz , are conserved. Theeigenfunctions are found to be

φν(r) = ei(ky y+kz z)χn(x) (1.91)

where χ(x) obeys the one-dimensional harmonic oscillator equation,

1

2m∗∗ χ ′′ − m∗∗ω2

2(x − xky )

2χ =(

α + k2z

2m∗∗

)χ. (1.92)

Here xky = −ky/(m∗∗ω) is the equilibrium position, ω = e∗Hc2/m∗∗ is thefrequency of the oscillator and ν ≡ (ky, kz, n) are the quantum numbers. Thewell-behaved normalized eigenstates are found to be

χn(x) =(

m∗∗ωπ

)1/4 Hn[(x − xky )(m∗∗ω)1/2]√

2nn! e−m∗∗(x−xky )2ω/2 (1.93)

and the eigenvalues areEn = ω(n + 1/2). (1.94)

Here

Hnξ = (−1)neξ2 dne−ξ2

dξn(1.95)

Ginzburg–Landau theory 25

Figure 1.5. Phase diagram of bulk type I (a) and type II (b) superconductors.

are the Hermite polynomials, n = 0, 1, 2, 3, . . . . The maximum value of ky

is determined by the requirement that the equilibrium position of the harmonicoscillator xky is within the sample,

−m∗∗ω2

Lx < ky <m∗∗ω

2Lx (1.96)

where Lx,y,z is the length of the sample along x , y and z, respectively. Thenumber d of allowed values of ky in this range is

d = L y

2πm∗∗ωLx = Lx L y

πeHc2 (1.97)

so that every energy level is d-fold degenerate. We are interested in the highestvalue of H = Hc2, which is found from

ω(n + 1/2) = −(

α + k2z

2m∗∗

)(1.98)

with kz = n = 0,

Hc2 = −2m∗∗αe∗ . (1.99)

We see that Hc2(T ) allows for a direct measurement of the superconductingcoherence length ξ(T ), because Hc2 = �0/2πξ2(T ). Near Tc, the upper criticalfield is linear in temperature Hc2(T ) � (Tc − T ) in GL theory, where α ∝ T − Tc.Hc2(0) at zero temperature is normally below the Clogston–Chandrasekhar [26]limit, which is also known as the Pauli pair-breaking limit given by Hp � 1.84Tc(in tesla, if Tc is in kelvin). The limit can be exceeded due to the spin–orbitcoupling [27] or triplet pairing but in any case Hc2(0) remains finite in theframework of BCS theory. Nonetheless, it might exceed the thermodynamic field

26 Phenomenology

by several orders of magnitude in superconductors with a large GL parameter, asfollows from the relation

Hc2 = 21/2κ Hc. (1.100)

At the border between type II and I superconductors, where κ = 2−1/2, allcritical fields should be the same, Hc1 = Hc2 = Hc, in agreement withequation (1.100). The H –T phase diagram of type I and II bulk superconductorsis shown in figure 1.5. The finite size of the sample and its surface might affectthe magnetic field penetration and the superconducting nucleation. In particular,even type I superconductors could be in an intermediate state with normal andsuperconducting domains, if the field H < Hc, while the superconductingnucleation in type II superconductors could take place at the surface in a fieldHc2 < H < Hc3.

1.6.5 Vortex lattice

Vortex configuration in the field Hc1 < H < Hc2 can be found by numericalintegration of nonlinear GL equations. If H is only slightly less than Hc2,the profile of the order parameter can be established analytically [24]. Inthe superconducting state close to Hc2(T ) line the order parameter φ(x, y) isrepresented by a linear combination of degenerate solutions of the linearized GLequation with n = kz = 0, equation (1.91),

φ(x, y) =∑ky

c(ky)eiky y exp

[− (x − xky )

2

2ξ2

]. (1.101)

We can expect the minimum of the free energy to correspond to a regulararrangement of the vortex lines. This happens because vortices repel each other.To minimize their repulsive energy, the lines will take a regular arrangement sothat φ(x, y) is periodic in the x and y directions. To see how the repulsion occurs,let us consider two p = 1 vortices with their cores at ρ1 and ρ2 in the planeperpendicular to B in a type II superconductor with κ � 1. The free energy isgiven by the London expression:

� ≈ 1

8π

∫dr (B2 + λ2

L|∇ × B|2)

= Lλ2L H 2

c

4π

∫dρ(h2 + |∇ × h|2) (1.102)

where the integral is taken in the plane outside the vortex cores. The magneticfield h(ρ) is the superposition of the fields h1(ρ) and h2(ρ) of each vortex,

h(ρ) = 1

κ[K0(|ρ − ρ1|) + K0(|ρ − ρ2|)]. (1.103)

Ginzburg–Landau theory 27

���

���

Figure 1.6. Two parallel vortex lines repel each other.

Integrating equation (1.102) by parts and taking the limit κ → ∞, one obtains inthe leading order

� = 2�1 + U12. (1.104)

The interaction energy U12 is given by

U12 = Lλ2L H 2

c

2π

∮dl2 · h1 × ∇ × h2 (1.105)

where the contour integral is taken over the surface of the second vortex core.Integrating, one obtains the positive interaction energy

U12 = Lλ2L H 2

c

κ2K0(|ρ1 − ρ2|). (1.106)

The free energy of the vortex state with two vortex lines is larger than twice thesingle vortex energy. Hence, two vortices repel each other. The supercurrents ofthe two vortices are added to each other in the outer regions but they are subtractedfrom each other in the region between two vortices, figure 1.6. According tothe Bernoulli law, the pressure is higher where the velocity of an ideal liquid islower. Hence, the pressure is higher in the region between two vortices, creatinga repulsive force between them,

f 12 = −∇U12. (1.107)

The magnetic field in the core of the first vortex, created by the second vortexis B12 = 21/2 HcK0(|ρ1 − ρ2|)/κ . Applying the Maxwell equation, ∇ × B12 =4π j12, the repulsion force applied to the first vortex is expressed through thecurrent j12 of the second vortex as

f 12 = L�0 j12 × n (1.108)

where n is a unit vector parallel to the field.Let us now assume that the order parameter in the vicinity of the Hc2(T ) line

is periodic with period a. The translation in the y direction by a does not changeφ(x, y) if ky = 2πl/a in the sum (1.101) with the integer l = 0,±1,±2, . . . .Hence, for H slightly less than Hc2, we have

φ(x, y) =∑

l

cle2π ily/a exp

[−[x − πl/(ea Hc2)]2

2ξ2

]. (1.109)

28 Phenomenology

In order that φ(x, y) also be periodic in x , it is necessary to impose a periodicityon the coefficients cl . A square lattice is obtained if cl+1 = cl . Then φ(x, y)

is periodic with period a = (2π)1/2ξ . A triangular lattice is obtained using thecondition cl+2 = cl and c1 = ic0. The latter is more stable in a perfect crystal.With a further lowering of the magnetic field, the density of vortices drops and theperiod of the vortex lattice becomes larger than the coherence length. The vortexlattice in type II superconductors is directly observed using ferromagnetic powdersputtered on the sample surface and in neutron scattering experiments.

1.6.6 Critical current

If the current density is too high, the superconducting state is destroyed. The GLphenomenology allows us to calculate the critical current density jc. The simplestcase is the uniform current distribution across the sample. This condition is foundin a thin superconducting film, when the thickness of the film is small comparedwith the magnetic field penetration depth and the coherence length. The magneticfield due to the current in the film is proportional to the film thickness. Hence, ifthe film is sufficiently thin, we can neglect the magnetic energy in the free energydensity fs ,

fs ≈ fn + αns + βn2s

2+ nsv

2s

2m∗∗ . (1.110)

Minimizing fs as a function of the condensate density ns , we obtain

ns = − 1

β

(α + v2

s

2m∗∗

). (1.111)

Hence, the supercurrent density j = e∗vsns is a nonlinear function of vs ,

j = −e∗vs

β

(α + v2

s

2m∗∗

)(1.112)

with a maximum at vs = (2m∗∗|α|/3)1/2. The maximum (i.e. critical) currentdensity is

jc = 4e(m∗∗)1/2|α/3|3/2

β. (1.113)

The Landau criterion of superfluidity allows us to understand jc at a moremicroscopic level. Let us consider a superfluid, which flows with a constantvelocity v. In a homogeneous system, the elementary excitations of the liquidhave well-defined momenta k, so that their energy εk is a function of k. Becauseof the friction, the kinetic energy of the condensate may dissipate and the flowwould gradually stop. This process creates at least one elementary excitation.In the coordinate frame moving with the liquid, the momentum and energy

Josephson tunnelling 29

conservation in a ‘collision’ with an object (such as the retaining wall) takes thefollowing form:

−Mv = −Mv′ + k (1.114)

andMv2

2= Mv′2

2+ εk. (1.115)

Here M is the mass of the object and v′ is the liquid velocity after the collision.Combining these two equations, one obtains

εk + k · v + k2

2M= 0. (1.116)

The critical value of the velocity is obtained for M = ∞ as

vc = min[εk

k

]. (1.117)

The liquid is superfluid, if vc �= 0 and the flow is slow enough, v � vc. InBCS theory (chapter 2), the excitation spectrum has a gap near the Fermi surface(k = kF) and min εk = � > 0. Hence, there is a non-zero critical velocity inthe BCS superconductor, vc = �/kF. Flow with a higher velocity leads to pairbreaking and a loss of superconductivity. In the ideal Bose gas εk = k2/2m∗∗and vc = min[k/2m∗∗] = 0. Hence, Bose–Einstein condensation alone is notsufficient for superfluidity. According to Bogoliubov [4], the repulsion betweenbosons modifies their excitation spectrum so that the repulsive Bose gas is asuperfluid (chapter 4).

The critical current in the vortex state of bulk type II superconductors doesnot reach the pair-breaking limit. The current j in the direction perpendicular tothe vortex lines creates the Lorenz force applied to the vortex core, as followsfrom equation (1.108) with j12 = j . As a result, vortices move across thecurrent and the current inevitably flows through their normal cores. This vortexmotion leads to the energy dissipation. Hence, an ideal type II superconductorhas zero critical current in the vortex state, when Hc1 < H < Hc2. However, ifthe external current is not very large, different defects of the crystal lattice ‘pin’vortices preventing their motion. That is why the critical current of real type IIsuperconductors depends on the sample quality. Disordered samples can carry thecritical current density of about 107 A cm−2 or even higher [28].

1.7 Josephson tunnelling

Let us consider a bulk superconductor separated into two parts by a thin contactlayer with different properties from those of the bulk. The layer might bean insulator, a normal metal or any weak link with a reduced condensatedensity or a small cross section. Its thickness is supposed to be smallcompared with the coherence length (figure 1.7). Josephson [29] predicted

30 Phenomenology

that a supercurrent can flow through the weak link without any voltage andit oscillates as a function of time if there is a voltage drop on the contact.Following Feynman [30], we can understand this effect in the framework of theLondon description of superconductivity in terms of the condensate wavefunctionφ(x, t) = n1/2

s exp[i�(x, t)] (x is the direction perpendicular to the contact). Ifwe suppose, that there is a supercurrent j = (e∗/m∗∗)ns d�/dx along x in thesample, the condensate phase should depend on x (figure 1.7). To keep the samecurrent density along the sample, the phase gradient should be larger in the weaklink, where ns is suppressed. Hence, there is a finite phase shift ϕ = �2 − �1 ofthe condensate wavefunction on the right- (φ2) and left-hand (φ1) ‘banks’ of thecontact, x = ±0. The carriers change their quantum state from φ1(t) to φ2(t) inthe contact with a transition amplitude K , which is a property of the weak link.Hence, one can write

iφ1(t) = Kφ2(t) (1.118)

andiφ2(t) = Kφ1(t) + e∗V φ2(t) (1.119)

where e∗V is a change in the electrostatic energy at the transition due to voltageV . Taking time derivatives and separating the real and imaginary parts of theseequations, one obtains

ns1 = 2K (ns1ns2)1/2 sin ϕ (1.120)

ns2 = − 2K (ns1ns2)1/2 sin ϕ

andϕ = e∗V + K [(ns1/ns2)

1/2 − (ns2/ns1)1/2] cos ϕ. (1.121)

The current I through the contact is proportional to the number of carrierstunnelling per second, I ∝ ns1, so that

I = Ic sin ϕ (1.122)

where Ic � K (ns1ns2)1/2. If both banks of the Josephson contact are made from

the same superconductor, ns1 = ns2 = ns and

ϕ = e∗V .

As a result, we obtainI = Ic sin(ϕ0 + e∗V t) (1.123)

where ϕ0 is the phase difference in the absence of voltage.We conclude that some current I < Ic can flow through the insulating thin

layer between two bulk superconductors with no applied voltage, if ϕ0 is non-zero. When the voltage is applied, the current oscillates with a frequency (in theordinary units)

ωJ = 2eV

�. (1.124)

Josephson tunnelling 31

�

�

Φ(��

�

�

�

�

Figure 1.7. Josephson’s weak link between two superconductors.

Figure 1.8. Magnetic field dependence of the Josephson current.

The Josephson current also oscillates as a function of the applied magnetic field.Indeed, let us introduce a magnetic field B into the dielectric layer parallel to thecontact (figure 1.7). The magnetic field penetrates into the superconductor. Thesize of the region along x with the magnetic field and the supercurrent is aboutw = t + 2λH, where t is the thickness of the layer.

Using the same arguments as in section 1.3 we obtain∮C

dl · A(r) = δ�

e∗ (1.125)

where the contour C crosses the contact twice as shown in figure 1.7. Here wetake into account that the contributions of the transport current flowing along x aremutually cancelled. The contribution of the screening current is negligible if thelength of the contour is much larger than w. The left-hand side of equation (1.125)is an elementary flux d�B = Bw dy across the area of the contour, while δ� isthe change of the phase shift along the contact area, so that

d�B = ϕ(y + dy) − ϕ(y)

e∗ . (1.126)

We see that, in the presence of the magnetic field, the phase shift and Josephsoncurrent density j = jc sin ϕ(y) are not uniform within the contact area because

32 Phenomenology

(from equation (1.126)) ϕ(y) = e∗Bwy + ϕ0. The total current through thecontact is

I = jcL∫ L/2

−L/2dy sin(e∗Bwy + ϕ0) = Ic(sin ϕ0)

[sin (π�B/�0)

π�B/�0

](1.127)

where Ic = jcS and S = L2 is the contact cross section. The Josephson currentoscillates as a function of the magnetic flux revealing a characteristic quantuminterference pattern (figure 1.8). The Josephson effect laid the groundwork fordesigning squids (Superconducting Quantum Interference Devices). Squids canmeasure magnetic fields and voltages as low as 10−11 G and 10−15 V, respectively.

Chapter 2

Weak coupling theory

The phenomenological Bose-gas picture rendered no quantitative account for thecritical parameters of conventional low-temperature superconductors. Its failurestemmed from the very large diameter ξ of pairs in conventional superconductorswhich can be estimated using the uncertainty principle, ξ ≈ 1/δk. Theuncertainty in the momentum δk is estimated using the uncertainty in the kineticenergy δE � vFδk, which should be of the order of Tc, the only characteristicenergy of the superconducting state. Therefore, ξ ≈ vF/Tc turns out to be about1 µm for simple superconducting metals where Tc � 1 K and the Fermi velocityvF � 108 cm s−1. This means that pairs in conventional superconductors stronglyoverlap and the Ogg–Shafroth model of real space pairs cannot be applied.

An ultimately convincing theory of conventional superconductors wasformulated by Bardeen, Cooper and Schrieffer [2]. In this and the next chapterswe describe the essentials of BCS theory using the Bogoliubov transformation [4]in the weak-coupling regime and Green’s functions for the intermediate coupling(see also the excellent books by Schrieffer [31], Abrikosov et al [32] and DeGennes [33]).

2.1 BCS Hamiltonian

The BCS theory of superconductivity [2] relies on Frohlich’s observation [9]that conduction electrons could attract each other due to their interaction withvibrating ions of the crystal lattice. In a more complete analysis by Bardeenand Pines [34] in which Coulomb effects and collective plasma excitations wereincluded, the interaction between electrons and the phonon field was shown todominate over the matrix element of the Coulomb interaction near the Fermisurface. The key point is the fact discovered by Cooper [35] that any attractionbetween degenerate electrons leads to their pairing no matter how weak theattraction is. One can understand the Cooper phenomenon by transforming thecorresponding Schrodinger equation for a pair into the momentum space. Becauseof the Pauli principle, a pair of electrons can only ‘move’ along the Fermi surface

33

34 Weak coupling theory

in the momentum space. But it is well known that in two dimensions a boundstate exists for any attraction, however weak. The Cooper solution of the two-particle problem in the presence of the Fermi sea demonstrated the instability ofan attractive Fermi liquid versus pair formation with a possible Bose–Einsteincondensation in the momentum space. Because of the Pauli principle, twoelectrons with parallel spins tend to be at larger distances than two electrons withantiparallel spins. Hence, following BCS, one can expect electrons to be pairedinto singlets rather than into triplets with the zero centre-of-mass momentum ofthe pairs. For these reasons, BCS introduced a model (truncated) Hamiltonian inwhich the interaction term contains only the attraction between electrons with theopposite spins and momenta,

H =∑

k,s=↑,↓ξkc†

kscks +∑

k

c†k↑c†

−k↓�(k) (2.1)

where�(k) =

∑k′

V (k, k′)c−k′↓ck′↑. (2.2)

The first term describes independent Bloch electrons, where cks is the electronannihilation operator with the momentum k and spin s, ξk = Ek − µ is theband energy dispersion referred to the chemical potential µ. It is convenient toconsider an open system with a fixed chemical potential rather than with a fixedtotal number of particles Ne � 1 to avoid some artificial difference between oddand even Ne. 〈H − µNe〉 has a minimum in the ground state for the open system.That is why the electron energy in equation (2.1) is referred to as µ. We recall(appendix C) that annihilation and creation operators anticommute for fermions:

{cks, ck′s ′ } ≡ cksck′s ′ + ck′s ′cks = 0 (2.3)

and{cks , c†

k′s ′ } ≡ cksc†k′s ′ + c†

k′s ′ck′s ′ = δkk′δss ′. (2.4)

They lower or raise the number of fermions nks in a single-particle state |k, s〉by one and multiply a many-particle wavefunction of non-interacting fermions by±nks or ±(1 − nks) as

cks |nk1s1,nk2s2, . . . , nks, . . .〉 = ± nks |nk1s1,nk2s2, . . . , nks − 1, . . .〉 (2.5)

c†ks |nk1s1,nk2s2, . . . , nks, . . .〉 = ± (1 − nks)|nk1s1,nk2s2, . . . , nks + 1, . . .〉.

Here + or − depends on the evenness or oddness of the number of occupiedstates, respectively, which precede the state |k, s〉 in an adopted ordering of single-particle states. Modelling the attractive potential V (k, k′), BCS took into accountthe retarded character of the attraction mediated by lattice vibrations. The ionsof the lattice must have sufficient time to react, otherwise there would be nooverscreening of the conventional Coulomb repulsion between electrons. In

Ground state and excitations 35

other words, the characteristic time τs for the scattering of one electron withthe energy ξk by another one should be longer than the characteristic time oflattice relaxation which is about the inverse (Debye) frequency of vibrationsω−1

D . Using the energy–time uncertainty principle, we estimate the scatteringtime as τs ≈ |ξk − ξk′ |−1. Hence, V (k, k′) could be negative (i.e. attractive),if |ξk − ξk′ | < ωD. To describe the essential physics of superconductors, BCSintroduced a simple approximation for the attractive interaction:

V (k, k′) = −2E p (2.6)

if the condition|ξk|, |ξk′ | < ωD (2.7)

is satisfied, and zero if otherwise. Here E p is a positive energy depending onthe strength of the electron–phonon coupling. Multiplying it by the density ofelectron states per unit cell at the Fermi level N(EF) (appendix B), we obtain adimensionless constant

λ = 2E p N(EF) (2.8)

which conveniently characterizes the strength of the coupling. The BCS theorywas originally developed for weakly coupled electrons and phonons with λ � 1;extended towards the intermediate coupling (λ � 1) by Eliashberg [36]; and tothe strong coupling (λ � 1) by us [10, 11].

2.2 Ground state and excitations

The operators �, �† in the BCS Hamiltonian (2.1) annihilate and create pairswith total momentum K = 0. One can expect that the pairs condense inthe ground state with K = 0 like bosons below some critical temperature Tc(appendix B). The number of condensed pairs Ns should be macroscopicallylarge, if T < Tc. The matrix elements of � and �† should be large aswell, like the matrix elements of the annihilation and creation boson operators,〈Ns − 1|b0|Ns 〉 = 〈Ns + 1|b†

0|Ns 〉 = (Ns)1/2 � 1 (appendix C). That is all

commutators of �, �† should be much smaller than the operators themselvesbecause [b0b†

0] = 1 � (Ns )1/2. Hence, we can neglect the fact that � does not

commute with �† and c†ks . It is the same as replacing these operators by their

expectation values in the open system,

�(k) ≈ �k (2.9)

where

�k = −2E p�(ωD − |ξk|)∑

k′�(ωD − |ξk′ |)〈〈c−k′↓ck′↑〉〉 (2.10)

36 Weak coupling theory

and �(x) = 1 for positive x and zero otherwise. The wavefunction of theopen system is a superposition of different eigenfunctions of the total numberoperator and the average equation (2.10) is not zero but macroscopically largein the superconducting state. Here the double brackets mean both quantum andstatistical averages (appendix B). The substitution of equation (2.9) transformsthe BCS Hamiltonian into a quadratic form with respect to the electron operators,

H =∑

k

[ξk(c†k↑ck,↑ + c†

−k↓c−k↓) + �kc†k↑c†

−k↓ + �∗kc−k↓ck↑] + |�|2

2E p(2.11)

where the last term is added to make sure that the ground-state energies of theexact BCS Hamiltonian and of the approximation (2.11) are the same. Here

� = −2E p

∑k′

�(ωD − |ξk′ |)〈〈c−k′↓ck′↑〉〉 (2.12)

does not depend on k. Now, following Bogoliubov [4], we can replace the electronoperators by new fermion operators:

ck↑ = ukαk + vkβ†k (2.13)

c−k↓ = ukβk − vkα†k. (2.14)

This transformation reduces the whole problem of correlated electrons to an idealFermi gas comprising two types of new non-interacting fermions (i.e. quasi-particles) α and β, if we choose

u2k = 1

2

(1 + ξk

εk

)(2.15)

v2k = 1

2

(1 − ξk

εk

)(2.16)

and

ukvk = − �k

2εk(2.17)

where

εk =√

ξ2k + |�k|2. (2.18)

Then new annihilation and creation operators anticommute like fermionoperators,

{αkα†k′ } = {βkβ

†k′ } = δkk′

{αkαk′ } = {βkβk′ } = {αkβ†k′ } = {αkβk′ } = 0.

and the transformed Hamiltonian becomes diagonal,

H = E0 +∑

k

εk(α†kαk + β

†kβk) (2.19)

Ground state and excitations 37

where

E0 = 2∑

k

(ξkv2k + �kukvk) + |�|2

2E p. (2.20)

The order parameter � is determined from the self-consistent equation (2.12)replacing the electron operators by quasi-particle ones,

� = E p

∑k′

�

εk′(1 − 2 fk′) (2.21)

wherefk = 〈〈α†

kαk〉〉 = 〈〈β†kβk〉〉

is the quasi-particle distribution function. Unlike the case of bare electrons thetotal average number of quasi-particles is not fixed. Therefore, their chemicalpotential is zero in the thermal equilibrium. They do not interact, so that(appendix B)

fk = 1

exp(εk/T ) + 1. (2.22)

We see that there are no quasi-particles in the ground state, fk = 0, atT = 0. Hence, E0 in equation (2.19) is the ground-state energy of the BCSsuperconductor. One can replace the sum in equation (2.21) by the integral usingthe definition of the density of states N(E) in the Bloch band (appendix A). Inconventional metals, the Debye frequency is small (ωD � µ) and the density ofstates (DOS) is practically constant in the narrow energy region ±ωD around theFermi energy, N(E) � N(EF). As a result, we obtain

� = λ�

∫ ωD

0

tanh√

ξ2+|�|22T√

ξ2 + |�|2 dξ. (2.23)

There is a trivial solution of this equation: � = 0. Above some Tc, this isthe only solution corresponding to the normal state. However, below Tc thereare two solutions: � = 0 and a real � �= 0. The system prefers to be in thesuperconducting (condensed) state below Tc because the condensation energy Ecis negative. This energy is the ground-state energy referred to the normal stateenergy. At T = 0, we have

Ec = E0 − 2∑ξk<0

ξk. (2.24)

Using the definition of E0 (equation (2.20)), we obtain

Ec = 2N(EF)

∫ ωD

0dξ

(ξ − ξ2 + �2(0)/2√

ξ2 + �2(0)

)(2.25)

38 Weak coupling theory

∆���

�

ε�

��

µ

Figure 2.1. Excitation spectrum of the BCS superconductor.

where �(0) is the order parameter at T = 0. Indeed, this integral is negative:

Ec ≈ − 12 N(EF)�2(0) < 0. (2.26)

Far away from the Fermi surface, the quasi-particles, α and β, are electronswith spins ‘up’ and ‘down’, respectively, if ξk > 0 and holes if ξk < 0. Inthe vicinity of the Fermi surface, they are a mixture of both and their energydispersion εk is remarkably different from that of the non-interacting electronsand holes (see figure 2.1). The quasi-particle energy spectrum satisfies the Landaucriterion (chapter 1) of superfluidity and the critical velocity vc � �/kF. Thedistribution of ‘bare’ electrons is of the form (T = 0)

nk = 〈c†k↑ck↑〉 = v2

k (2.27)

which has a zero step (Z = 0) which differs from the Fermi distribution at T = 0with Z = 1 (figure 2.2). This is a clear manifestation of a breakdown of theFermi-liquid description of attractive fermions at low temperatures. The ‘mean-field’ approximation (equation (2.9)) replacing the pair operators �(k) by theanomalous averages �k is perfectly self-consistent. Indeed, let us calculate thecommutator

[�(k), �†(k)] (2.28)

to show that its expectation value is zero. Using

[c1c2, c†3c†

4] = δ23c1c†4 − δ13c2c†

4 + δ24c†3c1 − δ14c†

3c2 (2.29)

we obtain

[�(k), �†(k)] = 4E2p�(ωD − |ξk|)

∑k′

�(ωD − |ξk′ |)(1 − 2c†k′↑ck′↑). (2.30)

Ground state and excitations 39

���

���������

��� �����������

����

��

���

���

Figure 2.2. Distribution of electrons at T = 0.

Replacing the electron operators by the quasi-particle ones and taking the average,we obtain

〈[�(k), �†(k)]〉 = 4E2p�(ωD −|ξk|)

∑k′

�(ωD −|ξk′ |)(u2k′ −v2

k′)(1−2 fk′) = 0

(2.31)because the function under the integral is odd with respect to ξk′ .

The ground state of the BCS Hamiltonian �0 is a vacuum with respect toquasi-particles. Quasi-particles are fermions and their vacuum state is obtainedfrom the electron vacuum |0〉 by applying the quasi-particle annihilation operatorsfor every momentum k:

�0 = A∏

k

αkβk|0〉. (2.32)

Indeed, every single particle state k in �0 is free from quasi-particles, so thatapplying their annihilation operator to �0, we obtain zero. The eigenfunctionshould be normalized, which is the case if

A = 〈0|∏

k

β†kα

†kαkβk|0〉−1/2. (2.33)

In terms of the electron operators, we have

αk = ukck↑ − vkc†−k↓

βk = vkc†k↑ + ukc−k↓

and

�0 = A∏

k

(ukvk − v2kc†

−k↓c†k↑)|0〉 (2.34)

A =(∏

k

vk

)−1

. (2.35)

40 Weak coupling theory

Thus the ground-state wavefunction is

�0 =∏

k

(uk − vkc†−k↓c†

k↑)|0〉. (2.36)

It is a superposition of the eigenfunctions of the total number operator N ≡∑ks c†

kscks .

2.3 Meissner–Ochsenfeld effect

BCS reduced the many-body problem to a non-interacting gas of quasi-particlesallowing for an analytical description of thermodynamic and kinetic propertiesof conventional superconductors. Let us first discuss the BCS theory of themagnetic flux expulsion. We apply the perturbation theory for a linear interactionof electrons with the vector potential A(r) taking the BCS Hamiltonian as a zero-order one,

Hint = − e

m

∑k,q,s

(k · Aq)c†k+qscks (2.37)

where Aq is the Fourier component of A(r). This form of the interactionfollows from the velocity operator [−i∇ − e A(r)]/m in the effective mass (m)approximation for the band energy dispersion and the gauge, where ∇ · A(r) = 0.The field distribution in the sample is determined by the average of the currentdensity operator, which follows as the symmetrized form of the velocity operatorin the second quantization

j(r) = j p(r) + jd (r) (2.38)

wherej p(r) = − e

2m

∑k,q,s

c†k+qscks(2k + q)e−iq·r (2.39)

is the paramagnetic part and

jd(r) = −e2

m

∑k,q,s

c†k+qscks A(r)e−iq·r (2.40)

the diamagnetic part. The perturbed many-particle state � in the first order in Ais given by

� = |nα, nβ 〉 +∑

n′α,n′

β

|n′α, n′

β 〉 〈n′α, n′

β |Hint|nα, nβ 〉Enαnβ − En′

αn′β

(2.41)

where |nα, nβ 〉 and Enαnβ are the eigenstates and eigenvalues of the zero-

order Hamiltonian H , respectively. Applying the Bogoliubov transformation

BCS gap, critical temperature and single-electron tunnelling 41

(equations (2.13) and (2.14)) for Hint yields the average current density at T = 0:

j p(r) = 2e2

m2

∑k,q

uk+qvk(2k + q)(k · Aq)

εk+q + εk

× exp(−iq · r)(uk+qvk − ukvk+q) (2.42)

jd = − ne2

mA(r). (2.43)

Let us assume that the magnetic field varies over the characteristic length λH,which is large compared with the coherence length ξ . In this case one can takethe limit q → 0 in equation (2.42). In this limit uk+qvk − ukvk+q = 0, whilethe denominator remains finite: εk+q + εk > 2�(0). Therefore, the paramagneticcontribution vanishes and we obtain the London equation (chapter 1),

j(r) = −ne2

mA(r). (2.44)

In the opposite limit (λH < ξ ), the Pippard non-local theory of the flux expulsionfollows from equations (2.42) and (2.43). In the normal state the denominator inequation (2.42) turns out to be zero at the Fermi level and the paramagnetic currentappears to be finite. Actually, one can show that it cancels the diamagnetic partso that the normal state current is zero in a permanent magnetic field.

2.4 BCS gap, critical temperature and single-electrontunnelling

The BCS theory introduces the order parameter � (equation (2.21)) which is alsoa gap in the quasi-particle spectrum (figure 2.1) for a homogeneous system. Thevalue of the gap at T = 0 should be of the order of Tc. In fact, BCS theory predictsa universal ratio 2�(0)/Tc � 3.5 as follows from the master equation (2.21). AtT = 0, the non-trivial solution is determined from

1

λ=

∫ ωD

0

dξ√ξ2 + �2(0)

. (2.45)

Integration yields

�(0) � 2ωD exp

(− 1

λ

)(2.46)

for λ � 1, the limit to which the theory is applied. This is a remarkableresult which demonstrates the instability of the Fermi liquid for any value of theattraction λ in agreement with the Cooper two-particle solution. The exponent inequation (2.46) cannot be expanded in a series of λ. Thus the superconductingground state cannot be derived by using the perturbation theory with respect tothe pairing potential up to any order.

42 Weak coupling theory

At T = Tc, the gap should be zero so that Tc is determined by

1

λ=

∫ ωD

0

dξ tanh(ξ/2Tc)

ξ. (2.47)

Integrating by parts and replacing the upper limit by infinity, we have

Tc = 2eCωD

πexp

(− 1

λ

)(2.48)

where C � 0.577 is the Euler constant. The numerical coefficient inequation (2.48) is ≈1.14, so that

2�(0)

Tc� 3.5. (2.49)

To calculate the temperature dependence of �(T ) at low temperatures, T � Tcwe rewrite the master equation using its zero-temperature form (equation (2.45)):

ln�(T )

�(0)= −2

∫ ∞

0

dξ√ξ2 + �2(T )

f (ξ). (2.50)

Here we have replaced the upper limit in the integral for infinity which is justifiedbecause the quasi-particle distribution function falls down exponentially at lowtemperatures,

f (ξ) ≈ exp

[−

√ξ2 + �2(T )

T

]. (2.51)

Replacing the integration over ξ by the integration over the energy ε =√ξ2 + �2(T ) yields

ln�(T )

�(0)= −2

∫ ∞

�(T )

dεe−ε/T√

ε2 − �2(T ). (2.52)

The remaining integral is exponentially small at low temperatures, so that wecan replace �(T ) by �(0) on the right-hand side and expand ln in powers of�1(T ) = �(T ) − �(0) on the left-hand side with the following result:

�1(T ) = −√2πT�(0) exp

(−�(0)

T

). (2.53)

The temperature correction to the gap appears to be exponentially small at lowtemperatures. In the vicinity of Tc, where (Tc − T )/Tc � 1, the whole gap issmall compared with the temperature. However, a direct expansion in powers of� cannot be applied in equation (2.23), because every term of such an expansionwould render a divergent integral. Instead we use

tanh x

x=

∞∑n=−∞

1

x2 + [π(n + 1/2)]2 (2.54)

BCS gap, critical temperature and single-electron tunnelling 43

so that1

λ= 2T

∑n

∫ ωD

0

dξ

ξ2 + ω2n + �2(T )

(2.55)

where ωn = πT (2n + 1) are the so-called Matsubara frequencies, n =0,±1,±2, . . . . The last equation can be expanded in powers of �(T ) as

lnTc

T= 2�2(T )Tc

∑n

∫ ∞

0

dξ

(ξ2 + ω2n)2

. (2.56)

Calculating the integral over ξ and the sum over n (appendix B, equation (B.27))we obtain

�(T ) = πTc

[8

7ζ(3)

]1/2√

1 − T

Tc≈ 3.06Tc

√1 − T

Tc. (2.57)

There is a discontinuity in the temperature derivative of �(T ) at Tc which leadsto a jump of the specific heat (section 2.6).

The gap can be measured directly in tunnelling experiments where oneapplies voltage V to a thin dielectric layer between the normal metal and thesuperconductor (figure 2.3) [37]. The current running through the dielectric isproportional to the number of electrons tunnelling under the barrier per second.The electron, tunnelling from the normal metal, becomes a quasi-particle in thesuperconductor. Applying the Fermi–Dirac golden rule, we obtain the transitionrate as

I (V ) �∑

k(ξk<0),k′T 2

kk′δ(εk′ − ξk − eV ). (2.58)

The matrix element Tkk′ is almost independent of the momentum k in thenormal metal and of k′ in the superconductor, if the voltage is not very higheV ∼ � � µ. The δ-function in equation (2.58) takes into account the differenceeV in the normal and superconducting chemical potentials. Replacing the sum bythe integral we obtain

I (V ) �

∫ eV

−∞dξ

∫ +∞

−∞dξ ′ δ

[√ξ ′2 + �2(T ) − ξ

]. (2.59)

Then the conductance σ = dI/dV is found:

σ �

∫ +∞

−∞dξ ′ δ

[√ξ ′2 + �2(T ) − eV

]. (2.60)

Calculating the remaining integral, we obtain

σ

σN= eV√

(eV )2 − �2(T )(2.61)

44 Weak coupling theory

µ

2∆

�

�

��������

Figure 2.3. Tunnelling from the normal metal (N) to the superconductor (S) through adielectric barrier. Shaded areas correspond to occupied states.

where σN is the normal state conductance of the barrier above Tc.There is no current if |eV | < � because there are no states inside the

gap in the superconductor (figure 2.3). Just above the threshold, eV = �, theconductance has a maximum because the quasi-particle density of states ρ(ε)

diverges at ε = �,

ρ(ε) ≡ ∂|ξ |∂ε

= ε√ε2 − �2

. (2.62)

The typical experimental ratio σ/σN as a function of the voltage and thetemperature follows the BCS prediction (2.61) rather well in conventionalsuperconductors.

2.5 Isotope effect

The origin of the electron–electron attraction in superconductors can be testedby isotope substitution, when an ion mass M is varied without any change ofthe electronic configuration of the ion. There are two parameters in the BCSexpression for Tc (equation (2.48)) which depend on the mechanism of theinteraction. The characteristic phonon frequency ωD is proportional to 1/

√M

as a frequency of any harmonic oscillator. However, the coupling constant λ isindependent of the ion mass (section 3.3). Hence, the isotope exponent is foundas

α = − d ln Tc

d ln M= 0.5. (2.63)

In fact, the isotope exponent α could be lower than 0.5 in a BCS superconductorbecause of the Coulomb repulsion and the anharmonicity of phonons. But, inany case, the finite value of α measured experimentally proves that phonons are

Heat capacity 45

involved in the pairing mechanism. The isotope effect has been observed in manyconventional superconductors and in high-temperature superconductors (Part II).

2.6 Heat capacity

Only the electrons near the Fermi surface can absorb heat in a metal because ofthe Pauli principle. The number of these electrons is proportional to temperature.Therefore, their specific heat Ce in the normal state is linear as the function oftemperature (appendix B). The temperature dependence of Ce changes drasticallyin the superconducting state due to the gap. The quasi-particle energies depend onthe temperature in the self-consistent BCS approximation which should be takeninto account in the calculations of temperature derivatives of the thermodynamicpotential. We apply the definition of the specific heat as Ce = T dS/dT , wherethe quasi-particle entropy is defined as

S = −〈ln P(UQ , N)〉. (2.64)

Here P(UQ , N) = Z−1e−βUQ is the statistical probability of finding the Fermigas of N quasi-particles with energy UQ (appendix B),

UQ =∑

k

εk(nkα + nkβ) (2.65)

where nkα,β = 0, 1 are the quasi-particle occupation numbers; and

Z =∏

k

[1 + eβεk ]2 (2.66)

is the quasi-particle grand partition function. Here we take into account thefact that the quasi-particle chemical potential is zero. Calculating the statisticalaverage in equation (2.64), we obtain the entropy of the ideal Fermi gas of quasi-particles as

S = −2∑

k

[ fk ln fk + (1 − fk) ln(1 − fk)] (2.67)

where the distribution function fk = 〈nkα〉 = 〈nkβ〉 is defined in equation (2.22).The temperature derivative of the distribution function includes the derivative ofεk as

d fk

dT= fk(1 − fk)

(εk

T 2− 1

T

dεk

dT

)

so that

Ce = 2∑

k

fk(1 − fk)

(ε2

k

T 2− εk

T

dεk

dT

). (2.68)

46 Weak coupling theory

At low temperatures, the number of quasi-particles is exponentially small fk �e−�(0)/T and so is the specific heat (equation (2.68)). Above Tc, εk = |ξ | and weobtain Ce = CN, where

CN = N(EF)

T 2

∫ ∞

0dξ

ξ2

cosh2(ξ/2T )= 2π2

3N(EF)T (2.69)

as expected for the ideal Fermi gas (appendix B). However, just below Tc, thesecond term in the brackets of equation (2.68) appears to be finite:

εk

T

dεk

dT= �

T

d�

dT= −π2

2

(8

7ζ(3)

)(2.70)

and the specific heat has a discontinuity,

Ce = CN + 8π2

7ζ(3)N(EF)Tc (2.71)

if T = Tc − 0. Here ζ(3) � 1.202. The relative value of the jump is

Ce(Tc − 0) − Ce(Tc + 0)

Ce(Tc + 0)= 12

7ζ(3)� 1.43 (2.72)

which agrees with the value measured in many conventional superconductors.The phase transition turns out to be second order as in the GL phenomenology(chapter 1).

2.7 Sound attenuation

The interaction of ultrasound waves with electrons is described by the followingHamiltonian:

Hint = V (q)eiνt∑k,s

c†ksck−qs + H.c. (2.73)

where q = ν/s and ν are the wavevector and frequency of the sound, respectively(s is the sound velocity), V (q) is proportional to the sound amplitude and H.c. isthe Hermitian conjugate.

Applying the Bogoliubov transformation, we obtain four terms in theinteraction (2.73). Two of them correspond to the annihilation and creation oftwo different quasi-particles and the other two correspond to their scattering. Thesound frequency is low (ν � � � Tc) and only the scattering terms are relevant:

Hint �∑

k

M(−)kq (α

†kαk−q + β

†kβk−q) (2.74)

whereM(−)

kq = ukuk−q − vkvk−q (2.75)

Sound attenuation 47

���

���

����

������� �

� �����

���

���

���

Figure 2.4. Temperature dependence of sound attenuation and thermal conductivity (1)compared with the nuclear spin relaxation rate (2).

is a so-called coherence factor.The rate of the sound absorption is given by the Fermi–Dirac golden rule as

Wabs �∑

k

[M(−)kq ]2 fk−q(1 − fk)δ(εk−q − εk + ν) (2.76)

and the emission rate is

Wemi �∑

k

[M(−)

kq ]2 fk−q(1 − fk)δ(εk−q − εk − ν). (2.77)

The sound attenuation �, which is the difference between these two rates, is givenby

� �∑

k

[M(−)kq ]2( fk − fk−q)δ(εk−q − εk − ν). (2.78)

If ν � T,�, we have fk − fk−q ≈ ν∂ f/∂ε. Replacing the integration over k andover the angle between k and q by the integration over ξ = ξk and ξ ′ = ξk−q ,respectively, we obtain

� � −∫

dξ

∫dξ ′ [u(ξ)u(ξ ′) − v(ξ)v(ξ ′)]2 ∂ f

∂εδ(ε − ε′) (2.79)

where ε = √ξ2 + �2 and ε′ = √

ξ ′2 + �2. Only transitions with ξ = ξ ′contribute to the integral over ξ ′, so that

� � −∫ ∞

�

dε

(∂ξ

∂ε

)2

[u2(ξ) − v2(ξ)]2 ∂ f

∂ε. (2.80)

Here we can see that the large density of quasi-particle states ∂|ξ |/∂ε =ε/

√ε2 − �2 is cancelled by the small coherence factor in the integral in (2.80) as

� � −∫ ∞

�

dεε2 − �2

ε2 − �2

∂ f

∂ε= 1

exp(�/T ) + 1. (2.81)

48 Weak coupling theory

The sound attenuation in the superconducting state depends exponentially ontemperature (figure 2.4). Using its ratio to the normal-state attenuation,

�s

�n= 2

exp(�(T )/T ) + 1(2.82)

one can measure the temperature dependence of the BCS gap.

2.8 Nuclear spin relaxation rate

The measurement of the linewidth of the nuclear magnetic resonance (NMR) isanother powerful method of determining �(T ). The linewidth depends on theinverse time of the relaxation of nuclear magnetic moment due to the spin-flipscattering of carriers off nuclei, 1/T1. This scattering is described by the hyperfineinteraction of the nucleus with the electron spin:

Hint �∑k,k′

c†k′↓ck↑ + H.c. (2.83)

The NMR frequency is very small and the spin-flip scattering is practicallyelastic. That is why only the scattering of quasi-particles contribute to 1/T1 as inthe case of sound attenuation,

Hint �∑

k

M(+)

kk′ (β†k′αk + α

†kβk′). (2.84)

However, here the coherence factor

M(+)

kk′ = ukuk′ + vkvk′ (2.85)

is different. Applying the Fermi–Dirac golden rule, we obtain

1/T1 �∑k,k′

[M(+)

kk′ ]2( fk − fk′)δ(εk′ − εk − ν) (2.86)

ν → 0. Replacing the sums by the integrals yields a divergent integral,

1/T1 � −T∫ ∞

�

dεε2 + �2

ε2 − �2

∂ f

∂ε. (2.87)

In fact, the divergency is cut by some damping of excitations τ−1, for exampledue to the inelastic electron–phonon scattering. Above Tc, where � = 0, therelaxation rate is proportional to T . This linear temperature dependence of 1/T1in a normal metal is known as the Korringa law. It has the same origin as thelinear specific heat. Both are due to the Pauli principle. Only electrons in anarrow energy region around the Fermi surface can exchange their spin with

Thermal conductivity 49

nuclei and absorb heat. Well below Tc, the NMR relaxation rate is exponentiallysmall because of the gap:

1/T1 � e−�/T ln(�τ) (2.88)

as in sound attenuation. However, just below Tc, it has a maximum (seefigure 2.4). The maximum in 1/T1 is one of the most interesting and importantfeatures of BCS theory. This result is distinctly different from the result for soundattenuation. They differ because the coherence factors are different:

M(+,−) = (uu′ ± vv′). (2.89)

A simple energy-gap form of a two-fluid model could account for the drop inthe sound attenuation but not for the rapid rise of 1/T1 just below Tc. Theobservation by Hebel and Slichter [38] of the peak in 1/T1 was one of the firstin the body of evidence for the detailed nature of the pairing correlations in BCSsuperconductors.

2.9 Thermal conductivity

Important information about the excitation spectrum of the superconducting statecan also be obtained from thermal conductivity [39]. Quasi-particles contributeto heat transfer in a superconductor. Their contribution Q to the heat flow isdetermined by the deviation f of the distribution function from the equilibrium,

Q =∑

k

f vεk (2.90)

where v = ∂εk/∂k is the quasi-particle group velocity. The distribution functionobeys the Boltzmann equation, which for a small deviation from the equilibriumhas the form:

v · ∂ f

∂ r− ∂εk

∂ r· ∂ f

∂k= − f

τ str

(2.91)

where the transport relaxation rate for elastic scattering is obtained using theFermi–Dirac golden rule:

1

τ str

= |ξ |ε

1

τ ntr

. (2.92)

In the superconducting state, the transport relaxation rate is diminished by a factor|ξ |/ε < 1 compared with that in the normal state (1/τ n

tr) because of the squareof the coherence factor M(−) in the probability of scattering. The second termon the left-hand side of the Boltzmann equation accounts for the driving force,which acts on a quasi-particle due to the temperature dependence of its energy,ε = √

ξ2 + |�(T (r)|2. The solution of equation (2.91) is

f = τ ntr

ε2

T |ξ |∂ f

∂εv · ∇T . (2.93)

50 Weak coupling theory

Substituting it into equation (2.90) yields the thermal conductivity K =| Q|/|∇T | as

Ks � − 1

T

∫ ∞

�

dε ε2 ∂ f

∂ε. (2.94)

Hence, the ratio to the normal-state thermal conductivity is

Ks

Kn=

∫ ∞�/2T dx x2/cosh2(x)∫ ∞

0 dx x2/ cosh2(x). (2.95)

The normalized thermal conductivity (equation (2.95)) drops exponentially withthe temperature lowering below Tc just like sound attenuation (figure 2.4).

2.10 Unconventional Cooper pairing

In BCS theory electrons are paired with the opposite momenta on the Fermisurface and the opposite spins. The simple approximation for the pairing potential(equation (2.6)) leads to the momentum-independent gap �k = �, which isuniform along the Fermi surface. This is an s-wave order parameter becausethe pair wavefunction is isotropic in real space. It depends only on the distancebetween two correlated electrons. BCS theory describes an anisotropic Cooperpairing with non-zero orbital momentum of pairs l �= 0 as well, if the potentialallows for such pairing [40]. In the general case, the mean-field Hamiltonian is

H =∑

k,s=↑,↓

[ξkc†

kscks + 1

2

∑s ′

(�ss ′k c†

ksc†−ks ′ + H.c.)

](2.96)

where we omit the c-number term, which does not contain fermionic operators.The order parameter should be antisymmetric:

�ss ′k = −�ss ′

−k (2.97)

according to the Pauli exclusion principle. Applying the Bogoliubovtransformation, the master equation for any pairing potential V (k, k′) takes thefollowing form:

�ss ′k = −

∑k′

V (k, k′)�ss ′

k′

2εk′(1 − 2 fk′). (2.98)

Let us assume that V (k, k′) depends on the value of the momentum transferq = |k − k′|, which is the case for an isotropic system. In BCS theory,the states near the Fermi surface contribute to the sum in equation (2.97) andq ≈ 21/2kF

√1 − cos �′, where �′ is the angle between k and k′. Then all

functions in the master equation depend on the angles of k and k′ with the absolute

Unconventional Cooper pairing 51

values of the momenta equal to the Fermi momentum, |k| = |k′| = kF. We canexpand the order parameter and the potential in the polynomial series:

�ss ′k =

∞∑l=0

�(l)Pl (cos �) (2.99)

V (q) =∞∑

l=0

V (l)Pl(cos �′)

where Pl(cos �) are the Legendre polynomials,

Pl(x) = 1

2ll!dl

dxl(x2 − 1)l (2.100)

which are orthogonal and normalized as

∫ 1

−1dx Pl(x)Pl′(x) = 2δll′

2l + 1. (2.101)

This expansion is instrumental due to the summation theorem:

Pl (cos �′) = Pl(cos �)Pl(cos �′′)

+ 2l∑

m=1

(l − m)!(l + m)! Pm

l (cos �)Pml (cos �′′) cos[m(φ − φ′′)]

(2.102)

where Pml (x) are associated Legendre functions defined as

Pml (cos �) = sinm �

dm Pl(cos �)

(d cos�)m

and �′ is the angle between any two directions determined by the spherical angles(�, φ) and (�′′, φ′′), respectively. Substitution of the series into the masterequation yields a system of coupled nonlinear equations for the amplitudes �(l).These equations are linearized and decoupled at T = Tc. Using the orthogonalityof the Legendre polynomials and the summation theorem (equation (2.102)), weobtain, for an isotropic system with parabolic energy dispersion,

�(l) = − V (l)N(EF)

2l + 1�(l)

∫ ωD

0

dξ tanh(ξ/2Tc)

ξ. (2.103)

Then the critical temperature

Tc = 2eCωD

πexp

(− 1

λl∗

)(2.104)

52 Weak coupling theory

where λl∗ ≡ |V (l∗)|N(EF)/(2l∗ + 1) for negative V (l∗). The solution ofequation (2.103) is trivial: �(l) = 0 for all l except l = l∗ with a maximumvalue of λl∗ and negative V (l∗). Hence, if the Fourier transform of the pairingpotential has an attractive component with l �= 0, the system condenses intothe state with non-zero orbital momentum of the pair wavefunction. The single-particle gap, �k � Pl(cos �), has nodes on the Fermi surface if l �= 0. It is odd forl = 1, 3, 5, . . . and even for even l, �−k = (−1)l�k . The order parameter, �k , isproportional to the Fourier transform of the pair wavefunction. The change in signof k corresponds to the permutation of real-space coordinates of the two fermionsof the pair. According to the Pauli exclusion principle, the total wavefunctionof the pair should change its sign after the permutation of both the orbital andspin coordinates of two fermions. Hence, the spin component of the paired statewith an odd orbital momentum should be symmetric under the permutation of thespin coordinates of two particles, while the spin component of even orbital statesshould be antisymmetric. Therefore, the even l states (s, d, . . .) are singlets (totalspin S is zero) and the odd l states (p, f, . . .) are triplets (S = 1).

2.11 Bogoliubov equations

The Bogoliubov transformation of the BCS Hamiltonian can be readilygeneralized for inhomogeneous superconductors. Let us introduce field operators(appendix C)

�s(r) =∑

k

cks exp(ik · r). (2.105)

Here and further on, we take the volume of the system as V = 1. Then the BCSHamiltonian can be written as follows:

H =∫

dr[∑

s

�†s (r)h(r)�s(r) + �(r)�†

↑(r)�†↓(r) + �∗(r)�↓(r)�↑(r)

](2.106)

where

h(r) = −[∇ + ie A(r)]2

2m+ U(r) − µ (2.107)

is the one-electron Hamiltonian in the external magnetic (A(r)) and electric(U(r)) fields. We apply the effective mass (m) approximation for the banddispersion (see appendix A) and drop c-number terms in the total energy. Thecoordinate-dependent order parameter is given by

�(r) = −2E p〈〈�↓(r)�↑(r)〉〉. (2.108)

Superfluid properties of inhomogeneous superconductors can be studied by theuse of the Bogoliubov equations, fully taking into account the interaction of quasi-

Bogoliubov equations 53

particles with the condensate. To derive these equations, we introduce the time-dependent Heisenberg operators (appendix D) as

ψs(r, t) = eiHt�s(r)e−iH t . (2.109)

The equations of motion for these operators are readily derived:

i∂ψ↑(r, t)

∂ t= − [H , ψ↑(r, t)]= h(r)ψ↑(r, t) + �(r)ψ†

↓(r, t) (2.110)

and

−i∂ψ

†↓(r, t)

∂ t= [H , ψ

†↓(r, t)]

= h∗(r)ψ†↓(r, t) − �∗(r)ψ↑(r, t). (2.111)

Here we have applied commutation relations for the field operators:

{�s(r),�†s ′(r ′)} =

∑k,k′

{cks , c†k′s ′ } exp[ik · r − ik′ · r ′)

= δss ′∑

k

exp[ik · (r − r ′)] = δss ′δ(r − r ′)

and

[�†↑(r ′)�†

↓(r ′),�↑(r)] = − δ(r − r ′)�†↓(r) (2.112)

[�†↑(r ′)�↑(r ′),�↑(r)] = − δ(r − r ′)�†

↑(r).

The linear Bogoliubov transformation of ψ-operators has the form

ψ↑(r, t) =∑

n

[un(r, t)αn + v∗n (r, t)β†

n ] (2.113)

ψ↓(r, t) =∑

n

[un(r, t)βn − v∗n (r, t)α†

n ] (2.114)

where αn and α†n are the fermion operators, which annihilate and create quasi-

particles in a quantum state n. Using this transformation in equations (2.110)and (2.111), we obtain two coupled Schrodinger equations for the wavefunctionsu(r, t) and v(r, t):

id

dtu(r, t) = h(r)u(r, t) − �(r)v(r, t), (2.115)

−id

dtv(r, t) = h∗(r)v(r, t) + �∗(r)u(r, t).

54 Weak coupling theory

There is also the sum rule,∑n

[un(r, t)u∗n(r ′, t) + vn(r, t)v∗

n (r ′, t)] = δ(r − r ′) (2.116)

which retains the Fermi commutation relations for all operators. When themagnetic and electric fields are stationary, these equations are reduced to thesteady-state ones:

εnun(r) ={

−[∇ + ie A(r)]2

2m+ U(r) − µ

}un(r) − �(r)vn(r) (2.117)

εnvn(r) = −{

−[∇ − ie A(r)]2

2m+ U(r) − µ

}vn(r) − �∗(r)un(r).

Using the same transformation, the quantum and statistical averages inequations (2.108) yield the self-consistent equation for the order parameter:

�(r) = −2E p

∑n

un(r)v∗n (r)(1 − 2 fn) (2.118)

where fn = 〈〈α†nαn〉〉 = 〈〈β†

nβn〉〉 = (1 + exp εn/T )−1 is the equilibrium quasi-particle distribution in the quantum states n with energy εn . We can readilysolve the set of Bogoliubov equations (2.117) in the homogeneous case, whenA(r) = U(r) = 0. In this case the excitation wavefunctions are plane waves,

uk(r) = ukeik·r (2.119)

vk(r) = vkeik·r (2.120)

and the order parameter �(r) is r-independent, �(r) = �. Substitutingequations (2.119) and (2.120) into equations (2.117), we obtain

εkuk = ξkuk − �vk (2.121)

εkvk = − ξkvk − �∗uk (2.122)

and, from equation (2.116),

|uk|2 + |vk|2 = 1. (2.123)

As a result, we find

|uk|2 = 1

2

(1 + ξk

εk

)(2.124)

|vk|2 = 1

2

(1 − ξk

εk

)(2.125)

ukv∗k = − �

2εk. (2.126)

Landau criterion and gapless superconductivity 55

The elementary excitation energy is

εk =√

ξ2k + |�|2 (2.127)

as it should be (see equation (2.18)), where the gap is found using the masterequation,

� = 2E p

∑k

�

2εk(1 − 2 fk). (2.128)

2.12 Landau criterion and gapless superconductivity

The Bogoliubov equations are coupled nonlinear integra-differential equations.They are mathematically transparent but the analytical solution is possible only ina few simple cases. As an example, we consider a uniform flow with the superfluidvelocity vs . This state is described by an oscillating complex order parameter

�(r) = � exp(2iq · r) (2.129)

where 2q = 2mvs is the centre-of-mass momentum of the Cooper pair, and � isthe real amplitude. Let us examine how the flow destroys superconductivity. Ifthe flow is uniform, the system remains translation invariant and the momentumk is a quantum number. The solution has the form

un(r) = uk exp[i(k + q) · r] (2.130)

vn(r) = vk exp[i(k − q) · r]where the coefficients uk and vk are found from

(Ek − ξk+q)uk+�vk = 0 (2.131)

(Ek + ξk−q)vk+�uk = 0.

Taking into account the normalization condition u2k + v2

k = 1, we obtain

u2k, v

2k = 1

2

(1 ±

ξk+q + ξ k−q

2εk

)(2.132)

ukvk = − �

2εk

and the excitation spectrum

Ek =ξk+q − ξ k−q

2+

√(ξk+q + ξ k−q)2

4+ �2. (2.133)

If we apply the Landau criterion (1.117) to the spectrum equation (2.133), thecritical velocity will be vc = �/kF. Hence, we can keep only the terms of the

56 Weak coupling theory

first order in q as long as q � mvc � kF in equation (2.133). Then the excitationspectrum becomes

Ek ≈ εk + k · vs (2.134)

and the equation for the order parameter is

� = 2E p

∑k

�

2εk[1 − 2 f (εk − k · vs)]. (2.135)

The excitation energy (equation (2.133)) is negative for some directions of kand quasi-particles appear even at zero temperature, when the superfluid velocityexceeds vc. The Landau criterion tells us that superconductivity should disappearat vs > vc. Let us see how it happens by solving the BCS equation (2.135) in twoand three dimensions. For T = 0, we have

f (εk−k · vs) = θ(−εk + kvs cos ϕ)

where ϕ is the angle between vs and k. If vs < vc, there are no quasi-particles( fk = 0) and

� = �(0). (2.136)

Here �(0) is the BCS gap (equation (2.46)) in the absence of the flow. However, ifvs > vc, there is an interval of |ϕ| � arccos[εk/(kFvs)], where f (εk−k · vs) = 1and the gap should be different from �(0). Integrating over the angle, we obtain

� ln�(0)

�= 2�

π

∫ kFvs

�

dεarccos(ε/kFvs)√

ε2 − �2. (2.137)

in two dimensions. Calculating the remaining integral, we arrive at

� ln�(0)

�= � ln

kFvs

�. (2.138)

This equation has only the trivial solution � = 0. Hence, the Landau criterioncannot be compromised in two-dimensional s-wave BCS superconductors, thatis superconductivity disappears precisely at vs � vc. The superflow in three-dimensional superconductors is different [41]. In this case, integrating over theangle yields

� ln�(0)

�= �

∫ kFvs

�

dε1 − (ε/kFvs)√

ε2 − �2. (2.139)

There is a non-trivial solution � �= 0 even if vs > vc, which is found from

lnkFvs

�(0)=

√1 −

(�

kFvs

)2

− ln

1 +

√1 −

(�

kFvs

)2 . (2.140)

It disappears only at vs � evc/2, where e = 2.718. The excitation spectrum(equation (2.133)) has no gap but the order parameter is still non-zero if vc < vs <

Andreev reflection 57

evc/2. The superconductivity in this region is gapless. Normal excitations fill thenegative part of their energy spectrum in k-space resulting in two components(normal and superfluid), as at finite temperatures. The quasi-particles do notfully destroy the order parameter, they renormalize it while vs < evc/2. Thetwo-fluid situation at T = 0 is possible due to the Fermi statistics of quasi-particles in the BCS superconductor because the Pauli exclusion principle limitstheir density. When quasi-particles are bosons (chapter 4), the kinetic energy of amoving condensate entirely dissipates into quasi-particles as soon as vs � vc.

There are other examples of the gapless superconductivity like unconven-tional Cooper pairs (section 2.10) and ‘dirty’ superconductors with magnetic im-purities [42].

2.13 Andreev reflection

The Bogoliubov equations are particularly instrumental in handling the interfacebetween normal and superconducting metals (NS interface) and between differentsuperconductors (SS interface). They allow us to calculate the I–V characteristicsof tunnelling structures in the same fashion as in the conventional single-particletunnelling problem in quantum mechanics. If there is a potential barrier due to adielectric layer, NS conductance can be readily calculated using the Fermi–Diracgolden rule (section 2.4). It shows the gap structure. In the absence of the barrier,a new phenomenon is observed, in which an incoming electron from the normalside of the normal/superconducting contact is reflected as a hole along the sametrajectory [46]. The Andreev reflection results in an increase in the tunnellingconductance in the voltage range |eV | � � in sharp contrast to its suppression inthe case of the barrier.

A simple theory of NS tunnelling in a metallic (no-barrier) regime [47]follows from the one-dimensional Bogoliubov equations, which can be writtenin the matrix form:

Eψ(x) =(−(1/2m) d2/dx2 − µ(x) �(x)

�(x) (1/2m) d2/dx2 + µ(x)

)ψ(x).

(2.141)The gap �(x) and the chemical potential µ(x) depend on the coordinate xperpendicular to the contact area. In the normal state (x � 0, �(x) = 0,

µ(x) = µn), equation (2.141) is the free-particle Schrodinger equation (first row)or its time-reversed version (second row). The two-component wavefunction ofthe normal metal is given by

ψn(x < 0) =(

10

)eiq+x + b

(10

)e−iq+x + a

(01

)e−iq−x (2.142)

where momenta associated with the energy E are q± = [2m(µn ± E)]1/2. Herethe first and second terms describe the incident and reflected electron plane waves,respectively. The third term describes the reflected (Andreev) hole. The hole

58 Weak coupling theory

appears when the electron with the momentum q ≈ kF tunnels from a normalmetal into a superconducting condensate together with its ‘partner’ having theopposite momentum −q . This simultaneous two-electron tunnelling results in thehole excitation on the hole branch of the excitation energy spectrum near −kFin the normal metal (figure 2.5). In the superconductor (x � 0, �(x) = �,

µ(x) = µs) the incoming electron can produce only outgoing particles (i.e. withthe positive group velocity ∂εk/∂k) (figure 2.5). The solution in this region isgiven by

ψs(x > 0) = c

(1�

E + ξ

)eik+x + d

(1�

E − ξ

)e−ik−x (2.143)

Here the momenta associated with the energy E are k± = [2m(µs ± ξ)]1/2,where ξ = √

E2 − �2. The energy of the incident electron is defined in thewhole positive region (E � 0), so that ξ is not necessary real.

The coefficients a, b, c, d are determined from the boundary conditions,which are the continuity of ψ(x) and its first derivative at x = 0, as inthe conventional single-particle tunnelling problem. Applying the boundaryconditions, we obtain

1 + b = c + d (2.144)

a = c�

E + ξ+ d

�

E − ξ(2.145)

q+(1 − b) = ck+ − dk− (2.146)

and

q−a = ck+ �

E + ξ− dk− �

E − ξ. (2.147)

For simplicity, we now take q± ≈ k± ≈ kF because the Fermi energy is hugecompared with the gap in the BCS superconductors. Then we find b = d = 0,

which, physically speaking, means that all reflection is the Andreev reflection andall transmission occurs without branch crossing.

The transmission coefficient, which determines the conductance, is given by

T (E) = 1 + |a|2 − |b|2. (2.148)

With b = d = 0, we obtain c = 1 and

a = �

E + ξ(2.149)

from equations (2.144) and (2.145), respectively. If the incident energy (or thevoltage V = E/e) is larger than the gap (E � �), ξ is real and the transmissionis obtained as

T (E) = 2E

E + √E2 − �2

. (2.150)

Green’s function formulation of the BCS theory, T = 0 59

�

������

��� ������ ���

�����

������ �� ��� ����������

��� ����

Figure 2.5. Schematic plot of excitation energies versus k at an NS interface. The opencircle denotes the Andreev hole, the closed circles denote an incident at (1), reflected (b)electrons, and transmitted (c), (d) quasi-particles.

It tends to one in the high energy limit (E � �). Remarkably, when the energyis below the gap (E � �), the transmission is doubled compared with the normalstate limit:

T (E) = 1 + �2

(E + ξ)(E + ξ∗)= 2. (2.151)

Hence, the Andreev reflection is observed as an enhancement of the tunnellingconductance of NS metallic contacts in the gap region V � �/e. Thephenomenon serves as a powerful tool in the gap determination (part 2).

2.14 Green’s function formulation of the BCS theory, T = 0

There is yet another elegant formalism introduced by Gor’kov [43], which allowsfor an economic derivation of BCS results and an extension of the theory tothe intermediate coupling regime (chapter 3). Using the Heisenberg operators,equation (2.109), let us define the time-dependent ‘normal’ G(r, r ′, t) and‘anomalous’ F±(r, r ′, t) one-particle Green’s functions (GF) as

iG(r, r ′, t) = �(t)〈ψs(r, t)ψ†s (r ′, 0)〉 − �(−t)〈ψ†

s (r ′, 0)ψs(r, t)〉 (2.152)

iF+(r, r ′, t) = �(t)〈ψ†↓(r, t)ψ†

↑(r ′, 0)〉 − �(−t)〈ψ†↑(r ′, 0)ψ

†↓(r, t)〉

iF(r, r ′, t) = �(t)〈ψ↓(r, t)ψ↑(r ′, 0)〉 − �(−t)〈ψ↑(r ′, 0)ψ↓(r, t)〉.Here the quantum averages are calculated in the ground state of the system at zerotemperature. The operators ψ↑(r, t) and ψ↓(r ′, t ′) anticommute if t ′ = t . Hence,F(r, r ′, 0) and F+(r, r ′, 0) are connected by the relation

F∗(r, r ′, 0) = −F+(r, r ′, 0). (2.153)

60 Weak coupling theory

The first time derivatives of the GFs are calculated using the equations of motion(2.110) and (2.111). Taking into account that d�(t)/dt = δ(t), we obtain asystem of two coupled equations:

i∂G(r, r ′, t)

∂ t= δ(t)δ(r − r ′) + h(r)G(r, r ′, t) + �(r)F+(r, r ′, t) (2.154)

and

i∂ F+(r, r ′, t)

∂ t= −h∗(r)F+(r, r ′, t) + �∗(r)G(r, r ′, t). (2.155)

The order parameter is expressed in terms of the anomalous GF as

�(r) = −2iE p[F+(r, r,+0)]∗. (2.156)

We can readily solve these self-consistent equations in the absence of anexternal field. In a homogeneous superconductor, G(r, r ′, t) = G(r − r ′, t),F+(r, r ′, t) = F+(r − r ′, t) and �(r) is a real constant. Applying the Fouriertransforms

G(r − r ′, t) = 1

2π

∑k

∫ ∞

−∞dω G(k, ω) exp(ik · r − iωt) (2.157)

F+(r − r ′, t) = 1

2π

∑k

∫ ∞

−∞dω F+(k, ω) exp(ik · r − iωt)

we obtain

(ω − ξk)G(k, ω) − �F+(k, ω) = 1 (2.158)

(ω + ξk)F+(k, ω) − �G(k, ω) = 0.

Then the Fourier components are found to be

G(k, ω) = ω + ξk

ω2 − ε2k

= u2k

ω − εk+ v2

k

ω + εk(2.159)

F+(k, ω) = �

ω2 − ε2k

.

These expressions are well defined for any ω but not for ω = ±εk . They cannotbe used for integration with respect to ω because their simple poles are just on thereal axis of ω. We have to define a way of bypassing the poles in evaluating theintegral. The solution of the first-order differential equations (2.154) and (2.155)are not well defined because we have not applied any initial condition so far.The easiest way to apply the condition is to consider the normal state limit ofequation (2.159) with � = 0,

G(0)(k, ω) = �(ξk)

ω − ξk+ �(−ξk)

ω − ξk(2.160)

[F+(k, ω)](0) = 0.

