Models of Itinerant Ordering in Crystals: An Introduction

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Models of

ITINERANT ORDERING IN CRYSTALS

This page intentionally left blank

Models of

ITINERANT ORDERING IN CRYSTALS An Introduction

By

JERZY MIZIA and

GRZEGORZ GÓRSKI

Amsterdam • Boston • Heidelberg • London • New York • Oxford Paris • San Diego • San Francisco • Singapore • Sydney • Tokyo

Elsevier Linacre House, Jordan Hill, Oxford OX2 8DP, UK 84 Theobald’s Road, London WC1X 8RR, UK First edition 2007 Copyright © 2007 Elsevier Ltd. All rights reserved No part of this publication may be reproduced, stored in a retrieval system or transmitted in any form or by any means electronic, mechanical, photocopying, recording or otherwise without the prior written permission of the publisher Permissions may be sought directly from Elsevier’s Science & Technology Rights Department in Oxford, UK: phone (+44) (0) 1865 843830; fax (+44) (0) 1865 853333; email: [email protected]. Alternatively you can submit your request online by visiting the Elsevier web site at http://elsevier.com/locate/permissions, and selecting Obtaining permission to use Elsevier material Notice No responsibility is assumed by the publisher for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions or ideas contained in the material herein. Because of rapid advances in the medical sciences, in particular, independent veriﬁcation of diagnoses and drug dosages should be made British Library Cataloguing in Publication Data Mizia, Jerzy Models of itinerant ordering in crystals: an introduction 1. Crystals – Mathematical models 2. Crystallography, Mathematical I. Title II. Górski, Grzegorz 548.7 Library of Congress Number: 2007925017 ISBN: 978-0-08-044647-9

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To my sons: big Tom and little Mike Jerzy Mizia To my wife Ann and my children: Alexandra and Jacob Grzegorz Górski

v

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CONTENTS

Preface

xi

Part One. Introduction to Theory of Solids

1

1. Periodic Structures

3

1.1 1.2

Fundamental Types of Lattices Diffraction of Waves by a Crystal and the Reciprocal Lattice 1.2.1 Reciprocal lattice vectors 1.3 Brillouin Zones References

2. Various Statistics Appendix 2A: Fermi–Dirac and Bose–Einstein Distribution Functions References

3. Paramagnetism and Weiss Ferromagnetism 3.1 Paramagnetism 3.2 Weiss Ferromagnetism References

4. Electron States 4.1

The Nearly Free Electron Model 4.1.1 General result for 4.1.2 Use of DOS for evaluating lattice sums in momentum space 4.1.3 Heat capacity of the free electron gas: an introduction 4.2 The Tight-Binding Method 4.2.1 Cohesion energy 4.3 Bloch Theorem Appendix 4A: Nearly Free Electrons, Two-Plane Waves Model References

3 4 7 8 10

11 14 17

19 19 25 26

27 27 31 32 34 35 40 43 44 48

Part Two. Models of Itinerant Ordering in Crystals

49

5. The Hubbard Model

51

5.1 Simple Hubbard Model 5.2 Extended Hubbard Model References

51 54 58 vii

viii

Contents

6. Different Approximations for Hubbard Model 6.1 Chain Equation for Green Functions 6.2 Hartree–Fock Approximation 6.3 Hubbard I Approximation 6.3.1 Atomic limit 6.3.2 Finite bandwidth limit 6.4 Extended Hubbard III Approximation 6.5 Coherent Potential Approximation 6.5.1 Relation between CPA, Hubbard III and extended Hubbard III approximations 6.5.2 Different applications of the CPA 6.6 Spectral Density Approach 6.7 Modified Alloy Analogy 6.8 Dynamical Mean-Field Theory 6.9 Hubbard Model Extended by Inter-site Interactions 6.9.1 Modified Hartree–Fock approximation 6.9.2 Coherent potential approximation for the extended Hubbard model Appendix 6A: Equation of Motion for the Green Functions Appendix 6B: Hubbard Solution for the Scattering and Resonance Broadening Effects 6B.1 The scattering effect 6B.2 The resonance broadening effect Appendix 6C: Modified Hartree–Fock Approximation for the Inter-site Interactions References

7. Itinerant Ferromagnetism 7.1 7.2 7.3 7.4

7.5 7.6

7.7 7.8

Periodic Table – Ferromagnetic Elements 7.1.1 Ferromagnetic elements Introduction to Stoner Model 7.2.1 Static magnetic susceptibility Stoner Model for Ferromagnetism Stoner Model for Rectangular and Parabolic Band 7.4.1 Rectangular band 7.4.2 Parabolic nearly free electron band Modified Stoner Model 7.5.1 Modified Stoner Model for a semi-elliptic band Beyond Hartree–Fock Model 7.6.1 General formalism 7.6.2 Enhancement of magnetic susceptibility 7.6.3 Critical values of interactions 7.6.4 Numerical results The Critical Point Exponents Spin Waves in Ferromagnetism 7.8.1 Energy of spin-wave excitations

59 60 63 64 64 66 69 72 80 81 81 86 88 90 90 94 95 97 97 100 106 113

115 115 118 125 129 131 134 134 136 138 141 144 144 148 149 150 154 157 161

Contents

7.8.2 Dynamic susceptibility of ferromagnets 7.8.3 Curie temperature References

8. Itinerant Antiferromagnetism 8.1 8.2 8.3 8.4 8.5

Phenomenological Introduction Simple Model of Itinerant Antiferromagnetism Free Energy and the Magnetic Susceptibility Antiferromagnetism Induced by On-site and Inter-site Correlations Free Energy and the Magnetic Susceptibility Including Correlation Effects 8.5.1 Longitudinal and transversal susceptibility 8.6 Onset of Antiferromagnetism 8.6.1 The case of zero Coulomb correlation: U = 0 8.6.2 The case of the strong correlation: U >> D 8.7 Numerical Results for Magnetization and Néel’s Temperature 8.8 Spin-Density Waves Appendix 8A: Antiferromagnetism in the Presence of On-site and Inter-site Coulomb Correlation References

9. Alloys, Disordered Systems 9.1 Introduction 9.2 Order–Disorder Transformation and Bragg–Williams Approximation 9.2.1 Bragg–Williams approximation 9.3 Relation with the Band Model 9.4 Transition Metal Alloys 9.5 Different Types of Disorder in Bragg–Williams Approximation References

10. Itinerant Superconductivity 10.1 Phenomenological Introduction and Historical Background 10.2 Physical Properties of the High-Temperature Superconductors 10.2.1 General properties 10.2.2 Crystal structure of the HTS 10.2.3 Symmetry of the energy gap 10.2.4 Dependence of the critical temperature on concentration 10.2.5 Phase diagrams of the ordering 10.3 Classic (BCS) Model for Superconductivity 10.4 Electron–Electron Interaction as a Source of Superconductivity 10.4.1 Introduction 10.4.2 Single-band model 10.4.2.1 Model Hamiltonian 10.4.2.2 Moments method for the superconductivity equation

ix 161 164 165

167 167 168 174 175 180 182 186 187 189 191 193 199 202

203 203 205 206 208 213 219 224

227 228 230 230 231 232 235 236 237 240 240 243 243 245

x

Contents

10.4.2.3 Analysis of the solution: critical temperature dependence on concentration 10.4.2.4 Effect of internal pressure on superconductivity 10.4.2.5 Symmetry of the energy gap 10.4.3 Three-band model 10.4.3.1 Introduction 10.4.3.2 The Model Hamiltonian 10.4.3.3 Hamiltonian diagonalization and the pairing interaction 10.4.3.4 Results Appendix 10A: Transformation of Superconductivity Hamiltonian to Momentum Space Appendix 10B: Green Function Equations for Singlet Superconductivity Appendix 10C: Effective Pairing Potential in the Single-Band Model Appendix 10D: Bogoliubov Transformation References

11. The Coexistence Between Magnetic Ordering and Itinerant Electron Superconductivity Experimental Evidence of Coexistence Between Magnetic Ordering and Superconductivity 11.2 Coexistence of Ferromagnetism and High-Temperature Superconductivity 11.2.1 Model Hamiltonian 11.2.2 Green function solutions 11.2.2.1 General equations 11.2.2.2 Ferromagnetism coexisting with singlet superconductivity 11.2.2.3 Ferromagnetism coexisting with triplet opposite spins pairing superconductivity 11.2.2.4 Ferromagnetism coexisting with triplet parallel (equal) spins pairing superconductivity 11.2.3 Comparison with experimental results (for UGe2 ZrZn2 , URhGe) 11.3 Coexistence of Antiferromagnetism and High-Temperature Superconductivity 11.3.1 Model Hamiltonian 11.3.2 Formalism of the model 11.3.3 Numerical examples 11.4 Superconducting Gap in Stripe States Appendix 11A: Coexistence of Ferromagnetism and Singlet Superconductivity Appendix 11B: Coexistence of Antiferromagnetism and Singlet Superconductivity References

248 252 257 259 259 259 263 266 269 274 277 280 282

287

11.1

Subject Index

288 297 297 300 300 301 304 304 306 309 310 312 313 315 318 321 323 325

PREFACE Over the last few years there has been extensive new research in the ﬁeld of superconductivity (SC) interacting with antiferromagnetism (AF), stimulated by the experimental discoveries of high-temperature SC in cuprates (YBa2 Cu3 O7– , La2–x (Ba,Sr)x CuO4 . Similarly there has been equally extensive research in the ﬁeld of SC interacting with ferromagnetism (F), following the discovery of new superconducting materials, e.g. UGe2 , ZrZn2 , URhGe, showing weak ferromagnetic properties and SC under high pressure. In the more established ﬁeld of itinerant F also, there have been slow but steady developments recently. There are similarities between all these processes, which have resulted in their close existence under experimental conditions. This should be reﬂected by the similarities in their formalism. The aim of this book is to present a uniﬁed itinerant model of these phenomena. Such a model will be an introduction to the ﬁeld of itinerant F, itinerant AF and electronic SC (i.e. driven by electron–electron interactions). Prompted by experimental evidence, this book also includes the areas of interaction between F and AF on one side and electronic SC on the other. Since we undertook this rather ambitious task, we apologize to the reader if in some places we did not rise to this difﬁcult project. The book came to fruition during the preparation of our students for their M.Sc. dissertations in this ﬁeld during a two-semester course. These students had rather weak preparation in elementary quantum mechanics and statistical physics. Therefore we included general introductory chapters on solid-state physics (Chapters 1–4) and the Hubbard model (Chapter 5), which will allow this book to be understood with minimal prerequisites; also the mathematical techniques are explained thoroughly. We hope that due to its tutorial nature, the book or parts of it will have the potential to become a textbook for teaching the course on a more introductory level. This book is primarily intended for undergraduate as well as graduate students and young scientists working in the ﬁeld. The reader will be taken gradually, step by step from the rudiments of solidstate physics to the basics of the many-body theory and on to the understanding of the different approximations. We have considered (see Chapter 6) the Hartree—Fock (H–F) and the modiﬁed H–F approximations, the classic CPA, the modiﬁed CPA which includes the inter-site correlation, the spectral density approximation (SDA) and the dynamical mean ﬁeld theory (DMFT). These approximations are used to evaluate, quantitatively, the minimum energy of a system and the values of critical interactions for F, AF and SC types of ordering. xi

xii

Preface

This book also covers the close relation between the BCS (Bardeen, Cooper and Schrieffer) formalism of SC and the itinerant model for AF (Chapter 8). The comparison will start from the simple basic models inter-relating quantities describing the AF and SC. Next the Green function formalism for SC in the presence of AF or F is developed (Chapter 11) and the conclusions are put forward for the simultaneous appearance of SC and AF or F in the H–F approximation and beyond the H–F approximation to include the strong on-site Coulomb correlation. Due to the rapid development in this ﬁeld, a concerted effort has been made to include and relate the results of the most recent research in the areas of electronic SC, itinerant F and AF. The authors realize that these matters are not covered wholly satisfactorily since there is a great deal of new literature published every month in this ﬁeld. Rzeszów

Jerzy Mizia Grzegorz Górski

PART

1 Introduction to Theory of Solids

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CHAPTER

1 Periodic Structures

Contents

1.1 1.2

Fundamental Types of Lattices Diffraction of Waves by a Crystal and the Reciprocal Lattice 1.2.1 Reciprocal lattice vectors 1.3 Brillouin Zones References

3 4 7 8 10

1.1 FUNDAMENTAL TYPES OF LATTICES In a solid crystal, the atoms are located in the same positions with respect to each other. Their relative location depends on the character of chemical bonding and the conditions for minimum energy. For crystals built of identical atoms the energy minimum is reached when all atoms have the same surrounding. Atomic positions are called lattice points and the whole lattice is called the crystal structure. In the case of a compound, the lattice points of the crystal structure are formed by molecules of the compound. The smallest part of the crystal structure is the primitive (elementary) cell. It can be created in many ways and is repeated translationally in each direction. The translational symmetry allows the description of the whole crystal by deﬁning primitive axes a1 a2 a3 . Every lattice point can be described by the multiplicity of these axes: r = n 1 a1 + n2 a2 + n3 a3

(1.1)

The choice of primitive basis vectors generating a given lattice is to some extent arbitrary but they have been selected to have the smallest possible length. 3

4

Introduction to Theory of Solids

a2 a1 (a)

(b)

FIGURE 1.1 Primitive cells: (a) defined by the primitive axes a1 a2 ; (b) Wigner–Seitz primitive cell.

This is the ideal crystal lattice. Every real lattice has limited dimensions and defects caused by impurities of other elements, dislocations and vacancies which inﬂuence the mechanical, electric and thermal properties of the lattice. For example, in good conductors the current carriers can be scattered on impurities, decreasing the conductivity, but in semiconductors impurities donating charges (donors) can increase the conductivity. In a real lattice, three primitive basis vectors create the primitive cell, which is the smallest of all elementary cells. Its volume is V = a1 × a2 · a3 . A primitive cell, containing only one lattice site, may also be chosen by drawing lines connecting a given lattice point with all nearby lattice points and then, at the midpoint of all these lines, drawing planes normal to these lines (see Fig. 1.1). The smallest volume enclosed in this way is the Wigner–Seitz primitive cell. The most popular structures are: sc: simple cubic; bcc: body-centred cubic; fcc: face-centred cubic; hcp: hexagonal close packed. The simple cubic cells are characterized by a1 = a2 = a3 and a1 · a2 = a2 · a3 = a1 · a3 = 0. These cells are shown in Fig. 1.2. In Table 1.1 are listed the most common crystal structures and lattice structures of the elements. For a crystal structure of different elements it is advised to consult Wyckoff [1.1].

1.2 DIFFRACTION OF WAVES BY A CRYSTAL AND THE RECIPROCAL LATTICE Diffraction is the main method of investigating crystal structures. On the other hand the diffraction of electron waves of electrons belonging to the crystal

Periodic Structures

sc

5

bcc

fcc

hcp

FIGURE 1.2 The cubic space lattices (sc, bcc, fcc) and the hexagonal lattice (hcp).

is the origin of the Brillouin zones (see Section 1.3) and of electron bands in crystals (see Chapter 4). The diffraction of a beam on a crystal is the reﬂection of waves on the periodic structure of the crystal followed by their interference. In diffractional analysis, one uses radiation with a wavelength comparable with inter-atomic distances or, in other words, with the lattice constant. Most commonly, X-ray radiation, electron and neutron beams are used. Neutron radiation, being only weakly absorbed, is used for larger samples, while the electron beams, which are strongly absorbed, are used mostly for surface analysis. W.L. Bragg described the diffraction of beams from a crystal. The crystal is a periodic set of parallel atomic planes, each of which reﬂects a very small fraction of the incident beam (see Fig. 1.3). The distance of the planes is d. The incident angle is deﬁned as in Fig. 1.3. For coherent diffraction the extra path 2d sin must be an integral number of wavelengths: 2d sin = n This is Bragg’s law.

(1.2)

6

Table 1.1 Crystal structure of the elements He

hcp

hcp

Li

Be

B

C

N

O

bcc

hcp

rhom.

diam.

cubic (N2)

comp. (O2)

Na

Mg

Al

Si

P

S

bcc

hcp

fcc

diam.

comp.

comp.

comp. (Cl2)

K

Ca

Sc

Ti

V

Cr

Se

Br

bcc

fcc

hcp

hcp

bcc

bcc

cubic comp.

hex. chains

comp. (Br2)

Rb

Mn

Fe

Co

Ni

Cu

Zn

Ga

Ge

As

bcc

hcp

fcc

fcc

hcp

comp.

diam.

rhom.

F

Ne

mon.

fcc

Cl

Sr

Y

Zr

Nb

Mo

Tc

Ru

Rh

Pd

Ag

Cd

In

Sn

Sb

Te

I

bcc

fcc

hcp

hcp

bcc

bcc

hcp

hcp

fcc

fcc

fcc

hcp

tetr.

diam.

rhom.

hex. chains

comp. (I2)

Cs

Ba

Ta

W

Re

Os

Ir

Pt

Au

Hg

Tl

Pb

Bi

Po

bcc

La– Lu

Hf

bcc

hcp

bcc

bcc

hcp

hcp

fcc

fcc

fcc

rhom.

hcp

fcc

rhom.

sc

Fr

Ra

Ar fcc

Kr fcc

Xe fcc

Rn

At

Ac– Lr

Pm

Sm

Eu

Gd

Tb

Dy

Ho

Er

Tm

Yb

Lu

comp.

bcc

hcp

hcp

hcp

hcp

hcp

hcp

fcc

hcp

U

Np

Pu

Am

Cm

Bk

Cf

comp.

comp.

comp.

hex

hcp

hcp

hcp

Es

Fm

Md

No

Lr

La

Ce

Pr

Nd

hex

fcc

hex

hex

Ac

Th

Pa

fcc

fcc

tetr.

sc, simple cubic; bcc, body-centred cubic; fcc, face-centred cubic; hex, hexagonal; hcp, hexagonal close packed; diam., diamond; rhom., rhombic; tetr., tetragonal; comp., complex; mon., monoclinic.

Introduction to Theory of Solids

H

7

Periodic Structures

θ

θ d

θ d sin θ

FIGURE 1.3 The reflection of an incident beam (wave) on atomic planes in crystal.

1.2.1 Reciprocal lattice vectors The Bragg condition can be expressed in general terms. Let k and k be the wave vectors of an incident and reﬂected beam, respectively k = 2/. Vector k = k − k is normal to the reﬂection plane, and its length (see Fig. 1.4) is k = k − k =

4 sin

(1.3)

The Bragg’s law (1.2) (for n = 1) can be written as k = k − k =

2 d

(1.4)

To proceed further with the Bragg’s law one has to introduce the reciprocal lattice. One deﬁnes the axis vectors b1 b2 b3 of the reciprocal lattice as being orthogonal to the axis vectors a1 a2 a3 of the real lattice: b1 = 2

a 2 × a3

a1 · a2 × a3

b2 = 2

a 3 × a1

a1 · a2 × a3

b3 = 2

a1 × a2 a1 · a2 × a3

(1.5)

Δk –k

θ

k′

θ

k

FIGURE 1.4 The relation between incident k, reflected k and the wave vector k = k − k.

8

Introduction to Theory of Solids

where a1 · a2 × a3 = V is the volume of the elementary cell of the real lattice. Given the primitive (basic) vectors of the reciprocal lattice b1 b2 b3 one can write each vector of reciprocal lattice as G = hb1 + kb2 + lb3

(1.6)

where h k l are integers. The scalar product of an arbitrary vector in a real lattice, r, and an arbitrary vector in the reciprocal lattice, G, is the multiplicity of the factor 2: r · G = n1 h + n2 k + n3 l2

(1.7)

therefore their product fulﬁls the relation expir · G = 1

(1.8)

Vectors in the reciprocal lattice have the dimension of [1/length], and the volume of the elementary reciprocal lattice cell is 23 /V . Having deﬁned the reciprocal lattice one can return to the Bragg’s law. It can be proved that the spacing between parallel lattice planes that are normal to the direction G = hb1 + kb2 + lb3 is dhkl = 2/G (see [1.2]). Thus one can write the Bragg condition (1.4) as k = k − k = G

(1.9)

Condition (1.9) is the basic condition for the reﬂection of scattered waves by the crystal. In elastic scattering of electrons, the magnitudes k and k are equal, and k2 = k2 . Therefore we have k − k = G

k − G2 = k2

2k · G = G2

(1.10)

1.3 BRILLOUIN ZONES The primitive cell of the reciprocal lattice can be spanned on the primitive axes b1 b2 b3 . It can also be created by the Wigner–Seitz method explained above. The Wigner–Seitz primitive cell is bound by planes normal to the vectors connecting the origin with the nearest-neighbour points of the reciprocal lattice and drawn at their midpoints. This cell is called a Brillouin zone. An example of the ﬁrst Brillouin zone for the two-dimensional (2D) rectangular lattice is shown in Fig. 1.5. For complicated structures the shape of the ﬁrst Brillouin zone becomes spherical. The second Brillouin zone is the space between the ﬁrst zone and the planes drawn at the midpoints of vectors pointing to the second neighbours and so on

9

Periodic Structures

b2

First Brillouin zone

b1 Second Brillouin zone

FIGURE 1.5 First and second Brillouin zones for a two-dimensional rectangular lattice. kz 0,0,2 –1,–1,1 –1,1,1

1,–1,1 1,1,1

–2,0,0

0,–2,0

Γ 2,0,0

0,0,0 0,2,0 ky

–1,–1,–1

kx 1,–1,–1

–1,1,–1 1,1,–1 0,0,–2

FIGURE 1.6 First Brillouin zone for the face-centred cubic (fcc) lattice. The square and hexagonal limiting planes come from the points (2,0,0) and (1,1,1) of the reciprocal lattice, respectively.

for subsequent Brillouin zones, see Fig. 1.5. In Fig. 1.6, the ﬁrst Brillouin zone is shown for the fcc lattice, which by itself is the bcc lattice in the reciprocal space. The Brillouin zone gives a physical interpretation of diffraction condition (1.10). After dividing both sides of (1.10) by 4 we obtain

2 1 1 k· G = G 2 2

(1.11)

10

Introduction to Theory of Solids

This relation states that the reﬂected wave with the wave vector on the boundary of the Brillouin zone fulﬁls the Bragg condition. In the process of interference with incoming wave, it forms the standing wave (see Chapter 4) and, in consequence, generates an energy gap on the Brillouin zone boundary. We can see how the internal diffraction of crystal electrons by obeying the Bragg’s law creates the Brillouin zones and in effect the electron bands in crystals.

REFERENCES [1.1] W.G. Wyckoff, Crystal Structures, Krieger, Florida (1981). [1.2] H. Ibach and H. Lüth, Solid-State Physics. An Introduction to Principles of Materials Science, Springer, Berlin (1995).

CHAPTER

2 Various Statistics

Contents

Appendix 2A: Fermi–Dirac and Bose–Einstein Distribution Functions References

14 17

The occupation numbers of quantum particles in each one-particle state are strongly restricted by a general principle of quantum mechanics. The wave function of a system of identical particles must be either symmetrical (Bose) or antisymmetrical (Fermi) in permutation of all particle coordinates (including spin) [2.1]. Generally there are such particles as fermions, which have half-integral spin, and bosons, which have integral spin. Examples of fermions are electrons (e), positrons e+ , protons (p) and neutrons (n). Bosons are, e.g. H = p + e, photons. The fermions obey the Pauli exclusion, which states that there can be only one particle in each particle state, while the bosons may have occupation: 0 1 2 . For the half-spin particle system in equilibrium we have the Fermi–Dirac (F–D) statistics for fermions: f =

1 e −/kB T + 1

(2.1)

which gives the probability that an orbital at energy will be occupied by the particle (electron). For the bosons we have the Bose–Einstein (B–E) statistics given by f =

1

(2.2) −1 Both these distributions will be derived below in Appendix 2A. The kinetic energy of the electron gas described by (2.1) increases as the temperature is increased: some energy levels are occupied which were vacant at absolute zero (Fig. 2.1). e −/kB T

11

12

Introduction to Theory of Solids

1

0K

10 000 K

f (ε)

0.8 0.6 0.4 0.2 0

0

4

2

6

8

10

μ /kB ε /kB in units of 104 K FIGURE 2.1 The Fermi–Dirac distribution function for 0 and 10 000 K at /kB = 50 000 K. The results apply to a gas in three dimensions. The total number of particles is constant, independent of temperature. The chemical potential at all temperatures may be read off the graph as the energy at which f =05.

The function (2.2) called the Bose–Einstein distribution function is shown in Fig. 2.2; f becomes inﬁnite as → . Therefore if the lowest value of the one-particle energy is chosen to be zero one must have ≤ 0

(2.3)

As will be seen below when the condition >> kB T is satisﬁed, the system is so non-degenerate and the Bose statistics may be replaced by Boltzmann statistics. The quantity in the Fermi distribution (2.1) is the chemical potential. It is a function of temperature and is chosen in such a way that the total number of particles in the system is N . The Fermi distribution function, f, at absolute zero changes discontinuously from the value 1 (ﬁlled) to the value 0 (empty) at =F =, where F is the Fermi energy. At all temperatures f is equal to 1/2 when =, since then the denominator of (2.1) has the value of 2. The classical limit (Boltzmann statistics) can be obtained from both (2.1) and (2.2) when f << 1

or

− >> kB T

(2.4)

This limiting case is called the Boltzmann (or sometimes Maxwell) statistics and its distribution is given by f = e− /kB T

(2.5)

13

Various Statistics

5

4

5 × 104 K 2.5 × 104 K

f (ε)

3

1 × 104 K

2

1

0

μ /kB –4

–2

0

4

2

ε /kB in units of 104 K

6

8

10

FIGURE 2.2 The Bose distribution function. The physically available energies are positive, > 0.

One can write for the number of particles N=

f = e/kB T

e− /kB T

(2.6)

and to calculate this sum classically N = e/kB T

V −p2 /2mkB T 2 mkB T 3/2 /kB T e dp = e V 3 3

(2.7)

so that e/kB T =

N 3 V 2 mkB T 3/2

(2.8)

Therefore in order to satisfy (2.4) by all ≥ 0 one must have

V N

1/3 >>

=√B 2

2 mkB T

(2.9)

i.e. it is necessary (for the Boltzmann statistics) that the average de Broglie wavelengths be much smaller than the mean distance between particles. Once the conditions for the validity of classical Boltzmann statistics are established, the formula (2.5) can be simpliﬁed as follows: f = e− /kB T = e/kB T e− /kB T = Ce− /kB T

(2.10)

14

Introduction to Theory of Solids

where the constant C can be found from the normalization condition 1= fd = C e−/kB T d 0

(2.11)

0

giving C = 1/kB T and hence f =

1 −/kB T e kB T

(2.12)

APPENDIX 2A: FERMI–DIRAC AND BOSE–EINSTEIN DISTRIBUTION FUNCTIONS The Fermi–Dirac and Bose–Einstein distribution functions may be derived using statistical mechanics. We will use the notation S for the conventional entropy and for the fundamental entropy, which is given by S = kB ; T is the Kelvin temperature related to the fundamental temperature B by B = kB T , where kB is the Boltzmann constant with the value 1 38 × 10−23 J K−1 . For the system with g accessible states the entropy is deﬁned as = log g. It is a function of the energy U , the number of particles N and the volume of the system V . Its total differential can be expressed by the following formula [2.2]: dS =

dU p + dV − dN T T T

(2.A1)

Expression (2.A1) is obtained from the ﬁrst principles of thermodynamics: dU = dQ + dA + dN (ﬁrst law of thermodynamics), dQ = T dS (second law of thermodynamics) and the expression dA = −p dV for the element of work. To derive a very simple formula of the Boltzmann factor one has to consider a small system with two states, one at energy 0 and one at energy , placed in thermal contact with a large system, called the reservoir. The total energy of the combined system is U ; when the small system is in energy 0, the reservoir has energy U and will have gU states accessible to it. When the small system is in energy , the reservoir has energy U − and will have gU − states accessible to it. From (2.A1) with dV = dN ≡ 0 and dU = − one has d = U − − U =

dU 1 dS = =− kB B B

By fundamental assumption we can write, for entropy, that U − = log gU −

and U = log gU

(2.A2)

Various Statistics

15

After inserting this into (2.A2) one obtains log

gU − =− gU B

(2.A3)

The ratio of probability of ﬁnding the small system with energy to the probability of ﬁnding it with energy 0 is P gU − = = exp − P0 gU B

(2.A4)

This is the Boltzmann result. To demonstrate its use the Planck distribution of a set of identical oscillators (e.g. phonons, magnons) in thermal equilibrium will be calculated later. We return now to problem of the Fermi–Dirac distribution. This is the case of a system that can transfer particles as well as energy with the reservoir. From (2.A1) with dV ≡ 0 one has the following formula: d =

dS dU = − dN kB B B

(2.A5)

By analogy with (2.A2), for the system which exchanges one particle of energy with the reservoir, we can write d = U − N − 1 − U N =

dU − −1 = − + B B B B

(2.A6)

Using this result we can extend (2.A4) to the ratio of probability that the system is occupied by one particle at energy to the probability that the system is unoccupied at energy 0: P1 gU − N − 1 exp U − N − 1 = = = exp − /B P0 0 gU N exp U N

(2.A7)

The expression (2.A7) after normalization P1 +P0 0 = 1 readily gives P1 =

1 exp − /B + 1

This is the Fermi–Dirac distribution function.

(2.A8)

16

Introduction to Theory of Solids

For bosons the system in one state can exchange an arbitrary number of particles with the reservoir. When n particles are exchanged, instead of (2.A7) we have Pn n = exp n − /B P0 0

(2.A9)

Using (2.A9) one obtains for the ratio of the probability of n + 1 particles state to the state with n particles Nn+1 = exp − − /B Nn

(2.A10)

Thus the fraction of the total number of states occupied by n particles is N exp −n − /B Pn n n = Ns exp −s − / B s

(2.A11)

s=0

which leads to the expression for the average occupation number of the n particles state

s exp −n − /B n = Pn n = exp −s − /B n

(2.A12)

s

The summations in (2.A12) are s

xs =

1 1−x

s

sxs = x

d s x x = dx s 1 − x2

(2.A13)

with x = exp − − /B . Thus one can rewrite (2.A12) as n =

x 1 = 1 − x exp − /kB T − 1

(2.A14)

This is the Bose–Einstein distribution function. After substituting into the Bose–Einstein distribution (2.A14), − ⇒ , where is the energy of identical oscillators (e.g. phonons, magnons) in thermal equilibrium, one obtains the Planck distribution function: n =

1 exp/B − 1

(2.A15)

Various Statistics

17

In the case of magnons, which travel over the crystal lattice with momentum k, one should replace the energy by k .

REFERENCES [2.1] L.I. Schiff, Quantum Mechanics, McGraw-Hill Book Company, New York (1968). [2.2] R. Kubo, Thermodynamics, North-Holland Publishing Company, Amsterdam (1968).

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CHAPTER

3 Paramagnetism and Weiss Ferromagnetism

Contents

3.1 Paramagnetism 3.2 Weiss Ferromagnetism References

19 25 26

3.1 PARAMAGNETISM The magnetization M is deﬁned as the magnetic moment per unit volume. The magnetic susceptibility per unit volume is deﬁned as =

M H

(3.1)

where H is the macroscopic magnetic ﬁeld intensity. Substances with a negative magnetic susceptibility are called diamagnetic. Substances with a positive susceptibility are called paramagnetic. Electronic paramagnetism (positive contribution to ) is found in different classes of materials which have permanent electronic dipole moments disordered in the absence of an external magnetic ﬁeld. These materials are [3.1]: (i) atoms, molecules and lattice defects possessing an odd number of electrons, as here the total spin of the system is not zero; (ii) free atoms and ions with a partly ﬁlled inner shell: transition elements; rare earth and actinide elements; (iii) some compounds with an even number of electrons, including molecular oxygen and organic biradicals; (iv) metals. 19

20

Introduction to Theory of Solids

The experimental results show that the magnetic susceptibility for paramagnetic substances is inversely proportional to the absolute temperature T . This dependence is well known as the Curie law. It can be written as =

C T

(3.2)

where C is the Curie constant. Equation (3.2) can be obtained on the basis of the quantum theory of paramagnetism. As will be shown later the Curie constant is proportional to the total orbital momentum J [see (3.23) below]. Taking into account the quantum mechanical results the magnetic moment of an atom or ion in free space can be written as = J ≡ −gJ B J

(3.3)

where the total angular momentum J is the vector sum of the orbital L and spin S angular momenta. The constant is the ratio of the magnetic moment to the angular momentum; is called the gyromagnetic ratio. For electronic systems a quantity, gJ , called the gJ factor or the spectroscopic splitting factor is deﬁned by (3.3). It is worth noting here that for an electron spin, gJ = 20023 (usually taken as 2), and the expression gJ J reduces to unity. For a free atom with one or more electrons, the gJ factor is given by the Landé equation: g = 1+

JJ + 1 + SS + 1 − LL + 1 2J + 1

(3.4)

This equation can be easily obtained with the aid of (3.3) and the vector model of the LS coupling in atoms [3.2, 3.3]. The Bohr magneton, B , appearing in (3.3) is equal to e/2mc in the CGS system. It is almost equal to the spin magnetic moment of a free electron. The CGS unit system will be used in the considerations presented here. To describe the interaction of the magnetic sample with the magnetic ﬁeld, H, the equation for the energy of the magnetic dipole in the magnetic ﬁeld will be used. The energy levels of the system in a magnetic ﬁeld are U = − · H = −z H = mJ gJ B H

(3.5)

where mJ is the azimuthal quantum number and has the values J J −1 −J ; is the magnetic moment [see (3.3)]; and z is the component of this moment along the direction of the applied magnetic ﬁeld, H. For a single spin with no orbital moment we have mJ = ±1/2 and g1/2 = 2, hence U = ±B H. This splitting of energy is shown in Fig. 3.1. In the general case of an atom with an angular momentum quantum number, J , located in the magnetic ﬁeld, there will be 2J + 1 equally spaced energy levels. The magnetization of the sample depends on the temperature to which it is subjected.

21

Paramagnetism and Weiss Ferromagnetism

ms

µz

1

–µ B

2

ΔU = 2µ BH

–

1 2

µB

FIGURE 3.1 Energy level splitting for one electron in a magnetic field H directed along the positive z axis. For an electron the magnetic moment is opposite in sign to the spin S, so that = −gJ B S. In the low-energy state the magnetic moment is parallel to the magnetic field.

At zero temperature all N participating atoms will occupy the lowest energy level [see (3.5)]. In this case, the magnetization M0 of the sample takes on the following form: M0 = NJgJ B

(3.6)

because the lowest energy level has the value of mJ = −J . In relation (3.6), N is the number of magnetic moments per unit volume. The magnetization M0 is also called the saturation magnetization, i.e. the maximum value of a magnetic moment which can be achieved for the material considered. When the sample considered is placed at a ﬁnite temperature T > 0 K the situation is much more complicated since higher lying levels become occupied. As will be shown below, the reduced magnetization (i.e. the ratio of magnetization at temperature T to the saturation magnetization) is equal to the so-called Brillouin function, BJ (Fig. 3.2) and is expressed as M = BJ x M0

x ≡ gJ JB H/kB T

(3.7)

To obtain (3.7) one has to bear in mind that the probability Pn of ﬁnding an atom in a state with energy En is equal to 1 E Pn = exp − n (3.8) Z kB T where Z is the so-called partition function and is deﬁned by E Z = exp − n kB T n

(3.9)

Taking into account the statistical average over all possible states of mJ one has to insert in the above En → EmJ = mJ gJ B H. The magnetization is obtained

22

Introduction to Theory of Solids

7 Gd3+

Magnetic moment (µ B /ion)

6

5 Fe3+ 4

3 Cr3+ 1.30 K

2

2.00 K 3.00 K

1

4.21 K Brillouin functions

0

0

10

30

20

40

B /T (kG/K)

FIGURE 3.2 Plot of magnetic moment versus B/T for spherical samples of (i) potassium chromium alum, (ii) ferric ammonium alum and (iii) gadolinium sulfate octahydrate. Over 99.5% magnetic saturation is achieved at 1.3 K and about 50 000 G (5 T). After W.E. Henry [3.4]. Reprinted with permission from W.E. Henry, Phys. Rev. 88, 559 (1952). Copyright 2007 by the American Physical Society.

by weighting the magnetic moment z of each state by the probability that this state is occupied and summing up over all states, i.e. − M = N z = N

J mJ =−J

gJ mJ B exp−aJ mJ

J mJ =−J

(3.10)

exp−aJ mJ

where aJ = gJ B H/kB T . Equation (3.10) can be rewritten as − M = NgJ B

J mJ =−J J mJ =−J

mJ exp−aJ mJ exp−aJ mJ

J d = NgJ B log exp−aJ mJ (3.11) daJ mJ =−J

Paramagnetism and Weiss Ferromagnetism

23

Since J

exp−aJ mJ = eJaJ 1 + e−aJ + · · · + e−2JaJ

mJ =−J

one can write that M = NgJ B

d e−J +1aJ − eJaJ log daJ e−aJ − 1

(3.12)

where the expression for the sum of a geometric series was used. After multiplying the numerator and denominator of the bracket in (3.12) by eaJ /2 one obtains sinhJ + 1/2aJ d eJ +1/2aJ − e−J +1/2aJ d log = NgJ B log M = NgJ B sinh aJ /2 daJ daJ eaJ /2 − e−aJ /2 (3.13) Next, after differentiating the expression in the bracket of (3.13), one obtains sinhJ + 1/2aJ aJ d 1 1 + 2J 1 + 2J log = coth aJ − coth (3.14) daJ sinh aJ /2 2 2 2 2 Taking into account (3.6), the deﬁnition of x [see (3.7)] and multiplying the numerator and denominator of (3.13) by J , one may write M = BJ x M0

(3.15)

where x = aJ J = gJ JB H/kB T . Finally the Brillouin function BJ x is given by 2J + 1 2J + 1 1 x BJ x = coth x − coth (3.16) 2J 2J 2J 2J As we can see, the relation (3.15) is the desired result for reduced magnetization of a system in the magnetic ﬁeld. It is worth noting here that reduced magnetization depends only on the total momentum J and the H/T ratio. The J dependence BJ x is shown in Fig. 3.3. There are some well-known limits of the Brillouin function, which according to (3.15) represents the temperature dependence of normalized magnetization. In transition metals, one has the effect of the so-called quenching of the electron’s orbital momentum. As a result the total angular momentum J is reduced to the spin S angular momentum. In this case, the normalized magnetization is described as M = B1/2 x = tanhx M0

M0 = NnJgJ B NnB

(3.17)

24

Introduction to Theory of Solids

1

BJ (x)

0.8 0.6 0.4 J = 1/2 J = 3/2 J = 5/2 J = 50

0.2 0

0

0.5

1

1.5

2

2.5

3

x

FIGURE 3.3 The small x dependence of the Brillouin functions for different J.

where n is the number of electrons per atom, each with the moment of JgJ B B . The magnetization at a given temperature is written as M = NmJgJ B = NmB

m = tanhx n

(3.18)

where m is the dimensionless magnetization (at the given temperature) per atom in Bohr’s magnetons. On the opposite side of the spectra lie the so-called superparamagnetic particles, which are the large clusters of magnetic atoms commanding huge magnetic moments. For these systems one has, from (3.17) and (3.16) for BJ x with J → , the following expression: M 1 = B x = cothx − ≡ Lx M0 x

(3.19)

where Lx is the Langevin function. To analyse the magnetic susceptibility we need the low x expansions of the Brillouin function: 1 J +1 lim BJ x = x ≡ fJ x x→0 3 J

(3.20)

For J = S = 1/2 one has f1/2 x = x, which is also the result of (3.17) since tanhx ≈ x. For J → one has from (3.20) f x = x/3, which is also the x→0

low x expansion of (3.19). Therefore the slopes of all the Brillouin functions in Fig. 3.3 at the Curie point (T/TC ≈ 1 and x → 0) are contained between those corresponding to J → : slope 1/3 and J = 1/2: slope 1.

Paramagnetism and Weiss Ferromagnetism

25

Using expansion (3.20) for small magnetizations of the localized moments J we can write M = NgJ JB fJ x

(3.21)

and the Curie law is given by ≡

gJ2 2B fJ x 1 M C = NgJ JB = NJJ + 1 = H H 3 kB T T

(3.22)

hence the Curie constant is gJ2 2B 1 C = NJJ + 1 3 kB

(3.23)

This is the well-known Curie law for the susceptibility which holds for those materials with permanent dipole moments (of magnitude gJ JJ + 11/2 B ), but which do not become magnetic at low temperatures. For transition metals J = 1/2 gJ = 2 N ⇒ Nn, and from (3.22) one has C = Nn

2B kB

(3.24)

where the number of magnetic moments per unit volume in the localized model, N , has been replaced by the number of itinerant electrons per unit volume, Nn (n is the number of itinerant electrons in the band per atom), each of which carries the independent itinerant magnetic moment B .

3.2 WEISS FERROMAGNETISM For magnetic materials which eventually become ferromagnetic at some temperatures, one has to replace the magnetic ﬁeld H in (3.21) by H + M (see also Section 7.2), where the second term is the Weiss ﬁeld proportional to the existing magnetization. Assuming the existence of this ﬁeld one has to replace (3.22) by M C = H + M T

(3.25)

Recalling that the susceptibility = M/H one obtains from (3.25) the Curie– Weiss law =

C T − TC

with TC = C

(3.26)

26

Introduction to Theory of Solids

Inserting here the Curie constant for localized moments, (3.23), we obtain gJ2 2B 1 TC = NJJ + 1 3 kB

(3.27)

and inserting the Curie constant for itinerant moments, (3.24), we have TC =

Nn2B kB

(3.28)

This result is identical to the result (3.27) under the following substitutions: J = S = 1/2, gJ = 2, N ⇒ Nn. The relation (3.28) will be obtained again from the Stoner model in Section 7.2.

REFERENCES [3.1] [3.2] [3.3] [3.4]

Ch. Kittel, Introduction to Solid State Physics, Wiley, New York (1996). P. Mohn, Magnetism in the Solid State, Springer, Berlin (2003). K. Yosida, Theory of Magnetism, Springer, Berlin (1996). W.E. Henry, Phys. Rev. 88, 559 (1952).

CHAPTER

4 Electron States

Contents

4.1

The Nearly Free Electron Model 4.1.1 General result for 4.1.2 Use of DOS for evaluating lattice sums in momentum space 4.1.3 Heat capacity of the free electron gas: an introduction 4.2 The Tight-Binding Method 4.2.1 Cohesion energy 4.3 Bloch Theorem Appendix 4A: Nearly Free Electrons, Two-Plane Waves Model References

27 31 32 34 35 40 43 44 48

4.1 THE NEARLY FREE ELECTRON MODEL The nearly free electron method deals with electrons, which spend most of their time outside the atomic cores where they propagate as plane waves. This is on the opposite side of the spectrum of electrons closely bound with their atoms which spend most of their time inside atoms. The periodic table (see Table 7.1) tells us that there are numerous examples of elements with nearly free electrons. These elements are listed in Table 4.1. In addition to these elements, the transition metal elements of groups 3d, 4d, 5d also have nearly free electrons of 4s, 5s and 6s orbitals, respectively. The difference with the simple nearly free electron elements is that in the case of transition elements, there are tightly bound d electrons simultaneously with nearly free s electrons. For nearly free electrons the wave function is the plane wave [3.1, 4.1], which satisﬁes the Bloch condition (see Section 4.3) 1 k r = √ eik·r V

(4.1) 27

28

Introduction to Theory of Solids

This wave function is normalized to unity over the space of the crystal which contains an electron: 1 −ik·r ik·r k∗ rk rdr = e e dr = 1 (4.2) V V V The dispersion relation of an electron in the lattice, k , is its energy dependence on the wave vector k. Inserting the wave function (4.1) into the Schrödinger equation one obtains k =

2 k2 p2 ≡ 2m∗ 2m∗

(4.3)

where the de Broglie relation between particle momentum p and its wave number k, p = m∗ v = k = 2/ , was used. The mass of an electron in the lattice, which appears in (4.3), is the effective mass. The quasi-free electron in the lattice moves differently than in the free space (see Fig. 4.4). Its motion is affected by the attractive potentials of the ions, which are trying to bind it. Speaking metaphorically it feels like a train moving through the rough junctions on the railway track. This is why its mass may be much heavier than in the free space. This is not the only feature which is different for an electron in the crystal than in the open space. Another difference is that the electron moving as the plane wave in the crystal undergoes Bragg’s reﬂection from the crystal planes. At the zone boundary the incoming and the reﬂected electron waves form a standing wave. This is the physical condition for creating the boundaries of the Brillouin zone (BZ) analysed in Section 1.3. The standing wave at zone boundaries means that the group velocity of the electron wave goes to zero. A time-varying frequency of an electron wave is

k =

1 k

(4.4)

The group velocity of a wave packet near this frequency would be vk = k k =

1 k k

(4.5)

For a nearly free electron, using (4.3), one has vk =

k p = ∗ ∗ m m

(4.6)

which is the velocity of an electron considered as a wave packet moving freely in the crystal. In mathematical language, the velocity in (4.5) is proportional to the tangent of the dispersion relation. Hence the slope of the k curve must go to zero at the zone boundaries as the velocity of standing waves goes to

29

εk

εk

Electron States

–3π/a

–2π/a

–π/a

0

π/a

2π/a

3π/a

k (a)

–π /a

0 k

π /a

(b)

FIGURE 4.1 The dispersion relation, k , for nearly free electrons. At zone boundaries it deviates from the free energy parabola to accommodate for the zero slopes at these points. (a) Extended zone scheme; (b) reduced zone scheme.

zero. As a result we can draw the k model dependence initially for small k s as a parabolic curve [see (4.3)] but later with a slope decreasing to zero at the zone boundary, see Fig. 4.1. Quantitatively the dispersion relation for nearly free electrons can be investigated on the basis of the simpliﬁed two-plane waves model, which is considered in Appendix A. The results conﬁrm the qualitative predictions drawn above. First, the effective mass near k = 0 is the free electron mass m. Secondly, at the zone boundary one has the energy gap, which is equal to 2UG , where UG is the Fourier transform of the lattice potential. Thirdly, the slope of the dispersion relation k , proportional to the wave velocity, is zero at the zone boundary, where the incoming wave function interfering with the reﬂected wave function forms the standing wave. There is one more difference between the electron wave in a free space and a nearly free electron wave in a crystal. It is the quantization of the possible values of vector k in the crystal. The wave function in a crystal has to be periodic. This is the cyclic condition. One can also use the Born–von Karman boundary condition which states that the wave function vanishes on the edges of the crystal. The results are equivalent within a factor of two. Using the periodicity condition in the x direction over the distance L one has eikx x+L+iky y+ikz z = eikx x+iky y+ikz z

30

Introduction to Theory of Solids

hence eikx L = 1

or

n kx = 2 L

(4.7)

where n is any integer number. Applying the same periodicity condition over distance L in y and z directions one obtains the conditions ky = 2

m L

l and kz = 2 L

(4.8)

From (4.7) and (4.8) we can see that there is only one wave vector deﬁned by the triplet of quantum numbers: kx , ky , kz – for the volume element 2/L3 of k space. Thus in the sphere of volume 4k3 /3 the total number of orbitals, including two possible spin directions, is as follows: 2

4k3 /3 V 3 = k = N 3 2/L 3 2

(4.9)

hence

3 2 N V

k=

1/3 (4.10)

depends only on the particle concentration. Using (4.3) we have k =

2 2m∗

3 2 N V

2/3

(4.11)

Now we can ﬁnd an expression for the total number of orbitals per unit energy range, , called the density of states (DOS). The expression (4.11) for the total number of orbitals of energy less or equal to k = gives the result V N= 3 2

2m∗ 2

3/2

(4.12)

which for the DOS produces the formula dN V

≡ = d 2 2

2m∗ 2

3/2 1/2 = C1/2

corresponding to the parabolic dispersion relation.

(4.13)

Electron States

31

4.1.1 General result for () Now we want to ﬁnd a general expression for the density of states, (), for any electron dispersion relation k . The number of allowed values of k, for which the electron energy is between and + d, is V

d = 2 d3 k (4.14) 23 shell where V = L3 is the volume of the crystal, the integral is extended over the volume of the shell in k space bounded by two surfaces of constant energy, one surface on which the energy is and the other on which the energy is + d. The factor two on the right comes from the two allowed values of the spin quantum number. The problem is how to evaluate the volume of this shell. Let dS denote an element of area (Fig. 4.2) on the surface in k space of the selected constant energy . The element of volume between the constant energy surfaces and + d is a cylinder of base dS and altitude dk⊥ , so that d3 k = dS dk⊥ (4.15) shell

Here dk⊥ is the perpendicular distance (Fig. 4.2) between the surface constant and the surface + d constant. The value of dk⊥ will vary from one point to another on the surface. The gradient of , which is k , is also normal to the surface constant, and the quantity k dk⊥ = d kz dk⊥

ε + dε = const dSε k

ε = const ky

kx

FIGURE 4.2 Element of constant energy area, dS , in k space. The volume between two surfaces of constant energy at and + d is equal to dS /k d. The quantity dk⊥ is the perpendicular distance between two constant energy surfaces in k space, one at energy and the other at energy + d.

32

Introduction to Theory of Solids

is the difference in energy between the two surfaces connected by dk⊥ . Thus the element of the volume in k space is d 1 d = dS k g where, in agreement with (4.5), g = k is the magnitude of the group velocity of an electron. From (4.14) one has V 1 dS

d = 2 d 23

g dS dk⊥ = dS

Dividing both sides by d the result for DOS is V 1 dS V dS

= 2 = 2 23

g 23 k

(4.16)

The integral is taken over the area of the surface constant, in k space. One can also use this result to evaluate the phonon density of states. As an example this general expression will now be used to calculate DOS for nearly free electrons, given by (4.13) above. Since the electron dispersion relation (4.3) is isotropic one can write from (4.16) that ∗ 3/2 2V dS 2Vkm∗ V 2m 2V 4k2

= = = 1/2 (4.17) = 23 k 23 2 k/m∗ 2 2 2 2 2 2 In the next section, the results of using formula (4.16) for calculating DOS in the tight-binding approximation for different types of crystal structure will be reported.