Green’s function formulation of the BCS theory, T = 0 61

Integrating the Fourier transform G(0)(k, ω) should yield the normal state GF,which can be directly calculated using its definition. The free-electron Heisenbergoperators are found as

ψs(r, t) =∑

k

cks exp[i(k · r − ξkt)]

so that

iG(0)(r − r ′, t) = �(t)∑k,k′

〈cksc†k′s〉 exp[i(k · r − k′ · r ′) − iξkt]

− �(−t)〈c†k′scks〉 exp[i(k · r − k′ · r ′) − iξkt] (2.161)

where 〈c†k′scks〉 = δkk′�(−ξk) is the Fermi–Dirac distribution at T = 0 and

〈cksc†k′s〉 = δkk′�(ξk). Hence, we have

iG(0)(r − r ′, t) =∑

k

exp[ik · (r − r ′) − iξkt]{�(t)�(ξk) − �(−t)�(−ξk)}.(2.162)

We obtain the same result using the Fourier transform equation (2.160), if wechoose the following way of bypassing the poles:

G(0)(k, ω) = �(ξk)

ω − ξk + iδ+ �(−ξk)

ω − ξk − iδ(2.163)

where δ = +0 is an infinitesimal positive constant. Indeed let us calculate theintegral∫ ∞

−∞dω G(0)(k, ω) exp(−iωt) = �(ξk)

∫ ∞

−∞dω

e−iωt

ω − ξk + iδ

+ �(−ξk)

∫ ∞

−∞dω

e−iωt

ω − ξk − iδ. (2.164)

When t is positive the contour in both integrals of the right-hand side shouldbe chosen in the lower half-plane of the complex variable. The pole in thefirst term is below the real axis, while the pole in the second term is found inthe upper half-plane. The first integral yields −2π i�(t)�(ξk) and the secondintegral is zero. When t is negative, the first integral is zero, while the secondone is 2π i�(−t)�(−ξk). As a result, we recover the ideal Fermi-gas GF(equation (2.162)). Hence, the superconducting GFs, which provide a correctnormal state limit, are

G(k, ω) = u2k

ω − εk + iδ+ v2

k

ω + εk − iδ(2.165)

F+(k, ω) = ukvk

[1

ω + εk − iδ− 1

ω − εk + iδ

].

62 Weak coupling theory

We note that the poles of the Fourier transform of GF yields the excitationspectrum of the superconductor.

Let us show that these GFs also provide the BCS results for the electrondistribution function and the gap. The electron density is calculated as

ne = −iG(r, r,−0) = − i

2π

∑k

[ ∫ ∞

−∞dω e−iωt G(k, ω)

]t→−0

. (2.166)

Here t is negative and the contour should be taken in the upper half-plane. Onlythe second term of the normal GF, equation (2.165), contributes with the followingresult:

ne =∑

k

v2k (2.167)

so that the electron distribution function is v2k, as it should be (see

equation (2.27)). The gap is found to be

� = −2iE p[F+(r, r,+0)]∗ = −2iE p

2π

∑k

∫ ∞

−∞dω eiωt {F+(k, ω)}∗ (2.168)

where t = +0. Calculating the integral with the Fourier transformequation (2.165), we obtain the BCS equation at T = 0,

� = −2E p

∑k

ukvk. (2.169)

2.15 Green’s functions of the BCS superconductor at finitetemperatures

At finite temperatures, the BCS theory can be formulated with the ‘temperature’GFs (appendix D). Following Matsubara [44], we replace time t in the definitionof the Heisenberg operators by a ‘thermodynamic time’ τ = it . Then thetemperature GF is defined as

�(r, r ′, τ1, τ2) = −〈〈Tτψs(r, τ1)ψ†s (r ′, τ2)〉〉 (2.170)

where ψs(r, τ ) = exp(H τ )�s(r) exp(−H τ ) and Gor’kov’s temperature GF is

�+(r, r ′, τ1, τ2) = −〈〈Tτψ†↓(r, τ1)ψ

†↑(r ′, τ2)〉〉.

Here the thermodynamic ‘times’, τ1, τ2, are real and positive, varying in theinterval 0 < τ1, τ2 < 1/T . The double angular brackets correspond to quantumas well as statistical averages of any operator A with the Gibbs distribution(appendix B),

〈〈A〉〉 =∑ν

e(�−Eν )/T 〈ν| A|ν〉 ≡ Tr{e(�−H)/T A} (2.171)

Green’s functions of the BCS superconductor at finite temperatures 63

where � is the thermodynamic potential and |ν〉 are the eigenstates of H with theeigenvalues Eν . The operation Tτ performs the ‘time’ ordering according to thefollowing definition:

Tτψs(r, τ1)ψ†s ′(r ′, τ2) ≡ �(τ1 − τ2)ψs(r, τ1)ψ

†s ′(r ′, τ2)

− �(τ2 − τ1)ψ†s ′(r ′, τ2)ψs(r, τ1). (2.172)

Actually the temperature GFs depend on the difference τ = τ1 − τ2 because ofthe trace in their definition. Indeed let us consider τ1 < τ2, so that

�(r, r ′, τ1, τ2) = exp(�/T ) Tr{e−H/T eτ2 H�†s (r ′)eτ H�s(r)e−τ1 H }

= exp(�/T ) Tr{e−(τ+1/T )H�†s (r ′)eτ H�s(r)} (2.173)

where we have performed the cyclic permutation under the trace. Differing frompositive τ1 and τ2, the variable τ is defined in the domain

−1/T � τ � 1/T . (2.174)

There is a connection between GFs with negative and positive τ . Following thedefinition in equation (2.170), we find, for τ > 0, that

�(r, r ′, τ ) = − exp(�/T ) Tr{e(τ−1/T )H�s(r)e−τ H�†s (r ′)}. (2.175)

Here replacing the positive τ by a negative τ as τ = τ + 1/T , we obtain (τ < 0)

�(r, r ′, τ + 1/T ) = − exp(�/T ) Tr{eτ H �s(r)e−(τ+1/T )H�†s (r ′)}

= − �(r, r ′, τ ). (2.176)

The Fourier transform theory states that if a function F(τ ) is defined over theinterval −1/T � τ � 1/T , then its Fourier expansion is

F(τ ) = T∞∑

n=−∞fn exp(−iπnT τ ) (2.177)

where n = 0,±1,±2, . . . , and

fn = 12

∫ 1/T

−1/Tdτ F(τ ) exp(iπnT τ ). (2.178)

In our case, F(τ ) = −F(τ + 1/T ) for a negative τ and

fn = 12 (1 − eiπn)

∫ 1/T

0dτ F(τ ) exp(iπnT τ ). (2.179)

64 Weak coupling theory

Hence the Fourier components fn of the fermionic GFs are non-zero only forodd n. As a result we can expand the temperature GFs of a homogeneoussuperconductor into a Fourier series as follows:

�(r, r ′, τ ) = T∑ωn

�(k, ωn) exp[ik · (r − r ′) − iωnτ ] (2.180)

�+(r, r ′, τ ) = T∑ωn

�+(k, ωn) exp[ik · (r − r ′) − iωnτ ] (2.181)

where the discrete Matsubara frequencies are ωn = πT (2n + 1), n =0,±1,±2, . . . . Differentiating the Matsubara operators with respect to τ , weobtain the equations of motion,

−∂ψ↑(r, τ )

∂τ= h(r)ψ↑(r, τ ) + �(r)ψ†

↓(r, τ ) (2.182)

∂ψ†↓(r, τ )

∂τ= h∗(r)ψ†

↓(r, τ ) − �∗(r)ψ↑(r, τ )

and the equations for temperature GFs,

−∂�(r, r ′, τ )

∂τ= δ(τ )δ(r − r ′) + h(r)�(r, r ′, τ )

+ �(r)�+(r, r ′, τ ), (2.183)∂�+(r, r ′, τ )

∂τ= h∗(r)�+(r, r ′, τ ) − �∗(r)�(r, r ′, τ ). (2.184)

The order parameter is given by

�∗(r) = −2E p�+(r, r, 0). (2.185)

In the absence of an external field, the Fourier transforms of these equations are:

(iωn − ξk)�(k, ωn) − ��+(k, ωn) = 1 (2.186)

(iωn + ξk)�+(k, ωn) − �∗�(k, ωn) = 0

and�∗ = −2E pT

∑k

∑ωn

�+(k, ωn). (2.187)

The solution is

�(k, ωn) = − iωn + ξk

ω2n+ε2

k

(2.188)

�+(k, ωn) = − �∗

ω2n+ε2

k

Green’s functions of the BCS superconductor at finite temperatures 65

and the equation for the order parameter takes the following form:

� = 2E pT∑

k

∑ωn

�

ω2n+ε2

k

. (2.189)

The sum over frequencies in equation (2.189) is calculated using the expansion oftanh(x) (equation (2.54)). As a result, we obtain the BCS equation for the orderparameter:

1 = λ

2

∫dξ√

ξ2 + �(T )2tanh

√ξ2 + �(T )2

2T(2.190)

where the integral should be cut at |ξ | � ωD.There is no direct physical meaning of poles of the Fourier transforms of

temperature GFs, which are found on the imaginary frequency axis. Nonetheless,they allow for a direct calculation of the thermodynamic properties of the system,for example of the gap (equation (2.190)). The kinetic properties are expressed interms of real-time GFs. In fact, the Matsubara GFs lead directly to real-time GFs.A one-particle real-time GF is defined at finite temperatures as

G(k, t) = −i〈〈Tt cks(t)c†ks〉〉 (2.191)

with the real time t . There are also retarded GR and advanced GA GFs:

GR(k, t) = − i�(t)〈〈{ck(t)c†k}〉〉 (2.192)

GA(k, t) = i�(−t)〈〈{ck(t)c†k}〉〉. (2.193)

Their Fourier components are analytical in the upper or lower half-plane of ω,respectively. There is a simple connection between the Fourier components of Gand GR,A (appendix D):

GR,A(k, ω) = Re G(k, ω) ± i coth( ω

2T

)Im G(k, ω) (2.194)

and between those of GR,A and � ,

GR(k, iωn) = �(k, ωn) (2.195)

for ωn > 0 and�(k,−ωn) = �∗(k, ωn). (2.196)

In our case the temperature GF is

�(k, ωn) = u2k

iωn − εk+ v2

k

iωn + εk(2.197)

where u2k, v

2k = (εk ± ξk)/2εk and εk =

√ξ2

k + �2. The analytical continuationof this expression to the upper half-plane yields

GR(k, ω) = u2k

ω − εk + iδ+ v2

k

ω + εk + iδ. (2.198)

66 Weak coupling theory

Then using equation (2.194), we obtain

G(k, ω) = P

(u2

k

ω − εk+ v2

k

ω + εk

)− iπ tanh

( ω

2T

)[u2

kδ(ω−εk)+v2kδ(ω+εk)]

(2.199)where we have applied the relation 1/(ω+ iδ) = P(1/ω)− iπδ(ω). Here the firstterm is understood as the principal value of the integral when integrating withrespect to ω. For zero temperature, tanh(ω/2T ) = sign(ω) and

G(k, ω) = u2k

ω − εk + iδ+ v2

k

ω + εk − iδ(2.200)

as it should (see equation (2.165)).

2.16 Microscopic derivation of the Ginzburg–Landauequations

Using the Green’s function formalism near Tc, Gor’kov [45] derived theGinzburg–Landau equations from BCS theory. Let us consider the BCSsuperconductor in a stationary magnetic field with the vector potential A(r).The Fourier transform of equations (2.183) and (2.184) with respect to thethermodynamic time yields{

iωn + [∇ + ie A(r)]2

2m+ µ

}�ωn (r, r ′) = δ(r − r ′) + �(r)�+

ωn(r, r ′),

{−iωn+[∇ − ie A(r)]2

2m+ µ

}�

+ωn

(r, r ′) = − �∗(r)�ωn (r, r ′) (2.201)

where

�ωn (r, r ′) =∫ 1/T

0dτ �(r, r ′, τ ) exp(iπnT τ )

�+ωn

(r, r ′) =∫ 1/T

0dτ �+(r, r ′, τ ) exp(iπnT τ )

and�∗(r) = −2E pT

∑ωn

�+ωn

(r, r).

If the temperature is close to Tc, the order parameter is small. Then we canexpand the GFs in powers of �(r). The zero-order GF is a temperature GF ofan ideal Fermi gas, �(n)

ωn (r, r ′) in the magnetic field, which satisfies the followingequation:{

iωn + [∇ + ie A(r)]2

2m+ µ

}�

(n)ωn

(r, r ′) = δ(r − r ′). (2.202)

Microscopic derivation of the Ginzburg–Landau equations 67

If there is no magnetic field, �(n)ωn (r, r ′) = �

(0)ωn (r − r ′) and its Fourier transform

is found as

�(0)(k, ωn) = 1

iωn − ξk. (2.203)

Transforming back to real space we obtain

�(0)ωn

(ρ) = 1

(2π)3

∫dk

eik·ρ

iωn − ξk(2.204)

where ρ = r − r ′. Calculating the integral over the angles in the polar sphericalcoordinates yields

�(0)ωn

(ρ) = 1

(2π)2iρ

∫ ∞

0k dk

eikρ − e−ikρ

iωn − ξ. (2.205)

We can replace k by k = kF + ξ/vF because only the states near the Fermi surfacecontribute to the integral in equation (2.205), when ωn is of the order of Tc � EF.The result is

�(0)ωn

(ρ) = m

(2π)2iρ

∫ ∞

−∞dξ

eikFρeiξρ/vF − e−ikρe−iξρ/vF

iωn − ξ. (2.206)

Using the contour in the upper or lower half-plane, we obtain

�(0)ωn

(ρ) = − m

2πρexp[i sign(ωn)kFρ − |ωn |ρ/vF]. (2.207)

The Fourier component of the normal state GF falls exponentially with thecharacteristic length

ρ ≈ ξ0 (2.208)

where ξ0 = vF/(2πTc) is the zero temperature coherence length. GFs in amagnetic field are not translation invariant and the exact calculation of the normal�

(n)ωn (r, r ′) using equation (2.202) is a challenging problem. But we can assume

that the spatial variations of the vector potential are small over the characteristicdistance ξ0. If we neglect these variations, then the field operator in the magneticfield can be readily expressed via the free-electron operator. It satisfies theequation of motion

∂ψs(r, τ )

∂τ=

{[∇ + ie A(r)]2

2m+ µ

}ψs(r, τ ). (2.209)

If A(r) is a constant then the solution is

ψs(r, τ ) =∑

k

cks exp[i(k − e A) · r − ξkτ )] (2.210)

68 Weak coupling theory

which is verified by direct substitution into equation (2.209). Hence, if the vectorpotential varies slowly, the GF differs from the zero-field GF only by the phase:

�(n)ωn

(r, r ′) = e−ie A(r)·ρ�(0)ωn

(ρ). (2.211)

This quasi-classical approximation with respect to the magnetic field is appliedif the phase change over the distance ξ0 is small. |A(r)| varies by Bξ0 overthis distance, and the phase changes by eBξ2

0 . The maximum field is (seesection 1.6.4)

B � Hc2 ≈ �0

2πξ20

Tc − T

Tc(2.212)

so that the quasi-classical approximation is applied, if

Tc − T

Tc� 1. (2.213)

It is the very same region where the Ginzburg–Landau theory is applied. Usingthe normal state GF in the magnetic field (equation (2.211)), we can transformGor’kov equations (2.201) into the integral form:

�ωn (r, r ′) = �(n)ωn

(r, r ′) +∫

dx�(n)ωn

(r, x)�(x)�+ωn

(x, r ′) (2.214)

�+ωn

(r, r ′) = −∫

dx �(n)−ωn

(x, r)�∗(x)�ωn (x, r ′).

Now we expand �ωn (r, r ′) to the terms of the second order and �ωn (r, r ′) to theterms of the third order in �(r) inclusive:

�ωn (r, r ′) = �(n)ωn

(r, r ′) −∫

dx∫

d y �(n)ωn

(r, x)�(x)

× �(n)−ωn

(y, x)�∗(y)�(n)ωn

(y, r ′) (2.215)

�+ωn

(r, r ′) = −∫

dx�(n)−ωn

(x, r)�∗(x)�(n)ωn

(x, r ′)

+∫

dx∫

d y∫

dz �(n)−ωn

(x, r)�∗(x)�(n)ωn

(x, y)�(y)

× �(n)−ωn

(z, y)�∗(z)�(n)ωn

(z, r ′).

Using this expansion, we obtain the following equation for the order parameter:

�∗(r) = 2E pT∑ωn

∫dx �(n)

−ωn(x, r)�∗(x)�(n)

ωn(x, r)

− 2E pT∑ωn

∫dx

∫d y

∫dz �(n)

−ωn(x, r)�∗(x)�(n)

ωn(x, y)�(y)

× �(n)−ωn

(z, y)�∗(z)�(n)ωn

(z, r). (2.216)

Microscopic derivation of the Ginzburg–Landau equations 69

Let us consider the kernel in the first term

K (x − r) = T∑ωn

�(n)−ωn

(x, r)�(n)ωn

(x, r) = K (0)(x − r)e−2ie A(r)·(x−r) (2.217)

where

K (0)(x − r) = Tm2

(2πρ)2

∑ωn

exp[−2|ωn|ρ/vF]

= m2T

(2πρ)2 sinh(2πTρ/vF)(2.218)

and ρ = |x − r|. The kernel K (0)(ρ) has a characteristic radius about ξ0, while�(x) changes over a much larger distance of the order of ξ(T ) � ξ0 near Tc.Therefore in the integral over x we can expand all quantities near the point x = rup to the second order in ρ inclusive:∫

dx K (0)(x − r)e−2ie A(r)·(x−r)�∗(x)

≈ �∗(r)∫

dρ K (0)(ρ) +∫

dρ K (0)(ρ)ρ · [∇r − 2ie A(r)]�∗(r)

+ 1

2

3∑i, j=1

∫dρ K (0)(ρ)ρiρ j

(∂

∂ri− 2ieAi

)(∂

∂r j− 2ieA j

)�∗(r).

(2.219)

Here the second term is zero because the function under the integral is odd. Theremaining integrals are:∫

dρ K (0)(ρ) = T∑ωn

∫dρ �

(0)−ωn

(ρ)�(0)ωn

(ρ)

= T∑

k

∑ωn

1

ω2n + ξ2

k

= N(EF)

∫ ωD

0dξ

tanh(ξ/2T )

ξ(2.220)

and∫dρ K (0)(ρ)ρiρ j = δi j

3

∫dρ K (0)(ρ)ρ2

= δi j

3

v2F

4(πT )2N(EF)

∫ ∞

0dx

x2

sinh x= δi j

3

7ζ(3)v2F

8(πT )2N(EF).

(2.221)

We cut the divergent integral in equation (2.220) at |ξ | = ωD as usual andintroduce the DOS at the Fermi level: N(EF) = mkF/(2π2). The second term

70 Weak coupling theory

on the right-hand side of equation (2.216) is cubic in �(r). Here we can neglectthe space variations of �(x) and the phase due to the vector potential under theintegrals. Then applying the Fourier transform of the zero-field ideal gas GF, weobtain the following expression for this term:

−2E pT �∗(r)|�(r)|2∑

k

∑ωn

1

(ω2n + ξ2

k )2= −λ

7ζ(3)

8(πT )2�∗(r)|�(r)|2.

(2.222)We note that Tc is determined from the linearized BCS equation:

1 = λ

∫ ωD

0dξ

tanh(ξ/2Tc)

ξ.

Then close to Tc, we can expand it as∫ ωD

0dξ

tanh(ξ/2T )

ξ≈ 1

λ+ Tc − T

Tc(2.223)

where we use the integral, equation (B.28) from the appendix. Introducing the‘condensate wavefunction’ φ(r) as

φ(r) = �(r)

√7ζ(3)ne

8π2T 2c

(2.224)

and collecting all integrals in equation (2.216), we obtain the phenomenologicalGinzburg–Landau equation (section 1.6.1):

− 1

4m[∇ + i2e A(r)]2φ(r) + β|φ(r)|2φ(r) = −αφ(r).

Now the coefficients α and β are determined microscopically as

α = 12π2Tc

7ζ(3)mv2F

(T − Tc) (2.225)

β = 12π2T 2c

7ζ(3)mv2Fne

.

Here ne = k3F/(3π2) is the electron density. Finally let us show that the second GL

equation for supercurrent follows from the expansion of GFs (2.215) in powersof �(r). The current–density operator can be expressed in terms of the fieldoperators:

(r) =∑

s

[ie

2m(∇r − ∇r ′)|r ′→r�

†s (r ′)�s(r) − e2

mA(r)�†

s (r)�s(r)

].

(2.226)

Microscopic derivation of the Ginzburg–Landau equations 71

Its expectation value is

j(r) = ie

m(∇r − ∇r ′)|r ′→r T

∑ωn

�ωn (r, r ′) − e2ne

mA(r). (2.227)

There is no current in the normal state in the stationary magnetic field, and thefirst (normal) term in the expansion of GFs (equation (2.215)) cancels the secondterm on the right-hand side of equation (2.227). The remaining supercurrent isquadratic with respect to the order parameter:

j(r) = ie

m(∇r ′ − ∇r )|r ′→r T

×∑ωn

∫dx

∫d y �(n)

ωn(r, x)�(x)�

(n)−ωn

(y, x)�∗(y)�(n)ωn

(y, r ′).

(2.228)

The normal state GF in the magnetic field �(n)ωn (r, r ′) = exp(−ie A · ρ)�

(0)ωn (ρ)

is the product of a slowly varying phase exponent and the zero-field GF, whichoscillates in real space with the electron wavelength (≈1/kF). Differentiating theoscillating part yields

j(r) = ie

mT

∑ωn

∫dx

∫d y �(x)�∗(y)e2iA·(x−y)�

(0)−ωn

(| y − x|)

× [�(0)ωn

(|r − x|)∇r�(0)ωn

(| y − r|) − �(0)ωn

(| y − r|)∇r�(0)ωn

(|r − x|)].(2.229)

Expanding the slowly varying order parameter and the magnetic phase exponentin powers of | y − r| and |x − r|, which are of the order of ξ0,

�(x) ≈ �(r) + (x − r) · ∇�(r)

�∗(y) ≈ �(r) + (y − r) · ∇�∗(r)

exp[2ie A · (x − y)] ≈ 1 + 2ie A · (x − y)

we obtain

j(r) = ie

mC[�∗(r)∇�(r) − �(r)∇�∗(r)] − 4e2C

m|�(r)|2 A(r). (2.230)

Here the constant C is given by

C = T

3

∑ωn

∫dx

∫d y �(0)

−ωn(| y − x|)�(0)

ωn(|r − x|)[(x − y) · ∇r ]�(0)

ωn(| y − r|).

(2.231)

72 Weak coupling theory

Using the Fourier transform of the ideal gas GF in calculating the integrals overx and y in C leads to

C = T

3

∑ωn

∑k

1

iωn + ξk

∂

∂k· k

1

(iωn − ξk)2

= T

3m

∑ωn

∑k

k2

(iωn + ξk)(iωn − ξk)3. (2.232)

Integrating with respect to k yields

C = 2k2F

3mN(EF)T

∑ωn

∫ ∞

−∞dξ

ω2n − ξ2

(ω2n + ξ2)3 (2.233)

and we obtain

C = k2Fπ

6mN(EF)T

∑ωn

1

ω3n

= 7ζ(3)ne

16π2T 2c

.

Replacing the order parameter in equation (2.230) by the ‘condensatewavefunction’ leads to the second GL equation (see equation (1.40)):

j(r) = ie∗

2m∗∗ [φ∗(r)∇φ(r) − φ(r)∇φ∗(r)] − e∗2

m∗∗ A(r)|φ(r)|2 (2.234)

where e∗ = 2e and m∗∗ = 2m are the charge and mass of the Cooper pair,respectively.

The BCS theory is a mean-field approximation, which is valid if the volumeoccupied by the correlated electron pair is large compared with the volume perelectron. As an example, in aluminium the size of the pair (i.e. the coherencelength at T = 0) is about ten thousand times larger than the distance betweenelectrons. The Cooper pairs in the BCS superconductor disappear above Tcand the one-particle excitations are fermions. When the coupling constant λ

increases, the critical temperature increases and the coherence length becomessmaller. Hence, one can expect some deviations from the BCS behaviour in theintermediate coupling regime. At the first stage, deviations from BCS theory arisein two ways: (1) the BCS approximation for the interaction between electronsdoes not provide an adequate representation of the retarded nature of the phononinduced attraction; and (2) the damping rate becomes comparable with the quasi-particle energy. Both the retardation effect and the damping are taken into accountin the Eliashberg extension of the BCS theory to the intermediate coupling regime[36], which is discussed in the next chapter. However, if λ is larger than 1, theFermi liquid becomes unstable even in the normal state above Tc because of thepolaron collapse of the electron band (chapter 4). Here we encounter qualitativelydifferent physics [11].

Chapter 3

Intermediate-coupling theory

3.1 Electron–phonon interaction

The attraction between electrons in BCS theory is the result of an ‘overscreening’of their Coulomb repulsion by vibrating ions. When the interaction betweenion vibrations and electrons (i.e. the electron–phonon interaction) is strong, theelectron Bloch states are affected even in the normal phase. Phonons are alsoaffected by conduction electrons. In doped insulators (like high-temperaturesuperconductors), ‘bare’ phonons are well defined in insulating parent compoundsbut a separation of electron and phonon degrees of freedom might be a problemin a metal. Here we have to start with the first-principle Hamiltonian describingconduction electrons and ions coupled by the Coulomb forces:

H = −∑

i

∇2i

2me+ e2

2

∑i �=i ′

1

|r i − r i ′ | − Ze2∑

i j

1

|r i − R j |

+ Z2e2

2

∑j �= j ′

1

|R j − R j ′ | −∑

j

∇2j

2M(3.1)

where r i , R j are the electron and ion coordinates, respectively, i = 1, . . . , Ne;j = 1, . . . , N ; ∇i = ∂/∂ r i , ∇ j = ∂/∂ R j , Ze is the ion charge and Mis the ion mass. The system is neutral and Ne = Z N . The inner electronsare strongly coupled to the nuclei and follow their motion. Hence, the ionscan be considered as rigid charges. To account for their high-energy electrondegrees of freedom we can replace the elementary charge in equation (3.1) bye/

√ε∞, where ε∞ is the phenomenological high-frequency dielectric constant.

One cannot solve the corresponding Schrodinger equation perturbatively becausethe Coulomb interaction is strong. The ratio of the characteristic Coulomb energyto the kinetic energy is rs = mee2/(4πne/3)1/3 ≈ 1 for the electron densityne = Z N/V = 1023 cm−3 (further we take the volume of the system as V = 1,

unless specified otherwise). However, we can take advantage of the small value

73

74 Intermediate-coupling theory

of the adiabatic ratio me/M < 10−3. The ions are heavy and the amplitudes〈|u|〉 � √

1/MωD of their vibrations near the equilibrium R0 ≡ l are muchsmaller than the lattice constant a = N−1/3:

〈|u|〉a

≈(

me

Mrs

)1/4

� 1. (3.2)

In this estimate we take the characteristic vibration frequency ωD of the orderof the ion plasma frequency ωi = √

4π N Z2e2/M . Because the vibrationamplitudes are small we can expand the Hamiltonian in powers of |u| up toquadratic terms inclusive. Any further progress requires a simplifying physicalidea, which is to approach the ground state of the many-electron system viaa one-electron picture. This is called the local density approximation (LDA),which replaces the Coulomb electron–electron interaction by an effective one-body potential �(r):

�(r) = −Ze2∑

j

1

|r − R j | + e2∫

dr ′ n(r ′)|r − r ′| + µex[n(r)] (3.3)

where µex[n(r)] is the exchange interaction, usually calculated numericallyor expressed as µex[n(r)] = −βn1/3(r) with the constant β in a simpleapproximation. �(r) is the functional of the electron density n(r) =∑

s〈�†s (r)�s(r)〉. As a result, the Hamiltonian takes the form in the second

quantization:H = He + Hph + He−ph + He−e (3.4)

where

He =∑

s

∫dr �†

s (r)

[− ∇2

2me+ V (r)

]�s(r) (3.5)

is the electron energy in a periodic crystal field V (r) = ∑l v(r − l) which is �(r)

calculated at R j = l and with the periodic electron density n(0)(r + l) = n(0)(r),

Hph =∑

l

[− ∇2

u

2M+ ul · ∂

∂ l

∫dr n(0)(r)V (r)

]+ 1

2

∑l,m,α,β

u lαumβ Dαβ(l − m)

(3.6)is the vibration energy. Here α, β = x, y, z and

Dαβ(l − m) = ∂2

∂lα∂mβ

[Z2e2

2

∑l ′ �=m′

1

|l ′ − m′| +∫

dr n(0)(r)V (r)]

(3.7)

is a dynamic matrix. The electron–phonon interaction is given by

He−ph =∑

l

ul · ∂

∂ l

∫dr

[∑s

�†s (r)�s(r) − n(0)(r)

]V (r)

Electron–phonon interaction 75

+ 1

2

∑l,m,α,β

u lαumβ∂2

∂lα∂mβ

∫dr

[∑s

�†s (r)�s(r) − n(0)(r)

]V (r)

(3.8)

and the electron–electron correlations are described by

He−e = 1

2

∫dr

∫dr ′ e2

|r − r ′|[∑

ss ′�†

s (r)�†s ′(r ′)� ′

s ′(r ′)�s(r)]

−∫

dr[ ∫

dr ′ e2n(0)(r ′)|r − r ′| +µex[n(0)(r)]

]∑s

�†s (r)�s(r)

+ Z2e2

2

∑l �=m

1

|l − m| . (3.9)

We include the electrostatic repulsive energy of nuclei in He−e, so that the averageof He−e is zero in the Hartree approximation.

The vibration Hamiltonian Hph is a quadratic form and, therefore, canbe diagonalized with a linear canonical transformation for the displacementoperators

ul =∑q,ν

eqν√2N Mωqν

dqν exp(iq · l) + H.c. (3.10)

∂

∂ul=

∑q,ν

eqν

√Mωqν

2Ndqν exp(iq · l) − H.c.

where q is the phonon momentum, dqν is the phonon (Bose) annihilationoperator, eqν and ωqν are the unit polarization vector and the phonon frequency,respectively, of the phonon mode ν. Then Hph takes the following form

Hph =∑q,ν

ωqν(d†qνdqν + 1/2) (3.11)

if the eigenfrequencies ωqν and the eigenstates eqν satisfy

Mω2qνeα

qν =∑β

Dαβq eβ

qν (3.12)

and ∑q

e∗αqν eβ

qν = Nδαβ . (3.13)

The last equation and the bosonic commutation rules [dqνd†q′ν′ ] = δνν ′δqq ′ follow

from (∂/∂uαl )uβ

l − uβ

l (∂/∂uαl ) = δαβ . Here

Dαβq =

∑m

exp(iq · m)Dαβ(m) (3.14)

76 Intermediate-coupling theory

is the Fourier transform of the second derivative of the ion potential energy. Thefirst derivative in equation (3.6) is zero in crystals with a centre of symmetry.Different solutions of equation (3.12) are classified with the phonon branch(mode) quantum number ν, which is 1, 2, 3 for a simple lattice and 1, . . . , 3kfor a lattice with k ions per unit cell.

The periodic part of the Hamiltonian He is diagonal in the Blochrepresentation (appendix A):

�s(r) =∑k,n,s

ψnks(r)cnks (3.15)

where cnks are the fermion annihilation operators. The Bloch function obeys theSchrodinger equation(

− ∇2

2me+ V (r)

)ψnks(r) = Enksψnks(r). (3.16)

One-particle states are sorted with the momentum k in the Brillouin zone, bandindex n and spin s. The solutions of this equation allow us to calculate the periodicelectron density n(0)(r), which determines the crystal field potential V (r). TheLDA can explain the shape of the Fermi surface of wide-band metals and gapsin narrow-gap semiconductors. The spin-polarized version of LDA can explain avariety of properties of many magnetic materials. This is not the case in narrowd- and f-band metals and oxides (and other ionic lattices), where the electron–phonon interaction and Coulomb correlations are strong. These materials displaymuch less band dispersion and wider gaps compared with the first-principle bandstructure calculations. Using the phonon and electron annihilation and creationoperators, the Hamiltonian is written as

H = H0 + He−ph + He−e (3.17)

whereH0 =

∑k,n,s

ξnksc†nkscnks +

∑q,ν

ωqν(d†qνdqν + 1/2) (3.18)

describes independent Bloch electrons and phonons, ξnks = Enks − µ is the bandenergy spectrum with respect to the chemical potential. The part of the electron–phonon interaction, which is linear in phonon operators, can be written as

He−ph = 1√2N

∑k,q,n,n′,ν,s

γnn′ (q, k, ν)ωqνc†nkscnk−qsdqν + H.c. (3.19)

where

γnn′(q, k, ν) = − N

M1/2ω3/2qν

∫dr (eqν · ∇v(r))ψ∗

nks(r)ψn′k−qs(r) (3.20)

Phonons in metal 77

is the dimensionless matrix element. Low-energy physics is often described by asingle-band approximation with the matrix element γnn(q, k, ν) depending onlyon the momentum transfer q (the Frohlich interaction):

γnn′(q, k, ν) = γ (q, ν). (3.21)

The terms of He−ph which are quadratic and higher orders in the phonon operatorsare small. They have a role to play only for those phonons which are not coupledwith electrons by the linear interaction (3.20). The electron–electron correlationenergy of a homogeneous electron system is often written as

He−e = 12

∑q

Vc(q)ρ†qρq (3.22)

where Vc(q) is a matrix element, which is zero for q = 0 because ofelectroneutrality and

ρ†q =

∑k,s

c†ksck+qs (3.23)

is the density fluctuation operator. H0 should also include a random potential indoped semiconductors and amorphous metals.

3.2 Phonons in metal

In wide-band metals such as Na or K, the correlation energy is relatively small(rs ≤ 1) and carriers are almost free. Core electrons together with nuclei formcompact ions with an effective Z . The carrier wavefunction outside the core canbe approximated by a plane wave

ψnks(r) � eik·r (3.24)

and the carrier density n(0)(r) is a constant. Therefore, the only relevantinteraction in the dynamic matrix (equation (3.7)) is the Coulomb repulsionbetween ions, which yields

Dαβ(l − m) = Z2e2

2

∂2

∂lα∂mβ

∑l ′ �=m′

1

|l ′ − m′| . (3.25)

The electron–ion interaction is a pure Coulomb attraction v(r) = −Ze2/r, whichis expanded in the Fourier series as

1

r= 4π lim

κ→0

∑q

1

q2 + κ2eiq·r . (3.26)

Substituting this expansion into equations (3.25) and (3.20), we obtain

Dαβ(m) = 4π limκ→0

∑q

qαqβ

q2 + κ2 cos(q · m) (3.27)

78 Intermediate-coupling theory

a

+

+

+ +

b

c d e

= +f

D(q,ω)

Figure 3.1. Second (a) and fourth order (b), (c), (d), (e) corrections to the phonon GF. Thephonon GF in the Migdal approximation (f ).

and

γ (q, ν) = i4π N Ze2√

Mω3q

limκ→0

eqν · q

q2 + κ2. (3.28)

Calculating the Fourier transform of equation (3.27), one obtains the followingequation for phonon frequencies and polarization vectors:

ω2q eqν = ω2

i qeqν · q

q2 . (3.29)

A longitudinal mode with e ‖ q is the ion plasmon

ωq = ωi (3.30)

and two shear (transverse) modes with e ⊥ q have zero frequencies, which isthe result of our approximation considering ions as rigid charges. In fact, coreelectrons undergo a polarization, when ions are displaced from their equilibriumpositions, which yields a finite shear mode frequency.

According to equation (3.28), the carriers interact only with longitudinalphonons. The interaction gives rise to a significant renormalization of thebare phonon frequency (equation (3.30)). We apply the Green’s function (GF)formalism (appendix D) to calculate the renormalized phonon frequency. TheFourier component of the free-electron GF is given by equation (2.163) withξk = k2/2me −µ. For interacting electrons, the electron self-energy is introducedas

�(k, ω) = [G(0)(k, ω)]−1 − [G(k, ω)]−1 (3.31)

and

G(k, ω) = 1

ω − ξk − �(k, ω). (3.32)

Phonons in metal 79

The phonon GF is defined as

D(q, t) = −iωq

2〈Tt (dq(t)d†

q + d†q (t)dq)〉, (3.33)

and its Fourier transform for free phonons is a dimensionless even function offrequency,

D(0)(q, ω) = ω2q

ω2 − ω2q + iδ

. (3.34)

The phonon self-energy is

�(q, ω) = [D(0)(q, ω)]−1 − [D(q, ω)]−1. (3.35)

The Feynman diagram technique is convenient, see figure 3.1. Thin straightand dotted lines correspond to G(0) and D(0), respectively, a vertex (circle)corresponds to the interaction matrix element γ (q)

√ωq/N and bold lines

represent G and D. The Frohlich interaction is the sum of two operatorsdescribing the emission and absorption of a phonon. Both events are taken intoaccount in the definition of D. Therefore wavy lines have no direction. There areno first- or higher-odd orders corrections to D because the Frohlich interaction isoff-diagonal with respect to phonon occupation numbers. The second-order termin D (figure 3.1(a)) includes the so-called polarization bubble, �

(0)e , which is a

convolution of two G(0). Among different fourth-order diagrams the diagram infigure 3.1(b) with two polarization loops is the most ‘dangerous’ one. Differingfrom others, it is proportional to 1/q2, which is large for small q . However, thesingularity of internal vertices is ‘integrated out’ in the diagrams in figures 3.1(c)–(d). The sum of all dangerous diagrams is given in figure 3.1(f ) which is

�(q, ω) = |γ (q)|2ωq

N�e(q, ω) (3.36)

where�e(q, ω) = �(0)

e (q, ω) (3.37)

and

�(0)e (q, ω) = − 2i

(2π)4

∫dk dε G(0)(k + q, ε + ω)G(0)(k, ε). (3.38)

The additional factor 2 in the phonon self-energy is due to a contribution of twoelectron spin states. It is convenient to integrate over frequency in equation (3.38)first with the following result:

�(0)e (q, ω) = 1

4π3

∫dk

�(ξk) − �(ξk+q)

ω + ξk − ξk+q + iδ sign(ξk+q). (3.39)

80 Intermediate-coupling theory

� �

����

Figure 3.2. Screened polarization bubble �e(q, ω).

The phonon frequency ω is small, ω � µ. Thus, we can take the limit ω → 0 in�

(0)e and obtain

Re �0e(q, ω) = − mekF

2π2h

(q

2kF

)(3.40)

Im �0e(q, ω) = − m2

e

2πq|ω|�(2kF − q) (3.41)

where

h(x) = 1 + 1 − x2

2xln

∣∣∣∣1 + x

1 − x

∣∣∣∣ .We should also take the Coulomb electron–electron interaction into accountbecause the corresponding vertex is singular in the long-wavelength limit,Vc(q) = 4πe2/q2. This leads to a drastic renormalization of the long wave-length behaviour of �e. In the ‘bubble’ or random phase approximation (RPA),we obtain figure 3.2 as in the case of the electron–ion plasmon interaction,figure 3.1(f ), but with the Coulomb (dashed-dotted) line instead of the dottedphonon line. In the analytical form, we have

�e(q, ω) = �(0)e (q, ω)

1 − (4πe2/q2)�0e(q, ω)

. (3.42)

As a result, in the long-wavelength limit q � qs one obtains

�e(q, ω) = −mekF

π2

q2

q2s

(3.43)

where qs = √4mekFe2/π is the inverse (Debye) screening radius.

While �(0)e is finite at q → 0, the screened �e(q, ω) is zero in this

limit. Using the RPA expressions (equations (3.40)–(3.42)) and γq determinedin equation (3.28) for ωq = ωi , we obtain the phonon GF as

D(q, ω) = ω2i

ω2 − ω2q. (3.44)

Electrons in metal 81

The poles of D determine a new phonon dispersion and a damping � due tointeraction with electrons:

ωq = ωi

ε(q, ωq )1/2(3.45)

where

ε(q, ω) = 1 − 4πe2

q2�(0)

e (q, ω) (3.46)

is the electron dielectric function. In the long-wavelength limit,

ε(q, 0) = 1 + q2s

q2 (3.47)

and we obtain the sound wave as the real part of ω:

ωq = sq (3.48)

where s = ZkF/√

3Mme is the sound velocity. The imaginary part of ω

determines the damping of the sound,

� �s

vFω. (3.49)

Because the ratio of the sound velocity to the Fermi velocity (vF) is adiabaticallysmall (s/vF � √

me/M), the damping is small (� � ω). Electrons screen thebare ion–ion Coulomb repulsion and the residual short-range dynamic matrix hasthe sound-wave linear dispersion of the eigenfrequencies in the long-wavelengthlimit.

3.3 Electrons in metal

The lowest contribution to the electron self-energy is given by two second-orderdiagrams, see figures 3.3(a) and (b). The diagram in figure 3.3(b) is proportionalto |γ (q)|2 with q ≡ 0, which is zero according to equation (3.28).

Higher-order diagrams are taken into account by replacing the bare ionicplasmon GF by a renormalized one, equation (3.44), and the bare electron–phonon interaction γ (q) by a screened one, γsc(q, ω), as shown in figure 3.4.Presented analytically, the diagram in figure 3.4 corresponds to

γsc(q, ω) = γ (q) + 4πe2

q2�(0)

e (q, ω)γsc(q, ω) (3.50)

so that

γsc(q, ω) = γ (q)

ε(q, ω). (3.51)

82 Intermediate-coupling theory

Figure 3.3. Second-order electron self-energy.

� �

Figure 3.4. Screened electron–phonon interaction (dark circle).

For low-energy excitations (ω � µ) a static approximation of the dielectricfunction (3.47) is appropriate. Instead of the D and γsc given by equations (3.44)and (3.51), respectively, we introduce the acoustic phonon GF

D(q, ω) = ω2q

ω2 − ω2q

(3.52)

and the electron–acoustic phonon vertex γ (q)√

ωq/N , where

γ (q) = γ (q)

ε(q, 0)

(ωi

ωq

)3/2

. (3.53)

Finally we obtain the diagram in figure 3.5 for the electron self-energy as aresult of the summation of the most divergent diagrams, which is

�(k, ε) = 2i

(2π)4 N

∫dq dω E pG(k − q, ε − ω)D(q, ω) (3.54)

where

2E p = |γ (q)|2ωq = 2µ

Z(1 + q2/q2s )

. (3.55)

Because Z is of the order of one, the electron–acoustic phonon interactionE p is generally of the order of the Fermi energy. Therefore, one has to consider

Electrons in metal 83

Figure 3.5. Electron self-energy in the Migdal approximation.

Figure 3.6. Adiabatically small corrections to the electron self-energy.

fourth- and higher-order diagrams with the crossing phonon lines as in figure 3.6,which are absent in figure 3.5. These diagrams are known as vertex corrections.Fortunately, as shown by Migdal [48], their contribution is adiabatically small(∼s/vF) compared with equation (3.54). This result is known as Migdal’s‘theorem’.

The electron energy spectrum, renormalized by the electron–phononinteraction, is determined as the pole of the electron GF,

ξk ≡ Ek − µ = vF(k − kF) + δEk (3.56)

whereδEk = �(k, ξk) − �(kF, 0) (3.57)

and µ = µ + �(kF, 0) is the renormalized Fermi energy. The region of largeq � kF � qs contributes mostly to the integral in equation (3.54). Thedimensionless coupling constant λ = 2E p N(EF) is small in this region,

λ ≈ rs � 1 (3.58)

and the second order is sufficient in our calculations of �. Hence, we canreplace the exact GF in the integral equation (3.54) by the free GF. This isappropriate for almost all quasi-particle energies. The difference between G andG(0) appears to be important only for the damping calculations in a very narrowregion |ξk| � ωDs/vF near the Fermi surface. As a result, we obtain

δEk = 2iE p

(2π)4 N

∫dq dω D(q, ω)[G(0)(k − q, ξk − ω) − G(0)(kF − q,−ω)].

(3.59)To simplify the calculations, we take E p as a constant and apply the Debyeapproximation ωq = sq for q < qD where qD � π/a is the Debye momentum.We also consider a half-filled band, kF ≈ π/2a with the energy-independent DOSnear the Fermi level N(EF) = mea2/4π .

The main contribution to the integral in equation (3.59) comes from themomentum region close to the Fermi surface,

|k − q| � kF. (3.60)

84 Intermediate-coupling theory

It is convenient to introduce a new variable k ′ = |k − q| instead of the angle �

between k and q, and extend the integration to ±∞ for ξ = vF(k ′ − kF). Thenthe angular integration in equation (3.59) yields∫

d� sin �(. . .) ∼∫ ∞

−∞dξ

ξ

[ξ − ω − ξ + iδ sign(ξ)][ω + ξ − iδ sign(ξ)] .(3.61)

This integral is non-zero only if ξ > ω > 0 or ξ < ω < 0. It is −2π i in thefirst region and 2π i in the second region. Taking into account that D is an evenfunction of ω, we obtain

δEk = 2E p

(2π)2vF N

∫ qD

0dq q

∫ |ξ |

0dω sign(ξ )

ω2q

ω2 − ω2q + iδ

. (3.62)

The real and imaginary parts of equation (3.62) determine the renormalizedspectrum and the lifetime of quasi-particles, respectively:

Re(δEk) = E p

4π2vF N

∫ qD

0dq qωq ln

∣∣∣∣∣ ωq − ξ

ωq + ξ

∣∣∣∣∣ (3.63)

Im(δEk) = E p

4πvF N

∫ qm

0dq qωq sign(ξ ) (3.64)

with qm = |ξ |/s if |ξ | < ωD and qm = qD if |ξ | > ωD. For excitations far awayfrom the Fermi surface (|ξ | � ωD), we find

Re(δEk) = −λω2

D

2ξ(3.65)

and for low-energy excitations with |ξ | � ωD,

Re(δEk) = −λξ . (3.66)

This means an increase in the effective mass of the electron due to the electron–phonon interaction,

ξ = kF

m∗ (k − kF) (3.67)

where the renormalized mass is

m∗ = (1 + λ)me. (3.68)

Hence, the excitation spectrum of metals has two different regions with twodifferent values of the effective mass. The thermodynamic properties of a metal atlow temperatures T � ωD involve m∗ but the optical properties in the frequencyrange ν � ωD are determined by high-energy excitations, where, according toequation (3.65), corrections are small and the mass is equal to the band mass me.

Eliashberg equations 85

Damping shows just the opposite behaviour. The integral in equation (3.64),yields

Im(δEk) = sign(ξ )πλωD

3(3.69)

if |ξ | > ωD and

Im(δEk) = sign(ξ )πλ|ξ |3

3ω2D

(3.70)

if |ξ | � ωD.These expressions describe the rate of decay of quasi-particles due to the

emission of phonons. In the immediate neighbourhood of the Fermi surface,|ξ | � ωD, the decay is small compared with the quasi-particle energy |ξ | even fora relatively strong coupling λ ∼ 1 and the concept of well-defined quasi-particleshas a definite meaning. Within the Migdal approximation the electron–phononinteraction does not destroy the Fermi-liquid behaviour of electrons. The Pauliexclusion principle is responsible for the stability of the Fermi liquid. In theintermediate-energy region |ξ | ∼ ωD, the decay is comparable with the energyand the quasi-particle spectrum loses its meaning. In the high-energy region|ξ | � ωD, the decay becomes small again in comparison with |ξ | and the quasi-particle concept recovers its meaning.

Going beyond the Migdal approximation, we have to consider adiabaticallysmall higher-order diagrams, that is to solve the Hamiltonian of free electrons andacoustic phonons coupled by an interaction:

He−ph = 1√2N

∑k,q,s

γ (q)ωqc†ksck−qs dq + H.c. (3.71)

where dq is the acoustic-phonon annihilation operator. From our consideration,it follows that, in applying this Hamiltonian to electrons, one should not considerthe acoustic phonon self-energy. Acoustic phonons in a metal appear as a resultof the electron-plasmon coupling and the Coulomb screening, so their frequencyalready includes the self-energy effect (section 3.2).

3.4 Eliashberg equations

Based on Migdal’s theorem, Eliashberg [36] extended BCS theory towards theintermediate-coupling regime (λ � 1), applying the Gor’kov formalism. Thecondensed state is described by a classical field, which is the average of theproduct of two annihilation field operators � � 〈ψψ〉 or two creation operators�+ � 〈ψ†ψ†〉. These averages are macroscopically large below Tc. Theappearance of anomalous averages cannot be seen perturbatively but they shouldbe included in the self-energy diagram, figure 3.5, from the very beginning. This

86 Intermediate-coupling theory

can be done in a compact form by introducing the matrix GF [49]:

�s(k, τ ) = −( 〈〈Tτ ck↑(τ )c†

k↑〉〉 〈〈Tτ ck↑(τ )c−k↓〉〉〈〈Tτ c†

−k↓(τ )c†k↑〉〉 〈〈Tτ c†

−k↓(τ )c−k↓〉〉)

(3.72)

and the matrix self-energy

�(k, ωn) = (�(0)(k, ωn))−1 − �−1s (k, ωn) (3.73)

where �(0)(k, ωn) = (iωnτ0−ξkτ3)−1 is the normal-state matrix GF. Here τ0,1,2,3

are the Pauli matrices,

τ0 =(

1 00 1

)τ1 =

(0 11 0

)

τ2 =(

0 −ii 0

)τ3 =

(1 00 −1

).

The generalized equation for the matrix � is given by the same diagram as in thenormal state (figure 3.5) but replacing γ (q) by γ (q)τ3 and the integral over ω bythe sum over the Matsubara frequencies:

�(k, ωn) = − T

(2π)3 N

∑ωn′

∫dq γ 2(q)ωqτ3�s(k − q, ωn′ )τ3�(q, ωn − ωn′).

(3.74)Here the temperature GF of phonons is given by

�(q,�n) = − ω2q

�2n + ω2

q(3.75)

where �n = 2πT n are the Matsubara frequencies for bosons. The mostimportant difference between equation (3.74), figure 3.7, and the normal stateequation (3.54), figure 3.5, is a finite value of the anomalous GF in theself-consistent solution. If we replace �s by �(0) on the right-hand side ofequation (3.74), we do not find any anomalous� . Hence, there are no anomalousaverages and no phase transition in the second or in any finite order of theperturbation theory. However, if we solve equation (3.74) self-consistently, wefind the finite anomalous averages.

The diagrams in figure 3.7 are expressed in analytical form as a setof Eliashberg equations, similar to Gor’kov’s equations (2.186) but with amicroscopically determined pairing potential. They include all non-crossing(‘ladder’) diagrams in every order of the perturbation theory. For simplicity, letus consider a momentum-independent γ 2(q)ωq as in the previous section andapproximate the phonon GF in equation (3.74) as

�(q,�n) ≈ −1 (3.76)

Eliashberg equations 87

� � �

� �

�

��

Figure 3.7. Normal � and anomalous �+ GFs of the BCS superconductor in theEliashberg theory.

if |�n| < ωD, and zero otherwise. � is the sum of three Pauli matrices τ0,1,3with the coefficients (1 − Z)iωn , � and χ, respectively, which in this case arefunctions of frequency alone,

�(k, ωn) = i(1 − Z)ωnτ0 + �τ1 + χτ3 (3.77)

and�−1

s (k, ωn) = iZωnτ0 − �τ1 − ξkτ3. (3.78)

Transforming the inverse matrix (equation (3.78)) back into the original one yields

�s(k, ωn) = − iZωn + �τ1 + ξkτ3

Z2ω2n + ξ2

k + |�|2 (3.79)

where ξk = ξk + χ . Substituting equations (3.76) and (3.79) into the masterequation (3.74) leads to the following simplified Eliashberg equations:

[1 − Z(ωn)]iωn = − λT∫ ∞

−∞dξ

∑ωn′

�(ωD − |ωn − ωn′ |)iωn′ Z

Z2ω2n′ + ξ2 + |�|2 (3.80)

χ(ωn) = − λT∑ωn′

∫ ∞

−∞dξ

�(ωD − |ωn − ωn′ |)ξZ2ω2

n′ + ξ2 + |�|2 (3.81)

and

�(ωn) = λT∑ωn′

∫ ∞

−∞dξ

�(ωD − |ωn − ωn′ |)�(ωn′)

Z2ω2n′ + ξ2 + |�|2 . (3.82)

We can satisfy equations (3.80) and (3.81) with Z = 1 and χ = 0. In the lastequation, we extend the summation over frequencies to infinity but cut the integral

88 Intermediate-coupling theory

over |ξ | at ωD. Then applying equation (2.54) for tanh yields the familiar BCSequation for the order parameter,

1 = λ

∫ ωD

0

dξ√ξ2 + |�|2

tanh

√ξ2 + |�|2

2T. (3.83)

Hence the Migdal–Eliashberg theory reproduces the BCS results, if a similarapproximation for the attraction between electrons is adopted. The criticaltemperature and the BCS gap are adiabatically small (� ωD) compared withthe Fermi energy and we could worry about the adiabatically small crossingdiagrams in figure 3.6, which are neglected in the master equation, (3.74).However, the BCS state is essentially the same as the normal state outside thenarrow momentum region around the Fermi surface. The outside regions mainlycontribute to the integrals of the crossing diagrams, which makes them small asin the normal state. As a result, the vertex corrections are small and Migdal’stheorem holds in the BCS state as well.

Within a more general consideration the master equation (3.74) takesproperly into account the phonon spectrum, retardation and realistic matrixelements of the electron–phonon interaction in metals [50–52]. The importantfeature of the intermediate-coupling theory is the explicit frequency dependenceof the order parameter �(ωn), which is transformed into the energy-dependentgap in the quasi-particle DOS as

ρ(ε) = ε√ε2 − |�(ε)|2 . (3.84)

The DOS (equation (3.84)) can be measured in the tunnelling experiments(section 2.4). As the gap depends on the phonon spectrum, phonons affect thetunnelling I–V characteristics. Let us define the phonon density of states

F(ω) = 1

N

∑qν

δ(ω − ωqν) (3.85)

and the so-called Eliashberg function α2 F,

α2(ω)F(ω) = 1

N(EF)N

∑k,k′

γ 2(k − k′, ν)ω2k−k′,νδ(ω − ωk−k′,ν)δ(ξk)δ(ξk′)

(3.86)which is an average over the Fermi surface of the interaction matrix elementsquared multiplied by the phonon spectral density. The quasi-particle DOS(equation (3.84)) can be obtained in a direct fashion using the tunnellingconductivity of the superconducting tunnel junctions. Then the function α2 Fcan be calculated to fit the experimental energy dependence of the gap �(ε)

[54] and compared with the phonon density of states measured independently,

Coulomb pseudopotential 89

for example, in neutron scattering experiments. The comparison shows thatfrequency dependence of the Eliashberg function and the phonon density ofstates are similar in many low-temperature superconductors with a weak orintermediate electron–phonon coupling, for example in lead. The observationof the characteristic phonon frequencies in tunnelling (usually in the secondderivative of the current versus voltage) and the isotope effect are used to verifythe phonon-mediated pairing mechanism.

3.5 Coulomb pseudopotential

The theory of superconductivity has to take account of the Coulomb repulsivecorrelations between electrons, which might be much stronger than the attractioninduced by phonons. There is no adiabatic parameter for this interaction becausethe electron plasma frequency ωe = √

4πnee2/me has about the same order ofmagnitude as the Fermi energy in metals. Nevertheless, we can account for theCoulomb repulsion in the same fashion as for the electron–phonon interaction butreplacing γ 2(q)ωq D(q, ωn − ωn′) in equation (3.74) by the Fourier componentof the Coulomb potential Vc. The Coulomb interaction is non-retarded forfrequencies less than ωe. Then the kernel K (ωn − ωn′) in the BCS equation,

�(ωn) = T∫

dξ∑ωn′

K (ωn − ωn′ )�(ωn′ )

ω2n′ + ξ2 + |�(ωn′)|2

(3.87)

can be parametrized as

K (ωn − ωn′) = λ�(ωD − |ωn − ωn′ |) − µc�(ωe − |ωn − ωn′ |) (3.88)

where µc = Vc N(EF). At T = Tc, we neglect second and higher powers of theorder parameter and integrating over ξ obtain

�(ωn) = πTc

∑ωn′

K (ωn − ωn′ )�(ωn′)

|ωn′ | . (3.89)

Let us adopt the BCS-like approximation of the kernel:

K (ωn − ωn′ ) � λ�(2ωD − |ωn |)�(2ωD − |ωn′ |)− µc�(2ωe − |ωn |)�(2ωe − |ωn′ |)

and replace the summation in equation (3.89) by the integral:

πTc

∑≈

∫ ∞

πTc

dω (3.90)

because Tc � ωD � ωe. Then the solution is found in the form

�(ω) = �1�(2ωD − |ω|) + �2�(2ωe − |ω|)�(|ω| − 2ωD) (3.91)

90 Intermediate-coupling theory

with constant but different values of the order parameter �1 and �2 below andabove the cut-off energy 2ωD, respectively. Substituting equation (3.91) intoequation (3.89) yields the following equations for �1,2:

�1

[1 − (λ − µc) ln

2ωD

πTc

]+ �2µc ln

ωe

ωD= 0 (3.92)

�1µc ln2ωD

πTc+ �2

[1 + µc ln

ωe

ωD

]= 0. (3.93)

A non-trivial solution of these coupled equations is found, if

Tc = 2ωD

πexp

(− 1

λ − µ∗c

)(3.94)

whereµ∗

c = µc

1 + µc ln(ωe/ωD)(3.95)

is the so-called Coulomb pseudopotential [53]. This is a remarkable result.It shows that even a large Coulomb repulsion µc > λ does not destroy theCooper pairs because its contribution is suppressed down to the value aboutln−1(ωe/ωD) � 1. The retarded attraction mediated by phonons acts well aftertwo electrons meet each other. This time delay is sufficient for two electrons tobe separated by a relative distance, at which the Coulomb repulsion is small.The Coulomb correlations also lead to a damping of excitations of the orderof ξ2/µ, which is relevant only in a narrow region around the Fermi surface|ξ | � ωD

√me/M . The damping due to the Frohlich interaction dominates outside

this region.Computational analysis of the Eliashberg equations led McMillan [54] to

suggest an empirical formula for Tc, which works well for simple metals andtheir alloys,

Tc = ωD

1.45exp

(− 1.04(1 + λ)

λ − µ∗c(1 + 0.62λ)

). (3.96)

However, in materials with a moderate Tc � 20 K (like Nb3Sn, V3Si and in otherA-15 compounds), the discrepancy between the value of λ, estimated from thisequation and from the first-principle band-structure calculations, exceeds the limitallowed by the experimental and computation accuracy by several times [55].

In the original papers, Migdal [48] and Eliashberg [36] restricted theapplicability of their approach to the intermediate region of coupling λ < 1. Withthe typical values of λ = 0.5 and µ∗

c = 0.14 and with the Debye temperatureas high as ωD = 400 K, McMillan’s formula predicts Tc ≈ 2 K, clearly too lowto explain high Tc values in novel superconductors. One can formally computeTc and the gap using the Eliashberg equations (3.74) also in the strong-couplingregime λ > 1. In particular, Allen and Dynes [56] found that in the extremestrong-coupling limit (λ � 1), the critical temperature may rise as

Tc ≈ ωDλ1/2

2π. (3.97)

Cooper pairing of repulsive fermions 91

However, the Migdal–Eliashberg theory is based on the assumption that the Fermiliquid is stable and the adiabatic condition µ � ωD is satisfied. As we shall laterdiscuss in chapter 4, this assumption cannot be applied in the strong-couplingregime and the proper extension of the BCS theory to λ > 1 inevitably involvessmall polarons and bipolarons.

3.6 Cooper pairing of repulsive fermions

The phenomenon of superconductivity is due to the interaction of electrons withvibrating ions, which mediates an effective attractive potential V (k, k′) < 0 inequation (2.98). In recent years, great attention has been paid to a possibility ofsuperconductivity mediated by strong electron–electron correlations without anyinvolvement of phonons. Some time ago, Kohn and Luttinger [57] pointed outthat such possibility exists at least theoretically. They found that a dilute Fermigas cannot remain normal down to absolute zero of temperature even in the caseof purely repulsive short-range interaction between the particles. A system offermions with purely repulsive short-range forces will inevitably be superfluid atzero temperature.

To understand what is involved, one should consider the screening of acharge placed in a metal. It has long been known [58] that if fermions aredegenerate (i.e. their Fermi surface is well defined), the screening produces anoscillatory potential of the form cos(2kFr + ϕ)/r3 at the distance r from thecharge (here ϕ is a constant) which has attractive regions. Using these regionsof screened interaction, unconventional Cooper pairs can form with non-zeroorbital momentum (section 2.10). Following Kohn and Luttinger, let us consider asimplified model of spin- 1

2 fermions with a weak short-range repulsive interactionbetween them. The critical temperature of unconventional Cooper pairing withthe orbital momentum l is found as (see equation (2.104))

Tc ≈ µ exp

(− 1

λl

). (3.98)

Here we replaced the Debye temperature by the Fermi energy µ becausethe fermion–fermion interaction is non-retarded. The coupling constant λl isexpressed in terms of spherical harmonics V (l) of the Fourier transform of theinteraction potential,

λl ≡ − V (l)mkF

2π(2l + 1)(3.99)

on condition that V (l) is negative. For simplicity, we choose the repulsive barepotential as V (r − r ′) = U with a positive U, if |r − r ′| � r0, and V (r − r ′) = 0,

if |r − r ′| > r0. We also assume that the gas is diluted, that is the radius of thepotential is small compared with the characteristic wave-length of fermions,

kFr0 � 1. (3.100)

92 Intermediate-coupling theory

It is easy to estimate V (l) for such a potential,

V (l) = 2l + 1

2

∫ π

0d� sin �Pl(cos �)V (q) (3.101)

where q2 = 2kF(1 − cos �) is about 2kF or less and

V (q) = U∫

r�adr exp(iq ·r) ≈ 4π

3Ur3

0

{1 − (qr0)

2

10+ (qr0)

4

40+�[(kFr0)

6]}

.

(3.102)Substituting the Fourier transform (3.102) into equation (3.101), we obtain apositive V (l) of the order of

V (l) � Ur30 (kFr0)

2l . (3.103)

We conclude that the repulsive bare potential does not produce any pairing. Let usnow take screening into account. It is sufficient to consider it perturbatively, if thepotential is weak, U � µ. The diagrams contributing to the effective interactionup to the second order in U are shown in figure 3.8. In the analytical form, theFourier transform of the effective interaction of two electrons on the Fermi surfaceis given by

V (k, k′) = V (q) − T

(2π)3

∑ωn

∫d p[2V 2(q) − 2V (q)V (k′ − p)]�(0)

ωn( p, ωn)

× �(0)ωn

( p + q, ωn)

+ T

(2π)3

∑ωn

∫d p V (k − p)V (k′ − p)�(0)

ωn( p, ωn)

× �(0)ωn

( p − k′ − k, ωn). (3.104)

Because the density is low (equation (3.100)) we can neglect the q-dependence of V (q) and take V (q) = 4πUr3

0 /3 ≡ v in the second-order terms ofequation (3.104). Then the diagrams (b) and (c) + (d) cancel each other and thesecond term in the right-hand side of equation (3.104) vanishes. The remaininglast term yields the familiar polarization bubble (equation (3.40)) which finallyleads to

λl ≡ − V (l)mkF

2π(2l + 1)

= λl − λ(−1)l∫ π

0d� sin �Pl(cos �)

[1 + 4k2

F − q2

4kFqln

∣∣∣∣2kF + q

2kF − q

∣∣∣∣]

(3.105)

where λ = (akF/π)2 � 1. Here the first repulsive term is about(akF)(kFr0)

2l, where a = vme/(2π) is the s-wave scattering amplitude in

Cooper pairing of repulsive fermions 93

�

� � � �

�

� �

� �

�

Figure 3.8. The first (a) and second-order diagrams (b–e) which contribute to the screenedelectron–electron interaction. Note that the non-crossing second-order diagram (f) as wellas all higher-order non-crossing diagrams are fully taken into account in the BCS equation(section 3.4) and should not be included in the irreducible scattering vertex.

the Born approximation. The second ‘polarization’ contribution to effectiveinteraction is of the order of (akF)

2. Hence, if the gas parameter is sufficientlysmall,

(kFr0)2l−3 � U

µ(3.106)

the polarization contribution overcomes direct repulsion for all l � 2 andit is attractive. For example, for l = 2 (d-wave pairing), the integral inequation (3.105) yields

λ2 = λ4

105(8 − 11 ln 2) ≈ 0.015λ. (3.107)

The corresponding critical temperature is

Tc ≈ µ exp

(− 1

0.015λ

)(3.108)

which is practically zero at any λ < 1. The situation is slightly better whenthe repulsive potential U is strong (U � µ) (i.e. for hard-core spheres) [59].Partial scattering amplitudes fl in a vacuum are of the order of r0(kFr0)

2l in thislimit. They are small compared with the attractive polarization contribution to theamplitudes (�r0(kFr0)) starting from l = 1. Hence, at T = 0, hard-core fermionswith repulsive scattering in a vacuum are necessarily in a superfluid p-wave state.Calculating the integral in equation (3.105) for l = 1, we obtain

λ1 = λ2

5(2 ln 2 − 1) ≈ 0.15λ (3.109)

94 Intermediate-coupling theory

and

Tc ≈ µ exp

(− 1

0.15λ

). (3.110)

The Fermi energy could be as large as µ = 104 K but Tc still remains very lowbecause λ � 1 in the dilute limit.

Chapter 4

Strong-coupling theory

The electron–phonon coupling constant λ is about the ratio of the electron–phonon interaction energy E p to the half-bandwidth D � N(EF)−1 (appendix A).We expect [11] that when the coupling is strong (λ > 1), all electrons in theBloch band are ‘dressed’ by phonons because their kinetic energy (<D) is smallcompared with the potential energy due to a local lattice deformation, E p , causedby an electron. If phonon frequencies are very low, the local lattice deformationtraps the electron. This self-trapping phenomenon was predicted by Landau [60].It has been studied in greater detail by Pekar [61], Frohlich [62], Feynman [63],Devreese [64] and other authors in the effective mass approximation, which leadsto the so-called large polaron. The large polaron propagates through the latticelike a free electron but with the enhanced effective mass. In the strong-couplingregime (λ > 1), the finite bandwidth becomes important, so that the effectivemass approximation cannot be applied. The electron is called a small polaronin this regime. The self-trapping is never ‘complete’, that is any polaron cantunnel through the lattice. Only in the extreme adiabatic limit, when the phononfrequencies tend to zero, is the self-trapping complete and the polaron motionno longer translationally continuous (section 4.2). The main features of the smallpolaron were understood by Tjablikov [65], Yamashita and Kurosava [66], Sewell[67], Holstein [68] and his school [69, 70], Lang and Firsov [71], Eagles [72]and others and described in several review papers and textbooks [13, 64, 73–76].The exponential reduction of the bandwidth at large values of λ is one of thosefeatures (section 4.3). The small polaron bandwidth decreases with increasingtemperature up to a crossover region from the coherent small polaron tunnellingto a thermally activated hopping. The crossover from the polaron Bloch states tothe incoherent hopping takes place at temperatures T ≈ ω0/2 or higher, where ω0is the characteristic phonon frequency. In this chapter, we extend BCS theory tothe strong-coupling regime (λ > 1) with the itinerant (Bloch) states of polaronsand bipolarons.