4.1.2 Use of DOS for evaluating lattice sums in momentum space Many textbooks [3.1, 4.1] have recorded that the lattice sum of any function of k : Z k can be calculated by interchanging the sum to the integral over energy in the expression of the following form: Zk fk = Z fd (4.18) k∈BZ

BZ

where f is the Fermi–Dirac function (see Chapter 2). One possible application of this general law is to ﬁnd the electron occupation at ﬁnite temperatures. Expressed as the lattice sum this quantity is given by n=

k∈BZ

fk

with fk =

1 expk − /kB T + 1

(4.19)

Electron States

33

ε

ρ (ε)ƒ(ε) 2k BT

ρ (ε)

2

εF

1

ρ (εF)

ρ (ε)

FIGURE 4.3 The density of states as a function of energy, for a free electron gas in three dimensions. The dashed curves represent the density f of filled orbitals at a finite temperature, but such that kB T is small in comparison with F . The shaded area represents the filled orbitals at absolute zero. The average energy is increased when the temperature is increased from 0 to T, as electrons are thermally excited from region 1 to region 2.

Hence, using (4.18) with Zk ≡ 1k , one has

fd n=

(4.20)

BZ

The DOS of nearly free electrons (4.13) is shown in Fig. 4.3 together with the product of f, which shows the thermal diffusion of the DOS around the Fermi energy in the range of kB T . One has to bear in mind that the thermal energy of kB T , even for temperatures of the order of 1000 K, is much smaller than the Fermi energy. As a result the transient interval of energy in Fig. 4.3, 2kB T , for the Fermi energy of the order of 4 eV at 1000 K is only of the order of 1/23 eV (since 1 eV ≈ 11 604 K × kB ). Assuming (after [3.1]) different valencies, Va , for different metals which are equal to the number of nearly free electrons per atom, knowing the atomic number in grams, A, and the weight density, W , we can calculate the total electron concentration, Ntot /V , as N Ntot = Va Av W V A

34

Introduction to Theory of Solids

Table 4.1 Calculated free electron Fermi energy for metals in the nearly free electron model Electron concentration 1022 cm−3

Fermi energy (eV)

Fermi temperature TF = F /kB 104 K

Ratio of T/TF × 10−3 at T = 300 K

Valency

Metal

1

Li Na K Rb Cs Cu Ag Au

470 265 140 115 091 845 585 590

472 323 212 185 158 700 548 551

548 375 246 215 183 812 636 639

547 8 1219 1395 1639 369 471 469

2

Be Mg Ca Sr Ba Zn Cd

242 860 460 356 320 1310 928

1414 713 468 395 365 939 746

1641 827 543 558 524 1090 866

183 363 552 537 572 275 346

3

Al Ga In

1806 1530 1149

1163 1035 860

1349 1201 998

222 249 301

4

Pb Sn

1320 1448

937 1003

1087 1164

276 258

Having the electron concentration, the Fermi energy can be found from (4.11) with k on the left replaced by the Fermi energy F : 2 F = 2m∗

3 2 Ntot V

2/3

2/3 2 NAv 2 3 Va =

2m∗ A W

(4.21)

The results are collected in Table 4.1, showing how large the Fermi energy is compared to the thermal activation energy (see the last two columns).

4.1.3 Heat capacity of the free electron gas: an introduction Explaining this phenomenon is important for acquiring an intuitive understanding of the quantum electron gas model. In a classic ideal gas, the internal energy present in 1 mol would be 3/2kB TVa NAv , where 1/2kB T is the energy per degree of freedom of one electron, Va NAv is the number of electrons in 1 mol. Hence the speciﬁc heat, calculated according to classical physics, would be cV = U / T = 3/2Va R, where R is the gas constant, R = kB NAv . This value is

Electron States

35

constant and very large compared to the experimental data, which show that the speciﬁc heat is linear with the temperature and vanishing at absolute zero. In a quantum model according to Fig. 4.3, when the specimen is heated from absolute zero, only those electrons within the energy range kB T of the Fermi level are excited thermally, gaining energy which is itself of the order of kB T . From the total number of electrons, only the fraction T/TF (TF is the Fermi temperature; TF ≡ /kB ) can be excited thermally, as only these electrons lie within an energy range of the order of kB T , at the top of the energy distribution. This fraction of electrons taking part in the heating process is very small even at room temperature; see the last column of Table 4.1. Each of these Va NAv T/TF electrons has a thermal energy of the order of kB T . The total electronic thermal kinetic energy U is of the order of U ≈ kB T Va NAv T/TF and the electronic speciﬁc heat per mol is given by U (4.22) ≈ 2Va RT/TF T which is directly proportional to T , in agreement with the experimental results. At room temperature cel is smaller than the classic value 3/2R by a factor of the order of 0.004 for TF ∼ 5 × 104 K. One could follow this line of reasoning and explain quantitatively the electronic speciﬁc heat and other properties, such as electrical conductivity, Ohm’s law, Hall effect and thermal conductivity. However, these properties have already been perfectly well explained in other textbooks – see [3.1] – and there is no need to repeat it here since the basic intuitive background of the nearly free electrons model has already been compiled for further applications within the scope of this book. cel =

4.2 THE TIGHT-BINDING METHOD The nearly free electron method deals with electrons, which spend most of their time outside the atomic cores where they propagate as plane waves. On the opposite side of the spectrum there are some groups of electrons closely bound with their atoms where they spend most of their time. A look at the periodic table with outer electron shells (see Table 7.1) tells us that there are numerous examples of elements with tightly bound electrons. These are transition elements of 3d, 4d and 5d groups, elements of 4f and 5f groups. For these electrons the function can be constructed [4.1, 4.2], which looks like an atomic orbital within atomic cores and satisﬁes the Bloch condition (see Section 4.3): 1 ik·l k r = e a r − l Na l

(4.23)

36

Introduction to Theory of Solids

where a r − l is an atomic orbital for a free atom in position l and Na is the number of atoms in the crystal. This function is a series of strongly localized atomic orbitals, multiplied by the wave phase factor expik · l. Within each atom the local orbital predominates and is a good solution to the local Schrödinger equation, which has the following form:

2 2 Ha a r = − + Va r a r = a a r 2m

(4.24)

Using (4.23) one arrives at the expression for the eigenvalue of Ha with respect to k : 1 ik·l Ha k r = e Ha a r − l = a k r Na l

(4.25)

In the crystal, the positive ions attract electrons on neighbouring lattice sites resulting in lowering the effective potential from Va r to Vr, see Fig. 4.4. The crystal Hamiltonian can be written as H = Ha + H − Ha = Ha + Vr − Va r

(4.26)

with Va r > Vr. As a ﬁrst approximation let us calculate the expected value of the energy for the wave function given by (4.23): k =

k∗

2 2 − + Vr k dr N 2m ∗ = k Dk k k dr

(4.27)

V (r ) Va(r ) r Va(r ) V (r )

+

+

+

FIGURE 4.4 Atomic (broken line) and ionic (solid line) potentials in crystal.

Electron States

37

The numerator of (4.27) is given by Nk = a + k∗ rH − Ha k r dr = a − a∗ rVa r − Vra r dr

c

−

eik·rm −rl

c

lm

= a − −

a∗ r − rl Va r − Vra r − rm dr

eik·h th

(4.28)

h

where = th =

a∗ rVa r − Vra r dr

c

a∗ r − hVa r − Vra r dr

c

h = r l − rm

(4.29)

and c is the unit cell volume. th deﬁned above is called the hopping integral. The denominator of (4.27) is equal to Dk = 1 +

eik·h Sh

Sh = a∗ r − ha r dr

h

Putting these results together to the ﬁrst power of the overlap integral S (S ≡ Sh , t ≡ th for vector h pointing to the nearest neighbour) one arrives at a − − t eik·h t eik·h − h h a k = = − 1 + S eik·h 1 + S eik·h 1 + S eik·h h

h

(4.30)

h

When S is small the last equation can be expressed as [4.3] k ≈ a − 1 − zS − t

eik·h + tz2 S

(4.31)

h

The behaviour of k , following (4.31), can be depicted schematically in the function of the atomic spread for the 3d and 4s orbitals in Fig. 4.5. For a large enough distance, the overlap of the atomic orbitals S can be ignored and one has the simpliﬁed, frequently used expression (although it will be shown below that this expression would lead to the collapse of the lattice) k = a − − t

h

eik·h

(4.32)

38

Introduction to Theory of Solids

εk

4s 3d

a

FIGURE 4.5 The change in atomic levels as the atoms move closer together.

For a simple cubic (sc) lattice with lattice constant a one arrives at

k = a − − 2t100 cosakx + cosaky + cosakz

(4.33)

Between the centre of the BZ k = 0 (-point) and the surface of the ﬁrst BZ, the energy k varies between a − ∓ 6t100 = a − ∓ D, D = 6t = zt, where D is a half of the bandwidth. For the superconducting cuprates, one has in the CuO2 plane a simple 2D cubic lattice. For such a lattice one obtains from (4.32)

k = a − − 2t100 cosakx + cosaky

(4.34)

where a is the distance of neighbouring oxygen atoms in the CuO2 plane. Now the energy varies between 0 − ∓ 4t100 . In Fig. 4.6, the graphic depicture of the electron dispersion relation from (4.34) is shown within the ﬁrst BZ for kx = ky changing between 0 and /a. One can see that the group velocity of an electron wave at the zone boundary, which is proportional to the slope of the relation k [see (4.5)], is zero. This is evidence of the existence of a standing wave at the zone boundary, which physically creates the BZ. Similarly for a body-centred cubic (bcc) lattice one can calculate from (4.32) k = a − − 8t 12 12 12 cos

aky akx ak cos cos z 2 2 2

(4.35)

with 0 − − 8t 12 12 12 ≤ k ≤ 0 − + 8t 12 12 12 , and for a face-centred cubic (fcc) lattice

εk – (εa – Δε)

Electron States

4t100

39

π/a

0

π/2a

–4t100 –π/a

0 –π/2a kx

ky

–π/2a

0 π/2a π/a

–π/a

(a)

εk – (εa – Δε)

4t100

0

–4t100 –π/a

–π/2a

0 kx = ky

π/2a

π/a

(b)

FIGURE 4.6 Electron dispersion relation for the 2D simple cubic lattice from (4.34). (a) 2D representation; (b) cross-section for kx = ky in the first Brillouin zone; (c) equipotential energies in kx ky plane.

k = a − − 4t 12 12 0

aky aky akx akx akz akz cos cos + cos cos + cos cos 2 2 2 2 2 2 (4.36)

with 0 − − 12t 12 12 0 ≤ k ≤ 0 − + 12t 12 12 0 . Generally in all these cases, for each type of lattice, one has a − − D ≤ k ≤ a − + D where D = zt-half bandwidth, z being the number of nearest neighbours (nn), which is 6 for sc lattice, 4 for 2D sc lattice, 8 for bcc lattice and 12 for fcc lattice.

40

Introduction to Theory of Solids

π/a

π/2a

ky

0

–π/2a

–π/a –π/a

–π/2a

0 kx

π/2a

π/a

(c)

FIGURE 4.6 Continued.

Evaluating the density of states, with the help of the general equation (4.16) for the dispersion relations corresponding to sc, bcc and fcc lattices [(4.33), (4.35) and (4.36), respectively], is a numerical task already calculated by Jelitto [4.4], and the results are presented in Fig. 4.7. DOS for an fcc lattice has a singular maximum, called the Van Hove singularity, at the upper end of the spectrum. The Fermi level is located near this maximum for the last elements of the 3d row. A high DOS on the Fermi level favours ferromagnetism according to the Stoner condition [see (7.41)]. The bcc lattice has the maximum DOS in the middle, which favours magnetism in the middle of the band or in the middle of the 3d row. At these electron occupations, the minimum critical interaction for the antiferromagnetism falls almost to zero and well below the minimum interaction for ferromagnetism (see Fig. 8.5). This is why elements like Cr and Mn are antiferromagnetic or have helical types of order.

4.2.1 Cohesion energy In the ﬁrst approximation, ignoring the overlap of atomic orbitals, S ≡ 0 as in relation (4.32), one can calculate the cohesion energy of a solid at T = 0 K in the paramagnetic state as F Ec = k − na = 2 d − n (4.37) k
−

41

Electron States

ε D

D/2

0

–D/2

–D 0

1/D

2/D

ρ (ε)

FIGURE 4.7 The density of states for the sc (solid line), bcc (dashed line) and fcc (dotted-dashed line) lattices (after [4.4]).

where n is the number of electrons per atom in the band, F is the Fermi level,

is the DOS per spin centred around the atomic level a . The factor 2 stands for two possible electron spin projections. For simplicity the constant DOS will be assumed now: ⎧ ⎨ 1 for − D ≤ ≤ D

= 2D (4.38) ⎩0 otherwise The half bandwidth of this density is: D = zt, with z given above for different lattice types. Obviously this is an oversimpliﬁcation since the DOS will not remain constant for different lattice types, see Fig. 4.7. The condition for the Fermi energy at zero temperature in the paramagnetic state, n F =

d 2 −D

(4.39)

gives us F = Dn − 1. Using this result one obtains from (4.37) the relation Ec = −n − Dn

n =

n 2 − n 2

(4.40)

where n is the cohesive energy as a function of the band ﬁlling in units of D. This is a well-known parabolic relation between the energy of cohesion and the occupation number of electrons [4.3] (Fig. 4.8). It holds well for the

42

Introduction to Theory of Solids

0.6

Ec [D ]

0.4

0.2

0

0

0.5

1 n

1.5

2

FIGURE 4.8 The cohesive energy as a function of the band filling in units of D for weak Coulomb correlations.

E c (eV/atom)

8

6

4

2

3d series

Sc

Ti

V

Cr

Mn

Fe

Co

Ni

Cu

Zn

4d series

Y

Zr

Nb

Mo

Tc

Ru

Rh

Pd

Ag

Cd

5d series

La

Hf

Ta

W

Re

Os

Ir

Pt

Au

Hg

FIGURE 4.9 Experimental cohesive energy for 3d (solid line), 4d (dashed line) and 5d (dotted line) transition metals.

d electrons in transition metal series 4d and 5d. The case of the 3d series is different than that of the 4d and 5d series because the 3d band is split in the middle, probably by the Coulomb repulsion, U , which plays an important role here [4.5] (Fig. 4.9). What will happen according to the formula (4.40) when atomic distance a is decreasing? For a given element the electron concentration n is ﬁxed; the cohesion Ec ∼ D = zt will keep increasing, since t is increasing with decreasing a.

Electron States

43

As a result one will have increasing cohesion until the lattice completely collapses at a → 0. The growing overlap of atomic wave functions, S, due to the decrease of the atomic distance a can correct this non-physical feature. The dispersion relation, which includes this overlap, is given by the expression (4.31). After inserting expression (4.31) into the formula for cohesion energy (4.37), one arrives at [4.3] k ≈ a − 1 − zS − t

eik·h + tz2 S

h

(4.41)

Ec = −n1 − zS + nzSD − nD It can be seen from formula (4.41) that with decreasing atomic distance a, there is an increasing repulsion term in energy, since Sa increases as a decreases. This correction is depicted schematically in Fig. 4.5 by the bands curving upwards with a decreasing lattice constant. Using this method one can perform more detailed calculations of equilibrium atomic distance in crystals, for example in 3d, 4d, 5d transition metals. The equilibrium value will give the atomic volumes of transition metals and in further calculations their mechanical properties such as compressibility, stiffness, etc. For further details on calculations the reader is referred to [4.6, 4.7].

4.3 BLOCH THEOREM According to Bloch [3.1, 4.1], the wave function in crystals fulﬁls the relation k r + T = eik·T k r

(4.42)

where eik·T is the phase factor by which a Bloch function is multiplied when there is a translation T in a crystal lattice. The main signiﬁcance of this statement is that the electron momentum k is a good quantum number in the crystal and can label the electron states. It is convenient to make an electron wave function of a given value of k to look as much as possible like a free electron wave. If one puts k r = eik·r uk r

(4.43)

then it may be proved at once from (4.42) that uk r + T = uk r

(4.44)

The form (4.43) with condition (4.44) is sometimes used as an alternative statement of the Bloch theorem.

44

Introduction to Theory of Solids

Inserting the plane wave (4.1) into Bloch’s theorem (4.42) one obtains 1 k r + T = √ eik·T eik·r = eik·T k r V

(4.45)

meaning that the plane wave fulﬁls the Bloch law. Inserting the tight binding wave function (4.23) into (4.42) one has the relation 1 ik·l 1 ik·l k r + T = e a r + T − l = eik·T e a r − l Na l Na l

(4.46)

= eik·T k r = k r We see that the tight binding function (4.23) also satisﬁes the Bloch law. Crystal momentum of an electron, p = k, labels the different states and wave functions of an electron in the crystal. If an electron with momentum k collides with a phonon of wave vector q one has the momentum conservation law, which states that k + q = k + G. In this process, an electron is scattered from the state k to a state k , with G a reciprocal lattice vector. Any arbitrariness in labelling the Bloch function can be absorbed in G without changing the physics of the process.

APPENDIX 4A: NEARLY FREE ELECTRONS, TWO-PLANE WAVES MODEL We assume the existence of a small potential Ur acting on the electrons moving in a crystal. For the periodic atomic location in the lattice this potential has to be invariant under a crystal transformation Ur = Ur + a

(4.A1)

thus it can be expanded as a Fourier series in the reciprocal lattice vector G: Ur = UG eiG·r (4.A2) G

The potential energy Ur is a real function, therefore one has UG∗ = U−G = UG , and the potential can be written as Ur = UG eiG·r + U−G e−iG·r = 2 UG cosG · r (4.A3) G>0

G>0

The wave function fulﬁlling the Bloch law has the form k r = eik·r uk r

(4.A4)

Electron States

45

where uk r + a = uk r

(4.A5)

The potential (4.A3) and the function (4.A4) are inserted into the Schrödinger equation: Hk = k , where H is the Hamiltonian and is the energy eigenvalue. The Hamiltonian can be written as H = H0 + Vpert = p2 /2m + Ur, where H0 = p2 /2m = −i d/dr2 /2m is the kinetic energy, and the Schrödinger equation takes the form 2

p + Ur k r = k r (4.A6) 2m The periodic wave function k r may be expressed as a Fourier series summed over all values of the reciprocal lattice vector G : k r = Ck−G eik−G ·r (4.A7) G

There are two dominant terms in this expansion: the unperturbed term Ck−0 = Ck and the term for G = G; Ck−G . Since there are only two Fourier components the relation (4.A7) will take on the form k r = Ck−G eik−G ·r = Ck eik·r + Ck−G eik−G·r (4.A8) G

On substituting (4.A8) into (4.A6) the kinetic term can be expressed as p2 1 d 2 r = Ck−G eik−G ·r = k Ck eikr + k−G Ck−G eik−G·r (4.A9) −i 2m k 2m dr G with the notation k =

2 k2 2m

(4.A10)

The potential energy term is given by Urk r = UG eiG ·r Ck eik·r + Ck−G eik−G·r G

=

G

UG Ck eik+G ·r + UG Ck−G eik+G −G·r

(4.A11)

Using (4.A8), (4.A9) and (4.A11) in the wave equation (4.A6) one obtains k Ck eik·r + k−G Ck−G eik−G·r + UG Ck eik+G ·r + UG Ck−G eik+G −G·r G

= Ck e

ik·r

+ Ck−G e

ik−G·r

(4.A12)

46

Introduction to Theory of Solids

Multiplying both sides by e−ik·r and integrating over the space of the crystal one has k − Ck + UCk−G = 0

(4.A13)

and after multiplying both sides by e−ik−G·r and integrating over the space of the crystal one has k−G − Ck−G + UCk = 0

(4.A14)

where U ≡ UG = U−G . Equations (4.A13) and (4.A14) have a non-zero solution for Ck , Ck−G , if the energy satisﬁes the condition k − U U k−G − = 0

(4.A15)

hence 2 − k−G + k + k−G k − U 2 = 0, which gives two roots for the energy

1/2 1 1 = k−G + k ± k−G − k 2 + U 2 2 4

(4.A16)

On a zone boundary one has k = 1/2G and k = 12 G k−G = − 12 G

12 G = − 12 G

(4.A17)

and from (4.A16) one obtains 2 = 12 G ± U = 2m

1 G 2

2 ± U ≡ ±

(4.A18)

Expression (4.A18) shows that the energy has one root − , lower than the free electron kinetic energy by U , and one + , higher by U . The difference between both roots, 2U , is the energy gap on the zone boundary. Dependence of both roots from (4.A16) on the wave vector is shown in Fig. 4.10. The ratio of the C’s may be calculated from (4.A13) or (4.A14) as − 12 G C−G/2 ±U = = = ±1 CG/2 U U

(4.A19)

where the last step uses (4.A18). Thus the Fourier expansion of k r from (4.A8) at the zone boundary has two solutions

Electron States

47

h 2G 2/4m + (h 4G 4/16m 2 + U 2)1/2

4 3

2

ε

Second band

1

εk

2U First band

0 0.5

1/2 G

1.5

2

k h 2G 2/4m – (h 4G 4/16m 2 + U 2)1/2

FIGURE 4.10 Solution of (4.A16) in the periodic zone scheme. The units are such that U = 05, G = 2 and 2 /m = 1. The free electron curve is drawn for comparison. The energy gap at the zone boundary is 2U = 1. A large value of U has deliberately been chosen for this illustration, too large for the two-plane waves approximation to be accurate.

r ≡ ± ∼ expiG · r/2 ± exp−iG · r/2

(4.A20)

or + ∼ cos r/a

− ∼ sin r/a

(4.A21)

These are two standing waves. For the standing wave + k r at k = G = 2/a, and + sign in (4.A19)] one has

+ = + 2 ∼ cos2 r/a

(4.A22)

and for the standing wave − k r at k = G = 2/a, and – sign in (4.A19)] one has

− = − 2 ∼ sin2 r/a

(4.A23)

The probability distribution, + , corresponds to the lower energy + = 12 G + U in (4.A18) (U is negative), and the probability distribution, − , corresponds to the higher energy − = 12 G − U in (4.A18). The function + piles up electrons (negative charge) on the positive ions centred at x = 0 a 2a , where the potential is lowest. That is why it has lower energy + . The other function − concentrates electrons away from the ion cores, where the potential is higher. It has higher energy − .

48

Introduction to Theory of Solids

REFERENCES [4.1] J.M. Ziman, Principles of the Theory of Solids, Cambridge University Press, Cambridge (1995). [4.2] J. Kübler, Theory of Itinerant Electron Magnetism, International Series of Monographs in Physics, Clarendon Press, Oxford (2000). [4.3] M. Cyrot, Solid State Commun. 22, 171 (1977). [4.4] R.J. Jelitto, J. Chem. Solids 30, 609 (1969). [4.5] F. Kajzar and J. Mizia, J. Phys. F 7, 1115 (1977). [4.6] J. Friedel, The Physics of Metals (ed. J.M. Ziman), Cambridge University Press, London (1969). [4.7] F. Ducastelle, J. Phys. 31, 1055 (1970).

PART

2 Models of Itinerant Ordering in Crystals

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CHAPTER

5 The Hubbard Model

Contents

5.1 Simple Hubbard Model 5.2 Extended Hubbard Model References

51 54 58

5.1 SIMPLE HUBBARD MODEL This model describes the s-like narrow energy band, the s-like meaning the band which can accommodate up to two electrons. The general model for interacting electrons moving in crystals reads [5.1] H =−

+ tij ci cj +

ij

1 U c+ c+ c c 2 ijkl ijkl i j l k

(5.1)

The ﬁrst term is the kinetic energy of electrons, K, in the tight-binding approximation (see Section 4.2 for the classical expression) and the second term is the potential energy, V . The electron hopping is controlled by the tij – + hopping integral between the ith and jth lattice site. The operator ci is creating an electron with spin on the ith lattice site, cj is annihilating an electron + with spin on the jth lattice site. The product ci cj is responsible for the process in which the electron is moving from jth to ith lattice site. In potential energy, the sum over the spin means = ±. Uijkl is the def

Coulomb integral, Uijkl = ij1/rkl, and in the real space notation Uijkl =

∗ r − ri ∗ r − rj

e2 r − rl r − rk dr dr r − r

(5.2)

where r − ri is the Wannier wave function localized on the ith atom, which is approximately equal to the atomic wave function for a given orbital forming 51

52

Models of Itinerant Ordering in Crystals

the narrow s-like band r − ri ≈ a r − ri . These functions were used for electrons in the crystal in the tight-binding approximation of Section 4.2. The hopping integral between the ith and jth lattice sites, tij , is given by (4.29), which is th = a∗ r − hVa r − Vr a rdr with h = ri − rj The basic approximation introduced by Hubbard in the Hamiltonian (5.1) [5.1–5.3] ignores in the potential energy the electron–electron interaction on different lattice sites, which is supposed to be much smaller than the on-site Coulomb repulsion; U = Uiiii : Uiiii = U =

∗ r − ri ∗ r − ri

e2 r − ri r − ri dr dr r − r

(5.3)

Experimental data support this assumption. Usually one has U = 3–5 eV (see the data for transition metals [5.5]). Sometimes when the band is quite narrow, which makes screening more difﬁcult, the on-site Coulomb repulsion is as much as 10 eV, while Uij for i = j is of the order of the fraction of 1 eV, which is much less than U . After ignoring all inter-site Coulomb interactions and taking into account the Pauli exclusion principle, which for electrons on the same lattice site, i, gives the condition of = −, one obtains H =−

+ tij ci cj − 0

ij

i

nˆ i +

U nˆ nˆ 2 i i i−

(5.4)

+ ci is the electron number operator for electrons with spin on where nˆ i = ci the ith lattice site, for which eigenvalues are 0 or 1. The term with the chemical potential 0 multiplied by the number of electrons per atom is added in here. The operator product, nˆ i nˆ i− , has an eigenvalue of one, only when two electrons with opposite spins − meet on the same lattice site, and zero if there is one or zero electrons on this site. The eigenvalue of one multiplied by U gives the repulsion between two electrons with opposite spins when they meet on the same lattice site. The kinetic energy will be transformed to the momentum space with the help of the following Fourier transformations:

1 + ik·ri + =√ ck e ci N k

1 cj = √ ck e−ik ·rj N k

(5.5)

+ where N is the number of atoms, ck is the operator creating electron with momentum k and spin and ck is the operator annihilating electron with momentum k and spin . The above operators are the fermion operators. Fermions are the particles with spins being the multiplicity of one half. They obey the Pauli exclusion

The Hubbard Model

53

principle. The consequences of this principle are the fermions anticommutation rules which are fulﬁlled by fermion operators + + + ck ck + = ck ck + ck ck = kk = kk

+ ck ck+

+

+ + + = ck ck + ck+ ck = 0

(5.6)

ck ck + = ck ck + ck ck = 0 Using Fourier transforms (5.5) for the kinetic energy of electrons and the property of translational symmetry, one has K=−

+ tij ci cj = −

ij

ij

1 ik·ri + 1 −ik ·rj tij √ e ck √ e ck N k N k

=−

1 + tij eik·ri −ik ·rj +ik·rj −ik·rj ck ck N ijkk

=−

1 1 + + tij eik−k ·rj eik·ri −rj ck ck = − th eik·h eik−k ·rj ck ck N ijkk N jhkk

where h = ri − rj . Applying above the relation

1 ik−k ·rj 0 where k = k e = kk = N j 1 where k = k

one obtains K=−

hkk

+ th eik·h kk ck ck =

k

+ k ck ck

k = −

th eik·h

h

where k is the dispersion relation for electrons, the same as discussed in Section 4.2 within the tight-binding method. The only difference with (4.32) is that in here it was assumed for simplicity that the energy of the atomic level, a , and its shift, , are zero. Thus K = k nˆ k (5.7) k

Finally the simple Hubbard Hamiltonian, in the momentum representation for the kinetic energy and space representation for the potential energy, has the form U H = k nˆ k − 0 nˆ k + nˆ nˆ (5.8) 2 i i i− k k

54

Models of Itinerant Ordering in Crystals

There is no exact solution to this Hamiltonian in 3D. Therefore one has to look for the approximate solutions. Following Hubbard [5.3] and Velický et al. [5.4] the motion of + in the ﬁeld of − electrons, which are frozen and distributed randomly among the lattice sites, will be considered. The random ni− = Vi will be assumed, which takes on the value 0 potential U nˆ i− ≈ U ni− is the with the probability 1 − ni− and U with probability ni− . Here probability that the incoming electron with spin will meet on the ith lattice site the electron with spin −, which is given approximately by the average number of electrons with spin − on the ith lattice site, ni− ni− . In order to decrease the perturbation, a term with the coherent potential (complex and the same on each lattice site), , will be added to the kinetic energy part. The same expression will be deducted from the perturbation term H=

k nˆ k − 0

k

nˆ k +

nˆ i +

i

k

Vi − nˆ i

(5.9)

i

Finally one can write H = H0 + Vpert

(5.10)

where the unperturbed part of the Hamiltonian is H0 =

k nˆ k − 0

k

k

nˆ k +

nˆ k

(5.11)

k

and the perturbed part is Vpert =

Vi − nˆ i

(5.12)

i

5.2 EXTENDED HUBBARD MODEL The Hamiltonian for one degenerate band can be written in the form given by Hubbard [5.2]: H =−

ij

+ tij ci cj − 0

i

nˆ i +

+ + i j1/rk lci cj cl ck (5.13)

ijkl

Here, in addition to the Hubbard’s approach [5.3], the nearest-neighbours hopping integral, tij , depends on the occupation of sites i and j. Quantity 0 + is the chemical potential, ci ci is creating (destroying) an electron of spin + ci is the electron in a Wannier orbital on the ith lattice site, nˆ i = ci number operator of spin in a Wannier orbital on the ith lattice site, the

The Hubbard Model

55

indices numerate the orbitals (sub-bands) in the degenerated single band. Taking into account in the Hamiltonian (5.13) only single-site i = j = k = l and two-site interactions k l = i j, as well as single sub-band ( = = = and two sub-band interactions = , one obtains and retains the following matrix elements: • single-site, single sub-band interaction: U0 = i i1/ri i

(5.14)

• single-site (subscript “in”), two sub-band interactions (for = ): Vin = i i1/ri i

Jin = i i1/ri i

Jin = i i1/ri i (5.15)

• two-site interactions: V0 = i j1/ri j

J0 = i j1/rj i

J0 = i i1/rj j

t0 = i i1/rj i

(5.16)

For simplicity the fully degenerate band, i.e. the band composed of identical orbitals [5.5] will be assumed. Keeping the above interactions in the Hamiltonian (5.13) and assuming one fully degenerate band, the following form [5.6] for the inter-site interactions in this Hamiltonian can be obtained: H =−

+ tij ci cj − 0

i

nˆ i − F

n nˆ i +

i

U nˆ nˆ 2 i i i−

+ + V J + + + nˆ nˆ + c c c c + J ci↑ ci↓ cj↓ cj↑ 2 i j 2 i j i j

(5.17)

+ ci creates (destroys) an electron of spin on the ith lattice site, where ci + nˆ i = ci ci is the electron number operator for electrons with spin on the ith lattice site, nˆ i = nˆ i + nˆ i− is the operator of the total number of electrons on the ith lattice site, n is the probability of ﬁnding the electron with spin in a given band. The new interaction constants, U , V , J , J , for the degenerate band are

• single-site interaction: U = i i1/ri i

(5.18)

• two-site interactions: V = i j1/ri j

J = i j1/rj i

J = i i1/rj j

(5.19)

56

Models of Itinerant Ordering in Crystals

The two-site interactions, V J J , can be called the charge–charge interaction, inter-site exchange interaction and pair hopping interaction, respectively. Since a fully degenerate single band was assumed, these interactions will be given by the single orbital interactions deﬁned previously [(5.14) and (5.16)] as U = dU0

J = dJ0

J = dJ0

V = dV0

where d is the number of degenerated orbitals in the band. The F constant is the intra-atomic Hund’s interaction, which can be expressed as the interaction between different orbitals on the same lattice site in a multi-orbital single-band model: F = d − 1Jin + Jin + Vin

(5.20)

As a result our model is a quasi-single-band model. The intra-atomic Hund ﬁeld in (5.17) is already written in the Hartree–Fock approximation. This can be justiﬁed only for small values of this interaction compared to the kinetic energy represented by the parameter of bandwidth, F << D. In the Hamiltonian (5.17) one has the spin-dependent correlation hopping tij , which can be expressed by the occupation of sites i and j in the operator form: tij = t1 − nˆ i− 1 − nˆ j− + t1 nˆ i− 1 − nˆ j− + nˆ j− 1 − nˆ i− + t2 nˆ i− nˆ j− (5.21) where t is the hopping amplitude for an electron of spin when both sites i and j are empty. Parameter t1 is the hopping amplitude for an electron of spin when one of the sites i or j is occupied by an electron with opposite spin. Parameter t2 is the hopping amplitude for an electron of spin when both sites i and j are occupied by electrons with opposite spin. Including the occupationally dependent hopping given by (5.21) in the Hamiltonian (5.17) and after performing some algebra, one obtains the following result + t − tnˆ i− + nˆ j− + 2tex nˆ i− nˆ j− ci cj − 0 nˆ i − F n nˆ i H =−

+

i

i

+ + U V J + + nˆ i nˆ i− + nˆ i nˆ j + ci cj ci cj + J ci↑ ci↓ cj↓ cj↑ 2 i 2 2 (5.22)

where the hopping interaction, t, and the exchange-hopping interaction, tex , are deﬁned as

t = t − t1

tex =

t + t2 − t1 2

(5.23)

57

The Hubbard Model

In this form, it is quite easy to see that t and tex are also inter-site interactions. The sign and magnitude of interactions t and tex , according to (5.23), depend on the hopping amplitudes t, t1 and t2 . For simplicity it will be assumed that t1 /t = S

and t2 /t1 = S1

In general, these parameters are different and they both fulﬁl the condition S < 1 and S1 < 1, which is equivalent to t > t1 > t2 [5.7]. However, in his paper Hirsch [5.7] has pointed out that for the hydrogen molecule, H2 , these integrals depend strongly on the inter-atomic distance and for a distance large enough one can even have the reverse relation t < t1 < t2 . The heavier elements (e.g. 3d or 4f) possess larger inter-atomic distances, therefore they may have growing hopping integrals with increasing occupation. Gunnarsson and Christensen [5.8] observed such dependence for 4f transition elements. In analysing the inﬂuence of interactions t and tex on magnetism, both negative and positive values will be considered. Taking into account the relations deﬁned above between hopping integrals one can write that

t = t1 − S and

tex = t1 + SS1 − 2S/2

(5.24)

For simplicity one can assume [5.7] that S1 = S. Then (5.24) for tex can be simpliﬁed even further tex /t = 1 − S2 /2

(5.25)

and the whole kinetic energy term will take on the following form: + t1 − 1 − Snˆ i− + nˆ j− + 1 − S2 nˆ i− nˆ j− ci cj K=−

(5.26)

which will be used later on. With this kinetic energy the Hamiltonian (5.22) will take on the form: + H =− t − 1 − Snˆ i− + nˆ j− + 1 − S2 nˆ i− nˆ j− ci cj − 0 nˆ i − F n nˆ i

+

i

i

+ + U V J + + ci cj ci cj + J ci↑ ci↓ cj↓ cj↑ nˆ i nˆ i− + nˆ i nˆ j + 2 i 2 2 (5.27)

This form or the form (5.22) will be used later on throughout this book to investigate the inﬂuence of electron correlations on such phenomena as ferromagnetism, antiferromagnetism and superconductivity in Chapters 7, 8, 10 and 11.

58

Models of Itinerant Ordering in Crystals

REFERENCES [5.1] [5.2] [5.3] [5.4] [5.5] [5.6] [5.7] [5.8]

J. Hubbard, Proc. R. Soc. A 276, 238 (1963). J. Hubbard, Proc. R. Soc. A 277, 237 (1964). J. Hubbard, Proc. R. Soc. A 281, 401 (1964). B. Velický, S. Kirkpatrick and H. Ehrenreich, Phys. Rev. 175, 747 (1968). J. Mizia, Phys. Stat. Sol. (b) 74, 461 (1976). J.C. Amadon and J.E. Hirsch, Phys. Rev. B 54, 6364 (1996). J.E. Hirsch, Phys. Rev. B 48, 3327 (1993). O. Gunnarsson and N.E. Christensen, Phys. Rev. B 42, 2363 (1990).

CHAPTER

6 Different Approximations for Hubbard Model

Contents

6.1 Chain Equation for Green Functions 6.2 Hartree–Fock Approximation 6.3 Hubbard I Approximation 6.3.1 Atomic limit 6.3.2 Finite bandwidth limit 6.4 Extended Hubbard III Approximation 6.5 Coherent Potential Approximation 6.5.1 Relation between CPA, Hubbard III and extended Hubbard III approximations 6.5.2 Different applications of the CPA 6.6 Spectral Density Approach 6.7 Modified Alloy Analogy 6.8 Dynamical Mean-Field Theory 6.9 Hubbard Model Extended by Inter-Site Interactions 6.9.1 Modified Hartree–Fock approximation 6.9.2 Coherent potential approximation for the extended Hubbard model Appendix 6A: Equation of Motion for the Green Functions Appendix 6B: Hubbard Solution for the Scattering and Resonance Broadening Effects 6B.1 The scattering effect 6B.2 The resonance broadening effect Appendix 6C: Modified Hartree–Fock Approximation for the Inter-Site Interactions References

60 63 64 64 66 69 72 80 81 81 86 88 90 90 94 95 97 97 100 106 113

59

60

Models of Itinerant Ordering in Crystals

6.1 CHAIN EQUATION FOR GREEN FUNCTIONS The simple Hubbard Hamiltonian in the real space representation has, according to relation (5.4), the following form:

H =−

+ tij ci cj +

U nˆ nˆ − 0 N 2 i i i−

(6.1)

The term with chemical potential, 0 , will be ignored since it will appear in the Fermi–Dirac statistics. As already mentioned in Section 5.1, there is no exact solution to this Hamiltonian in 3D. The whole Hamiltonian is split into a solvable (unperturbed) part, H0 , and into a perturbation, Vpert : H = H0 + Vpert

(6.2)

where the unperturbed and the perturbed parts of the Hamiltonian are (see Section 5.1 for the transformation of kinetic energy from real to momentum space) + cj = k nˆ k (6.3) H0 = − tij ci

Vpert =

k

U nˆ nˆ 2 i i i−

(6.4)

It is possible to ﬁnd the explicit form of the Green function for the unperturbed Hamiltonian H0 . For this purpose we use the deﬁnition of the Green function in the operator form in the energy and momentum representation: 0 Gkk = ck ck+ (6.5) Since H0 does not depend on , the Green function, being the solution for H0 , should not depend on the spin index either, therefore one can write 0

0 Gkk = 0 G− kk = Gkk

(6.6)

The equation of motion for this Green function, which is the solution to the Hamiltonian H0 , has been derived in Appendix 6A as (6.A15) and has the following form [5.1, 6.1]: A B = A B+ + A H0 − B

(6.7)

where for the Green function, given by (6.5), one has A ≡ ck and B ≡ ck+ . Using these deﬁnitions in (6.7) one can write the equation of motion for the Green function as (6.8) ck ck+ = ck ck+ + + ck H0 − ck+

61

Different Approximations for Hubbard Model

In this section, the equation of motion for the Green function, corresponding to the unperturbed part of the Hamiltonian, H0 , will be sought, therefore equation (6.7) or (6.8) is solvable (see later). For the full Green function corresponding to the full Hamiltonian H, the equation of motion will lead to the chain equation relating the considered Green function to the higher order Green functions. This technique was used originally by Hubbard [5.1, 5.3], and it will be described in detail in Sections 6.3 and 6.4 devoted to the Hubbard I and Hubbard III solutions. Inserting the unperturbed Hamiltonian, H0 , given by (6.3) into (6.8) one has

+ + + ck ck = ck ck + + ck (6.9) k nˆ k ck k

−

With the help of fermion anticommutation rules [they are the result of Pauli’s exclusion principle, see (5.6)] + + ck ck+ + = kk

ck ck + = 0 ck ck + = 0 (6.10) one can write for the second term on the right-hand side of (6.9):

+ + + + ck k nˆ k ck = k ck ck ck − ck ck ck ck k

−

k

=

k

k k k ck ck+

= k ck ck+

(6.11) Grouping expressions (6.9)–(6.11) one obtains G0kk = kk + k G0kk G0kk =

kk − k

(6.12) (6.13)

which for k = k becomes G0k =

1 − k

(6.14)

The Green function in (6.14) is written in a simpliﬁed way. The precise notation for all these Green functions is G0k =

1 + − k

where + = + i0+

and the symbol 0+ stands for the inﬁnitesimal positive value.

(6.15)

62

Models of Itinerant Ordering in Crystals

Using the following identity 1 1 =P − i − k − k + i0+ − k

(6.16)

where P is the principal value of the integral, one obtains from (6.15) the following relation for the density of states (DOS): 1 1 1 1 1 0 = − k = − Im = − Im F0 N k

N k + − k

(6.17)

The quantity F0 is the unperturbed Slater–Koster function deﬁned as 1 1 N k + − k

F0 =

(6.18)

In Section 6.5, formulas (6.17) and (6.18) will be extended to the general case of self-energy with a ﬁnite imaginary part. To ﬁnd the solution to the problem with perturbation, (6.4), one has to use the same chain equation for Green functions (6.7) with the full Hamiltonian (6.1), which includes the perturbation. Let us consider the Green function + Gij = ci cj (6.19)

+ and Hamiltonian (6.1) with in a real space and use (6.7) with A ≡ ci , B ≡ cj 0 = 0. In this case, one has

ci H− = −

tij cj + Uci nˆ i−

(6.20)

+ + tij ci cj − cj ci

(6.21)

j

nˆ i H− = −

j

Using these relations one obtains the following equation of motion: Gij = ij −

til Glj + Uij

(6.22)

l

where + ij = nˆ i− ci cj

(6.23)

Different Approximations for Hubbard Model

63

For the Green function, ij , one obtains the following relation from the equation of motion: + ij = ij n− + t0 ij + Uij − til nˆ i− cl cj

l=i

−

l=i

+ + + + til ci− cl− ci cj − cl− ci− ci cj

(6.24)

where n− = nˆ i− . As we can see, the Green function Gij in (6.22) is expressed by the Green function of the higher order, ij , which in turn couples to the higher order functions. Therefore the set (6.22) and (6.24) does not have an exact solution, as was mentioned earlier.

6.2 HARTREE–FOCK APPROXIMATION One can truncate this set of equations by assuming in (6.22) that + + ij = nˆ i− ci cj ≈ nˆ i− ci cj = n− Gij

(6.25)

which brings Gij = ij −

til Glj + Un− Gij

(6.26)

l

This equation will be solved by the Fourier transformation Gij =

1 G exp ik · ri − rj N k k

(6.27)

where Gk is the Green function in the momentum representation. In addition, one has to use the relations −tij = ij =

1 − t exp ik · ri − rj N k k 0

(6.28)

1 exp ik · ri − rj N k

(6.29)

obtaining ﬁnally Gk =

1 − k − Un−

(6.30)

64

Models of Itinerant Ordering in Crystals

In (6.28), t0 is the energy of atomic level. The energy t0 is the same as the energy a − from Chapter 4. This result causes, as can be seen from (6.17), the shift in the DOS, , by the amount of the average ﬁeld Un− : = 0 − Un−

(6.31)

For electron concentration at absolute zero, one has from (4.20) n =

F n + m F = d = 0 − Un− d 2 − −

(6.32)

where F is the Fermi energy. Differentiating both sides of this equation over m one obtains the Stoner condition for ferromagnetism: 1 = U0 F − Un/2

(6.33)

Thus, for some U , the condition U > 1/0 F − Un/2 will always be satisﬁed, and the Hartree–Fock (H–F) theory predicts that the system will become ferromagnetic. It will be found in Chapter 7 that when correlation effects are taken into account, one obtains a more restrictive condition for ferromagnetism. It is worthwhile mentioning here that the H–F approximation is equivalent to the Weiss ﬁeld assumption in magnetism, see (7.3). The approximation given by (6.30) may be obtained by performing it directly on the Hamiltonian (6.1): nˆ i nˆ i− nˆ i nˆ i− + nˆ i nˆ i− − nˆ i nˆ i− = n nˆ i− + n− nˆ i − n− n (6.34) The result is equivalent to (6.30), after dropping the last term as nonmagnetic, since it is proportional to m2 and not to m.

6.3 HUBBARD I APPROXIMATION 6.3.1 Atomic limit The Hubbard I approximation [5.1] is built on the atomic limit of the Hubbard model. The following simpliﬁcations are introduced to the Hamiltonian (6.1): −tij = t0 for i = j and tij = 0 for i = j: Hatomic =

i

t0 nˆ i + U nˆ i↑ nˆ i↓

nˆ i = nˆ i + nˆ i−

(6.35)

Different Approximations for Hubbard Model

Number of electrons

Energy

0

vacuum

0

1

or

t0

2

and

65

2t0 + U

FIGURE 6.1 The energy levels of an atom.

As a result one has three possible conﬁgurations on each atom: vacuum, one electron and two electrons with opposite spins (doublons), with energies 0, t0 and 2t0 + U , respectively (see Fig. 6.1). Further analysis is carried out using the equation of motion for the Green function (6.7) in the space representation. Using Hamiltonian (6.35) one can write

ci Hatomic − = t0 ci + Uci nˆ i−

(6.36)

which, in combination with (6.7), will give us + + + = 1 + t0 ci ci + U nˆ i− ci ci ci ci

(6.37)

Using the relations

+ nˆ i− ci ci

+

= nˆ i−

nˆ i Hatomic − = 0

(6.38)

and the commutator deﬁnition

nˆ i− ci Hatomic − = nˆ i− ci Hatomic − Hatomic nˆ i− ci = nˆ i− ci Hatomic − + nˆ i− Hatomic ci − Hatomic nˆ i− ci = nˆ i− ci Hatomic − + nˆ i− Hatomic − ci

(6.39)

= nˆ i− t0 ci + Uci nˆ i− one obtains from the equation of motion + + + = n− + t0 nˆ i− ci ci + U nˆ 2i− ci ci nˆ i− ci ci

(6.40)

66

Models of Itinerant Ordering in Crystals

For the fermions one has the relation nˆ 2i = nˆ i , which allows us to calculate the Green function appearing in (6.40): + nˆ i− ci ci =

n− − t0 − U

(6.41)

Inserting this result into (6.37) we ﬁnally arrive at 1 − n− n− + ci ci = + − t0 − t0 − U

(6.42)

This Green function can be written as + ci ci =

1 − t0 −

(6.43)

Comparing (6.42) with (6.43) one obtains the self-energy in the Hubbard I approximation: HI = Un−

− t0 − t0 − U1 − n−

(6.44)

The self-energy given above, HI , is real, spin dependent and independent on the wave vector k. Using the general relation between the Green function and the DOS 1 1 + = − Im ci ci

N i

(6.45)

one obtains the following expression: = 1 − n− − t0 + n− − t0 − U

(6.46)

According to this result the system may have only two energies: t0 and t0 + U , with 1 − n− and n− states per atom, respectively. Assuming zero temperature and the paramagnetic state, (n− = n = n/2), one obtains for n ≤ 1 that electrons will occupy only the lower state and the Fermi energy is equal to = t0 . When the lower level is ﬁlled n ≥ 1 the Fermi energy jumps to the higher level: = t0 + U .

6.3.2 Finite bandwidth limit The Hamiltonian (6.1), which has a ﬁnite bandwidth limit, will now be considered. The technique developed for the atomic limit will be extended to this case. In order to break the sequence of Green functions appearing in (6.24), the

Different Approximations for Hubbard Model

67

approximate expressions will be substituted for the last pair of terms in this equation: + n− Glj (6.47) nˆ i− cl cj

+ + + ci− cl− ci cj ci− cl− Gij

(6.48)

+ + + cl− ci− ci cj cl− ci− Gij

(6.49)

+ + ci− and ci− cl− one can use the relations For the averages cl− l l=i

+ 1 + til cl− ci− = til cl− ci− N li l=i

+ 1 + = til ci− cl− = til ci− cl− N li l l=i

(6.50)

l=i

which will reduce the last two terms in (6.24) to zero. This allows us to calculate the function ij as

n− (6.51) ij − til Glj ij = − t0 − U l=i Inserting (6.51) into (6.22) and separating in (6.22) the sum over l into the sum over l = i and l = i one obtains the equation

Un− (6.52) ij − til Glj Gij = t0 Gij + 1 + − t0 − U l=i which can be solved by the Fourier transformation (6.27)–(6.29). As a result one obtains the following expression: Un− (6.53) Gk = t0 Gk + 1 +

1 + k − t0 Gk − t0 − U The solution of this equation has the form Gk =

− t0 − U1 − n− − k − t0 − U + Un− t0 − k

(6.54)

Comparing the above function with the Green function given by the equation Gk =

1 − k − HI

(6.55)

68

Models of Itinerant Ordering in Crystals

one obtains, for the self-energy in Hubbard I approximation and the ﬁnite bandwidth, the same relation (6.44) as before in the atomic limit. To illustrate the changes of DOS in the Hubbard I approximation under the inﬂuence of the Coulomb correlation, U , at different carrier concentrations, n, one can use an initial semi-elliptic DOS 0 =

2 D2 − 2 1/2 D2

for < D

(6.56)

where D is the half bandwidth. The DOS perturbed by the self-energy HI from (6.44) will have the following form: 2 1/2 2 − t0 2 D − − Un− = 2 D − t0 − U1 − n−

(6.57)

for − Un− − t0 / − t0 − U1 − n− < D, and is shown in Fig. 6.2 for different values of U and in Fig. 6.3 for different values of n. At any non-zero Coulomb repulsion, the spin band is split into two sub-bands (see Fig. 6.2), with the same maxima of 2/ D. The width of the upper and lower bands depends on the electron concentration (see Fig. 6.3).

ε

U1 = 2D U2 = 0.8D U=0

t0 + U1

t0 + U2

t0

ρ (ε) FIGURE 6.2 The DOS calculated in the Hubbard I approximation from the initial semi-elliptic DOS for different values of U for n = 15.

Different Approximations for Hubbard Model

69

ε

t0 + U

n = 0.5 n = 1.5

t0

ρ (ε) FIGURE 6.3 The DOS calculated in the Hubbard I approximation from the initial semi-elliptic DOS for different values of electron concentration n, for U = 2D.