95

96 Strong-coupling theory

4.1 Electron–phonon and Coulomb interactions in theWannier representation

For doped semiconductors and metals with a strong electron–phonon (e–ph)interaction, it is convenient to transform the Bloch states to the site (Wannier)states using the canonical linear transformation of the electron operators(appendix C):

ci = 1√N

∑k

eik·mcks (4.1)

where i ≡ (m, s) includes both site m and spin s quantum numbers. In the newrepresentation, the periodic part of the Hamiltonian (3.17) takes the followingform:

He =∑i, j

[T (m − m′)δss ′ − µδi j ]c†i c j , (4.2)

where

T (m) = 1

N

∑k

Enkeik·m

is the ‘bare’ hopping integral (appendix A). Here i = (m, s) and j = (n, s′).The electron–phonon interaction and the Coulomb correlations acquire

simple forms in the Wannier representation, if their matrix elements inthe momentum representation depend only on the momentum transfer q,equation (3.21):

He−ph =∑q,ν,i

ωqν ni [ui (q, ν)dqν + H.c.] (4.3)

He−e = 12

∑i �= j

Vc(m − n)ni n j . (4.4)

Here

ui (q, ν) = 1√2N

γ (q, ν)eiq·m (4.5)

and

Vc(m) = 1

N

∑q

Vc(q)eiq·m (4.6)

are the matrix elements of the electron–phonon and Coulomb interactions,respectively, in the Wannier representation for electrons, and ni = c†

i ci is thedensity operator. Assuming the interaction matrix elements depend only on themomentum transfer, we neglect the terms in the electron–phonon and Coulombinteractions which are proportional to the overlap integrals of the Wannier orbitalson different sites. This approximation is justified for narrow-band materials,

Electron–phonon and Coulomb interactions 97

whose bandwidth 2D is less than the characteristic value of the crystal field. As aresult, in the Wannier representation, the Hamiltonian is

H =∑i, j

[T (m − m′)δss ′ − µδi j ]c†i c j +

∑q,ν,i

ωqν ni [ui (q, ν)dqν + H.c.]

+ 12

∑i �= j

Vc(m − n)ni n j +∑

q

ωqν(d†qνdqν + 1/2). (4.7)

This Hamiltonian should be treated as a ‘bare’ one for metals, where the matrixelements and phonon frequencies are ill defined. In contrast, the bare phonons ωqν

and the electron band structure Enk are well defined in doped semiconductors,which have their ‘parent’ dielectric compounds. Here, the effect of carriers on thecrystal field and on the dynamic matrix is small while the carrier density is muchless than the atomic one. Therefore, we can use the band structure and the crystalfield of the parent insulators to calculate the parameters of the Hamiltonian (4.7).Depending on the particular phonon branch, the interaction constant γ (q, ν) hasa different q-dependence. For example, in the long-wavelength limit (q � π/a),

γ (q, ν) �1

q� constant

�1√q

for optical, molecular (ωq � constant) and acoustic (ωq � q) phonons,respectively. We can transform the e–ph interaction further using the site-representation also for phonons. Replacing the Bloch functions in the definitionof γ (q, ν) (equation (3.20)) by their Wannier series yields

γ (q, ν) = − 1

M1/2ω3/2qν

∑n

e−iq·neqν · ∇nv(n). (4.8)

This result is obtained by neglecting the overlap integrals of the Wannierorbitals on different sites and by assuming that the single-ion potential v(r)varies over the distance, which is much larger than the radius of the orbital.After substituting equation (4.8) into equation (4.5) and using the displacementoperators (equation (3.10)), one arrives at the following expression:

He−ph =∑

m,n,s

nms un · ∇nv(m − n) (4.9)

which can also be derived by replacing the field operators by the Wannier seriesin equation (3.8). The site representation of He−ph (equation (4.9)) is particularlyconvenient for the interaction with dispersionless local modes whose ωqν = ων

and eqν = eν are q independent. Introducing the phonon site operators

dnν = 1√N

∑k

eiq·ndqν (4.10)

98 Strong-coupling theory

we obtain in this case

un =∑ν

eν√2Mων

(dnν + d†nν)

Hph =∑n,ν

ων(d†nνdnν + 1/2)

andHe−ph =

∑n,m,ν

ωνgν(m − n)(eν · em−n)nms(d†nν + dnν) (4.11)

where

gν(m) = 1

ων

√2Mων

dv(m)

dm

is a dimensionless force acting between the electron on site m and thedisplacement of ion n, and em−n ≡ (m − n)/|m − n| is the unit vector in thedirection from the electron m to the ion n. The ‘real space’ representation (4.11)is convenient in modelling the electron–phonon interaction in complex lattices.Atomic orbitals of an ion adiabatically follow its motion. Therefore, the electrondoes not interact with the displacement of the ion, whose orbital it occupies, thatis gν(0) = 0.

4.2 Breakdown of Migdal–Eliashberg theory in thestrong-coupling regime

The perturbative approach to the e–ph interaction fails when λ > 1. But onemight expect that the self-consistent Migdal–Eliashberg (ME) theory is still validin the strong-coupling regime because it sums the infinite set of particular (non-crossing) diagrams in the electron self-energy (chapter 3). One of the problemswith such an extension of ME theory is lattice instability. The same theory appliedto phonons yields the renormalized phonon frequency ω = ω(1 − 2λ)1/2 [48].The frequency turns out to be zero at λ = 0.5. Because of this lattice instability,Migdal [48] and Eliashberg [36] restricted the applicability of their approach toλ < 1. However, it was then shown that there was no lattice instability but only asmall renormalization of the phonon frequencies of the order of the adiabatic ratio,ω/µ � 1, for any value of λ, if the adiabatic Born–Oppenheimer approach wasproperly applied [77]. The conclusion was that the Frohlich Hamiltonian (4.3)correctly describes the electron self-energy for any value of λ but it should notbe applied to further renormalize phonons (see also chapter 3). As a result, manyauthors used ME theory with λ much larger than 1 [50].

In fact, the Migdal–Eliashberg theory cannot be applied at λ > 1 fora reason, which has nothing to do with lattice instability. The inverse (1/λ)expansion technique [11] showed that the many-electron system collapses intoa small polaron regime at λ ≈ 1 for any adiabatic ratio. This regime is beyond

Breakdown of Migdal–Eliashberg theory 99

ME theory. It cannot be described by a summation of the standard Feynman–Dyson perturbation diagrams even including the vertex corrections (section 3.3),because of the broken translation symmetry in the strong-coupling limit. Themajor problem with the extension of ME theory to strong coupling originates fromits basic assumption that the electron and phonon GFs are translation invariants.One assumes that G(r, r ′, τ ) = G(r − r ′, τ ) prior to solving the self-energyequation. This assumption excludes the possibility of the breakdown of thelocal translation symmetry due to lattice deformation, similar to the absenceof the anomalous Bogoliubov averages in any order of the perturbation theory(section 3.4). To enable the electron relaxation into the lowest polaron states,one has to introduce an infinitesimal translation-non-invariant potential in theHamiltonian which should be set equal to zero only in the final solution for theGFs [78]. As in the case of the off-diagonal superconducting order parameter,a small translation symmetry-breaking potential drives the system into a newground state, if the e–ph coupling is sufficiently strong (λ � 1). Setting thepotential equal to zero in the solution of the equations of motion restores thetranslation symmetry but in a new polaron band (section 4.3) rather than in thebare electron band.

To illustrate the point let us compare the Migdal solution of the molecular-chain (Holstein) model of the e–ph interaction [68] with the exact solution in theadiabatic limit, ω/µ → 0. The Hamiltonian of the model is

H = − t∑〈i j 〉

c†i c j + H.c. + 2(λkt)1/2

∑i

xic†i ci

+∑

i

(− 1

2M

∂2

∂x2i

+ kx2i

2

)(4.12)

where t is the nearest-neighbour hopping integral, xi is the normal coordinate ofthe molecule (site) i and k = Mω2. The Migdal theorem is exact in this limit.Hence, in the framework of ME theory, one would expect Fermi-liquid behaviourabove Tc and the BCS ground state below Tc at any value of λ. In fact, the exactground state is a self-trapped insulator at any filling of the band if λ � 1.

First, we consider a two-site case (zero-dimensional limit), i, j = 1, 2with one electron and then generalize the result for an infinite lattice with manyelectrons. The transformation X = (x1 + x2), ξ = x1 − x2 allows us to eliminatethe coordinate X , which is coupled only with the total density (n1 + n2 = 1).That leaves the following Hamiltonian to be solved in the extreme adiabatic limitM → ∞:

H = −t (c†1c2 + c†

2c1) + (λkt)1/2ξ(c†1c1 − c†

2c2) + kξ2

4. (4.13)

The solution isψ = (αc†

1 + βc†2)|0〉 (4.14)

100 Strong-coupling theory

where

α = t

[t2 + ((λkt)1/2ξ + (t2 + λktξ2)1/2)2]1/2(4.15)

β = − (λkt)1/2ξ + (t2 + λktξ2)1/2

[t2 + ((λkt)1/2ξ + (t2 + λktξ2)1/2)2]1/2(4.16)

and the energy is

E = kξ2

4− (t2 + λktξ2)1/2. (4.17)

In the extreme adiabatic limit the displacement ξ is classical, so the ground-state energy E0 and the ground-state displacement ξ0 are obtained by minimizingequation (4.17) with respect to ξ . If λ ≥ 0.5, we obtain

E0 = −t

(λ + 1

4λ

)(4.18)

and

ξ0 =[

t (4λ2 − 1)

λk

]1/2

. (4.19)

The symmetry-breaking ‘order’ parameter is

� ≡ β2 − α2 = [2λ + (4λ2 − 1)1/2]2 − 1

[2λ + (4λ2 − 1)1/2]2 + 1. (4.20)

If λ < 0.5, the ground state is translationally invariant, � = 0 and E0 = −t,ξ = 0, β = −α. Precisely this state is the ‘Migdal’ solution of the Holsteinmodel. Indeed, the Fourier transform of the GF should be diagonal in the Migdalapproximation, G(k, k′, τ ) = G(k, τ )δk,k′ . The site operators are transformedinto momentum space as

ck = N−1/2∑

j

c j exp(ika j) (4.21)

where k = 2πn/Na, −N/2 < n ≤ N/2. Then the off-diagonal GF with k = 0and k ′ = π/a of the two-site chain (N = 2) is given by

G(k, k ′,−0) = i

2〈(c†

1 − c†2)(c1 + c2)〉 (4.22)

at τ = −0. Calculating this average, we obtain

G(k, k ′,−0) = i

2(α2 − β2) (4.23)

which should vanish in the Migdal theory. Hence, the theory provides only asymmetric (translation invariant) solution with |α| = |β|. When λ > 0.5, this

Breakdown of Migdal–Eliashberg theory 101

0.5 0.7 0.9 1.1 1.3Λ

�1.6

�1.2

�0.8

0.4

0.0

0.4

0.8

E�t

,ord

erpa

ram

eter

Figure 4.1. The ground-state energy (in units of t , full line) and the order parameter (thinfull line) of the adiabatic Holstein model. The Migdal solution is shown as the broken line.

solution is not the ground state of the system (figure 4.1). The system collapsesinto a localized adiabatic polaron trapped on the ‘right-hand’ or ‘left-hand’ sitedue to the finite lattice deformation ξ0. Alternatively, when λ < 0.5, the Migdalsolution is the only solution.

The generalization to a multi-polaron system on an infinite lattice of anydimension is straightforward in the extreme adiabatic regime. The adiabaticsolution of the infinite one-dimensional chain with one electron was obtained byRashba [79] in a continuous (i.e. effective mass) approximation and by Holstein[68] and Kabanov and Mashtakov [80] for a discrete lattice. The latter authorsalso studied the Holstein two-dimensional and three-dimensional lattices in theadiabatic limit. According to [80], the self-trapping of a single electron occurs atλ ≥ 0.875 and at λ ≥ 0.92 in two and three dimensions, respectively. The radiusof the self-trapped adiabatic polaron, rp , is readily derived from its continuouswavefunction [79]

ψ(x) � 1/ cosh(λx/a). (4.24)

It becomes smaller than the lattice constant (rp = a/λ) for λ ≥ 1. Thatis why a multi-polaron system remains in a self-trapped insulating state in thestrong-coupling adiabatic regime, no matter how many polarons it has. Theonly instability which might occur in this regime is the formation of on-site self-trapped bipolarons (section 4.6), if the on-site attractive interaction, 2λzt , is largerthan the repulsive Hubbard U [14]. On-site self-trapped bipolarons form a charge-ordered state due to a weak repulsion between them [10]. The asymptoticallyexact many-particle ground state of the half-filled Holstein model in the strong-coupling limit (λ → ∞) is

ψ =∏j∈B

c†j↑c†

j↓|0〉 (4.25)

for any value of the adiabatic ratio, ω/zt [10, 81]. Here the js are B-sites of a

102 Strong-coupling theory

bipartite lattice A + B . It is an insulating state rather than a Fermi liquid, whichis expected in the Migdal approximation at any value of λ in the adiabatic limit,ω → 0.

The non-adiabatic corrections (phonons) allow polarons and bipolarons topropagate as Bloch states in a new narrow band (sections 4.3 and 4.6). Thus,under certain conditions, the multi-polaron system is metallic with polaronicor bipolaronic carriers rather than with bare electrons. There is a qualitativedifference between the ordinary Fermi liquid and the polaronic one. In particular,the renormalized (effective) mass of electrons is independent of the ion mass Min ordinary metals (equation (3.68)) because λ does not depend on the isotopemass. In contrast, the polaron effective mass m∗ will depend on M (section 4.3).Hence, there is a large isotope effect on the carrier mass in polaronic metals [82](section 4.7.5) while there is no carrier mass isotope effect in ordinary metals.Likewise, the bipolaron superconducting state is essentially different from theBCS superconductor (section 4.5).

In the last years, quite a few numerical and analytical studies have confirmedthese conclusions (see, for example, [80–96]). In particular, Takada [86, 88]applied the gauge-invariant self-consistent method neglecting the momentumdependence of the vertex. Benedetti and Zeyher [91] applied the dynamicalmean-field theory in infinite dimensions. As in the 1/λ expansion technique,both approaches avoided the problem of broken translation symmetry by usingthe non-dispersive vertex and GFs as the starting point. As a result, they arrivedat the same conclusion about the applicability of the Migdal approach (in [91] thecritical value of λ was found to be 1.3 in the adiabatic limit).

The transition into the self-trapped state due to the broken translationalsymmetry is expected at 0.5 < λ < 1.3 (depending on the lattice dimensionality)for any electron–phonon interaction conserving the on-site electron occupationnumbers. For example, Hiramoto and Toyozawa [97] calculated the strength ofthe deformation potential, which transforms electrons into small polarons andbipolarons. They found that the transition of two electrons into a self-trappedsmall bipolaron occurs at the electron–acoustic phonon coupling λ � 0.5, that ishalf of the critical value of λ at which the transition of the electron into the smallacoustic polaron takes place in the extreme adiabatic limit, sqD � zt . The effectof the adiabatic ratio sqD/zt on the critical value of λ was found to be negligible.The radius of the acoustic polaron and bipolaron is about the lattice constant, sothat the critical value of λ does not very much depend on the number of electronsin this case either.

4.3 Polaron dynamics

4.3.1 Polaron band

The kinetic energy is smaller than the interaction energy as long as λ > 1. Hence,a self-consistent approach to the many-polaron problem is possible with the ‘1/λ’

Polaron dynamics 103

expansion technique [98], which treats the kinetic energy as a perturbation. Thetechnique is based on the fact, known for a long time, that there is an analyticalexact solution of a single polaron problem in the strong-coupling limit λ → ∞[71]. Following Lang and Firsov, we apply the canonical transformation eS todiagonalize the Hamiltonian. The diagonalization is exact if T (m) = 0 (orλ = ∞):

H = eS H e−S (4.26)

whereS = −

∑q,ν,i

ni [ui (q, ν)dqν − H.c.] (4.27)

is such that S† = −S. The electron and phonon operators are transformed as

ci = ci exp

[∑q,ν

ui (q, ν)dqν − H.c.

](4.28)

anddqν = dqν −

∑i

ni u∗i (q, ν) (4.29)

respectively (appendix E). It follows from equation (4.29) that the Lang–Firsovcanonical transformation shifts the ions to new equilibrium positions. In amore general sense, it changes the boson vacuum. As a result, the transformedHamiltonian takes the following form:

H =∑i, j

[σi j − µδi j ]c†i c j − E p

∑i

ni +∑q,ν

ωqν(d†qνdqν + 1/2) + 1

2

∑i �= j

vi j ni n j

(4.30)where

σi j = T (m − n)δss ′ exp

(∑q,ν

[u j (q, ν) − ui (q, ν)]dqν − H.c.

)(4.31)

is the renormalized hopping integral depending on the phonon operators and

vi j ≡ v(m − n)

= Vc(m − n) − 1

N

∑q,ν

|γ (q, ν)|2ωqν cos[q · (m − n)] (4.32)

is the interaction of polarons comprising their Coulomb repulsion and theinteraction via a local lattice deformation. In the extreme infinite-coupling limit(λ → ∞) we can neglect the hopping term of the transformed Hamiltonian. Therest has analytically determined eigenstates and eigenvalues. The eigenstates|N〉 = |ni , nqν〉 are sorted by the polaron nms and phonon nqν occupationnumbers. The energy levels are

E = −(µ + E p)∑

i

ni + 12

∑i �= j

vi j ni n j +∑

q

ωqν(nqν + 1/2) (4.33)

104 Strong-coupling theory

where ni = 0, 1 and nqν = 0, 1, 2, 3, . . . ,∞.The Hamiltonian (4.30), in zero order with respect to the hopping describes

localized polarons and independent phonons, which are vibrations of ions relativeto new equilibrium positions, which depend on the polaron occupation numbers.The phonon frequencies remain unchanged in this limit. The middle of theelectron band falls by the polaron level-shift E p due to a potential well created bylattice deformation,

E p = 1

2N

∑q,ν

|γ (q, ν)|2ωqν. (4.34)

Now let us discuss the 1/λ expansion. First we restrict the discussion toa single-polaron problem with no polaron–polaron interaction and µ = 0. Thefinite hopping term leads to the polaron tunnelling because of degeneracy of thezero-order Hamiltonian with respect to the site position of the polaron. To seehow the tunnelling occurs, we apply the perturbation theory using 1/λ as a smallparameter, where

λ ≡ E p

D. (4.35)

Here D = zT (a), z is the coordination lattice number and T (a) is the nearest-neighbour hopping integral. The proper Bloch set of N-degenerate zero-ordereigenstates with the lowest energy (−E p) of the unperturbed Hamiltonian is

|k, 0〉 = 1√N

∑m

c†ms exp(ik · m)|0〉 (4.36)

where |0〉 is the vacuum. By applying the textbook perturbation theory, onereadily calculates the perturbed energy levels. Up to the second order in thehopping integral, they are given by

E(k) = −E p + εk −∑

k′,nqν

|〈k, 0|∑i, j σi j c†i c j |k′, nqν〉|2∑

q,ν ωqνnqν

(4.37)

where |k′, nqν〉 are the excited states of the unperturbed Hamiltonian with oneelectron and at least one real phonon. The second term in equation (4.37), which islinear with respect to the bare hopping T (m), describes a polaron-band dispersion,

εk =∑

m

T (m)e−g2(m) exp(−ik · m) (4.38)

where

g2(m) = 1

2N

∑q,ν

|γ (q, ν)|2[1 − cos(q · m)] (4.39)

is the band-narrowing factor at zero temperature. The third term inequation (4.37), quadratic in T (m), yields a negative k-independent correction

Polaron dynamics 105

��

�

�

Figure 4.2. ‘Back and forth’ virtual transitions of the polaron without any transfer of thelattice deformation from one site to another. These transitions shift the middle of the banddown without any real charge delocalization.

to the polaron level-shift of the order of 1/λ2. The origin of this correction,which could be much larger than the first-order contribution (equation (4.38))(containing a small exponent), is understood in figure 4.2. The polaron localizedin the potential well of depth E p on site m hops onto a neighbouring site n withno deformation around and comes back. As in any second-order correction, thistransition shifts the energy down by an amount of about −T 2(m)/E p. It has littleto do with the polaron effective mass and the polaron tunnelling mobility becausethe lattice deformation around m does not follow the electron. The electron hopsback and forth many times (about eg2

) waiting for a sufficient lattice deformationto appear around the site n. Only after the deformation around n is created doesthe polaron tunnel onto the next site together with the deformation.

4.3.2 Damping of the polaron band

The polaron band is exponentially narrow, see equation (4.38). Hence, onecan raise a concern about its existence in real solids. At zero temperaturethe perturbation term of the transformed Hamiltonian conserves the momentumbecause all off-diagonal matrix elements vanish:

〈k, 0|∑i, j

σi, j c†i c j |k′, 0〉 = 0 (4.40)

if k �= k′. The emission of a single high-frequency phonon is impossible for any kbecause of the energy conservation. The polaron half-bandwidth is exponentiallyreduced,

w ≈ De−g2(4.41)

and it is usually less than the optical phonon energy ω0 (g2 is of the order ofDλ/ω0). Hence, there is no damping of the polaron band at T = 0 caused byoptical phonons, no matter how strong the interaction is. These phonons ‘dress’the electron and coherently follow its motion. However, at finite temperatures, thesimultaneous emission and absorption of phonons (figure 4.3) become possible.

106 Strong-coupling theory

� ��

� �������

∼γ�

�

�

Figure 4.3. Two-phonon scattering responsible for the damping of the polaron band.

Moreover, the polaron bandwidth shrinks with increasing temperature because thephonon-averaged hopping integrals depend on temperature (appendix E):

〈σi j 〉ph = T (m − n)δss ′ exp

(− 1

2N

∑q,ν

|γ (q, ν)|2[1 − cos(q · m)] cothωqν

2T

).

For high temperatures (T � ω0/2) the band narrows exponentially:

w ≈ De−T/T0 (4.42)

where

T −10 = 1

N

∑q,ν

|γ (q, ν)|2ω−1qν [1 − cos(q · m)].

However, the two-phonon scattering of polarons (figure 4.3) becomes moreimportant with increasing temperature.

We can estimate the scattering rate by applying the Fermi–Dirac golden rule:

1

τ= 2π

⟨∑q,q ′

|Mqq ′ |2δ(εk − εk+q−q ′)

⟩ph

(4.43)

where the corresponding matrix element is

Mqq′ =∑i, j

〈k + q − q ′, nq − 1, nq ′ + 1|σi, j c†i c j |k, nq , nq ′ 〉.

For simplicity we drop the phonon branch index ν and consider the momentumindependent γ (q) = γ0 and ωq = ω0. Expanding σi j -operators in powers of thephonon creation and annihilation operators, we estimate the matrix element of thetwo-phonon scattering as

Mqq′ ≈ 1

Nwγ 2

0

√nq(nq ′ + 1). (4.44)

Using this estimate and the polaron density of states,

Np(ξ) ≡ 1

N

∑k

δ(ξ − εk) ≈ 1

2w(4.45)

Polaron dynamics 107

we obtain1

τ≈ wγ 4

0 nω(1 + nω) (4.46)

where nω = [exp(ω0/T ) − 1]−1 is the phonon distribution function. The polaronband is well defined if

1

τ< w (4.47)

which is satisfied for a wide temperature range

T ≤ Tmin ≈ ω0

ln γ 40

(4.48)

about half of the characteristic phonon frequency for relevant values of γ 20 .

Phonons dominate in the scattering at finite temperatures if the number ofimpurities is sufficiently low. Therefore, the polaron mobility decreases when thetemperature increases from zero to Tmin due to an increasing number of phonons.At higher temperatures, the incoherent thermal activated hopping dominates inthe polaron dynamics [66–68, 71] and the polaron states are no longer the Blochstates. Hence, the mobility increases above Tmin, where it is at minimum, dueto thermal activated hopping. There is a temperature range around Tmin wherethe thermal activated hopping still makes a small contribution to the conductivitybut the uncertainty in the polaron band is already significant [71]. The polarontransport theory requires a special diagrammatic technique in this region [74,75].The optical phonon frequencies are exceptionally high, about 1000 K or evenhigher and polarons are in Bloch states in the whole relevant range of temperaturesin novel superconductors.

4.3.3 Small Holstein polaron and small Frohlich polaron

The narrowing of the band and the polaron effective mass strongly depend onthe radius of the electron–phonon interaction [100]. Let us compare the smallHolstein polaron (SHP) formed by the short-range e–ph interaction and a smallpolaron formed by the long-range (Frohlich) interaction, which we refer as thesmall Frohlich polaron (SFP). Introducing a normal coordinate at site n as

ξn =∑

q

(2N Mωq )−1/2eiq·ndq + H.c. (4.49)

and a ‘force’ between the electron at site m and the normal coordinate ξn,

f (m) = N−1∑

q

γ (q)(Mω3q )1/2eiq·m (4.50)

we rewrite the e–ph interaction, equation (4.3), as

He−ph =∑n,i

f (m − n)ξnni . (4.51)

108 Strong-coupling theory

For simplicity we consider the interaction with a single phonon branch andγ (−q) = γ (q). In general, there is no simple relation between the polaron levelshift E p and the exponent g2 of the mass enhancement. This relation depends onthe form of the electron–phonon interaction. Indeed, for dispersionless phonons,ωq = ω0, using equations (4.34) and (4.39) we obtain

E p = 1

2Mω20

∑m

f 2(m) (4.52)

and

g2 = 1

2Mω30

∑m

[ f 2(m) − f (m) f (m + a)] (4.53)

where a is the primitive lattice vector. In the nearest-neighbour approximation,the effective mass renormalization is given by

m∗/m = eg2

where m is the bare band mass and 1/m∗ = ∂2εk/∂k2 at k → 0 is the inversepolaron mass.

If the interaction is short range, f (m) = κδm,0 (the Holstein model), theng2 = E p/ω. Here κ is a constant. In general, we have g2 = γ E p/ω with thenumerical coefficient

γ = 1 − ∑m f (m) f (m + a)∑

n f 2(n)(4.54)

which might be less than 1. To estimate γ , let us consider a one-dimensional chainmodel with a long-range Coulomb interaction between the electron on the chain(×) and ion vibrations of the chain (◦), polarized in a direction perpendicular tothe chains [94] (figure 4.4). The corresponding force is given by

f (m − n) = κ

(|m − n|2 + 1)3/2 . (4.55)

Here the distance along the chains, |m − n|, is measured in units of the latticeconstant, a, the inter-chain distance is also a and we take a = 1. For this long-range interaction, we obtain E p = 1.27κ2/(2Mω2), g2 = 0.49κ2/(2Mω3) and

g2 = 0.39E p/ω. (4.56)

Thus the effective mass renormalization is much smaller than in the Holsteinmodel, roughly m∗

SFP ∝ (m∗SHP)

1/2.Not only does the small polaron mass strongly depend on the radius of the

electron–phonon interaction but the range of the applicability of the analytical1/λ expansion theory also does. The theory appears almost exact in a wideregion of parameters for the Frohlich interaction. The exact polaron mass in a

Polaron dynamics 109

���

�

�

Figure 4.4. A one-dimensional model of a small polaron on the chain interacting with theion displacements of another chain.

wide region of the adiabatic parameter ω/T (a) and coupling was calculated withthe continuous-time path-integral quantum Monte Carlo (QMC) algorithm [94].This method is free from any systematic finite-size, finite-time-step and finite-temperature errors and allows for an exact (in the QMC sense) calculation of theground-state energy and the effective mass of the lattice polaron for any electron–phonon interaction described by the Hamiltonian (4.51).

At large λ (>1.5), the SFP was found to be much lighter than the SHP, whilethe large Frohlich polaron (i.e. at λ < 1) was heavier than the large Holsteinpolaron with the same binding energy (figure 4.5). The mass ratio m∗

FP/m∗HP is

a non-monotonic function of λ. The effective mass of Frohlich polarons, m∗FP(λ)

is well fitted by a single exponent, which is e0.73λ for ω0 = T (a) and e1.4λ forω = 0.5T (a). The exponents are remarkably close to those obtained with theLang–Firsov transformation, e0.78λ and e1.56λ, respectively. Hence, in the caseof the Frohlich interaction the transformation is perfectly accurate even in themoderate adiabatic regime, ω/T (a) ≤ 1 for any coupling strength. This is not thecase for the Holstein polaron. If the interaction is short range, the same analyticaltechnique is applied only in the non-adiabatic regime ω/T (a) > 1.

Another interesting point is that the size of the SFP and the length over whichthe distortion spreads are different. In the strong-coupling limit, the polaron isalmost localized on one site m. Hence, the size of its wavefunction is the atomicsize. In contrast, the ion displacements, proportional to the displacement forcef (m − n), spread over a large distance. Their amplitude at a site n falls withdistance as |m − n|−3 in our one-dimensional model. The polaron cloud (i.e.lattice distortion) is more extended than the polaron itself [66, 72, 100]. Such apolaron tunnels with a larger probability than the Holstein polaron due to a smallerrelative lattice distortion around two neighbouring sites. For a short-range e–phinteraction, the entire lattice deformation disappears at one site and then forms atits neighbour, when the polaron tunnels from site to site. Therefore, γ = 1 andthe polaron is very heavy already at λ ≈ 1. In contrast, if the interaction is long-ranged, only a fraction of the total deformation changes every time the polarontunnels from one site to its neighbour and γ is smaller than 1. The fact that theLang–Firsov transformation is rather accurate for the long-range interaction ina wide region of parameters, allows us to generalize this result. Including all

110 Strong-coupling theory

Figure 4.5. Inverse effective polaron mass in units of 1/m = 2T (a)a2.

phonon branches in the three-dimensional lattice, we obtain m∗SFP = (m∗

SHP)γ ,

where the masses are measured in units of the band mass, and

γ =∑

q,ν γ 2(q, ν)[1 − cos(q · m)]∑q,ν γ 2(q, ν)

. (4.57)

For the Frohlich interaction (i.e. γ (q) � 1/q), we find γ = 0.57 in a cubiclattice and γ = 0.255 for the in-plane oxygen hole in cuprates (Part II). A lightermass of SFP compared with the non-dispersive SHP is a generic feature of anydispersive electron–phonon interaction. As an example, a short-range interactionwith dispersive acoustic phonons (γ (q) ∼ 1/q1/2, ωq ∼ q) also leads to a lighterpolaron in the strong-coupling regime compared with the SHP. Actually, Holstein[68] pointed out in his original paper that the dispersion is a vital ingredient of thepolaron theory.

4.3.4 Polaron spectral and Green’s functions

The multi-polaron problem has an exact solution in the extreme infinite-couplinglimit (λ = ∞) for any type of e–ph interaction conserving the on-site occupationnumbers of electrons (equation (4.3)). For finite coupling, the 1/λ perturbationexpansion is applied. The expansion parameter is actually [72, 74, 98, 99]

1

2zλ2 � 1

so that the analytical perturbation theory has a wider region of applicabilitythan one can expect using a semiclassical estimate E p > D. However,

Polaron dynamics 111

the expansion convergency is different for different e–ph interactions. Exactnumerical diagonalizations of vibrating clusters, variational calculations [83, 85,87, 89, 92, 93], dynamical mean-field approach in infinite dimensions [91], andquantum Monte Carlo simulations [94] revealed that the ground state energy(≈−E p) is not very sensitive to the parameters. In contrast, the effective mass, thebandwidth and the polaron density of states strongly depend on the adiabatic ratioω/T (a) and on the radius of the interaction. The first order in 1/λ perturbationtheory is practically exact in the non-adiabatic regime (ω/T (a) > 1) for anyvalue of the coupling constant and any type of e–ph interaction. However, itoverestimates the polaron mass by a few orders of magnitude in the adiabaticcase (ω/T (a) < 1), if the interaction is short-ranged [83]. A much lowereffective mass of the adiabatic Holstein polaron compared with that estimatedusing first-order perturbation theory is the result of the poor convergency of theperturbation expansion owing to the double-well potential [68] in the adiabaticlimit. The tunnelling probability is extremely sensitive to the shape of thispotential. However, the analytical theory is practically exact in a wider range ofthe adiabatic parameter and of the coupling constant for the long-range Frohlichinteraction.

Keeping this in mind, let us calculate the one-particle GF in the firstorder in 1/λ. Applying the canonical transformation we write the transformedHamiltonian as

H = Hp + Hph + Hint (4.58)

whereHp =

∑k

ξ(k)c†kck (4.59)

is the ‘free’ polaron contribution and

Hph =∑

q

ωq(d†qdq + 1/2) (4.60)

is the free phonon part. For simplicity, we drop the spin and phonon branchindexes. Here ξ(k) = Z ′E(k) − µ is the renormalized polaron band dispersion.The chemical potential µ includes the polaron level shift −E p . It might alsoinclude all higher orders in 1/λ corrections to the spectrum independent of k.E(k) = ∑

m T (m) exp(−ik · m) is the bare dispersion in the rigid lattice and

Z ′ =∑

m T (m)e−g2(m) exp(−ik · m)∑m T (m) exp(−ik · m)

(4.61)

is the band-narrowing factor (Z ′ = exp(−γ E p/ω)) as discussed earlier). Theinteraction term Hint comprises the polaron–polaron interaction, equation (4.32),and the residual polaron–phonon interaction

Hp−ph ≡∑i �= j

[σi j − 〈σi j 〉ph]c†i c j . (4.62)

112 Strong-coupling theory

We can neglect Hp−ph in the first order in 1/λ � 1. To better understand thespectral properties of a single polaron, let us also neglect the polaron–polaroninteraction. If Hint = 0, the energy levels are

E{m} =∑

k

ξknk +∑

q

ωq [nq + 1/2] (4.63)

and the transformed eigenstates |m〉 are sorted by the polaron Bloch-stateoccupation numbers, nk = 0, 1, and the phonon occupation numbers, nq =0, 1, 2, . . . ,∞. The spectral function (equation (D.10)) is defined by the matrixelement 〈n|ck|m〉. It can be written as

〈n|ck|m〉 = 1√N

∑m

e−ik·m〈n|ci Xi |m〉 (4.64)

by the use of the Wannier representation and the Lang–Firsov transformation.Here

Xi = exp

[∑q

ui (q)dq − H.c.

].

Now, applying the Fourier transform of the δ-function in equation (D.10),

δ(ωnm + ω) = 1

2π

∫ ∞

−∞dt ei(ωnm+ω)t

the spectral function is expressed as

A(k, ω) = 12

∫ ∞

−∞dt eiωt 1

N

∑m,n

eik·(n−m)

× {〈〈ci (t)Xi (t)c†j X†

j 〉〉 + 〈〈c†j X†

j ci (t)Xi (t)〉〉}. (4.65)

Here the quantum and statistical averages are performed for independent polaronsand phonons, therefore

〈〈ci (t)Xi (t)X†j c

†i 〉〉 = 〈〈ci (t)c

†j 〉〉〈〈Xi (t)X†

j 〉〉. (4.66)

The Heisenberg free-polaron operator evolves with time as

ck(t) = cke−iξkt (4.67)

and

〈〈ci (t)c†i 〉〉 = 1

N

∑k′,k′′

ei(k′·m−k′′·n)〈〈ck′(t)c†k′′ 〉〉

= 1

N

∑k′

[1 − n(k′)]eik′·(m−n)−iξk′ t (4.68)

〈〈c†i ci (t)〉〉 = 1

N

∑k′

n(k′)eik′·(m−n)−iξk′ t (4.69)

Polaron dynamics 113

where n(k) = [1 + exp ξk/T ]−1 is the Fermi–Dirac distribution function ofpolarons. The Heisenberg free-phonon operator evolves in a similar way,

dq(t) = dqe−iωq t

and

〈〈Xi (t)X†j 〉〉 =

∏q

〈〈exp[ui (q, t)dq − H.c.] exp[−u j (q)dq − H.c.]〉〉 (4.70)

where ui, j (q, t) = ui, j (q)e−iωq t . This average is calculated using the operatoridentity (appendix E)

e A+B = e AeBe−[ A,B]/2 (4.71)

which is applied for any two operators A and B, whose commutator [ A, B] is anumber. Because [dq, d†

q ] = 1, we can apply this identity in equation (4.70) toobtain

e[ui (q,t)dq−H.c.]e[−u j (q)dq−H.c.] = e(α∗d†q−αdq )

× e[ui (q,t)u∗j (q)−u∗

i (q,t)u j (q)]/2

where α ≡ u j (q, t) − ui (q). Applying once again the same identity yields

e[ui (q,t)dq−H.c.]e[−u j (q)dq−H.c.] = eα∗d†q e−αdq e−|α|2/2

× e[ui (q,t)u∗j (q)−u∗

i (q,t)u j (q)]/2. (4.72)

Now the quantum and statistical averages are calculated by the use of(appendix E)

〈〈eα∗d†q e−αdq 〉〉 = e−|α|2nω (4.73)

where nω = [exp(ωq/T ) − 1]−1 is the Bose–Einstein distribution function ofphonons. Collecting all multiplies in equation (4.72), we arrive at

〈〈Xi (t)X†j 〉〉 = exp

{− 1

2N

∑q

|γ (q)|2 fq(m − n, t)

}(4.74)

where

fq(m, t) = [1 − cos(q · m) cos(ωq t)] cothωq

2T+ i cos(q · m) sin(ωq t). (4.75)

Here we have used the symmetry of γ (−q) = γ (q), and, hence, the termscontaining sin(q · m) have disappeared. The average 〈〈X†

j X i (t)〉〉, which is amultiplier in the second term in the brackets of equation (4.65), is obtained byreplacing ui (q, t)� u j (q) in the previous expressions. The result is

〈〈X†j X i (t)〉〉 = 〈〈Xi (t)X†

j 〉〉∗. (4.76)

114 Strong-coupling theory

To proceed with the analytical results, we consider low temperatures, T � ωq ,

when coth(ωq/2T ) ≈ 1. Then expanding the exponent in equation (4.74) yields

〈〈Xi (t)X†j 〉〉 = Z

∞∑l=0

{∑q |γ (q)|2ei[q·(m−n)−ωq t ]}l

(2N)ll! (4.77)

where

Z = exp

[− 1

2N

∑q

|γ (q)|2]. (4.78)

Substituting equations (4.77) and (4.68) into equation (4.66) and performingsummation with respect to m, n, k′ and integration with respect to time inequation (4.65), we arrive at [101]

A(k, ω) =∞∑

l=0

[A(−)l (k, ω) + A(+)

l (k, ω)] (4.79)

where

A(−)l (k, ω) = π Z

∑q1,...,ql

∏lr=1 |γ (qr )|2(2N)ll!

×[

1 − n

(k −

l∑r=1

qr

)]δ

(ω−

l∑r=1

ωqr−ξ k−∑l

r=1 qr

)(4.80)

and

A(+)l (k, ω) = π Z

∑q1,...,ql

∏lr=1 |γ (qr )|2(2N)ll!

× n

(k +

l∑r=1

qr

)δ

(ω+

l∑r=1

ωqr−ξ k+∑l

r=1 qr

). (4.81)

Obviously, equation (4.79) is in the form of a perturbative multi-phononexpansion. Each contribution A(±)

l (k, ω) to the spectral function describes the

transition from the initial state k of the polaron band to the final state k±∑lr=1 qr

with the emission (or absorption) of l phonons. The 1/λ expansion result(equation (4.79)) is applied to low-energy polaron excitations in the strong-coupling limit. In the case of the long-range Frohlich interaction with high-frequency phonons, it is also applied in the weak-coupling and intermediateregimes (section 4.3.3). Differing from the canonical Migdal GF (chapter 3),there is no damping of polaronic excitations in equation (4.79). Instead the e–phcoupling leads to the coherent dressing of electrons by phonons because of the

Polaron dynamics 115

energy conservation (section 4.3.2). The dressing can be seen as the phonon ‘side-bands’ with l ≥ 1. While the major sum rule (equation (D.11)) is satisfied,

1

π

∫ ∞

−∞dω A(k, ω) = Z

∞∑l=0

∑q1,...,ql

∏lr=1 |γ (qr )|2(2N)ll!

= Z∞∑

l=0

1

l!{

1

2N

∑q

|γ (q)|2}l

= Z exp

[1

2N

∑q

|γ (q)|2]

= 1 (4.82)

the higher-momentum integrals,∫ ∞−∞ dω ωp A(k, ω) with p > 0, calculated using

equation (4.79), differ from the exact value by an amount proportional to 1/λ.The difference is due to a partial ‘undressing’ of high-energy excitations in theside-bands, which is beyond the first-order 1/λ expansion.

The spectral function of the polaronic carriers comprises two different parts.The first (l = 0) k-dependent coherent term arises from the polaron bandtunnelling,

Acoh(k, ω) = [A(−)0 (k, ω) + A(+)

0 (k, ω)] = π Zδ(ω − ξ k). (4.83)

The spectral weight of the coherent part is suppressed as Z � 1. However,in the case of the Frohlich interaction, the effective mass is less enhanced,ξk = Z ′Ek − µ, because Z � Z ′ < 1 (section 4.3.3). The second incoherentpart Aincoh(k, ω) comprises all the terms with l ≥ 1. It describes the excitationsaccompanied by emission and absorption of phonons. We note that its spectraldensity spreads over a wide energy range of about twice the polaron level shiftE p , which might be larger than the unrenormalized bandwidth 2D in the rigidlattice without phonons. In contrast, the coherent part shows a dispersion onlyin the energy window of the order of the polaron bandwidth, 2w = 2Z ′D. Itis interesting that there is some k dependence of the incoherent background aswell, if the matrix element of the e–ph interaction and/or phonon frequenciesdepend on q . Only in the Holstein model with the short-range dispersionlesse–ph interaction (γ (q) = γ0 and ωq = ω0) is the incoherent part momentumindependent. Replacing k ± ∑l

r=1 qr by k′ in equations (4.80) and (4.81), weobtain the following expression in this case:

Aincoh(k, ω) = πZ

N

∞∑l=1

γ 2l0

2ll!×

∑k′

{[1 − n(k′)]δ(ω − lω0 − ξk′) + n(k′)δ(ω + lω0 − ξk′)}

(4.84)

116 Strong-coupling theory

which has no k-dependence.As soon as we know the spectral function, polaron GFs are easily obtained

using their analytical properties (appendix D). For example, the temperaturepolaron GF is given by the integral (D.26). Calculating the integral with thespectral density equation (4.79), we find, in the Holstein model [102], that

�(k, ωn) = Z

iωn−ξ k+ Z

N

∞∑l=1

γ 2l0

2ll!∑

k′

{1 − n(k′)

iωn − lω0 − ξk′+ n(k′)

iωn + lω0 − ξk′

}.

(4.85)Here the first term describes the coherent tunnelling in the narrow polaron bandwhile the second k-independent sum is due to the phonon cloud ‘dressing’ theelectron.

4.4 Polaron-polaron interaction and bipolaron

Polarons interact with each other (equation (4.32)). The range of the deformationsurrounding the Frohlich polarons is quite large and their deformation fieldsoverlap at finite density. Taking into account both the long-range attraction ofpolarons owing to the lattice deformations and their direct Coulomb repulsion,the residual long-range interaction turns out to be rather weak and repulsive inionic crystals [13]. The Fourier component of the polaron–polaron interaction,v(q), comprising the direct Coulomb repulsion and the attraction mediated byphonons, is

v(q) = 4πe2

ε∞q2− |γ (q)|2ωq . (4.86)

In the long-wave limit (q � π/a), the Frohlich interaction dominates in theattractive part, which is described by [62]

|γ (q)|2ω0 = 4πe2(ε−1∞ − ε−10 )

q2. (4.87)

Here ε∞ and ε0 are the high-frequency and static dielectric constants, respectively,of the host ionic insulator which are usually well known from experiment. Fouriertransforming equation (4.86) yields the repulsive interaction in real space,

v(m − n) = e2

ε0|m − n| > 0. (4.88)

We see that optical phonons nearly nullify the bare Coulomb repulsion in ionicsolids, where ε0 � 1, but cannot overscreen it at large distances.

Considering the polaron–phonon interaction in the multi-polaron system, wehave to take into account the dynamic properties of the polaron response function.One can erroneously believe that the long-range Frohlich interaction becomesa short-range (Holstein) one due to the screening of ions by heavy polaronic

Polaron-polaron interaction and bipolaron 117

carriers. In fact, small polarons cannot screen high-frequency optical vibrationsbecause their renormalized plasma frequency is comparable with or even less thanthe phonon frequency. In the absence of bipolarons (see later), we can apply theordinary bubble approximation (chapter 3) to calculate the dielectric responsefunction of polarons at the frequency �:

ε(q,�) = 1 − 2v(q)∑

k

n(k + q) − n(k)

� − εk + εk+q. (4.89)

This expression describes the response of small polarons to any external field ofthe frequency � � ω0, when phonons in the polaron cloud follow the polaronmotion. In the static limit, we obtain the usual Debye screening at large distances(q → 0). For a temperature larger than the polaron half-bandwidth (T > w), wecan approximate the polaron distribution function as

n(k) ≈ n

2a3

(1 − (2 − n)εk

2T

)(4.90)

and obtain

ε(q, 0) = 1 + q2s

q2(4.91)

where

qs =[

2πe2n(2 − n)

ε0T a3

]1/2

and n is the number of polarons per a unit cell. For a finite but rather lowfrequency (ω0 � � � w), the polaron response becomes dynamic:

ε(q,�) = 1 − ω2p(q)

�2(4.92)

whereω2

p(q) = 2v(q)∑

k

n(k)(εk+q − εk) (4.93)

is the temperature-dependent polaron plasma frequency squared, which is about

ω2p(q) � ω2

emeε∞m∗ε0

� ω2e .

The polaron plasma frequency is very low due to the large static dielectric constant(ε0 � 1) and the enhanced polaron mass m∗.

Now replacing the bare electron–phonon interaction vertex γ (q) by ascreened one, γsc(q, ω0), as shown in figure 3.4, we obtain

γsc(q, ω0) = γ (q)

ε(q, ω0)≈ γ (q) (4.94)

118 Strong-coupling theory

because ω0 > ωp . Therefore, the singular behaviour of γ (q) ∼ 1/q isunaffected by screening. Polarons are too slow to screen high-frequency crystalfield oscillations. As a result, the strong interaction with high-frequency opticalphonons in ionic solids remains unscreened at any density of small polarons.

Another important point is the possibility of the Wigner crystallization ofthe polaronic liquid. Because the net long-range repulsion is relatively weak, therelevant dimensionless parameter rs = m∗e2/ε0(4πn/3)1/3 is not very large indoped cuprates. The Wigner crystallization appears around rs � 100 or larger,which corresponds to the atomic density of polarons n ≤ 10−6 with ε0 = 30 andm∗ = 5me. This estimate tells us that polaronic carriers are usually in the liquidstate.

At large distance, polarons repel each other (equation (4.88)). Neverthelesstwo large polarons can be bound into a large bipolaron by an exchange interactioneven with no additional e–ph interaction other than the Frohlich one [64, 103].When a short-range deformation potential and molecular-type (i.e. Jahn–Teller[104]) e–ph interactions are taken into account together with the Frohlichinteraction, they overcome the Coulomb repulsion at a short distance of aboutthe lattice constant. Then, owing to a narrow band, two heavy polarons easilyform a bound state, i.e. a small bipolaron. Let us estimate the coupling constantλ and the adiabatic ratio ω0/T (a), at which the small ‘bipolaronic’ instabilityoccurs. The characteristic attractive potential is V = D/(λ−µc), where µc is thedimensionless Coulomb repulsion (section 3.5), and λ includes the interactionwith all phonon branches. The radius of the potential is about a. In threedimensions, a bound state of two attractive particles appears, if

V ≥ π2

8m∗a2. (4.95)

Substituting the polaron mass, m∗ = [2a2T (a)]−1 exp(γ λD/ω0), we find

T (a)

ω0≤ (γ zλ)−1 ln

[π2

4z(λ − µc)

]. (4.96)

The corresponding ‘phase’ diagram is shown in figure 4.6, where t ≡ T (a) andω ≡ ω0. Small bipolarons form at λ ≥ µc + π2/4z almost independently of theadiabatic ratio. In the Frohlich interaction, there is no sharp transition betweensmall and large polarons, as one can see in figure 4.5 and the first-order 1/λ

expansion is accurate in the whole region of coupling, if the adiabatic parameteris not very small (down to ω/T (a) ≈ 0.5). Hence, we can say that the carriersare small polarons independent of the value of λ in this case. It means that theytunnel together with the entire phonon cloud no matter how ‘thin’ the cloud is.

Polaronic superconductivity 119

Figure 4.6. The ‘t/ω–λ’ diagram with a small bipolaron domain and a region of unboundsmall polarons for z = 6, γ = 0.4 and Coulomb potential µc = 0.5.

4.5 Polaronic superconductivity

The polaron–polaron interaction is the sum of two large contributions of theopposite sign (equation (4.32)). It is generally larger than the polaron bandwidthand the polaron Fermi energy, εF = Z ′EF (equation (4.61)). This condition isopposite to the weak-coupling BCS regime, where the Fermi energy is the largest.However, there is still a narrow window of parameters, where bipolarons are‘extended’ enough and pairs of two small polarons overlap similarly to Cooperpairs. Here the BCS approach is applied to non-adiabatic carriers with a non-retarded attraction, so that bipolarons are the Cooper pairs formed by two smallpolarons [11]. The size of the bipolaron is estimated as

rb ≈ 1

(m∗�)1/2(4.97)

where � is the binding energy of the order of an attraction potential |V |. The BCSapproach is applied if rb � n−1/3, which puts a severe constraint on the value ofthe attraction

|V | � εF. (4.98)

There is no ‘Tolmachev’ logarithm (section 3.5) in the case of non-adiabaticcarriers, because the attraction is non-retarded if εF � ω0. Hence, asuperconducting state of small polarons is possible only if λ > µc. Thisconsideration leaves a rather narrow crossover region from the normal polaronFermi liquid to a superconductor, where one can still apply the BCS mean-fieldapproach,

0 < λ − µc � Z ′ < 1. (4.99)

120 Strong-coupling theory

In the case of the Frohlich interaction, Z ′ is about 0.1–0.3 for typical values of λ

(figure 4.5). Hence, the region, equation (4.99), is on the border-line in figure 4.6.In the crossover region polarons behave like fermions in a narrow band with

a weak non-retarded attraction. As long as λ � 1/√

2z, we can drop their residualinteraction with phonons in the transformed Hamiltonian,

H ≈∑i, j

[(〈σi j 〉ph − µδi j )c†i c j + 1

2vi j c†i c†

j c j ci ] (4.100)

written in the Wannier representation. If the condition (4.99) is satisfied, we cantreat the polaron–polaron interaction approximately by the use of BCS theory(chapter 2). For simplicity, let us keep only the on-site v0 and the nearest-neighbour v1 interactions. At least one of them should be attractive to ensurethat the ground state is superconducting. Introducing two order parameters

�0 = − v0〈cm,↑cm,↓〉 (4.101)

�1 = − v1〈cm,↑cm+a,↓〉 (4.102)

and transforming to the k-space results in the usual BCS Hamiltonian,

Hp =∑k,s

ξkc†kscks +

∑k

[�kc†k↑c†

−k↓ + H.c.] (4.103)

where ξk = εk − µ is the renormalized kinetic energy and

�k = �0 − �1ξk + µ

w(4.104)

is the order parameter.Applying the Bogoliubov diagonalization procedure (chapter 2), one obtains

the following expressions:

〈ck,↑c−k,↓〉 = �k

2√

ξ2k + �2

k

tanh

√ξ2

k + �2k

2T(4.105)

and

�0 = − v0

N

∑k

�k

2√

ξ2k + �2

k

tanh

√ξ2

k + �2k

2T(4.106)

�1 = − v1

Nw

∑k

�k(ξk + µ)

2√

ξ2k + �2

k

tanh

√ξ2

k + �2k

2T. (4.107)

Polaronic superconductivity 121

These equations are equivalent to a single BCS equation for �k = �(ξk) butwith the half polaron bandwidth w cutting the integral, rather than the Debyetemperature:

�(ξ) =∫ w−µ

−w−µ

dη Np(η)V (ξ, η)�(η)

2√

η2 + �2(η)tanh

√η2 + �2(η)

2T. (4.108)

Here V (ξ, η) = −v0 − zv1(ξ + µ)(η + µ)/w2.The critical temperature Tc of the polaronic superconductor is determined by

two linearized equations (4.106) and (4.107) in the limit �0,1 → 0:

[1 + A

(v0

zv1+ µ2

w2

)]� − Bµ

w�1 = 0 (4.109)

− Aµ

w� + (1 + B)�1 = 0 (4.110)

where � = �0 − �1µ/w and

A = zv1

2w

∫ w−µ

−w−µ

dηtanh(η/2Tc)

η

B = zv1

2w

∫ w−µ

−w−µ

dηη tanh(η/2Tc)

w2.

These equations are applied only if the polaron–polaron coupling is small(|v0,1| < w). A non-trivial solution is found at

Tc ≈ 1.14w

√1 − µ2

w2 exp

(2w

v0 + zv1µ2/w2

)(4.111)

if v0 + zv1µ2/w

2< 0, so that superconductivity exists even in the case of on-

site repulsion (v0 > 0), if this repulsion is less than the total inter-site attraction,z|v1|. There is a non-trivial dependence of Tc on doping. With a constant densityof states in the polaron band, the Fermi energy εF ≈ µ is expressed via the numberof polarons per atom n as

µ = w(n − 1) (4.112)

so that

Tc � 1.14w√

n(2 − n) exp

(2w

v0 + zv1[n − 1]2

). (4.113)

Tc have two maxima as a function of n separated by a deep minimum in a half-filled band (n = 1), where the nearest-neighbour contributions to pairing canceleach other.

122 Strong-coupling theory

4.6 Mobile small bipolarons

The attractive energy of two small polarons is generally larger than the polaronbandwidth, λ − µc � Z ′. When this condition is fulfilled, small bipolarons arenot overlapped. Consideration of particular lattice structures shows that smallbipolarons are mobile even when the electron–phonon coupling is strong andthe bipolaron binding energy is large [100]. Hence the polaronic Fermi liquidtransforms into a Bose liquid of double-charged carriers in the strong-couplingregime, rather than into the BCS-like ground state of the previous section. TheBose liquid is stable because bipolarons repel each other (see later). Here weencounter a novel electronic state of matter, a charged Bose liquid, qualitativelydifferent from the normal Fermi liquid and from the BCS superfluid.

4.6.1 On-site bipolarons and bipolaronic Hamiltonian

The small parameter Z ′/(λ − µc) � 1 allows for a consistent treatmentof bipolaronic systems [10, 11]. Under this condition the hopping term inthe transformed Hamiltonian H is a small perturbation of the ground state ofimmobile bipolarons and free phonons:

H = H0 + Hpert (4.114)

whereH0 = 1

2

∑i, j

vi j c†i c†

j c j ci +∑q,ν

ωqν[d†qνdqν + 1

2 ] (4.115)

andHpert =

∑i, j

σi j c†i c j . (4.116)

Let us first discuss the dynamics of on-site bipolarons, which are the groundstate of the system with the Holstein non-dispersive e–ph interaction. The on-sitebipolaron is formed if

2E p > U (4.117)

where U is the on-site Coulomb correlation energy (the so-called Hubbard U ).The inter-site polaron–polaron interaction (4.32) is purely the Coulomb repulsionbecause the phonon-mediated attraction between two polarons on different sitesis zero in the Holstein model. Two or more on-site bipolarons as well as threeor more polarons cannot occupy the same site because of the Pauli exclusionprinciple. Hence, bipolarons repel single polarons and each other. Their bindingenergy, � = 2E p − U, is larger than the polaron half-bandwidth, � � w, sothat there are no unbound polarons in the ground state. Hpert (equation (4.116))destroys bipolarons in the first order. Hence, it has no diagonal matrix elements.Then the bipolaron dynamics, including superconductivity, is described by the use

Mobile small bipolarons 123

of a new canonical transformation, exp(S2) [10], which eliminates the first orderof Hpert:

(S2)fp =∑i, j

〈 f |σi j c†i c j |p〉

Ef − Ep. (4.118)

Here Ef,p and | f 〉, |p〉 are the energy levels and the eigenstates of H0. Neglectingthe terms of the orders higher than (w/�)2, we obtain

(Hb)ff′ ≡ (eS2 He−S2)ff′ (4.119)

(Hb)ff′ ≈ (H0)ff′ − 12

∑ν

∑i �=i ′ , j �= j ′

〈 f |σii ′ c†i ci ′ |p〉〈p|σ j j ′c†

j c j ′ | f ′〉

×(

1

Ep − Ef′+ 1

Ep − Ef

).

S2 couples a localized on-site bipolaron and a state of two unbound polaronson different sites. The expression (4.119) determines the matrix elements ofthe transformed bipolaronic Hamiltonian Hb in the subspace | f 〉, | f ′〉 with nosingle (unbound) polarons. However, the intermediate bra 〈p| and ket |p〉 inequation (4.119) refer to configurations involving two unpaired polarons and anynumber of phonons. Hence, we have

Ep − Ef = � +∑q,ν

ωqν(npqν − nf

qν) (4.120)

where nf,pqν are phonon occupation numbers (0, 1, 2, 3, . . . ,∞). This equation is

an explicit definition of the bipolaron binding energy � which takes into accountthe residual inter-site repulsion between bipolarons and between two unpairedpolarons. The lowest eigenstates of Hb are in the subspace, which has only doublyoccupied c†

msc†ms ′ |0〉 or empty |0〉 sites. On-site bipolaron tunnelling is a two-

step transition. It takes place via a single polaron tunnelling to a neighbouringsite. The subsequent tunnelling of its ‘partner’ to the same site restores theinitial energy state of the system. There are no real phonons emitted or absorbedbecause the bipolaron band is narrow (see later). Hence, we can average Hb withrespect to phonons. Replacing the energy denominators in the second term inequation (4.119) by the integrals with respect to time,

1

Ep − Ef= i

∫ ∞

0dt ei(Ef−Ep+iδ)t

we obtain

Hb = H0 − i∑

m �=m′,s

∑n �=n′,s ′

T (m − m′)T (n − n′)

× c†mscm′sc†

ns ′cn′s ′∫ ∞

0dt e−i�t�nn′

mm′(t). (4.121)

124 Strong-coupling theory

Here �nn′mm′(t) is a multi-phonon correlator,

�nn′mm′(t) ≡ 〈〈X†

i (t)Xi ′ (t)X†j X j ′ 〉〉 (4.122)

which is calculated using the commutation relations in section 4.3.4. X†i (t) and

Xi ′ (t) commute for any γ (q, ν) = γ (−q, ν) (appendix E). X†j and X j ′ commute

also, so that we can write

X†i (t)Xi ′ (t) =

∏q

e[ui′ (q,t)−ui (q,t)]dq−H.c.] (4.123)

X†j X j ′ =

∏q

e[u j ′ (q)−u j (q)]dq−H.c.] (4.124)

where the phonon branch index ν is dropped for simplicity. Applying the identity(4.71) twice yields

X†i (t)Xi ′ (t)X†

j X j ′ =∏

q

eβ∗d†q e−βdq e−|β|2/2

× e[ui′ (q,t)−ui (q,t)][u∗

j ′(q)−u∗j (q)]/2−H.c.

(4.125)

whereβ = ui (q, t) − ui ′ (q, t) + u j (q) − u′

j (q).

Finally using the average equation (4.73), we find

�nn′mm′(t) = e−g2(m−m′)e−g2(n−n′)

× exp

{1

2N

∑q,ν

|γ (q, ν)|2 Fq(m, m′, n, n′)cosh[ωqν((1/2T ) − it)]

sinh[ωqν/2T ]}

(4.126)

where

Fq(m, m′, n, n′) = cos[q · (n′ − m)] + cos[q · (n − m′)]− cos[q · (n′ − m′)] − cos[q · (n − m)]. (4.127)

Taking into account that there are only bipolarons in the subspace where Hb

operates, we finally rewrite the Hamiltonian in terms of the creation b†m =

c†m↑c†

m↓ and annihilation bm = cm↓cm↑ operators of singlet pairs as

Hb = −∑

m

[� + 1

2

∑m′

v(2)(m − m′)]

nm

+∑

m �=m′[t (m − m′)b†

mbm′ + 12 v(m − m′)nmnm′ ]. (4.128)

Mobile small bipolarons 125

There are no triplet pairs in the Holstein model because the Pauli exclusionprinciple does not allow two electrons with the same spin to occupy the samesite. Here nm = b†

mbm is the bipolaron site-occupation operator,

v(m − m′) = 4v(m − m′) + v(2)(m − m′) (4.129)

is the bipolaron–bipolaron interaction including a direct polaron–polaroninteraction v(m − m′) and a second order in T (m) repulsive correction:

v(2)(m − m′) = 2iT 2(m − m′)∫ ∞

0dt e−i�t�m′m

mm′(t). (4.130)

This additional repulsion appears because a virtual hop of one of two polaronsof the pair is forbidden if the neighbouring site is occupied by another pair. Thebipolaron transfer integral is of the second order in T (m):

t (m − m′) = −2iT 2(m − m′)∫ ∞

0dt e−i�t�mm′

mm′(t). (4.131)

The bipolaronic Hamiltonian (4.128) describes the low-energy physics of stronglycoupled electrons and phonons. We use the explicit form of the multi-phononcorrelator, equation (4.126), to calculate t (m) and v(2)(m). If the phononfrequency is dispersionless, we obtain

�mm′mm′(t) = e−2g2(m−m′) exp[−2g2(m − m′)e−iω0t ]

�m′mmm′(t) = e−2g2(m−m′) exp[2g2(m − m′)e−iω0t ]

at T � ω0. Expanding the time-dependent exponents in the Fourier series andcalculating the integrals in equations (4.131) and (4.130) yield [105]

t (m) = −2T 2(m)

�e−2g2(m)

∞∑l=0

[−2g2(m)]l

l!(1 + lω0/�)(4.132)

and

v(2)(m) = 2T 2(m)

�e−2g2(m)

∞∑l=0

[2g2(m)]l

l!(1 + lω0/�). (4.133)

When � � ω0, we can keep the first term only with l = 0 in the bipolaronhopping integral in equation (4.132). In this case, the bipolaron half-bandwidthzt (a) is of the order of 2w2/(z�). However, if the bipolaron binding energy islarge (� � ω0), the bipolaron bandwidth dramatically decreases proportionallyto e−4g2

� 1 in the limit � → ∞. However, this limit is not realistic because� = 2E p − Vc < 2g2ω0. In a more realistic regime, ω0 < � < 2g2ω0,equation (4.132) yields

t (m) ≈ 2√

2πT 2(m)√ω0�

exp

[−2g2 − �

ω0

(1 + ln

2g2(m)ω0

�

)]. (4.134)

126 Strong-coupling theory

In contrast, the bipolaron–bipolaron repulsion (equation (4.133)) has no smallexponent in the limit � → ∞, v(2) � D2/�. Together with the direct Coulombrepulsion, the second order v(2) ensures the stability of the bipolaronic liquidagainst clustering.

The high-temperature behaviour of the bipolaron bandwidth is just theopposite to that of the small polaron bandwidth. While the polaron band collapseswith increasing temperature (equation (4.42)), the bipolaron band becomes wider[106]:

t (m) ∝ 1√T

exp

[− E p + �

2T

](4.135)

for T > ω0.

4.6.2 Inter-site bipolaron in the chain model

On-site bipolarons are very heavy for realistic values of the on-site attractiveenergy 2E p and phonon frequencies. Indeed, to bind two polarons on a singlesite 2E p should overcome the on-site Coulomb energy, which is typically ofthe order of 1 eV or higher. Optical phonon frequencies are about 0.1–0.2 eVin novel superconductors like oxides and doped fullerenes. Therefore, in theframework of the Holstein model, the mass enhancement exponent of on-sitebipolarons in equation (4.134) is rather large (� exp(2E p/ω0) > 150), so thaton-site bipolarons could hardly account for high values of the superconductingcritical temperature [100].

But the Holstein model is not a typical model. The Frohlich interaction withoptical phonons, which is unscreened in polaronic systems (section 4.4), is muchstronger. This longer-range interaction leads to a lighter polaron in the strong-coupling regime (section 4.3.3). Indeed, the polaron is heavy because it has tocarry the lattice deformation with it, the same deformation that forms the polaronitself. Therefore, there exists a generic relation between the polaron stabilizationenergy, E p , and the renormalization of its mass, m ∝ exp (γ E p/ω0), wherethe numeric coefficient γ depends on the radius of the interaction. For a short-range e–ph interaction, the entire lattice deformation disappears and then forms atanother site, when the polaron moves between the nearest lattices sites. Therefore,γ = 1 and polarons and on-site bipolarons are very heavy for the characteristicvalues of E p and ω0. In contrast, in a long-range interaction, only a fraction ofthe total deformation changes every time the polaron moves and γ could be assmall as 0.25 (part 2). Clearly, this results in a dramatic lightening of the polaronsince γ enters the exponent. Thus the small polaron mass could be ≤10me wherea Holstein-like estimate would yield a huge mass of 10 000me. The lower masshas important consequences because lighter polarons are more likely to remainmobile and less likely to trap on impurities.

The bipolaron also becomes much lighter, if the e–ph interaction is longrange. There are two reasons for the lowering of its mass with an increasingradius for the e–ph interaction. The first one is the same as in the case of the

Mobile small bipolarons 127

Figure 4.7. Simplified chain model with two electrons on the chain interacting withnearest-neighbour ions of another chain. Second-order inter-site bipolaron tunnelling isshown by arrows.

single polaron discussed earlier. The second reason is the possibility to forminter-site bipolarons which, in certain lattice structures, already tunnel coherentlyin the first-order in T (m) [100] (section 4.6.3), in contrast with on-site bipolarons,which tunnel only in the second order, equation (4.134).

To illustrate the essential dynamic properties of bipolarons formed by thelonger-range e–ph interaction let us discuss a few simplified models. FollowingBonca and Trugman [107], we first consider a single bipolaron in the chain modelof section 4.3.3 (figure 4.7). One can further simplify the chain model by placingions in the interstitial sites located between the Wannier orbitals of one chainand allowing for the e–ph interaction only with the nearest neighbours of anotherchain, as shown in figure 4.7. The Coulomb interaction is represented by theon-site Hubbard U term.

The model Hamiltonian is

H = T (a)∑j,s

[c†j+1,sc j s + H.c.] + ω0

∑i, j,s

g(i, j)n j s(d†i + di )

+ ω0

∑i

[d†i di + 1/2] + U

∑j

n j↑n j↓ (4.136)

in the site representation for electrons and phonons (section 4.1), where

g(i, j) = g0[δi, j + δi, j+1]

and i , j are integers sorting the ions and the Wannier sites, respectively. Thismodel is referred to as the extended Holstein–Hubbard model (EHHM) [107].We can view the EHHM as the simplest model with a longer range than theHolstein interaction. In comparison with the Frohlich model of section 4.3.3, theEHHM lacks a long-range tail in the e–ph interaction but reveals similar physicalproperties. In the momentum representation, the model is a one-dimensional caseof the generic Hamiltonian (4.7), with

γ (q) = g0√

2(1 + eiqa) (4.137)

128 Strong-coupling theory

and ω(q) = ω0. Using equations (4.34), (4.39) and (4.32), we obtain

E p = g20ω0a

π

∫ π/a

−π/adq [1 + cos qa] = 2g2

0ω0 (4.138)

for the polaron level shift,

g2 = g20a

π

∫ π/a

−π/adq [1 − cos2 qa] = g2

0 (4.139)

for the mass enhancement exponent and

v(0) = U − 4g20ω0 (4.140)

v(a) = − 2g20ω0a

π

∫ π/a

−π/adq [1 + cos qa] cos qa = −2g2

0ω0

for the on-site and inter-site polaron–polaron interactions, respectively. Hence,the EHHM has the numerical coefficient γ = 1/2, and the polaron mass

m∗EHP � exp

(E p

2ω0

)(4.141)

scales as the square root of the small Holstein polaron mass, m∗SHP �

exp(E p/ω0). In the case when U < 2g20ω0, the on-site bipolaron has the lowest

energy because |v(0)| > |v(a)|. In this regime the bipolaron binding energy is

� = 4g20ω0 − U. (4.142)

Using expression (4.132) for the bipolaron hopping integral, we obtain thebipolaron mass as

m∗∗EHB � exp

(2E p

ω0

)(4.143)

if � � ω0. It scales as (m∗EHP/m)4 but is much smaller than the on-site bipolaron

mass in the Holstein model, m∗∗SHB � exp(4E p/ω0), which scales as (m∗

SHP/m)4.In the opposite regime, when U > 2g2

0ω0, the inter-site bipolaron has the lowestenergy. Its binding energy

� = 2g20ω0 (4.144)

does not depend on U . Differing from the on-site singlet bipolaron, the inter-sitebipolaron has four spin states, one singlet S = 0 and three triplet states, S = 1,

with different z-components of the total spin, Sz = 0,±1. In the chain model(figure 4.7), the inter-site bipolaron also tunnels only in the second order in T (a),

when one of the electrons within the pair hops to the left (right) and then the otherfollows. This tunnelling involves the multi-phonon correlation function �

j+2, j+1j+1, j

(equation (4.126)):

�j+2, j+1j+1, j = e−2g2

0 .

Mobile small bipolarons 129

Hence, the inter-site bipolaron mass enhancement is

m∗∗EHB � T −2(a) exp

(E p

ω0

)�

(m∗

EHP

m

)2

(4.145)

in the infinite Hubbard U limit (U → ∞). We see that the inter-site bipolaron inthe chain model is lighter than the on-site bipolaron but still remains much heavierthan the polaron.

4.6.3 Superlight inter-site bipolarons

Any realistic theory of doped ionic insulators must include both the long-range Coulomb repulsion between carriers and the strong long-range electron–phonon interaction. From a theoretical standpoint, the inclusion of the long-range Coulomb repulsion is critical in ensuring that the carriers would not formclusters. Indeed, in order to form stable bipolarons, the e–ph interaction has tobe strong enough to overcome the Coulomb repulsion at short distances. Sincethe e–ph interaction is long range, there is a potential possibility for clustering.The inclusion of the Coulomb repulsion Vc makes the clusters unstable. Moreprecisely, there is a certain window of Vc/E p inside which the clusters areunstable but bipolarons form nonetheless. In this parameter window, bipolaronsrepel each other and propagate in a narrow band. At a weaker Coulombinteraction, the system is a charge-segregated insulator and at a stronger Coulombrepulsion, the system is the Fermi liquid or the Luttinger liquid, if it is one-dimensional.

Let us now apply a generic ‘Frohlich–Coulomb’ Hamiltonian, whichexplicitly includes the infinite-range Coulomb and electron–phonon interactions,to a particular lattice structure [108]. The implicitly present infinite Hubbard Uprohibits double occupancy and removes the need to distinguish the fermionicspin. Introducing spinless fermion operators cn and phonon operators dmν , theHamiltonian is written as

H =∑n �=n′

T (n − n′)c†ncn′ +

∑n �=n′

Vc(n − n′)c†ncnc†

n′cn′

+ ω0

∑n�=m,ν

gν(m − n)(eν · em−n)c†ncn(d†

mν + dmν)

+ ω0

∑m,ν

(d†mνdmν + 1

2 ). (4.146)

The e–ph term is written in real space, which is more convenient when workingwith complex lattices.

In general, the many-body model equation (4.146) is of considerablecomplexity. However, we are interested in the limit of the strong e–ph interaction.In this case, the kinetic energy is a perturbation and the model can be grossly

130 Strong-coupling theory

simplified using the canonical transformation of section 4.3 in the Wannierrepresentation for electrons and phonons:

S =∑

m �=n,ν

gν(m − n)(eν · em−n)c†ncn(d†

mν − dmν).

The transformed Hamiltonian is

H = e−S H eS =∑n �=n′

σnn′c†ncn′ + ω0

∑mα

(d†mνdmν + 1

2 )

+∑n �=n′

v(n − n′)c†ncnc†

n′cn′ − E p

∑n

c†ncn. (4.147)

The last term describes the energy gained by polarons due to the e–ph interaction.E p is the familiar polaron level shift:

E p = ω0

∑mν

g2ν(m − n)(eν · em−n)2 (4.148)

which is independent of n. The third term on the right-hand side ofequation (4.147) is the polaron–polaron interaction:

v(n − n′) = Vc(n − n′) − Vph(n − n′) (4.149)

where

Vph(n − n′) = 2ω0

∑m,ν

gν(m − n)gν(m − n′)(eν · em−n)(eν · em−n′).

The phonon-induced interaction, Vph, is due to displacements of common ions bytwo electrons. Finally, the transformed hopping operator σnn′ in the first term inequation (4.147) is given by

σnn′ = T (n − n′) exp

[∑m,ν

[gν(m − n)(eν · em−n)

− gν(m − n′)(eν · em−n′)](d†mα − dmα)

]. (4.150)

This term is a perturbation at large λ. Here we consider a particular latticestructure (ladder), where bipolarons already tunnel in the first order in T (n), sothat σnn′ can be averaged over phonons. When T � ω0, the result is

t (n − n′) ≡ 〈〈σnn′ 〉〉ph = T (n − n′) exp[−g2(n − n′)] (4.151)

g2(n − n′) =∑m,ν

gν(m − n)(eν · em−n)

× [gν(m − n)(eν · em−n) − gν(m − n′)(eν · em−n′)].

Mobile small bipolarons 131

By comparing equations (4.151) and (4.149), the mass renormalization exponentcan be expressed via E p and Vph as follows:

g2(n − n′) = 1

ω0

[E p − 1

2Vph(n − n′)

]. (4.152)

Now phonons are ‘integrated out’ and the polaronic Hamiltonian is

Hp = H0 + Hpert (4.153)

H0 = − E p

∑n

c†ncn +

∑n �=n′

v(n − n′)c†ncnc†

n′cn′

Hpert =∑n �=n′

t (n − n′)c†ncn′ .

When Vph exceeds Vc, the full interaction becomes negative and the polarons formpairs. The real-space representation allows us to elaborate the physics behindthe lattice sums in equations (4.148) and (4.149) more fully. When a carrier(electron or hole) acts on an ion with a force f , it displaces the ion by some vectorx = f /s. Here s is the ion’s force constant. The total energy of the carrier–ionpair is − f 2/(2s). This is precisely the summand in equation (4.148) expressedvia dimensionless coupling constants. Now consider two carriers interacting withthe same ion, see figure 4.8(a). The ion displacement is x = ( f 1 + f 2)/s andthe energy is − f 2

1/(2s) − f 22/(2s) − ( f 1 · f 2)/s. Here the last term should be

interpreted as an ion-mediated interaction between the two carriers. It depends onthe scalar product of f 1 and f 2 and, consequently, on the relative positions of thecarriers with respect to the ion. If the ion is an isotropic harmonic oscillator, aswe assume here, then the following simple rule applies. If the angle φ betweenf 1 and f 2 is less than π/2 the polaron–polaron interaction will be attractive, ifotherwise it will be repulsive. In general, some ions will generate attraction andsome repulsion between polarons (figure 4.8(b)).

The overall sign and magnitude of the interaction is given by the latticesum in equation (4.149), the evaluation of which is elementary. One shouldalso note that, according to equation (4.152), an attractive interaction reduces thepolaron mass (and, consequently, the bipolaron mass), while repulsive interactionenhances the mass. Thus, the long-range nature of the e–ph interaction serves adouble purpose. First, it generates an additional inter-polaron attraction becausethe distant ions have small angle φ. This additional attraction helps to overcomethe direct Coulomb repulsion between the polarons. And, second, the Frohlichinteraction makes the bipolarons lighter.

The many-particle ground state of H0 depends on the sign of the polaron–polaron interaction, the carrier density and the lattice geometry. Here we considerthe zig-zag ladder in figure 4.9(a), assuming that all sites are isotropic two-dimensional harmonic oscillators. For simplicity, we also adopt the nearest-neighbour approximation for both interactions, gν(l) ≡ g, Vc(n) ≡ Vc, and for

132 Strong-coupling theory

Figure 4.8. The mechanism for the polaron–polaron interaction. (a) Together, the twopolarons (filled circles) deform the lattice more effectively than separately. An effectiveattraction occurs when the angle φ between x1 and x2 is less than π/2. (b) A mixedsituation. Ion 1 results in repulsion between two polarons while ion 2 results in attraction.

the hopping integrals, T (m) = TNN for l = n = m = a, and zero otherwise.Hereafter we set the lattice period a = 1. There are four nearest neighbours in theladder, z = 4. Then, the one-particle polaronic Hamiltonian takes the form

Hp = − E p

∑n

(c†ncn + p†

n pn)

+∑

n

[t ′(c†n+1cn + p†

n+1 pn) + t (p†ncn + p†

n−1cn) + H.c.] (4.154)

where cn and pn are polaron annihilation operators on the lower and upper sitesof the ladder, respectively (figure 4.9(b)). Using equations (4.148), (4.149) and(4.152), we find

E p = 4g2ω0 (4.155)

t ′ = TNN exp

(−7E p

8ω0

)

t = TNN exp

(−3E p

4ω0

).

The Fourier transform of equation (4.154) into momentum space yields

Hp =∑

k

(2t ′ cos k − E p)(c†kck + p†

k pk)+ t∑

k

[(1 + eik)p†kck + H.c.]. (4.156)

A linear transformation of ck and pk diagonalizes the Hamiltonian, so that theone-particle energy spectrum E1(k) is found from

det

∣∣∣∣ 2t ′ cos k − E p − E1(k) t (1 + eik)

t (1 + e−ik) 2t ′ cos k − E p − E1(k)

∣∣∣∣ = 0. (4.157)

Mobile small bipolarons 133

Figure 4.9. One-dimensional zig-zag ladder: (a) initial ladder with the bare hoppingamplitude T (a); (b) two types of polarons with their respective deformations; (c) twodegenerate bipolaron configurations A and B; and (d) a different bipolaron configuration,C, whose energy is higher than that of A and B.

There are two overlapping polaronic bands,

E1(k) = −E p + 2t ′ cos(k) ± 2t cos(k/2)

with effective mass m∗ = 2/|4t ′ ± t| near their edges.Let us now place two polarons on the ladder. The nearest-neighbour

interaction (equation (4.149)) is v = Vc − E p/2, if two polarons are on differentsides of the ladder, and v = Vc − E p/4, if both polarons are on the same side. Theattractive interaction is provided via the displacement of the lattice sites, whichare the common nearest neighbours to both polarons. There are two such nearestneighbours for the inter-site bipolaron of type A or B (figure 4.9(c)) but thereis only one common nearest neighbour for bipolaron C (figure 4.9(d)). WhenVc > E p/2, there are no bound states and the multi-polaron system is a one-dimensional Luttinger liquid. However, when Vc < E p/2 and, consequently,v < 0, the two polarons are bound into an inter-site bipolaron of types A or B.

It is quite remarkable that bipolaron tunnelling in the ladder already appears

134 Strong-coupling theory

in the first order with respect to a single-electron tunnelling. This case is differentfrom both the on-site bipolarons discussed in section 4.6.1 and from the inter-sitechain bipolaron of section 4.6.2, where the bipolaron tunnelling was of the secondorder in T (a). Indeed, the lowest-energy configurations A and B are degenerate.They are coupled by Hpert. Neglecting all higher-energy configurations, we canproject the Hamiltonian onto the subspace containing A, B and empty sites.

The result of such a projection is a bipolaronic Hamiltonian:

Hb = (Vc − 52 E p)

∑n

[A†n An + B†

n Bn]− t ′∑

n

[B†n An + B†

n−1 An + H.c.] (4.158)

where An = cn pn and Bn = pncn+1 are inter-site bipolaron annihilationoperators and the bipolaron–bipolaron interaction is dropped (see later). ItsFourier transform yields two bipolaron bands,

E2(k) = Vc − 52 E p ± 2t ′ cos(k/2) (4.159)

with a combined width 4|t ′|. The bipolaron binding energy in zero order withrespect to t, t ′ is

� ≡ 2E1(0) − E2(0) = E p

2− Vc. (4.160)

The bipolaron mass near the bottom of the lowest band, m∗∗ = 2/t ′, is

m∗∗ = 4m∗[

1 + 0.25 exp

(E p

8ω0

)]. (4.161)

The numerical coefficient 1/8 ensures that m∗∗ remains of the order of m∗even at large E p. This fact combines with a weaker renormalization of m∗,equation (4.155), providing a superlight bipolaron.

In models with strong inter-site attraction, there is a possibility ofclusterization. Similar to the two-particle case described earlier, the lowest energyof n polarons placed on the nearest neighbours of the ladder is found as

En = (2n − 3)Vc − 6n − 1

4E p (4.162)

for any n ≥ 3. There are no resonating states for an n-polaron configurationif n ≥ 3. Therefore, there is no first-order kinetic energy contribution to theirenergy. En should be compared with the energy E1+(n−1)E2/2 of far separated(n − 1)/2 bipolarons and a single polaron for odd n ≥ 3, or with the energy of farseparated n bipolarons for even n ≥ 4. ‘Odd’ clusters are stable if

Vc <n

6n − 10E p (4.163)

and ‘even’ clusters are stable if

Vc <n − 1

6n − 12E p. (4.164)

Bipolaronic superconductivity 135

As a result, we find that bipolarons repel each other and single polarons atVc > 3

8 E p. If Vc is less than 38 E p , then immobile bound clusters of three and

more polarons could form. One should note that at distances much larger than thelattice constant, the polaron–polaron interaction is always repulsive (section 4.4)and the formation of infinite clusters, stripes or strings is impossible. Combiningthe condition of bipolaron formation and that of the instability of larger clusters,we obtain a window of parameters

38 E p < Vc < 1

2 E p (4.165)

where the ladder is a bipolaronic conductor. Outside the window, the ladder iseither charge-segregated into finite-size clusters (small Vc) or it is a liquid ofrepulsive polarons (large Vc).

4.7 Bipolaronic superconductivity

In the subspace with no single polarons, the Hamiltonian of electrons stronglycoupled with phonons is reduced to the bipolaronic Hamiltonian written in termsof creation, b†

m = c†m↑c†

m↓, and annihilation, bm, bipolaron operators as

Hb =∑

m �=m′[t (m − m′)b†

mbm′ + 12 v(m − m′)nmnm′ ] (4.166)

where v(m − m′) is the bipolaron–bipolaron interaction, nm = b†mbm, and

the position of the middle of the bipolaron band is taken as zero. There areadditional (spin) quantum numbers S = 0, 1; Sz = 0,±1, which should beadded to the definition of bm for the case of inter-site bipolarons. Also in somelattice structures (section 4.6.3 and part 2), inter-site bipolarons tunnel via aone-particle hopping rather than via simultaneous two-particle tunnelling of on-site bipolarons. This ‘crab-like’ tunnelling (figure 4.9) results in a bipolaronbandwidth of the same order as the polaron one. Keeping this in mind, we canapply Hb (equation (4.166)) to both on-site and inter-site bipolarons, and evento more extended non-overlapping pairs, implying that the site index m is theposition of the centre of mass of a pair.

4.7.1 Bipolarons and a charged Bose gas

Bipolarons are not perfect bosons. In the subspace of pairs and empty sites, theiroperators commute as

bmb†m + b†

mbm = 1 (4.167)

bmb†m′ − b†

m′bm = 0 (4.168)

for m �= m′. This makes useful the pseudospin analogy [10],

b†m = Sx

m − iSym (4.169)

136 Strong-coupling theory

and

b†mbm = 1

2 − Szm (4.170)

with the pseudospin 12 operators Sx,y,z = 1

2τ1,2,3. Szm = 1

2 correspondsto an empty site m and Sz

m = − 12 to a site occupied by the bipolaron.

Spin operators preserve the bosonic nature of bipolarons, when they are ondifferent sites and their fermionic internal structure. Replacing bipolarons byspin operators, we transform the bipolaronic Hamiltonian into the anisotropicHeisenberg Hamiltonian:

Hb =∑

m �=m′[ 1

2 vmm′ SzmSz

m′ + tmm′(Sxm Sx

m′ + Sym Sy

m′)]. (4.171)

This Hamiltonian has been investigated in detail as a relevant form for magnetismand also for quantum solids like a lattice model of 4He. However, while in thosecases the magnetic field is an independent thermodynamic variable, in our casethe total ‘magnetization’ is fixed,

1

N

∑m

〈〈Szm〉〉 = 1

2− nb (4.172)

if the bipolaron density nb is conserved. The spin- 12 Heisenberg Hamiltonian

(4.171) cannot be solved analytically. The complicated commutation rules forbipolaron operators (equations (4.167) and (4.168)) make the problem hard butnot in the limit of low atomic density of bipolarons, nb � 1 (for a complete phasediagram of bipolarons on a lattice see [10, 109]). In this limit we can reduce theproblem to a charged Bose gas on a lattice [110]. Let us transform the bipolaronicHamiltonian to a representation containing only the Bose operators am and a†

mdefined as

bm =∞∑

k=0

βk(a†m)kak+1

m (4.173)

b†m =

∞∑k=0

βk(a†m)k+1ak

m (4.174)

where

ama†m′ − a†

m′am = δm,m′ . (4.175)

The first few coefficients βk are found by substituting equations (4.173) and(4.174) into equations (4.167) and (4.168):

β0 = 1 β1 = −1 β2 = 1

2+

√3

6. (4.176)

Bipolaronic superconductivity 137

We also introduce bipolaron and boson �-operators as

�(r) = 1√N

∑m

δ(r − m)bm (4.177)

�(r) = 1√N

∑m

δ(r − m)am. (4.178)

The transformation of the field operators takes the form:

�(r) =[

1 − �†(r)�(r)N

+ (1/2 + √3/6)�†(r)�(r)�(r)

N2 + · · ·]

�(r).

(4.179)Then we write the bipolaronic Hamiltonian as

Hb =∫

dr∫

dr ′ �†(r)t (r − r ′)�(r′) + Hd + HK + H (3) (4.180)

where

Hd = 12

∫dr

∫dr ′ v(r − r ′)�†(r)�†(r ′)�(r ′)�(r) (4.181)

is the dynamic part,

HK = 2

N

∫dr

∫dr ′ t (r − r ′)

× [�†(r)�†(r ′)�(r ′)�(r ′) + �†(r)�†(r)�(r)�(r ′)]. (4.182)

is the kinematic (hard-core) part due to the ‘imperfect’ commutation rules, andH (3) includes three- and higher-body collisions. Here

t (r − r ′) =∑

k

ε∗∗k eik·(r−r ′)

v(r − r ′) = 1

N

∑k

vkeik·(r−r ′)

vk = ∑m �=0 v(m) exp(ik ·m) is the Fourier component of the dynamic interaction

andε∗∗

k =∑m �=0

t (m) exp(−ik · m) (4.183)

is the bipolaron band dispersion. H (3) contains powers of the field operator higherthan four. In the dilute limit (nb � 1) only two-particle interactions are essentialwhich include the short-range kinematic and direct density–density repulsions.Because v already has a short-range part v(2) (equation (4.129)), the kinematic

138 Strong-coupling theory

contribution can be included in the definition of v. As a result Hb is reduced tothe Hamiltonian of interacting hard-core charged bosons tunnelling in a narrowband.

To describe the electrodynamics of bipolarons, we introduce the vectorpotential A(r) using the so-called Peierls substitution [111]:

t (m − m′) → t (m − m′)ei2e A(m)·(m−m′)

which is a fair approximation when the magnetic field is weak compared with theatomic field, eH a2 � 1 (see also section 2.16). It has the following form:

t (r − r′) → t (r, r ′) =∑

k

ε∗∗k−2e Aeik·(r−r ′) (4.184)

in real space. If the magnetic field is weak, we can expand ε∗∗k in the vicinity of

k = 0 to obtain

t (r, r ′) ≈ −[∇ + 2ie A(r)]2

2m∗∗ δ(r − r ′) (4.185)

where1

m∗∗ =(

d2ε∗∗k

dk2

)k→0

(4.186)

is the inverse bipolaron mass. Here we assume a parabolic dispersion near thebottom of the band, ε∗∗

k � k2. Finally, we arrive at

Hb ≈ −∫

dr �†(r)

{[∇ + 2ie A(r)]2

2m∗∗ + µ

}�(r)

+ 12

∫dr dr ′ v(r − r ′)�†(r)�†(r ′)�(r)�(r ′) (4.187)

where we include the bipolaron chemical potential µ. We note that the bipolaron–bipolaron interaction is the Coulomb repulsion, v(r) ∼ 1/(ε0r) at large distances(section 4.4) and the hard-core repulsion is irrelevant in the dilute limit. Hence,equation (4.187) describes a charged Bose gas (CBG) with the effective bosonmass m∗∗ and charge 2e.

4.7.2 Bogoliubov equations in the strong-coupling regime

Let us derive equations describing the order parameter and excitations of CBG bythe use of the Bogoliubov transformation similar to that in chapter 2. The equationof motion for the field operator, ψ(r, t), is derived using the Hamiltonian (4.187)as

id

dtψ(r, t) = [H, ψ(r, t)] =

[− (∇ − i2e A)2

2m∗∗ − µ

]ψ(r, t)

+∫

dr ′ v(r − r ′)ψ†(r ′, t)ψ(r ′, t)ψ(r, t). (4.188)

Bipolaronic superconductivity 139

A large value for the static dielectric constant (ε0 � 1) in ionic solids makes theCoulomb repulsion between bipolarons rather weak. If the interaction is weak, weexpect that some properties of CBG to be similar to the properties of an ideal Bosegas (appendix B). In particular, the state with zero momentum (k = 0) remainsmacroscopically occupied and the corresponding Fourier component of the fieldoperator ψ(r, t) has an anomalously large matrix element between the states withN + 1 and N bosons. Introducing the chemical potential µ we consider eachquantum state as a superposition of states with a slightly different total numberof bosons. The weight of each state in the superposition is a smooth function ofN which is practically unchanged in the window ±

√N near the average number

N . Hence, because ψ changes the number of particles only by one, its diagonalmatrix element is practically the same as the off-diagonal one, calculated for stateswith fixed N = N + 1 and N = N . Following Bogoliubov [4] we separate thelarge diagonal matrix element ψs from ψ treating ψs as a number and the rest, ψ,

as a small fluctuation

ψ(r, t) = ψs(r, t) + ψ(r, t). (4.189)

Substituting equation (4.189) into equation (4.188) and collecting c-number termsof ψs we obtain a set of Bogoliubov-type equations for the CBG [112]. Themacroscopic condensate wavefunction ψs(r, t), which plays the role of an orderparameter, obeys the following equation:

id

dtψs(r, t) =

[− (∇ + i2e A)2

2m∗∗ − µ

]ψs(r, t)+

∫dr ′v(r − r ′)ns(r ′, t)ψs (r, t)

(4.190)which is a generalization of the so-called Gross–Pitaevskii (GP) [113]equation applied to neutral bosons. Taking into account the interactionof ‘supracondensate’ bosons (described by ψ(r, t)) with the condensate,and neglecting the interaction between the supra-condensate bosons, fromequation (4.188) we also obtain

id

dtψ(r, t) =

[− (∇ + i2e A)2

2m∗∗ − µ

]ψ(r, t)

+∫

dr ′ v(r − r ′)[ns(r, t) + ψ∗s (r ′, t ′)ψs(r, t)]ψ(r ′, t)

+∫

dr ′ v(r − r ′)ψs(r ′, t)ψs (r, t)ψ†(r ′, t). (4.191)

Herens(r, t) = |ψs(r, t)|2 (4.192)

is the condensate density. If the Coulomb repulsion of bosons is not very large,

rs = 4m∗∗e2

ε0(4πnb/3)1/3� 1 (4.193)

140 Strong-coupling theory

the number of bosons n pushed up from the condensate by the repulsion issmall (see later). Therefore, the contribution of the terms nonlinear in ψ isnegligible in equation (4.191). Applying the linear Bogoliubov transformationof ψ (chapter 2),

ψ(r, t) =∑

n

[un(r, t)αn + v∗n (r, t)α†

n ] (4.194)

where now αn and α†n are Bose operators, we obtain two coupled Schrodinger

equations for the quasi-particle wavefunctions, u(r, t) and v(r, t), as

id

dtu(r, t) =

[− (∇ + i2e A)2

2m∗∗ − µ

]u(r, t)

+∫

dr ′ v(r − r ′)[|ψs(r ′, t)|2u(r, t) + ψ∗s (r ′, t)ψs (r, t)u(r ′, t)]

+∫

dr ′ v(r − r ′)ψ0(r ′, t)ψ0(r, t)v(r ′, t) (4.195)

and

−id

dtv(r, t) =

[− (∇ − i2e A)2

2m∗∗ − µ

]v(r, t)

+∫

dr ′ v(r − r ′)[|ψs(r ′, t)|2v(r, t) + ψs(r ′, t)ψ∗s (r, t)]v(r ′, t)

+∫

dr ′ v(r − r ′)ψ∗s (r ′, t)ψ∗

s (r, t)u(r ′, t). (4.196)

There is also a sum rule,∑n

[un(r, t)u∗n(r ′, t) − vn(r, t)v∗

n (r ′, t)] = δ(r − r ′) (4.197)

which retains the Bose commutation relations for new operators. The set ofequations (4.190), (4.195) and (4.196) plays the same role in the strong-couplingtheory as the Bogoliubov equations for the BCS superconductors (section 2.11).

4.7.3 Excitation spectrum and ground-state energy

For a homogeneous case with A(r) = 0, the quasi-particle wavefunctions areplane waves,

uk(r, t) = ukeik·r−iεkt (4.198)

vk(r, t) = vkeik·r−iεkt

Bipolaronic superconductivity 141

and the condensate wavefunction is (r, t)-independent, ψs(r, t) = √ns =

constant. Then a solution of equation (4.190) is

µ = 0 (4.199)

because the volume average of v(r) is zero due to the interaction with the ionbackground (i.e. because of the global electroneutrality). Then the Bogoliubovequations yield

εkuk = k2

2m∗∗ uk + ns vk[uk + vk] (4.200)

−εkvk = k2

2m∗∗ vk + ns vk[uk + vk] (4.201)

and|uk|2 − |vk|2 = 1. (4.202)

Here vk is the Fourier transform of v(r). As a result, we find

u2k = 1

2

(1 + ξk

εk

)(4.203)

v2k = − 1

2

(1 − ξk

εk

)(4.204)

ukvk = − vkns

2εk(4.205)

where ξk = k2/2m∗∗ + vkns . The quasi-particle energy is

εk =√

k4

4(m∗∗)2 + k2vkns

m∗∗ . (4.206)

Using the Fourier component of the Coulomb interaction yields [114]

εk =√

k4

4(m∗∗)2+ ω2

ps (4.207)

with a gap

ωps =√

16πe2ns

ε0m∗∗ (4.208)

which is the plasma frequency. The quasi-particle spectrum (equation (4.207))differs qualitatively from the BCS excitation spectrum (section 2.2). The BCSquasi-particles are fermions and their energy is of the order of the BCS gap, �k,which is well below the electron plasma frequency, �k � ωe. The quasi-particlesin CBG are bosons and their energy is about the (renormalized) plasma frequency

142 Strong-coupling theory

ωps. The density of bosons pushed up from the condensate by the Coulombrepulsion at T = 0 is

n = 〈ψ†(r)ψ(r)〉 =∑

k

v2k (4.209)

which is small compared with the total density nb as

n

nb≈ 0.2r3/4

s . (4.210)

The ground state |0〉 is a vacuum of quasi-particles, αk|0〉 = 0. The ground-state energy E0 is obtained by substituting equation (4.194) into the Hamiltonian(4.187) and neglecting terms of higher order than quadratic in αk ,

E0 ≡ 〈0|H |0〉 = 12

∑k

(εk − ξk). (4.211)

This can be written per particle as

E0

nb= 23/2

31/4πωpsr

3/4s

∫ ∞

0dk k2

[√k4 + 1 − k2 − 1/(2k2)

]≈ −0.23ωpsr

3/4s .

(4.212)The negative value of the ground-state energy is due to the opposite chargebackground. The value of |E0| is considered as the gain in the total energydue to the condensation of interacting bosons with respect to the ground-stateenergy (= 0) of an ideal Bose gas. Therefore, |E0| plays the same role as thecondensation energy in the BCS superconductor.

These results for three-dimensional charged bosons are readily generalizedfor any bosons on a lattice beyond the effective mass approximation. For example,the quasi-particle spectrum is given by

εk =√

(ε∗∗k )2 + 2ε∗∗

k vkns . (4.213)

Hence, if the free-boson dispersion ε∗∗k is anisotropic, the plasma gap is

anisotropic as well. In an extreme case of (quasi-)two-dimensional bosonswith a parabolic dispersion and a two-dimensional Coulomb repulsion, vk =8πe2/(ε0k), the Bogoliubov spectrum is gapless,

εk = Es

√k/qs + k4/q4

s . (4.214)

Here Es = q2s /2m∗∗ and qs = (32πe2ns/ε0)

1/3 is a two-dimensional screeningwave-number, ns is the number of condensed bosons per unit area.

4.7.4 Mixture of two Bose condensates

The quasi-particle spectrum (equation (4.207)) satisfies the Landau criterion ofsuperfluidity (chapter 1), therefore the CBG is a superconductor. The actual

Bipolaronic superconductivity 143

spectrum of bipolarons on a lattice is more complicated, because inter-sitebipolarons have ‘internal’ quantum numbers like the spin (section 4.6), orbitalmomentum and different symmetries of the one-particle Wannier orbitals boundinto the bipolaron. These internal degrees of freedom can affect the collectiveexcitations of a bipolaronic superconductor. Here we extend the Bogoliubov-typeequations of the previous subsection to the multi-component Bose condensate ofnon-converting charged bosons [115]. Let us consider a two-component (1 and 2)mixture of bosons described by

H =∑j=1,2

∫dr �

†j (r)

[− ∇2

2m j− µ j

]� j (r)

+ 12

∑j, j ′

∫dr

∫dr ′ v j j ′(r − r ′)�†

j (r)� j (r)�†j ′(r ′)� j ′(r ′)

where m j is the mass of the boson j .Using the displacement transformation (equation (4.189)) and the equations

of motion for the Heisenberg ψ-operators, the condensate wavefunctions arefound from two coupled GP equations:

i∂

∂ tψs j (r, t) =

(− ∇2

2m j− µ j

)ψs j (r, t)

+∑

j ′

∫dr ′ Vj j ′(r − r ′)|ψs j ′(r ′, t)|2ψs j (r, t). (4.215)

The supracondensate wavefunctions satisfy four Bogoliubov-type equations:

i∂

∂ tu j (r, t) =

(− ∇2

2m j− µ j

)u j (r, t) +

∑j ′

∫dr ′ Vj j ′(r − r ′)

× [|ψs j ′(r ′, t)|2u j (r, t) + ψ∗s j (r ′, t)ψs j ′ (r, t)u j ′(r ′, t)

+ ψs j (r ′, t)ψs j ′ (r, t)v j ′(r ′, t)]. (4.216)

and

−i∂

∂ tv j (r, t) =

(− ∇2

2m j− µ j

)v j (r, t) +

∑j ′

∫dr ′ Vj j ′(r − r ′)

× [|ψs j ′(r ′, t)|2v j (r, t) + ψs j (r ′, t)ψ∗s j ′(r, t)v j ′(r ′, t)

+ ψ∗s j (r ′, t)ψ∗

s j ′(r, t)u j ′(r ′, t)]. (4.217)

Here we applied the linear transformation of ψ,

ψ j (r, t) =∑

n

unj (r, t)(αn + βn) + v∗nj (r, t)(α†

n + β†n )

144 Strong-coupling theory

where αn, βn are bosonic operators annihilating quasi-particles in the quantumstate n. Solving two GP equations (4.215), we obtain the chemical potentials of ahomogeneous system as

µ1 = vn1 + wn2

µ2 = un2 + wn1

and solving the four Bogoliubov equations, we determine the excitation spectrum,E(k), from

Det

ξ1(k) − E(k) vkψ2s1 wkψs1ψ

∗s2 wkψs1ψs2

vkψ∗2s1 ξ1(k) + E(k) wkψ

∗s1ψ

∗s2 wkψ

∗s1ψs2

wkψ∗s1ψs2 wkψs1ψs2 ξ2(k) − E(k) ukψ

2s2

wkψ∗s1ψ

∗s2 wkψs1ψ

∗s2 ukψ

∗2s2 ξ2(k) + E(k)

= 0.

(4.218)Here ξ1(k) = k2/2m1 + vkn1, ξ2(k) = k2/2m2 + ukn2 and vk, uk, wk are theFourier components of v11(r), v22(r) and v12(r), respectively, v ≡ v0, u ≡ u0,w ≡ w0 and n j = |ψs j |2 are the condensate densities. There are two branches ofexcitations with dispersion

E(k)1,2 = 2−1/2

ε2

1(k) + ε22(k) ±

√[ε2

1(k) − ε22(k)]2 + 4k4

m1m2w2

kn1n2

1/2

(4.219)where ε1(k) = [k4/(4m2

1) + k2vkn1/m1]1/2 and ε2(k) = [k4/(4m22) +

k2ukn2/m2]1/2 are Bogoliubov modes of two components. If the interactionis purely the Coulomb repulsion, vk = 4πq2

1/k2, uk = 4πq22/k2 and wk =

4πq1q2/k2, the upper branch is the geometric sum of familiar plasmon modesfor k → 0,

E1(k) =√

4πq21n1

m1+ 4πq2

2n2

m2(4.220)

while the lowest branch is gapless,

E2(k) = k2

2(m1m2)1/2

√q2

1 n1m1 + q22 n2m2

q21 n1m2+q2

2 n2m1. (4.221)

Remarkably, this mode does not depend on the interaction at any charges ofthe components, if m1 = m2. It corresponds to a low-frequency oscillation inwhich two condensates move in anti-phase with one another, in contrast to theusual optical high-frequency plasmon (equation (4.220)) in which the componentsoscillate inphase. The mode is similar to the acoustic plasmon (AP) mode in theelectron–ion [116] and electron–hole [117] plasmas. However, differing fromthese normal-state APs with a linear dispersion, the AP of Bose mixtures has aquadratic dispersion in the long-wavelength limit. We conclude that while the

Bipolaronic superconductivity 145

0.0 0.5 1.00

1

2

E(k

)/ω

p

k/qs

Figure 4.10. Excitation energy spectrum of the two-component Bose–Einstein condensatewith the long-range repulsive and hard-core interactions. In this plot n1 = n2 = n,m1 = m2 and u = v = ωp , q1 = q2 = e. Different curves correspond to the differentvalues of w/v = 0.1 (dashes), 0.5 (dots), and 0.95 (full line), respectively. Excitationenergy is measured in units of the plasma frequency ωp = 4πne2/m and momentum k ismeasured in inverse screening length qs = (16πe2nm)1/4.

CBG is a superfluid (according to the Landau criterion), their mixture is not.In the case of bipolarons, the interaction includes both the long-range repulsionand the hard-core interaction (section 4.7.1). Combining both interactions, i.e.taking vk, uk, wk ∝ constant + 1/k2, transforms the lowest AP mode into theBogoliubov sound with a linear dispersion at k → 0, figure 4.10. Hence, thetwo-component condensate of bipolarons is a superconductor.

4.7.5 Critical temperature and isotope effect

Polarons and bipolarons differ from electrons in many ways. One of thedifferences is their effective mass. Electrons change their mass in solids due tointeractions with ions, spins and between themselves. Because λ does not dependon the ion mass, the renormalized mass of electrons is independent of the ionmass M in ordinary metals, where the Migdal adiabatic approximation is believedto be valid (chapter 3). However, if the interaction between electrons and ionvibrations is strong and/or the adiabatic approximation is not applied, electronsform polarons and their effective mass m∗ depends on M as m∗ = m exp(A/ω0)

[82] (section 4.3.1), where m is the band mass in the absence of the electron–phonon interaction, A is a constant independent of M and ω0 is the characteristic

146 Strong-coupling theory

phonon frequency, which depends on M as ω0 � M−1/2. Hence, there is asubstantial isotope effect on the carrier mass in polaronic and bipolaronic systems,in contrast to the zero isotope effect in ordinary metals.

The isotope exponent in m∗ is defined as αm∗ = ∑i d ln m∗/d ln Mi , where

Mi is the mass of the i th ion in a unit cell. Using this definition and the expressionfor the polaron mass m∗ mentioned above, one readily finds

αm∗ = 12 ln(m∗/m). (4.222)

The same isotope exponent is predicted for inter-site bipolarons, when theirmass m∗∗ is proportional to the polaron mass (section 4.6.3). The isotopeeffect on the polaron/bipolaron mass leads to an anomalous isotope effecton the superconducting critical temperature of polaronic and bipolaronicsuperconductors [82]. Tc of bipolarons is the Bose–Einstein condensation(BEC) temperature of CBG. It is approximately the Bose–Einstein condensationtemperature of ideal bosons (appendix B) as long as the Coulomb repulsion isweak (rs � 1):

Tc ≈ 3.31n2/3b

m∗∗ (4.223)

where nb is the volume density of bipolarons. Corrections to equation (4.223),caused by the Coulomb repulsion, are numerically small even at rs = 1 [118,119].Hence, the isotope exponent α in Tc of inter-site bipolarons is the same as αm∗∗ ,

α ≡ − d ln Tc

d ln M= αm∗∗ . (4.224)

In the crossover region from the normal polaron Fermi liquid to bipolarons, onecan apply the BCS-like expression (4.113) for Tc of non-adiabatic carriers whichwe write as

Tc � D exp

(−g2 − Z ′

λ − µc

). (4.225)

Here only g2 �√

M depends on M . When λ − µc � Z ′ = e−g2, this

expression also describes the isotope exponent in Tc of inter-site bipolarons(equation (4.223)) if their effective mass m∗∗ � m∗. In this case we canapply equation (4.225) to the whole region of the phase diagram (figure 4.6)including the BCS-like crossover regime and the Bose–Einstein condensation.Differentiating equation (4.225) with respect to M yields

α = αm∗(

1 − Z ′

λ − µc

). (4.226)

We see that the isotope exponent is negative in polaronic superconductors, whereλ−µc < Z ′ but it is positive in bipolaronic superconductors, where λ−µc > Z ′.Equation (4.225), interpolating between BCS- and BEC-type superconductivity,allows us to understand the origin of high values of Tc in comparison with the

Bipolaronic superconductivity 147

weak-coupling BCS theory (equation (2.48)). High Tc originates in the polaronnarrowing of the band. The exponentially enhanced DOS of the narrow polaronicband effectively eliminates the small exponential factor in equation (4.225). Butat a very strong e–ph coupling, Tc drops again because the carriers become veryheavy. Therefore, we conclude that the highest Tc is in the crossover region frompolaronic to bipolaronic superconductivity where λ − µc ≈ Z ′. The polaronhalf-bandwidth Z ′D is normally less or about the phonon frequency ω0, so thatthe maximum value of Tc is estimated to be Tc � ω0/3 [13]. In novel oxygenand carbon-based superconductors (part 2), the characteristic optical phononfrequency is about 500 to 2000 K. That is why Tc is remarkably high in thesecompounds.

4.7.6 Magnetic field expulsion

The linear response function is defined as

jα(q, ω) =∑

β=x,y,z

K αβ(q, ω)aβ(q, ω) (4.227)

where jα(q, ω) and aβ(q, ω) are the Fourier transforms of the current and ofthe vector potential, respectively. To calculate the response, we need to solveequation (4.190) with µ = 0 in the first order with respect to A(r, t),

id

dtψs(r, t) = −[∇ + i2e A(r, t)]2

2m∗∗ ψs(r, t)+∫

dr ′ v(r−r ′)|ψ∗s (r ′, t)|2ψs(r, t).

Using a perturbed wavefunction,

ψs(r, t) = √ns + φ(r, t) (4.228)

and keeping only the terms linear in A(r, t), we obtain for the Fourier transformφ(q, ω):

ωφ(q, ω) = q2

2m∗∗ φ(q, ω)+ ns vq{φ(q, ω)+φ∗(−q,−ω)}− 2e√

ns

2m∗∗ q · a(q, ω).

(4.229)The solution is

φ(q, ω) + φ∗(−q,−ω) = − 2e√

ns

m∗∗ω

ω2 − ε2q

q · a(q, ω) (4.230)

φ(q, ω) − φ∗(−q,−ω) = − 4e√

ns

q2

ε2q

ω2 − ε2q

q · a(q, ω). (4.231)

The expectation value of the current is given by

j(r, t) = ie

m∗∗ [ψ∗s (r, t)∇ψs (r, t) − ψs(r, t)∇ψ∗

s (r, t)] − 4e2ns

m∗∗ A(r, t).

(4.232)

148 Strong-coupling theory

Using the perturbed wavefunction, we obtain the Fourier transform of the currentas

j(q, ω) = e√

ns

m∗∗ q[φ(q, ω) − φ∗(−q,−ω)] − 4e2ns

m∗∗ a(q, ω) (4.233)

and

K αβ(q, ω) = 4e2ns

m∗∗

[δαβ ω2

ε2q − ω2 + (qαqβ − δαβq2)

ε2q

q2(ε2q − ω2)

]. (4.234)

This response function has been split into a longitudinal Kl (�δαβ ) and atransverse Kt (�(qαqβ − δαβq2)) part. The longitudinal response to the field(D ‖ q) is expressed in terms of the so-called external conductivity σex:

j l(q, ω) = σex(q, ω)D(q, ω) (4.235)

where D is the external electric field. Using equation (4.234), we find

σex(q, ω) = Kl

iω= iε0ωω2

ps

4π(ω2 − ε2q)

. (4.236)

The conductivity sum rule (1.18) is satisfied as

∫ ∞

0dω Re σex(ω) = ε0ω

2ps

8. (4.237)

The conductivity in the transverse electromagnetic field (D ⊥ q) is given by

σt = i

4πλ2Hω

(4.238)

where

λH =[

m∗∗

16πe2ns

]1/2

. (4.239)

This expression combined with the Maxwell equation describes the Meissner–Ochsenfeld effect in the CBG with the magnetic field penetration depth λH.

4.7.7 Charged vortex and lower critical field

The CBG is an extreme type II superconductor, as shown later. Hence, we cananalyse a single vortex in the CBG and calculate the lower (first) critical field Hc1by solving a stationary equation for the macroscopic condensate wavefunction[120]:{

− [∇ + 2ie A(r)]2

2m∗∗ −µ+ 4e2

ε0

∫dr ′ |ψs(r ′, t)|2 − nb

|r − r ′|}ψs(r, t) = 0. (4.240)

Bipolaronic superconductivity 149

Subtracting nb in the integral of equation (4.240), we explicitly take into accountthe Coulomb interaction with a homogeneous charge background with the samedensity as the density of charged bosons.

The integra-differential equation (4.240) is quite different from theGinsburg–Landau equation (1.43). While the CBG shares quantum coherencewith the BCS superconductors owing to the Bose–Einstein condensate (BEC),the long-range (non-local) interaction leads to some peculiarities. In particular,the vortex is charged in the CBG and the coherence length occurs just the sameas the screening radius.

Indeed, introducing dimensionless quantities f = |ψs |/n1/2b , ρ = r/λ(0)

and h = 2eξ(0)λ(0)∇ × A for the order parameter, length and magnetic field,respectively, equation (4.240) and the Maxwell equations take the following form:

1

κ2ρ

d

dρρ

d f

dρ− 1

f 3

(dh

dρ

)2

− φ f = 0 (4.241)

1

κ2ρ

d

dρρ

dφ

dρ= 1 − f 2 (4.242)

1

ρ

d

dρ

ρ

f 2

dh

dρ= h. (4.243)

The new feature compared with the GL equations of chapter 1 is the electric fieldpotential determined as

φ = 1

2eφc

∫dr ′ v(r − r ′)[|ψs(r ′)|2 − nB] (4.244)

with a new fundamental unit φc = em∗∗ξ(0)2. The potential is calculated usingthe Poisson equation (4.242). At T = 0, the coherence length is the same as thescreening radius,

ξ(0) = (21/2m∗∗ωps)−1/2 (4.245)

and the London penetration depth is

λ(0) =(

m∗∗

16πnBe2

)1/2

. (4.246)

There are now six boundary conditions in a single-vortex problem. Four of themare the same as in the BCS superconductor (section 1.6.3), h = dh/ρ = 0, f = 1for ρ = ∞ and the flux quantization condition, dh/dρ = −p f 2/κρ for ρ = 0,where p is an integer. The remaining two conditions are derived from the globalcharge neutrality, φ = 0 for ρ = ∞ and

φ(0) =∫ ∞

0ρ ln(ρ)(1 − f 2) dρ (4.247)

for the electric field at the origin, ρ = 0. We note that the chemical potential iszero at any point in the thermal equilibrium.

150 Strong-coupling theory

CBG is an extreme type II superconductor with a very large Ginsburg–Landau parameter (κ = λ(0)/ξ(0) � 1). Indeed, with the material parameterstypical for oxides, such as m∗∗ = 10me, nb = 1021 cm−3 and ε0 = 103, weobtain ξ(0) � 0.48 nm, λ(0) � 265 nm, and the Ginsburg–Landau ratio κ � 552.Owing to a large dielectric constant, the Coulomb repulsion remains weak evenfor heavy bipolarons, rs � 0.46. If κ � 1, equation (4.243) is reduced to theLondon equation with the familiar solution h = pK0(ρ)/κ , where K0(ρ) is theHankel function of imaginary argument of zero order (section 1.6.3). For theregion ρ ≤ p, where the order parameter and the electric field differ from unityand zero, respectively, we can use the flux quantization condition to ‘integrateout’ the magnetic field in equation (4.241). This leaves us with two parameter-free equations written for r = κρ:

1

r

d

drr

d f

dr− p2 f

r2− φ f = 0 (4.248)

and1

r

d

drr

dφ

dr= 1 − f 2. (4.249)

They are satisfied by regular solutions of the form f = cprp and φ =φ(0) + (r2/4), when r → 0. The constants cp and φ(0) are determined bycomplete numerical integration of equations (4.248) and (4.249). The numericalresults for p = 1 are shown in figures 4.11 and 4.12, where c1 � 1.5188 andφ(0) � −1.0515.

In the region p � r < pκ , the solutions are f = 1 + (4 p2/r4) andφ = −p2/r2. In this region, f differs qualitatively from the BCS order parameter,fBCS = 1 − (p2/r2) (see also figure 4.11). The difference is due to a local chargeredistribution caused by the magnetic field in the CBG. Being quite different fromthe BCS superconductor, where the total density of electrons remains constantacross the sample, the CBG allows for flux penetration by redistributing thedensity of bosons within the coherence volume. This leads to an increase of theorder parameter compared with the homogeneous case ( f = 1) in the region closeto the vortex core. Inside the core the order parameter is suppressed (figure 4.11)as in the BCS superconductor. The resulting electric field (figure 4.12) (togetherwith the magnetic field) acts as an additional centrifugal force increasing thesteepness (cp) of the order parameter compared with the BCS superfluid, wherec1 � 1.1664, figure 4.11(a).

The breakdown of the local charge neutrality is due to the absence of anyequilibrium normal-state solution in the CBG below the Hc2(T ) line. Bothsuperconducting (�k �= 0) and normal (�k = 0) solutions are allowed at anytemperature in the BCS superconductors (chapter 2). Then the system decideswhich of two phases (or their mixture) is energetically favourable but the localcharge neutrality is respected. In contrast, there is no equilibrium normal-statesolution (with ψs = 0) in CBG below the Hc2(T )-line because it does not respectthe density sum rule. Hence, there are no different phases to mix and the only

Bipolaronic superconductivity 151

Figure 4.11. Vortex core profile in the charged Bose gas, fBEC, (a) compared with thevortex in the BCS superconductor, fBCS, (b).

way to acquire a flux in the thermal equilibrium is to redistribute the local densityof bosons at the expense of their Coulomb energy. This energy determines thevortex free energy � = Ev − E0, which is the difference between the energy ofthe CBG with, Ev , and without, E0, a magnetic flux:

� =∫

dr

{1

2m∗ |[∇ + 2ie A(r)]ψs(r)|2 + eφcφ[|ψs(r)|2 − nb] + (∇ × A)2

8π

}.

(4.250)Using equations (4.241), (4.242) and (4.243), this can be written in dimensionlessform:

F = 2π

∫ ∞

0[h2 − 1

2φ(1 + f 2)]ρ dρ. (4.251)

In the large κ limit, the main contribution comes from the region p/κ < ρ < p,where f � 1 and φ � −p2/(κ2ρ2). The energy is, thus, the same as that inthe BCS superconductor, F � 2πp2 ln(κ)/κ2 (section 1.6.3). It can be seen thatthe most stable solution is the formation of the vortex with one flux quantum,p = 1, and the lower critical field is the same as in the BCS superconductor,hc1 ≈ ln κ/(2κ). However, differing from the BCS superconductor, where theGinsburg–Landau phenomenology is microscopically justified in the temperatureregion close to Tc (section 2.16), the CBG vortex structure is derived here in the

152 Strong-coupling theory

Figure 4.12. Electric field potential φ as a function of the distance (measured in units ofξ(0)) (lower curve) together with the CBG (upper curve) and BCS order parameters (a);its profile is shown in (b).

low-temperature region. In fact, the zero-temperature solution (figure 4.11) isapplied in a wide temperature region well below the Bose–Einstein condensationtemperature, where the depletion of the condensate remains small. The actual sizeof the charged core is about 4ξ (figure 4.12).

4.7.8 Upper critical field in the strong-coupling regime

If we ‘switch off’ the Coulomb repulsion between bosons, an ideal CBG cannot bebose-condensed at finite temperatures in a homogeneous magnetic field becauseof one-dimensional particle motion at the lowest Landau level [7]. However, aninteracting CBG condenses in a field lower than a certain critical value Hc2(T )

[121]. Collisions between bosons and/or with impurities and phonons make themotion three dimensional, and eliminate the one-dimensional singularity of thedensity of states, which prevents BEC of the ideal gas in the field. As we showlater, the upper critical field of the CBG differs significantly from the Hc2(T ) ofBCS superconductors (section 1.6.4). It has an unusual positive curvature near Tc(Hc2(T ) � (Tc − T )3/2) and diverges at T → 0, if there is no localization due toa random potential. The localization can drastically change the low-temperaturebehaviour of Hc2(T ) (part 2).

In line with the conventional definition (section 1.6.4), Hc2(T ) is a field,where the first non-zero solution of the linearized stationary equation for the

Bipolaronic superconductivity 153

macroscopic condensate wavefunction occurs:[− 1

2m∗∗ [∇ − 2ie A(r)]2 + Vscat(r)]

ψs(r) = µψs(r). (4.252)

Here we include the ‘scattering’ potential Vscat(r) caused, for example, byimpurities. Let us first discuss non-interacting bosons, Vscat(r) = 0. Their energyspectrum in a homogeneous magnetic field is (section 1.6.4)

εn = ω(n + 1/2) + k2z

2m∗∗ (4.253)

where ω = 2eHc2/m∗∗ and n = 0, 1, 2, . . . ,∞. BEC occurs when the chemicalpotential ‘touches’ the lowest band edge from below, i.e. µ = ω/2 (appendix B).Hence, quite different from the GL equation (1.90), the Schrodinger equation(4.252) does not allow for a direct determination of Hc2, In fact, it determines thevalue of the chemical potential. Then using this value, the upper critical field isfound from the density sum rule,∫ ∞

Ec

f (ε)N(ε, Hc2) dε = nb (4.254)

where N(ε, Hc2) is the DOS of the Hamiltonian (4.252), f (ε) = [exp(ε−µ)/T −1]−1 is the Bose–Einstein distribution function and Ec is the lowest band edge.For ideal bosons, we have µ = Ec = ω/2 and

N(ε, Hc2) =√

2(m∗∗)3/2ω

4π2Re

∞∑n=0

1√ε − ω(n + 1/2)

. (4.255)

Substituting equation (4.255) into equation (4.254) yields√

2(m∗∗)3/2ω

4π2

∫ ∞

0

dx

x1/2

1

exp(x/T ) − 1= nb − n(T ) (4.256)

where

n(T ) =√

2(m∗∗)3/2ω

4π2

∫ ∞

0

dx

exp(x/T ) − 1

∞∑n=1

1√x − ωn

(4.257)

is the number of bosons occupying the levels from n = 1 to n = ∞. This numberis practically the same as in zero field, n(T ) = nb(T/Tc)

3/2 (see equation (B.61)),if ω � Tc. In contrast, the number of bosons on the lowest level, n = 0, is givenby a divergent integral on the left-hand side of equation (4.256). Hence the onlysolution to equation (4.256) is Hc2(T ) = 0.

The scattering of bosons effectively removes the one-dimensional singularityin N0(ε, Hc2) � ω(ε − ω/2)−1/2 leading to a finite DOS near the bottom of thelowest level,

N0(ε, Hc2) ∝ Hc2√�0(Hc2)

. (4.258)

154 Strong-coupling theory

Using the Fermi–Dirac golden rule, the collision broadening of the lowest level�0(Hc2) is proportional to the same DOS:

�0(Hc2) ∝ N0(ε, Hc2) (4.259)

so that �0 scales with the field as �0(Hc2) � H 2/3c2 . Then the number of bosons

at the lowest level is estimated to be

n0 =√

2(m∗∗)3/2ω

4π2

∫ ∞

�0

dx

x1/2

1

exp(x/T ) − 1� T H 2/3

c2 (4.260)

as long as T � �0. Here we apply the one-dimensional DOS but cut the integralat �0 from below. Finally we arrive at

Hc2(T ) = H0(t−1 − t1/2)3/2 (4.261)

where t = T/Tc and H0 is a temperature-independent constant. The scalingconstant, H0, depends on the scattering mechanism. If we write H0 =�0/(2πξ2

0 ), then the characteristic length is

ξ0 ≈(

l

nb

)1/4

(4.262)

where l is the zero-field mean-free path of low-energy bosons. The upper criticalfield has a nonlinear behaviour:

Hc2(T ) ∝ (Tc − T )3/2

in the vicinity of Tc and diverges at low temperatures as

Hc2(T ) � T −3/2.

These simple scaling arguments are fully confirmed by DOS calculationswith impurity [121] and boson–boson [122] scattering. For impurities, the energyspectrum of the Hamiltonian (4.252) consists of discrete levels (i.e. localizedstates) and a continuous spectrum (extended states). The density of extendedstates N (ε, Hc2) and their lowest energy Ec (the so-called mobility edge, whereN (Ec, Hc2) = 0) can be found, if the electron self-energy �(ε) is known. Wecalculate �(ε) in the non-crossing approximation of section 3.3 considering theimpurity scattering as the elastic limit of the phonon scattering:

�ν(ε) =∑ν ′

V 2νν ′

ε − εν ′ − �ν ′(ε). (4.263)

To obtain analytical results, we choose the matrix elements of the scatteringpotential as Vνν ′ = V δnn′ . Here ν ≡ (n, kx , kz) are the quantum numbers ofa charge particle in the magnetic field. Then the DOS is found to be

N (ε, Hc2) = 1

πV 2

∑n

Im �n(ε). (4.264)

Bipolaronic superconductivity 155

Integrating over kz in equation (4.263) yields a cubic algebraic equation for theself-energy �n(ε) of the nth level:

�n(ε) = dV 2

2π

∫ ∞

−∞dkz

1

ε − ω(n + 1/2) − k2z /(2m∗∗) − �n(ε)

= eHc2V 2√

m∗∗

π√

2

i√ε − ω(n + 1/2) − �n(ε)

. (4.265)

Solving this equation, we obtain

N (ε, H ) = eHc2

4π2

√6m∗∗�0

×∞∑

n=0

ε3

n

27+ 1

2+

√ε3

n

27+ 1

4

1/3

− ε3

n

27+ 1

2−

√ε3

n

27+ 1

4

1/3

(4.266)

and the mobility edge

Ec = ω

2− 3�0

22/3 . (4.267)

Here

�0 = 1

2

(2V 2eH

√m∗∗

π

)2/3

is the collision broadening of levels and εn = [ε − ω(n + 1/2]/�0.With this DOS in the sum rule (equation (4.254)), we obtain the same Hc2(T )

for T � �0 as through the scaling, equation (4.261), and

ξ0 ≈ 0.8

(l

nb

)1/4

. (4.268)

The zero-field mean free path l is expressed via microscopic parameters asl = π/(V m∗∗)−2. The ‘coherence’ length ξ0 of the CBG (equation (4.268))depends on the mean free path l and the inter-particle distance n−1/3

b . It has littleto do with the size of the bipolaron and could be as large as the coherence lengthof weak-coupling BCS superconductors.

Thus Hc2(T ) of strongly-coupled superconductors has a ‘3/2’ curvaturenear Tc which differs from the linear weak-coupling Hc2(T ) of section 1.6(figure 4.13). The curvature is a universal feature of the CBG, which does notdepend on a particular scattering mechanism and on any approximations. Anotherinteresting feature of strongly-coupled superconductors is a breakdown of thePauli paramagnetic limit given by Hp � 1.84Tc in the weak-coupling theory(section 1.6.4). The Hc2(T ) of bipolarons exceeds this limit because the singlet

156 Strong-coupling theory

�������

���������

����

��

�

����

������

Figure 4.13. BEC critical field of a CBG compared with the Hc2(T ) of BCSsuperconductors.

bipolaron binding energy � is much larger than their Tc. Bosons are condensedat T = 0 no matter what their energy spectrum is. Hence, in the CBG model,Hc2(0) = ∞ (figure 4.13). For composed bosons, like bipolarons, the pair-breaking limit is given by µB Hc2(0) ≈ �, so that Hc2(0) � Hp.

4.7.9 Symmetry of the order parameter

The anomalous Bogoliubov average

Fss ′(r1, r2) = 〈〈�s(r1)�s ′(r2)〉〉is the superconducting order parameter both in the weak- and strong-couplingregimes. It depends on the relative coordinate ρ = r1 − r2 of two electronsof the pair and on the centre-of-mass coordinate R = (r1 + r2)/2. Hence, itsFourier transform, f (k, K ), depends on the relative momentum k and on thecentre-of-mass momentum, K . In the BCS theory, K = 0 (in a homogeneoussuperconductor) and the Fourier transform of the order parameter is proportionalto the gap in the quasi-particle excitation spectrum, f (k, K ) � �k (section 2.2).Hence, the symmetry of the order parameter and the symmetry of the gap are thesame in the weak-coupling regime. Under the rotation of the coordinate system,�k changes its sign if the Cooper pairing is unconventional (section 2.10). In thiscase, the BCS quasi-particle spectrum is gapless.

In the bipolaron theory, the symmetry of the BEC is not necessarily thesame as the ‘internal’ symmetry of a pair [123]. While the latter describes thetransformation of f (k, K ) with respect to the rotation of k, the former (‘external’)symmetry is related to the rotation of K . Therefore, it depends on the bipolaronband dispersion but not on the symmetry of the bound state. As an example, letus consider a tight-binding bipolaron spectrum comprising two bands on a square

Bipolaronic superconductivity 157

�

�

�

��

�

Figure 4.14. d-wave condensate wavefunction, in the Wannier representation. The orderparameter has different signs in the shaded cells and is zero in the blank cells.

lattice with the period a = 1:

E xK = t cos(Kx) − t ′ cos(Ky) (4.269)

E yK = − t ′ cos(Kx) + t cos(Ky).

They transform into one another under π/2 rotation. If t, t ′ > 0, ‘x’ bipolaronband has its minima at K = (±π, 0) and the y-band at K = (0,±π). Thesefour states are degenerate, so that the condensate wavefunction ψs(m) in the sitespace, m = (mx , my), is given by

ψs(m) = N−1/2∑

K=(±π,0),(0,±π)

bK e−iK ·m. (4.270)

where bK = ±√ns are c-numbers at T = 0. The superposition (4.270) respects

the time-reversal and parity symmetries, if

ψ±s (m) = √

ns[cos(πmx) ± cos(πmy)]. (4.271)

The two order parameters (equation (4.271)) are physically identical becausethey are related by the translation transformation, ψ+

s (mx , my) = ψ−s (mx , my +

1). Both have a d-wave symmetry-changing sign, when the lattice is rotated byπ/2 (figure 4.14). The d-wave symmetry is entirely due to the bipolaron energydispersion with four minima at K �= 0. When the bipolaron spectrum is notdegenerate and its minimum is located at the � point of the Brillouin zone, thecondensate wavefunction is s-wave with respect to the centre-of-mass coordinate.The symmetry of the gap has little to do with the symmetry of the order parameterin the strong-coupling regime. The one-particle excitation gap is half of the

158 Strong-coupling theory

bipolaron binding energy, �/2, and does not depend on any momentum in zeroorder of the polaron bandwidth, i.e. it has an ‘s’-wave symmetry. In fact, dueto a finite dispersion of polaron and bipolaron bands (sections 4.3 and 4.6), theone-particle gap is an anisotropic s-wave. A multi-band electron structure caninclude bands only weakly coupled with phonons which could overlap with thebipolaronic band (see also part 2). In this case, the CBG coexists with the Fermigas just as 4He bosons coexist with 3He fermions in the mixture of He-4 and He-3.The one-particle excitation spectrum of such mixtures is gapless.

PART 2

APPLICATIONS TO HIGH-T c

SUPERCONDUCTORS

Chapter 5

Competing interactions in unconventionalsuperconductors

5.1 High-T c superconductors: different concepts

Nowadays there are many complex high-Tc superconductors including copper[1, 127] and a few other oxides, doped fullerenes MxC60 [128, 129] (M is analkali metal) and the more recently discovered magnesium diborade MgB2 [130]with a critical temperature above 30 K, figure 5.1.

These discoveries have broken all constraints on the maximum Tc predictedby conventional theory of low-temperature superconducting metals and theiralloys. As highlighted by Simon [132], the canonical theory has not provideda ‘materials’ aspect in the search for new high-Tc superconductors. Nevertheless,in a phenomenological sense, any superconductivity can just be treated as aconsequence of the formation of pairs and their condensation—even within theBCS framework, as was noted by Ginzburg [133] in 1968 (chapter 1). At anyvalue of the e–ph interaction superconductivity is a correlated many-body state ofpairs which is well described by BCS theory in the weak-coupling regime λ � 1,

where pairing takes place in the momentum space due to the Pauli exclusionprinciple (collective pairing) and by the bipolaron theory in the strong couplingregime, λ � 1, where pairing is individual (upper half-plane in figure 5.2). Hence,knowledge of the carrier-pairing mechanism and of the nature of the normal stateis central to an understanding of high-Tc supeconductivity.

In general, the bosonic field, which ‘glues’ two carriers together, can not onlybe ‘phononic’, as in BCS and bipolaron theory, but also ‘excitonic’ [133, 134],‘plasmonic’ [135, 136] and/or of magnetic origin [137, 138]. BCS theory likeany mean-field theory is rather universal, so that it describes the cooperativequantum phenomenon of superconductivity well even with these non-phononicattraction mechanisms, if the coupling is weak (see left-hand side of figure 5.2).If the coupling is strong, magnetic interaction could result in the spin–bipolaronformation as suggested by Mott [139]. The main motivation behind these concepts

161

162 Competing interactions in unconventional superconductors

Figure 5.1. Towards a century of superconductivity. A plot of the evolution of thesuperconducting transition temperature, from 1911, to the present situation. We alsoinclude the characteristic temperatures of a variety of cryogenic liquids, as well as thelowest recorded ground temperature on Earth (−89.2 ◦C) [131].

High-Tc superconductors: different concepts 163

Figure 5.2. BSC-like theories of high-Tc superconductors with the electron–phonon (top)and electron–electron (bottom) interactions (left-hand side). Right-hand side: non-Fermiliquid theories with electron–phonon (top) and electron–electron (bottom) correlations.

is that high Tc might be achieved by replacing low-frequency acoustic phononsin conventional BCS theory by higher-frequency bosonic modes, such as theelectron plasmon (chapter 3) or by spin waves (pseudomagnons). However, theCoulomb pseudopotential creates a serious problem for any of these non-phononicmechanisms. With the increasing energy scale of the bosonic field, the retardationof the effective attraction vanishes and the Tolmachev logarithm (section 3.5)cannot be applied. Then the direct Coulomb repulsion would definitely preventany pairing in these models because the exchange electron–electron interaction isalways weaker than direct repulsion. The problem is particularly grave in novelsuperconductors, where the carrier density is low, and the screening of the directCoulomb repulsion is poor.

Therefore, some authors [140, 141] dogmatized that the interaction in novelsuperconductors is essentially repulsive and unretarded and that it also provideshigh Tc without any phonons. A motivation for this concept can be foundin the earlier work by Kohn and Luttinger [57], who showed that the Cooperpairing of repulsive fermions is possible. But the same work clearly showedthat the Tc of repulsive fermions is extremely low, well below the mK scale(section 3.6). Nevertheless, BCS and BCS-like theories (including the Kohn–Luttinger consideration) rely heavily on the Fermi-liquid model of the normalstate. This model fails in many high-temperature superconductors (chapter 6).There are no obvious a priori reasons for discarding the dogma, if the normalstate is not the Fermi liquid. There is little doubt that strong on-site repulsivecorrelations (Hubbard U ) are an essential feature of the cuprates. Indeed allundoped parent compounds are insulators with an insulating gap of about 2 eVor so. But if the repulsive correlations are weak, one would expect a metallicbehaviour for the half-filled d-band of copper in cuprates or, at most, a muchsmaller gap caused by lattice and spin distortions (i.e. due to charge and/or spindensity waves [142]). Therefore, it is the strong on-site repulsion of d-electronsin cuprates which results in their parent ‘Mott’ insulating state (section 5.2).

164 Competing interactions in unconventional superconductors

Differing from conventional band-structure insulators with completely filled orempty Bloch bands, the Mott insulator arises from a potentially metallic half-filledband due to the Coulomb blockade of electron tunnelling to neighbouring sites,if U > zT (a) [143]. The insulator is antiferromagnetic with one hole and spin- 1

2per site. In using this model, we have to realize that the insulating properties ofthe Mott insulator do not depend on the ordering of the spins; they persist abovethe Neel temperature and arise because the on-site Coulomb repulsion is largerthan the half-bandwidth.

When on-site correlations are strong and dimensionality is low, there is analternative to the usual Fermi-liquid description proposed by Anderson [140].In Anderson’s resonating-valence-bond (RVB) theory, the ground state supports‘topological solitons’ (the so-called spinons and holons), such as occur in one-dimensional models like the one-dimensional Hubbard model (see later). Themain idea is that an electron injected into a two-dimensional layer decaysinto a singlet charge e component (holon) and a spin- 1

2 component (spinon)and, conversely, must form again in order to come out. This is the casefor one-dimensional repulsive electrons, which form the so-called Luttingerliquid in one dimension. Bose quasi-particles imply a condensate. However,there is no experimental evidence for a charge e superfluid. Therefore, inthe so-called interlayer RVB extension of the model [144], it was suggestedthat the superconductivity of copper-based high-Tc materials is due to holon-pair tunnelling between the copper–oxygen planes. There is no single-particlecoherent tunnelling between two spinon–holon planes above Tc. However,there is a coherent two-particle tunnelling between them below Tc. Then thecorresponding kinetic energy should be responsible for the BCS-like pairing attemperatures below Tc ≈ t2⊥/t and for the plasma-like gap, observed in the c-axis conductivity. Here t and t⊥ are the in-plane and out-of-plane renormalizedhopping integrals, respectively. Anderson argued that the existence of the upperHubbard band (section 5.2) would necessarily lead to the Luttinger liquid even intwo dimensions, as opposed to the Fermi liquid. While the interlayer RVB modelwas found to be incompatible with experiments [145], the basic idea of spin andcharge separation had been worked out in great detail [146]. The microscopicHubbard Hamiltonian [147]

H = T (a)∑

〈mn〉,s[c†

mscns + H.c.] + U∑

m

nm↑nm↓ (5.1)

was proposed to justify the RVB concept, where 〈mn〉 are the nearest neigbours.The Hamiltonian describes the antiferromagnetic Mott insulator at half fillingwhen U > D. In the strong correlation limit, U � T (a), the doubly occupiedsites take a large Coulomb energy and the Hubbard Hamiltonian can be reducedto the so-called t–J model [148]

H = T (a)∑

〈mn〉,s[c†

ms cns + H.c.] + J∑〈mn〉

(Sm Sn − 14 nmnn) (5.2)

High-Tc superconductors: different concepts 165

where the projected electron operators cm↑ = cm↑(1 − nm↓) act in the subspacewithout double occupancy, nm = nm↑ + nm↓,

Sαm = 1

2

∑ss ′

c†ms(τα)ss ′ cms ′

are three components of the on-site spin- 12 operator (α = 1, 2, 3) and τα are the

Pauli matrices as in equation (3.73). The second term in H describes the spin- 12

Heisenberg antiferromagnet with the exchange energy

J = 4T 2(a)

U(5.3)

for the nearest neigbours. The exchange energy leads to spin polarons similar tolattice polarons but dressed with magnetic fluctuations rather than with phonons.However, the non-fermionic commutation relations for the projected electronoperators lead to an additional kinematic interaction between carriers doped intothe Mott insulator. As a result an analytical solution of the t–J model is a veryhard problem but not in the dilute limit. In this limit a doped hole dressed byantiferromagnetic spin fluctuations propagates coherently in a narrow band ofthe order of J � T (a) like a small polaron. One could believe that the samespin fluctuations, which dress a single hole, mediate a superconducting pairing ofspin polarons due to an effective attractive interaction of the order of J . To beconsistent with the Neel temperature of parent undoped cuprates and with theirspin-wave excitation spectrum, J should be of the order of 0.1 eV. The magneticsinglet pairing might be effective in the d-channel (section 2.10), where thedirect Coulomb repulsion is diminished due to the d-wave symmetry of the BCSorder parameter �k . The possibility of d-wave BCS-like superconductivity inthe two-dimensional Hubbard model near half-filling was suggested by Scalapinoet al [149] concurrently with the discovery of novel superconductors. There isnow a variety of phase-sensitive experiments, which support the unconventionalsymmetry of the order parameter in some cuprates, while other experimentsappear to contradict this symmetry (see section 8.2).