6.4 EXTENDED HUBBARD III APPROXIMATION The Hubbard I approximation described earlier produces the band split into two sub-bands separated by an energy gap for arbitrarily small Coulomb repulsion. The additional odd feature of this approximation is an inﬁnite lifetime of the pseudo-particles caused by the real value of the self-energy. These two negative results were caused by the assumption that the dominant correlation takes place only between two electrons on the same lattice site, and all Green functions involving more than two atomic sites can be approximated in terms of the single-site average multiplied by the two-site Green function. As aresult + was the last two terms of (6.24) were ignored and the function nˆ i− cl cj + approximated to nˆ i− cl cj . To obtain a better result Hubbard intro duced the approximation called the Hubbard III approximation [5.3]. This approximation will be described in detail here together with the correction, + cl− , which comes from including the inter-site correlation function I− = ci− originally ignored in the Hubbard approach. Following the Hubbard III approximation the following notation is introduced: ˆ i nˆ + i ≡ n ˆ i nˆ − i ≡ 1 − n

(6.58)

70

Models of Itinerant Ordering in Crystals

which has the property

nˆ i = 1

(6.59)

=±

The same notation can be introduced for the average occupations, nˆ i : + ˆ i− ≡ n n+ = n (6.60) ˆ− n− = n i− ≡ 1 − n and for the two resonant energies: + ≡ t0 + U (6.61)

− ≡ t0

(6.60) and (6.61), the equation (6.24) for the function Using (6.58), + nˆ i− cl cj = ± can be written as

+ + + nˆ i− ci cj = n− ij − til cl cj + nˆ i− ci cj

l=i

−

til

l=i

−

+ nˆ i− − n− cl cj

l=i

(6.62)

til

+ + + + ci− cl− ci cj − cl− ci− ci cj

where ± = ±1. The ﬁrst two terms on the right-hand side of (6.62) give the Hubbard I approximation. Including the third term in (6.62) (this term comes from the commutator ci H− in the equation of motion) leads to the scattering effect. The last term (which comes from the commutator nˆ i− H− in the equation of motion) gives the resonance broadening effect. In further considerations of both the scattering +effect and the resonance broadening effect, the new averages of the cl− will be kept. This will result in corrections to the Hubbard type I− = ci− scattering and resonance broadening effects depending on the generalized ∓ ± ± . = cl− ci− average: Ili− The solution of (6.62) including the scattering and resonance broadening effects corrected for the inter-site correlations is described in Appendix 6B. Using the Fourier transformation to the momentum space of (6.B47) we arrive at Appendix 6B at the ﬁnal result for the Green function Gk =

1 T T FH − k − t0 − Bk + S−

(6.63)

71

Different Approximations for Hubbard Model

with Hubbard’s deﬁned function 1 FH

=

1 FH0

− T

=

− T − n+ − − + n− + − T − T − +

− − − n+ − − + − n− − n− n−

T 2 (6.64)

or

− FH = − n+ − + + n− − −

+ 2 n− − n− + − − − T − n+ − − + n− + −

(6.65)

T The bandwidth correction, Bk , and the bandshift correction, ST , appearing above are given by

1 T = C− − T FH C− FH0 −tlm nˆ l− nˆ m− − n2− Bk N lm + + + + − cl cm− cl− cm − cl cl− cm− cm exp ik · rl − rm (6.66)

ST

=

FH C− +

− T

−C− FH0

l=im

til Wlmi tmi nˆ i− nˆ l− − n2−

− + − − tmi tin Wlmi Iln− − Wlmi − + + − Iln−

(6.67)

lmn

−FH0 C−

l=i

+ − + − 2til nˆ l cl− ci− − FH0 C+ til cl− ci− l=i

The deﬁnitions of C± , FH0 , Wlmi and T are included in Appendix 6B. T = 0 and ST = 0 in (6.63), it reduces this result to the classic By making Bk Hubbard III approximation given by

Gk =

1 FH − k − t0

(6.68)

It will be shown later (see Section 6.5) that the standard Hubbard III approximation (both scattering and resonance broadening effects) is equivalent to the coherent potential approximation (CPA) under the appropriate change of variables between the Hubbard method of this section and the CPA approximation of Section 6.5.

72

Models of Itinerant Ordering in Crystals

T The bandwidth correction, Bk , and the bandshift correction, ST , will be compared in Section 6.6 with the result of the spectral density approach T and ST in analysing the ferro(SDA). The consequences of including Bk magnetism will be presented later in Chapter 7, and they will be compared with the results of the standard CPA approach.

6.5 COHERENT POTENTIAL APPROXIMATION We follow the approach of Hubbard [5.3] and Velický et al. [5.4] and consider the motion of + electrons in the ﬁeld of − electrons, which are frozen and distributed randomly among the lattice sites. This is equivalent to replacing the product U nˆ i− in (6.4) by the stochastic potential, Vi , taking on two values with the corresponding probabilities 0 1 − ni− with probabilities Pi = (6.69) Vi = U ni− where ni− = nˆ i− . As before, the whole Hamiltonian is split into the solvable (unperturbed) part, H0 , and the perturbation, Vpert [see (6.2)]. These parts are now given by H0 = k nˆ k + nˆ k = k + nˆ k = Ek nˆ k (6.70) k

k

k

with Ek = k + and Vpert =

Vi − nˆ i

k

(6.71)

i

The coherent potential, , is added to the unperturbed part in order to maximize the part of the Hamiltonian for which one can ﬁnd the solution and to minimize the perturbation, from which it is deducted. The solution is given in powers of Vpert /H0 and will contain the condition (prescription) of how to calculate the unknown coherent potential, . It has to be emphasized that the Hamiltonian with the perturbation given by (6.71) is already approximate, since the product of two operators multiplied by U was approximated by the stochastic potential, Vi . As will be mentioned later in this section the problem described by the potentials and probabilities given by (6.69) in the CPA does not produce the ferromagnetic ground state. Therefore to consider ferromagnetic elements and their alloys, the researchers added to this stochastic potential the exchange interaction between different orbitals within the same band in the mean ﬁeld or H–F approximation (see Section 5.2), obtaining −Fni 1 − ni− with (the same) probabilities Pi = (6.72) Vi = U − Fni ni−

Different Approximations for Hubbard Model

73

As mentioned above there is no exact solution to the Hamiltonian (6.1), not even with the perturbation approximated by (6.71), but one can solve the problem described by the unperturbed Hamiltonian (6.70). This is done in a similar way to calculating the Green function corresponding to H0 given by (6.3). The result is 0

Gk =

1 − k −

(6.73)

To derive the CPA method one can start from the Dyson identity, which is now introduced. The Green function in the operator form for the whole Hamiltonian (6.2) can be written as G =

1 −H

or

G =

1 − H0 − Vpert

(6.74)

while the Green function in the operator form for the unperturbed Hamiltonian, H0 , is G0 =

1 − H0

(6.75)

In the momentum and energy representation this Green function is given by G0 =

1 1 ⇒ 0 Gk = − H0 − k −

(6.76)

Using (6.74) and (6.75) we can check the following identity called the Dyson’s identity: G = G0 + G0 Vpert G

(6.77)

The Dyson equation is only the formal solution, since the unknown full Green function, G, also appears on the right-hand side of this equation. Iterating the Dyson equation (6.77) one obtains the Feynman–Dyson perturbation series in growing powers of the perturbation Vpert ≡ V: G1 = G0 + G0 VG0 G2 = G0 + G0 VG0 + G0 VG0 VG0 (6.78) G = G0 + G0 VG0 + G0 VG0 VG0 + G0 VG0 VG0 VG0 + · · ·

74

Models of Itinerant Ordering in Crystals

Gkk ′ Gk0δkk ′ Gk0 Gk0′ × + = i

Gk0 +

× i

Gk0″

× j

Gk0′

Gk0 +

× i

Gk0″

× j

Gk0″′

× l

Gk0′

+…

FIGURE 6.4 Schematic diagram for the full Green function as the sum of free electron propagators scattered on the perturbation potential.

The full Green function, which is represented in Fig. 6.4 by the bold line with an arrow, will be the sum of the diagrams corresponding to subsequent terms in the above equation. The function G0k ≡ 0 Gk in the momentum representation, also called the free electron propagator, is represented by the thin line with an arrow (Fig. 6.4). Each scattering on the potential is depicted by the cross with the lattice site index. Using this series different approximate calculations of the full Green function, Gkk ≡ Gkk , will be introduced. As a ﬁrst step one can re-establish the Hartree–Fock approximation from Section 6.2. This approximation is obtained in the ﬁrst-order expansion of the Dyson equation: G1 = G0 + G0 VG0

(6.79)

Assuming that the average of the full Green function is equal to the unperturbed Green function, G1 ≈ G0 , one has 0 ≈ G0 V G0

or

0 ≈ V = Vi −

(6.80)

where the potentials and probabilities for the Hubbard model are given by (6.69). Hence one has ≈ Vi = Uni−

(6.81)

which is the well-known result for the effective ﬁeld in the H–F approximation [5.1]. The self-energy obtained in this way is a constant, ≡ , and it is real. This will cause, as was seen earlier [relation (6.31)], the DOS, , to be only shifted in energy by the amount of the average ﬁeld Uni− . The shape of the DOS remains unchanged by this approximation. As a result even a strong Coulomb repulsion U cannot split the band into the lower and higher subbands. It is an obvious oversimpliﬁcation of the effect of electron correlations in the itinerant band. Mathematically, as was seen above, the H–F approximation came as a result of using the ﬁrst term in Dyson equation. Now the CPA will be derived, as the solution of an inﬁnite series of terms in the Dyson equation (6.78). The approximation, used so far, was the replacement of the electron number operator by the stochastic two-value potential. Inserting the perturbation potential, V=

1

V − N i i

Different Approximations for Hubbard Model

75

into the series (6.78) and transforming it into the momentum space one obtains the expression Gkk = G0k + G0k + G0k

1

Vi − G0k N i

1

Vi − eik−k ·ri G0k 2 N ijk

(6.82)

× Vj − eik −k·rj G0k + · · · Comparing this expression with Fig. 6.4 one can assign into each propagator line, entering the given vertex i, the phase factor eik·ri and into the line leaving this vertex, the factor eik·ri ∗ = e−ik·ri . Assuming (after [5.4]) the single-site approximation (SSA), meaning that each time the scattering takes place on the same lattice site: i = j = k = l = · · · , one obtains Gkk G0k + G0k + G0k + G0k

1

Vi − G0k N i

1

V − G0k Vi − G0k N 2 ik i

(6.83)

1

V − G0k N 3 ik k i

× Vi − G0k Vi − G0k + · · · This expression can be simpliﬁed to the form 2 1 2 1 2 Gkk G0k + G0k

Vi − + G0k

Vi − F N i N i (6.84) 0 2 1 3 2 + Gk

Vi − F + · · · N i where the Slater–Koster function, F , is deﬁned as F =

1 0 1 0 1 1 Gk ≡ Gk = N k N k N k − k −

(6.85)

76

Models of Itinerant Ordering in Crystals

The self-energy in (6.84), , is adjusted to approximately fulﬁl the equality relation between the average of the full Green function and the unperturbed Green function, Gkk ≈ G0k , obtaining 0=

1 1 2

Vi − +

Vi − F N i N i 1 3 +

Vi − F 2 + · · · N i

(6.86)

After inserting Vi − F = qi one has 0=

1

Vi − 1 + qi + qi2 + qi3 · · · N i

(6.87)

For the geometrical series one can write that 1 + qi + qi2 + · · · = 1/1 − qi (assuming that qi < 1), hence 1 Vi − = 0 N i 1 − Vi − F

(6.88)

If the probability of ﬁnding a given potential Vi is equal to Pi [as in (6.72)], then the last expression can be written as i

Pi

Vi − = 0 1 − Vi − F

(6.89)

For i = 2 this equation has the form P1

V1 − V2 − + P = 0 2 1 − V1 − F 1 − V2 − F

(6.90)

where, in the case of the electron correlation in the pure itinerant band and after adding to the stochastic potential the H–F ﬁeld, Vi and Pi are given by (6.72). Equation (6.90) allows us to ﬁnd the self-energy, and later on, the density of electron states in the presence of perturbation. Its applications can be different. It can be easily recast into the original form developed by Soven [6.2]: = ¯ − V1 − F V2 −

(6.91)

where ¯ is the average energy given by ¯ = P1 V1 + P2 V2

(6.92)

Different Approximations for Hubbard Model

77

or to the form obtained by Velický et al. [5.4]: = ¯ +

V2 − V1 2 P1 P2 F 1 + + ¯ − V1 + V2 F

(6.93)

The self-energy calculated on the basis of the above equations allows us to ﬁnd the change in DOS due to perturbation. The CPA approximation was introduced by Soven [6.2], Taylor [6.3] and Velický et al. [5.4]. Its broadest application was to the disordered binary alloys and to the description of the electron correlation in pure itinerant systems. How to ﬁnd the density of states in the general case of self-energy having a ﬁnite imaginary part? It will be proved below that 1 = − Im F

(6.94)

Proof: The complex self-energy can be written as the sum of the real and imaginary parts: = R − iI From the deﬁnition of Slater–Koster function (6.85) one has F =

=

1 1 1 1 = N k − k − N k − k − R + iI 1 − k − R − iI N k − k − R 2 + I 2

hence 1 1 1 I − Im F =

N k − k − R 2 + I 2 =

1 L − k − R = N k

where L − k − R is the Lorentzian function normalized to unity. In particular, when ⇒ i0+ [i.e. R = 0 and I ⇒ 0+ ], one obtains the previously introduced (6.17): I 1 1 1 1 ⇒ − k = 0 = − ImF0 2 2

N k − k − R + I N k (6.95) where F0 is the unperturbed Slater–Koster function given by (6.18).

78

Models of Itinerant Ordering in Crystals

Comparing (6.18) with (6.85) one can write another important relation: F = F0 −

(6.96)

This expression allows a physical comparison of the perturbed DOS (6.94) with the unperturbed DOS (6.17) after calculating the self-energy. Example of electron correlation in pure elements: Inserting potentials Vi and probabilities Pi given by (6.69) into (6.91), one obtains the relation − + n− U + U − F = 0

(6.97)

The above formula can be easily cast into another popular form [6.4] =

Un− 1 − F U −

(6.98)

Both these forms are simpliﬁed due to the assumption that V1 = 0 and V2 = U . To illustrate changes in DOS, created by the correlation U treated in the CPA approximation, the initial (unperturbed) semi-elliptic DOS given by (6.56) is used, for which the Slater–Koster function has the form 2

− 2 − D2 1/2 (6.99) D2 where D is the half bandwidth of the unperturbed band. Using relation (6.96) and this DOS one obtains F0 =

= −

D2 1 F − 4 F

(6.100)

Inserting the last relation into (6.97) one formulates a third-order algebraic equation for the Slater–Koster function, F : D4 D2 D2 3 2 2 − F + 2 − U F + U − − F + n− − 1U + = 0 16 4 4 (6.101) This equation may be solved for real energy, , resulting either in three real roots or in one real root and a complex pair of functions F . The complex function, F , with the negative imaginary part will give a positive DOS calculated from (6.94). The equation (6.101) is solved for a given electron occupation in the paramagnetic case. Dependence of the DOS per spin on Coulomb interaction, U , and carrier concentration, n, is shown in Figs 6.5 and 6.6, respectively. At small U the band is not split but is deformed, showing two local maxima around energy 0 and U . For U exceeding the half bandwidth the band is split into two sub-bands localized around energy 0 and U . The band’s width and height depend on electron concentration, as shown in Fig. 6.6. In the paramagnetic state, the total capacity of the lower spin band is 1 − n/2 and of the upper spin band is n/2.

Different Approximations for Hubbard Model

ε

U1 = 2D U2 = 0.8D U=0

U1

U2

0

ρ (ε) FIGURE 6.5 The DOS dependence on Coulomb correlation U for the initial semi-elliptic DOS, calculated in the coherent potential approximation for n = 15.

ε

U

n = 0.5 n = 1.5

0

ρ (ε) FIGURE 6.6 The DOS calculated in the coherent potential approximation for the initial semi-elliptic DOS and different values of electron concentration n at U = 2D.

79

80

Models of Itinerant Ordering in Crystals

6.5.1 Relation between CPA, Hubbard III and extended Hubbard III approximations It was shown by Velický et al. [5.4] that the CPA approximation is equivalent to the Hubbard III approximation under the following changes of variables between Section 6.4 on Hubbard’s solution and this section: Gk → 0 Gk FH → − k − t0 → k

(6.102)

n− − → P1

− → V1

n+ − → P2

+ → V2

U ≡ + − − = V2 − V1 With this identiﬁcation it is straightforward to demonstrate that the Hubbard’s relations (6.68) and (6.65) are identical to the CPA relations (6.73) and (6.93) or (6.91). It has been shown [6.4] that the CPA approximation with potentials and probabilities given by (6.69) does not bring about the ferromagnetic ground state. Therefore the above identiﬁcation of CPA with Hubbard III approximation shows that the Hubbard III approximation is also paramagnetic. The Hubbard III approximation with correction for the inter-site correlation I = 0 is given by relation (6.B50). Using the equivalence (6.102) it can be translated into the following expression in the CPA language: =

Un− + UST F 1 − F U −

(6.103)

This expression is an extension of the standard CPA relation (6.98). It has an extra term in the numerator, responsible for the band shift, T ST . The bandwidth correction, Bk , is included in F through (6.167), derived below in Section 6.9.2 for the case of CPA with a variable bandwidth. The result of formula (6.103) for ferromagnetism should be analysed with great care. It will be compared with the SDA described in Section 6.6. It will also be covered brieﬂy in Chapter 7 on itinerant ferromagnetism, where it will be shown that at strong electron correlation U >> D formula (6.103) leads to ferromagnetic instability in large intervals of concentrations.

Different Approximations for Hubbard Model

81

6.5.2 Different applications of the CPA One example is the description of the correlation effects in the pure material presented above. The stochastic potential, according to (6.69), takes on values 0 and U with probabilities 1 − ni− and ni− , respectively, where in the case of an uniform magnetic ordering one has ni = n . Unfortunately, as already shown [5.4, 6.4], the ground state of this model is not magnetic. This has forced researchers to introduce an additional interaction in the H–F approximation [see Section 5.2 and the expression (6.72)], which has the physical interpretation of the Weiss ﬁeld (see Section 7.2). Of course after adding the Weiss ﬁeld one can always obtain the ferromagnetic state, but the question is at what values of the interaction constant, and whether these values are small enough to justify the use of H–F approximation. This problem will be analysed in great detail in Chapter 7. Another example to which one may apply the Green function decoupling given by (6.90) is the binary substitutional alloy Ax B1−x (see Chapter 9). In some of these alloys, the electronic density of both components is so similar that it can be treated as identical, which implies the same electron dispersion relation, k , and the same unperturbed part of the Hamiltonian, H0 . The components of the alloy have different average energies of the band: A B . To these energies the Coulomb interaction in the H–F approximation is added. As a result the perturbation takes on the form of the following stochastic potential: − UA nA− x with probabilities Pi = (6.104) Vi = A B − UB nB− y where x and y = 1 − x are the concentrations of A and B atoms in the Ax B1−x alloy, respectively. We can now use the same equation (6.90) with the above potentials and probabilities to calculate the self-energy and later the DOS in the alloy, after assuming the density for pure components. Yet another example of CPA is the use of relation (6.90) for binary alloys with the electron correlation described also within the CPA decoupling. The straightforward extension of the relation to this problem, with stochastic potential taking on four values with corresponding probabilities, is not the right approach since the alloy potentials and stochastic Coulomb potentials have to be treated differently. By the correct approach one ﬁnds ﬁrst the coherent potential describing correlation on each component of an alloy, Ui , i = A B and in the next step this potential is added to the alloy’s stochastic potential, Vi = i + Ui , i = A B . The details of the method can be found in [6.4].

6.6 SPECTRAL DENSITY APPROACH The Hubbard model is the basic model used to describe strongly correlated systems. The H–F approximation of the Hubbard model yields simple results,

82

Models of Itinerant Ordering in Crystals

but it overestimates the ordering and the results may be valid only for systems with a weak interaction constant. The Hubbard I approximation may be used for systems with strong correlation but it fails at U < D, since it produces an energy gap at any U . Among the relatively simple approximations, the best seems to be the CPA approximation which describes relatively well these systems with strong and weak correlations. Unfortunately this approximation (without additional H–F ﬁeld coming from either on-site or inter-site interactions) does not bring magnetic ordering [5.5, 6.4]. The approximation best describing the strongly correlated systems and bringing magnetic ordering is the SDA [6.5, 6.6] introduced by Nolting and co-workers. The basis of this approximation is the Roth’s two-pole approximation [6.7], which gives the two-pole ansatz for the single-particle spectral density function. Such an approach is well justiﬁed for systems with strong correlation. In the Hubbard I approximation for systems with D = 0 [see (6.46)], the spectral density consists of two weighted -functions localized at the two energies: t0 and t0 + U . The SDA method is based on the two-pole ansatz for the spectral density: SDA Sk =

2

i k − SDA i k

(6.105)

i=1

Using spectral weights, (6.105), the spectral moments can be calculated as n

Mk

+ −

SDA n Sk d

(6.106)

This will allow the calculation of the free parameters i k and SDA i k by ﬁtting the results to the moments calculated exactly from the expression below:

1 −ik·ri −rj n + Mk = e

ci H− H− H H cj − − (6.107) N + n−p-times p-times where p is an integer between 0 and n. For the Hubbard model expressed by Hamiltonian (6.1), the ﬁrst four moments, given by (6.107), will be equal to 0

(6.108)

1

(6.109)

2

(6.110)

Mk = 1 Mk = k + Un− Mk = 2k + 2Un− k + U 2 n−

3 Mk = 3k + 3Un− 2k + U 2 2n− + n2− k + U 2 n− 1 − n− Bk− + U 3 n− (6.111)

Different Approximations for Hubbard Model

83

The term Bk− in the third moment consists of higher correlation functions − n− 1 − n− Bk− − t0 = BS− + BD k

(6.112)

Function BS− (index S stands for the band shift) depends on the electron spin and is not dependent on the wave vector k: BS− =

+ 1 −tij ci− cj− 2nˆ i − 1 N

(6.113)

i=j

− BD k

(index D stands for the band deformation) depends also on Function the wave vector k and is given by 1 + + + + − BD k = −tij e−ik·ri −rj nˆ i− nˆ j− − n2− − cj cj− ci− ci − cj ci− cj− ci N i=j

(6.114) BS− ,

called the spin-dependent band shift, causes exchange splitThe term − ting between the spin-up and spin-down spectrum. The term BD k , called the bandwidth correction, leads to a change in the width of the spin sub-bands with respect to each other. It has three parts which are interpreted as density correlation, double hopping and spin exchange. The mean value of the translationally invariant function + + + + nˆ i− nˆ j− − n2− − cj cj− ci− ci − cj ci− cj− ci = fi − j

(6.115)

is constant

+ + + + BD− = fi − j = nˆ i− nˆ j− − n2− − cj cj− ci− ci − cj ci− cj− ci (6.116)

and the k-dependence in (6.114) can be separated as − BD k = k − t0 BD−

(6.117)

The analogous expressions for bandwidth and band shift were also obtained from the extended Hubbard III approximation of Section 6.4. For the band shift the following formula [relation (6.67)] was arrived at:

T T S = FH C− − −C− FH0 til Wlmi tmi nˆ i− nˆ l− − n2− l=im

+

− + − − tmi tin Wlmi Iln− − Wlmi − + + − Iln−

lmn

−FH0 C−

l

+ − + − 2til nˆ l cl− ci− − FH0 C+ til cl− ci− l

(6.118)

84

Models of Itinerant Ordering in Crystals

Assuming that FH C− − T ≈ −1, FH0 C− ≈ −1, FH C+ ≈ +1, and ignoring the ﬁrst two terms one has ST = −

1 + − 1 + − 2til nl cl− ci− + t c c N li N li il l− i−

+ − 1 = −til cl− ci− 2nˆ l − 1 N li

(6.119)

This formula for the band shift is identical to relation (6.113) of SDA under exchange: ST ≡ BS− . In the case of the bandwidth correction, the formula derived by the extended Hubbard III approximation [see (6.66)] has the form

T Bk = FH C− − T C− FH0 −tlm nm− nl− − n2− lm

+ + + + − cl cm− cl− cm − cl cl− cm− cm e−ik·rm −rl Using, formula, the same approximation as in this FH C− − T ≈ −1, FH0 C− ≈ −1, one has

+ + T = −tlm nˆ m− nˆ l− − n2− − cl Bk cm− cl− cm lm

+ + cl− cm− cm e−ik·rm −rl − cl

(6.120) before:

(6.121)

This relation is identical to (6.114) of SDA for the bandwidth broadening T − assuming that Bk ≡ BD k . The dependence of the effective band shift, BS− /1 − n− , and the effective bandwidth correction, BD− /1 − n− , on electron occupation, n (Fig. 6.7) was calculated in the SDA approximation by Herrmann and Nolting [6.8] in the strong correlation limit for 3D bcc lattice with D = 2 eV. Note that BS− /1 − n− is the effective band shift of the lower Hubbard sub-band with spin . The same holds for the effective bandwidth BD− /1 − n− . Comparing moments from relations (6.108)–(6.111) with those from (6.105) and (6.106) one obtains the quasi-particle energies 1/2 ! 1 − 2 − − SDA (6.122) = B + U + ∓ + U − + 4Un − B B k k − k 12 k k 2 k with the corresponding spectral weights 1 k =

− SDA 1 k − Bk − U1 − n− SDA 1 k − SDA 2 k

2 k = −

− SDA 2 k − Bk − U1 − n− SDA SDA 1 k − 2 k

(6.123)

(6.124)

85

Different Approximations for Hubbard Model

1.0 BS–σ /(1 – n–σ) PM

BS–σ /(1 – n–σ) FM σ =

BS–σ /(1 – n–σ) (eV)

0.8

–σ

BS /(1 – n–σ) FM σ =

0.6

U = 5.0 eV

0.4

0.2

BD–σ /(1 – n–σ)

0.0

BD–σ /(1 – n–σ) PM

–0.1

BD–σ /(1 – n–σ) FM σ = –σ

BD /(1 – n–σ) FM σ =

–0.2 0.0

0.2

0.4

0.6

0.8

1.0

n

FIGURE 6.7 Effective bandshift, B− S /1 − n− , and effective bandwidth correction, B− D /1 − n− , as a function of band occupation, n, for the paramagnetic (PM) and the ferromagnetic (FM) phases, for 3D bcc lattice with D = 2 eV, U = 5 eV and T = 0 K.