Independent of any experimental evidence, the Hubbard U and t–J modelsshare an inherent difficulty in determining the order. While some groupsclaimed that they describe high-Tc superconductors at finite doping, other authorscould not find any superconducting instabilty without an additonal (i.e. e–ph)interaction [150]. Therefore, it has been concluded that models of this kindare highly conflicting and confuse the issue by exaggerating the magnetismrather than clarifying it [151]. In our view [12], the problem with the RVB-likeconcepts of high-Tc is that for them to be valid, the diffractive scattering betweenholes of cuprate superconductors needs to be absent. A known case wherediffractive scattering is absent is that of one-dimensional interacting fermionsystems. But in one-dimensional interacting fermion systems the scattering isnon-diffractive due to topological reasons. In two and higher dimensions, one

166 Competing interactions in unconventional superconductors

does not have such a topological constraint. On the experimental side, there aremeasurements of out-of-plane resistivity ρc [152, 153] which show the metallic-like temperature dependence of ρc and the mean-free path in the c-directioncomparable with the interplane spacing in a number of high-Tc cuprates. Well-developed interlayer tunnelling invalidates these and any other low-dimensionalconcepts (see figure 5.2), which definitely fail to account for dρc/dT > 0 andhigh-Tc of highly oxygenated cuprates, doped fullerenes and MgB2.

Another problem of the microscopic Hamiltonians (equations (5.1) and(5.2)), which are behind the RVB concept, is that they neglect the long-range Coulomb and electon-phonon interactions, which are essential in novelsuperconductors (section 5.2). As a result, some exact solutions of various limitsand numerical studies on t–J and Hubbard models led to the conclusion thatthe electronic structure in cuprates is much more prone to inhomogeneity andintermediate-scale structures such as stripes of hole-rich domains separated byinsulating antiferromagnetic domains. Moreover, it has been proposed that thestripes are essential to a high-Tc mechanism [141], especially in the underdopedregime. In this scenario, stripe formation permits hole delocalization in onedirection, but hole motion transverse to the stripe is still restricted. It is thusfavourable for the holes to pair so that the pairs can spread out somewhat intothe antiferromagnetic neighbourhood of the stripe, where their interaction isallegedly attractive due to spin fluctuations. The proximity effect in conventionalsuperconductor–normal metal structures is a prototypical example of such amechanism of pairing: when the BCS superconductor and a normal metal areplaced in contact with each other, electrons in the metal pair even if the interactionbetween them is repulsive. As we discuss in section 8.6, this ‘stripe’ scenario isincompatible with Coulomb’s law: there are no stripes if the long-range Coulombrepulsion is properly taken into account. On the experimental side, recent neutron[154] and X-ray [155] spectroscopic studies did not find any bulk charge/spinsegregation in the normal state of a few cuprates suggesting an absence of in-plane carrier density modulations in these materials above Tc.

Strong repulsive correlations between holes could provide another novelmechanism for Cooper pairing, i.e. the kinetic-energy-driven mechanism ofsuperconductivity proposed by Hirsch [156] (see figure 5.2). Such a mechanismdoes not require any dynamic attraction between holes. The qualitativeexplanation is as follows: electrons in metals are ‘dressed’ by a cloud of otherelectrons with which they interact and form the so-called electronic polarons.The dressing causes an increase in the electron’s effective mass and when thedressing is large, the metal is a poor conductor. If, however, the electronsmanage to ‘undress’ when the temperature is lowered, their effective mass will bereduced and electricity will flow easily. A model Hamiltonian, which describesthe ‘undressing’, is that of small polarons with a nonlinear interaction with abackground bosonic degree of freedom which gives rise to an effective massenhancement that depends on the local charge occupation. The undressing processcan only occur if the carriers are ‘holes’ rather than electrons and when two hole

High-Tc superconductors: different concepts 167

carriers form a pair. High-precision optical conductivity measurements [157]observed a temperature-dependent change in the conductivity sum rule (1.18)in the optimally doped Bi-cuprate integrated up frequencies a few orders ofmagnitude greater than Tc. According to [157], this observation implies a changein the kinetic energy anticipated in the theory of ‘hole’ superconductivity [156]and in the bipolaron theory [13].

Less exotic concepts are based on the Fermi-liquid and BCS-like approachbut take into account the van Hove singularities of the electron density of states[158], and/or a singular interaction between holes due to a closeness to a quantumphase transition of any type [138, 142, 159, 160]. In particular, Pines and hisco-workers argued that the magnetic interaction between planar quasi-particlesin cuprate superconductors leads to a new quantum state of matter, the nearlyantiferromagnetic Fermi liquid (NAFL). It possesses a well-defined Fermi surfacebut it is not the ordinary Landau–Fermi liquid, because of a singular interactionbetween electrons with their Fourier component peaked at some momentumcaused by soft magnetic fluctuations. The model clearly contradicts the neutrondata by Smith et al [161], obtained on the high flux reactor at Grenoble. Thequasi-elastic peak and diffuse magnetic scattering are virtually absent in themetallic phase of YBa2Cu3O7−δ in a wide energy range up to 30 meV. It followsfrom these data that the local magnetic moments are practically absent fromoptimally doped cuprates. Some authors attribute high Tc and non-Fermi liquidfeatures of the cuprates to the proximity of the Fermi level to a van Hovesingularity in the density of states. However, other authors [162] concludein the framework of the Eliashberg theory (chapter 3), that this model yieldsTc � 10 K for both La2−xSrxCuO4 and YBa2Cu3O7−δ , when its parameters areconstrained by neutron-scattering and transport measurements. Hence, the vanHove singularity scenario enhances Tc much less effectively than weak-couplingad hoc calculations would suggest.

There are several semi-phenomelogical concepts for the behaviour ofa non-Fermi liquid of cuprates. In particular, it is conceivable that novelsuperconductors are marginal Fermi liquids (MFL) [163] in the sense that theenergy interval near the Fermi level in which the Landau quasi-particles exist issmall compared with Tc. Outside the interval a linear energy dependence of thequasi-particle scattering rate has been postulated, which allows for a descriptionof optical responce in some cuprates. A linear frequency dependence of theimaginary part of the electron self-energy does not follow from the e–ph orCoulomb interactions (see chapter 3), where it is cubic or quadratic, respectively.Therefore, an MFL needs a singular interaction like NAFL, which might becaused by a closeness to a quantum phase transition of any type.

Other authors [141,164] dismiss any real-space pairing claiming that pairingis collective even in underdoped cuprates. They believe in a large Fermi sur-face with the number of holes (1 + x) rather than x in superconducting cuprates,where x is the doping level like in La2−xSrxCuO4. As an alternative to a three-dimensional Bose–Einstein condensation of bipolarons (chapter 4), these authors

168 Competing interactions in unconventional superconductors

suggest a collective pairing (i.e. the Cooper pairs in the momentum space) at sometemperature T ∗ > Tc but without phase ordering. In this concept, the phase coher-ence and superconducting critical temperature Tc are determined by the superfluiddensity, which is proportional to doping x due to a low dimensionality, rather thanto the total density of carriers (1 + x). However, a large Fermi surface is clearlyincompatible with a great number of thermodynamic, magnetic and kinetic mea-surements, which show that only holes doped into a parent insulator are carriers inthe normal state (chapter 6). On theoretical grounds this preformed Cooper-pair(or phase-fluctuation) scenario contradicts a theorem [165], which proves that thenumber of supercarriers (at T = 0) and normal-state carriers is the same in anyclean superfluid. It also contradicts a parameter-free estimate of the Fermi energyand Tc in the cuprates, as explained in sections 5.3 and 7.1, respectively.

For although high-temperature superconductivity has not yet been targetedas ‘the shame and despair of theoretical physics’—a label attributed tosuperconductivity during the first half-century after its discovery—the parlousstate of current theoretical constructions has led to a current consensus that thereis no consensus on the theory of high-Tc superconductivity. Our view, whichwe discuss in the rest of the book, is that the extension of BCS theory towardsthe strong interaction between electrons and ion vibrations (chapter 4) describesthe phenomenon naturally and that high-temperature superconductivity exists inthe crossover region of the electron–phonon interaction strength from the BCS-like to bipolaronic superconductivity as was predicted before [11], and exploredin greater detail after discovery [13, 64, 166–169]. Experimental [170–177]and theoretical [64, 84, 178–182] evidence for an exceptionally strong electron–phonon interaction in high temperature superconductors is now so overwhelming,that even advocates of a non-phononic mechanism [141] accept this fact.

5.2 Band structure and essential interactions in cuprates

In this book we take the view that cuprates and related transition metal oxidesare charge transfer Mott insulators at any level of doping [183]. As establishedin a few site-selective experiments [184] and in first-principle (the so-called‘LDA+U’) band-structure calculations, their one-particle DOS is schematicallyrepresented by figure 5.3. Here a d-band of a transition metal is split into thelower and upper Hubbard bands due to on-site repulsive interaction (the HubbardU), while the first band to be doped is the oxygen band lying within the Hubbardgap. The oxygen band is less correlated and completely filled in parent insulators.As a result, a single oxygen hole has well-defined quasi-particle properties in theabsence of interactions with phonons and fluctuations of d-band spins. A strongcoupling with phonons, unambiguously established for many oxides, should leadto a high-energy spectral weight in the spectral function of an oxygen hole atenergies about twice the Franck–Condon (polaron) level shift, E p , and to a band-narrowing effect (section 4.3).

Band structure and essential interactions in cuprates 169

Figure 5.3. Density of states in cuprates. The chemical potential is inside the chargetransfer gap due to bipolaron formation (a) and bipolarons coexist with thermally excitednon-degenerate polarons. It might enter the oxygen band in overdoped cuprates (b), wherebipolarons may coexist with unpaired degenerate polarons.

The e–ph interaction also binds holes into inter-site oxygen bipolarons thesize of a lattice constant (section 4.4). The bipolaron density remains relativelylow (below 0.15 per cell) at any relevant level of doping and the residual repulsiveinteraction of bipolarons is strongly suppressed by the lattice polarization owingto a large static dielectric constant (section 4.4). That is why bipolarons in oxidesare fairly well described by the charged Bose-gas model of section 4.7. Oneof their roles in the one-particle DOS is to pin the chemical potential inside thecharge transfer gap, half the bipolaron binding energy above the oxygen bandedge, shifted by E p, figure 5.3. This binding energy as well as the singlet–triplet bipolaron exchange energy are thought to be the origin of normal-statepseudogaps, as first proposed by us [185] (sections 6.4 and 6.5). In overdopedsamples, carriers screen part of the e–ph interaction with low-frequency phonons.Hence, the binding energy and the hole spectral function depend on doping. Inparticular, the bipolaron and polaron bands could overlap because the bipolaronbinding energy becomes smaller [186], so the chemical potential could enter theoxygen band in the overdoped cuprates, figure 5.3(b). Then a Fermi-level crossingcould be seen in the angle-resolved photoemission spectra (ARPES) (section 8.4).

To assess quantitatively the role of the Frohlich e–ph interaction, we applyan expression for E p, which depends only on the experimentally measured high-frequency, ε∞, and static, ε0, dielectric constants of the host insulator as [187]

E p = 1

2κ

∫BZ

d3q

(2π)3

4πe2

q2 (5.4)

where κ−1 = ε−1∞ − ε−10 . Here the size of the integration region, which is the

Brillouin zone (BZ) is determined by the lattice constants a, b, c. As shown in

170 Competing interactions in unconventional superconductors

Table 5.1. Polaron shift E p due to the Frohlich interaction. Data taken from the 1997Handbook of Optical Constants of Solids ed E D Palik (New York: Academic). The valueε∞ = 5 for WO3 is an estimate.

System ε∞ ε0 a × b × c (A3) Ep (eV)

SrTiO3 5.2 310 3.9053 0.852BaTiO3 5.1–5.3 1499 3.9922 × 4.032 0.842BaBiO3 5.7 30.4 4.342 × 4.32 0.579La2CuO4 5.0 30 3.82 × 6a 0.647LaMnO3 3.9b 16b 3.863 0.884La2−2x Sr1+2x -Mn2O7 4.9b 38b 3.862 × 3.9c 0.807NiO 5.4 12 4.183 0.429TiO2 6–7.2 89–173 4.592 × 2.96 0.643MgO 2.964 9.816 4.21473 0.982CdO 5.4 21.9 4.73 0.522WO3 5 100–300 7.31 × 7.54 × 7.7 0.445NaCl 2.44 5.90 5.6433 0.749EuS 5.0 11.1 5.9683 0.324EuSe 5.0 9.4 6.19363 0.266

a Distance between CuO2 planes.b Ishikawa T private communication.c Distance between MnO2 planes.

table 5.1, the Frohlich interaction alone provides the binding energy of two holes(≈2E p) almost by one order of magnitude larger than the magnetic interactionJ ∼ 0.1 eV of the t–J model. The data in the Table represent lower boundariesfor the polaron shift, since the deformation potential and/or the molecular-type(Jahn–Teller) e–ph interaction are not included in equation (5.4). There isvirtually no screening of c-axis polarized optical phonons in cuprates becausean upper limit for the out-of-plane plasmon frequency (�200 cm−1 [188]) is wellbelow the phonon frequency (section 4.4). Hence, equation (5.4) is perfectlyapplied at any doping.

The polaron shift about 1 eV results in the formation of small polarons nomatter which criterion for their formation is applied. The bare half-bandwidthis about 1 eV or less in cuprates, so even a naive variational criterion of thesmall polaron formation (E p > D) could be satisfied. Moreover, optical phononfrequencies are high in oxides, fullerenes and MgB2, ω0 � 0.1 eV. In thisnon-adabatic or intermediate regime, the variational criterion for small polaronformation fails. The correct criterion for small polaron formation is determinedby the convergence of the 1/λ perturbation expansion as discussed in chapter 4.

Exact Monte Carlo calculations [94] show that the polaron theory based on

Low Fermi energy: pairing is individual in many cuprates 171

this expansion describes quantitatively both strong-coupling (small polaron) andweak-coupling (large polaron) regimes without any restriction on the value ofE p for a long-range electron–phonon interaction with high-frequency phonons.The effective polaron–polaron attraction due to the overlap of the deformationfields is about 2E p (section 4.4). It is sufficient to overcome the inter-siteCoulomb repulsion at short distances, in particular if a weaker interaction withthe deformation potential is taken into account. We show later that the adiabaticratio in all novel superconductors is of the order or even larger than one for mostessential optical phonons.

5.3 Low Fermi energy: pairing is individual in many cuprates

First-principle band-structure calculations show that copper, alkali metals andmagnesium donate their outer electrons to oxygen, C60, and boron in cuprates,doped fullerenes, and in MgB2, respectively. In cuprates and MgB2, the bandstructure is quasi-two-dimensional with a few degenerate hole pockets. Applyingthe parabolic approximation for the band dispersion we obtain the renormalizedFermi energy as

εF = πni d

m∗i

(5.5)

where d is the interplane distance, and ni , m∗i are the density of holes and their

effective mass in each of the hole sub-bands i renormalized by the electron–phonon (and by any other) interaction. One can express the renormalized band-structure parameters through the in-plane magnetic-field penetration depth atT � 0, measured experimentally:

λ−2H = 4πe2

∑i

ni

m∗i. (5.6)

As a result, we obtain a parameter-free expression for the ‘true’ Fermi energy as

εF = d

4ge2λ2H

(5.7)

where g is the degeneracy of the spectrum, which is g = 2 in MgB2. g maydepend on doping in cuprates. One expects four hole pockets inside the Brillouinzone (BZ) due to the Mott–Hubbard gap in underdoped cuprates. If the hole bandminima are shifted with doping to BZ boundaries, all their wavevectors wouldbelong to the stars with two or more prongs. The groups of wavevectors for thesestars have only one-dimensional representations. It means that the spectrum willbe degenerate with respect to the number of prongs which the star has, i.e. g � 2.The only exception is the minimum at k = (π, π) with one prong and g = 1.Hence, in cuprates the degeneracy is 1 � g � 4. Because equation (5.7) doesnot contain any other band-structure parameters, the estimate of εF using this

172 Competing interactions in unconventional superconductors

equation does not depend very much on the parabolic approximation for the banddispersion.

Generally, the ratios n/m∗ in equations (5.5) and (5.6) are not necessarily thesame. The ‘superfluid’ density in equation (5.6) might be different from the totaldensity of delocalized carriers in equation (5.5). However, in a translationallyinvariant system they must be the same [165]. This is also true even in theextreme case of a pure two-dimensional superfluid, where quantum fluctuationsare important. One can, however, obtain a reduced value for the zero-temperaturesuperfluid density in the dirty limit, l � ξ(0), where ξ(0) is the zero-temperaturecoherence length. The latter was measured directly in cuprates as the size of thevortex core. It is about 10 A or even less. In contrast, the mean free path wasfound to be surprisingly large at low temperatures, l ∼ 100–1000 A. Hence, it israther probable that all novel superconductors, including MgB2 are in the cleanlimit, l � ξ(0), so that the parameter-free expression for εF, equation (5.7), isperfectly applicable.

A parameter-free estimate of the Fermi energy obtained by usingequation (5.7) is presented in table 5.2. The renormalized Fermi energy inmagnesium diboride and in more than 30 cuprates is less than 100 meV, if thedegeneracy g ≥ 2 is taken into account. This should be compared with thecharacteristic phonon frequency, which is estimated as the plasma frequency ofboron or oxygen ions,

ω0 =√

4π Z2e2 N

M. (5.8)

With Z = 1, N = 2/Vcell, M = 10 au, one obtains ω0 � 69 meV for MgB2, andω0 = 84 meV with Z = 2, N = 6/Vcell, M = 16 au for YBa2Cu3O6. Here Vcellis the volume of the (chemical) unit cell. The estimate agrees with the measuredphonon spectra. As established experimentally in cuprates (section 5.1), thehigh-frequency phonons are strongly coupled with carriers. The parameter-freeexpression (5.7) does not apply to doped fullerenes with their three-dimensionalenergy spectrum. However, it is well established that they are also in the non-adiabatic regime [190], εF � ω0.

A low Fermi energy is a serious problem for the Migdal–Eliashberg approach(chapter 3). Since the Fermi energy is small and phonon frequencies are high,the Coulomb pseudopotential µ∗

c is of the order of the bare Coulomb repulsion,µ∗

c � µc � 1 because the Tolmachev logarithm is ineffective. Hence, to get anexperimental Tc, one has to have a strong coupling, λ > µc. However, one cannotincrease λ without accounting for the polaron collapse of the band (chapter 4).Even in the region of the applicability of the Eliashberg theory (i.e. at λ ≤ 0.5),the non-crossing diagrams cannot be treated as vertex corrections as in [191],since they are comparable to the standard terms, when ω0/εF � 1. Because novelsuperconductors are in the non-adiabatic regime, interaction with phonons mustbe treated within the multi-polaron theory (chapter 4) at any value of λ.

In many cases (table 5.2), the renormalized Fermi energy is so small that

Low Fermi energy: pairing is individual in many cuprates 173

Table 5.2. The Fermi energy (multiplied by the degeneracy) of cuprates and MgB2 [189].

Compound Tc (K) λH,ab (A) d (A) gεF (meV)

La1.8Sr0.2CuO4 36.2 2000 6.6 112La1.78Sr0.22CuO4 27.5 1980 6.6 114La1.76Sr0.24CuO4 20.0 2050 6.6 106La1.85Sr0.15CuO4 37.0 2400 6.6 77La1.9Sr0.1CuO4 30.0 3200 6.6 44La1.75Sr0.25CuO4 24.0 2800 6.6 57YBa2Cu3O7 92.5 1400 4.29 148YBaCuO(2%Zn) 68.2 2600 4.29 43YBaCuO(3%Zn) 55.0 3000 4.29 32YBaCuO(5%Zn) 46.4 3700 4.29 21YBa2Cu3O6.7 66.0 2100 4.29 66YBa2Cu3O6.57 56.0 2900 4.29 34YBa2Cu3O6.92 91.5 1861 4.29 84YBa2Cu3O6.88 87.9 1864 4.29 84YBa2Cu3O6.84 83.7 1771 4.29 92YBa2Cu3O6.79 73.4 2156 4.29 62YBa2Cu3O6.77 67.9 2150 4.29 63YBa2Cu3O6.74 63.8 2022 4.29 71YBa2Cu3O6.7 60.0 2096 4.29 66YBa2Cu3O6.65 58.0 2035 4.29 70YBa2Cu3O6.6 56.0 2285 4.29 56HgBa2CuO4.049 70.0 2160 9.5 138HgBa2CuO4.055 78.2 1610 9.5 248HgBa2CuO4.055 78.5 2000 9.5 161HgBa2CuO4.066 88.5 1530 9.5 274HgBa2CuO4.096 95.6 1450 9.5 305HgBa2CuO4.097 95.3 1650 9.5 236HgBa2CuO4.1 94.1 1580 9.5 257HgBa2CuO4.101 93.4 1560 9.5 264HgBa2CuO4.101 92.5 1390 9.5 332HgBa2CuO4.105 90.9 1560 9.5 264HgBa2CuO4.108 89.1 1770 9.5 205MgB2 39 1400 3.52 122

pairing is certainly individual, i.e. the bipolaron radius (equation (4.97)), issmaller than the inter-carrier distance. Indeed, this is the case, if

εF � π�. (5.9)

The bipolaron binding energy is thought to be twice the so-called pseudogapexperimentally measured in the normal state of many cuprates (section 6.5),� � 100 meV, so that equation (5.9) is well satisfied in underdoped and even

174 Competing interactions in unconventional superconductors

in a few optimally and overdoped cuprates. One should note that the coherencelength in the charged Bose gas has nothing to do with the size of the boson. Itdepends on the inter-particle distance and the mean-free path, equation (4.262),and might be as large as in the BCS superconductor. Hence, it would be incorrectto apply the ratio of the coherence length to the inter-carrier distance as a criterionof the BCS–Bose liquid crossover. The correct criterion is given by equation (5.9).

5.4 Bipolaron bands in high-T c perovskites

Consideration of particular lattice structures shows that small inter-site bipolaronsare perfectly mobile even when the electron–phonon coupling is strong and thebipolaron binding energy is large (section 4.6). Let us analyse the important caseof copper-based high-Tc oxides. They are doped charged-transfer ionic insulatorswith narrow electron bands. Therefore, the interaction between holes can beanalysed using computer simulation techniques based on a minimization of theground-state energy of an ionic insulator with two holes, the lattice deformationsand the Coulomb repulsion fully taken into account but neglecting the kineticenergy terms.

Using these techniques net inter-site interactions of an in-plane oxygen holewith an apex hole, figure 5.4, and of two in-plane oxygen holes were found to beatractive (the binding energies � = 119 meV and � = 60 meV, respectively), inLa2CuO4 [192]. All other interactions were found to be repulsive. The Blochbands of those bipolarons in the perovskites are obtained using the canonicaltransformations of chapter 4 [100, 108].

5.4.1 Apex bipolarons

An apex bipolaron can tunnel from one unit cell to another via a direct single-polaron tunnelling from one apex oxygen to its apex neighbour as in a ladder(section 4.6.3). Oxides are highly polarizable materials, so that we can apply theFrohlich interaction with the coupling constant

γ (q) �1

q

to calculate the bipolaron mass. Lattice polarization is coupled with the electrondensity, therefore the interaction is diagonal in the site representation and thecoupling constant does not depend on a particular orbital state of the hole. Thecanonical displacement transformation (section 4.3.1) eliminates an essential partof the electron–phonon interaction. The transformed Hamiltonian is given by

H = eS H e−S = (Tp − E p)∑i(p)

ni(p) + (Td − Ed)∑i(d)

ni(d)

+∑i �= j

σi j c†i c j +

∑q

ωq(d†qdq + 1/2)

Bipolaron bands in high-Tc perovskites 175

�

Figure 5.4. Apex bipolaron tunnelling in perovskites.

− 12

∑q,i, j

(2ωqui (q)u∗j (q) − Vij )ni n j . (5.10)

The oxygen (p) and the copper (d) diagonal terms include the polaron level shift,which is the same for oxygen and copper ions

E p = Ed =∑

q

|u j (q)|2ωq . (5.11)

If the charge-transfer gap is large enough, Eg � ω0, the bandwidth narrowing

factors are the same for the direct σpp′ and the second-order (via copper σ(2)

pp′)oxygen–oxygen polaron hopping integrals (T = 0),

σpp′ ≡ 〈0|σpp′ |0〉 = Tpp′e−g2

pp′ (5.12)

σ(2)

pp′ ≡∑ν

〈0|σpd |ν〉〈ν|σdp′ |0〉E0 − Eν

≈ − T 2pd

Ege−g2

pp′ (5.13)

where |ν〉, Eν are eigenstates and eigenvalues of the transformed Hamiltonian(5.10) without the third hopping term, |0〉 is the phonon vacuum and the reductionfactor is

g2pp′ = 1

2N

∑q

|γ (q)|2(1 − cos[q · (m p − m p′)]). (5.14)

These expressions are the result of straightforward calculations described later.Taking into account that Eν − E0 = Eg + ∑

q ωqnq , the second-order indirecthopping (equation (5.13)) is written as

σ(2)

pp′ = −i∫ ∞

0dt e−iEgt 〈0|σpd(t)σdp′ |0〉 (5.15)

where

σpd (t) = Tpd exp

(∑q

u∗d(q, t)d†

q − H.c.

)exp

(∑q

u p(q, t)dq − H.c.

).

(5.16)

176 Competing interactions in unconventional superconductors

Here u j (q, t) ≡ u j (q) exp(−iωq t) and nq = 0, 1, 2, . . . the phonon occupationnumbers. Calculating the bracket in equation (5.15), one obtains

〈. . .〉 = T 2pde−g2

pd e−g2

dp′ exp

(−

∑q

[ud(q) − u p(q)][u∗p′(q) − u∗

d (q)]e−iωq t)

.

(5.17)If ωq is q-independent the integral in equation (5.15) is calculated by theexpansion of the exponent in equation (5.17) as

σ(2)

pp′ = − T 2pd

Ege−g2

pd e−g2

dp′∞∑

k=0

(−1)k(∑

q [ud(q) − u p(q)][u∗p′(q) − u∗

d(q)])k

k!(1 + kω0/Eg).

(5.18)Then equation (5.13) is obtained in the limit Eg � ω0. Equation (5.14) yields

g2pp′ = E p

ω0

(1 − Si(qdm)

qdm

), (5.19)

if we approximate the Brillouin zone by a sphere of the radius qd (the Debyeapproximation). Here

Si(x) =∫ x

0dt

sin t

t

m = a/√

2 and m = a for the in-plane oxygen–oxygen, and for apex–apexnarrowing factors, respectively. In cuprates with qd ≈ 0.7 A−1 and a � 3.8 Aone obtains g2

pp′ ≈ 0.2E p/ω0 for an in-plane hopping, and g2 ≈ 0.3E p/ω0 foran apex–apex one. Hence, both band-narrowing factors are much less than onecan expect in the Holstein model.

The bipolaron hopping integral, t , is obtained by projecting the Hamiltonian(5.10) onto a reduced Hilbert space containing only empty or doubly occupiedelementary cells and averaging the result with respect to phonons. Thewavefunction of an apex bipolaron localized, let us say, in the cell m is written as

|m〉 =4∑

i=1

Ai c†i c†

apex|0〉 (5.20)

where i denotes the px,y orbitals and spins of the four plane oxygen ions in thecell (figure 5.4) and c†

apex is the creation operator for the hole on one of the threeapex oxygen orbitals with the spin, which is the same or opposite of the spin ofthe in-plane hole depending on the total spin of the bipolaron. The probabilityamplitudes Ai are normalized by the condition |Ai | = 1/2, if four plane orbitalspx1, py2, px3 and py4 are involved or by |Ai | = 1/

√2 if only two of them are

relevant.The matrix element of the Hamiltonian (5.10) of first order with respect to the

transfer integral responsible for the bipolaron tunnelling to the nearest-neighbourcell m + a is

t = 〈m|H |m + a〉 = |Ai |2T apexpp′ e−g2

(5.21)

Bipolaron bands in high-Tc perovskites 177

where T apexpp′ e−g2

is a single-polaron hopping integral between two apex ions. Asa result, the hole bipolaron energy spectrum consists of two bands E x,y(K ),

formed by an overlap of px and py apex polaron orbitals, respectively, as inequation (4.269). In this equation t is the renormalized hopping integral betweenp orbitals of the same symmetry elongated in the direction of the hopping (ppσ )and t ′ is the renormalized hopping integral in the perpendicular direction (ppπ).Their ratio t/t ′ = T apex

pp′ /T ′apexpp′ = 4 as follows from the tables of hopping

integrals in solids. Two different bands are not mixed because T apexpx ,p′

y= 0

for the nearest neighbours. The random potential does not mix them either ifit varies smoothly on the lattice scale. Hence, we can distinguish ‘x’ and ‘y’bipolarons with a lighter effective mass in the x or y direction, respectively. Theapex z bipolaron, if formed is approximately four times less mobile than x andy bipolarons. The bipolaron bandwidth is of the same order as the polaron one,which is a specific feature of the inter-site bipolarons (section 4.6). For a largepart of the Brillouin zone near (0, π) for ‘x’ and (π, 0) for ‘y’ bipolarons, onecan adopt the effective mass approximation

E x,yK = K 2

x

2m∗∗x,y

+ K 2y

2m∗∗y,x

(5.22)

with Kx,y taken relative to the band bottom positions and m∗∗x = 1/t , m∗∗

y =4m∗∗

x .

5.4.2 In-plane bipolarons

Here we consider a two-dimensional lattice of ideal octahedra that can beregarded as a simplified model of the copper–oxygen perovskite layer as shown infigure 5.4. The lattice period is a = 1 and the distance between the apical sites andthe central plane is h = a/2 = 0.5. For mathematical transparency, we assumethat all in-plane atoms, both copper and oxygen, are static but apical oxygens areindependent three-dimensional isotropic harmonic oscillators, so that the modelHamilonian of section 4.6.3 is applied. Due to poor screening, the hole–apicalinteraction is purely Coulombic:

gν(m − n) = κν

|m − n|2where ν = x, y, z. To account for the experimental fact that z-polarized phononscouple to holes stronger than others [172], we choose κx = κy = κz/

√2. The

direct hole–hole repulsion is

Vc(n − n′) = Vc√2|n − n′|

so that the repulsion between two holes in the nearest-neighbour (NN)configuration is Vc. We also include the bare NN hopping TNN, the next nearest

178 Competing interactions in unconventional superconductors

neighbour (NNN) hopping across copper TNNN and the NNN hopping betweenthe pyramids T ′

NNN. The polaron shift is given by the lattice sum (4.148) which,after summation over polarizations, yields

E p = 2κ2x ω0

∑m

(1

|m − n|4 + h2

|m − n|6)

= 31.15κ2xω0 (5.23)

where the factor two accounts for two layers of apical sites. For reference, theCartesian coordinates are n = (nx + 1

2 , ny + 12 , 0), m = (mx , my, h) and

nx , ny, mx , my are integers. The polaron–polaron attraction is

Vph(n − n′) = 4ω0κ2x

∑m

h2 + (m − n′) · (m − n)

|m − n′|3|m − n|3 . (5.24)

Performing the lattice summations for the NN, NNN and NNN′ configurationsone finds Vph = 1.23E p, 0.80E p and 0.82E p, respectively. As a result, we obtaina net inter-polaron interaction as vNN = Vc − 1.23E p, vNNN = Vc√

2− 0.80E p,

v′NNN = Vc√

2− 0.82E p and the mass renormalization exponents as g2

NN =0.38(E p/ω0), g2

NNN = 0.60(E p/ω0) and (g′NNN)2 = 0.59(E p/ω0).

Let us now discuss different regimes of the model. At Vc > 1.23E p, nobipolarons are formed and the system is a polaronic Fermi liquid. The polaronstunnel in the square lattice with NN hopping t = TNN exp(−0.38E p/ω0) andNNN hopping t ′ = TNNN exp(−0.60E p/ω0). (Since g2

NNN ≈ (g′NNN)2 one can

neglect the difference between NNN hoppings within and between the octahedra.)A single-polaron spectrum is, therefore,

E1(k) = −E p − 2t ′[cos kx + cos ky] − 4t cos(kx/2) cos(ky/2). (5.25)

The polaron mass is m∗ = 1/(t + 2t ′). Since in general t > t ′, the mass is mostlydetermined by the NN hopping amplitude t .

If Vc < 1.23E p then inter-site NN bipolarons form. The bipolarons tunnel inthe plane via four resonating (degenerate) configurations A, B, C and D, as shownin figure 5.5.

In the first order in the renormalized hopping integral, one should retainonly these lowest-energy configurations and discard all the processes that involveconfigurations with higher energies. The result of such a projection is a bipolaronHamiltonian:

Hb = (Vc − 3.23E p)∑

l

[A†l Al + B†

l Bl + C†l Cl + D†

l Dl ]

− t ′∑

l

[A†l Bl + B†

l Cl + C†l Dl + D†

l Al + H.c.]

− t ′∑

n

[A†l−x Bl + B†

l+yCl + C†l+x Dl + D†

l−y Al + H.c.] (5.26)

Bipolaron bands in high-Tc perovskites 179

Figure 5.5. Four degenerate in-plane bipolaron configurations A, B, C and D. Somesingle-polaron hoppings are indicated by arrows.

where l numbers octahedra rather than individual sites, x = (1, 0) and y = (0, 1).A Fourier transformation and diagonalization of a 4×4 matrix yields the bipolaronspectrum:

E2(k) = Vc − 3.23E p ± 2t ′[cos(kx/2) ± cos(ky/2)]. (5.27)

There are four bipolaronic sub-bands combined in the band of width 8t ′. Theeffective mass of the lowest band is m∗∗ = 2/t ′. The bipolaron binding energyis � ≈ 1.23E p − Vc. The bipolaron already moves in the first order of polaronhopping. This remarkable property is entirely due to the strong on-site repulsionand long-range electron–phonon interaction that leads to a non-trivial connectivityof the lattice. This fact combines with a weak renormalization of t ′ yielding asuperlight bipolaron with the mass m∗∗ ∝ exp(0.60E p/ω). We recall that in theHolstein model m∗∗ ∝ exp(2E p/ω) (section 4.6.1). Thus the mass of the Frohlichbipolaron in the perovskite layer scales approximately as a cubic root of that ofthe Holstein one.

With an even stronger e–ph interaction, Vc < 1.16E p, NNN bipolaronsbecome stable. More importantly, holes can now form three- and four-particleclusters. The dominance of the potential energy over kinetic in the transformedHamiltonian enables us to readily investigate these many-polaron cases. Threeholes placed within one oxygen square have four degenerate states with the energy2(Vc − 1.23E p) + Vc/

√2 − 0.80E p. The first-order polaron hopping processes

mix the states resulting in a ground-state linear combination with the energyE3 = 2.71Vc − 3.26E p − √

4t2 + t ′2. It is essential that between the squaressuch triads could move only in higher orders of polaron hopping. In the first order,

180 Competing interactions in unconventional superconductors

they are immobile. A cluster of four holes has only one state within a square ofoxygen atoms. Its energy is E4 = 4(Vc − 1.23E p) + 2(Vc/

√2 − 0.80E p) =

5.41Vc − 6.52E p. This cluster, as well as all bigger ones, are also immobilein first-order polaron hopping. We would like to stress that at distances muchlarger than the lattice constant the polaron–polaron interaction is always repulsive(section 4.4) and the formation of infinite clusters, stripes or strings is strictlyprohibited (section 8.6). We conclude that at Vc < 1.16E p, the system quicklybecomes a charge segregated insulator.

The fact that within the window, 1.16E p < Vc < 1.23E p, there are no threeor more polaron bound states means that bipolarons repel each other. The systemis effectively a charged Bose gas, which is a superconductor (section 4.7). Thesuperconductivity window, that we have found, is quite narrow. This indicatesthat the superconducting state in cuprates requires a rather fine balance betweenelectronic and ionic interactions.

5.5 Bipolaron model of cuprates

The considerations set out in sections 5.2, 5.3 and 5.4 leads us to a simple modelof cuprates [179]. The main assumption is that all electrons are bound into smallsinglet and triplet inter-site bipolarons stabilized by e–ph and spin–fluctuationinteractions. As the undoped plane has a half-filled Cu3d9 band, there is nospace for bipolarons to move if they are inter-site. Their Brillouin zone is half theoriginal electron one and is completely filled with hard-core bosons. Hole pairs,which appear with doping, have enough space to move, and they are responsiblefor low-energy kinetics. Above Tc a material such as YBa2Cu3O6+x contains anon-degenerate gas of hole bipolarons in singlet and triplet states. Triplets areseparated from singlets by a spin-gap J and have a lower mass due to a lowerbinding energy (figure 5.6). The main part of the electron–electron correlationenergy (Hubbard U and inter-site Coulomb repulsion) and the electron–phononinteraction are taken into account in the binding energy of bipolarons �, and intheir band-width renormalization as described in chapter 4. When the carrierdensity is small (nb � 1 (as in cuprates)), bipolaronic operators are almostbosonic (section 4.7.1). The hard-core interaction does not play any role inthis dilute limit, so only the Coulomb repulsion is relevant. This repulsion issignificantly reduced due to a large static dielectric constant in oxides (ε0 �1). Hence, carriers are almost-free charged bosons and thermally excited non-degenerate fermions, so that the canonical Boltzmann kinetics (section 1.1) andthe Bogoliubov excitations of the charged Bose gas (section 4.7.3) are perfectlyapplied in the normal and superconducting states, respectively.

The population of singlet, ns , triplet nt and polaron, n p bands is determinedby the chemical potential µ ≡ T ln y, where y is found using the thermalequilibrium of singlet and triplet bipolarons and polarons:

2ns + 2nt + n p = x . (5.28)

Bipolaron model of cuprates 181

Figure 5.6. Bipolaron picture of high-temperature superconductors. A indicatesdegenerate singlet and triplet inter-site bipolarons. B shows non-degenerate singlet andtriplet inter-site bipolaron, which naturally includes the addition of an extra excitationband. The crosses are copper sites and the circles are oxygen sites. w is the half-bandwidthof the polaron band, t is the half-bandwidth of the bipolaronic bands, �/2 is the bipolaronbinding energy per polaron and J is the exchange energy per bipolaron.

Applying the effective-mass approximation for quasi-two-dimensional energyspectra of all particles, we obtain for 0 � y < 1:

−m∗∗s ln(1 − y) − 3m∗∗

t ln(1 − ye−J/T ) + m∗ ln(1 + y1/2e−�/(2T )) = πx

T(5.29)

in the normal state, and y = 1 in the superconducting state. Here x is the totalnumber of holes per unit area. If the polaron energy spectrum is (quasi-)one-dimensional (as in section 6.5), an additional T −1/2 appears in front of thelogarithm in the third term on the left-hand side of equation (5.29).

We should also take into account localization of holes by a random potential,because doping inevitably introduces some disorder. The Coulomb repulsion

182 Competing interactions in unconventional superconductors

restricts the number of charged bosons in each localized state, so that thedistribution function will show a mobility edge Ec [194]. The number of bosons ina single potential well is determined by the competition between their long-rangeCoulomb repulsion (�4e2/ξ ) and the binding energy, Ec − ε. If the localizationlength diverges with the critical exponent ν < 1 (ξ ∼ (Ec − ε)−ν), we canapply a ‘single-well–single-particle’ approximation assuming that there is onlyone boson in each potential well (see also section 8.1). Within this approximationlocalized charged bosons obey the Fermi–Dirac statistics, so that their density isgiven by

nL(T ) =∫ Ec

−∞NL(E) dE

y−1exp(E/T ) + 1(5.30)

where NL(E) is the density of localized states. Near the mobility edge, it remainsconstant NL(E) � nL/γ, where γ is of the order of the binding energy in a singlerandom potential well, and nL is the number of localized states per unit area. Thenumber of empty localized states turns out to be linear as a function of temperaturein a wide temperature range T � γ from equation (5.30). Then the conservationof the total number of carriers yields for the chemical potential:

π(x − 2nL)

T= − m∗∗

s ln(1 − y) − 3m∗∗t ln(1 − ye−J/T )

+ m∗ ln(1 + y1/2e−�/(2T )) − 2πnL

γln(1 + y−1). (5.31)

If the number of localized states is about the same as the number of pairs(nL ≈ x/2), a solution of this equation does not depend on temperature in awide temperature range above Tc. With y to be a constant (y ≈ 0.6 in a widerange of parameters in equation (5.31)), the number of singlet bipolarons in theBloch states is linear in temperature:

ns(T ) = (x/2 − nL) + TnL

γln(1 + y−1). (5.32)

The numbers of triplets and polarons are exponentially small at low temperatures,T � J,�.

The model suggests a phase diagram of the cuprates as shown in figure 5.7.This phase diagram is based on the assumption that to account for the highvalues of Tc in cuprates, one has to consider electron–phonon interactionslarger than those used in the intermediate-coupling theory of superconductivity(chapter 3). Regardless of the adiabatic ratio, the Migdal–Eliashberg theory ofsuperconductivity and the Fermi-liquid theory break at λ ≈ 1 (section 4.2). Amany-electron system collapses into a small (bi)polaron regime at λ ≥ 1 withwell-separated vibration and charge-carrier degrees of freedom. Even thoughit seems that these carriers should have a mass too large to be mobile, theinclusion of the on-site Coulomb repulsion and a poor screening of the long-rangeelectron–phonon interaction leads to mobile inter-site bipolarons (section 4.6).

Bipolaron model of cuprates 183

Figure 5.7. Phase diagram of superconducting cuprates in the bipolaron theory (courtesyof J Hofer).

Above Tc, the Bose gas of these bipolarons is non-degenerate and below Tc, theirphase coherence sets in and hence superfluidity of the double-charged 2e bosonsoccurs. Of course, there are also thermally excited single polarons in the model(figure 5.6).

There is much evidence for the crossover regime at T ∗ and normal-statecharge and spin gaps in cuprates (chapter 6). These energy gaps could beunderstood as half of the binding energy � and the singlet–triplet gap ofpreformed bipolarons, respectively [185] and T ∗ is a temperature, where thepolaron density compares with the bipolaron one. Further evidence for bipolaronscomes from a parameter-free estimate of the renormalized Fermi energy εF, whichyields a very small value as discussed in section 5.3. There might be a crossoverfrom the Bose–Einstein condensation to a BCS-like polaronic superconductivity(section 4.5) across the phase diagram [179]. If the Fermi liquid does exist atoverdoping, then it is likely that the heavy doping causes an ‘overcrowding effect’where polarons find it difficult to form bipolarons due to a larger number ofcompeting holes. Many experimental observations can be explained using thebipolaron model (chapters 6, 7 and 8).

Chapter 6

Normal state of cuprates

6.1 In-plane resistivity and Hall ratio

Thermally excited phonons and (bi)polarons are well decoupled in the strong-coupling regime of the electron–phonon interaction (chapter 4), so that thestandard Boltzmann equation of section 1.1 for renormalized carriers is applied.We make use of the τ approximation [193] in an electric field E = ∇φ, atemperature gradient ∇T and in a magnetic field B ⊥ E, ∇T . Bipolaron andsingle-polaron non-equilibrium distributions are found to be

f (k) = f0(E) + τ∂ f0

∂ Ev · {F + �n × F} (6.1)

where

F = (E − µ)∇T/T + ∇(µ − 2eφ)

f0(E) = [y−1exp(E/T ) − 1]−1

for bipolarons with the energy E = k2/(2m∗∗) and with the Hall angle � =�b = 2eBτb/m∗∗ and

F = (E + �/2 − µ/2)∇T/T + ∇(µ/2 − eφ)

f0(E) = {y−1/2exp[(E + �/2)/T ] + 1}−1

E = k2/(2m∗) and � = �p = eBτp/m∗ for thermally excited polarons. Heren = B/B is a unit vector in the direction of the magnetic field. The Hall anglesare assumed to be small (� � 1), because the polarons and bipolarons are heavy.Equation (6.1) is used to calculate the electrical and thermal currents induced bythe applied thermal and potential gradients:

jα = aαβ∇β(µ − 2eφ) + bαβ∇βT (6.2)

wα = cαβ∇β(µ − 2eφ) + dαβ∇βT . (6.3)

184

In-plane resistivity and Hall ratio 185

Equation (6.2) defines the current with the polaronic conductivity σp =e2τpn p/m∗, where kinetic coeffficents are given by

ax x = ayy = 1

2eσp(1 + 4A) (6.4)

ayx = − axy = 1

2eσp(�p + 4A�b)

bx x = byy = σp

e[�p + � − µ

2T+ 2A(�b − µ/T )]

byx = − bxy = σp

e

[�p

(�p + � − µ

2T

)+ 2A�b(�b − µ/T )

].

Equation (6.3) defines the heat flow with coefficients given by

cx x = cyy = σp

2e2 [T�p + �/2 + eφ + 2A(T�b + 2eφ)] (6.5)

cyx = − cxy = σp

2e2[�p(T�p + �/2 + eφ) + 2A�b(T �b + 2eφ)]

dx x = dyy = σp

e2

{T γp + �p(� − µ/2 + eφ) + (�/2 + eφ)

� − µ

2T

+ A[Tγb + �b(2eφ − µ) − 2eφµ/T ]}

dyx = − dxy = σp

e2

{�p

[Tγp + �p(� − µ/2 + eφ) + (�/2 + eφ)

� − µ

2T

]

+ A�b[Tγb + �b(2eφ − µ) − 2eφµ/T )]}.

Here

� =∫ ∞

0 dE E2∂ f0/∂ E

T∫ ∞

0 dE E∂ f0/∂ E= 2�(z, 2, 1)

�(z, 1, 1)(6.6)

and

γ =∫ ∞

0 dE E3∂ f0/∂ E

T 2∫ ∞

0 dE E∂ f0/∂ E= 6�(z, 3, 1)

�(z, 1, 1)

are numerical coefficients, expressed in terms of the Lerch transcendent

�(z, s, a) =∞∑

k=0

zk

(a + k)s

with z = y in �b, γb and z = −y1/2 exp[−�/(2T )] in �p , γp,

A = m∗τbnb

m∗∗τpn p(6.7)

186 Normal state of cuprates

is the ratio of bipolaron and polaron contributions to the transport, which stronglydepends on temperature. Here, for simplicity, we neglect the spin gap, which issmall in optimally doped cuprates [195]. Then the bipolaron singlet and tripletstates are nearly degenerate, so that bipolaron and polaron densities are expressedas

nb = 2m∗∗T

π| ln[1 − y]| (6.8)

n p = m∗T

πln

[1 + y1/2 exp

(− �

2T

)]. (6.9)

Using the kinetic coefficients (equations (6.4) and (6.5)), we obtain

ρ = 1

σp(1 + 4A)(6.10)

and

RH = 1 + 4A�b/�p

en p(1 + 4A)2(6.11)

for the in-plane resistivity and the Hall ratio, respectively.According to [194], the main scattering channel above Tc is due to particle–

particle collisions with the relaxation time τb,p � 1/T 2. In case of the Bose–Einstein statistics, umklapp scattering can be neglected, so the scattering betweenbosons in extended states does not contribute to the resistivity in a clean sample.However, the inelastic scattering of an extended boson by bosons localized inthe random potential and the scattering of extended bosons by each other make acontribution because the momentum is not conserved in two-particle collisions inthe presence of the impurity potential. In the framework of the ‘single-well–single-particle approximation’, the repulsion between localized bosons playsthe same role as the Pauli exclusion principle in fermionic systems. Then therelaxation rate is proportional to the temperature squared because only bosonswithin the energy shell of the order of T near the mobility edge contribute tothe scattering and because the number of final states is proportional to T in anon-degenerate gas. The chemical potential is pinned near the mobility edge, sothat y ≈ 0.6 in a wide temperature range, if the number of localized states inthe random potential is about the same as the number of bipolarons (section 5.5).This could be the case in YBa2Cu3O6+x , where every excess oxygen ion x canlocalize the bipolaron. Hence, neglecting the polaron contribution to the in-planetransport which is exponentially small below T ∗, we find that

ρ � T (6.12)

and

RH �1

T(6.13)

In-plane resistivity and Hall ratio 187

Figure 6.1. In-plane resistivity and the Hall ratio of bipolarons and thermally excitedpolarons for �/2 = 600 K and �b/�p = 2m∗τb/(m∗∗τp) = 0.44 (inset), and the HallLorenz number as a function of temperature compared with the experimental data by Zhanget al in the optimally doped YBa2Cu3O6.95 [196]. The second inset shows the ratio of theHall Lorenz number to the Lorenz number.

as observed in many cuprates. Taking into account the polaron contribution inequations (6.10) and (6.11) does not change this result, if the binding energyis sufficiently large, as shown in figure 6.1. One can also take into accountthe transport relaxation rate due to a two-dimensional boson–acoustic phononscattering,

1

τb−ac� T (6.14)

and the elastic relaxation rate due to the scattering by unoccupied impurity wellswith their density proportional to temperature,

1

τb−im� T . (6.15)

Hence, neglecting the polaron contribution, we finally obtain [194]

RH ≈ 1

2e(x/2 − nL + bnLT )(6.16)

188 Normal state of cuprates

ρ ≈ CT + σbT 2

x/2−nL + bnLT(6.17)

where σb is the relative boson–boson scattering cross sections, C is a temperature-independent constant, proportional to the carrier mass squared and b is about 1/γ ,where γ was introduced in section 5.5. These expressions contain importantinformation about the number of bosons, localized states and the relativestrength of the different scattering channels. They fit well the experimentaldata in a number of cuprates [194, 197]. Importantly, they predict the Hallratio and resistivity scaling with doping as 1/x because nL � x . This scalingwas experimentally observed in underdoped and optimally doped cuprates[198]. There is a characteristic change in the linear slope of ρ(T ) (see, forexample, [199]) below the temperature, where the spin gap is observed in NMR(section 6.4). This change was explained by Mott [200]: the spin gap is thesinglet–triplet separation energy in the bipolaron model (figure 5.6). Becausetriplets are lighter than singlets, they lead to a smaller C in equation (6.17) and toa smaller temperature gradient of the in-plane resistivity at temperatures above J,

where their density becomes comparable with the singlet density.

6.2 Normal-state resistivity below T c

The scaling of the in-plane resistivity and of the Hall ratio with doping tells us thatcuprates are doped semiconductors, which might be metallic Fermi liquids. Butthe linear (in temperature) resistivity and the temperature-dependent Hall ratio,discussed earlier, are only a tiny part of the non-Fermi-liquid characteristics ofcuprates. Hence, one can ask a fundamental question: ‘What are high-Tc cupratesin their normal state—metals or something else?’. The answer to this vexing ques-tion clearly depends on how precisely one can define a metal. To arrive at one sin-gle definition, one would require a measurement of the normal-state conductivitydown to very low temperatures. Thus, metals show a non-zero low-temperatureconductivity, while the electrical conductivity is zero in non-metals as T → 0 K.On the basis of this simple—but potent—definition, of course, any superconduc-tor must clearly be a metal, since the conductivity at T = 0 K is infinite. However,one can suppress the superconducting state by applying a large external magneticfield to measure a ‘true’ normal-state conductivity down to a very low tempera-ture well below a zero field Tc. This important experiment has been carried outby Boebinger et al [201] by applying an ultra-high magnetic field (up to 60 T) todestroy the superconducting state. This then allows one to scrutinize the nature ofthe electronic structure of cuprates without the ‘complication’ of superconductiv-ity at elevated temperatures. Remarkably, many high-Tc cuprates studied to dateby this technique appear to be doped three-dimensional anisotropic insulators,with the anticipated divergent resistivity and temperature-independent anisotropyat low temperatures. These observations have been explained [202] as a result ofthe resonance scattering of bipolarons by lattice defects.

Normal-state resistivity below Tc 189

As we discussed in the previous section, there is no need to abandon theBoltzmann kinetics to explain the linear in-plane resistivity and the temperature-dependent Hall effect above Tc in cuprates in the framework of the bipolarontheory. A fraction of bipolarons is localized by disorder, so that the number ofdelocalized carriers is proportional to T at sufficiently high temperatures, whilethe boson–boson inelastic scattering rate is proportional to T 2. This allows us toexplain that both the in-plane resistivity and the inverse Hall ratio are proportionalto T . Let us now extend the theory towards low temperatures, where the transportrelaxation time of bipolarons is determined by elastic boson–impurity scatteringand single polarons are frozen out.

The number of extended bosons nb(T ) above the mobility edge is determinedin the ‘single-well–single-particle’ approximation (equation (5.30)) as

nb(T ) = 12 x − nL(T ) (6.18)

where 12 x is the total number of pairs and nL(T ) � nL − NL(0)T is the number

of bosons localized by the random potential with NL(0) the density of localizedstates near the mobility edge. The Hall ratio, RH, measures the inverse carrierdensity, so that

RH(T )

RH(0)= 1

1 + T/TL(6.19)

where TL = (x − 2nL)/2NL(0). This simple expression fits the Hall ratio inLa2−xSrxCuO4 at optimum doping (x = 0.15) (figure 6.2) with TL = 234 K andthe constant RH(0) = 2 × 10−3 cm3 C−1. If the total number of carriers 1

2 x islarger than the number of the potential wells nL, the carrier density is practicallytemperature independent at low temperatures.

The characteristic kinetic energy of non-degenerate bipolarons in the normalstate is of the order of the temperature rather than the Fermi energy in metalsand in conventional semiconductors. The most effective scattering at lowtemperatures is then caused by attractive shallow potential wells. For slowparticles, this is described by the familiar Wigner resonance cross section as

σ(E) = 2π

m∗∗1

E + |ε| (6.20)

where

ε = −π2

16Umin

(U

Umin− 1

)2

(6.21)

is the energy of a shallow virtual (U < Umin) or real (U > Umin) localized level,U is the well depth and Umin = π2/8m∗∗a2 with the well size a. The transportrelaxation rate is the sum of the scattering cross sections from different potentialwells within the unit volume multiplied by the velocity v = √

2E/m∗∗. Thereis a wide distribution of potential wells with respect to both U and a in cuprates.Therefore, one has to integrate the Wigner cross section (equation (6.20)) over U

190 Normal state of cuprates

Figure 6.2. The Hall ratio in La1.85Sr0.15CuO4 (triangles [198]) described byequation (6.19).

and over a. Performing the integration over U , we take into account only shallowwells with U < Umin because the deeper wells are occupied by localized carriersand cannot yield a resonant scattering. The result is:

〈σ(E)〉 ≡ γ −1∫ Umin

0σ(E) dU = 4π

m∗∗γ a√

2m∗∗Etan−1

(π2

8a√

2m∗∗E

)(6.22)

where the width of the U -distribution, γ , is supposed to be large compared withUmin. Integrating over the size a, one has to take into account the fact that theWigner formula (equation (6.20)) is applied only to slow particles with a ≤�/

√2m∗∗E . However, because the U -averaged cross section (equation (6.22))

behaves like 1/a2 at a large a, we can extend the integration over a up to infinity.As a result, we obtain the inverse mean free path:

l−1(E) = nL

A

∫ ∞

amin

〈σ(E)〉 da � π2 NL(0)

m∗∗ A√

2m∗∗Eln

E0

E(6.23)

for E � E0. Here A is the width of the size distribution of the random potentialwells, NL(0) � nL/γ , E0 = π4/128m∗∗a2

min and amin is the minimum size.We expect a very large value of A of the order of 10 A or more due to thetwin boundaries and impurity clusters, which are not screened. However, singleimpurities are screened. A simple estimate of the screening radius by the use

Normal-state resistivity below Tc 191

Figure 6.3. In-plane resistivity of La1.87Sr0.13CuO4 (diamonds [201]) described byequation (6.26).

of the classical expression, rD = √T ε0/(16πnbe2) yields a value of amin of

the order of the interatomic spacing (∼2 A), which corresponds to quite a largeE0 = 1500 K if m∗∗ = 10me. Hence, at low temperatures, we arrive at thelogarithmic transport relaxation rate as

1

τ≡ vl−1(E) = 1

τ0ln

E0

E(6.24)

where τ0 = m∗∗2 A/π2 NL(0) is a constant.The low-temperature resistivity is now derived by the use of the Boltzmann

theory:

ρ(T ) = ρ0 lnE0

T(6.25)

where ρ−10 = 2(x − 2nL)e2τ0/m∗∗. Combining both elastic (equation (6.25))

and inelastic (section 6.1) scattering rates and taking into account the temperaturedependence of the extended boson density nb(T ) (equation (6.18)), we find that

ρ(T )

ρ0= ln(E0/T ) + (T/Tb)

2

1 + T/TL(6.26)

with a constant Tb. The full curve in figure 6.3 is a fit to the experimental datawith ρ0 = 7.2 × 10−5 � cm, E0 = 1900 K and Tb = 62 K which appears to beremarkably good.

The value of E0 agrees well with the estimate. Because the bipolaronenergy spectrum is three dimensional at low temperatures, there is no temperaturedependence of the anisotropy ρc/ρab at low T as observed.

192 Normal state of cuprates

6.3 Lorenz number: evidence for bipolarons

Kinetic evidence for 2e bipolarons in cuprates (sections 6.1 and 6.2) is strongbut direct evidence that these materials contain a charged 2e Bose liquid in theirnormal state is highly desirable. Mott and the author [203] discussed the thermalconductivity κ . The contribution from the carriers, given by the Wiedemann–Franz ratio, depends strongly on the elementary charge as ∼(e∗)−2 and shouldbe significantly suppressed in the case of e∗ = 2e compared with the Fermi-liquid contribution. As a result, the Lorenz number L (= (e/kB)2κe/(T σ)) differssignificantly from the Sommerfeld value Le (= π2/3) of standard Fermi-liquidtheory, if the carriers are bipolarons. Here κe, σ and e are the electronic thermalconductivity, the electrical conductivity, and the elementary charge, respectively.Reference [203] predicted a very low Lorenz number Lb of bipolarons—Lb =6Le/(4π2) ≈ 0.15Le—due to the double charge of carriers and also due to theirnearly classical distribution function above Tc.

Unfortunately, the extraction of the electron thermal conductivity has provendifficult since both the electron term, κe, and the phonon term, κph, are comparableto each other in cuprates. Some experiments have attemped to get around thisproblem in a variety of ways [204,205]. In particular, Takenaka et al [204] foundthat κe is constant or weakly T -dependent in the normal state of YBa2Cu3O6+x .This approximately T -independent κe, therefore, implies the violation of theWiedemann–Franz law (since resistivity is found to be a nonlinear function oftemperature) in the underdoped region. The breakdown of the Wiedemann–Franzlaw has also been seen in other cuprates [206, 207].

A new way to determine the Lorenz number has been realized by Zhang etal [196] based on the thermal Hall conductivity. The thermal Hall effect allowedfor an efficient way to separate the phonon heat current even when it is dominant.As a result, the ‘Hall’ Lorenz number (LH ≡ Lxy = (e/kB)2κyx/(T σyx)) hasbeen directly measured in YBa2Cu3O6.95 because the transverse thermal κxy andthe electrical σxy conductivities involve only electrons. Remarkably, the valueof LH just above Tc was found to be about the same as that predicted by thebipolaron model (LH ≈ 0.15Le). However, the experimental LH showed astrong temperature dependence which violates the Wiedemann–Franz law. Thisexperimental observation is hard to explain in the framework of any Fermi-liquid model. Here we demonstrate that the Wiedemann–Franz law breaks downbecause of the interference of polaron and bipolaron contributions to the heattransport [209]. When thermally excited polarons are included, the bipolaronmodel explains the violation of the Wiedemann–Franz law in cuprates and theHall Lorenz number as seen in the experiment.

There is no electric current ( j = 0) in the thermal conductivitymeasurements. This constraint allows us to express the electric and chemicalpotential gradients ∇(µ − 2eφ) via the temperature gradient ∇T usingequations (6.4) and (6.5). Then the thermal conductivity, κ , and the thermal Hall

Lorenz number: evidence for bipolarons 193

conductivity, κxy , are found to be

κ = dx x − cx xbx x

dx x(6.27)

κyx = dyx − cyxbx x

ax x+ cx x(ax xbyx − ayxbx x)

a2x x

and the Lorenz and Hall Lorenz numbers to be

L = L p + 4ALb

1 + 4A+ A[2�p − �b + �/T ]2

(1 + 4A)2(6.28)

LH = L p + 4ALb�b/�p

1 + 4A�b/�p

+ A(4A + �b/�p)[2�p − �b + �/T ]2

(1 + 4A)2(1 + 4A�b/�p). (6.29)

Here L p = (γp − �2p) and Lb = (γb − �2

b)/4 are the polaron and bipolaronLorenz numbers. In the limit of a purely polaronic system (i.e. A = 0), theLorenz numbers (equations (6.28) and (6.29)) are L = LH = L p . In theopposite limit of a purely bipolaronic system (i.e. A = ∞), we obtain a reducedLorenz number [203]: L = LH = Lb. However, in general, equations (6.28)and (6.29) yield temperature-dependent Lorenz numbers that differ significantlyfrom both limits. The main difference originates in the second terms in the right-hand side of equations (6.28) and (6.29), which describe an interference betweenthe polaron and bipolaron contributions in the heat flow. In the low-temperatureregime (T � �), this contribution is exponentially small because the number ofunpaired polarons is small. However, it is enhanced by the factor (�/T )2 andbecomes important in the intermediate-temperature range Tc < T < T ∗. Thecontribution appears as a result of the recombination of a pair of polarons into abipolaronic bound state at the cold end of the sample, which is reminiscent of thecontribution of the electron-hole pairs to the heat flow in semiconductors [193].These terms are mainly responsible for the breakdown of the Wiedemann–Franzlaw in the bipolaronic system.

The bipolaron model, which fits the in-plane resistivity and the Hall ratio,also fits the Hall Lorenz number measured by Zhang et al [196]. To reduce thenumber of fitting parameters, we take the charge pseudogap �/2 = 600 K, asfound by Mihailovic et al [195] for nearly optimally doped YBa2Cu3O6+x in theirsystematic analysis of charge and spin spectroscopies. As discussed in section 6.1,the main scattering channel above Tc is due to particle–particle collisions with arelaxation time τb,p � 1/T 2. The chemical potential is pinned near the mobilityedge, so that y ≈ 0.6 in a wide temperature range, if the number of localizedstates in the random potential is about the same as the number of bipolarons.As a result, there is only one fitting parameter in LH (equation (6.29)) whichis the ratio of the bipolaron and polaron Hall angles �b/�p. The model gives

194 Normal state of cuprates

quite a good fit (figure 6.1) with a reasonable value of �b/�p = 0.44. It alsodescribes well the (quasi-)linear in-plane resistivity and the inverse Hall ratio, asobserved in cuprates (inset in figure 6.1). The Hall Lorenz number appears tobe slightly larger then the Lorenz number (figure 6.1, lower inset). Because thethermal Hall conductivity directly measures the Lorenz number, it can be usedto measure the lattice contribution to the heat flow as well. When we subtractthe electronic contribution determined by using the Lorenz number in figure 6.1,the lattice contribution to the diagonal heat flow appears to be much higher thananticipated in the framework of any Fermi-liquid model.

6.4 Spin pseudogap in NMR

Pairing of holes into singlets well above Tc should be seen as a drop of magneticsusceptibility. Indeed, a rapid decrease in the uniform magnetic susceptibilityand of the nuclear magnetic relaxation rate 1/T1 with temperature lowering is acommon feature of the normal state of novel superconductors. Let us see how thebipolaron model describes the temperature dependence of 1/T1 [185].

The conventional contact hyperfine coupling of nuclear spin on a site i withelectron spins is described by the following Hamiltonian in the site representation:

Hi = Ai

∑j

c†j↑c j↓ + H.c. (6.30)

where Ai is an operator acting on the nuclear spin and j is its nearest-neighboursite. Performing transformations to polarons and bipolarons as described inchapter 4, we obtain the effective spin–flip interaction of triplet bipolarons withthe nuclear spin:

Hi �∑j,l �=l′

b†j,lb j,l′ + H.c. (6.31)

Here l, l ′ = 0,±1 are the z-components of spin S = 1. The NMR width due tothe spin-flip scattering of triplet bipolarons on nuclei is obtained using the Fermi–Dirac golden rule as in section 2.8,

1

T1= − B

t2

∫ 2t

0dE

∂ f (E)

∂ E(6.32)

where f (E) = [exp(E + J − µ)/T −]−1 is the distribution function and 2t is thebandwidth of triplet bipolarons. For simplicity, we take their DOS as a constant(= 1/(2t)). As a result, we obtain

1

T1= BT sinh(t/T )

t2[cosh[(t + J )/T − ln y] − cosh(t/T )] (6.33)

where B is a temperature-independent hyperfine coupling constant.

c-axis transport and charge pseudogap 195

Figure 6.4. Temperature dependence of the nuclear spin relaxation rate 1/T1 [209]compared with the theory for J = 150 K and t = 250 K showing the absence of theHebel–Slichter peak of Cu NMR in YBa2Cu4O8.

Equation (6.33) describes all the essential features of the nuclear spinrelaxation rate in copper-based oxides: the absence of the Hebel–Slichter coherentpeak below Tc, the temperature-dependent Korringa ratio (1/T T1) above Tc anda large value of 1/T1 due to the small bandwidth 2t . It fits the experimental datawith reasonable values of the parameters, t and J (figure 6.4). NMR is almostunaffected by the superconducting transition in the bipolaron model, becausethere are no coherence factors (section 2.8) for triplets. There is only a tiny changein the slope in 1/T1(T ) near Tc due to a kink in the temperature dependence ofthe chemical potential µ(T ) caused by the Bose–Einstein condensation of singlets(appendix B), as shown in the inset. A similar unusual behaviour of the NMR wasfound in underdoped YBa2Cu3O6+x , and in many other cuprates. The Knightshift, which measures the spin susceptibility of carriers, also drops well aboveTc in many copper oxides, in agreement with the bipolaron model. The value ofthe singlet–triplet bipolaron exchange energy J , determined from the fit to theexperimental NMR width, is close to the ‘spin’ gap observed above and below Tcin YBa2Cu3O6+x with unpolarized [210], and polarized [211] neutron scattering.