It is also interesting to look at the self-energy, SDA k , which is related to the spectral density throughout the relation SDA = − SDA k Sk

(6.125)

Comparing (6.125) with (6.105) and using (6.122)–(6.124) one obtains the self-energy given by the formula SDA k = Un−

− Bk− − U1 − n−

− Bk−

(6.126)

Relations (6.122)–(6.124) and (6.126) were simpliﬁed, after assuming that − the band deformation term BD k is negligible in the magnetic problems [6.6, 6.8]. This brought the following expression for the self-energy: SDA ≡ SDA k = Un−

− BS− / n− 1 − n− − BS− / n− 1 − n− − U1 − n−

(6.127)

86

Models of Itinerant Ordering in Crystals

where the band shift factor, BS− , can be calculated from the equation 1 2 SDA BS− = k − t0 Sk− (6.128) − k − 1 fd N k U − with f being the Fermi function. The approximate self-energy (6.127) is real, spin dependent and does not depend on the wave vector. Assuming that BS− = 0, one obtains Hubbard I self-energy (6.44). − The assumption of BD k ≡ 0 was oversimpliﬁcation, since it is now known [5.6] that the deformation of the band shape can signiﬁcantly inﬂuence ferromagnetism. The full problem is described by (6.105), (6.122)–(6.124) and (6.128), which form a close system of self-consistent equations. Use of the SDA approximation for the magnetism gives at some concentrations and effective Coulomb coupling U/D solution with a band shift different for + and − electrons, yielding spontaneous magnetization. The defect of the SDA is the real self-energy which completely ignores the quasi-particle damping.

6.7 MODIFIED ALLOY ANALOGY Classical CPA approximation applied to the standard Hubbard model (see Section 6.5) does not describe the magnetic ordering since the self-energy from (6.90) or (6.91) does not depend on the spin. Another defect of the CPA is the inability to reproduce the exact strong-coupling limit (U → ) of Harris and Lange [6.9] and Potthoff et al. [6.10]. The SDA approximation (see Section 6.6) gives the magnetic results in the strong-coupling limit, but as was mentioned above, it neglects the quasiparticle damping. Utilizing ideas from both the CPA and SDA approximations Nolting and co-workers proposed the modiﬁed alloy analogy (MAA) method. MAA In this method, they used CPA equations with two centres of gravity, Vi , SDA which are the approximate SDA energies, i k, in the limit of k → t0 . The self-energy in the MAA approximation is calculated from the classic CPA equation (6.90) with the potentials Vi and probabilities Pi obtained from the SDA method in the atomic limit of k → t0 : SDA 1 BS− MAA Vi = i k →t = + U + t0 + −1i k 0 2 n− 1 − n− ⎫ (6.129)

2 1/2 ⎬ − − BS BS × + U − t0 + 4Un− t0 − ⎭ n− 1 − n− n− 1 − n− MAA

P1

= 1 kk →t0 =

V1 − BS− / n− 1 − n− − U1 − n− MAA = 1 − P2 V1 − V2 (6.130)

Different Approximations for Hubbard Model

87

These expressions include the band shift BS− . Inserting relations (6.129) and (6.130) into the CPA equation (6.90) one obtains the MAA self-energy MAA =

Un− 1 − F BS− 1 − F U + BS− − MAA

(6.131)

with the band shift, BS− , given by the expression BS−

1 2 = Im f − 1 − MAA − t0 F − 1 d (6.132)

U MAA −

If BS− is replaced by 0, the MAA formula (6.131) reduces to the conventional CPA of (6.98). In all other cases, the MAA is different from the CPA. In particular, via the band shift the atomic levels might now become spin dependent. While the MAA is correct in the strong-correlation regime, there is a severe drawback of the method: it fails to reproduce the Fermi-liquid properties for small interactions U . The same defect is inherent in the conventional alloy analogy. According to Herrmann and Nolting [6.11], the MAA method has a selfconsistent ferromagnetic solution (see Fig. 6.8), but in a rather small region of the band ﬁlling for which the chemical potential is located in the vicinity of high quasi-particle DOS (see Fig. 6.9).

2 MAA MAA CPA

ρ (ε) (1/eV)

1.5 1 0.5 0 –1 –0.5

0

0.5

εFCPA εFMAA

1

ε (eV)

9.5

10

10.5

11

FIGURE 6.8 Quasi-particle density of states calculated in the MAA, as a function of energy for band occupation n = 066 and magnetic moment m = 014 for T = 0 K. Solid lines are for up-spin spectra, broken lines are for down-spin spectra. Bars on the -axis mark the Fermi edge. For comparison the conventional alloy analogy results are plotted as dotted lines. Further parameters: U = 10 eV, D = 1 eV, bcc-Bloch density of states from [4.4]. After [6.11]. Reprinted with permission from T. Herrmann and W. Nolting, Phys. Rev. B 53, 10579 (1996). Copyright 2007 by the American Physical Society.

88

Models of Itinerant Ordering in Crystals

1.0

0.8

U = 5 eV U = 10 eV U = 20 eV U = 30 eV

SDA

m

0.6 MAA

0.4

0.2

0.0 0.4

0.5

0.6

0.7 n

0.8

0.9

1.0

FIGURE 6.9 Magnetic moment m as a function of the band occupation n, for various values of the Coulomb repulsion U. Parameters: bcc lattice, D = 1 eV, T = 0 K. Reprinted with permission from T. Herrmann and W. Nolting, Phys. Rev. B 53, 10579 (1996). Copyright 2007 by the American Physical Society.

As can be seen from Fig. 6.8, contrary to the normal CPA results for the Hubbard model, the MAA produces a self-consistent ferromagnetic solution. The results of the MAA formula (6.131) for magnetism should be compared with the results of the extended Hubbard III formula (6.98). The last one includes the inﬂuence of both band shift and bandwidth change. Further analytical and numerical development in this area is very interesting, but it exceeds the scope of this textbook and will not be described below or in Chapter 7.

6.8 DYNAMICAL MEAN-FIELD THEORY The dynamical mean-ﬁeld theory (DMFT) approximation is used in ﬁnding the solution to the original Hubbard model (6.1). The main point of this method is to formulate and solve the single-site problem. This initial model is mapped as the effective impurity model, which describes a single correlated impurity orbital embedded in an uncorrelated bath of conduction band states. This mapping is a self-consistent one, namely that the bath parameters depend on the on-site lattice Green function. The DMFT method is exact in the non-trivial limit of inﬁnite spatial dimensions d = . Transition to d = requires scaling of the hopping integral as t∗ t= √ d

(6.133)

Different Approximations for Hubbard Model

89

In the d-dimensional cubic lattice, the electron dispersion relation can be written as d 2t∗ k = √ cos k a d =1

(6.134)

Corresponding to this dispersion is the Gaussian density of electron states: 2 1 0 = √ ∗ exp − 2t∗ 2 t

(6.135)

In the case of d = , the self-energy obtained in the DMFT method becomes a purely local quantity (a single-site) ij = ij

(6.136)

which, after the Fourier transform, gives the momentum-independent selfenergy k =

(6.137)

The electron Green’s function in (k representation Gk =

Gij exp ik · ri − rj

(6.138)

ij

can be written as Gk =

1 − k −

(6.139)

To compute the self-energy one considers an auxiliary impurity problem with the effective single-site action: Seff = −

0

0

+ c0 G−1 0 − c0 d d + U

0

n0↑ n0↓ d

(6.140)

Here G0 plays the role of a bare Green’s function for the local effective action Seff . This function contains the information from all the other sites which have been integrated. G0 does not coincide with the non-interacting site-diagonal Green’s function of the Hubbard model. Solving the effective impurity problem one obtains the Green’s function, G , given by the expression G = c+ cSeff =

1 G−1 0 −

(6.141)

90

Models of Itinerant Ordering in Crystals

Dyson equation

Σ

G0

Impurity problem

FIGURE 6.10 The schematic diagram of the DMFT self-consistency solution of a many-body problem.

where G =

1 G N k k

(6.142)

The single-site problem (6.140) can be solved by different analytical methods, such as the iterated perturbation theory and non-crossing approximation, or using the numerical techniques such as quantum Monte-Carlo simulations or the exact diagonalization. For the review of these methods we advise reading paper [6.12], and for the applications to superconductivity, magnetism, and Mott transition the papers [6.13, 6.14]. The single-site problem (6.140) solved self-consistently with the use of Dyson’s equation (6.141) gives the DMFT solution of a given many-body problem. The schematic diagram of this procedure is the following (see Fig. 6.10): (1) (2) (3) (4) (5) (6)

Choose an initial Weiss function G0 . Calculate the impurity effective action (6.140). Calculate G from G = c+ cSeff . Calculate the self-energy from (6.142). Using the Dyson relation (6.141) calculate the new Weiss function G0 . Iterate until the self-energy will reach convergence, and G ≈ G0 .

6.9 HUBBARD MODEL EXTENDED BY INTER-SITE INTERACTIONS 6.9.1 Modified Hartree–Fock approximation This approximation is applied only to the inter-site interactions, therefore it is used in the case of the Hubbard Hamiltonian from Section 5.2 extended to the

91

Different Approximations for Hubbard Model

inter-site interactions. The Hamiltonian for the one-band model can be written in the form [6.15] (see Section 5.2) H =−

+ tij ci cj − 0

nˆ i − F

i

n nˆ i +

i

U nˆ nˆ 2 i i i− (6.143)

+ + V J + + + nˆ i nˆ j + ci cj ci cj + J ci↑ ci↓ cj↓ cj↑ 2 2 with the generalized hopping integral given by

tij = t1 − nˆ i− 1 − nˆ j− + t1 nˆ i− 1 − nˆ j− + nˆ j− 1 − nˆ i− + t2 nˆ i− nˆ j− (6.144) Including the occupationally dependent hopping given by (6.144) into the Hamiltonian (6.143) one obtains [see (5.22)] the following result: + H =−

t − tnˆ i− + nˆ j− + 2tex nˆ i− nˆ j− ci cj − 0 nˆ i − F n nˆ i

i

i

(6.145) + + U V J + + + nˆ nˆ + nˆ nˆ + c c c c +J ci↑ ci↓ cj↓ cj↑ 2 i i i− 2 i j 2 i j i j where as in Chapter 5 one deﬁnes the hopping interaction, t, and the exchange-hopping interaction, tex , as t + t2 − t1 (6.146) 2 In the form (6.145), it is quite obvious that the interactions t, tex , V , J , J are the inter-site interactions. The on-site Coulomb repulsion will be set aside in this section and only inter-site terms will be considered in the % Hamiltonian (6.145). The on-site exchange interaction term, −F i n nˆ i , is already expressed in the H–F approximation with the constant of the ﬁeld, F , given as the sum of different on-site interactions by (5.20). For all the above interactions, which are of the inter-site type, the classic H–F approximation is given by (6.34) t = t − t1

tex =

nˆ i nˆ j ≈ ni nˆ j + nˆ i nj − ni nj

(6.147)

In the modiﬁed H–F approximation ([5.6, 6.16] and Appendix 6C), which also includes the inter-site averages, one has the following form: + + nˆ i nˆ j = ci ci cj cj ≈ nˆ i nˆ j + nˆ i nˆ j − nˆ i nˆ j

+ + + + + + − cj ci ci cj − cj ci ci cj + cj ci ci cj + + = ni nˆ j + nˆ i nj − I ci cj − I cj ci + const

(6.148)

92

Models of Itinerant Ordering in Crystals

where + the on-site and inter-site averages are deﬁned as ni = nˆ i and I = ci cj . As one can see, this approximation will contribute not only to the H–F ﬁeld, but also to the bandwidth (the terms with constant I ). To make practical use of this approximation needs the prescription of how to calculate the + one inter-site average I = ci cj . The method is as follows. The parameter I , according to its deﬁnition, is proportional to the average kinetic energy of electrons with spin : + + K = −t ci cj = −tz ci cj = −DI (6.149) ij

The average kinetic energy, K , can be also written as K =

D −D

fd

(6.150)

Comparing the above equations the parameter I can be written as

I =

d − d D 1 + eb −+M /kB T −D D

(6.151)

This expression can be simpliﬁed further by assuming zero temperature and the rectangular DOS, 0 = 1/2D. One then obtains I± = 0

± F −D

−

d D

where n± = 0

± F

−D

d

(6.152)

Hence one can prove that I± = n± 1 − n±

(6.153)

In this simpliﬁed form, the parameter I± gains the physical interpretation of the probability for the electron with spin ± moving from ith to jth lattice site. More precisely we should write that I = ni 1 − nj nj 1 − ni 1/2

(6.154)

which in the case of ferromagnetism will produce relation (6.153), and in the case of antiferromagnetism the relation IAF =

2n − n2 − m2 n n 1− 2 41 − m 2 2

(6.155)

In Sections 7.6 and 8.4, this physical quantity will be estimated more precisely in the case of the band being split by the strong Coulomb correlation U >> D, for the ferromagnetic and antiferromagnetic ordering.

Different Approximations for Hubbard Model

93

After applying the modiﬁed H–F approximation to all the inter-site interactions, t tex J J V (see Appendix 6C), one obtains the following simpliﬁed Hamiltonian [see (6.C10)]: + + + H =− teff ci cj − 0 nˆ i + Mi nˆ i + U nˆ i↑ nˆ i↓ + a1 ci↑ ci↓ + hc

+

i

i

i

+ + + + + + + + a2 ci↑ cj↓ − ci↓ cj↑ + hc + a3 ci↑ cj↓ + ci↓ cj↑ + hc

+

i

(6.156)

+ + a4 ci cj + hc

= tb is the effective hopping integral, with the bandwidth modiﬁwhere cation factor b [see (6.C11)] given by teff

b = 1 −

1 2 tni− + nj− − 2tex ni− nj− − I− − 2I I− t 2

(6.157)

− 0 2 − ij 2 + − + J − VI + J + J I− The molecular ﬁeld Mi for electrons with spin is expressed as [see (6.C12)] Mi = − Fni − J j nj + V j nj + nj− + 2ztI− (6.158) − 2tex j 2I− nj + ij ∗0 + ∗ij 0 % where z is the number of the nearest neighbours, j is the sum over the nearest neighbours of the lattice site i. The molecular ﬁeld is the sum of the on-site contribution, F , and the inter-site contributions, which will be different for F and AF. The last four terms in (6.156) describe the on-site singlet, inter-site singlet, opposite spin triplet and equal spin triplet superconductivity. The total energy gaps for the on-site singlet, inter-site singlet, opposite spin triplet and equal spin triplet superconductivity are, respectively, equal to (see Appendix 6C) a1 = 2ztSij + zJ 0 − 2ztex nSij − I− + I 0 (6.159) 1 a2 = t0 + V + J Sij − tex n0 − I− + I Sij 2 1 a3 = V − J + tex I− + I Tij 2 1 a4 = V − J − 2tex I− 2

(6.160) (6.161)

(6.162)

94

Models of Itinerant Ordering in Crystals

where 0 = ci↓ ci↑ ij = cj↓ ci↑

1 Sij = cj↓ ci↑ − cj↑ ci↓ 2

1 Tij = cj↓ ci↑ + cj↑ ci↓ 2

= cj ci

and

(6.163)

6.9.2 Coherent potential approximation for the extended Hubbard model In this case, one has to refer back to the full Hamiltonian (6.156) and consider the on-site Coulomb repulsion as creating the stochastic atomic potential of values 0 and U , with the corresponding probabilities given below [see (6.72)]. All inter-site interactions, which are weaker, are treated in the modiﬁed H–F approximation (see previous section). As before [see (6.2) and (6.70), (6.71)], the Hamiltonian can be split into its unperturbed and perturbed parts: H = H0 + Vpert which are now given by

H0 = −

Vpert =

+ teff ci cj +

(6.164)

nˆ i

i

Vi − nˆ i

(6.165)

i with teff = tb [see (6.156)], and the stochastic potential and probabilities given by the following expression: Mi 1 − ni− with probabilities Pi = (6.166) Vi = U + Mi ni−

The molecular ﬁeld for + electrons on site i, Mi , is given by (6.158). This is the result of applying the modiﬁed H–F approximation to all interactions other than U . The formula (6.166) is the direct extension of (6.72) in the case of the Weiss ﬁeld resulting from different interactions. To calculate the self-energy, , the classic CPA equation (6.90) is used with the stochastic potential given by (6.166). The Slater–Koster function in the modiﬁed CPA approach has the form 1 1 1 − F = = F (6.167) N k − k b − b 0 b where F0 is given by (6.18). The spin DOS, , is calculated as before from the relation 1 = − ImF

(6.168)

Different Approximations for Hubbard Model

95

APPENDIX 6A: EQUATION OF MOTION FOR THE GREEN FUNCTIONS For two operators At and Bt in the Heisenberg representation, At = eiHt A0e−iHt (for simplicity it is assumed that = 1), one can write the Green function at zero temperature as GA B t − t = At Bt = −iT AtBt

(6.A1)

where the symbol is used for the ground state. The symbol T is the Dyson’s time-ordering operator, which acting on the time-dependent operators orders them in decreasing time: T Vˆ t1 Vˆ t2 Vˆ t3 = Vˆ t3 Vˆ t1 Vˆ t2

for t3 > t1 > t2

(6.A2)

Introducing the step function ⎧ ⎪ for x > 0 ⎨1 x = 0 for x < 0 ⎪ ⎩ 1/2 for x = 0

(6.A3)

one can write T AtBt = t − t AtBt − t − tBt At

(6.A4)

The retarded Green’s function can be written as GR A B t − t = At Bt + = −it − t At Bt

(6.A5)

and the advanced Green’s function as GA A B t − t = At Bt − = it − t At Bt

(6.A6)

where A B = AB + BA, with = +1 for fermions and = −1 for bosons. To transfer from the time-dependent to the energy-dependent Green functions one has to use the Fourier transform GRA A B = A B± =i

−

At B0± eit dt

(6.A7)

Since the Green functions (6.A1), (6.A5) and (6.A6) depend on the time difference t − t , one can assume that t = 0. Differentiating Green functions

96

Models of Itinerant Ordering in Crystals

(6.A5) and (6.A6) over time t and using the relation dt/dt = t one obtains dGRA A B t dAt i (6.A8) = t At B0 ∓ i±t i B0 dt dt The operator At in the Heisenberg representation fulﬁls the equation of motion: i

dAt = At H− dt

(6.A9)

Using (6.A9) one can write (6.A8) in the form i

dGRA A B t = t At B0 ∓ i±t

At H− B0 dt

(6.A10)

dAt B0± = t At B0 + At H− B0± dt

(6.A11)

or i

One can deﬁne for real the Fourier transforms 1 A B± = At B0± eit dt √ 2 −

(6.A12)

+

The retarded function, A B , is a regular function of in the upper − half of the complex plane. Similarly the advanced Green’s function, A B , is a regular function in the lower half of the complex -plane. One may deﬁne * + A B if > 0 * (6.A13) A B = − A B if < 0 which will be a regular function throughout the whole complex -plane except on the real axis. From (6.A11) it can be shown that A B satisﬁes A B = A B + A H− B

(6.A14)

For fermions = +1 the equation of motion in the energy representation is A B = A B+ + A H− B

(6.A15)

where the subscripts + and − are used for the anticommutation and commutation relations, respectively. This equation is frequently used throughout this book.

97

Different Approximations for Hubbard Model

APPENDIX 6B: HUBBARD SOLUTION FOR THE SCATTERING AND RESONANCE BROADENING EFFECTS 6B.1 The scattering effect To consider this effect we ignore the last term in (6.62) and search for the + solution of function nˆ i− − n− cl cj . Using (6.59) one can write this Green function as + + = (6.B1) nˆ i− − n− nˆ l− cl cj nˆ i− − n− cl cj

=±

+ The equation of motion for nˆ i− − n− nˆ l− cl cj has the following form: + = jl nˆ i− − n− nˆ l− nˆ i− − n− nˆ l− cl cj

+ + + nˆ i− − n− nˆ l− cl cj − tml nˆ i− − n− nˆ l− cm cj

(6.B2)

m

+ other terms where “other terms” come from the commutators nˆ i− H− and nˆ k− H , − which are ignored in the scattering effect. The average in the ﬁrst term on the right-hand side of (6.B2) can be written as

(6.B3) nˆ i− − n− nˆ l− = nˆ i− nˆ l− − n2− = Bil− where Bil− = nˆ i− nˆ l− − n2−

(6.B4)

In the original Hubbard model [6.3], this term was ignored. It will stay in here, since one can write the simple approximation + + + + 2 Bil− = nˆ i− nˆ l− − n2− ci− ci− cl− cl− − ci− cl− cl− ci− − n2− −I− (6.B5) + where I− = ci− cl− , and the quantity I− can be calculated in the simple way as in Section 6.9.1. + For the function nˆ i− − n− nˆ l− cm cj appearing in (6.B2), the follow ing approximation will be used + + n− nˆ i− − n− cm cj (6.B6) nˆ i− − n− nˆ l− cm cj

98

Models of Itinerant Ordering in Crystals

Inserting the approximations (6.B3) and (6.B6) into (6.B2) one obtains the relation + − nˆ i− − n− nˆ l− cl cj = Bil− jl

+ − tml n− nˆ i− − n− cm cj

(6.B7)

m

Dividing both sides of this equation by − and summing over = ± one has + = C− Bil− jl − nˆ i− − n− cl cj

1 + ˆ n − n c c t i i− − j FH0 il (6.B8) + tml nˆ i− − n− cm cj

−

1 FH0 m=i

where =

n+ n− − + − − + − −

(6.B9)

C± =

1 1 ± − + − −

(6.B10)

1 FH0

Using relation (6.52) one can write that Glj + jl = FH0

m=l

tml Gmj

(6.B11)

Inserting (6.B11) into (6.B8) one obtains + − C− FH0 Bil− Glj = nˆ i− − n− cl cj

1

+ − C− FH0 Bil− Gij nˆ i− − n− ci cj

−

t FH0 il

−

t FH0 m=i ml

1

(6.B12)

+ − C− FH0 Bil− Gmj nˆ i− − n− cm cj

Introducing notation + Xmi = nˆ i− − n− cm cj − C− FH0 Bil− Gmj

(6.B13)

Different Approximations for Hubbard Model

99

the relation (6.B12) will take on the following form: 1 Xli =− t X + t X FH0 il ii m=i ml mi

(6.B14)

which is analogous to equation (A1) from [5.3]. Hence one can apply Hubbard’s solution (Appendix A in [5.3]), which is Xli =−

Wlmi tmi Xii

(6.B15)

m

where Wlmi = glm −

gli gim gii

(6.B16)

and gij =

exp ik · ri − rj k

FH0 − k − t0

(6.B17)

Taking into account the deﬁnition (6.B13) and the solution (6.B15) one + the following form: obtains for the Green function nˆ i− − n− cl cj

+ = C− FH0 Bil− Glj nˆ i− − n− cl cj

−

Wlmi tmi

+ − C− FH0 Bil− Gij nˆ i− − n− ci cj

(6.B18)

m

The equation (6.B18) differs from the Hubbard’s expression for scattering effect [(26) in Hubbard III] by the ﬁrst and third terms on the right-hand side, which include the Bil− factor. Inserting the Green function from (6.B18) into (6.62), still ignoring the last term in (6.62), which will be dealt with in the next section on resonance broadening effect, one obtains the following equation for + the Green function nˆ i− ci cj

+

− − nˆ i− ci cj = n− ij − til Glj − Gij

l=i

+ SS Gij +

l=i

(6.B19) S Bil Glj

100

Models of Itinerant Ordering in Crystals

where =

til Wlmi tmi

(6.B20)

lm SS = −C− FH0

l=im

til Wlmi tmi nˆ i− nˆ l− − n2−

S Bil = C− FH0 −til nˆ i− nˆ l− − n2−

(6.B21) (6.B22)

Using the relation

+

+ − − + ˆ i− + nˆ − ˆ i− n− − nˆ − nˆ i− − n− = nˆ i− n+ − + n− − n− n i− = n i− n− (6.B23) one can write relation (6.B19) as + + − ˆ − nˆ i− ci cj + n+ n c c − i− i j

− n− −

+ nˆ + c c = n− ij − til Glj i− i j

+ SS Gij +

l=i

(6.B24)

l=i

S Bil Glj

Before proceeding further we will add to this equation the resonance broadening effect, which comes from the last term of (6.62).