6.5 c-axis transport and charge pseudogap

The bipolaron theory quantitatively explains the c-axis transport and anisotropyof cuprates [186, 197, 212, 213]. The crucial point is that polarons dominate inthe c-axis transport at intermediate and high temperatures because they are muchlighter than bipolarons in the c-direction. The latter can propagate across the

196 Normal state of cuprates

planes only due to a simultaneous two-particle tunnelling, which is much lessprobable than a single-polaron tunnelling (chapter 4). Along the planes, polaronsand inter-site bipolarons propagate with about the same effective mass. Hence,the polaron contribution to the in-plane transport is small at low temperatures dueto their low density compared with the bipolaron density. As a result, we have amixture of non-degenerate quasi-two-dimensional bosons and thermally excitedfermions, which are capable of propagating along the c-axis. Polarons mainlycontribute to c-axis transport, if the temperature is not very low, which leads to afundamental relation between anisotropy and magnetic susceptibility [186].

To illustrate the point, let us consider a system containing only singletpairs and thermally excited polarons. Quite generally the in-plane bipolaronand c-axis polaron, conductivities (σab and σc respectively) and the uniform spinsusceptibility (χs) are expressed as follows:

σab(T, x) = − 4∫ ∞

0dE σb(E)

∂ fb

∂ E(6.34)

σc(T, x) = − 2∫ ∞

0dE σpc(E)

∂ f p

∂ E(6.35)

χs(T, x) = − 2µ2B

∫ ∞

0dE Np(E)

∂ f p

∂ E(6.36)

where fb = [y−1 exp(E/T ) − 1]−1 and f p = [y−1/2 exp(E/T + �/2T ) + 1]−1

are the bipolaron and polaron distribution functions, respectively, and µB is theBohr magneton. Polarons are not degenerate at any temperature neither arebipolarons degenerate above Tc, so that

fb ≈ y exp

(− E

T

)(6.37)

and

f p ≈ y1/2 exp

(− E + �/2

T

). (6.38)

If the scattering mechanism is the same for polarons and bipolarons, the ratioof the differential bipolaron and polaron conductivities (σb(E) and σpc(E),respectively) is independent of the energy and doping

σb(E)

σpc(E)≡ A = constant. (6.39)

There is a large difference in the values of the ppσ and ppπ hoppingintegrals between different oxygen sites (section 5.4). Therefore, we expecta highly anisotropic polaron energy spectrum with a quasi-one-dimensionalpolaron density of states as also observed in high-resolution ARPES experiments(section 8.4):

Np(E) � 1

4πda

(m∗

2E

)1/2

(6.40)

c-axis transport and charge pseudogap 197

Figure 6.5. (a) Experimental anisotropy [214] compared with equation (6.41) and (b)using experimental magnetic susceptibilities of La2−x Srx CuO4 [215].

where a is the in-plane lattice constant. Then the anisotropy and the spinsuceptibility are expressed as

ρc(T, x)

ρab(T, x)= 2Ay1/2 exp

(�

2T

)(6.41)

and

χs(T, x) = µ2B

2da

(ym∗

2πT

)1/2

exp

(− �

2T

). (6.42)

The chemical potential, y = 2πnb(T, x)/(T m∗∗ab ), is calculated taking

into account the localization of bipolarons in a random potential (section 5.5).The bipolaron density (nb(T, x) per cm2) is linear in temperature and doping

198 Normal state of cuprates

(nb(T, x) ≈ xT/γ a2) in a wide temperature range above Tc which fitsthe temperature and doping dependence of the Hall ratio RH � 1/(2enb)

(section 6.2). As a result, we find the temperature-independent y � x and

ρc(T, x)

ρab(T, x)�

x

T 1/2χs(T, x)(6.43)

χs(T, x) �( x

T

)1/2exp

(− �

2T

). (6.44)

We expect some dependence of the binding energy on doping because ofscreening. Bipolarons are heavy non-degenerate particles, which well screen theelectron–phonon interaction with low-frequency phonons. In a wide range ofdoping near the optimum value (x = 0.15), different experiments are consistentwith

� = �0

x(6.45)

where �0 is doping independent.One can describe all qualitative features of c-axis resistivity and magnetic

susceptibility of La2−xSrxCuO4 using equations (6.43), (6.44) and (6.45) withoutany fitting parameters. Anisotropy is quantitatively described by equation (6.43)using the experimental values of χs(T, x), as shown in figure 6.5. Thetemperature-independent anisotropy of a heavily doped sample, x = 0.3 infigure 6.5 is explained by the contribution of polarons to the in-plane transport.If the binding energy � is below 100 K, the polaron contribution flattensthe temperature dependence of the anisotropy. An exponential temperaturedependence for the c-axis resistivity and anisotropy was also interpretedwithin the framework of the bipolaron model in other cuprates, in particularin Bi2Sr2CaCu2O8+δ [197], YBa2Cu3O6+x [213] and HgBa2CuO4+δ [212].Importantly, the uniform magnetic susceptibility above Tc increases with doping(figure 6.5(b)). It proves once again that cuprates are doped insulators, where low-energy charge and spin degrees of freedom are due to holes doped into a parentinsulating matrix with no free carriers and no free spins. A rather low magneticsusceptibility of parent insulators in their paramagnetic phase is presumably dueto a singlet pairing of copper spins as discussed in section 5.5.

6.6 Infrared conductivity

Studies of photoinduced carriers in dielectric ‘parent’ compounds like La2CuO4,YBa2Cu3O6, and others demonstrated the formation of self-localized smallpolarons or bipolarons. In particular, Mihailovic et al [170] described thespectral shape of the photoconductivity with the small polaron transport theory.They also argued that a similar spectral shape and systematic trends in boththe photoconductivity of optically doped dielectric samples and the infraredconductivity of chemically doped high-Tc oxides indicate that carriers in a

Infrared conductivity 199

Figure 6.6. (a) Photoinduced infrared conductivity in the insulator precursors forTl2Ba2CaCu2O8 (top), YBa2Cu3O7 (middle) and La2−x Srx CuO4 (bottom) comparedwith the polaron infrared conductivity (broken curve); (b) the infrared conductivity forsuperconducting samples (broken curves) compared with (a).

concentrated (metallic) regime retain much of the character of carriers in a dilute(photoexcited) regime. The measured photoinduced infrared conductivity σ(ν)

(full curves) in insulating parent compounds is shown in figure 6.6(a). Itscharacteristic maximum is due to the incoherent spectral weight of the polaronGreen function (section 4.5) as follows from a comparison with a theoreticalsmall-polaron infrared conductivity (broken curves) in figure 6.6(a). The infraredconductivity of high-Tc superconductors (broken curves) is compared with thephotoinduced infrared conductivity (full curves) in respective insulator precursorsin figure 6.6(b). One of the qualitative observations, which follow from acomparison of the infrared conductivities of insulating and superconductingmaterials is that in all perovskites exhibiting superconductivity, the peakfrequency shifts toward lower values as the Tc of the material increases. Thepeak position in σ(ν) is related to the polaron level shift and, therefore, to the(bi)polaron mass. The critical temperature of the superconducting transition isproportional to 1/m∗ in bipolaron theory (section 4.7.5). A lower peak frequencycorresponds to a lower mass of bipolaronic carriers and, therefore, to a higher Tc.

Chapter 7

Superconducting transition

7.1 Parameter-free description of T c

An ultimate goal of the theory of superconductivity is to provide an expression forTc as a function of some well-defined parameters characterizing the material. Inthe framework of BCS theory, the Eliashberg equation (3.74) for the gap functionproperly takes into account a realistic phonon spectrum and retardation of theelectron–phonon interaction. Tc is fairly approximated by McMillan’s formula(3.96), which works well for simple metals and their alloys. But applying a theoryof this kind to high-Tc cuprates is problematic. Since bare electron bands arenarrow, strong correlations result in the Mott insulating state of undoped parentcompounds. As a result, µ∗ and λ are ill defined in doped cuprates and polaroniceffects are important as in many doped semiconductors [13]. Hence, an estimateof Tc in cuprates within BCS theory appears to be an exercise in calculating µ∗rather than Tc itself. Also, one cannot increase λ without accounting for a polaroncollapse of the band (chapter 4). This appears at λ � 1 for uncorrelated electrons(holes) and even at a smaller value of bare electron–phonon coupling in stronglycorrelated models [216].

However, the bipolaron theory provides a parameter-free expression forTc [217], which fits the experimentally measured Tc in many cuprates for anylevel of doping. Tc is calculated using the density sum rule as the Bose–Einsteincondensation (BEC) temperature of 2e charged bosons on a lattice. Just beforethe discovery [1], we predicted Tc would be as high as ∼100 K using an estimateof the bipolaron effective mass [105]. Uemura [218] established a correlationof Tc with the in-plane magnetic field penetration depth measured by the µs Rtechnique in many cuprates as Tc � 1/λ2

ab. The technique is based on theimplantation of spin-polarized muons. It monitors the time evolution of themuon spin polarization. He concluded that cuprates are neither BCS nor BECsuperfluids but that they exist in the crossover region from one to the other,because the experimental Tc was to be found about three or more times lowerthan the BEC temperature.

200

Parameter-free description of Tc 201

Here we calculate the Tc of a bipolaronic superconductor taking properlyinto account the microscopic band structure of bipolarons in layered cuprates asderived in section 5.4. We arrive at a parameter-free expression for Tc, which incontrast to [218] involves not only the in-plane, λab, but also the out-of-plane, λc,magnetic field penetration depth and a normal state Hall ratio RH just above thetransition. It describes the experimental data for a few dozen different samplesclearly indicating that many cuprates are in the BEC rather than in the crossoverregime.

The energy spectrum of bipolarons is twofold degenerate in cuprates(section 5.4). One can apply the effective mass approximation at T � Tc,equation (5.22), because Tc should be less than the bipolaron bandwidth. Also athree-dimensional correction to the spectrum is important for BEC (appendix B)which is well described by the tight-binding approximation:

E x,yK = �2K 2

x,y

2m∗∗x

+ �2K 2y,x

2m∗∗y

+ 2t⊥[1 − cos(Kzd)] (7.1)

where d is the interplane distance and t⊥ is the inter-plane bipolaron hoppingintegral. Substituting the spectrum (equation (7.1)) into the density sum rule,∑

K ,i=(x,y)

[exp(EiK/Tc) − 1]−1 = nb (7.2)

one readily obtains Tc (in ordinary units):

kBTc = f

(t⊥

kBTc

)× 3.31�2(nB/2)2/3

(m∗∗x m∗∗

y m∗∗c )1/3

(7.3)

where the coefficient f (x) ≈ 1 is shown in figure 7.1 as a function of theanisotropy, t⊥/(kBTc) and m∗∗

c = �2/(2|t⊥|d2).This expression is rather ambiguous because the effective mass tensor as well

as the bipolaron density nb are not well known. Fortunately, we can express theband-structure parameters via the in-plane magnetic field penetration depth

λab =[

m∗∗x m∗∗

y

8πnBe2(m∗∗x + m∗∗

y )

]1/2

and out-of-plane penetration depth,

λc =[

m∗∗c

16πnbe2

]1/2

(we use c = 1). The bipolaron density is expressed through the in-plane Hall ratio(above the transition) as

RH = 1

2enb× 4m∗∗

x m∗∗y

(m∗∗x + m∗∗

y )2 (7.4)

202 Superconducting transition

Figure 7.1. Correction coefficient to the three-dimensional Bose–Einstein condensationtemperature as a function of anisotropy.

which leads to

Tc = 1.64 f

(t⊥

kBTc

)(eRH

λ4abλ

2c

)1/3

. (7.5)

Here Tc is measured in kelvin, eRH in cm3 and λ in cm. The coefficient f isabout unity in a very wide range of t⊥/(kBTc) ≥ 0.01, figure 7.1. Hence, thebipolaron theory yields a parameter-free expression, which unambiguously tellsus how near the cuprates are to the BEC regime:

Tc ≈ Tc(3D) = 1.64

(eRH

λ4abλ

2c

)1/3

. (7.6)

We compare the last two expressions with the experimental Tc of more than 30different cuprates, for which both λab and λc have been measured along withRH(Tc + 0) in table 7.1 and figure 7.2. The Hall ratio has a strong temperaturedependence above Tc (section 6.1). Therefore, we use the experimental Hall ratiojust above the transition. In a few cases (mercury compounds), where RH(Tc + 0)

is unknown, we take the inverse chemical density of carriers (divided by e) asRH. For almost all samples, the theoretical Tc fits experimental values within anexperimental error bar for the penetration depth (about ±10%). There are a fewZn-doped YBCO samples (figure 7.2) whose critical temperature is higher thanthe theoretical one. If we assume that the degeneracy of the bipolaron spectrum isremoved by the random potential of Zn, then the theoretical Tc would be almostthe same as the experimental values for these samples as well.

Parameter-free description of Tc 203

Table 7.1. Experimental data on Tc (K), ab and c penetration depth (nm), Hall coefficient(10−3 (cm3 C−1)), and calculated values of Tc, respectively, for La2−x Srx CuO4 (La),YBaCuO (x%Zn) (Zn), YBa2Cu3O7−x (Y) and HgBa2CuO4+x (Hg) compounds.

Compound T expc λab λc RH Tc (3D) Tc TKT

La(0.2) 36.2 200 2 540 0.8 38 41 93La(0.22) 27.5 198 2 620 0.62 35 36 95La(0.24) 20.0 205 2 590 0.55 32 32 88La(0.15) 37.0 240 3 220 1.7 33 39 65La(0.1) 30.0 320 4 160 4.0 25 31 36La(0.25) 24.0 280 3 640 0.52 17 19 47Zn(0) 92.5 140 1 260 1.2 111 114 172Zn(2) 68.2 260 1 420 1.2 45 46 50Zn(3) 55.0 300 1 550 1.2 35 36 38Zn(5) 46.4 370 1 640 1.2 26 26 30Y(0.3) 66.0 210 4 530 1.75 31 51 77Y(0.43) 56.0 290 7 170 1.45 14 28 40Y(0.08) 91.5 186 1 240 1.7 87 88 98Y(0.12) 87.9 186 1 565 1.8 75 82 97Y(0.16) 83.7 177 1 557 1.9 83 89 108Y(0.21) 73.4 216 2 559 2.1 47 59 73Y(0.23) 67.9 215 2 630 2.3 46 58 73Y(0.26) 63.8 202 2 740 2.0 48 60 83Y(0.3) 60.0 210 2 880 1.75 43 54 77Y(0.35) 58.0 204 3 890 1.6 35 50 82Y(0.4) 56.0 229 4 320 1.5 28 42 65Hg(0.049) 70.0 216 16 200 9.2 23 60 115Hg(0.055) 78.2 161 10 300 8.2 43 92 206Hg(0.055) 78.5 200 12 600 8.2 28 69 134Hg(0.066) 88.5 153 7 040 6.85 56 105 229Hg(0.096) 95.6 145 3 920 4.7 79 120 254Hg(0.097) 95.3 165 4 390 4.66 61 99 197Hg(0.1) 94.1 158 4 220 4.5 66 105 216Hg(0.101) 93.4 156 3 980 4.48 70 107 220Hg(0.101) 92.5 139 3 480 4.4 88 127 277Hg(0.105) 90.9 156 3 920 4.3 69 106 220Hg(0.108) 89.1 177 3 980 4.2 58 90 171

One can argue that, due to a large anisotropy, cuprates may belong to a two-dimensional ‘XY ’ universality class with the Kosterlitz–Thouless (KT) criticaltemperature TKT of preformed bosons [219, 220] or Cooper pairs [164]. Shouldthis be the case, one could hardly discriminate the Cooper pairs with respect tobipolarons. The KT critical temperature can be expressed through the in-plane

204 Superconducting transition

Figure 7.2. Theoretical critical temperature compared with the experiment (the theoryis exact for samples on the straight line) for LaSrCuO compounds (squares), forZn-substituted YBa2Cu1−x Znx O7 (circles), for YBa2Cu3O7−δ (triangles), and forHgBa2CuO4+δ (diamonds). Experimental data for the London penetration depth are takenfrom Xiang T et al 1998 Int. J. Mod. Phys. B 12 1007 and Janossy B et al 1991 PhysicaC 181 51 in YBa2Cu3O7−δ and YBa2Cu1−x Znx O7; from Grebennik V G et al 1990Hyperfine Interact. 61 1093 and Panagopoulos C Private communication in underdopedand overdoped La2−x Srx CuO4, respectively, and from Hofer J et al 1998 Physica C 297103 in HgBa2CuO4+δ . The Hall coefficient above Tc is taken from Carrington A et al1993 Phys. Rev. B 48 13 051 and Cooper J R Private communication (YBa2Cu3O7−δ

and YBa2Cu1−x Znx O7) and from Hwang H Y et al 1994 Phys. Rev. Lett. 72 2636(La2−x Srx CuO4).

penetration depth alone [164]

kBTKT ≈ 0.9d�2

16πe2λ2ab

. (7.7)

It appears significantly higher than the experimental values in many cases (seetable 7.1). There are also quite a few samples with about the same λab and thesame d but with very different values of Tc, which proves that the phase transitionis not the KT transition. In contrast, our parameter-free fit of the experimentalcritical temperature and the critical behaviour (sections 7.3 and 7.4) favour 3DBose–Einstein condensation of charged bosons as the mechanism for high Tcrather than any low-dimensional phase-fluctuation scenario.

The fluctuation theory [221] further confirms the three-dimensionalcharacter of the phase transition in cuprates. However, it does not mean that

Isotope effect on Tc and on supercarrier mass 205

Table 7.2. Mass enhancement in cuprates.

Compound mab mc

La(0.2) 22.1 3558La(0.15) 15.0 2698La(0.1) 11.3 1909Y(0.0) 7.2 584Y(0.12) 8.3 600Y(0.3) 10.6 1994

all cuprates are in the BEC regime with charged bosons as carriers. Some ofthem, in particular overdoped samples, might be in the BCS or intermediateregime, which makes the BCS–BEC crossover problem relevant. Starting fromthe pioneering works by Eagles [72] and Legget [222], this problem receivedparticular attention in the framework of a negative Hubbard U model [223, 224].Both analytical (diagrammatic [225], path integral [226]) and numerical [227]studies have addressed the intermediate-coupling regime beyond a variationalapproximation [223], including two-dimensional systems [227–229]. However,in using the negative Hubbard U model, we have to realize that this model, whichpredicts a smooth BCS-BEC crossover, cannot be applied to a strong electron–phonon interaction and polaron–bipolaron crossover (section 4.5). An essentialeffect of the polaron band-narrowing is missing in the Hubbard model. Asdiscussed in section 4.7.5, the polaron collapse of the bandwidth is responsiblefor high Tc. It strongly affects the BCS–BEC crossover, significantly reducing thecrossover region.

It is interesting to estimate the effective mass tensor using the penetrationdepth and the Hall ratio. These estimates for in-plane and out-of-plane bosonmasses are presented in table 7.2. They well argee with the inter-site bipolaronmass (sections 4.6 and 5.4). We note, however, that an absolute value of theeffective mass in terms of the free electron mass does not describe the actualband mass renormalization if the bare (band) mass is unknown. Nevertheless, anassumption [141] that the number of carriers is determined by Luttinger’s theorem(i.e. n = 1 + x) would lead to much heavier carriers with m∗ about 100me.

7.2 Isotope effect on T c and on supercarrier mass

The advances in the fabrication of the isotope substituted samples made it possibleto measure a sizeable isotope effect, α = −d ln Tc/d ln M in many high-Tcoxides. This led to a general conclusion that phonons are relevant for high Tc.Moreover, the isotope effect in cuprates was found to be quite different from theBCS prediction: α = 0.5 (or less) (section 2.5). Several compounds showed

206 Superconducting transition

Figure 7.3. The isotope effect on the magnetic field penetration depth in two samples ofLa2−x Srx CuO4 [173] (courtesy of J Hofer).

α > 0.5 [230], and a small negative value of α was found in Bi-2223 [231].

Essential features of the isotope effect, in particular large values in low-Tccuprates, an overall trend to lower value as Tc increases [232] and a small or evennegative α in some high-Tc cuprates were understood using equations (4.225) and(4.226) for the isotope exponents of (bi)polaronic superconductors [82]. Withincreasing ion mass, the bipolaron mass increases and the BEC temperature Tcdecreases in the bipolaronic superconductor. In contrast, an increase in the ionmass leads to band narrowing and an enhancement of the polaron density ofstates and to an increase in Tc in polaronic superconductors. Hence, the isotopeexponent in Tc can distinguish the BCS-like polaronic superconductivity withα < 0 and the BEC of small bipolarons with α > 0. Moreover, underdopedcuprates, which are definitely in the BEC regime (section 5.3), could have α > 0.5(equation (4.226)) as observed.

Another prediction of the bipolaron theory is an isotope effect on the carriermass (equation (4.222)) which is linked with the isotope effect on Tc, accordingto equations (4.224) and (4.226).

Specific heat anomaly 207

Remarkably, this prediction was experimentally confirmed by Zhao et al[173] providing compelling evidence for polaronic carriers in doped cuprates.The effect was observed in the London penetration depth of isotope-substitutedcuprates (figure 7.3). The carrier density is unchanged with the isotopesubstitution of O16 by O18, so that the isotope effect on λab directly measuresthe isotope’s effect on the carrier mass. In particular, the carrier mass isotopeexponent αm = d ln m∗∗/d ln M was found to be as large as αm = 0.8 inLa1.895Sr0.105CuO4. Then, according to equation (4.222), the polaron massenhancement should be m∗∗/m � 5 in this material. Using equation (5.27),we obtain an in-plane bipolaron mass as large as m∗∗ ≈ 10me with the barehopping integral T (NNN) = 0.2 eV. The in-plane magnetic field penetrationdepth, calculated with this mass is λab = [m∗∗/8πne2]1/2 ≈ 316 nm, where n isthe hole density. This agrees well with the experimental one, λab � 320 nm.Using the measured values of λab = 320 nm, λc = 4160 nm and RH =4×10−3 cm3 C−1 (just above Tc), we obtain Tc = 31 K in astonishing agreementwith the experimental value Tc = 30 K in this compound.

7.3 Specific heat anomaly

Bose liquids (or, more precisely, 4He) show the characteristic λ-point singularityin their specific heat but superfluid Fermi liquids like BCS superconductorsexhibit a sharp second-order phase transition accompanied by a finite jump inthe specific heat (section 2.6). It was established beyond doubt [233–237] that theanomaly in high-Tc cuprates differs qualitatively from the BSC prediction. As wasstressed by Salamon et al [238], the heat capacity is logarithmic near the transitionand, consequently, cannot be adequately treated by mean-field BCS theory even ifthe Gaussian fluctuations are included. In particular, estimates using the Gaussianfluctuations yield an unusually small coherence volume (table 7.3), comparablewith the unit cell volume [234] and Gi number (equation (1.53)) of the order ofone.

Table 7.3. Coherence volume � in A3, the in-plane ξab and out-of-plane ξc coherencelengths derived from a Ginzburg–Landau analysis of the specific heat [234].

Compound � ξ2ab (A2) ξc (A)

YBa2Cu3O7 400 125 3.2YBa2Cu3O7−0.025 309 119 2.6YBa2Cu3O7−0.05 250 119 2.1YBa2Cu3O7−0.1 143 119 1.2Ca0.8Y0.2Sr2Tl0.5Pb0.5Cu2O7 84 70 1.2Tl1.8Ba2Ca2.2Cu3O10 40 <0.9

208 Superconducting transition

2.5

3.0

3.5

4.0

4.5

.4 .6 .8 1.0 1.2 1.4

a

ωH/Tc= 00.010.030.10.20.3

T/Tc

C/T

-1.0

-.8

-.6

-.4

-.2

0

.2

.4 .6 .8 1.0 1.2 1.4

H*

b

0.010.030.10.20.3

ωH/Tc=

Hc2(T)

T/Tc

C(H

,T)-

C(0

,T)

Figure 7.4. Temperature dependence of the specific heat divided by temperature (arbitraryunits) of the charged Bose gas scattered off impurities for several fields (ωH = 2eB/m∗∗).(b) shows two anomalies, the lowest one traces the resistive transition, while the highestanomaly is the normal-state feature.

The magnetic field dependence of the anomaly [239] is also unusual butit can be described by the bipolaron model [240, 241]. Calculations of thespecific heat of charged bosons in a magnetic field require an analytical DOS,N(ε, B) of a particle, scattered by other particles and/or by a random potential ofimpurities. We can use the DOS in a magnetic field with an impurity scatteringas in section 4.7.8, which allows for an analytical result (equation (4.266)). Thespecific heat coefficient

C(T, B)

T= d

T dT

∫dε

N(ε, B)ε

exp[(ε − µ)/T ] − 1

calculated with this DOS and with µ determined from nb = ∫dε N(ε, B) f (ε) is

shown in figure 7.4.The broad maximum at T � Tc is practically the same as in an ideal Bose

gas without scattering [240] (appendix B.4.2). It barely shifts in the magneticfield. However, there is a second anomaly at lower temperatures, which is absentin the ideal gas. This shifts with the magnetic field, tracing precisely the resistivetransition (section 7.4), as clearly seen from the difference between the specificheat in the field and zero-field curve, figure 7.4(b). The specific heat (figure 7.4)has a striking resemblance with the Geneva group’s experiments on DyBa2Cu307and on YBa2Cu3O7 [239], where both anomalies were observed. Within thebipolaron model, when the magnetic field is applied, it hardly changes the

Universal upper critical field of unconventional superconductors 209

temperature dependence of the chemical potential near the zero-field Tc becausethe energy spectrum of thermally excited bosons is practically unchanged. This isbecause their characteristic energy (of the order of Tc) remains huge comparedwith the magnetic energy of the order of 2eB/m∗∗. In contrast, the energyspectrum of low-energy bosons is strongly perturbed even by a weak magneticfield. As a result the chemical potential ‘touches’ the band edge at lowertemperatures, while having almost the same ‘kink’-like temperature dependencearound Tc as in a zero field. While the lower anomaly corresponds to the true long-range order due to BEC, the higher one is just a ‘memory’ about the zero-fieldtransition. This microscopic consideration shows that a genuine phase transitioninto a superconducting state is related to a resistive transition (section 7.4) and tothe lower specific heat anomaly, while the broad higher anomaly is the normal-state feature of the bosonic system in an external magnetic field. Differing fromthe BCS superconductor, these two anomalies are well separated in the bosonicsuperconductor at any field but zero.

7.4 Universal upper critical field of unconventionalsuperconductors

The upper critical field (Hc2(T ) = �0/2πξ(T )2) is very different in a BCSsuperconductor (section 1.6.4) and in a charged Bose gas (section 4.7.8). WhileHc2(T ) is linear in temperature near Tc in the Landau theory of second-orderphase transitions, it has a positive curvature (Hc2(T ) � (Tc − T )3/2) in a CBG.Also at zero temperature, Hc2(0) is normally below the Pauli pair-breaking limitgiven by Hp � 1.84Tc (in tesla) in the BCS theory but the limit can be exceededby many times in CBG.

In cuprates [241–247], spin-ladders [248] and organic superconductors[249], high-magnetic field studies revealed a non-BCS upward curvature ofresistive Hc2(T ). When measurements were performed on low-Tc unconventionalsuperconductors [243, 244, 246, 248, 249], the Pauli limit was exceeded severaltimes. A nonlinear temperature dependence in the vicinity of Tc wasunambigously observed in a few samples [241, 245–247]. Importantly, athermodynamically determined Hc2 turned out to be much higher than theresistive Hc2 [250] due to the contrasting magnetic field dependences of thespecific heat anomaly and of resistive transition.

We believe that many unconventional superconductors are in the ‘bosonic’limit of preformed real-space bipolarons, so their resistive Hc2 is actually acritical field of the BEC of charged bosons [121]. Calculations carried out forthe heat capacity of a CBG in section 7.3 led to the conclusion that the resistiveHc2 and the thermodynamically determined Hc2 are very different in bosonicsuperconductors. While the magnetic field destroys the condensate of idealbosons, it hardly shifts the specific heat anomaly as observed.

A comprehensive scaling of resistive Hc2 measurements in unconventional

210 Superconducting transition

0

1

2

0 .2 .4 .6 .8 1.0

T/Tc

Hc2

(T)/

H0

.001

.01

.1

1

.01 .1 1

Tl-2201Bi-2201Sr2Ca12Cu24O41(TMTSF)2PF6LSCOYBCO+ZnL2-xCexCuO4-yBi-2212YBCO-123

1-T/Tc

Figure 7.5. Resistive upper critical field (determined at 50% of the transition) ofcuprates, spin-ladders and organic superconductors scaled according to equation (7.9). Theparameter b is 1 (dots), 0.02 (chain), 0.0012 (full curve), and 0 (dashes). The inset showsa universal scaling of the same data near Tc on the logarithmic scale.

superconductors is shown in figure 7.5 [241] in the framework of the microscopicmodel of charged bosons scattered off impurities (section 4.7.8). The generalizedequation (4.261), accounting for a temperature dependence of the number ofdelocalized bosons nb(T ), can be written as follows:

Hc2(T ) = H0

[nb(T )

tnb(Tc)− t1/2

]3/2

(7.8)

where Tc is the zero-field critical temperature and t = T/Tc. As shownin section 4.7.8, the scaling constant H0 depends on the mean-free path l,H0 = �0/2πξ2

0 , with the characteristic (coherence) length ξ0 � (l/nb(Tc))1/4.

In the vicinity of Tc, one obtains a parameter-free Hc2(T ) ∝ (1 − t)3/2

using equation (7.8) but the low-temperature behaviour depends on a particularscattering mechanism and a detailed structure for the density of localized states.

Universal upper critical field of unconventional superconductors 211

As suggested by the normal-state Hall measurements in cuprates (section 6.1),nb(T ) can be parametrized as nb(T ) = nb(0) + constant × T , so that Hc2(T ) canbe described by a single-parameter expression:

Hc2(T ) = H0

[b(1 − t)

t+ 1 − t1/2

]3/2

. (7.9)

The parameter b is proportional to the number of delocalized bosons at zerotemperature. We expect that this expression is applied in the whole temperatureregion except at ultra-low temperatures, where the Fermi–Dirac golden rule inthe scaling fails. Exceeding the Pauli pair-breaking limit readily follows from thefact that the singlet-pair binding energy is related to the normal-state pseudogaptemperature T ∗, rather than to Tc. T ∗ is higher than Tc in bosonic superconductorsand cuprates (figure 5.6).

The universal scaling of Hc2 near Tc is confirmed by resistive measurementsof the upper critical field of many cuprates, spin-ladders, and organicsuperconductors, as shown in figure 7.5. All measurements reveal a universal(1 − t)3/2 behaviour in a wide temperature region (inset), when they are fitted byequation (7.9). The low-temperature behaviour of Hc2(T )/H0 is not universal butwell described using the same equation with the single fitting parameter, b. Theparameter is close to one in high-quality cuprates with a very narrow resistivetransition [247]. It naturally becomes rather small in overdoped cuprates whererandomness is more essential, so almost all bosons are localized (at least in onedimension) at zero temperature.

Chapter 8

Superconducting state of cuprates

Independent observations of normal-state pseudogaps in a number of magneticand kinetic measurements (chapter 6) and unusual critical phenomena (chapter 7)tell us that many cuprates may not be BCS superconductors. Indeed theirsuperconducting state is as anomalous as the normal one. In particular,there is strong evidence for a d-like order parameter (changing sign whenthe CuO2 plane is rotated by π/2) in cuprates [251]. A number of phase-sensitive experiments [252] provide unambiguous evidence in this direction;furthermore, the low-temperature magnetic penetration depth [253, 254] wasfound to be linear in a few cuprates as expected for a d-wave BCS superconductor.However, superconductor–insulator–normal metal (SIN) and superconductor–insulator–superconductor (SIS) tunnelling studies (sections 8.2 and 8.3), the c-axis Josephson tunnelling [255] and some high-precision magnetic measurements[256] show a more usual s-like symmetry or even reveal an upturn in thetemperature dependence of the penetration depth below some characteristictemperature [257]. Also both angle-resolved photoemission spectroscopy(ARPES) [264] and scanning tunnelling microscopy (STM) [265] have shownthat the maximum energy gap is many times larger and the 2�/Tc ratio is wellabove that expected in weak-coupling BCS theory (∼3.5) or in its intermediate-coupling generalization (chapter 3).

Strong deviations from the Fermi/BCS-liquid behaviour are suggestive of anew electronic state in cuprates, which is a charged Bose liquid of bipolarons.In this chapter we discuss the low-temperature London penetration depth [258],tunnelling [259–261] and ARPES [262], the symmetry of the order parameter andsuperconducting stripes [263] in the framework of the bipolaron theory.

8.1 Low-temperature penetration depth

If the total number of bipolarons in one unit cell is x/2 of which nL are in localizedstates and nb are in delocalized states, then the number in the condensate nc is

nc = 12 x − nL − nb (8.1)

212

Low-temperature penetration depth 213

and the London penetration depth λ ∝ 1/√

nc. Taking the delocalized bipolaronsto be a three-dimensional gas, we have nb ∝ T 3/2. Thus in the low-temperaturelimit, we can neglect nb and make the approximation

nc ≈ x/2 − nL. (8.2)

In this limit, λ(T ) − λ(0) is small and so

λ(T ) − λ(0) ∝ nL(T ) − nL(0) (8.3)

i.e. λ has the same temperature dependence as nL. For small amounts of disorderdelocalized bipolarons may contribute to the low-temperature dependence ofλ(T ) as well; for non-interacting bipolarons moving in d dimensions, this wouldgive λ ∝ T d/2.

In our picture, interacting bosons fill up all the localized single-particlestates in a random potential and Bose-condense into the first extended state atthe mobility edge, Ec. For convenience we choose Ec = 0. When two or morecharged bosons are in a single localized state of energy E , there may be significantCoulomb energy and we take this into account as follows. The localization lengthξ is thought to depend on E via

ξ ∝ 1

(−E)ν(8.4)

where ν > 0. The Coulomb potential energy of p charged bosons confined withina radius ξ can be expected to be

potential energy � 4 p(p − 1)e2

ε0ξ. (8.5)

Thus, the total energy of p bosons in a localized state of energy E is taken to be

w(E) = pE + p(p − 1)κ(−E)ν (8.6)

where κ > 0. We can thus define an energy scale E1:

E1 = κ1/(1−ν). (8.7)

From here on we choose our units of energy such that E1 = 1. We take the totalenergy of charged bosons in localized states to be the sum of the energies of thebosons in individual potential wells. The partition function Z for such a systemis then the product of the partition functions z(E) for each of the wells,

z(E) = eαp20

∞∑p=0

e−α(p−p0)2

(8.8)

214 Superconducting state of cuprates

where

p0 = 12 {1 + (−E)1−ν} (8.9)

α = (−E)ν

θ(8.10)

and

θ = T

E1. (8.11)

The average number of bosons in localized states nL is

nL =∫ 0

−∞dE 〈p〉NL(E) (8.12)

where the mean occupancy 〈p〉 of a single localized state is taken to be

〈p〉 =∑∞

p=0 pe−α(p−p0)2

∑∞p=0 e−α(p−p0)

2 (8.13)

and NL(E) is the one-particle density of localized states per unit cell below themobility edge (section 5.5).

We now focus on the temperature dependence of nL at low temperature(θ � 1) for the case where the (dimensionless) width of the impurity tail γ islarge (γ > 1). We note that the parameter γ is the ratio of the width of the tail tothe characteristic Coulomb repulsion, E1. In the following, we consider the caseν > 1 first and then ν < 1. If ν > 1, we can approximate nL as

nL

NL≈ 1 + ν − 1

2(2 − ν)γ+ 2θ

(2 − ν)γln 2. (8.14)

So we expect nL to be close to the total number of wells NL and to increaselinearly with temperature. Figure 8.1(a) compares this analytical formula withan accurate numerical calculation for the case ν = 1.5, γ = 20. We also notethat even when γ < 1, nL(θ) will still be linear with the same slope providedθ � γ . If ν < 1 we obtain, keeping only the lowest power of θ (valid providedθ1/ν � θ )

nL

NL= 1

2+ �(2 − ν)γ 1−ν

2+ 1 − ν

2(2 − ν)γ− θ

γln 2. (8.15)

Hence, in this case, nL decreases linearly with increasing temperature (in thelow-temperature limit). Figure 8.1(b) compares this analytical formula with thenumerical calculation for the case ν = 0.65, γ = 20. We note that sucha value of ν is typical for amorphous semiconductors [266]. A large valueof γ is also expected in disordered cuprates with their large static dielectricconstant.

Low-temperature penetration depth 215

Figure 8.1. Dependence of the density of localized bosons nL on temperature θ forγ = 20: (a) ν = 1.5, (b) ν = 0.65. The full lines correspond to the low-temperaturepredictions from equations (8.14) and (8.15) while the broken lines are derived from anaccurate numerical calculation.

Figure 8.2. Fit to the London penetration depth measured by Walter et al [257] for aYBCO film. The parameter values from the fit were E1 = 74 K, γ = 20 and ν = 0.67.

Figure 8.2 shows that the low-temperature experimental data [257] on theLondon penetration depth λ of YBCO films can be fitted very well by this theorywith ν < 1. It is more usual to see λ increase linearly with temperature [253,254]and this would correspond to ν > 1 (or to the predominance of the effect ofdelocalized bipolarons moving in two dimensions). We believe that ν < 1 ismore probable for a rapidly varying random potential while ν ≥ 1 is morelikely for a slowly varying one. Both ν < 1 and ν ≥ 1 are observed in dopedsemiconductors [266]. Hence, it is not surprising that a drastically different low-temperature dependence of the London penetration depth is observed in differentsamples of doped cuprates. In the framework of this approach, λ(T ) is relatedto the localization of carriers at low temperatures rather than to any energy scalecharacteristic of the condensate or to its symmetry. The excitation spectrum of thecharged Bose liquid determines, however, the temperature dependence of λ(T ) athigher temperatures.

The key parameter of the temperature dependence of the London penetrationdepth is the exponent ν of the localization length of bosons. Experimentally, themore usual linear increase of λ with temperature is observed in samples whicheither have no disorder or a shallow and smooth random potential profile. Here we

216 Superconducting state of cuprates

expect ν ≥ 1 and a substantial contribution from delocalized bipolarons movingprincipally in two dimensions and thus, that λ increases linearly with temperature,as observed [254]. However, heavy-ion bombardment [257] introduces ratherdeep and narrow potential wells for which we might expect ν < 1. This wouldexplain the upturn in the temperature dependence of λ in the disordered films[257].

8.2 SIN tunnelling and Andreev reflection

There is compelling experimental evidence that the pairing of carriers takes placewell above Tc in cuprates (chapter 6), the clearest being uniform susceptibility[186, 267] and tunnelling [265]. The gap in tunnelling and photoemissionis almost temperature independent below Tc [265, 268] and exists above Tc[265, 269–271]. Kinetic [198] and thermodynamic [272] data suggest that thegap opens in both charge and spin channels at any relevant temperature in a widerange of doping. At the same time, reflection experiments, in which an incomingelectron from the normal side of a normal/superconducting contact is reflectedas a hole along the same trajectory (section 2.13), revealed a much smaller gapedge than the bias at the tunnelling conductance maxima in a few underdopedcuprates [273]. Other tunnelling measurements [274, 275] also showed distinctlydifferent superconducting- and normal-state gaps.

In the framework of bipolaron theory, we can consider a simplified model,which describes the temperature dependence of the gap and tunnelling spectrain cuprates and accounts for two different energy scales in the electron-holereflection [260]. The assumption is that the attraction potential in cupratesis large compared with the (renormalized) Fermi energy of polarons. Themodel is a generic one-dimensional Hamiltonian including the kinetic energy ofcarriers in the effective mass (m∗) approximation and a local attraction potential,V (x − x ′) = −Uδ(x − x ′), as

H =∑

s

∫dx ψ†

s (x)

(− 1

2m∗d2

dx2− µ

)ψs(x)

− U∫

dx ψ†↑(x)ψ

†↓(x)ψ↓(x)ψ↑(x) (8.16)

where s = ↑,↓ is the spin. The first band in cuprates to be doped is theoxygen band inside the Hubbard gap (section 5.2). This band is quasi-one-dimensional as discussed in section 5.4, so that a one-dimensional approximation(equation (8.16)) is a realistic starting point. Solving a two-particle problemwith the δ-function potential, one obtains a bound state with the binding energy2�p = m∗U2/4 and with the radius of the bound state r = 2/(m∗U). Weassume that this radius is less than the inter-carrier distance in cuprates. It is thenthat real-space bipolarons are formed. If three-dimensional corrections to theenergy spectrum of pairs are taken into account, the ground state of the system is

SIN tunnelling and Andreev reflection 217

the Bose–Einstein condensate (BEC). The chemical potential is pinned below theband edge by about �p both in the superconducting and normal states, so that thenormal state single-particle gap is �p .

Now we take into account that in the superconducting state (T < Tc)single-particle excitations interact with the condensate via the same potential U .Applying the Bogoliubov approximation (section 2.2) we reduce the Hamiltonian(8.16) to a quadratic form:

H =∑

s

∫dx ψ†

s (x)

(− 1

2m∗d2

dx2− µ

)ψs(x)+

∫dx [�cψ

†↑(x)ψ

†↓(x)+ H.c.]

(8.17)where the coherent pairing potential, �c = −U〈ψ↓(x)ψ↑(x)〉, is proportional

to the square root of the condensate density, �c = constant × n1/2c (T ). The

single-particle excitation energy spectrum, E(k), is found using the Bogoliubovtransformation as

E(k) = [(k2/2m∗ + �p)2 + �2

c]1/2 (8.18)

if one assumes that the condensate density does not depend on position. Thisspectrum is quite different from the BCS quasi-particles because the chemicalpotential is negative with respect to the bottom of the single-particle band,µ = −�p . The single-particle gap, �/2, defined as the minimum of E(k), isgiven by

�/2 = [�2p + �2

c(T )]1/2. (8.19)

It varies with temperature from �(0)/2 = [�2p + �2

c(0)]1/2 at zero temperaturedown to the temperature-independent �p above Tc. The theoretical temperaturedependence, equation (8.19), describes well the pioneering experimentalobservation of the anomalous gap in YBa2Cu3O7−δ in the electron-energy-lossspectra by Demuth et al [277] (figure 8.3) with �2

c(T ) = �2c(0) × [1 − (T/Tc)

n]below Tc and zero above Tc, and n = 4. We note that n = 4 is an exponent whichis expected in the two-fluid model of any superfluid [276].

A normal metal–superconductor (SIN) tunnelling conductance via adielectric contact, dI/dV is proportional to the density of states, ρ(E), of thespectrum (equation (8.18)). Taking into account the scattering of single-particleexcitations by a random potential, as well as thermal lattice and spin fluctuations(see section 8.4.3), one finds, at T = 0

dI/dV = constant ×[ρ

(2eV − �

�0

)+ Aρ

(−2eV − �

�0

)](8.20)

with

ρ(ξ) = 4

π2× Ai(−2ξ)Ai′(−2ξ) + Bi(−2ξ)Bi′(−2ξ)

[Ai2(−2ξ) + Bi2(−2ξ)]2(8.21)

A is an asymmetry coefficient, explained in [259], Ai(x) and Bi(x) the Airyfunctions and �0 is the scattering rate.

218 Superconducting state of cuprates

Figure 8.3. Temperature dependence of the gap (equation (8.19) (line)) compared with theexperiment [277] (dots) for �p = 0.7�(0).

We compare the conductance (equation (8.20)) with one of the bestSTM spectra measured in Ni-substituted Bi2Sr2CaCu2O8+x single crystals byHancottee et al [268] in figure 8.4(a). This experiment shows anomalouslylarge �/Tc > 12 with the temperature dependence of the gap similar tothat in figure 8.3 below Tc. The theoretical conductance (equation (8.20))describes well the anomalous gap/Tc ratio, injection/emission asymmetry, zero-bias conductance at zero temperature and the spectral shape inside and outsidethe gap region. There is no doubt that the gap (figure 8.4(a)) is s-like. Theconductance (equation (8.20)) also fits well the conductance curve obtained on‘pure’ Bi-2212 single crystals, as shown in figure 8.4(b), while a simple d-waveBCS density of states cannot describe the excess spectral weight in the peaks andthe shape of the conductance outside the gap region. We note that the scatteringrate, �0, is apparently smaller in the ‘pure’ sample than in the Ni-substitutedsample, as expected.

A simple theory of tunnelling into a bosonic (bipolaronic) superconductorin a metallic (no-barrier) regime follows from this model. As in the canonicalBCS approach applied to normal metal–superconductor tunnelling by Blonderet al [47], the incoming electron produces only outgoing particles in thesuperconductor (x > l), allowing for a reflected electron and (Andreev) holein the normal metal (x < 0) (section 2.13). There is also a buffer layer of thethickness l at the normal metal–superconductor boundary (x = 0), where thechemical potential with respect to the bottom of the conduction band changesgradually from a positive large value µ in metal to a negative value −�p ina bosonic superconductor. We approximate this buffer layer by a layer with aconstant chemical potential µb (−�p < µb < µ) and with the same strength ofthe pairing potential �c as in a bulk superconductor. The Bogoliubov equationsmay be written as usual (section 2.13), with the only difference being that thechemical potential with respect to the bottom of the band is a function of the

SIN tunnelling and Andreev reflection 219

Figure 8.4. Theoretical tunnelling conductance (equation (8.20) (line)) compared with theexperimental STM conductance (dots) in (a) Ni-substituted Bi2Sr2CaCu2O8+x [268] with� = 90 meV, A = 1.05, �0 = 40 meV and (b) in ‘pure’ Bi-2212 [268] with � = 43 meV,A = 1.2 and �0 = 18 meV.

coordinate x :

Eψ(x) =(−(1/2m) d2/dx2 − µ(x) �c

�c (1/2m) d2/dx2 + µ(x)

)ψ(x). (8.22)

Thus, the two-component wavefunction in normal metal is given by

ψn(x < 0) =(

10

)eiq+x + b

(10

)e−iq+x + a

(01

)e−iq−x (8.23)

220 Superconducting state of cuprates

Figure 8.5. Transmission versus voltage (measured in units of �p/e) for �c = 0.2�p ,µ = 10�p , µb = 2�p and l = 0 (bold line), l = 1 (bold dashes), l = 4 (thin line), andl = 8 (thin dashes) (in units of 1/(2m�p)1/2).

while in the buffer layer, it has the form

ψb(0 < x < l) = α

(1�c

E+ξ

)eip+x + β

(1�c

E−ξ

)e−ip−x

+ γ

(1�c

E+ξ

)e−ip+x + δ

(1�c

E−ξ

)eip−x (8.24)

where the momenta associated with the energy E are q± = [2m(µ ± E)]1/2 andp± = [2m(µb ± ξ)]1/2 with ξ = (E2 − �2

c)1/2. A well-behaved solution in the

superconductor with a negative chemical potential is given by

ψs(x > l) = c

(1�c

E+ξ

)eik+x + d

(1�c

E−ξ

)eik−x (8.25)

where the momenta associated with the energy E are k± = [2m(−�p ± ξ)]1/2.The coefficients a, b, c, d, α, β, γ, δ are determined from the boundary

conditions, which are the continuity of ψ(x) and its derivatives at x = 0 andx = l. Applying the boundary conditions and carrying out an algebraic reduction,

SIN tunnelling and Andreev reflection 221

Figure 8.6. Transmission versus voltage (measured in units of �c/e) for �p = 0.2�c,µ = 10�c, µb = 2�c and l = 0 (bold line), l = 1 (bold dashes), l = 4 (thin line), andl = 8 (thin dashes) (in units of 1/(2m�c)

1/2).

we find

a = 2�cq+(p+ f −g+ − p− f +g−)/D (8.26)

b = − 1 + 2q+

D[(E + ξ) f +(q− f − − p−g−) − (E − ξ) f −(q− f + − p+g+)]

(8.27)

with

D = (E + ξ)(q+ f + + p+g+)(q− f − − p−g−)

− (E − ξ)(q+ f − + p−g−)(q− f + − p+g+) (8.28)

and

f ± = p± cos(p±l) − ik± sin(p±l) (8.29)

g± = k± cos(p±l) − ip± sin(p±l).

The transmission coefficient in the electrical current, 1+|a|2−|b|2 is shownin figure 8.5 for different values of l, when the coherent gap �c is smaller than thepair-breaking gap �p and in figure 8.6 in the opposite case, �p < �c. In the first

222 Superconducting state of cuprates

case, figure 8.5, we find two distinct energy scales, one is �c in the subgap regiondue to the electron-hole reflection and the other one is �/2, which is a single-particle band edge. However, there is only one gap, �c, which can be seen in thesecond case, figure 8.6. We note that the transmission has no subgap structure ifthe buffer layer is absent (l = 0) in both cases. In the extreme case of a widebuffer layer, l � (2m�p)

−1/2 (figure 8.5) or l � (2m�c)−1/2 (figure 8.6), there

are some oscillations of the transmission due to the bound states inside the bufferlayer.

We expect that �p � �c in underdoped cuprates (figure 8.5) while �p ≤�c in optimally doped cuprates (figure 8.6). Thus, the model accounts for twodifferent gaps experimentally observed in Giaver tunnelling and electron-holereflection in underdoped cuprates and for a single gap in optimally doped samples[273]. An oscillating structure, observed in underdoped YBa2Cu3O7−δ [273], isalso found in the theoretical conductance at finite l (figure 8.5). The transmissions(figures 8.5 and 8.6) are due to a coherent tunnelling into the condensate andinto a single-particle band of the bosonic superconductor. There is an incoherenttransmission into localized single-particle impurity states and into incoherent(‘supracondensate’) bound pair states as well, which might explain a significantfeatureless background in the subgap region [273].

8.3 SIS tunnelling

Within the standard approximation [76], the tunnelling current, I (V ), betweentwo parts of a superconductor separated by an insulating barrier is proportionalto a convolution of the Fourier component of the single-hole retarded Green’sfunction, GR(k, ω), with itself as

I (V ) ∝∑k, p

∫ ∞

−∞dω Im GR(k, ω) Im GR( p, e|V | − ω) (8.30)

where V is the voltage in the junction.The problem of a hole on a lattice coupled with the bosonic field of lattice

vibrations has a solution in terms of the coherent (Glauber) states in the strong-coupling limit, λ > 1, where the Migdal–Eliashberg theory cannot be applied(section 4.3.4),

GR(k, ω) = Z∞∑

l=0

∑q1,...,ql

∏lr=1 |γ (qr )|2

(2N)l l!(ω − ∑lr=1 ωqr

− ε(k + ∑lr=1 qr ) + iδ)

.

(8.31)The hole energy spectrum, ε(k), is renormalized due to the polaron narrowing ofthe band and (in the superconducting state) also due to interaction with the BECof bipolarons as discussed in section 8.2,

ε(k) = [ξ2(k) + �2c]1/2. (8.32)

SIS tunnelling 223

Here ξ(k) = Z ′E(k) − µ is the renormalized polaron band dispersion withthe chemical potential µ and E(k) = ∑

m T (m) exp(−ik · m) is the bare banddispersion in a rigid lattice.

Quite differently from the BCS superconductor, the chemical potential µ isnegative in the bipolaronic system, so that the edge of the single-hole band isfound above the chemical potential at −µ = �p . Near the edge the parabolicone-dimensional approximation for the oxygen hole is applied, compatible withthe ARPES data [262],

εk � k2x

2m∗ + �/2. (8.33)

Differently from the canonical Migdal–Eliashberg Green function (chapter 3),there is no damping (‘dephasing’) of low-energy polaronic excitations inequation (8.31) due to the electron–phonon coupling alone. This coupling leadsto a coherent dressing of low-energy carriers by phonons, which is seen in theGreen fucntion as phonon sidebands with l ≥ 1.

However, elastic scattering by impurities yields a finite life-time for theBloch polaronic states. For the sake of analytical transparency, here we model thisscattering as a constant imaginary self-energy, replacing iδ in equation (8.31) bya finite i�/2. In fact, the ‘elastic’ self-energy can be found explicitly as a functionof energy and momentum (see section 8.4.3). Its particular energy/momentumdependence is essential in the subgap region of tunnelling, where it determinesthe value of zero-bias conductance. However, it hardly plays any role in the peakregion and higher voltages.

Substiting equation (8.33) into equations (8.31) and (8.30), and performingthe intergration with respect to frequency and both momenta, we obtain for thetunnelling conductance, σ(V ) = dI/dV ,

σ(V ) ∝∞∑

l,l′=0

∑q,q ′

∏lr=1

∏l′r ′=1 |γ (qr )|2|γ (q′

r ′)|2(2N)l+l′ l!l ′!

× L

[e|V | − � −

l∑r=1

ωqr−

l′∑r ′=1

ω(q ′r ′ ), �

](8.34)

where L[x, �] = �/(x2 + �2). To perform the remaining integrations andsummations, we introduce a model analogue of the Eliashberg spectral function(α2 F(ω) (section 3.4)) by replacing the q-sums in equation (8.34) by

1

2N

∑q

|γ (q)|2 A(ωq) = g2

π

∫dω L[ω − ω0, δω]A(ω) (8.35)

for any arbitrary function of the phonon frequency A(ωq). In this way weintroduce the characteristic frequency, ω0, of phonons strongly coupled withholes, their average number g2 in the polaronic cloud and their dispersion δω.

224 Superconducting state of cuprates

As soon as δω is less than ω0, we can extend the integration over phononfrequencies from −∞ to ∞ and obtain

σ(V ) ∝∞∑

l,l′=0

g2(l+l′)

l!l ′! L[e|V | − � − (l + l ′)ω0, � + δω(l + l ′)]. (8.36)

By replacing the Lorentzian in equation (8.36) by the Fourier integral, weperform the summation over l and l ′ with the final result for the conductance asfollows:

σ(V ) ∝∫ ∞

0dt exp[2g2e−δωt cos(ω0t) − �t]

× cos[2g2e−δωt sin(ω0t) − (e|V | − �)t]. (8.37)

From the isotope effect on the carrier mass, phonon densities of states,experimental values of the normal-state pseudogap and the residual resistivity(chapters 6 and 7), one estimates the coupling strength g2 to be of the order of one,the characteristic phonon frequency about 30–100 meV, the phonon frequencydispersion about a few tens meV, the gap �/2 about 30–50 meV and the impurityscattering rate of the order of 10 meV.

The SIS conductance (equation (8.37)) calculated with parameters in thisrange is shown in figure 8.7 for three different values of coupling. Theconductance shape is remarkably different from the BCS density of states, boths-wave and d-wave. There is no Ohm’s law in the normal region, e|V | > �,the dip/hump features (due to phonon sidebands) are clearly seen already in thefirst derivative of the current, there is a substantial incoherent spectral weightbeyond the quasi-particle peak for the strong coupling, g2 ≥ 1, and there isan unusual shape for the quasi-particle peaks. All these features as well as thetemperature dependence of the gap are beyond BSC theory no matter what thesymmetry of the gap is. However, they agree with the experimental SIS tunnellingspectra in cuprates [265,268,269,274,275]. In particular, the theory quantitativelydescribes one of the best tunnelling spectra obtained on Bi2Sr2CaCu2O8+δ singlecrystals by the break-junction technique [268] (figure 8.8). Some excess zero-bias conductance compared with the experiment is due to our approximation ofthe ‘elastic’ self-energy. The exact (energy-dependent) self-energy provides anagreement in this subgap region as well, as shown in figure 8.9. The dynamicconductance of Bi-2212 mesas [274] at low temperatures is almost identical tothe theoretical one in figure 8.7 for g2 = 0.5625.

The unusual shape of the main peaks (figure 8.8) is a simple consequenceof the (quasi-)one-dimensional hole density of states near the edge of the oxygenband. The coherent (l = l ′ = 0) contribution to the current with no elasticscattering (� = 0) is given by

I0 ∝∫ ∞

�

dε

(ε − �/2)1/2

∫ ∞

�

dε′

(ε′ − �/2)1/2δ(ε + ε′ − e|V |) (8.38)

SIS tunnelling 225

Figure 8.7. SIS tunnelling conductance in a bipolaronic superconductor for differentvalues of the electron–phonon coupling, g2, and �/2 = 29 meV, ω0 = 55 meV,δω = 20 meV, � = 8.5 meV.

so that the conductance is a δ function:

σ0(V ) ∝ δ(e|V | − �). (8.39)

Hence, the width of the main peaks in the SIS tunnelling (figure 8.8) measures theelastic scattering rate directly.

The disappearance of the quasi-particle sharp peaks above Tc in Bi-cupratescan also be explained in the framework of bipolaron theory. Below Tc, bipolaroniccondensate provides an effective screening of the long-range (Coulomb) potentialof impurities, while above Tc the scattering rate might increase by many times(section 8.5). This sudden increase in � in the normal state washes out sharp peaksfrom tunnelling and ARPES spectra. The temperature-dependent part, �c(T ),of the total gap (equation (8.19)) is responsible for the temperature shift of the

226 Superconducting state of cuprates

Figure 8.8. Theoretical conductance of figure 8.7 for g2 = 0.5625 (full line) comparedwith the tunnelling spectrum obtained on Bi2Sr2CaCu2O8+δ single crystals by thebreak-junction technique [268] (dots).

main peaks in the superconducting state of an SIS junction, while the incoherenttemperature-independent part �p survives well above Tc as observed [274, 275].

8.4 ARPES

Let us discuss ARPES in doped charge-transfer Mott insulators in the frameworkof bipolaron theory [262] to describe some unusual ARPES features of high-TcYBa2Cu3O7−δ (Y123), YBa2Cu4O8 (Y124) and several other materials.

Perhaps the most intriguing feature of ARPES in cuprates is an extremelynarrow and intense peak lying below the Fermi energy, which is most clearlyseen near the Y and X points in Y124 [278], and Y123 [279]. Its angulardependence and spectral shape as well as the origin of the featureless (butdispersive) background remain unclear. Some authors [278] refer to the peakas an extended van Hove singularity (evHs) arising from a plane (CuO2) stronglycorrelated band. They also implicate the resulting (quasi-)one-dimensional DOSsingularity as a possible origin for the high transition temperature. However,polarized ARPES studies of untwinned Y123 crystals of exceptional quality [279]unambiguously show that the peak is a narrow resonance arising primarily fromthe quasi-one-dimensional CuO3 chains in the buffer layers rather than from theplanes. Interestingly, a very similar narrow peak was observed by Park et al [280]in high-resolution ARPES near the gap edge of the cubic semiconductor FeSi withno Fermi surface at all.

As discussed in section 5.2, cuprates and many other transition metalcompounds are charge transfer Mott–Hubbard insulators, where the first band tobe doped is the oxygen band lying within the Hubbard gap, as shown in figure 5.3.

ARPES 227

A single photoexited oxygen hole is described by the polaron spectral function ofsection 4.3.4. Its low-energy part is affected by low-frequency thermal lattice,spin and random fluctuations. The latter can be described as a ‘Gaussian whitenoise’ potential. Also p-hole polarons in oxides are almost one-dimensional dueto a large difference in the ppσ and ppπ hopping integrals. This allows usto explain the shape of the narrow peaks in ARPES using the spectral densityA(k, E) of a one-dimensional particle in a Gaussian white noise potential [281].

8.4.1 Photocurrent

The interaction of the crystal with the electromagnetic field of frequency ν isdescribed by the following Hamiltonian (in the dipole approximation):

Hint = (8π I )1/2 sin(νt)∑k,k′

(e · dkk′)c†kh†

k′ + H.c. (8.40)

where I is the intensity of the radiation with polarization e, k is the momentumof the final state (i.e. of the photoelectron registered by the detector), k′ is the(quasi-)momentum of the hole remaining in the sample after the emission andc†

k and h†k′ are their creation operators, respectively. For simplicity, we suppress

the band index in h†k′ . Due to the translational symmetry of the Bloch states,

|k′〉 ≡ u−k′(r) exp(−ik′ · r) (appendix A), there is a momentum conservation inthe dipole matrix element,

dkk′ = d(k)δk+k′,G (8.41)

with

d(k) = ie(N/v0)1/2∇k

∫v0

e−iG·ruk−G(r) dr (8.42)

and v0 is the unit cell volume (G is a reciprocal-lattice vector). The Fermi–Diracgolden rule gives the photocurrent to be

I (k, E) = 4π2 I |e ·d(k)|2∑i, f

e�+µNi −Ei |〈 f |h†k−G |i〉 f |2δ(E + Ef − Ei ) (8.43)

where E is the binding energy, Ei, f is the energy of the initial and final states and�,µ, Ni are the thermodynamic and chemical potentials and number of holes,respectively. By definition, the sum in equation (8.43) is n(E)A(k − G,−E),

where the spectral function

A(k − G, E) = − 1

πIm GR(k − G, E) (8.44)

is proportional to the imaginary part of the retarded GF (appendix D) andn(E) = [exp(E/T )+1]−1 is the Fermi distribution. In the following, we consider

228 Superconducting state of cuprates

temperatures well below the experimental energy resolution, so that n(E) = 1, ifE is negative, and zero otherwise, and we put G = 0.

The spectral function depends on essential interactions of a single holewith the rest of the system. The most important interaction in oxides isthe Frohlich electron–phonon interaction with c-axis polarized high-frequencyphonons (section 5.2), which leads to the polaron spectral function (4.79). As aresult, we obtain

I (k, E) ∼ |d(k)|2n(E)Zδ(E + ξk) + Iincoh(k, E) (8.45)

where Iincoh(k, E) is an incoherent part, which spreads from about −ω0 down to−2E p. There might be some multi-phonon structure in Iincoh(k, E) as observedin tunnelling (section 8.3).

Here we concentrate on the angular, spectral and polarization dependence ofthe first coherent term in equation (8.45). The present experimental resolution[264] allows the intrinsic damping of the coherent quasi-particle excitations to beprobed. The damping appears due to a random field and low-frequency latticeand spin fluctuations described by the polaron self-energy, �(k, E), so that thecoherent part of the spectral function is given by

Ac(k, E) = − Z

π

Im �(k, E)

[E + Re �(k, E) − ξk)]2 + [Im �(k, E)]2. (8.46)

Hence, the theory of narrow ARPES peaks is reduced to determining the self-energy of a hole.

8.4.2 Self-energy of one-dimensional hole in a non-crossing approximation

Due to energy conservation, small polarons exist in the Bloch states attemperatures below the optical phonon frequency T < ω0/2 (section 4.3.2). Afinite polaron self-energy appears due to (quasi-)elastic scattering off impurities,a low-frequency deformation potential and spin fluctuations. First we apply thesimplest non-crossing (ladder) approximation (chapter 3) to define an analytical�(k, E). Within this approximation the self-energy is k-independent for a short-range scattering potential like a deformation or a screened impurity potential, sothat

�(E) �∑

k

GR(k, E) (8.47)

where GR(k, E) = [E − ξk − �(E)]−1.The hole energy spectrum is parametrized in a tight-binding model as

ξx,yk = 2t cos(kx,ya) − 2t ′ cos(ky,xa) − µ. (8.48)

We assume that the minima of two polaron bands (equation (8.48)), are found atthe Brillouin zone boundary in X (π, 0) and Y (0, π).

ARPES 229

As previously mentioned, the oxygen hole is (quasi-)one-dimensional due toa large difference between the oxygen hopping integrals for the orbitals elongatedparallel to and perpendicular to the oxygen–oxygen hopping t ′ � t . This allowsus to apply a one-dimensional approximation, reducing equation (8.48) to twoone-dimensional parabolic bands near the X and Y points, ξ

x,yk = k2/2m∗ − µ

with m∗ = 1/2ta2 and k taking relative to (π, 0) and (0, π), respectively. Then,equation (8.47) for the self-energy in the non-crossing approximation takes thefollowing form:

�(ε) = −2−3/2[�(ε) − ε]−1/2 (8.49)

for each doublet component. Here we introduce a dimensionless energy (andself-energy), ε ≡ (E + µ)/�0 using �0 = (D2m∗)1/3 as the energy unit.The constant D is the second moment of the Gaussian white noise potential,comprising thermal and random fluctuations as D = 2(V 2

0 T/M + nimα2), whereV0 is the amplitude of the deformation potential, M is the elastic modulus, nim isthe impurity density and α is the coefficient of the δ-function impurity potential(i.e. the strength of the scattering potential). The solution is

�(ε) = ε

3−

(1 + i31/2

2

) 1

16+ ε3

27+

(1

256+ ε3

216

)1/2

1/3

−(

1 − i31/2

2

) 1

16+ ε3

27−

(1

256+ ε3

216

)1/2

1/3

. (8.50)

While the energy resolution in the present ARPES studies is almost perfect[264], the momentum resolution remains finite in most experiments, δ > 0.1π/a.Hence, we have to integrate the spectral function (equation (8.46)) with aGaussian momentum resolution to obtain the experimental photocurrent:

I (k, E) �

∫ ∞

−∞dk ′ Ac(k

′,−E) exp

[− (k − k ′)2

δ2

]. (8.51)

The integral is expressed in terms of �(ε) and the error function w(z) as follows:

I (k, E) � −2Z

δIm{�(ε)[w(z1) + w(z2)]} (8.52)

where z1,2 = [±k − i/2�(ε)]/δ, w(z) = e−z2erfc(−iz) and ε = (−E + µ)/�0.

This photocurrent is plotted as broken lines in figure 8.9 for two momenta,k = 0.04π/a (almost Y or X points of the Brillouin zone) and k = 0.3π/a.The chemical potential is placed in the charge transfer gap below the bottomof the hole band, µ = −20 meV, the momentum resolution is taken as δ =0.28π/a and the damping �0 = 19 meV. The imaginary part of the self-energy(equation (8.50)) disappears below ε = −3/25/3 � −0.9449. Hence, this

230 Superconducting state of cuprates

Figure 8.9. The polaron spectral function, integrated with the momentum resolutionfunction for two angles, k = 0.04π/a (upper curves) and k = 0.30π/a with the damping�0 = 19 meV, the momentum resolution δ = 0.28π/a and the polaron mass m∗ = 9.9me.The bipolaron binding energy 2|µ| = 40 meV. The broken curves are the spectral densityintegrated with the momentum resolution in the non-crossing approximation.

approximation gives a well-defined gap rather than a pseudogap. Actually, thenon-crossing approximation fails to describe the localized states inside the gap(i.e. a tail of the density of states). We have to go beyond a simple ladder todescribe the single-electron tunnelling inside the gap and the ARPES spectra atsmall binding energies.

8.4.3 Exact spectral function of a one-dimensional hole

The exact spectral function for a one-dimensional particle in a random Gaussianwhite noise potential was calculated by Halperin [281] and the density of statesby Frisch and Lloyd [282]. Halperin derived two pairs of differential equationsfrom whose solutions the spectral function may be calculated:

Ac(k, ε) = 4∫ ∞

−∞p0(−z) Re p1(z) dz. (8.53)

Here p0,1(z) obeys two differential equations:

[d2

dz2+ d

dz(z2 + 2ε)

]p0 = 0 (8.54)

and [d2

dz2 + d

dz(z2 + 2ε) − z − ik

]p1 + p0 = 0 (8.55)

ARPES 231

with the boundary conditions

limz→∞ z2−n pn(z) = lim

z→−∞ z2−n pn(z) (8.56)

where k is measured in units of k0 = (D1/2m∗)2/3. The first equation may beintegrated to give

p0(z) = exp(−z3/3 − 2zε)∫ z−∞ exp(u3/3 + 2uε) du

π1/2∫ ∞

0 u−1/2 exp(−u3/12 − 2uε) du. (8.57)

The equation for p1(z) has no known analytic solution and, hence, must be solvednumerically. There is, however, an asymptotic expression for Ac(k, ε) in the tailwhere |ε| � 1:

Ac(k, ε) � 2π(−2ε)1/2 exp

[−4

3(−2ε)3/2

]cosh2

[πk

(−8ε)1/2

]. (8.58)

The result for Ac(k,−E) integrated with the Gaussian momentum resolutionis shown in figure 8.9 for two values of the momentum (full lines). In contrastto the non-crossing approximation, the exact spectral function (averaged withthe momentum resolution function) has a tail due to the states localized bydisorder within the normal-state gap. However, besides this tail, the non-crossingapproximation gives very good agreement and for a binding energy greater thanabout 30 meV, it is practically exact.