6B.2 The resonance broadening effect To consider this effect one has to return to (6.62). In considering the scattering effect, we have kept the last but one term and ignored the last term. In the case of the resonance broadening effect, one does just the opposite; neglect the last but one term and include the last term in (6.62), for which it is assumed that + + + + ci− cl− ci cj = − cl− ci− ci cj (6.B25)

There will be no changes to the Hubbard’s procedure [5.3], which is reported below, other than retaining this term. After [5.3], the following notation is introduced − ci− destruction operator + ci− creation operator

(6.B26)

101

Different Approximations for Hubbard Model

∓ + ± To ﬁnd the Green function cl− ci− ci cj we have to use (6.59), allowing us to write that ± ∓ ∓ + + ± ci− ci cj = nˆ l cl− ci− ci cj cl−

(6.B27)

=±

∓ + ± The function nˆ l cl− fulﬁls the following equation of motion: ci− ci cj

+ ± ∓ ∓ ∓ + ± ± nˆ l cl− ci− ci cj = ij nˆ l cl− ci− − jl cl cl− ci− ci

−

∓ ∓ + + + ± + ± tml cl cm cl− ci− ci cj − cm cl cl− ci− ci cj

m

±

∓ + + ± ± ∓ tml nˆ l cm− ∓ tim nˆ l cl− ci− ci cj cm− ci cj

m

−

(6.B28)

m

∓ ∓ + + ± ± tim nˆ l cl− ci− + ± ± − ∓ nˆ l cl− cm cj ci− ci cj

m

The terms in (6.B28) will be approximated as follows: + ∓ ∓ + + ± ± cl cm cl− ci− ci cj cl ci cl− ci− Gmj

(6.B29)

+ ∓ ∓ + + ± ± cm Glj cl cl− ci− ci cj cm ci cl− ci−

(6.B30)

∓ + ± nˆ l cm− ci− ci cj

+ ∓ ∓ + ± ± n cm− ci− ci cj − cl ci cm− ci− Glj

(6.B31)

+ ± ∓ nˆ l cl− cm− ci cj

+ ± ± ∓ n Ilm− Gij − cl ci cl− cm− Glj

(6.B32)

+ ∓ ∓ ∓ + ± ± ± nˆ l cl− ci− cm cj nˆ l cl− ci− Gmj − cl cm cl− ci− Glj (6.B33)

where ± ∓ ± Ili− = cl− ci−

(6.B34)

Additionally the approximation (6.B11) will be used. Equation (6.B28), after including (6.B29)–(6.B34) and (6.B11), will take on the form

102

Models of Itinerant Ordering in Crystals

∓ ∓ + ± ±

− ± ± − ∓ nˆ l cl− ci− ci cj = nˆ l cl− ci− FH0 Gij

+ + ∓ ∓ ± ± − cl ci cl− ci− FH0 Glj − tlm cm ci cl− ci− Glj m

∓

+ ± ± ∓ tim n Ilm− Gij − cl ci cl− cm− Glj

(6.B35)

m

±

+ ∓ ∓ + ± ± Glj tlm n cm− ci− ci cj − cl ci cm− ci−

m

+

+ ∓ ± tim cl cm cl− ci− Glj

m

Dividing both sides of (6.B35) by − ± ± − ∓ and using the property (6.B27) one obtains % % ± ∓ + ± ci− ci cj ∓ tim Ilm− Gij ± tlm cm− m m ∓ + ± cl− ci− ci cj = − FH0 − ± ± ∓ +

± ∓ nˆ l cl− ci− F Gij − ± ± − ∓ H0

+ ∓ ± − C− − ± ± ∓ cl ci cl− ci− FH0 Glj

− C− − ± ± ∓

(6.B36)

+ + ∓ ∓ ± ± ± cl Glj tlm cm ci cl− ci− ci cm− ci−

m

−

+ ∓ + ± ± ∓ tim cl cm cl− ci− ± cl ci cl− cm− Glj

m

The last term in (6.B36) can be written as + + ∓ ∓ ± ± tlm cm ci cl− ci− ± cl ci cm− ci− Glj m

−

+ + ∓ ± ± ∓ tim cl cm cl− ci− ± cl ci cl− cm− Glj

m

(6.B37)

+ + ∓ ∓ ± ± ± tli n cl− ± cl Glj ci− ci n± − − til n cl− ci− ± cl ci n−

+

± + + + ± ± tlm Imi − tim Ilm Ili− ± tlm Imi− − tim Ilm− Ili Glj = 0

m=il

m=il

103

Different Approximations for Hubbard Model

what simpliﬁes (6.B36) to the form

∓ + ± ci− ci cj cl−

±

m=ik

+

=

1 − FH0 − ± ± ∓

∓ + ± ci− ci cj ±tli ci−

∓ + ± ± tlm cm− ci− ci cj ∓ til n± tim Ilm− Gij − Gij ∓

m=il

± ∓ nˆ l cl− ci− F Gij − ± ± − ∓ H0

+ ∓ ± − C− − ± ± ∓ cl ci cl− FH0 Glj ci−

(6.B38)

This equation has the same structure as (6.B14), hence its solution is given by ∓ + ± ci− ci cj =∓ cl−

−

± ∓ nˆ l cl− ci− F Gij − ± ± − ∓ H0

− Wlmi − ± ± ∓ tmi

m

tin ± + ± ± nˆ i− ci cj − n− + Gij I t ln− n=il il

+ ∓ ± ± C− − ± ± ∓ cl ci cl− ci− FH0 Glj (6.B39) ∓ + ± is used in (6.62) allowing us to ci− ci cj The above expression for cl− + + obtain two equations for the functions nˆ + and nˆ − [see i− ci cj i− ci cj (6.B40)]. Insertingthis result into (6.62) one arrives at two equations for functions + + + − nˆ i− ci cj and nˆ i− ci cj , which can be written in the matrix form as

B − − − n+ − − B n+ − −

− nˆ = i− nˆ + i−

⎡ ⎤ + nˆ − i− ci cj ⎢ ⎥ ⎦ ⎣ − B + + − + − n− − nˆ i− ci cj B n− − −

(6.B40) −1 B −1 B ij − til Glj + Bil Glj + S− Gij +1 +1 l=i l=i

where B− =

− − til tmi Wlmi − Wlmi − + + − = − − − − + + −

lm

(6.B41)

104

Models of Itinerant Ordering in Crystals

+ − − + + B − Bil = −til FH0 C− cl ci cl− ci− − cl− ci− B S− =

(6.B42)

− + − − tmi tin Wlmi Iln− − Wlmi − + + − Iln−

lmn − FH0 C−

l=i

+ − (6.B43) + − 2til nˆ l cl− ci− − FH0 C+ til cl− ci− l=i

In this and all subsequent relations, the new correction terms contain the B B or S− . factor Bli Finally the scattering effect is now combined with the resonance broadening effect. Equations (6.B24) for scattering effect and (6.B40) for resonance broadening effect have the same form. Therefore they can be written as one equation in the matrix form ⎡ ⎤

+ T − T nˆ − ci cj i− n − − − n+ − − − − ⎢ ⎥ ⎦ ⎣ + T − T + + n− − − + − n− − nˆ i− ci cj

− nˆ = i− nˆ + i− +

ij −

l=i

til Glj

−1 S B + B + Bil Glj +1 l=i il

(6.B44)

−1 S B S− + S− Gij +1

where (6.B45) T = + − − − − + + − + + − ˆ Solving (6.B44) one ﬁnds the functions nˆ + c c and n c c , i i i− j j i− and after using the identity + + nˆ i− ci cj = ci cj = Gij (6.B46)

=±

one has FH Gij = ij −

l

where

til Glj +

T T Bli Glj + S− Gij

(6.B47)

l

T Bli = FH C− − T C− FH0 −til + − − + + − × nˆ i− nˆ l− − n2− + cl ci cl− ci− − cl− ci−

(6.B48)

105

Different Approximations for Hubbard Model

ST = FH C− − T SS + SB

= FH C− − T +

−C− FH0

l=im

til Wlmi tmi nˆ i− nˆ l− − n2−

− + − − tmi tin Wlmi Iln− − Wlmi − + + − Iln−

(6.B49)

lmn

−FH0 C−

l=i

+ − + − 2til nˆ l cl− ci− − FH0 C+ til cl− ci− l=i

1 1 = FH FH0

− T =

− T − n+ − − + n− + − 2 T − T − + T

− − − n+ − − + − n− − n− n−

(6.B50)

Similarly as in the case of the Hubbard I approximation in the ﬁnite bandwidth limit, the relation (6.B47) may be solved in the momentum representation by applying Fourier transforms (6.27)–(6.29) and the relation l=i

T Bli Glj =

T Bk Gk exp ik · ri − rj

(6.B51)

k

where

1 T = C− − T FH C− FH0 −tlm nˆ l− nˆ m− − n2− Bk N lm + + + + − cl cm− cl− cm − cl cl− cm− cm exp ik · rl − rm

(6.B52)

Using all these transformations one obtains from (6.B47) the following form: FH

Gk exp ik · ri − rj =

k

+

1 + k − t0 Gk exp ik · ri − rj

k

(6.B53)

T T Bk + S− Gk exp ik · ri − rj

k

from which one arrives at the ﬁnal result Gk =

1 T T FH − k − t0 − Bk + S−

(6.B54)

106

Models of Itinerant Ordering in Crystals

APPENDIX 6C: MODIFIED HARTREE–FOCK APPROXIMATION FOR THE INTER-SITE INTERACTIONS The full extended Hubbard Hamiltonian, used frequently throughout this textbook, has the following form (see Section 5.2 and [6.15]): H =−

+

t − tnˆ i− + nˆ j− + 2tex nˆ i− nˆ j− ci cj − 0

+U

i

nˆ i − F

i

ni nˆ i

i

+ + V J + + nˆ i↑ nˆ i↓ + nˆ nˆ + c c c c + J ci↑ ci↓ cj↓ cj↑ 2 i j 2 i j i j

(6.C1)

In this form, one has ﬁve inter-site interactions: t, tex , V , J , J , to which the modiﬁed H–F approximation is applied. There are terms with four and six operators. The terms with four operators are approximated by the average of two of +them multiplied by the remaining two. The averages with the spin ﬂip type ci cj− are ignored. Standing at tex the six-operator term is approximated by the product of two averages of two operators multiplied by the remaining two operators. This approximation was introduced by Foglio and Falicov [6.17] and extended to superconductivity by Aligia and co-workers [6.16, 6.18]. + The two remaining operators are in all cases of either kinetic, ci cj , or the + single site, nˆ i , type. The expressions at ci cj will contribute to the bandwidth change, while the expressions at nˆ i will contribute to the effective molecular ﬁeld. In addition to these terms, there are four terms describing superconduct+ + ing ordering. They have operators ci↑ ci↓ , corresponding to the on-site singlet + + + + cj↑ , corresponding to the inter-site sinsuperconductivity; operators ci↑ cj↓ − ci↓ + + + + glet superconductivity; operators ci↑ cj↓ + ci↓ cj↑ , corresponding to the inter-site + + opposite spin triplet superconductivity; and operators ci cj , corresponding to the equal spin triplet superconductivity. As a result the modiﬁed H–F approximation for the hopping (inter-site) interaction, t, will take on the following form: + + + nˆ i− + nˆ j− ci cj ≈ nˆ i− + nˆ j− ci cj + nˆ i− + nˆ j− ci cj + + + + + ci cj− cj− cj + ci ci− ci− cj

(6.C2)

+ + + + + ci cj− cj− cj + ci ci− ci− cj + const The following notation is introduced:

+ cj ci = I

0 = ci↓ ci↑

1 Sij = cj↓ ci↑ − cj↑ ci↓ 2

and nˆ j− = nj− (6.C3)

107

Different Approximations for Hubbard Model

using which the t term is given by t

+ nˆ i− + nˆ j− ci cj t

+ 2zt

+ ni− + nj− ci cj + 2zt

I− nˆ i

i

+ +

+ + + + Sij ci↑ ci↓ + hc + t 0 ci↑ cj↓ − ci↓ cj↑ + hc + const

i

(6.C4)

where z is the number of the nearest neighbours. In this expression, the ﬁrst term is the bandwidth change, the second is the effective molecular ﬁeld, the third describes the on-site singlet superconductivity and the last term describes the inter-site singlet superconductivity (with the parameter 0 = ci↓ ci↑ describing the on-site singlet superconductivity). The modiﬁed H–F approximation for the exchange-hopping interaction, tex , is calculated in the following way: + + + + nˆ i− nˆ j− ci ci− cj− cj = ci− cj− ci cj

+ + + + + ≈ ci cj nˆ i− nˆ j− − ci− cj− cj− ci− + ci− cj− cj− ci− + − ci− cj− + − cj− ci−

+ + + + cj− ci− ci cj + ci cj− ci− cj

+ + + + ci− cj− ci cj + ci ci− cj− cj

+ + + cj + ci cj− cj− cj + nˆ i− nˆ j− ci

+ + + + nˆ j− nˆ i− ci cj + ci ci− ci− cj + + + ci ci−

(6.C5)

+ ci− cj nˆ j− − cj− cj cj− ci−

+ + + + ci cj− nˆ i− cj− cj − ci− cj ci− cj− + ci− cj + cj− cj

+ + + + + ci ci− nˆ j− − ci cj− ci− cj−

+ + + + + ci cj− nˆ i− − ci ci− cj− ci−

+ + + + + + ci− cj− cj− ci− + cj− ci− ci− cj− ci cj + const

108

Models of Itinerant Ordering in Crystals

As a result the tex term takes on the form

− 2tex

+ nˆ i− nˆ j− ci cj

− 2tex

+ 2 ni− nj− − I− − 2I I− − 0 2 − ij 2 + − 2 ci cj

− 2tex

2nj I− + ij ∗0 + ∗ij 0 nˆ i

− 2tex

nj Sij − I + I− 0

+ + ci↑ ci↓ + hc

(6.C6)

− tex

n0 − I↑ + I↓ Sij

+ + + + ci↑ cj↓ − ci↓ cj↑ + hc

+ tex

+ + + + + + I↑ + I↓ Tij ci↑ cj↓ + ci↓ cj↑ + hc − 2tex I− ci cj + hc

where ij = cj↓ ci↑ is the inter-site superconductivity parameter, Tij = 1 c c + cj↑ ci↓ is the inter-site opposite spin triplet superconducting 2 j↓ i↑ parameter, and the equal spin pairing parameter is given by = cj ci . In this expression, the ﬁrst four terms are of the same type as in (6.C4) and the last term is the inter-site triplet superconductivity. The modiﬁed H–F approximation for the inter-site exchange interaction J is calculated as follows: + + For = − one has ci cj− ci− cj , and the operators average will take on the form

+ + + + + + ci cj− ci− cj = ci↑ cj↓ ci↓ cj↑ + ci↓ cj↑ ci↑ cj↓

+ + ci cj cj− ci−

+

+ + 1 + + + + ci cj cj− ci− + ci↑ cj↓ − ci↓ cj↑ cj↓ ci↑ − cj↑ ci↓ 2

1 1 + + + + + + + + + cj↓ ci↑ − cj↑ ci↓ ci↑ cj↓ − ci↓ cj↑ − cj↓ + ci↓ cj↑ ci↑ 2 2 1 + + + + × cj↓ ci↑ + cj↑ ci↓ − cj↓ ci↑ + cj↑ ci↓ ci↑ cj↓ + ci↓ cj↑ + const 2

109

Different Approximations for Hubbard Model

+ + For = one has ci cj ci cj , and taking averages of two different operators one obtains + + + + + + + + + + ci cj ci cj ci cj cj ci + ci cj cj ci − ci ci cj cj − ci ci cj cj

+ + + + − ci cj cj ci − ci cj cj ci + const As a result the term J takes on the form J + + + ci cj ci cj J I + I− ci cj − J nj nˆ i j 2 i +

J J S + + + + + + + + ij ci↑ cj↓ − ci↓ cj↑ + hc − Tij ci↑ cj↓ + ci↓ cj↑ + hc 2 2

−

J + + ci cj + hc 2

(6.C7)

% where j is the sum over nearest neighbours of the lattice site i. The different terms in this expression are interpreted as for the previous interactions. The modiﬁed H–F approximation for the inter-site pair hopping interaction J is + + + + + + ci↓ cj↓ cj↑ ci↑ cj↑ ci↓ cj↓ + ci↑ cj↑ ci↓ cj↓ ci↑ + + + + + ci↑ ci↓ cj↓ cj↑ + ci↑ ci↓ cj↓ cj↑ + const As a result the term J takes on the form J

+ + ci↑ ci↓ cj↓ cj↑ J

+ +

+ I− ci cj + hc + zJ 0 ci↑ ci↓ + hc

(6.C8)

i

The modiﬁed H–F approximation for the inter-site charge–charge interaction V : for = + + + + + + + + ci cj cj −ci cj cj ci − ci cj cj ci + ci ci cj cj ni nj = ci + + + + + + + ci ci cj cj − ci cj ci cj − ci cj ci cj + const

110

Models of Itinerant Ordering in Crystals

for = − + + + + + + + + ni nj− = ci↑ ci↑ cj↓ cj↓ + ci↓ ci↓ cj↑ cj↑ ci ci cj− cj− + ci ci cj− cj−

1 + + + + 1 ci↑ cj↓ − ci↓ cj↑ cj↓ ci↑ − cj↑ ci↓ + cj↓ ci↑ 2 2 1 + + + + + + + + −cj↑ ci↓ ci↑ cj↓ − ci↓ cj↑ + ci↑ cj↓ + ci↓ cj↑ cj↓ ci↑ + cj↑ ci↓ 2 1 + + + + + cj↓ ci↑ + cj↑ ci↓ ci↑ cj↓ + ci↓ cj↑ + const 2

+

As a result the term V takes on the form V + nˆ i nˆ j − V I ci cj + V n + n nˆ i j j− j 2 i +

V S + + + + ij ci↑ cj↓ − ci↓ cj↑ + hc 2

+

V V T + + + + + + ij ci↑ cj↓ + ci↓ cj↑ + hc + ci cj + hc 2 2

(6.C9)

Now the terms from all different interactions multiplied by the hopping + operators product, ci cj , are collected, while the terms at the single-site particle number operator, nˆ i , and the superconducting on-site and inter-site singlet and triplet terms are collected separately. In this way, the following simpliﬁed Hamiltonian is obtained: + H =− teff ci cj + hc − 0 nˆ i + Mi nˆ i + U nˆ i↑ nˆ i↓

+

i

i

+ + + + + + a1 ci↑ a2 ci↑ cj↓ − ci↓ ci↓ + hc + cj↑ + hc

i

+

i

(6.C10)

+ + + + + + a3 ci↑ cj↓ + ci↓ cj↑ + hc + a4 ci cj + hc

= tb is the effective hopping integral, with the bandwidth factor b where teff given by

1 2 − 2I I− tni− + nj− − 2tex ni− nj− − I− t − 0 2 − ij 2 + − 2 + J − V I + J + J I−

b = 1 −

(6.C11)

Different Approximations for Hubbard Model

111

The spin-dependent modiﬁed molecular ﬁeld for electrons with spin , Mi , is expressed as Mi = − Fni − J − 2tex

j

nj + V

j

nj + nj− + 2ztI−

∗ ∗ 2I n + + − j ij 0 ij 0 j

(6.C12)

This ﬁeld is the sum of the on-site contribution with Hund constant F and the inter-site contributions (all other terms above), which will be different for F and AF ordering. In the Hamiltonian (6.C10): the total on-site singlet energy gap is a1 = 2ztSij + zJ 0 − 2ztex nSij − I− + I 0

(6.C13)

the total inter-site singlet energy gap is 1 a2 = t0 + V + JSij − tex n0 − I− + I Sij 2 the total opposite spin triplet energy gap is 1 a3 = V − J + tex I− + I Tij 2 and the total equal spin triplet energy gap is 1 a4 = V − J − 2tex I− 2

(6.C14)

(6.C15)

(6.C16)

In the ferromagnetic state, the electron concentration is the same on each lattice site, ni = nj = n , and the parameter I is spin dependent. In this case, all the superconducting ordering terms are assumed to be zero: 0 = ij = ≡ 0. Inserting those values into (6.C11) and (6.C12) one has the effective bandwidth factor given by b = 1 − 2

J −V t t J + J 2 n− + 2 ex n2− − I− − 2I I− − I − I t t t t −

(6.C17)

and the effective molecular ﬁeld equal to Mi ≡ M = −F + zJ n + zVn + 2zI− t − 2tex n

(6.C18)

The total band shift is F 2E = M − − M ≡ Ftot m

(6.C19)

112

Models of Itinerant Ordering in Crystals

hence the total Stoner ﬁeld for ferromagnetism is F Ftot = F + z J + tex n2 − m2 + 2t1 − n

(6.C20)

where I was replaced by the simplest expression (6.153). In the antiferromagnetic state, the crystal lattice is divided into two interpenetrating sub-lattices with opposite spins, , and with the average electron numbers equal to n± = n± =

n±m 2

n± = n∓ =

n∓m 2

(6.C21)

where m is the antiferromagnetic moment per atom in Bohr’s magnetons. The magnetic moment on the nearest lattice sites is opposite with respect to each other [see (6.C21)], the quantity I = I− = IAF and the bandwidth reduction parameter b from (6.C11) is spin independent: b = b− = bAF = 1 −

t t n + 2 ex t t

2J + J − V n 2 − m2 2 − − 3IAF IAF 4 t

(6.C22)

The generalized (modiﬁed) molecular ﬁeld in the AF case is given by (6.C12). It depends on the spin index and sub-lattice indices: or . For the sub-lattice one has M = −Fn − zJ + 4tex IAF n− + zVn + 2ztIAF

(6.C23)

and for the sub-lattice M = −Fn− − zJ + 4tex IAF n + zVn + 2ztIAF

(6.C24)

There is also the relation between different molecular ﬁelds: M± ≡ M∓

(6.C25)

The total Stoner ﬁeld for antiferromagnetism is calculated in a similar way to the case of ferromagnetism as AF Ftot =

M+ − M− M− − M+ = F − zJ + 4tex IAF = m m

(6.C26)

where the expression (6.155) for IAF should now be used. Finally the Hamiltonian (6.C10) for the superconducting ordering in the paramagnetic state (n = n− = n/2 and I = I− = I) can be written. Adding

113

Different Approximations for Hubbard Model

the spin-independent part of the molecular ﬁeld into the chemical potential one obtains + +

+ H =− teff ci cj + hc − nˆ i + U nˆ i↑ nˆ i↓ + a1 ci↑ ci↓ + hc

+

i

i

+ + + + + + + + a2 ci↑ cj↓ − ci↓ cj↑ + hc + a3 ci↑ cj↓ + ci↓ cj↑ + hc

+

i

(6.C27)

+ + a4 ci cj + hc

where the effective chemical potential is n = 0 − 2zV − zJ − F − J j nj − 2ztI 2 + 2ztex In + ij ∗0 + ∗ij 0

(6.C28)

and the effective hopping integral is spin-independent teff = tb, with the bandwidth factor b given by 1 b = 1 − tn − 2tex n2 /4 − 3I 2 − 0 2 − ij 2 + − 2 + 2J + J − VI (6.C29) t The total energy gap for the on-site singlet, inter-site singlet, opposite spin triplet and equal spin triplet superconductivities in this case is given by a1 = 2ztSij + zJ 0 − 2ztex nSij − I− + I 0 1 a2 = t0 + V + J Sij − tex n0 − I− + I Sij 2 1 a3 = V − J + tex I− + I Tij 2 1 a4 = V − J − 2tex I− 2

(6.C30) (6.C31) (6.C32)

(6.C33)

REFERENCES [6.1] [6.2] [6.3] [6.4]

D.N. Zubarev, Usp. Fiz. Nauk 71, 71 (1960) [Translation Sov. Phys. Usp. 3, 320 (1960)]. P. Soven, Phys. Rev. 156, 809 (1967). D.W. Taylor, Phys. Rev. 156, 1017 (1967). H. Fukuyama and H. Ehrenreich, Phys. Rev. B 7, 3266 (1973).

114 [6.5] [6.6] [6.7] [6.8] [6.9] [6.10] [6.11] [6.12] [6.13] [6.14] [6.15] [6.16] [6.17] [6.18]

Models of Itinerant Ordering in Crystals G. Geipel and W. Nolting, Phys. Rev. B 38, 2608 (1988). W. Nolting and W. Borgieł, Phys. Rev. B 39, 6962 (1989). L.M. Roth, Phys. Rev. 184, 451 (1969). T. Herrmann and W. Nolting, J. Magn. Magn. Mater. 170, 253 (1997). A.B. Harris and R.V. Lange, Phys. Rev. 157, 295 (1967). M. Potthoff, T. Herrmann and W. Nolting, Eur. Phys. J. B 4, 485 (1998). T. Herrmann and W. Nolting, Phys. Rev. B 53, 10579 (1996). A. Georges, G. Kotliar, W. Krauth and M.J. Rozenberg, Rev. Mod. Phys. 68, 13 (1996). A.-M.S. Tremblay, B. Kyung and D. Sénéchal, Low Temp. Phys. 32, 424 (2006). J. Wahle, N. Blümer, J. Schlipf, K. Held and D. Vollhardt, Phys. Rev. B 58, 12749 (1998). J.E. Hirsch, Phys. Rev. B 59, 6256 (1999). L. Arrachea and A.A. Aligia, Physica C 289, 70 (1997). M.E. Foglio and L.M. Falicov, Phys. Rev. B 20, 4554 (1979). A.A. Aligia, E. Gagliano, L. Arrachea and K. Hallerg, Eur. Phys. J. B 5, 371 (1998).