The cumulative DOS (appendix B)

K0(ε) = (2π)−1∫ ε

−∞dε′

∫ ∞

−∞dk A p(k, ε′) (8.59)

is expressed analytically [282] in terms of the tabulated Airy functions Ai(x) andBi(x) as

K0(ε) = π−2[Ai2(−2ε) + Bi2(−2ε)]−1. (8.60)

DOS ρ(ε) = dK0(ε)/dε fits well the voltage–current tunnelling characteristics ofcuprates, as discussed in section 8.2.

8.4.4 ARPES in Y124 and Y123

With the polaronic doublet (equation (8.48)) placed above the chemical potential,we can quantitatively describe high-resolution ARPES in Y123 [279] and Y124[278]. The exact one-dimensional polaron spectral function, integrated withthe experimental momentum resolution (shown in figure 8.10), provides aquantitative fit to the ARPES spectra in Y124 along the Y–� direction. Theangular dispersion is described with the polaron mass m∗ = 9.9me. The spectralshape is reproduced well with �0 = 19 meV. There is also quantitative agreement

232 Superconducting state of cuprates

Figure 8.10. Theoretical ARPES spectra for the Y–� direction (b). Parameters are thoseof figure 8.9. The theory provides a quantitative fit to experiment (a) [278] in this scanningdirection.

Figure 8.11. Theoretical ARPES spectra in Y124 for the Y–S direction (b). The theoreticalfit agrees well with experiment (a) [278] in a restricted range of kx near the Y-point and,outside this range, the theoretical peaks are somewhat higher than in experiment.

ARPES 233

Figure 8.12. Energy-integrated ARPES intensity in Y124 in the Y–� (a) and Y–S (b)directions. Momenta are measured relative to the Y-point of the Brillouin zone.

between theory and experiment in the perpendicular direction Y–S, in a restrictedregion of small kx , where almost no dispersion is observed around Y (figure 8.11).

However, there is a significant loss of the energy-integrated intensity alongboth directions (figure 8.12) which the theoretical spectral function alone cannotaccount for. The energy-integrated ARPES spectra obey the sum rule,∫ ∞

−∞dE I (k, E) � |d(k)|2 (8.61)

if the chemical potential is pinned inside the charge-transfer gap. Therefore,we have to conclude that the dipole matrix element depends on k. The rapidloss of the integrated intensity in the Y–S direction was interpreted by someauthors [283] as a Fermi-surface crossing. While the Fermi-surface crossingmight be compatible with our scenario (see figure 5.3(b)), it is hard to believethat it has been really observed in Y124. Indeed, the peaks in the Y–S directionare all 15 meV or more below the Fermi level—at a temperature of 1 meV. If theloss of spectral weight were due to a Fermi-surface crossing, one would expectthe peaks to approach much closer to the Fermi level.

Also the experimental spectral shape of the intensity at k = kF isincompatible with any theoretical scenario, including different marginal Fermi-liquid models, as shown in figure 8.13. The spectral function on the Fermi surfaceshould be close to a simple Lorentzian:

Ac(kF, E) ∼ |E |βE2 + constant × E2β

(8.62)

because the imaginary part of the self-energy behaves as |E |β with 0 ≤ β ≤ 2in a Fermi or in a marginal Fermi liquid. In contrast, the experimental intensity

234 Superconducting state of cuprates

Figure 8.13. The experimental ARPES signal (full line) on the alleged Fermi surface doesnot correspond to a Fermi-liquid spectral function (broken line). We assume particle–holesymmetry to obtain the spectral function for negative binding energy.

shows a pronounced minimum at the alleged Fermi surface (figure 8.13). If thereis indeed no Fermi-surface crossing, why then do some determinations point toa large Fermi surface in cuprates, which is drastically incompatible with theirkinetic and thermodynamic properties? It might be due to the fact that the oxygenhole band has its minima at large k inside or even on the boundary of the Brillouinzone. Then ARPES show intense peaks near large k imitating a large Fermi-surface.

8.5 Sharp increase of the quasi-particle lifetime below T c

It has been observed at certain points in the Brillouin zone that the ARPES peak inbismuth cuprates is relatively sharp at low temperatures in the supercondunctingstate but that it almost disappears into the background above the transition [284,285]. A very sharp increase in the quasi-particle lifetime in the superconductingstate has been observed in the thermal Hall conductivity measurements [286].The abrupt appearence of the quasi-particle state below Tc is also implied in thetunnelling I–V characteristics [274, 275].

Here we show that a large increase in the quasi-particle lifetime below Tcis due to the screening of scatterers by the Bose–Einstein condensate of chargedbipolarons [203, 287]. To illustrate the point, let us calculate the scattering crosssection of a charged particle (mass m, charge e) scattered by a static Coulombpotential V (r) screened by a charged Bose gas. The general theory of potentialscattering in terms of phase shifts was developed in the earliest days of quantummechanics [288]. While, in principle, this allows scattering cross sections to becalculated for an arbitary potential, in practice the equations for the radial part

Sharp increase of the quasi-particle lifetime below Tc 235

of the wavefunction may only be solved analytically for a few potentials and, inthe standard formulation, are not in a suitable form for numerical computation.The ‘variable phase’ approach [289] solves this problem by taking the phase-shift functions of the radial coordinate. The Schrodinger equation for each radialcomponent of the wavefunction then reduces to a first-order differential equationfor the corresponding phase shift.

In dimensionless units (� = 2m = 1), the Schrodinger equation for the radialpart of the angular momentum component (l) of the wavefunction of a particlewith wavevector k undergoing potential scattering is

u′′l (r) + [k2 − l(l + 1)/r2 − V (r)]ul(r) = 0. (8.63)

The scattering phase shift, δl , is obtained by comparison with the asymptoticrelation

ul(r)r→∞−→ sin(kr − lπ/2 + δl) (8.64)

and the scattering cross section is then

σ = 4π

k2

∞∑l=0

sin2 δl . (8.65)

In the variable phase method [289], we must satisfy the condition that

V (r)r→0−→ V0r−n (8.66)

with n < 2. The angular momentum phase shift is then

δl = limr→∞ δl(r) (8.67)

where the phase function, δl(r), satisfies the phase equation

δ′l(r) = −k−1V (r)[cos δl(r) jl(kr) − sin δl(r)nl(kr)]2 (8.68)

with

δl(r)r→0−→ − V0r−n

k2

(kr)2l+3

(2l + 3 − n)[(2l + 1)!!]2(8.69)

and jl(x) and nl (x) are the Riccati–Bessel functions. In the l = 0 case, the phaseequation reduces to

δ′0(r) = −k−1V (r) sin2[kr + δ0(r)]. (8.70)

In the slow-particle limit, we may neglect higher-order contributions to thescattering cross section, so that

σ = 4π

k2sin2 δ0. (8.71)

236 Superconducting state of cuprates

The effective potential about a point charge in the charged Bose gas (CBG)was calculated by Hore and Frankel [290]. Its static dielectric function is:

ε(q, 0) = 1 + 4π(e∗)2

q2ε0�

∑k

(fk − fk−q

q2/2m∗∗ − k · q/m∗∗

)(8.72)

in which e∗ = 2e the boson charge, � is the volume of the system and

fk = 1

exp(

k2/2m∗∗−µT

)− 1

.

Eliminating the chemical potential, for small q the dielectric function for T < Tcis

ε(q, 0) = 1 + 4(m∗∗ωp)2

q4

[1 −

(T

Tc

)3/2]

+ O

(1

q3

)(8.73)

and, for T → ∞,

ε(q, 0) = 1 + 1

q2

m∗∗ω2p

T

[1 + ζ(3/2)

23/2

(Tc

T

)3/2

+ · · ·]

+ O(q0) (8.74)

with ω2p = 4π(e∗)2nb/(ε0m∗∗) and nb the boson density. If the unscreened

scattering potential is the Coulomb potential (V (r) = V0/r ), then performingthe inverse Fourier transforms, one finds that, for T < Tc [290],

limr→∞ V (r) = V0

rexp[−Ksr ] cos[Ksr ] ≡ Vs(r) (8.75)

with

Ks = (m∗∗ωp)1/2

[1 −

(T

Tc

)3/2]1/4

(8.76)

and for T → ∞,

limr→∞ V (r) = V0

rexp[−Knr ] ≡ Vn(r) (8.77)

with

Kn =(

m∗∗ω2p

T

)1/2 [1 + ζ(3/2)

23/2

(Tc

T

)3/2

+ · · ·]1/2

. (8.78)

The T < Tc result is exact for all r at T = 0.There are two further important analytical results; the first (Levinson’s

theorem [289]) states that for ‘regular’ potentials (which include all those withwhich we shall be concerned), the zero-energy phase shift is equal to π multipliedby the number of bound states of the potential. The second is the well-known

Sharp increase of the quasi-particle lifetime below Tc 237

Figure 8.14. (a) Plots of zero-energy scattering cross sections (i) σn and (ii) σs

against screening wavevector K for the potentials (i) Vn(r) = −(1/r)e−Kn r and (ii)Vs(r) = −(1/r)e−Ksr cos(Ksr). (b) Plot of σn/σs for a range of K = Kn = Ks inwhich both potentials have no bound states. In each case the units are those used to derivethe phase equation.

Wigner resonance scattering formula, which states that for slow-particle scatteringof a particle with energy E off a potential with a shallow bound state of thebinding energy ε � E , the total scattering cross section is

σ = 2π

m

1

E + |ε| . (8.79)

The zero-energy scattering cross sections for the potentials Vn(r) =

238 Superconducting state of cuprates

−(V0/r)e−K r and Vs(r) = −(V0/r)e−K r cos(Kr) are shown in figure 8.14(a).These graphs are plotted for V0 = 1; in each case, the equivalent graph forarbitary V0 may be found by rescaling σ and K . According to the Wigner formula(equation (8.79)) as K decreases, when a new bound state appears there should bea peak in the cross section, as there will then be a minimum in the binding energyof the shallowest bound state. This is the origin of the peaks in figure 8.14(a),which may be checked using Levinson’s theorem. It can also be seen that as Kdecreases, the first few bound states appear at higher K in the ordinary Yukawapotential; this agrees with the intuitive conclusion that the bound states should,in general, be deeper in a non-oscillatory potential. Another intuitive expectationwhich is also borne out is that for a given V0 and K , the non-oscillatory potentialshould be the stronger scatterer. In figure 8.14(b) it may be seen that this is thecase when K is large enough, so that both potentials have no bound states (thedifference in cross sections is then, in fact, about three orders of magnitude).

At zero temperature, the screening wavevector is K0 = (m∗∗ωp)1/2, andat a temperature T = Tc + 0 just above the transition, KTc = (m∗∗ω2

p/Tc)1/2.

Substituting ωp and Tc � 3.3n2/3b /m∗∗, we obtain

KTc

K0=

(2.1e

√m∗∗

ε1/20 n1/6

b

)1/2

. (8.80)

From this, we see that the ratio is only marginally dependent on the boson density,so substituting for nb = 1021 cm−3, e and me, we obtain

KαTc

K0= 3.0

(m∗∗

meε0

)1/4

. (8.81)

With a realistic boson mass m∗∗ = 10me and dielectic constant ε0 = 100,KTc and K0, while different, are of the same order of magnitude. If the screeningwavevectors are such that neither the normal state nor condensate impuritypotentials have bound states, with these parameters it would then follow that thequasi-particle lifetime is much greater in the superconducting state, figure 8.14(b).This effect could then explain the appearance of sharp quasi-particle peaks inARPES and tunnelling, and the enhancement of the thermal conductivity incuprates below Tc [203] due to a many-fold increase in the quasi-particle lifetime.

8.6 Symmetry of the order parameter and stripes

In bipolaron theory, the symmetry of the BEC on a lattice should be distinguishedfrom the ‘internal’ symmetry of a single-bipolaron wavefunction and from thesymmetry of a single-particle excitation gap. As described in section 4.7.9, aBose condensate of bipolarons might be d-wave, if bipolaron bands have theirminima at finite K in the centre-of-mass Brillouin zone. The d-wave condensate

Symmetry of the order parameter and stripes 239

reveals itself as a checkerboard modulation of the hole density and of the globalgap (equation (8.19)) in the real (Wannier) space (figure 4.14). At the same time,the single-particle excitation spectrum might be an anysotropic s-wave providingan explanation of conflicting experimental observations.

The two-dimensional pattern (figure 4.14) is oriented along the diagonals,i.e. the d-wave bipolaron condensate is ‘striped’. Hence, there is a fundamentalconnection between stripes detected by different techniques [292,293] in cupratesand the symmetry of the order parameter [123]. Originally antiferromagneticinteractions were thought to give rise to spin and charge segregation (stripes)[291]. However, the role of long-range Coulomb and Frohlich interactions hasnot been properly addressed. Here we show that the Frohlich electron–phononinteraction combined with the direct Coulomb repulsion does not lead to chargesegregation like strings or stripes in doped insulators and the antiferromagneticexchange interaction is not sufficient to produce long stripes either [294].However, the Frohlich interaction significantly reduces the Coulomb repulsion,and allows much weaker short-range electron–phonon and antiferromagneticinteractions to bound carriers into small bipolarons. Then the d-wave Bosecondensate of bipolarons naturally explains superstripes in cuprates.

As discussed in section 4.4. the extention of the deformation surrounding(Frohlich) polarons is large, so their deformation fields overlap at a finite density.However, taking into account both the long-range attraction of polarons due tothe lattice deformations and the direct Coulomb repulsion, the net long-rangeinteraction is repulsive. At distances larger than the lattice constant (|m − n| ≥a ≡ 1), this interaction is significantly reduced to

vi j = e2

ε0|m − n| . (8.82)

Optical phonons reduce the bare Coulomb repulsion at large distances in ionicsolids if ε0 � 1, which is the case in oxides.

Let us first consider a non-adibatic and intermediate regime when thecharacteristic phonon energy is comparable with the kinetic energy of holes.In this case the problem is reduced to narrow-band fermions with a repulsiveinteraction (equation (8.82)) at large distances and a short-range attraction atatomic distancies. Because the net long-range repulsion is relatively weak, therelevant dimensionless parameter rs (= m∗e2/ε0(4πn/3)1/3) is not very large indoped cuprates and the Wigner crystallization does not appear at any physicallyinteresting density. In contrast, polarons could be bound into small bipolaronsand/or into small clusters as discussed in sections 4.6.3 and 5.4.2 but, in anycase, the long-range repulsion would prevent any clustering in infinitely chargeddomains.

In the opposite adiabatic limit, one can apply a discrete version [80] ofthe continuous nonlinear Pekar equation [61], taking into account the Coulombrepulsion and lattice deformation for a single-polaron wavefunction, ψn (the

240 Superconducting state of cuprates

amplitude of the Wannier state |n〉):−

∑m �=0

t (m)[ψn − ψn+m] − eφnψn = Eψn. (8.83)

The potential φn,k acting on a fermion, k, at the site n is created by thepolarization of the lattice, φl

n,k , and by the Coulomb repulsion with the otherM − 1 fermions, φc

n,k :

φn,k = φln,k + φc

n,k . (8.84)

Both potentials satisfy the discrete Poisson equation:

κ�φln,k = 4πe

M∑p=1

|ψn,p|2 (8.85)

and

ε∞�φcn,k = −4πe

M∑p=1,p �=k

|ψn,p|2 (8.86)

with �φn = ∑m(φn − φn+m). Then the functional J [61], describing the total

energy in this self-consistent Hartree approximation, is given by

J = −∑

n,p,m �=0

ψ∗n,pt (m)[ψn,p − ψn+m,p] − 2πe2

κ

∑n,p,m,q

|ψn,p|2�−1|ψm,q |2

+ 2πe2

ε∞

∑n,p,m,q �=p

|ψn,p|2�−1|ψm,q |2.

The single-particle function of a hole trapped in a string of the length N isψn = N−1/2 exp(ikn) with periodic boundary conditions, so that the functional Jis expressed as J = T + U , where T = −2t (N − 1) sin(π M/N)/[N sin(π/N)]is the kinetic energy, proportional to t , and

U = −e2

κM2 IN + e2

ε∞M(M − 1)IN (8.87)

corresponds to the polarization and Coulomb energies. Here the integral IN isgiven by

IN = π

(2π)3

∫ π

−π

dx∫ π

−π

dy∫ π

−π

dzsin(Nx/2)2

N2 sin(x/2)2(3 − cos x − cos y − cos z)−1.

It has the following asymptotics at N � 1:

IN = 1.31 + ln N

N. (8.88)

Symmetry of the order parameter and stripes 241

If we split the first (attractive) term in equation (8.87) into two parts byreplacing M2 for M + M(M − 1), then it becomes clear that the net interactionbetween polarons remains repulsive in the adiabatic regime as well becauseκ > ε∞. And this shows the absence of strings or stripes. The energy of Mwell-separated polarons is lower than the energy of polarons trapped in a string.

When a short-range e–ph (or exchange) interaction is taken into account,a string of finite length can appear (sections 4.6.3 and 5.4.2). We can readilyestimate its length by the use of equation (8.87) for any type of short-rangeinteraction. For example, dispersive phonons, ωq = ω + δω(cos qx + cos qy +cos qz) with a q-independent matrix element γ (q) = γ , yield a short-rangepolaron–polaron attraction:

vatt(n − m) = −Eatt(δω/ω)δ|n−m|,1 (8.89)

where Eatt = γ 2ω/2. Taking into account the long-range repulsion as well, thepotential energy of the string with M = N polarons is now

U = e2

ε0N2 IN − N Eattδω

ω. (8.90)

Minimizing this energy yields the length of the string as

N = exp

(ε0 Eattδω

e2ω− 2.31

). (8.91)

Actually, this expression provides a fair estimate of the string length forany kind of attraction (not only generated by phonon dispersion) but also byantiferromagnetic exchange and/or by Jahn–Teller type of e–ph interactions. Dueto the numerical coefficient in the exponent in equation (8.91), one can expectonly short strings (if any) with realistic values of Eatt (about 0.1 eV) and with astatic dielectric constant ε0 ≤ 100.

We conclude that there are no strings in ionic narrow-band dopedinsulators with only the Frohlich interaction. Short-range electron–phonon and/orantiferromagnetic interactions could give rise to a bipolaronic liquid and/or shortstrings only because the long-range Frohlich interaction significantly reduces theCoulomb repulsion in highly polarizable ionic insulators. However, with thetypical values of ε0 = 30, a = 3.8 A, one obtains N � 2 in equation (8.91)even with Eatt as high as 0.3 eV. Hence, the short-range attractive forces are notstrong enough to segregate charges into strings of any length, at least not in allhigh-Tc cuprates.

If (bi)polaronic carriers in many cuprates are in a liquid state, one canpose the key question of how one can see stripes at all. In fact, the bipolaroncondensate might be striped owing to the bipolaron energy band dispersion,as previously discussed. In this scenario, the hole density, which is abouttwice that of the condensate density at low temperatures, is striped, with the

242 Superconducting state of cuprates

characteristic period of stripes determined by inverse wavevectors correspondingto bipolaron band minima. Such an interpretation of stripes is consistent withthe inelastic neutron scattering in YBa2Cu3O7−δ, where the incommensuratepeaks were observed only in the superconducting state [154]. The vanishingat Tc of the incommensurate peaks is inconsistent with any other stripepicture, where a characteristic distance needs to be observed in the normalstate as well. In contrast, with the d-wave striped Bose–Einstein condensate,the incommensurate neutron peaks should disappear above Tc, as observed.Importantly, a checkerboard modulation (figure 4.14) has been observed in STMexperiments with [295] and without [296] an applied magnetic field in Bi cuprates.

Chapter 9

Conclusion

The main purpose of this book was to lead the reader from the basic principlesthrough detailed derivations to a description of the many facinating phenomenain conventional and unconventional (high Tc) superconductors. The seminal workby Bardeen, Cooper and Schrieffer taken further by Eliashberg to intermediatecoupling solved the major scientific problem of Condensed Matter Physics inthe first half of the 20th century. High-temperature superconductors present achallenge to the conventional theory. While the BCS theory gives a qualitativelycorrect description of some novel superconductors like magnesium diborade anddoped fullerenes, if the phonon dressing of carriers (i.e. polaron formation)is properly taken into account, cuprates remain a real problem. Here strongantiferromagnetic and charge fluctuations and the Frohlich and Jahn–Tellerelectron–phonon interactions have been identified as an essential piece of physics.We have discussed the multi-polaron approach to the problem based on ourrecent extension of BCS theory to the strong-coupling regime. The low-energyphysics in this regime is that of small bipolarons, which are real-space electron(hole) pairs dressed by phonons. They are itinerant quasi-particles existingin the Bloch states at temperatures below the characteristic phonon frequency.We have discussed a few applications of the bipolaron theory to cuprates, inparticular the bipolaron theory of the normal state, of the superconductingcritical temperature and the upper crtitical field, the isotope effect, normal andsuperconducting gaps, the magnetic-field penetration depth, tunnelling and theAndreev reflection, angle-resolved photoemission, stripes and the symmetry ofthe order parameter. These and some other experimental observations have beensatisfactorily explained using this particular approach, which provides evidencefor a novel state of electronic matter in layered cuprates. This is the charged Boseliquid of bipolarons. A direct measurement of a double elementary charge 2e oncarriers in the normal state could be decisive.

243

Appendix A

Bloch states

A.1 Bloch theorem

If we neglect the interaction of carriers with impurities and ion vibrations, andtake the Coulomb interaction between carriers in the mean-field (Hartree–Fock)approximation (similar to the central-field approximation for atoms) into account,we arrive at a one-electron Hamiltonian with a periodic (crystal field) potentialV (r). The wavefunction obeys the Schrodinger equation[

− ∇2

2me+ V (r)

]ψ(r) = Eψ(r). (A.1)

The Bloch theorem states that one-particle eigenstates in the periodic potentialare sorted by the wavevector k in the first Brillouin zone and by the band index nand have the form

ψnk(r) = unk(r)eik·r . (A.2)

To prove the theorem, let us define the translation operator Tl , which shifts theargument by the lattice vector l while acting upon any function F(r):

Tl F(r) = F(r + l). (A.3)

Since the Hamiltonian has the translation symmetry, H (r + l) = H (r),it commutes with Tl . Hence, the eigenstates of H can be chosen to besimultaneously eigenstates of all the Tl :

Tlψ(r) = c(l)ψ(r) (A.4)

where c(l) is a number depending on l . The eigenvalues of the translationoperators are related as follows:

c(l + l ′) = c(l)c(l ′) (A.5)

244

Bloch theorem 245

since shifting the argument by l + l ′ leads to the same function as two successivetranslations, Tl+l ′ = Tl Tl ′ . This relation is satisfied if

c(l) = exp(ik · l) (A.6)

with any k. Imposing an appropriate boundary condition on the wavefunctionmakes the allowed wavevectors k real and confined to the first Brillouin zone. Themost convenient condition is the Born–von Karman periodic boundary condition

ψ(r + N j a j ) = ψ(r) (A.7)

where j = 1, 2, 3, the a j are three primitive vectors of the crystal lattice and N j

are large integers, such that N = N1 N2 N3 is the total number of primitive cellsin the crystal (N � 1023 cm−3). This requires that

exp(iN j k · a j ) = 1. (A.8)

Therefore, the general form for allowed wavevectors is

k = n1

N1b1 + n2

N2b2 + n3

N3b3. (A.9)

Here, ni are integers and bi are primitive vectors for a reciprocal lattice, whichsatisfy

bi a j = 2πδi j (A.10)

where δi j is the Kronecker delta symbol (δi j = 1, if i = j, and zero otherwise).Replacing k by k + G does not change any of the c(l) if G is a reciprocal latticevector (i.e. a linear combination of the bi with integer coefficients). Hence, alleigenstates of the periodic Hamiltonian can be sorted by the wavevectors confinedto the primitive cell of the reciprocal lattice, for example to the region

−π < ka j � π (A.11)

which is the first Brillouin zone. In particular for a simple cubic lattice with perioda,

k = 2π

a

∑i=1,2,3

ni

Niei (A.12)

where −Ni /2 < ni � Ni /2 and ei are unit vectors parallel to the primitive latticevectors. The number of allowed wavevectors in the first Brillouin zone is equal tothe number of cells, N , in the crystal. The volume, �k, of the reciprocal k-spaceper allowed value of k is just the volume of a little parallelepiped with edgesbi/Ni :

�k = 1

Nb1 · b2 × b3. (A.13)

246 Bloch states

Since the volume of the reciprocal lattice primitive cell is (2π)3 N/V , where V isthe volume of the crystal, �k can be written in the alternative form:

�k = (2π)3

V. (A.14)

This allows us to replace the sum over all allowed k by an integral over the firstBrillouin zone,

∑k

F(k) = V

(2π)3

∑k

F(k)�k = V

(2π)3

∫BZ

F(k) dk (A.15)

for any analytical function F(k). Different eigenstates of the periodicHamiltonian are also the eigenstates of the translation operator. Hence, theytransform under translations as

ψk(r + l) = eik·lψk(r). (A.16)

If we write ψ(r) in the form

ψk(r) = uk(r)eik·r (A.17)

then uk(r) should be periodic (uk(r + l) = uk(r)) as follows fromequation (A.16). Substituting ψk(r) into the Schrodinger equation yields thefollowing equation for the periodic part of the wavefunction:

(− 1

2me(∇2 + 2ik · ∇) + V (r)

)uk(r) =

(Ek − k2

2me

)uk(r). (A.18)

This equation should be solved within the primitive unit cell and extendedperiodically for the rest of the crystal. Just like for particles in a box, theeigenstates of the equation are each identified by a discrete quantum numbern = 0, 1, 2, . . . . Hence, the eigenstates of an electron in a crystal are describedby almost continuous wavevectors k in the first Brillouin zone, and by a discreteband index n. The wavevector, k, plays a fundamental role in the electron energyband structure and dynamics, similar to that of the free electron momentum p.However, although the free-electron wavevector and the momentum are the same(if � = 1), the Bloch k is not proportional to the electron momentum on a lattice.When acting on ψk(r), the momentum operator p = −i∇ yields

−i∇ψk(r) = kψk(r) − ieik·r∇uk(r) (A.19)

which is not a constant multiplied by ψk(r) because of the second term. Hence,ψk(r) is not a momentum eigenstate.

Effective mass approximation 247

A.2 Effective mass approximation

Electron wavefunctions in semiconductors and narrow-band metals differsignificantly from plane waves because the periodic part of the Bloch function isstrongly modulated with a characteristic length of the order of the lattice constant.In general, the band structure should be calculated numerically. However, indoped semiconductors, states in the vicinity of special points of the Brillouinzone matter only at low temperatures and doping. These are the points where theenergy dispersion, Enk , of the conduction (empty) band has a minimum and thevalence band dispersion has a maximum. While their positions and band edgesare determined experimentally or numerically, the energy dispersion nearby canbe calculated analytically by applying the so-called ‘k · p’ perturbation theory. Toillustrate the theory, we consider a simple cubic semiconductor with the gap Eg,which has a non-degenerate conduction band of s or d-like symmetry at k = 0(� point) and three p-like valence bands degenerate at k = 0 and transforming atthis point like x , y and z under the rotation of the crystal space group. There is nospin–orbit interaction in this example, so that spin is irrelevant. According to theBloch theorem,

ψnk(r) = eik·runk(r) (A.20)

where unk(r) is periodic in r . It satisfies equation (A.18). At the point k = 0,this equation for un0(r) is(

− 1

2me∇2 + V (r)

)un0(r) = En0un0(r). (A.21)

Hence, un0(r) has the symmetry of the crystal space group. For a small k, onecan expand unk in the series,

unk(r) =∑

n′=s,x,y,z

an′kun′0(r) (A.22)

to obtain a secular equation:

det

∣∣∣∣∣∣∣∣∣∣

Eg + k2

2me− Ek

kx pme

ky pme

kz pme

kx pme

k2

2me− Ek 0 0

ky pme

0 k2

2me− Ek 0

kz pme

0 0 k2

2me− Ek

∣∣∣∣∣∣∣∣∣∣= 0 (A.23)

where p ≡ 〈s| − i∇x |x〉 = 〈s| − i∇y |y〉 = 〈s| − i∇z |z〉. Different bras and ketscorrespond to the four different Bloch functions un0(r). There are four solutionsto this equation. Two of them correspond to the conduction (c) and light hole (lh)valence band:

Ec,lhk = k2

2me+ Eg

2±

√E2

g

4+ k2 p2

m2e

(A.24)

248 Bloch states

�

��

���

���

Figure A.1. ‘k · p’ energy bands near the �-point of the Brillouin zone in the cubic lattice.

and the other two, which are degenerate, correspond to heavy holes (hh):

Ehhk = k2

2me(A.25)

One can split two degenerate heavy-hole bands and change the sign of theireffective mass if one includes the states at the �-point split-off from those underconsideration by an energy larger than Eg as shown by a thin line in figure A.1.

The effective mass approximation follows from equation (A.24) for smallk � me Eg/p,

Eck − Eg � k2

2mc(A.26)

and

Elhk � − k2

2mlh(A.27)

where the effective mass is

mc,lh

me= 1

p2/(me Eg) ± 1. (A.28)

In many semiconductors, the interband dipole moment p is large and the bandgapis small, p2 � me Eg. Therefore, the electron and hole masses can be significantlysmaller than the free-electron mass:

mc

me� mlh

me� 1. (A.29)

As a result, when experiment does not show heavy carriers, it does not necessarilyfollow that the electron–phonon interaction is small and the renormalization of the

Tight-binding approximation 249

band mass is absent. The effective mass approximation is applied if the externalfield varies slowly in time and smoothly in space. If the external or internal fieldsare strong and contain high-frequency and (or) short-wave Fourier components,they involve large momenta of the order of the reciprocal lattice constant makingall states of the electron band relevant. In this case, a tight binding approximationis more appropriate.

A.3 Tight-binding approximation

For narrow-band semiconductors and metals, it is convenient to replace the Blochstates by the site (Wannier) wavefunctions, defined as

wm(r) = 1√N

∑k

e−ik·mψk(r) (A.30)

where m = ∑j m j a j is the lattice vector with the integer m j and we drop the

band quantum number n. The Wannier wavefunctions are orthogonal,∫dr w∗

m(r)wn(r) = δm,n (A.31)

because the Bloch functions are orthogonal. Indeed calculating the integral inequation (A.31), one obtains∫

dr w∗m(r)wn(r) = 1

N

∑k

eik·(m−n). (A.32)

The sum is

∑k

eik·(m−n) =N1/2∑

n1=−N1/2+1

e2π in1l1/N1

N2/2∑n2=−N2/2+1

e2π in2l2/N2

×N3/2∑

n3=−N3/2+1

e2π in3l3/N3 (A.33)

where l j ≤ N j − 1 are integers. Here each of the multipliers is the sum of ageometric progression:

N j /2∑n j =−N j /2+1

exp(2iπn j l j /N j ) = eiπli

N j∑n j =1

exp(2iπn j l j/N j )

= eiπli(e2iπn j − 1) exp(2iπl j/N j )

e2iπl j /N j − 1. (A.34)

250 Bloch states

The denominator of this expression is not zero for any allowed l j , except l j = 0.However, the numerator is zero because n j is an integer. Hence, the sumequation (A.32) is non-zero only for m = n. In this case the sum is triviallyequal to N which proves the orthogonality of the Wannier functions. In the sameway, one can prove that

1

N

∑m

exp[i(k − k′) · m] = δkk′ . (A.35)

Then multiplying equation (A.30) by exp(ik · m) and performing the sum withrespect to m, we express the Bloch state as a linear combination of the Wannierstates:

ψk(r) = 1√N

∑m

eik·mwm(r). (A.36)

If we substitute this sum into the Schrodinger equation and multiply it by w∗n(r),

then, after integrating with respect to the electron coordinate r, the energy banddispersion, Ek, is expressed via the hopping integrals T (m − n) as

Ek = 1

N

∑m

T (m)eik·m (A.37)

where

T (m − n) =∫

dr w∗n(r)

[− ∇2

2me+ V (r)

]wm(r). (A.38)

The idea behind the tight-binding approximation is to fit the band dispersioncalculated numerically with a finite number of hopping integrals. Many electronicstructures, in particular perovskite ones, can be fitted with only the nearest-neighbour matrix elements between s-, p- and d-like orbitals. The hoppingintegrals could not be calculated by using tabulated atomic wavefunctions andpotentials estimated for various solids. True atomic orbitals are not orthogonal fordifferent sites and they do not provide a quantitative description of bands in solids.However, atomic-like Wannier orbitals (equation (A.30)) can provide a very gooddescription already in a tight-binding nearest-neighbour approximation. For anon-degenerate band in a cubic lattice, the approximation yields

Ek = 1

N

∑|m|=a

T (m)eik·m = 2T (a)[cos(kxa) + cos(kya) + cos(kza)] (A.39)

if the middle of the band is taken as zero (T (0) = 0). If the nearest-neighbourhopping integral, T (a), is negative, the bottom of the band is found at k = 0.Near the bottom the dispersion is parabolic with the effective band mass

m = 1

2a2|T (a)| . (A.40)

Tight-binding approximation 251

Half the bandwidth is D = z|T (a)|, where z = 6 is the number of nearestneighbours. Many physical properties of solids depend on the density of states(DOS) per unit cell,

N(E) = 1

N

∑k

δ(E − Ek) (A.41)

rather than on a particular dispersion Ek . Replacing the sum by the integral, weobtain

N(E) = 1

4π2|T (a)|

√E

|T (a)| (A.42)

if the dispersion is parabolic, Ek ∝ k2. If the dispersion is parabolic but two-dimensional, Ek ∝ (k2

x + k2y), the DOS is energy independent:

N(E) = 1

4π |T (a)| . (A.43)

Finally, for a one-dimensional parabolic spectrum (Ek ∝ k2x ), the DOS has a

square-root singularity near the bottom of the band:

N(E) = 1

2π |T (a)|√ |T (a)|

E. (A.44)

Appendix B

Quantum statistics and Boltzmann kinetics

B.1 Grand partition function

If the number of particles is large, the life history of a single particle is notso important. What really determines the physical properties of a macroscopicsystem is the average distribution of particles in real and momentum space.A statistical approach combined with quantum mechanics allow for a fulldescription of the macroscopic system. Within this approach, we first assumethat we know the exact eigenstates |Q〉 and energy levels UQ of the many-particle system. The particles interact with the walls of their container (i.e. witha thermostat) so as to establish a thermal equilibrium but not to such an extentas to affect the whole set of quantum numbers Q and energy levels. Due to theinteraction with the thermostat, the energy and the total number of particles (N)

fluctuate. The system takes some time staying in every allowed quantum state Q.The statistical probability P of finding the system in a particular quantum stateQ with the total number of particles N is proportional to this time. It depends onUQ and N : P = P(UQ , N).

It is not hard to find the probability P(UQ , N) in thermal equilibrium. Letus consider a system containing two independent parts 1 and 2 with the energiesU1,2 and the number of particles N1,2, respectively. If the two parts do not interactwith each other, the probability P1(U1, N1) that system 1 should be in the stateU1, N1 is independent of the probability P2(U2, N2). The probability for twoindependent events to occur is equal to the product of their separate probabilities.This means that the probability P(U, N) of finding the whole system in the stateU , N is the product of P1(U1, N1) and P2(U2, N2),

P(U, N) = P1(U1, N1)P2(U2, N2). (B.1)

The thermal equilibrium is described by a universal probability, P1(U, N) =P2(U, N) = P(U, N), which depends only on U and N but not on the particularsystem. Then taking the logarithm of both parts of equation (B.1), we obtain

ln P(U, N) = ln P(U1, N1) + ln P(U2, N2). (B.2)

252

Fermi–Dirac and Bose–Einstein distribution functions 253

The energy of non-interacting systems and the number of particles are additive:

U = U1 + U2 (B.3)

N = N1 + N2 (B.4)

and the only possibility to satisfy equation (B.2) is given by

ln P(U, N) = − ln Z − β(U − µN) (B.5)

where Z , β and µ are independent of U and N . Hence, the equilibrium densitymatrix is given by

P(UQ , N) = Z−1e−β(UQ−µN). (B.6)

The constant Z (the grand partition function) depends on β and µ because theprobability is normalized by the condition

∞∑N=0

∑Q

P(UQ , N) = 1. (B.7)

Hence, the grand partition function is obtained as

Z =∞∑

N=0

∑Q

e−β(UQ−µN). (B.8)

The quantity β = 1/T is the inverse temperature. It describes the interaction withthe thermostat and determines the average energy of the macroscopic system.Finally, the constant µ is the chemical potential. It determines the averageequilibrium number of particles in an open system.

B.2 Fermi–Dirac and Bose–Einstein distribution functions

The quantum statistics of non-interacting identical particles is readily derivedusing the grand partition function. The quantum state of an ideal Fermi or Bosegas is fully described by the numbers nν of identical particles in every single-particle quantum state |ν〉. This means that proper quantum numbers of thewhole gas are different sets of the occupation numbers of single-particle states,Q = {nν}, where nν = 0, 1 for fermions and nν = 0, 1, 2, . . . ,∞ for bosons.The energy levels are given by

UQ =∑ν

Eνnν (B.9)

and the total number of particles is

N =∑ν

nν . (B.10)

254 Quantum statistics and Boltzmann kinetics

Here Eν are the single-particle energy levels. The average energy and the averagenumber of particles of a gas depend on the average values nν of the occupationnumbers of each single-particle state. The distribution function

nν =∑

Q

nν P(UQ , NQ ) (B.11)

can be expressed as

nν = − 1

Zβ

∞∑N=0

∑Q

∂

∂ Eν

exp

[− β

∑ν

nν(Eν − µ)

]= −∂ ln Z

∂ Eν

. (B.12)

The partition function is readily obtained by using

Z =∑{nν }

∏ν

e−βnν (Eν−µ) =∏ν

∑nν

e−βnν (Eν−µ). (B.13)

In the case of fermions, there are only two terms in the sum with nν = 0 andnν = 1, and

Z =∏ν

[1 + e−β(Eν−µ)]. (B.14)

Hence the (Fermi–Dirac) distribution of ideal fermions is

nFν = 1

eβ(Eν−µ) + 1. (B.15)

The partition function of ideal bosons is determined by the sum of the geometricprogression as

∞∑nν=0

e−βnν (Eν−µ) = 1

1 − e−β(Eν−µ)(B.16)

and the (Bose–Einstein distribution) of ideal bosons is

nBν = 1

eβ(Eν−µ) − 1.

B.3 Ideal Fermi gas

B.3.1 Fermi energy

The density of states (DOS) (per unit of volume) of a single particle in amacroscopic box is proportional to its kinetic energy E = k2/(2m) in a powerdepending on the dimensionality d of the box (appendix A.3):

N(E) � E (d−2)/2. (B.17)

Ideal Fermi gas 255

If we place NF non-interacting fermions with the spin- 12 into a box of volume Ld ,

their total energy U and particle density n (= NF/Ld ) are given by

U = 2∫

dE N(E)E

eβ(E−µ) + 1(B.18)

and

n = 2∫

dE N(E)1

eβ(E−µ) + 1. (B.19)

Here, 2 takes into account the degeneracy of every orbital state due to the spin.Let us calculate these integrals at zero temperature. The chemical potential shouldbe positive at T = 0, otherwise the Fermi–Dirac distribution function would bezero for any positive E . For µ > 0 and β → ∞, the distribution is a step function

1

eβ(E−µ) + 1≈ �(EF − E) (B.20)

where EF ≡ µ(0) is the Fermi energy, and �(x) = 1 for x > 0 and zerootherwise. Calculating the integrals with a step-like distribution yields

EF = (3π2n)2/3

2m(B.21)

EF = πn

m

EF = π2n2

8m

for a three-, two- and one-dimensional box, respectively, and the total energyU � n(d+2)/d .

B.3.2 Specific heat

At almost all temperatures of interest, T remains much smaller than the Fermitemperature of ordinary metals, TF = EF/kB � 1000 K. Following Sommerfeld,we can expand the total energy in powers of T/TF to calculate the electroncontribution to the specific heat of a metal. Let us introduce a function of energydefined as

Kr (E) =∫ E

−∞dE ′ N(E ′)E ′r . (B.22)

In particular K0(E) represents the cumulative DOS, which is the number of stateswith an energy less than E . Then integrating by parts, the integral∫ ∞

−∞dE N(E)

Er

eβ(E−µ) + 1

becomesβ

4

∫ ∞

−∞dE

Kr (E)

cosh2[β(E − µ)/2] . (B.23)

256 Quantum statistics and Boltzmann kinetics

Since cosh−2(x/2) falls exponentially for |x | > 1, only a narrow region of theorder of a few T in the vicinity of the Fermi energy contributes to the integral.Provided that Kr (E) is non-singular in the neighbourhood of E = µ, we canexpand it in a Taylor series around E = µ to obtain∫ ∞

−∞dE N(E)

Er

eβ(E−µ) + 1=

∞∑p=0

T pap∂pKr (µ)

dµp . (B.24)

Here the dimensionless coefficients ap are given by

ap = 2p−1

p!∫ ∞

−∞dx

xp

cosh2(x). (B.25)

Only terms with even p contribute because ap = 0 for odd p. The coefficientswith p � 2 are usually written in terms of Riemann’s zeta function as

ap = [2 − 22−p]ζ(p) (B.26)

where

ζ(z) = 1

(1 − 21−z)�(z)

∫ ∞

0dt

t z−1

et + 1

= 1

�(z)

∫ ∞

0dt

t z−1

et − 1(B.27)

and �(z) is the gamma function. The zero-order coefficient of the Taylorexpansion is obtained by a straightforward integration:

a0 =∫ ∞

0dx

1

cosh2(x)= 1. (B.28)

At low temperatures, we can keep only this, p = 0, and the quadratic termswith p = 2 and a2 = π2/6. In particular, for the total energy (r = 1) in anydimensions, we obtain

U(T ) ≈ 2∫ µ

0dE N(E)E + π2

3T 2[N(µ) + µN ′(µ)] (B.29)

where N ′(µ) ≡ dN(µ)/dµ and for the particle density (r = 0)

n ≈ 2∫ µ

0dE N(E) + π2

3T 2 N ′(µ). (B.30)

The chemical potential differs from the Fermi energy by small terms of the orderof T 2, so we write

U(T ) ≈ U(0) + 2EFN(EF)(µ − EF) + π2

3T 2[N(EF) + EF N ′(EF)]

n ≈ n(0) + 2N(EF)(µ − EF) + π2

3T 2 N ′(EF) (B.31)

Ideal Fermi gas 257

where U(0) and n(0) are the total energy and the particle density at zerotemperature. To calculate the specific heat at constant density, we take n to betemperature independent (n = n(0)) and find that

µ = EF − π2

6T 2 N ′(EF)/N(EF). (B.32)

Then the total energy becomes

U(T ) ≈ U(0) + π2

3T 2 N(EF) (B.33)

and the specific heat of a degenerate electron gas is, therefore,

Ce(T ) = 2π2

3T N(EF). (B.34)

The lattice contribution to the specific heat falls off as the cube of the temperatureand at very low temperatures, it drops below the electronic contribution, which islinear in T . In practical terms, one can separate these two contributions plottingC(T )/T against T 2. Thus one can find γ ≡ 2π2 N(EF)/3 and the DOS at theFermi level by extrapolating the C(T )/T curve linearly down to T 2 = 0 andnoting where it intercepts the C(T )/T axis.

B.3.3 Pauli paramagnetism, Landau diamagnetism and de Haas–vanAlphen quantum oscillations

Let us consider a two-dimensional ideal electron gas with the parabolic energydispersion Ek = (k2

x +k2y)/2m on a lattice in an external magnetic field H directed

along z. The energy levels of electrons are quantized (see section 1.6.4) as

En = ω(n + 1/2) + µBσ H (B.35)

where each level is g-fold degenerate with g = L2eH/2π and the last termtakes into account the splitting of the levels due to the spin with σ = ±1. Hereω = eH/m is the Larmour frequency and µB = e/(2me) is the Bohr magneton.The band mass m might be different from the free-electron mass me. The DOSbecomes a set of narrow peaks:

N(E) = d∞∑

n=0

δ[E − ω(n + 1/2) ∓ µB H ] (B.36)

rather than a constant. This sharp oscillatory structure of the DOS imposed bythe quantization of levels in a magnetic field combined with a step-like Fermi–Dirac distribution at low temperatures leads to the famous de Haas and vanAlphen oscillations of magnetization of two- and three-dimensional metals with

258 Quantum statistics and Boltzmann kinetics

the magnetic field. We consider an open system (grand canonical ensemble) witha fixed chemical potential independent of the applied field. It is described by thethermodynamic potential � = F −µN = −T ln Z . The magnetization of the gasis

M(T, H ) = − ∂�

∂ H. (B.37)

For non-interacting fermions, the partition function Z is the productequation (B.14) and

� = −dT∞∑

n=0

∑σ=±1

ln

{1 + exp

µ − ω(n + 1/2) − µBσ H

T

}. (B.38)

We calculate this sum by applying the Poisson formula for an arbitrary realfunction F(n):

∞∑n=0

F(n) =∫ ∞

−δ

dx F(x) + 2∞∑

r=1

Re Fr (B.39)

where Fr is the transform of F(x),

Fr =∫ ∞

−δ

dx F(x)e2π irx (B.40)

and δ → +0. The Poisson formula is derived by Fourier transforming the sum ofδ-functions

�(x) =∞∑

n=−∞δ(x − n).

This sum is periodic as a function of x with the period 1. Hence, we can expandit into the Fourier series

�(x) =∞∑

r=−∞�r e2π ir x

where the Fourier coefficients are

�r =∫ 1/2

−1/2dx e−2π irx�(x) = 1.

Multiplying the Fourier expansion of �(x),

∞∑n=−∞

δ(x − n) =∞∑

r=−∞e2π irx (B.41)

by F(x) and integrating it with respect to x from −δ to +∞ we obtain the Poissonformula (B.39). Applying the formula for F(n) = −gT

∑σ=±1 ln{1 + exp[(µ −

ω(n + 1/2) − µBσ H )/T ]} yields

� = �P + �L (B.42)

Ideal Fermi gas 259

where

�P = −mL2

2πT

∑σ=±1

∫ ∞

0dE ln

{1 + exp

µ − E − µBσ H

T

}(B.43)

is the ‘classical’ part containing no orbital effect of the magnetic field but onlyspin splitting and

�L = −2 dT Re∞∑

r=1

∑σ=±1

∫ ∞

0dx e2π irx ln

{1 + exp

µ − ω(x + 1/2) − µBσ H

T

}(B.44)

is the ‘quantum’ part due to the quantization of the orbital motion. Takingthe derivative of �P, we obtain the Pauli paramagnetic contribution to themagnetization

MP = µB(N↓ − N↑) (B.45)

where

N↑↓ = L2m

2π

∫ ∞

0dE

1

exp(

E±µB H−µT

)+ 1

is the number of electrons with spin parallel (↑) or antiparallel (↓) to the magneticfield. Performing the integration for T � TF with the Fermi step-function, wefind that

MP = 2µ2B H N(EF) (B.46)

where N(EF) = L2m/(2π) is the two-dimensional zero-field DOS. Thespin susceptibility χP = ∂MP/∂ H is positive and essentially independent oftemperature. It is proportional to the DOS at the Fermi level:

χP = 2µ2B N(EF). (B.47)

This result is actually valid for any energy band spectrum of a metal. Now letus calculate the quantum part �L. Integrating equation (B.44) twice by parts, weobtain

�L = �LD + �dHvA (B.48)

where

�LD = dω

2π2Re

∞∑r=1

1

r2

∑σ=±1

1

1 + exp(

ω/2+µBσ H−µT

) (B.49)

and

�dHvA = dω2

8π2TRe

∞∑r=1

1

r2

∑σ=±1

∫ ∞

0dx

e2π irx

cosh2(

ω(x+1/2)+µBσ H−µ2T

) . (B.50)

260 Quantum statistics and Boltzmann kinetics

At all temperatures and fields of interest, µ � EF � T and µ � ω, so that

�LD = L2(eH )2

2mπ3ζ(2). (B.51)

Here we applied the series representation of the Riemann’s function ζ(2) =∑∞r=1 r−2 = π2/6. Differentiating this expression twice with respect to H yields

the diamagnetic (Landau) orbital susceptibility

χD = −∂2�LD

∂ H 2= −1

3

(me

m

)2χP (B.52)

which might be larger or smaller than the paramagnetic Pauli term depending onthe band mass. The remaining part �dHvA is responsible for the de Haas–vanAlphen (dHvA) effect. The main contribution to the integral in equation (B.50)yields the region of x � µ/ω � 1, so that we can replace the lower limit in theintegral by −∞. Then using∫ ∞

−∞dt

eiαt

cosh2(t/2)= 4πα

sinh(πα)

we obtain

�dHvA =∞∑

r=1

Ar cos

(r f

H+ πr

)(B.53)

where the amplitudes of the Fourier harmonics are

Ar = L2eH T

πr sinh(2π2r T/ω)cos

(πmr

me

)(B.54)

and f = 2πmµ/e is the frequency of oscillations with respect to the inversemagnetic field 1/H . The oscillating part �dHvA is small compared with the‘classical’ part �P as �dHvA/�P � (ω/µ)2 at any temperature. However, dueto the high frequency of oscillations, the oscillating part of magnetization at zerotemperature is larger than the monotonic part as µ/ω,

MdHvA = −∂�dHvA

∂ H≈ − f

H 2

∞∑r=1

r Ar sin

(r f

H+ πr

). (B.55)

The period of oscillations with the inverse magnetic field of the first harmonic(r = 1),

�(1/H ) = 2πe

S. (B.56)

It directly measures the cross section of the Fermi surface S = πk2F, where

kF = (2mµ)1/2 is the Fermi momentum. With increasing temperature, theoscillating part of magnetization falls exponentially as

Ar � e−2π2rT/ω. (B.57)

Ideal Bose gas 261

The scattering of electrons by impurities also washes away dHvA oscillations assoon as the scattering rate is large enough compared with the Larmour frequency,ωτ � 1. The ratio equation (B.52) of the Landau diamagnetism and the Pauliparamagnetism does not depend on a particular distribution function and remainsthe same even for non-degenerate electrons. However, the dHvA effect can beobserved only at low temperatures (T � ω) in clean metals with a well-definedFermi surface. The observation of dHvA oscillations is considered to be theultimate evidence for a Fermi liquid.

B.4 Ideal Bose gas

B.4.1 Bose–Einstein condensation temperature

Let us consider NB non-interacting Bose particles with zero spin in a cubic boxof the volume L3 under periodic boundary conditions. The one-particle energyspectrum is given by

Ek = 4π2

2mL2

3∑j=1

n2j (B.58)

where n j are positive integers including zero and k = (2π/L){n1, n2, n3} is thewavevector. The chemical potential of a gas µ(T, n) as a function of temperatureand boson density n = NB/L3 is determined using the ‘density sum rule’:

n = 1

L3

∑j

1

exp[β(Ek − µ)] − 1(B.59)

where β = 1/T . This equation has a negative solution for µ at any temperature.If µ is negative, every term of the sum is finite. When L, NB → ∞ but n is finite(the so-called thermodynamic limit), the one-particle energy spectrum becomescontinuous. In this limit the contribution of every individual term to the sum isnegligible (as 1/N) compared with n. Hence, we can replace the sum by theintegral using the DOS. However, if the chemical potential approaches zero frombelow, a relative contribution of the term with k = 0 and Ek = 0 could be finite.Taking into account this singular contribution we replace the sum by the integralfor all but the ground state. Let us take the number of bosons in the ground stateper unit of volume as ns(T ). Then equation (B.59) is written as

n = ns(T ) +∫ ∞

0dE

N(E)

exp[β(E − µ)] − 1(B.60)

where N(E) = m3/2E1/2/(π2√

2). If the value of µ is of the order of levelspacing 1/(mL2) → 0, one can put µ = 0 in this equation. Calculating theintegral, we obtain the density of bosons in the ground state (i.e. the condensatedensity),

ns(T ) = n[1 − (T/Tc)3/2] (B.61)

262 Quantum statistics and Boltzmann kinetics

where

Tc = 2π

[n

ζ(3/2)m3/2

]2/3

(B.62)

is the Bose–Einstein condensation temperature. Here ζ(3/2) = 2.612. Inordinary units, we have

kBTc = 3.313�2n2/3

m. (B.63)

B.4.2 Third-order phase transition

To satisfy equation (B.60), the condensate density ns(T ) should be zero attemperatures T > Tc, where the chemical potential is negative. In thisnormal state, the population of every individual level remains microscopic. Attemperatures far above Tc, the gas is classical. Here the Bose–Einstein distributionis almost the same as the Maxwell–Boltzmann distribution,

1

exp[β(E − µ)] − 1≈ exp[β(µ − E)] (B.64)

if T � Tc. The magnitude of the chemical potential is logarithmically largecompared with the temperature in this limit,

µ = −3

2T ln

Tc

T. (B.65)

When the temperature approaches Tc from above, the chemical potential becomessmall, and finally zero at and below Tc. To obtain its normal state value in thevicinity of Tc, we rewrite the density sum rule as

n[1 − (T/Tc)3/2] =

∫ ∞

0dE N(E)

[1

exp[β(E − µ)] − 1− 1

exp[β E] − 1

].

(B.66)The function under the integral can be expanded for small E and µ, so that theintegral becomes

n[1 − (T/Tc)3/2] = T µ

∫ ∞

0dE

N(E)

E(E − µ). (B.67)

The remaining integral yields

µ = − 9π2n2

2m3T 2c

[T − Tc

Tc

]2

(B.68)

for 0 < T − Tc � Tc. Now we can calculate the free energy and the specific heatnear the transition. The total energy of the gas per unit of volume is

U =∫ ∞

0dE N(E)

E

exp[β(E − µ)] − 1. (B.69)

Ideal Bose gas 263

In the condensed state below Tc, the chemical potential is zero and

U = Us(T ) = 3ζ(5/2)

2ζ(3/2)nT (T/Tc)

3/2 ≈ 0.128m3/2T 5/2. (B.70)

Remarkably the energy density of the condensed phase does not depend onthe particle density. This is because the macroscopic part of particles in thecondensate does not contribute to the energy. The free energy density

F(T ) = −T∫ T

0dT

U(T )

T 2(B.71)

is found to beFs(T ) = − 2

3Us(T ). (B.72)

The specific heat Cs(T ) = 5Us(T )/(2T ) is proportional to T 3/2 in the condensedphase. Differentiating equation (B.69) with respect to the chemical potential, oneobtains

∂Un

∂µ= −

∫ ∞

0dE N(E)E

∂

∂ E

1

exp[β(E − µ)] − 1(B.73)

in the normal state. We can take µ = 0 on the right-hand side of this equationbecause the chemical potential is small just above Tc. Then integrating by parts,we find that

∂Un

∂µ= 3

2n (B.74)

and

Un(T ) = Us(T ) + 3

2nµ = Us(T ) − 27π2n3

4m3T 2c

[T − Tc

Tc

]2

. (B.75)

Using equation (B.71), we arrive at the free-energy density in the normal stateclose to the transition as

Fn(T ) = Fs(T ) + 9π2n3

4m3T 2c

[T − Tc

Tc

]2

(B.76)

and the normal state specific heat

Cn(T ) = Cs(T ) − 27π2n3

2m3T 3c

[T − Tc

Tc

]. (B.77)

While the specific heat of the ideal Bose gas is a continuous function (seefigure 7.4), its derivative has a jump at T = Tc,(

∂Cs

∂T− ∂Cn

∂T

)= 27π2n3

2m3T 4c

≈ 3.66n

Tc. (B.78)

264 Quantum statistics and Boltzmann kinetics

B.5 Boltzmann equation

An external perturbation can drive a macroscopic system out of the thermalequilibrium. The single-particle distribution function f (r, k, t) in real r andmomentum k space is no longer a Fermi–Dirac or a Bose–Einstein distribution. Itsatisfies the celebrated Boltzmann kinetic equation. The equation can be derivedby considering a change in the number of particles in an elementary volume dr dkof the phase space, which is given at the time t by f (r, k, t) dr dk. Particles withthe momentum k move with the group velocity v = dEk/dk. Their number inthe elementary volume dr dk changes due to their motion. The change during aninterval of time dt due to the motion in real space is given by

δNr = [ f (x, y, z, k, t) − f (x + dx, y, z, k, t)]vx dt dy dz dk

+ [ f (x, y, z, k, t) − f (x, y + dy, z, k, t)]vy dt dx dz dk

+ [ f (x, y, z, k, t) − f (x, y, z + dz, k, t)]vz dt dx dy dk.

Not only do the coordinates but also the momentum k change under the influenceof a force F(r, t). The change in the number of particles during the interval dtdue to the ‘motion’ in the momentum space is

δNk = [ f (r, kx , ky, kz, t) − f (r, kx + dkx, ky, kz, t)]kx dt dky dkz dr

+ [ f (r, kx , ky, kz, t) − f (r, kx , ky + dky, kz, t)]ky dt dkx dkz dr

+ [ f (r, kx , ky, kz, t) − f (r, kx , ky, kz + dkz, t)]kz dt dkx dky dr

where k = F(r, t). As a result, the change in the number of particles in theelementary volume of the phase space is

δN = δNr + δNk . (B.79)

This change can also be expressed via a time derivative of the distribution functionas δN = (∂ f/∂ t) dt dr dk, which finally yields

∂ f (r, k, t)

∂ t+ v · ∇r f (r, k, t) + F(r, t) · ∇k f (r, k, t) = 0. (B.80)

The force acting upon a particle is the sum of the external force and the internalforce of other particles of the system. Splitting the last term in equation (B.80)into the external and internal contributions, we obtain equation (1.4), where thecollision integral describes the effect of the internal forces.

Appendix C

Second quantization

C.1 Slater determinant

Let us consider a quantum mechanical system of N identical particles witha mass m, a spin s and the interaction potential between any two of themV (r i − r j ). There is also an external field V (r). Here r i are three positioncoordinates {xi , yi , zi }, i, j = 1, 2, 3, . . . , N . The system is described bya many-particle wavefunction �(q1, q2, . . . , qN ), which satisfies the stationarySchrodinger equation:{ N∑

i=1

[− 1

2m�i + V (r i )

]+ 1

2

N∑i=1

∑j �=i

V (r i − r j )

}�(q1, q2, . . . , qN )

= E�(q1, q2, . . . , qN ) (C.1)

where qi ≡ (r i , σi ) is a set of position r i and spin σi coordinates. The spincoordinate (z-component of spin sz) describes the ‘internal’ state of the particle,which might be a composite particle like an atom. We can readily solve thisequation for non-interacting particles when V (r i − r j ) = 0. In this case, many-particle eigenstates are products of one-particle wavefunctions,

�Q(q1, q2, . . . qi , . . . q j , . . . qN ) = uk1(q1)uk2(q2) × · · · × ukN (qN ) (C.2)

where uk(q) is a solution of a one-particle Schrodinger equation[− 1

2m� + V (r)

]uk(r, σ ) = εkuk(r, σ ) (C.3)

k is a set of one-particle quantum numbers, for example the wavevector k and sz ,if the particles are free, or of n, k and sz, if V (r) is periodic (appendix A). Directsubstitution of equation (C.2) into equation (C.1) yields the total energy of thesystem, which is the sum of one-particle energies,

EQ =N∑

i=1

εki . (C.4)

265

266 Second quantization

The quantum number Q of the whole system is the set of single-particle quantumnumbers,

Q = {k1, k2, . . . ki , . . . k j , . . . kN }. (C.5)

If we swap the coordinates of any two particles, the new state �Q(q1, q2, . . .

q j , . . . qi , . . . qN ) will also be a solution of the Schrodinger equation with thesame total energy. Any linear combination of such wavefunctions solves theproblem as well. Importantly, only one of them is acceptable for identicalparticles. While in classical mechanics the existence of sharp trajectories makesit possible to distinguish identical particles by their paths, there is no way ofkeeping track of individual particles in quantum mechanics. This quantum‘indistinguishability’ of identical particles puts severe constraints on our choiceof a many-particle wavefunction. Let us introduce an operator Pi j , which swapsthe coordinates qi and q j ,

Pi j �(q1, q2, . . . qi , . . . q j , . . . qN ) = �(q1, q2, . . . q j , . . . qi , . . . qN ). (C.6)

The Hamiltonian (C.1) is symmetric with respect to the permutation of any pairof the coordinates, that is Pi j and H commute:

[Pi j , H ] = 0. (C.7)

Hence, the eigenfunctions of H are also the eigenfunctions of Pi j , so that

Pi j �(q1, q2, . . . qi , . . . q j , . . . qN ) = P�(q1, q2, . . . qi , . . . q j , . . . qN ). (C.8)

Since two successive permutations of qi and q j bring back the originalconfiguration, we have P2 = 1 and P = ±1. The eigenstates with the eigenvalueP = 1 are called symmetric and the eigenstates with P = −1 antisymmetric.There is an extreme case of a totally symmetric (or totally antisymmetric)state which does not change (or changes its sign) under any permutation Pi j .Only these extreme cases are realized for identical particles. Any state ofbosons is symmetric and any state of fermions is antisymmetric. Relativisticquantum mechanics connects the symmetry of the many-particle wavefunctionwith the spin. Bosons have integer spins, s = 0, 1, 2, 3, . . . and fermionshave half odd integer spins, s = 1/2, 3/2, 5/2, . . . . A totally antisymmetricwavefunction �A of non-interacting identical fermions is readily constructed aslinear superpositions of �Q(q1, q2, . . . qi , . . . q j , . . . qN ) (equation (C.2)) withcoefficients determined by the use of the Slater determinant:

�A = 1√N !Det

uk1(q1) uk2(q1) . . . ukN (q1)

uk1(q2) uk2(q2) . . . ukN (q2)

. . . . . . . . . . . .

uk1(qN ) uk2(qN ) . . . ukN (qN )

. (C.9)

This function is totally antisymmetric because the permutation of the coordinatesof any two particles (let us say q1 and q2) corresponds to the permutation of two

Annihilation and creation operators 267

rows in the matrix N × N which changes the sign of the whole determinant.The factor 1/

√N ! is introduced to make �A properly normalized. The totally

symmetric �S of bosons is obtained as

�S =√

N1!N2! . . .N !

∑�Q(q1, q2, . . . qi , . . . q j , . . . qN ) (C.10)

where the sum is taken over all permutations of different single-particle quantumnumbers ki in Q. N1, N2, . . . are the numbers of identical ki in Q, so thatN1 + N2 + · · · = N . For example the system of two identical non-interactingfermions, which can be only in two one-particle states uk(q) and u p(q), isdescribed by

�A(q1, q2) = 1√2{uk(q1)u p(q2) − uk(q2)u p(q1)} (C.11)

while two bosons are described by

�S(q1, q2) = 1√2{uk(q1)u p(q2) + uk(q2)u p(q1)} (C.12)

if k �= p, and by�S(q1, q2) = uk(q1)uk(q2) (C.13)

if k = p. These functions are normalized,∫dq1

∫dq2 |�S,A(q1, q2)|2 = 1

if uk,p(q) are normalized and orthogonal,∫dq u∗

k(q)u p(q) = δkp . (C.14)

It is apparent that if two or more single-particle quantum numbers ki are the same,the totally antisymmetric wavefunction vanishes, �A = 0. Only one fermion canoccupy a given one-particle quantum state. This statement expresses the Pauliexclusion principle formulated in 1925.

C.2 Annihilation and creation operators

The collection of identical particles which do not mutually interact is a trivialproblem. It is solved in the form of the Slater determinant for fermions(equation (C.9)) and of the sum (C.10) for bosons. However, the interactionterm makes the problem hard. Exact many-particle wavefunctions can beexpanded in a series of ‘non-interacting’ functions equation (C.9) or (C.10).Then a set of algebraic equations for the expansion coefficients may be treated

268 Second quantization

perturbatively, if all matrix elements of the many-particle Hamiltonian are known.These coefficients and matrices perfectly replace the original wavefunctions andoperators, respectively. The non-interacting eigenstates (C.9) and (C.10) are fullyidentified by a set {nk1 , nk2 , . . . nki , . . .} of the occupation numbers nki of eachsingle-particle quantum state. Differing from the original wavefunctions, whichare functions of the position and spin coordinates of particles qi , the expansioncoefficients are functions of the occupation numbers {nk1 , nk2 , . . . nki , . . .}. Arepresentation, where the initial coordinates {q1, q2, . . . qN } are replaced by new‘coordinates’ {nk1 , nk2 , . . . nki , . . .}, is known as the second quantization. Thewavefunction of non-interacting particles (equations (C.9) and (C.10)) is writtenin the second quantization using the Dirac notations as a ket,

�{ni } = |nk1 , nk2 , . . . nki , . . .〉 (C.15)

and its complex conjugate as a bra,

�∗{ni } = 〈nk1 , nk2 , . . . nki , . . . | (C.16)

where nki = 0, 1, 2, 3, . . . ,∞ for bosons and nki = 0, 1 for fermions. The matrixelements of any operator A(q1, q2, . . . , qN ) are written as

〈mk1 , mk2 , . . . mki , . . . | A|nk1 , nk2 , . . . nki , . . .〉≡

∫dq1

∫dq2 . . .

∫dqN �∗{mi }(q1,q2, . . . , qN ) A�{ni }(q1,q2, . . . , qN ).

(C.17)

There are two major types of many-particle symmetric Hermitian operators:

A(1) ≡N∑

i=1

h(qi ) (C.18)

A(2) ≡N∑

i, j=1

V (qi , q j ).

While A(1) is the sum of identical one-particle operators (like the kinetic energy),A(2) is the sum of identical two-particle operators (like the potential energy).Their matrix elements can be readily calculated by the use of annihilationbk(ak) and creation b†

k(a†k ) operators, whose matrices are defined in the second

quantization as

〈mk1 , mk2 , . . . mki , . . . |bki |nk1 , nk2 , . . . nki , . . .〉= √

nki δmk1 ,nk1δmk2 ,nk2

× · · · × δmki ,nki −1 × · · · × δmk j ,nk j× · · ·

(C.19)

〈mk1 , mk2 , . . . mki , . . . |b†ki|nk1 , nk2 , . . . nki , . . .〉

= √nki + 1δmk1 ,nk1

δmk2 ,nk2× · · · × δmki ,nki +1 × · · · × δmk j ,nk j

× · · ·

Annihilation and creation operators 269

for bosons and

〈mk1 , mk2 , . . . mki , . . . |aki |nk1 , nk2 , . . . nki , . . .〉= ±√

nki δmk1 ,nk1δmk2 ,nk2

× · · · × δmki ,nki −1 × · · · × δmk j ,nk j× · · ·

(C.20)

〈mk1 , mk2 , . . . mki , . . . |a†ki|nk1 , nk2 , . . . nki , . . .〉

= ±√nki + 1δmk1 ,nk1

δmk2 ,nk2× · · · × δmki ,nki +1 × · · · × δmk j ,nk j

× · · ·

for fermions. Here + or − depend on the evenness or oddness, respectively, of thenumber of one-particle occupied states, which precede the state ki . It follows fromthis definition that b†

k is the Hermitian conjugate of bk and a†k is the Hermitian

conjugate of ak . We see that bk(ak) decreases the number of particles in the state kby one and b†

k(a†k ) increases this number by one. The products b†

kbk and a†k ak have

diagonal matrix elements only, which are nk . These are the occupation numberoperators. The operators bkb†

k and aka†k are also diagonal and their eigenvalues

are 1 + nk and 1 − nk, respectively. As a result, the commutation rules for theseoperators are:

bkb†k − b†

kbk ≡ [bk, b†k ] = 1 (C.21)

and

aka†k + a†

k ak ≡ {aka†k } = 1. (C.22)

The bosonic annihilation and (or) creation operators for different single-particlestates commute and the fermionic annihilation and (or) creation operatorsanticommute:

[bk, b†p] = [bk, bp] = 0 (C.23)

{ak, a†p} = {ak, ap} = 0

if k �= p. For example, the matrix elements of fermionic annihilation and creationoperators are:

〈0k|ak |1k〉 = 〈1k |a†k |0k〉 = (−1)N(1,k−1) (C.24)

where N(1, k − 1) is the number of occupied states, which precede the state kin the adopted ordering of one-particle states and the occupation numbers of allother single-particle states are the same in both bra and ket. Then multiplying a†

p

and a†k , we obtain a non-zero matrix element

〈1p, 0k|a†pak|0p, 1k〉 = 〈1p, 0k |a†

p|0p, 0k〉〈0p, 0k |ak|0p, 1k〉= (−1)N(1,p−1)(−1)N(1,k−1) = (−1)N(p+1,k−1).

(C.25)

270 Second quantization

Multiplying these operators in the inverse order yields

〈1p, 0k |aka†p|0p, 1k〉 = 〈1p, 0k |ak|1p, 1k〉〈1p, 1k |a†

p|0p, 1k〉= (−1)N(1,k−1)(−1)N(1,p−1) = (−1)N(p+1,k−1)+1.

(C.26)

Equations (C.25) and (C.26) have opposite signs because the p-state of theintermediate bra and ket in equation (C.26) is occupied, while it is empty in theintermediate bra and ket in equation (C.25). Hence, we obtain aka†

p + a†pak = 0

for k �= p.Any operator of the first type is expressed in terms of the annihilation and

creation operators as

A(1) ≡∑k,p

hk′kb†k′bk (C.27)

A(1) ≡∑k,p

hk′ka†k′ak

for bosons and fermions, respectively. And an operator of the second type isexpressed as

A(2) ≡∑

k,p,k′,p′V p′k′

pk b†k′b

†p′bpbk (C.28)

A(2) ≡∑

k,p,k′,p′V p′k′

pk a†k′a

†p′apak .

Here, h pk and V p′k′pk are the matrix elements of h and V in the basis of one-particle

wavefunctions:

hk′k ≡∫

dq u∗k′(q)h(q)uk(q) (C.29)

V p′k′pk ≡

∫dq

∫dq ′ u∗

k′ (q)u∗p′(q ′)V (q, q ′)u p(q

′)uk(q).

For example, let us consider a system of two bosons, N = 2, which can be in thetwo different one-particle states um(q) and ul(q). If bosons do not interact, theeigenstates of the whole system are

|2m, 0l〉 = um(q1)um(q2) (C.30)

|0m, 2l〉 = ul(q1)ul(q2)

|1m, 1l〉 = 1√2[ul(q1)um(q2) + ul(q2)um(q1)].

�-operators 271

The diagonal matrix elements of A(1), calculated directly by the use of theorthogonality of um(q) and ul(q) for m �= l, are

〈2m, 0l | A(1)|2m, 0l〉 =∫ ∫

dq1 dq2 u∗m(q1)u

∗m(q2)[h(q1) + h(q2)]

× um(q1)um(q2) = 2hmm (C.31)

〈0m, 2l | A(1)|0m, 2l〉 = 2hll

〈1m, 1l | A(1)|1m, 1l〉 = hmm + hll .

The off-diagonal elements are:

〈0m, 2l | A(1)|2m, 0l〉 = 0 (C.32)

〈0m, 2l | A(1)|1m, 1l〉 = √2hlm

〈2m, 0l | A(1)|1m, 1l〉 = √2hml .

However, the matrix elements of b†k′bk , calculated by the use of the matrix

elements of the annihilation and creation operators (equation (C.19)), are:

〈2m, 0l |b†k′bk|2m, 0l〉 = 2δkmδk′m (C.33)

〈0m, 2l |b†k′bk |0m, 2l〉 = 2δklδk′l

〈1m, 1l |b†k′bk|1m, 1l〉 = δkmδk′m + δklδk′l

〈0m, 2l |b†k′bk|2m, 0l〉 = 0

〈0m , 2l |b†k′bk |1m, 1l〉 = √

2δkmδk′l

〈2m, 0l |b†k′bk|1m, 1l〉 = √

2δklδk′m .

Substituting these matrix elements into equation (C.27), we obtain the result ofdirect calculations (equations (C.31) and (C.32)). The expressions (C.27) and(C.28) allow us to replace the first (coordinate) representation by the secondquantization (occupation numbers) representation.

C.3 �-operators

The second quantization representation depends on our choice of a complete setof one-particle wavefunctions. In many cases we do not know the eigenstatesof the one-particle Hamiltonian h(q), and a representation in terms of the field�-operators is more convenient. These operators are defined as

�(q) =∑

k

uk(q)bk (C.34)

�†(q) =∑

k

u∗k(q)b†

k

272 Second quantization

for bosons and in the same way for fermions,

�(q) =∑

k

uk(q)ak (C.35)

�†(q) =∑

k

u∗k(q)a†

k .

Differing from the annihilation and creation operators, �-operators do not dependon a particular choice of one-particle states. Both A(1) and A(2) many-particleoperators are readily expressed in terms of �-operators:

A(1) =∫

dq �†(q)h(q)�(q) (C.36)

A(2) =∫

dq∫

dq ′ �†(q)�†(q ′)V (q, q ′)�(q ′)�(q).

Indeed, if we substitute equations (C.34) or (C.35) into equations (C.36) we obtainequations (C.27) and (C.28). Here the operators h(q) and V (q, q ′) act on thecoordinates, which are parameters (i.e. c-numbers), while the ‘true’ operators arethe field operators acting on the occupation numbers. The commutation rulesfor the field operators are derived using the commutators of the annihilation andcreation operators,

[�(q)�†(q ′)] =∑k,p

uk(q)u∗p(q

′)[aka†p]

=∑

k

uk(q)u∗k(q

′) = δ(r − r ′)δσσ ′ (C.37)

[�(q)�(q ′)] = 0 (C.38)

for bosons and anticommutators

{�(q)�†(q ′)} = δ(r − r ′)δσσ ′ (C.39)

{�(q)�(q ′)} = 0

for fermions. The Hamiltonian of identical interacting particles can be expressedin terms of �-operators:

H =∫

dq �†(q)h(q)�(q) +∫

dq∫

dq ′ �†(q)�†(q ′)V (q, q ′)�(q ′)�(q).

(C.40)Now we can use any complete set of one-particle states to express it in terms ofthe annihilation and creation operators. For example, for fermions on a lattice aconvenient set is the set of Bloch functions (appendix A),

�(q) =∑n,k,s

ψnk(r)χs(σ )anks (C.41)

�-operators 273

where we also include the spin component χs(σ ). If we drop the band index nin the framework of the single-band approximation, the Hamiltonian takes thefollowing form

H =∑k,s

ξka†ksaks +

∑k,k′,s

∑p, p′,s ′

V p′k′pk a†

k′sa†p′s ′a ps ′aks . (C.42)

Here V p′k′pk is the matrix element of the particle–particle interaction V (r, r ′)

calculated by the use of the Bloch states. We assume that V (r, r ′) does notdepend on the spin. One can equally use the Wannier functions (A.30) as anothercomplete set,

�(q) =∑m,s

wm(r)χs(σ )ams (C.43)

to express the same Hamiltonian in the site representation:

H =∑

m,n,s

[T (m − n) − µδm,n]a†msans +

∑m,m′,s

∑n,n′,s ′

V n′m′nm a†

m′sa†n′s ′ans ′ams .

(C.44)Here T (m − n) is the hopping integral (equation (A.38)) and V n′m′

nm are thematrix elements of the interaction potential, calculated by the use of the Wannierfunctions.

Appendix D

Analytical properties of one-particleGreen’s functions

Green’s functions were originally introduced to solve differential equations andlink them to the corresponding integral equations. Later on, their applicationsbecame remarkably broad in theoretical physics. Nowadays Green’s functions(GFs) serve as a very powerful tool in many-body theory. Regarded as adescription of the time development of a system, a one-particle GF is appliedto any interacting system of identical particles. We use the occupation numberrepresentation and suppose that the quantum state is described by |n〉. If at timet = 0 we add a particle with the quantum number ν, the system is describedimmediately after this addition by c†

ν |n〉, where c†ν is the creation operator for the

state |ν〉. Now the development of the system proceeds according to e−iH t c†ν |n〉.

However, ν is not usually an eigenstate of H , so the particle in the state ν

gets scattered. Thus, when at a later time we measure to see how much of aprobability amplitude is left in the state ν, the measurement provides informationabout the interaction in the system. If we require the probability amplitude forthe persistence of the added particle in the quantum state ν, we must take the

scalar product of e−iH t c†ν |n〉 with a function describing the quantum state plus

the particle added at the time t . At this time, the state is described by e−iHt |n〉,and immediately after the addition of the particle it is described by c†

νe−iHt |n〉.The quantum number ν can be anything depending on the problem of interest.Usually, it is assigned as the quantum number of a free particle, i.e. ν = (k, σ )

representing both the momentum and spin. Such is the idea behind the followingdefinition of the one-particle GF at any temperature:

G(r, r ′, t − t ′) = − i Tr{e(�−H)/T Ttψ(r, t)ψ†(r ′, t ′)} (D.1)

≡ − i〈〈Ttψ(r, t)ψ†(r ′, t ′)〉〉 (D.2)

whereψ(r, t) = eiHt�(r)e−iHt

274

Analytical properties of one-particle Green’s functions 275

is the time-dependent (Heisenberg) field operator and

Tt A(t)B(t) ≡ �(t − t ′) A(t)B(t ′) ± �(t ′ − t)B(t ′) A(t)

with + for bosons and − for fermions. In the following we consider fermionswith only one component of their spin and drop the spin index. The GF allowsus to calculate the single-particle excitation spectrum of a many-particle system.However, the perturbation theory is easier to formulate for the Matsubara GFdefined as

�(r, r ′, τ − τ ′) = −〈〈Tτ ψ(r, τ )ψ†(r ′, τ ′)〉〉. (D.3)

Here ψ(r, τ ) = exp(H τ )�(r) exp(−H τ ) and the thermodynamic ‘time’ τ isdefined within the interval 0 � τ � 1/T . If the system is homogeneous, GFsdepend only on the difference (r − r ′) and their Fourier transforms depend on asingle wavevector k:

G(k, t) = − i〈〈Tt ck(t)c†k〉〉 (D.4)

�(k, τ ) = − 〈〈Tτ ck(τ )c†k〉〉.

Let us connect the GF and the Matsubara GF. By the use of the exact eigenstates|n〉 and eigenvalues En of the Hamiltonian, we can write

G(k, t) = −ie�/T∑n,m

e−En/T eiωnmt |〈n|ck|m〉|2 (D.5)

for t > 0 and

G(k, t) = ie�/T∑n,m

e−Em/T eiωnmt |〈n|ck|m〉|2 (D.6)

for t < 0, where ωnm ≡ En − Em . The Fourier transform with respect to timeyields for the Fourier component

G(k, ω) = − ie�/T∑n,m

e−En/T |〈n|ck|m〉|2∫ ∞

0dt ei(ωnm+ω)t

+ ie�/T∑n,m

e−Em/T |〈n|ck|m〉|2∫ 0

−∞dt ei(ωnm+ω)t . (D.7)

Using the formula ∫ ∞

0dt eizt = lim

δ→+0

∫ ∞

0dt eizt−δt

= i

z + iδ

= iP1

z+ πδ(z) (D.8)

276 Analytical properties of one-particle Green’s functions

we obtain

G(k, ω) = e�/T∑n,m

e−En/T |〈n|ck|m〉|2

×{

P1 + eωnm/T

ωnm + ω− iπ[1 − eωnm/T ]δ(ωnm + ω)

}. (D.9)

Here P means the principal value of the integral, when integrating over ω. It isconvenient to introduce the spectral function A(k, ω):

A(k, ω) ≡ π(1 + e−ω/T )e�/T∑n,m

e−En/T |〈n|ck|m〉|2δ(ωnm + ω) (D.10)

which is real and strictly positive (A(k, ω) > 0). It obeys an important sum rule:

1

π

∫ ∞

−∞dω A(k, ω) = 1. (D.11)

Indeed, integrating equation (D.10) yields

1

π

∫ ∞

−∞dωA(k, ω) = e�/T

∑n,m

e−En/T

× (〈n|ck|m〉〈m|c†k|n〉 + 〈n|c†

k|m〉〈m|ck|n〉). (D.12)

The set of exact eigenstates is complete, i.e.∑m

|m〉〈m| = 1

so that we can eliminate the summation over m in equation (D.12),

1

π

∫ ∞

−∞dω A(k, ω) = e�/T

∑n,m

e−En/T 〈n|{ckc†k}|n〉

= e�/T∑

n

e−En/T = 1. (D.13)

The spectral function plays an important role in the theoretical description ofthe photoemission spectra (part 2). Using equation (D.9) we can connect theimaginary part of the GF with the spectral function as

A(k, ω) = − coth( ω

2T

)Im G(k, ω). (D.14)

The real part of the GF can also be expressed via A(k, ω) as an integral,

Re G(k, ω) = 1

πP

∫ ∞

−∞dω′ A(k, ω′)

ω − ω′ . (D.15)

Analytical properties of one-particle Green’s functions 277

Let us also introduce retarded and advanced GFs defined as

GR(k, t) = − i�(t)〈〈{ck(t)c†k}〉〉 (D.16)

GA(k, t) = i�(−t)〈〈{ck(t)c†k}〉〉

respectively. By the use of the exact eigenstates and eigenvalues of the many-particle Hamiltonian, we obtain, as before,

GR(k, ω) = e�/T∑n,m

e−En/T |〈n|ck|m〉|2(1 + eωnm/T )

×{

P1

ωnm + ω− iπδ(ωnm + ω)

}. (D.17)

We see thatIm GR(k, ω) = −A(k, ω) (D.18)

and

Re GR(k, ω) = 1

πP

∫ ∞

−∞dω′ A(k, ω′)

ω − ω′ . (D.19)

As a result, we connect the GF with the retarded (or advanced) GF:

GR(k, ω) = Re G(k, ω) + i coth( ω

2T

)Im G(k, ω) (D.20)

andGA(k, ω) = Re G(k, ω) − i coth

( ω

2T

)Im G(k, ω). (D.21)

From equations (D.18) and (D.19), we also have

GR(k, ω) = 1

π

∫ ∞

−∞dω′ A(k, ω′)

ω − ω′ + iδ(D.22)

and, in a similar way,

GA(k, ω) = 1

π

∫ ∞

−∞dω′ A(k, ω′)

ω − ω′ − iδ. (D.23)

It becomes clear that the retarded GF is analytical in the upper half-plane of ω andthe advanced GF is analytical in the lower half-plane. Finally, using the definition(equation (D.4)) of the Matsubara GF, we can write, for τ > 0,

�(k, τ ) = −e�/T∑n,m

e−En/T |〈n|ck|m〉|2eωnmτ . (D.24)

Transforming this function into a Fourier series yields

�(k, ωk) =∫ 1/T

0dτ �(k, τ ) exp(iωkτ )

= e�/T∑n,m

e−En/T |〈n|ck|m〉|2 1 + eωnm/T

ωnm + iωk(D.25)

278 Analytical properties of one-particle Green’s functions

where ωk = πT (2k + 1), k = 0,±1,±2, . . . . This Fourier transform can be alsoexpressed via the spectral function:

�(k, ωk) = 1

π

∫ ∞

−∞dω′ A(k, ω′)

iωk − ω′ . (D.26)

By comparison of equations (D.26) and (D.22), we obtain, for ωk > 0,

GR(k, iωk) = �(k, ωk) (D.27)

and, from equation (D.26),

�(k,−ωk) = �∗(k, ωk). (D.28)

These relations allow us to construct GR(k, ω) for real frequencies continuing�(k, ωk) analytically from a discrete set of points to the upper half-plane. Thenusing equation (D.20), one can obtain G(k, ω).

Appendix E

Canonical transformation

Various canonical transformations serve as a powerful tool in the polaron theory[124]. Using an appropriate transformation, we can approximately diagonalizethe Hamiltonian of strongly coupled electrons and phonons and then apply theperturbation theory with respect to residual off-diagonal terms. The philosophyof the method is simple. We are looking for a complete orthonormal set of multi-particle eigenstates |n〉, which obey the steady-state Schrodinger equation

H |n〉 = En|n〉. (E.1)

There exists a unitary transformation U such that the eigenstates |n〉 may begenerated from another arbitrary complete orthonormal set |n〉 such that

|n〉 = U |n〉. (E.2)

The requirement that the states generated by equation (E.2) form an eigenbaseof the given Hamiltonian H is equivalent to the condition that the transformedHamiltonian

H = U† HU (E.3)

is diagonal with respect to |n〉:H |n〉 = En|n〉. (E.4)

The orthogonality should conserve (〈n | n′〉 = 〈n|U†U |n′〉 = δnn′ ) so that

U† = U−1. (E.5)

A frequently imployed transformation in polaron theory is the displacementtransformation introduced by Lee et al [125] for a single polaron and by Langand Firsov [71] for a multi-polaron system. It displays ions to new equilibriumpositions depending on the electron coordinates,

U = e−S (E.6)

279

280 Canonical transformation

whereS =

∑q,ν,i

ni [u∗i (q, ν)d†

qν − H.c.] (E.7)

is such that S† = S−1 = −S. The electron and phonon operators are transformedas

ci = eSci e−S (E.8)

dqν = eSdqνe−S .

We can simplify equations (E8) scaling all matrix elements by the same amount,ui (q, ν) → ηui (q, ν), and differentiating the transformed operators with respectto the scaling parameter η as

∂ ci

∂η=

∑q,ν

eS[ni , ci ](u∗i (q, ν)d†

qν − ui (q, ν)dqν)e−S (E.9)

and∂ dqν

∂η=

∑i

eSni u∗i (q, ν)[d†

qν, dqν]e−S . (E.10)

Using commutators [ni , ci ] = −ci , [d†qν, dqν] = −1 and [ni , S] = 0, we find

∂ ci

∂η= − ci

∑q,ν

(u∗i (q, ν)d†

qν − ui (q, ν)dqν) (E.11)

∂ dqν

∂η= −

∑i

ni u∗i (q, ν).

The solutions of these differential equations, which respects the ‘boundary’conditions dqν = dqν and ci = ci when η = 0, are

ci = ci exp

[η∑q,ν

(ui (q, ν)dqν − u∗i (q, ν)d†

qν)

](E.12)

anddqν = dqν − η

∑i

ni u∗i (q, ν). (E.13)

By taking η = 1 in equations (E.12) and (E.13), we obtain equations (4.28) and(4.29), respectively.

Let us now calculate the statistical average of the multi-phonon operator,which determines the polaron bandwidth in equations (4.38) and (4.42):

〈〈X†i X j 〉〉 ≡

∏q,ν

〈〈exp[u∗i (q, ν)d†

qν − H.c.] exp[u j (q, ν)dqν − H.c.]〉〉. (E.14)

Canonical transformation 281

Here an operator identity

e A+B = e AeBe−[ A,B]/2 (E.15)

is instrumental. It is applied when the commutator [ A, B] is a number. To provethe identity, we use the parameter differentiation [126] as before. Let us assumethat

eη( A+B) = eη Aeη Beη2C (E.16)

where C is a number. Differentiating both sides of equation (E.16) with respectto the parameter η and multiplying it from the right by

e−η( A+B) = e−η2Ce−η Be−η A

one obtainsA + B = A + eη A Be−η A + 2ηC. (E.17)

The quantity eη A Be−η A is expanded by means of

eη A Be−η A =∞∑

r,q=0

(−1)qηr+q

r !q! Ar B Aq

= B + η[ A, B] + η2

2[ A, [ A, B]] + · · · . (E.18)

Because [ A, B] is a number, all terms, starting from the third one, vanish in theright-hand side of equation (E.18). Then using equation (E.17), one obtains

C = − 12 [ A, B]. (E.19)

The identity (E.15) allows us to write

e[u∗i (q,ν)d†

qν−H.c.]e[u j (q,ν)dqν−H.c.] = e(α∗d†qν−αdqν)

× e[ui (q,ν)u∗j (q,ν)−u∗

i (q,ν)u j (q,ν)]/2

where α ≡ ui (q, ν) − u j (q, ν). Applying once again the same identity yields

e[u∗i (q,ν)d†

qν−H.c.]e[u j (q,ν)dqν−H.c.] = eα∗d†q e−αdq e−|α|2/2

× e[ui (q,ν)u∗j (q,ν)−u∗

i (q,ν)u j (q,ν)]/2. (E.20)

Quantum and statistical averages are calculated by expanding the exponents in thetrace as

〈〈eα∗d†e−αd〉〉 = (1 − p)

∞∑N=0

N∑n=0

pN (−1)n |α|2n

(n!)2N(N − 1) × · · · × (N − n + 1)

(E.21)

282 Canonical transformation

where we have dropped the phonon and site quantum numbers for transparency.Here p = exp(−ωqν/T ), so that a single-mode phonon partition function(appendix B) is

Zph = 1

1 − p.

Equation (E.21) can be written in the form [71]

〈〈eα∗d†e−αd〉〉 = (1 − p)

N∑n=0

(−1)n |α|2n

(n!)2 pn dn

d pn

∞∑N=0

pN . (E.22)

Taking the sum over N ,∞∑

N=0

pN = 1

1 − p

and differentiating it n times yields n! in the numerator, after which the series overn turns out equal to

〈〈eα∗d†e−αd〉〉 = e−|α|2nω (E.23)

where nω = [exp(ωq/T ) − 1]−1 is the Bose–Einstein distribution function ofphonons. Now collecting all multipliers we obtain equations (4.38) and (4.42).

References

[1] Bednorz J G and Muller K A 1986 Z. Phys. B 64 189[2] Bardeen J, Cooper L N. and Schrieffer J R 1957 Phys. Rev. 108 1175[3] London F 1938 Phys. Rev. 54 947[4] Bogoliubov N N 1947 Izv. Acad. Sci. (USSR) 11 77[5] Landau L D 1947 J. Phys. (USSR) 11 91[6] Ogg Jr. R A 1946 Phys. Rev. 69 243[7] Schafroth M R 1955 Phys. Rev. 100 463[8] Blatt J M and Butler S T 1955 Phys. Rev. 100 476[9] Frohlich H 1950 Phys. Rev. 79 845

[10] Alexandrov A S and Ranninger J 1981 Phys. Rev. B 23 1796[11] Alexandrov A S 1983 Zh. Fiz. Khim. 57 273

Alexandrov A S 1983 Russ. J. Phys. Chem. 57 167Alexandrov A S 1998 Models and Phenomenology for Conventional and High-

temperature Superconductivity (Course CXXXVI of the Intenational School ofPhysics ‘Enrico Fermi’) ed G Iadonisi, J R Schrieffer and M L Chiofalo(Amsterdam: IOS) p 309

[12] Alexandrov A S and Mott N F 1994 High Temperature Superconductors and OtherSuperfluids (London: Taylor and Francis)

[13] Alexandrov A S and Mott N F 1995 Polarons and Bipolarons (Singapore: WorldScientific)

[14] Anderson P W 1975 Phys. Rev. Lett. 34 953[15] Chakraverty B K 1979 J. Physique Lett. 40 L99

Chakraverty B K 1981 J. Physique Lett. 42 1351[16] Bednorz J G and Muller K A 1988 Angew. Chem. Int. Ed. Engl. 27 735[17] Meissner W and Ochsenfeld R 1933 Naturw. 21 787[18] London F and London H 1935 Proc. R. Soc. A149 71[19] Deaver Jr. B S and Fairbank W M 1961 Phys. Rev. Lett. 7 43

Doll R and Nabauer M 1961 Phys. Rev. Lett. 7 51[20] Pippard A B 1953 Proc. R. Soc. A 216 547[21] Ginzburg V L and Landau L D 1950 Zh. Eksp. Teor. Fiz. 20 1064[22] Levanyuk A P 1959 Zh. Eksp. Teor. Fiz. 36 810

Pippard A B 1950 Proc. R. Soc. A 203 210[23] Landau L D and Lifshitz E M 1980 Statistical Mechanics 3rd edn (Oxford:

Pergamon)[24] Abrikosov A A 1957 Zh. Eksp. Teor. Fiz. 32 1442[25] Fock V 1928 Z. Phys. 47 446

283

284 References

Landau L D 1930 Z. Phys. 64 629[26] Clogston A M 1962 Phys. Rev. Lett. 9 266

Chandrasekhar B S 1962 Appl. Phys. Lett. 1 7[27] Werthamer N R, Helfand E and Hohenberg P C 1966 Phys. Rev. 147 295[28] Schmidt V V 1997 Physics of Superconductors ed P Muller and A V Ustinov

(Berlin: Springer)[29] Josephson B D 1962 Phys. Lett. 1 251[30] Feynman R P, Leighton R B and Sands M 1989 Feynman Lectures on Physics, The

Commemorative Issue (San Francisco, CA: Benjamin Cummings)[31] Schrieffer J R 1964 Theory of Superconductivity (Reading, MA: Benjamin)[32] Abrikosov A A, Gor’kov L P and Dzaloshinski I E 1963 Methods of Quantum Field

Theory in Statistical Physics (Englewood Cliffs, NJ: Prentice-Hall)[33] de Gennes P G 1989 Superconductivity of Metals and Alloys (New York: Addison-

Wesley)[34] Bardeen J and Pines D 1955 Phys. Rev. 99 1140[35] Cooper L N 1957 Phys. Rev. 104 1189[36] Eliashberg G M 1960 Zh. Eksp. Teor. Fiz. 38 966

Eliashberg G M 1960 Zh. Eksp. Teor. Fiz. 39 1437Eliashberg G M 1960 Sov. Phys.–JETP 11 696Eliashberg G M 1960 Sov. Phys.–JETP 12 1000

[37] Giaever I 1960 Phys. Rev. Lett. 5 147, 464[38] Hebel L C and Slichter C P 1959 Phys. Rev. 113 1504[39] Geilikman B T 1958 Zh. Eksp. Teor. Fiz. 34 1042[40] Lifshits E M and Pitayevski 1978 Statistical Physics Part 2 Theory of Condensed

State (Oxford: Butterworth–Heinemann)[41] Svidzinskii A V 1982 Space Inhomogeneous Problems in the Theory of

Superconductivity (Moscow: Nauka)[42] Abrikosov A A and Gor’kov L P 1960 Zh. Eksp. Teor. Fiz. 38 178[43] Gor’kov L P 1958 Zh. Eksp. Teor. Fiz. 34 735

Gor’kov L P 1958 Sov. Phys.–JETP 7 505[44] Matsubara T 1955 Prog. Theor. Phys. 14 351[45] Gor’kov L P 1959 Zh. Eksp. Teor. Fiz. 36 1918

Gor’kov L P 1959 Sov. Phys.–JETP 9 1364[46] Andreev A F 1964 Zh. Eksp. Teor. Fiz. 46 1823

Andreev A F 1964 Sov. Phys.–JETP 19 1228[47] Blonder G E, Tinkham M and Klapwijk T M 1982 Phys. Rev. B 25 4515[48] Migdal A B 1958 Zh. Eksp. Teor. Fiz. 34 1438

Migdal A B 1958 Sov. Phys.–JETP 7 996[49] Nambu Y 1960 Phys. Rev. 117 648[50] Scalapino D J 1969 Superconductivity ed R D Parks (New York: Marcel Dekker)[51] Allen P B and Mitrovic B 1982 Solid State Physics vol 37, ed H Ehrenreich, F Seitz

and D Turnbull (New York: Academic)[52] Ginzburg V L and Kirzhnits D A (ed) 1977 The Problem of High-Temperature

Superconductivity (Moscow: Nauka)[53] Tolmachev V V 1959 A new Method in the Theory of Superconductivity ed

N N Bogoliubov et al (New York: Consultant Bureau)[54] McMillan W J 1968 Phys. Rev. B 167 331[55] Klein B M, Boyer L L and Papaconstantopoulos D A 1978 Phys. Rev. B 18 6411

References 285

[56] Allen P B and Dynes R C 1975 Phys. Rev. B 12 905[57] Kohn W and Luttinger J M 1965 Phys. Rev. Lett. 15 524[58] Friedel H 1954 Adv. Phys. 3 446[59] Fay D and Layzer A 1968 Phys. Rev. Lett. 20 187[60] Landau L D 1933 J. Phys. (USSR) 3 664[61] Pekar S I 1946 Zh. Eksp. Teor. Fiz. 16 335[62] Frohlich H 1954 Adv. Phys. 3 325[63] Feynman R P 1955 Phys. Rev. 97 660[64] Devreese J T 1996 Encyclopedia of Applied Physics vol 14 (Weinheim: Wiley–

VCH) p 383 and references therein[65] Tjablikov S V 1952 Zh. Eksp. Teor. Fiz. 23 381[66] Yamashita J and Kurosawa T 1958 J. Phys. Chem. Solids 5 34[67] Sewell G L 1958 Phil. Mag. 3 1361[68] Holstein T 1959 Ann. Phys. 8 325

Holstein T 1959 Ann. Phys. 8 343[69] Friedman L and Holstein T 1963 Ann. Phys. 21 494[70] Emin D and Holstein T 1969 Ann. Phys. 53 439[71] Lang I G and Firsov Yu A 1962 Zh. Eksp. Teor. Fiz. 43 1843

Lang I G and Firsov Yu A 1963 Sov. Phys.–JETP 16 1301[72] Eagles D M 1963 Phys. Rev. 130 1381

Eagles D M 1969 Phys. Rev. 181 1278Eagles D M 1969 Phys. Rev. 186 456

[73] Appel J 1968 Solid State Physics vol 21, ed F Seitz, D Turnbull and H Ehrenreich(New York: Academic)

[74] Firsov Yu A (ed) 1975 Polarons (Moscow: Nauka)[75] Bottger H and Bryksin V V 1985 Hopping Conduction in Solids (Berlin: Academie)[76] Mahan G D 1990 Many Particle Physics (New York: Plenum)[77] Geilikman B T 1975 Usp. Fiz. Nauk. 115 403

Geilikman B T 1975 Sov. Phys.–Usp. 18 190[78] Alexandrov A S and Mazur E A 1989 Zh. Eksp. Teor. Fiz. 96 1773[79] Rashba E I 1957 Opt. Spectrosc. 2 75

Rashba E I 1985 Excitons ed E I Rashba and D M Struge (Moscow: Nauka)[80] Kabanov V V and Mashtakov O Yu 1993 Phys. Rev. B 47 6060[81] Aubry S 1995 Polarons and Bipolarons in High-Tc Superconductors and Related

Materials ed E K H Salje, A S Alexandrov and W Y Liang (Cambridge:Cambridge University Press) p 271

[82] Alexandrov A S 1992 Phys. Rev. B 46 14 932[83] Alexandrov A S, Kabanov V V and Ray D K 1994 Phys. Rev. B 49 9915[84] Bishop A R and Salkola M 1995 Polarons and Bipolarons in High-Tc

Superconductors and Related Materials ed E K H Salje, A S Alexandrov andW Y Liang (Cambridge: Cambridge University Press) p 353

[85] Marsiglio F 1995 Physica C 244 21[86] Takada Y and Higuchi T 1995 Phys. Rev. B 52 12 720[87] Fehske H, Loos J and Wellein G 1997 Z. Phys. B 104 619[88] Hotta T and Takada Y 1997 Phys. Rev. B 56 13 916[89] Romero A H, Brown D W and Lindenberg K 1998 J. Chem. Phys. 109 6504[90] La Magna A and Pucci R 1996 Phys. Rev. B 53 8449[91] Benedetti P and Zeyher R 1998 Phys. Rev. B 58 14 320

286 References

[92] Frank T and Wagner M 1999 Phys. Rev. B 60 3252[93] Bonca J, Trugman S A and Batistic I 1999 Phys. Rev. B 60 1633[94] Alexandrov A S and Kornilovich P E 1999 Phys. Rev. Lett. 82 807[95] Proville L and Aubry S 1999 Eur. Phys. J. B 11 41[96] Alexandrov A S 2000 Phys. Rev. B 61 12 315[97] Hiramoto H and Toyozawa Y 1985 J. Phys. Soc. Japan 54 245[98] Alexandrov A S 1992 Phys. Rev. B 46 2838[99] Gogolin A A 1982 Phys. Status Solidi B 109 95

[100] Alexandrov A S 1996 Phys. Rev. B 53 2863[101] Alexandrov A S and Sricheewin C 2000 Europhys. Lett. 51 188[102] Alexandrov A S and Ranninger J 1992 Phys. Rev. B 45 13 109

Alexandrov A S and Ranninger J 1992 Physica C 198 360[103] Vinetskii V L and Giterman M Sh 1957 Zh. Eksp. Teor. Fiz. 33 730

Vinetskii V L and Giterman M Sh 1958 Sov. Phys.–JETP 6 560Vinetskii V L and Pashitskii E A 1983 Fiz. Tverd. Tela (Leningrad) 25 1744Vinetskii V L and Pashitskii E A 1983 Sov. Phys. Solid State 25 1005

[104] Muller K A 2002 Phys. Scr. T 102 39 and references therein[105] Alexandrov(Aleksandrov) A S and Kabanov V V 1986 Fiz. Tverd. Tela 28 1129

Alexandrov(Aleksandrov) A S and Kabanov V V 1986 Sov. Phys. Solid State 28631

[106] Bryksin V V and Gol’tsev A V 1988 Fiz. Tverd. Tela 30 1476Bryksin V V and Gol’tsev A V 1988 Sov. Phys. Solid State 30 851

[107] Bonca J and Trugman S A 2001 Phys. Rev. B 64 094507[108] Alexandrov A S and Kornilovitch P E 2002 J. Phys.: Condens. Matter 14 5337[109] Alexandrov A S, Ranninger J and Robaszkiewicz S 1986 Phys. Rev. B 33 4526[110] Alexandrov A S, Samarchenko D A and Traven S V 1987 Zh. Eksp. Teor. Fiz. 93

1007Alexandrov A S, Samarchenko D A and Traven S V 1987 Sov. Phys.–JETP 66 567

[111] Peierls R E 1933 Z. Phys. 80 763[112] Alexandrov A S and Beere W H 1995 Phys. Rev. B 51 5887[113] Gross E P 1961 Lett. Nuovo Cimento 20 454

Pitaevskii L P 1961 Sov. Phys.–JETP 13 451[114] Foldy L L 1961 Phys. Rev. 124 649[115] Alexandrov A S and Kabanov V V 2002 J. Phys.: Condens. Matter 14 L327[116] Tonks L and Langmuir I 1929 Phys. Rev. 33 195[117] Pines D and Schrieffer J R 1961 Phys. Rev. 124 1387[118] Fetter A L 1971 Ann. Phys. 64 1[119] Bishop R F 1974 J. Low Temp. Phys. 15 601[120] Alexandrov A S 1999 Phys. Rev. B 60 14 573[121] Alexandrov A S 1984 Doctoral Thesis MIFI (Moscow)

Alexandrov A S 1993 Phys. Rev. B 48 10 571[122] Alexandrov A S, Beere W H and Kabanov V V 1996 Phys. Rev. B 54 15 363[123] Alexandrov A S 1998 Physica C 305 46[124] Wagner M 1986 Unitary Transformations in Solid State Physics (Amsterdam:

North-Holland)[125] Lee T D, Low F and Pines D 1953 Phys. Rev. 90 297[126] Wilcox R M 1967 J. Math. Phys. 8 962

References 287

[127] Wu M K, Ashburn J R, Torng C J, Hor P H, Meng R L, Gao L, Huang Z J, Wang YQ and Chu C W 1987 Phys. Rev. Lett. 58 908

[128] Hebard A F, Rosseinsky M J, Haddon R C, Murphy D W, Glarum S H, Palstra T T,Ramirez A P, and Kortan A R 1991 Nature 350 600

[129] Holczer K, Klein O, Huang S M, Kaner R B, Fu K J, Whetten R L and Diederich F1991 Science 252 1154

[130] Nagamatsu J, Nakagawa N, Muranaka T, Zenitani Y and Akimitsu J 2001 Nature410 63

[131] Alexandrov A S and Edwards P P 2000 Physica C 331 97[132] Simon A 1997 Angew: Chem. Int. Ed. Engl. 36 1788[133] Ginzburg V L 1968 Contemp. Phys. 9 355[134] Little W A 1964 Phys. Rev. A 134 1416[135] Frohlich H 1968 Phys. Lett. A 26 169[136] Pashitskii E A 1968 Zh. Eksp. Teor. Fiz. 55 2387

Pashitskii E A 1969 Sov. Phys.–JETP 28 1267[137] Schrieffer J R, Wen X G and Zhang S C 1989 Phys. Rev. B 39 11 663[138] Millis A J, Monien H and Pines D 1990 Phys. Rev. B 42 167[139] Mott N F 1990 Adv. Phys. 39 55[140] Anderson P W 1997 The Theory of Superconductivity in the Cuprates (Princeton

NJ: Princeton University Press)[141] Carlson E W, Emery V J, Kivelson S A and Orgad D 2002 Concepts in

high temperature superconductivity Preprint cond-mat/0206217 and referencestherein

[142] Gabovich A M, Voitenko A I, Annett J F and Ausloos M 2001 Supercond. Sci.Technol. 14 R1

[143] Mott N F 1990 Metal–Insulator Transitions 2nd edn (London: Taylor and Francis)[144] Wheatley J M, Hsu T C and Anderson P W 1988 Phys. Rev. B 37 5897[145] Schutzmann J, Somal H S, Tsvetkov A A, van der Marel D, Koops G E J,

Koleshnikov N, Ren Z F, Wang J H, Bruck E and Menovsky A A 1997 Phys.Rev. B 55 11 118

[146] Nagaosa N and Lee P A 2000 Phys. Rev. B 61 9166[147] Hubbard J 1963 Proc. R. Soc. A 276 238[148] For a review of the Hubbard and t–J models see Plakida N M 2001 Lectures on

the Physics of Highly Correlated Electron Systems V (AIP Conf. Proc. 580) edF Mancini (Melville, NY: American Institute of Physics) p 121

[149] Scalapino D J, Loh J E and Hirsch J E 1986 Phys. Rev. B 34 8190[150] Scherman A and Scrieber M 1995 Phys. Rev. B 52 10 621[151] Laughlin R B 2002 Preprint cond-mat/0209269[152] Xiang X D, Vareka W A, Zettl A, Corkill J L, Cohen M L, Kijima N and Gronsky

R 1992 Phys. Rev. Lett. 68 530[153] Forro L, Ilakovac V, Cooper J R, Ayache C and Henry J Y 1992 Phys. Rev. B 46

6626[154] Bourges P, Sidis Y, Fong H F, Regnault L P, Bossy J, Ivanov A and Keimer B 2000

Science 288 1234Bourges P, Keimer B, Regnault L P and Sidis Y 2000 J. Supercond. 13 735

[155] Abbamonte P, Venema L, Rusydi A, Sawatzky G A, Logvenov G and Bozovic I2002 Science 297 581

Cho A 2002 Science 297 499 and references therein

288 References

[156] Hirsch J E 2002 Science 295 2226Hirsch J E 2000 Phys. Rev. B 62 14 487 and references therein

[157] Molegraaf H J A, Presura C, van der Marel D, Kes P H and Li M 2002 Science 2952239

[158] Labbe J and Bok J 1987 Europhys. Lett. 3 1225Newns D M, Pattnaik P C and Tsuei C C 1991 Phys. Rev. B 43 3075Abrikosov A A 2000 Physica C 341 97

[159] Castellani C, DiCastro C and Grilli M 1995 Phys. Rev. Lett. 75 4650[160] Sachdev S, Chubukov A V and Ye J 1994 Phys. Rev. B 49 11 919[161] Smith T J, Andersen K H, Beck U, Capellmann H, Kremer R K, Neumann K U,

Scharpf O, Simon A and Ziebeck K R A 1998 J. Magnetism Magn. Mater. 177543

[162] Radtke R J, Levin K, Schuttler H B and Norman M R 1993 Phys. Rev. B 48 15 957[163] Varma C M, Littlewood P B, Schmitt-Rink S, Abrahams E and Ruckenstein A E

1989 Phys. Rev. Lett. 63 1996[164] Emery V J and Kivelson S A Nature 374 434[165] Leggett A J 1973 Physica Fennica 8 125

Leggett A J 1998 J. Stat. Phys. 93 927Popov V N 1987 Functional Integrals and Collective Excitations (Cambridge:

Cambridge University Press)[166] Alexandrov A S 1987 Pis’ma Zh. Eksp. Teor. Fiz. (Prilozh.) 46 128

Alexandrov A S 1987 JETP Lett. Suppl. 46 107[167] Prelovsek P, Rice T M and Zhang F C 1987 J. Phys. C: Solid State Phys. 20 L229[168] Emin D 1989 Phys. Rev. Lett. 62 1544[169] See different contributions in 1995 Polarons and Bipolarons in High-Tc

Superconductors and Related Materials ed E K H Salje, A S Alexandrov andW Y Liang (Cambridge: Cambridge University Press) and in 1995 Anharmonicproperties of High Tc cuprates ed D Mihailovic et al (Singapore: WorldScientific)

[170] Mihailovic D, Foster C M, Voss K and Heeger A J 1990 Phys. Rev. B 42 7989[171] Calvani P, Capizzi M, Lupi S, Maselli P, Paolone A, Roy P, Cheong S-W, Sadowski

W and Walker E 1994 Solid State Commun. 91 113[172] Timusk T, Homes C C and Reichardt W 1995 Anharmonic Properties of High Tc

Cuprates ed D Mihailovic D et al (Singapore: World Scientific) p 171[173] Zhao G, Hunt M B, Keller H and Muller K A 1997 Nature 385 236[174] 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 Zand Shen Z X 2001 Nature 412 510

[175] Egami T 1996 J. Low Temp. Phys. 105 791[176] Temprano D R, Mesot J, Janssen S, Conder K, Furrer A, Mutka H and Muller K A

2000 Phys. Rev. Lett. 84 1990[177] Shen Z X, Lanzara A, Ishihara S and Nagaosa N 2002 Phil. Mag. B 82 1349[178] Phillips J C 1989 Physics of high Tc Superconductors (New York: Academic)[179] Alexandrov A S and Mott N F 1993 Supercond. Sci. Technol. 6 215

Alexandrov A S and Mott N F 1994 J. Supercond. 7 599Alexandrov A S and Mott N F 1994 Rep. Prog. Phys. 57 1197

[180] Allen P B 2001 Nature 412 494[181] Gor’kov L P 1999 J. Supercond. 12 9

References 289

[182] Tempere J, Fomin V M and Devreese J T 1997 Solid State Commun. 101 661[183] Alexandrov A S 1998 Phil. Trans. R. Soc. A 356 197[184] Merz M, Nucker N, Schweiss P, Schuppler S, Chen C T, Chakarian V, Freeland J,

Idzerda Y U, Klaser M, Muller-Vogt G and Wolf Th 1998 Phys. Rev. Lett. 805192

Dow J D, Howard U, Blackstead A 1998 Bull. Am. Phys. Soc. 43 877[185] Alexandrov A S 1992 J. Low Temp. Phys. 87 721[186] Alexandrov A S, Kabanov V V and Mott N F 1996 Phys. Rev. Lett. 77 4796[187] Alexandrov A S and Bratkovsky A M 2000 Phys. Rev. Lett. 84 2043[188] Kim J H, Feenstra B J, Somal H S, van der Marel D, Lee W Y, Gerrits A M and

Wittlin A 1994 Phys. Rev. B 49 13 065[189] Alexandrov A S 2001 Physica C 363 231[190] Alexandrov A S and Kabanov V V 1996 Phys. Rev. B 54 3655[191] Pietronero L 1992 Europhys. Lett. 17 365

Pietronero L and Strassler S 1992 Europhys. Lett. 18 627Grimaldi C, Pietronero L and Strassler S 1995 Phys. Rev. Lett. 75 1158Pietronero L, Strassler S and Grimaldi C 1995 Phys. Rev. B 52 10 516

[192] Catlow C R A, Islam M S and Zhang X 1998 J. Phys.: Condens. Matter 10 L49[193] Anselm A 1981 Introduction of Semiconductor Theory (Englewood Cliffs, NJ:

Prentice-Hall)[194] Alexandrov A S, Bratkovsky A M and Mott N F 1994 Phys. Rev. Lett. 72 1734[195] Mihailovic D, Kabanov V V, Zagar K, and Demsar J 1999 Phys. Rev. B 60 6995 and

references therein[196] Zhang Y, Ong N P, Xu Z A, Krishana K, Gagnon R and Taillefer L 2000 Phys. Rev.

Lett. 84 2219[197] Chen X H, Yu M, Ruan K Q, Li S Y, Gui Z, Zhang G C and Cao L Z 1998 Phys.

Rev. B 58 14 219Chen W M, Franck J P and Jung J 2000 Physica C 341 1875

[198] Hwang H Y, Batlogg B, Takagi H, Kao H L, Kwo J, Cava R J, Krajewski J J andPeck W F 1994 Phys. Rev. Lett. 72 2636

Batlogg B, Hwang H Y, Takagi H, Cava R J, Kao H L and Kwo J 1994 Physica C235 130

[199] Bucher B, Steiner P, Karpinski J, Kaldis E and Wachter P 1993 Phys. Rev. Lett. 702012

[200] Mott N F 1993 Phil. Mag. Lett. 68 245[201] Boebinger G S, Ando Y, Passner A, Kimura T, Okuya M, Shimoyama J, Kishio K,

Tamasaku K, Ichikawa N and Uchida S 1996 Phys. Rev. Lett. 77 5417Ando Y, Boebinger G S, Passner A, Kimura T and Kishio K 1995 Phys. Rev. Lett.

75 4662[202] Alexandrov A S 1997 Phys. Lett. A 236 132[203] Alexandrov A S and Mott N F 1993 Phys. Rev. Lett. 71 1075[204] Takenaka K, Fukuzumi Y, Mizuhashi K, Uchida S, Asaoka H and Takei H 1997

Phys. Rev. B 56 5654[205] Yu R C, Salamon M B, Lu J P and Lee W C 1992 Phys. Rev. Lett. 69 1431[206] Hill R W, Proust C, Taillefer L, Fournier P and Greene R L 2001 Nature 414 711[207] Takeya J, Ando Y, Komiya S and Sun X F 2002 Phys. Rev. Lett. 88 077001[208] Proust C, Boakin E, Hill R W, Taillefer L and Mackenzie A P 2002 Preprint cond-

mat/0202101

290 References

[209] Machi T, Tomeno I, Miyatake T, and Tanaka S 1991 Physica C 173 32[210] Rossat-Mignod J, Regnault L P, Bourges P, Vettier C, Burlet P and Henry J Y 1992

Phys. Scr. 45 74[211] Mook H A, Yethiraj M, Aeppli G, Mason T E and Armstrong T 1993 Phys. Rev.

Lett. 70 3490[212] Hofer J, Karpinski J, Willemin M, Meijer G I, Kopnin E M, Molinski R, Schwer H,

Rossel C and Keller H 1998 Physica C 297 103[213] Zverev V N and Shovkun D V 2000 JETP Lett. 72 73[214] Nakamura Y and Uchida S 1993 Phys. Rev. B 47 8369[215] Nakano T, Oda M, Manabe C, Momono N, Miura Y and Ido M 1994 Phys. Rev. B

49 16 000[216] Fehske H, Roder H, Wellein G and Mistriotis A 1995 Phys. Rev. B 51 16 582[217] Alexandrov A S 1999 Phys. Rev. Lett. 82 2620

Alexandrov A S and Kabanov V V 1999 Phys. Rev. B 59 13 628[218] Uemura Y J 1995 Polarons and Bipolarons in High-Tc Superconductors and Related

Materials ed E K H Salje, A S Alexandrov and W Y Liang (Cambridge:Cambridge University Press) p 453

[219] Alder B J and Peters D S 1989 Europhys. Lett. 10 1[220] Pokrovsky V L 1988 Pis’ma Zh. Teor. Fiz. 47 5395[221] Schneider T and Singer J M 2000 Phase Transition Approach To High Temperature

Superconductivity (London: Imperial College Press)[222] Legget A J 1980 J. Physique 41 C7[223] Nozieres P and Schmitt-Rink S 1985 J. Low Temp. Phys. 59 195[224] Micnas R and Kostyrko T 1995 Recent developments in High Temperature

Superconductivity ed P W Klamut and M Kazimierski (Berlin: Springer) p 221and references therein

[225] Gyorffy B L, Staunton J B and Stocks G M 1991 Phys. Rev. B 44 5190Gorbar E V, Loktev V M and Sharapov S G 1996 Physica C 257 355Pistolesi F and Strinati G C 1996 Phys. Rev. B 53 15 168Marini M, Pistolesi F and Strinati G C 1998 Eur. Phys. J. B 1 158Maly J, Levin K and Liu D Z 1996 Phys. Rev. B 54 15 657

[226] Alexandrov A S and Rubin S G 1993 Phys. Rev. B 47 5141[227] Tiverdi N and Randeria M 1995 Phys. Rev. Lett. 75 312

Singer J M, Pedersen M H and Schneider T 1995 Physica B 230 955[228] Schmitt Rink S, Varma C M and Ruckenstein A E 1989 Phys. Rev. Lett. 63 445[229] Traven S V 1994 Phys. Rev. B 51 3242[230] Crawford M K, Farneth W E, McCarron E M III, Harlow R L and Moudden A H

1990 Science 250 1309[231] Bornemann H J, Morris D E, Liu H B, Sinha A P, Narwankar P and Chandrachood

M 1991 Physica C 185–189 1359[232] Franck J P, Jung J, Salomons G J, Miner W A, Mohamed M A K, Chrzanowski J,

Gygax S, Irwin J C, Mitchell D F and Sproule I 1989 Physica C 162 753Franck J P, Jung J, Mohamed M A K, Gygax S, and Sproule G I 1991 Phys. Rev. B

44 5318[233] Fisher R A, Kim S, Lacy S E, Phillips N E, Morris D E, Markelz A G, Wei J Y T

and Ginley D S 1988 Phys. Rev. B 38 11 942[234] Loram J W, Cooper J R, Wheatley J M, Mirza K A and Liu R S 1992 Phil. Mag. B

65 1405

References 291

[235] Inderhees S E, Salamon M B, Goldenfeld N, Rice J P, Pazol B G and Ginzberg D M1988 Phys. Rev. Lett. 60 1178

[236] Junod A, Eckert D, Triscone G, Lee V Y and Muller J 1989 Physica C 159 215[237] Schnelle W, Braun E, Broicher H, Domel R, Ruppel S, Braunisch W,

Harnischmacher J and Wohlleben D 1990 Physica C 168 465[238] Salamon M B, Inderhees S E, Rice J P and Ginsberg D M 1990 Physica A 168 283[239] Junod A, 1996 Studies of High Temperature Superconductors vol 19, ed A Narlikar

(New York: Nova Science, Commack) p 1Revaz B, Junod A and Erb A 1998 Phys. Rev. B 58 11 153Roulin M, Junod A and Walker E 1998 Physica C 296 137

[240] Alexandrov A S, Beere W H, Kabanov V V and Liang W Y 1997 Phys. Rev. Lett.79 1551

[241] Zavaritsky V N, Kabanov V V and Alexandrov A S 2002 Europhys. Lett. 60 127[242] Bucher B, Karpinski J, Kaldis E and Wachter P 1990 Physica C 167 324[243] Mackenzie A P, Julian S R, Lonzarich G G, Carrington A, Hughes S D, Liu R S and

Sinclair D C 1993 Phys. Rev. Lett. 71 1238[244] Osofsky M S, Soulen R J, Wolf S A, Broto J M, Rakoto H, Ousset J C, Coffe G,

Askenazy S, Pari P, Bozovic I, Eckstein J N and Virshup G F 1993 Phys. Rev.Lett. 71 2315

Osofsky M S, Soulen R J, Wolf S A, Broto J M, Rakoto H, Ousset J C, Coffe G,Askenazy S, Pari P, Bozovic I, Eckstein J N and Virshup G F 1994 Phys. Rev.Lett. 72 3292

[245] Alexandrov A S, Zavaritsky V N, Liang W Y and Nevsky P L 1996 Phys. Rev. Lett.76 983

[246] Lawrie D D, Franck J P, Beamish J R, Molz E B, Chen W M and Graf M J 1997 J.Low Temp. Phys. 107 491

[247] Gantmakher V F, Tsydynzhapov G E, Kozeeva L P and Lavrov A N Zh. Eksp. Teor.Fiz. 88 148

Gantmakher V F, Emel’chenko G A, Naumenko I G, and Tsydynzhapov G E 2000JETP Lett. 72 21

[248] Nakanishi T, Motoyama N, Mitamura H, Takeshita N, Takahashi H, Eisaki H,Uchida S and Mori N 2000 Int. J. Mod. Phys. 14 3617

[249] Lee I J, Chaikin P M and Naughton M J 2000 Phys. Rev. B 62 R14 669[250] Wen H H, Li S L and Zhao Z X 2000 Phys. Rev. B 62 716[251] Annett J, Goldenfeld N and Legget A J 1996 Physical Properties of High

Temperature Superconductors vol 5, ed D M Ginsberg (Singapore: WorldScientific) p 375

[252] Tsuei C C and Kirtley J R 1997 Physica C 282 4 and references therein[253] Bonn D A, Kamal S, Zhang K, Liang R X, Baar D J, Klein E and Hardy W N 1994

Phys. Rev. B 50 4051[254] Xiang T, Panagopoulos C and Cooper J R 1998 Int. J. Mod. Phys. B 12 1007[255] Li Q, Tsay Y N, Suenaga M, Klemm R A, Gu G D and Koshizuka N 1999 Phys.

Rev. Lett. 83 4160[256] Muller K A 2002 Phil. Mag. Lett. 82 279 and references therein[257] Walter H, Prusseit W, Semerad R, Kinder H, Assmann W, Huber H, Burkhardt H,

Rainer D and Sauls J A 1998 Phys. Rev. Lett. 80 3598[258] Alexandrov A S and Giles R T 1999 Physica C 325 35[259] Alexandrov A S 1998 Physica C 305 46

292 References

[260] Alexandrov A S and Andreev A F 2001 Europhys. Lett. 54 373[261] Alexandrov A S and Sricheewin C 2002 Europhys. Lett. 58 576[262] Alexandrov A S and Dent C J 1999 Phys. Rev. B 60 15 414[263] Alexandrov A S 2002 Phil. Mag. B 81 1397[264] For a review see Shen Z X 1998 Models and Phenomenology for Conventional

and High-temperature Superconductivity (Course CXXXVI of the IntenationalSchool of Physics ‘Enrico Fermi’) ed G Iadonisi, J R Schrieffer and M L Chiofalo(Amsterdam: IOS) p 141

[265] Renner Ch, Revaz B, Genoud J Y, Kadowaki K and Fischer Ø 1998 Phys. Rev. Lett.80 149

[266] Mott N F and Davis E 1979 Electronic Processes in Non-crystalline Materials2nd edn (Oxford: Oxford University Press)

[267] Muller K A, Zhao G M, Conder K and Keller H 1998 J. Phys.: Condens. Matter 10L291

[268] Hancotte H, Deltour R, Davydov D N, Jansen A G M and Wyder P 1997 Phys. Rev.B 55 R3410

[269] Mourachkine A 2000 Europhys. Lett. 49 86[270] Ding H, Yokoya T, Campuzano J C, Takahashi T, Randeria M, Norman M R,

Mochiku T, Kadowaki K and Giapintzakis J 1996 Nature 382 51[271] Shen Z X and Schrieffer J R 1997 Phys. Rev. Lett. 78 1771 and references therein[272] Loram J W, Mirza K A, Wade J M, Cooper J R and Liang W Y 1994 Physica C 235

134[273] Deutscher G 1999 Nature 397 410

Yagil Y, Hass N, Desgardin G and Monot I 1995 Physica C 250 59[274] Krasnov V M, Yurgens A, Winkler D, Delsing P and Claeson T 2000 Phys. Rev.

Lett. 84 5860[275] Suzuki M and Watanabe T 2000 Phys. Rev. Lett. 85 4787[276] Tinkham M 1996 Introduction to Superconductivity (Singapore: McGraw-Hill) p 20[277] Demuth J E, Persson B N J, Holtzberg F and Chandrasekhar C V 1990 Phys. Rev.

Lett. 64 603[278] Gofron K, Campuzano J C, Abrikosov A A, Lindroos M, Bansil A, Ding H, Koelling

D and Dabrowski B 1994 Phys. Rev. Lett. 73 3302[279] Schabel M C, Park C H, Matsuura A, Shen Z X, Bonn D A, Liang R and Hardy W

N 1998 Phys. Rev. B57 6090[280] Park C H, Shen Z X, Loeser A G, Dessau D S, Mandrus D G, Migliori A, Sarrao J

and Fisk Z 1995 Phys. Rev. B 52 R16 981[281] Halperin B I 1965 Phys. Rev. 139 A104[282] Frisch H L and Lloyd S P 1960 Phys. Rev. 120 1175[283] Randeria M and Campuzano J C 1998 Models and Phenomenology for Conventional

and High-temperature Superconductivity (Course CXXXVI of the IntenationalSchool of Physics ‘Enrico Fermi’) ed G Iadonisi, J R Schrieffer and M L Chiofalo(Amsterdam: IOS) p 115

[284] Kaminski A, Mesot J, Fretwell H, Campuzano J C, Norman M R, Randeria M, DingH, Sato T, Takahashi T, Mochiku T, Kadowaki K and Hoechst H 2000 Phys. Rev.Lett. 84 1788

[285] Fedorov A V, Valla T, and Johnson P D, Li Q, Gu G D and Koshizuka N 1999 Phys.Rev. Lett. 82 2179

References 293

[286] Zhang Y, Ong N P, Anderson P W, Bonn D A, Liang R and Hardy W N 2001 Phys.Rev. Lett. 86 890

[287] Alexandrov A S and Dent C J 2001 J. Phys.: Condens. Matter 13 L417[288] Landau L D and Lifshitz E N 1977 Quantum Mechanics (Non-relativistic Theory)

3rd edn (Oxford: Butterworth–Heinemann)[289] Calogero F 1967 Variable Phase Approach to Potential Scattering (New York:

Academic)[290] Hore S R and Frankel N E 1975 Phys. Rev. B 12 2619[291] Zaanen J and Gunnarsson O 1989 Phys. Rev. B 40 7391[292] Tranquada J M, Sternlieb B J, Axe J D, Nakamura Y and Uchida 1996 Nature 375

561[293] Bianconi A 1999 J. Physique IV 9 325 and references therein[294] Alexandrov A S and Kabanov V V 2000 JETP Lett. 72 569[295] Hoffman J E, Hudson E W, Lang K M, Madhavan V, Eisaki H, Uchida S and Davis

J C 2002 Science 295 466[296] Howald C, Eisaki H, Kaneko N and Kapitulnik A 2002 Preprint cond-mat/0201546

Index

adiabatic ratio, 98Andreev reflection

in BCS superconductors, 57in high-Tc cuprates, 219

anisotropy, 198anomalous averages, 36, 52, 85antiferromagnetic interaction, 239ARPES, 226attraction non-retarded, 120

Bardeen–Cooper–Schrieffer (BCS)–Bose liquid crossover, 174, 205excitation spectrum, 36gap, 41ground state, 39Hamiltonian, 36-like theories of high-Tc, 167master equation, 37, 86theory, 33

Bednorz and Muller discovery, 161bipolaron

apex, 174band, 174binding energy, 179hopping, 177in high-Tc oxides, 174in-plane, 177inter-site, 126, 129localized, 182model of cuprates, 180on-site, 122repulsion, 125self-trapped, 101singlet, 180, 181

size,119small, 119spectrum, 179superlight, 129theory, 243triplet, 180, 181

bipolaronicgas, 136Hamiltonian, 135instability, 119superconductivity, 135

Blochfunction, 76representation, 76theorem, 244

Bogoliubovequations, 52, 138transformation, 36, 53, 140

Boltzmannequation, 4, 264kinetics, 3

Born–Oppenheimer, 98Bose gas, 261boundary

conditions, 21, 58NS phases, 18

Brillouin zone, 245buble

approximation, 79, 80polarization, 79

c-axis transport, 195clusters, 135, 179coherence

294

Index 295

factor, 45length, 12volume, 16

collision integral, 4commutation rules, 75condensate

wavefunction, 8, 70, 148energy, 13, 16, 38, 142

conductivityHall, 5infrared, 199longitudinal, 5optical, 167sum rule, 6thermal, 49

Cooperpairing of repulsive fermions, 91pairing unconventional, 50pairs, 32phenomenon, 32

crossover regime, 205Coulomb

pseudopotential, 89repulsion, 73

criticalcurrent, 28field lower, 20, 151field thermodynamic, 17field upper, 23, 152velocity, 29

critical temperatureof BCS superconductors, 42of bipolaronic superconductors,

146of Bose–Einstein condensation,

262of high-Tc cuprates, 162, 203of polaronic superconductors,

121crystal field, 74current

density, 5diamagnetic, 40paramagnetic, 40

permanent, 6

Debyeapproximation, 83momentum, 83screening, 80

density of statesin cuprates, 169per unit cell, 251quasi-particle, 44with scattering, 154

dHvA oscillations, 257dielectric function

of CBG, 148high frequency, 73static, 81

distributionBose–Einstein, 254Fermi–Dirac, 254function, 4of bare electrons, 38of quasi-particles, 37

dirty superconductors, 12Drude model, 6dynamic matrix, 74dynamical mean-field theory, 102

effective mass, 248electons in metal, 81electron–electron correlations, 75electron–phonon

coupling, 35coupling constant, 35interaction, 74

Eliashberg:equations, 85extension of BCS theory, 72spectral function, 88

entropyof BCS quasi-particles, 45of ideal Fermi gas, 45

exchange interaction, 74, 165excitation spectrum

of BCS superconductors, 36

296 Index

of charged Bose gas, 141of two-component Bose gas, 145

Fermi energy, 173, 255Fermi liquid

breakdown, 38instability, 98theories of high Tc, 164

fluctuations, 15fluctuation theory, 204flux

expulsion, 40quantization, 10quantum, 10

free energyLondon expression, 9superfluid, 9vortex, 23

Frohlich–Coulomb Hamiltonian, 129Hamiltonian, 77, 97interaction, 77observation, 33polaron, 107

gapin BCS superconductors, 41in high Tc oxides, 218

gapless superconductivity, 55gauge transformation, 18Gibbs energy, 17Gi -number, 16Ginzburg

–Landau equation, 14–Landau parameter, 15–Landau theory, 13–Landau theory limitations, 15

golden rule, 43Gor’kov

equations, 60, 64formalism, 59

Green’s functionadvanced, 65, 277anomalous, 62, 85

electron, 78formulation of BCS theory, 59matrix, 86Matsubara, 275normal, 59one-particle, 274phonon, 79retarded, 65temperature, 277

Hall ratio, 5, 189harmonic oscillator, 24Hartree approximation, 75, 2404He, 1583He, 158heat capacity

of BCS superconductors, 45of high-Tc oxides, 208

Hebel–Slichter peakabsence in high-Tc oxides, 195in BSC superconductors, 48, 49

HeisenbergHamiltonian, 136operators, 53, 112, 275

high-Tc superconductors, 162Hosltein

–Hubbard model, 127model, 99polaron, 107

hopping, 95, 99Hubbard U

infinite, 129negative, 205repulsive, 164

hyperfine interaction, 48, 194

incoherent background, 115isotope effect

bipolaronic, 146, 206in BCS superconductors, 44on carrier mass, 206, 207on penetration depth, 206

Index 297

Josephsoncurrent, 30effect, 32

kinetic mechanism, 163, 166Knight shift, 195Kohn–Luttinger, 91Korringa law

breakdown, 195in metals, 48

k · p perturbation theory, 247Kosterlitz–Thouless temperature,

204

Landaucriterion, 28diamagnetism, 257theory, 13

Lang–Firsov transformation, 103LDA, 74Legendre polynomials

summation theorem, 51Levinson theorem, 236lifetime of quasi-particles, 234localization length, 213London

equation, 9penetration depth, 9theory, 6

Lorenz number, 192

massof small polaron, 111renormalized, 84tensor, 5

Matsubarafrequencies, 86operators, 62, 275

Maxwellequation, 9gauge, 8

McMillan formula, 90Meissner–Ochsenfeld

effect, 7

effect in CBG, 148experiment, 7

metal–ammonia solution, 11Migdal

–Eliashberg theory, 88–Eliashberg theory breakdown,

98theorem, 88

mixture of condensates, 142Monte Carlo algorithm, 109Mott insulator, 163multi-phonon correlator, 124

nearly Fermi liquid, 167Neel temperature, 165neutron scattering, 167, 195, 242non-Fermi-liquid theories

interlayer RVB, 164Luttinger liquid, 164marginal, 167RVB, 164

nuclearspin relaxation rate, 48, 194

Ogg’s pairs, 11Ogg–Schafroth phenomenology, 12operators

annihilation, 34, 268, 269creation, 34, 268, 269current density, 71displacement, 75field, 52, 271identity, 113, 281phonon, 75

order parameterBCS, 35, 50bipolaronic, 157symmetry, 156

overcrowding, 183overdoped cuprates, 183overscreening, 34

pairingcollective, 167

298 Index

individual, 171pairs

Bose, 11Cooper, 12

Pauliexclusion principle, 267matrices, 86pair-breaking limit, 25paramagnetism, 257

partition function, 254Peierls substitution, 138Pekar equation, 239perturbation expansion, 103phase diagram, 182, 183phase equation, 235phase shift, 235phonon frequency, 78phonons, 77phonon sidebands, 115Pippard

equation, 12limit, 13non-local relation, 12theory, 12

plasmaexcitations, 33frequency, 89, 117, 141

plasmonacoustic, 144ionic, 78optical, 144

polaronband, 102band narrowing factor, 104Bloch states, 95collapse, 72damping, 105dynamics, 102large, 95level shift, 104mass, 109, 110radius, 101small, 95spectral function, 110

–phonon interaction, 111–polaron interaction, 103, 116

polaronicFermi-liquid, 102superconductivity, 119

pseudogap, 194, 195pseudospin, 135

quantum phase transition, 167quasi-particle spectrum, 142

resistivitybelow Tc, 188in-plane, 183out-of-plane, 195

response function, 117, 147

scatteringboson–acoustic phonon, 187impurity, 187of bosons, 186resonance, 189two-phonon, 106

self-energyelectron, 83hole, 228phonon, 79second order, 82

self-trapping, 95Slater determinant, 266sound

attenuation, 46damping, 81velocity, 81

specific heatanomaly, 207fermions, 257jump, 46

spectral function, 110, 228, 230,276

spin bipolaron, 161spin-flip scattering, 48spin gap, 194strings, 241

Index 299

stripes, 166, 239strong coupling theory, 95sum rule, 6, 276supercurrent, 13superfluid

density, 13surface energy, 17symmetry, 156

susceptibilitydiamagnetic, 7spin, 196

thermal conductivityenhancement, 238Hall, 192of BCS superconductors, 49of high Tc cuprates, 192

thermally activated hopping, 95tight-binding approximation, 249time ordering, 63t–J model, 164Tolmachev logarithm, 163transformation

Bogoliubov, 36canonical, 279Lang–Firsov, 103

transitioncontinuous, 17second-order, 16third-order, 17, 262

transmission, 221tunnelling

conductance, 43in high-Tc cuprates, 219

Josephson, 29single-electron, 41SIS, 221

Type I, II superconductors, 20

uncertainty principle, 35upper critical field

curvature, 156of BCS superconductor, 23of ideal Bose gas, 154of interacting Bose gas, 154, 156unconventional, 210

Van Hove scenario, 167variational calculations, 111vertex

corrections, 83electron–phonon, 117

vortexcharged, 148, 152core, 22free energy, 23, 152interaction, 27lattice, 26line, 20pinning, 29

Wannierorbitals, 96, 249representation, 96

Wiedemann–Franz law, 192Wigner crystallization, 118Wigner cross section, 189, 237