CHAPTER

7 Itinerant Ferromagnetism

Contents

7.1

Periodic Table – Ferromagnetic Elements 7.1.1 Ferromagnetic elements 7.2 Introduction to Stoner Model 7.2.1 Static magnetic susceptibility 7.3 Stoner Model for Ferromagnetism 7.4 Stoner Model for Rectangular and Parabolic Band 7.4.1 Rectangular band 7.4.2 Parabolic nearly free electron band 7.5 Modified Stoner Model 7.5.1 Modified Stoner Model for a semi-elliptic band 7.6 Beyond Hartree–Fock Model 7.6.1 General formalism 7.6.2 Enhancement of magnetic susceptibility 7.6.3 Critical values of interactions 7.6.4 Numerical results 7.7 The Critical Point Exponents 7.8 Spin Waves in Ferromagnetism 7.8.1 Energy of spin-wave excitations 7.8.2 Dynamic susceptibility of ferromagnets 7.8.3 Curie temperature References

115 118 125 129 131 134 134 136 138 141 144 144 148 149 150 154 157 161 161 164 165

7.1 PERIODIC TABLE – FERROMAGNETIC ELEMENTS There are a few simple rules to understand the construction of the periodic table of elements. The atomic number of an element is equal to the number of electrons of this element. The reason is that the atomic number is equal to the number of protons in the nuclei. Every proton carries an elementary positive charge, which in the 115

116

Models of Itinerant Ordering in Crystals

neutral state of an atom has to be balanced by a negative elementary charge of electron in the electron shell. The electron orbit is characterized by four quantum numbers, which arise in the solution to the Schrödinger equation for the hydrogen atom, three quantum numbers arise from the space geometry of the solution and a fourth arises from electron spin. No two electrons can have an identical set of quantum numbers according to the Pauli exclusion principle, so the quantum numbers set limits on the number of electrons which can occupy a given state and therefore give insight into the building up of the periodic table of elements. Principal quantum number n = 1 2 3 ; Orbital quantum number l = 0 1 n − 1; Magnetic quantum number ml = −l −l + 1 0 l − 1 l or 2l + 1 values; Spin quantum number mS = +1/2 −1/2. Different orbital quantum numbers l = 0 1 2 3 correspond to different types of orbits, denoted by letters s, p, d, f. For example, the letter d means orbit with l = 2. On this orbit one has ml = −2 −1 0 1 2 and mS = +1/2 −1/2, which gives a maximum capacity of 10 electrons in this orbit. In a similar way, one obtains s2 p6 d10 f 14 , where the superscripts denote the maximum capacity of each orbit (Table 7.1). The elements are placed in seven rows corresponding to principal quantum number varying from 1 to 7. The superscript on the right denotes the actual number of electrons in the orbit. The periodic table shows us that the energetic sequence of the orbits is not that simple. Orbits with a high angular momentum have energy comparable with that of higher principal quantum numbers, but low angular momentum. For example, orbit 3d is ﬁlled together with 4s, orbit 4d with 5s and orbit 5d with 6s. These three rows are those of the transition metals (3d, 4d and 5d transition metals, total of 30 elements). Another example of this inversion is ﬁlling orbit 4f together with orbit 6s and even 5d. This row of 14 elements is called the rare earth row. The last row with an inversion is the row of 14 actinides where the 5f orbit is ﬁlled with orbits 7s and 6d. As a result, the sequence of electronic orbits will be changed from

to

2

1s 2s2 p6

1s2

3s2 p6 d10 4s2 p6 d10 f 14 5s2 p6 d10 f 14

3s2 p6 4s2 3d10 , 4p6 5s2 4d10 , 5p6

6s2 p6

6s2 5d10 , 6p6

7s2

7s2 , 6s2 4f 14

2s2 p6

7s2 5f

Table 7.1 Outer electron configurations of neutral atoms H1

He2 1 s2

1s 3

Li

4

Be

11

Mg

3s

3 s2

4s

Rb

37

12

23

V

4 s2

3d 4 s2

3d2 4 s2

3d3 4 s2

38

Ba56 6s

Y

39

Ra

3d5 4s

Nb

41

Mo

Mn

25

3d5 4 s2 42

43

26

Fe

Co

3d6 4 s2

3d7 4 s2

Tc

Ru

44

Rh

27

45

28

29

Ni

Cu

3d8 4 s2

3d10 4s

Pd

46

Ag

Zn

30

3d10 4 s2

47

Cd

48

31

O

2s22p3

2s22p4

14

15

16

Si

3s23p2 32

P

S

3s23p3 33

Ge

As

4s24p

4s24p2

4s24p3

In

F

2s22p2

Ga

49

8

Sn

50

Sb

51

9

Ne10

2s22p5 17

Cl

3s23p4 34

Se

4s24p4 52

Te

3s23p5

Br

35

4s24p5

I

53

2s22p6

A r 18 3s23p6

K r 36 4s24p6

Xe54

4d 5 s2

4d2 5 s2

4d4 5s

4d5 5s

4d6 5s

4d7 5s

4d8 5s

4d10 –

4d10 5s

4d10 5 s2

5s25p

5s25p2

5s25p3

5s25p4

5s25p5

5s25p6

La57– Lu71

H f 72

Ta73

W 74

Re75

Os76

I r 77

Pt78

A u 79

Hg80

Tl81

P b 82

Bi83

Po84

At85

Rn86

4f14 5d2 6 s2

5d3 6 s2

5d4 6 s2

5d5 6 s2

5d6 6 s2

5d9 –

5d9 6s

5d10 6s

5d10 6 s2

6s26p

6s26p2

6s26p3

6s26p4

6s26p5

6s26p6

Ce58

P r 59

Nd60

P m 61

Sm62

Eu63

G d 64

Tb65

Dy66

H o 67

Er68

Tm69

Y b 70

Lu71

4f2

4f3

4f 4

4f5

4f 6

4f7

4f11

4f12

4f13

4f14

6 s2

4f8 5d 6 s2

4f10

6 s2

4f7 5d 6 s2

6 s2

6 s2

6 s2

4f14 5d 6 s2

No102

Lr103

2

88

Zr

40

Cr

24

N

7

89

Ac – Lr103

7 s2

La57 5d 6 s2

6 s2 89

Th

6d 7 s2

6d2 7 s2

91

92

Pa

U

5f2 6d 7 s2

5f3 6d 7 s2

Np

6 s2 93

Pu

6 s2 94

Am

5f5

5f6

5f7

7 s2

7 s2

7 s2

95

Cm 5f7 6d 7 s2

96

Bk

97

Cf

98

Es

6 s2 99

Fm

100

Md

101

117

Ac

6 s2 90

Itinerant Ferromagnetism

7s

22

Ti

Cs55

Fr

21

Sc

5 s2

87

Al

Ca

5s

6s

13

3s23p 20

Sr

C

2s22p

Na

19

6

B

2 s2

2s

K

5

118

Models of Itinerant Ordering in Crystals

7.1.1 Ferromagnetic elements What happens when atoms get close together to form a solid? We found in Section 4.2 that when the lattice constant a decreases, the energy levels widen to form a band since the overlap integral or hopping constant t increases. This causes the itinerancy of electrons which do not any longer stay on one atom. In general, one may have fractional numbers of electrons as the average number in the band. For example, if three atoms have ﬁve electrons each and two atoms have six electrons, then on average one has 5.40 electrons in the band. When the atoms are getting close together, the bands formed from the orbits, which are close in energy, start overlapping. The overlapping bands are ﬁlled to the same energy, which is called the Fermi energy. It is like a liquid in connected vessels which are ﬁlled to the same level. This is why the itinerant electrons are sometimes called the Fermi liquid. The total number of electrons in overlapping bands is the same in the atomic state and in the solid state. For example, Fe26 , which in the atomic state has 3d6 + 4s2 = eight outer electrons, has eight outer electrons also in the solid state, but they are distributed differently between 3d and 4s bands leading to fractional numbers of electrons in the s and d band. Let us now consider the ferromagnetism of the transition metals Fe, Co and Ni. In the ferromagnetic state, the 3d sub-bands with spins up 3d↑ and down 3d↓ are shifted with respect to each other by the amount of energy 2E = Fm, where m is the magnetization in Bohr’s magnetons per atom = B m and F is the exchange integral. In the case of an exchange shift strong enough to push the entire majority spin sub-band, 3d↑ , below the Fermi energy, one has strong ferromagnetism, but if the exchange shift is weaker, then both spin sub-bands are crossed by the Fermi energy and one has a case of weak ferromagnetism. Both cases are shown in Fig. 7.1.

ε

ε εF

εF

2ΔE 2ΔE

ρ+σ

ρ–σ (a)

ρ+σ

ρ–σ (b)

FIGURE 7.1 Schematic density of states in the case of strong (a) and weak (b) ferromagnetism.

Itinerant Ferromagnetism

119

The broad 4s band is not magnetic since it has a low density of states (DOS), which does not fulﬁl the Stoner condition (see Section 7.3). The experimental data are shown in Table 7.2. Observed values of m are non-integral. The reason is, as mentioned above, the redistribution of electrons between s and d bands. The 3d metals in reverse order, Cu, Ni, Co, Fe, starting with the paramagnetic copper, will be analysed. The relationship between 4s and 3d bands in copper is shown in Fig. 7.2 [7.1, 7.2]. If one electron is removed from copper, one obtains nickel which has the possibility of a hole in the 3d band. In Fig. 7.3a, the DOS for majority and minority spins − of nickel [7.3] are shown. A relatively narrow d band is hybridized with the broad sp band. Magnetism of nickel is strong, meaning that the majority spin band is fully occupied and cannot get a larger magnetic moment per atom. In Fig. 7.3a, one can see also the Stoner gap, , which is the difference between the Fermi energy and the top of the spin up (majority Table 7.2 Ferromagnetic crystals Element

m(0 K) in Bohr’s magnetons per atom

Curie temperature (K)

2.22 1.72 0.606

1043 1388 627

Fe Co Ni

0

ε – εF (eV)

–2

–4

–6

–8

–10

3.0

2.0

1.0

ρ – σ (electrons/eV)

0 (a)

1.0

2.0

3.0

ρσ (electrons/eV)

FIGURE 7.2 (a) The density of states of copper [7.1]. Reprinted with permission from J.F. Janak, A.R. Williams and V.L. Moruzzi, Phys. Rev. B 11, 1522 (1975). Copyright 2007 by the American Physical Society. (b) Schematic relationship of 4s and 3d bands in metallic copper. The 3d band holds 10 electrons per atom and is filled in copper. The 4s band can hold two electrons per atom; it is shown halffilled, as copper has one valence electron outside the filled 3d shell.

120

Models of Itinerant Ordering in Crystals

εF

3d 5 electrons (b)

4s 1 electron

3d 5 electrons

FIGURE 7.2 (Continued)

s electrons

2.5

ε – εF (eV)

0

εF

Δ

–2.5 d electrons –5.0

–7.5 2.5

2.0

1.5

1.0

ρσ (electrons/eV)

0.5

0

0.5

1.0

1.5

2.0

2.5

ρ – σ (electrons/eV) (a)

FIGURE 7.3 (a) The density of states for majority and minority spins − of nickel [7.3]. Reprinted with permission from J. Callaway and S.C. Wang, Phys. Rev. B 7, 1096 (1973). Copyright 2007 by the American Physical Society. (b) Band relationship in nickel above the Curie temperature. The net magnetic moment is zero, as there are equal numbers of electrons in both 3d↓ and 3d↑ bands. (c) Schematic relationship of bands in nickel at absolute zero. The energies of the 3d↓ and 3d↑ bands are separated by an exchange interaction. The 3d↑ band is filled; the 3d↓ band contains 4.46 electrons and 0.54 hole. The 4s band is usually thought to contain approximately equal numbers of electrons in both spin directions, and so there is no problem in dividing it into sub-bands. The net atomic moment of 054 B per atom arises from the excess population of the 3d↑ band over the 3d↓ band.

Itinerant Ferromagnetism

121

0.27 hole

εF

4s

3d

3d

0.54 electron

4.73 electrons (b)

4.73 electrons

0.54 hole

εF

Δ

4s 0.54 electron

3d

3d 5 electrons (c)

4.46 electrons

FIGURE 7.3 (Continued)

spin) band. The schematic band structure of nickel is shown in Fig. 7.3b for T > TC , where 2 × 0 27 = 0 54 of an electron was taken away from the 3d band and 0.46 away from the 4s band, as compared with copper. The band structure of nickel at absolute zero is shown in Fig. 7.3c. Nickel is ferromagnetic, and at absolute zero it has m = 0 60 per atom. After allowing for the magnetic contribution of orbital electronic motion, nickel has an access of 0.54 electron per atom, having spin preferentially oriented in one direction. The energy difference between bands 3d↓ and 3d↑ is 2E = Fm. The next element is cobalt. In the atomic state it has total number of nine outer electrons in conﬁguration 3d7 4s2 (see Table 7.1). Its magnetic moment per atom (Table 7.2) is m = 1 72. It is also known as a strong ferromagnet, meaning that the whole majority spin sub-band 3d↑ is located below the Fermi energy and as a result n3d↑ ≡ nd↑ = 5. Hence one obtains that m = 1 72 = nd↑ − nd↓ = 5 − nd↓

or

nd↓ = 5 − 1 72 = 3 28

The total number of 3d electrons is n3d = 5 + 3 28 = 8 28. The number of 4s electrons is n4s = 9 − 8 28 = 0 72.

122

Models of Itinerant Ordering in Crystals

Figure 7.4 shows the schematic band structure for cobalt at absolute zero. The last element which will be analysed is iron. It has one less electron than nickel, since its atomic conﬁguration is 3d6 4s2 , and it has eight outer electrons. Iron is not a strong ferromagnet, which means that nd↑ < 5, since the exchange interaction separating the 3d↓ and 3d↑ sub-bands is not strong enough to shift the entire 3d↑ sub-band below the Fermi surface. From Table 7.2 one knows that m ≈ 2 2 = nd↑ − nd↓

(7.1)

What would be the maximum magnetic moment of iron in a state of strong ferromagnetism, when the entire majority spin sub-band is located below the Fermi level? The answer to this question comes from the experimental data on ferromagnetic alloys. The partial magnetic moments associated with particular components in the Fe–Co binary alloys are shown in Fig. 7.5. The magnetic moment on the cobalt atom is not affected by alloying, but that on the iron atom increases to approximately 2 8B as the cobalt concentration increases. This increasing moment of iron is typical for the weak ferromagnet which, in consequence of alloying, is gradually transformed into a strong ferromagnet. The magnetizations of different binary alloys are shown in Fig. 7.6. For the binary alloys which are strong ferromagnets, the magnetization is a linear function of concentration. This linearity will be explained later on in Section 9.4. Extrapolating linearly the Fe–Co curve to pure iron, one again gets moment in the vicinity of 2 8B – 2 9B for the magnetically strong iron. Assuming the value of the magnetic moment of Fe in a state of strong ferromagnetism to be approximately 2 8B , one can write 2 8 = 5 − nd↓

hence nd↓ = 2 2

and nd = 5 + nd↓ = 7 2

1.72 holes

εF

Δ

4s 0.72 electron

3d 5 electrons

3d 3.28 electrons

FIGURE 7.4 Schematic relationship of bands in cobalt at absolute zero temperature. The energies of the 3d↓ and 3d↑ bands are separated by an exchange interaction. The 3d↑ band is filled; the 3d↓ band contains 3.28 electrons and 1.72 hole. The 4s band is usually thought to be paramagnetic, and so it is not divided into sub-bands. The net atomic moment of 172 B per atom arises from the excess population of the 3d↑ band over the 3d↓ band.

Itinerant Ferromagnetism

123

3.2

Magnetic moment (μ B)

2.8

Iron

2.4

Bulk Fe

Bulk Co

2.0

1.6

Cobalt

1.2 0.0

0.2

0.4

0.6

0.8

1.0

x

FIGURE 7.5 Moments attributed to 3d electrons in Fe–Co alloys (after [7.5]) as a function of composition; experimental points are from neutron measurements by Collins and Forsyth [7.4]. Reprinted with permission from M. Liberati, G. Panaccione, F. Sirotti, P. Prieto, and G. Rossi, Phys. Rev. B 59, 4201 (1999). Copyright 2007 by the American Physical Society.

Atomic moment in Bohr magnetons

3.0 + 2.5

+

Ni–Zn

+

Ni–V Ni–Cr

+

1.0

0 Cr 6

Ni–Cu

+

1.5

0.5

Fe–Co Ni–Co

++

2.0

Ni–Mn Co–Cr Co–Mn Pure metals

+ +

Mn 7

Fe–V Fe–Cr Fe–Ni

Fe Co 8 9 Electron per atom

Ni 10

Cu 11

FIGURE 7.6 The Slater–Pauling curve: average magnetic moments of binary alloys of the 3d transition elements (after Bozorth [7.6]).

124

Models of Itinerant Ordering in Crystals

Assuming that the number of 3d electrons per iron atom in the weak pure ferromagnetic iron and in the Fe–Co alloy is the same, one obtains 7 2 = nd↑ + nd↓ This equation when combined with (7.1) gives us nd↑ = 4 7

and nd↓ = 2 5

The number of 4s electrons is the number of outer electrons, 8, minus the number of 3d electrons, 7.2, which is 0.8. It is also assumed here that this number stays unchanged during the transformation from the paramagnetic to ferromagnetic iron. In Fig. 7.7a b, the real and schematic band structure for the weakly ferromagnetic iron is shown with the above estimated numbers. 5.0 s electrons 2.5

εF

ε – εF (eV)

0

–2.5 d electrons –5.0

–7.5

–10.0

4.0

3.0

2.0

1.0

0

ρσ (electrons/eV)

1.0

2.0

3.0

4.0

ρ – σ (electrons/eV) (a)

FIGURE 7.7 (a) The density of states for majority and minority spins − of bcc iron [7.7]. Reprinted with permission from R.A. Tawil and J. Callaway, Phys. Rev. B 7, 4242 (1973). Copyright 2007 by the American Physical Society. (b) Schematic relationship of bands in iron at absolute zero. The energies of the 3d↓ and 3d↑ bands are separated by an exchange interaction. This exchange interaction is too weak for the 3d↑ band to be shifted entirely below the Fermi energy and to be completely filled. The 3d↑ band contains 4.7 electrons; the 3d↓ band contains 2.5 electrons. The net magnetic moment of 22 B per atom arises from the excess population of the 3d↑ band over the 3d↓ band. The 4s band is thought to be paramagnetic; therefore, it is not divided into sub-bands.

Itinerant Ferromagnetism

0.3 hole

125

2.5 holes

εF

3d

4s

3d

0.8 electron

4.7 electrons

2.5 electrons

(b)

FIGURE 7.7 (Continued)

7.2 INTRODUCTION TO STONER MODEL According to Weiss [7.8], in magnetic materials, the internal ﬁeld acts on the electrons inside atoms. Its value is given by the following expression: Hin =

F m 2

(7.2)

where the proportionality constant, F , is the material constant with the dimension of the magnetic ﬁeld. The energy of magnetic dipole in the magnetic ﬁeld is given by the following expression: E = B Hin cos Hin B where Hin B is the angle between the ﬁeld and the direction of the magnetic dipole, represented here by the electron moment or Bohr’s magneton, B . The magnetic dipole created by the electron spin is the quantum entity and can have only two directions with respect to the magnetic ﬁeld: parallel or antiparallel. Hence the shift in energy of electrons with spins parallel or antiparallel ± to the internal ﬁeld, or equivalently to the sample magnetization, is E = ±B

F F m = ± m 2 2

F = F B

(7.3)

where the internal exchange ﬁeld, F (also called the Stoner ﬁeld), has the unit of energy and the Weiss ﬁeld, F , has the unit of magnetic ﬁeld. The Weiss ﬁeld after multiplying by Bohr’s magneton, B , is equal to the atomic internal exchange ﬁeld or Hund’s constant.

126

Models of Itinerant Ordering in Crystals

In the language of second quantization and in the Hartree–Fock (H–F) approximation, the energy of electrons in the exchange ﬁeld, F , is given by −F i n nˆ i [see (5.22)], with the constant of the exchange ﬁeld, F , given as the sum of different on-site interactions by (5.20). With the energy difference between + and − electrons given by (7.3), the material at a given temperature will have net magnetization m=

m = BJ =1/2 x = tanhx m0

x=

E Fm = kB T 2kB T

(7.4)

where BJ =1/2 x is the Brillouin function introduced in Section 3.1 for the total electron moment J. In our case of itinerant electrons, this moment is equal to the spin moment J = L + S = 1/2, since the orbital moments are quenched in the ferromagnetic transition metals (see Section 3.1 and e.g. Kittel [3.1]). The result is that one has the orbital moment L = 0 and the total moment J = S = 1/2, for which value the Brillouin function is equal to tanhx. The result (7.4) can also be derived directly. The electron numbers with spin ± in thermal equilibrium are related as in the following equation: Eb + E Eb − E n exp = n− exp − (7.5) kB T kB T where Eb is the energy barrier between the two states in the absence of an internal ﬁeld. From (7.5) one has n = e2x n−

x=

E Fm = kB T 2kB T

(7.6)

Recalling that the total number of electrons n = n + n−

(7.7)

and magnetization per atom is mB , with m = n − n−

(7.8)

one ﬁnds that m=

mB m n − n− = = = tanhx ≡ B1/2 x nB n n + n−

(7.9)

To ﬁnd the magnetization in thermal equilibrium, one has to solve simultaneously the set of two simple equations. It is (7.9) and the linear equation obtained from (7.6): m=

2kB T x F

(7.10)

Itinerant Ferromagnetism

1

127

T < Tc

Tc

0.8

m

0.6 0.4 0.2 0 0

0.5

1 x

1.5

2

FIGURE 7.8 Graphic solution for magnetization, m = m/n.

In the temperature range between 0 and the Curie temperature, TC , the set of equations (7.9) and (7.10) has a non-zero solution for m. We can solve them graphically by searching for the intersection of both curves, as illustrated in Fig. 7.8. The curves of m versus T obtained in this way reproduce roughly the features of the experimental results, as shown in Fig. 7.9 for nickel. As T increases, the magnetization decreases smoothly to zero at T = TC . At the Curie temperature one has from (7.10) and (7.9), with tanhx x m=

m Fm ≈x= n 2kB TC

(7.11)

1.0

m/m0

0.8 0.6 0.4 0.2 0

0

0.2

0.4

0.6 T/Tc

0.8

1.0

FIGURE 7.9 Saturation magnetization of nickel as a function of temperature, together with the theoretical curve for S = 1/2 on the mean-field theory. Experimental data are quoted after [7.9].

128

Models of Itinerant Ordering in Crystals

hence TC =

Fn 2kB

(7.12)

The mean-ﬁeld theory does not give a good description of the variation of m at low temperatures. The experimental results show a much more rapid decrease of m at low temperatures. These results [3.1, 3.2, 4.2] ﬁnd a natural explanation in terms of the spin-wave theory. The quantum term, written in (5.17), is in the H-F approximation: −F i n nˆ i ; it decreases the energy of the spin system for the ferromagnetic ordering according to the quadratic formula: F W = const − m2 2

(7.13)

Assuming (after Slater [7.10]) that every pair of mutually parallel spins decreases the energy of the system to the amount of F↑↑ , or F↓↓ , we can prove that formula (7.13) holds for the decrease of the total potential energy of all spins, W , with F given below as F=

F↑↑ + F↓↓ − F↑↓ 2

(7.14)

The proof starts by counting the numbers of electrons on the same atom with a given spin W = −F↑↑ n2 − F↓↓ n2− − 2F↑↓ n n−

(7.15)

The electron numbers can be expressed as n =

n+m 2

n− =

n−m 2

(7.16)

After inserting (7.16) into (7.15) one obtains W = const −

F↑↑ − F↓↓ m2 nm − F 2 2

(7.17)

Equation (7.17) is equivalent to (7.13) if F↑↑ = F↓↓ . At the same time one gets the physical insight into the meaning of the on-site exchange ﬁeld constant, F , as the energy decrease of the pair with parallel spins with respect to the pair with antiparallel spins. Writing (7.13) or (7.17) for the potential energy of electrons is equivalent to assuming that the energy of ± electrons is shifted by the amount of ∓Fm/2.

Itinerant Ferromagnetism

129

This can be proved by considering the total energy in the ferromagnetic state, W , as follows: d (7.18) W=

where is the spin DOS in a ferromagnetic state. According to our assumption of the energy shift given by (7.3), one has =

0

Fm + 2

(7.19)

with = ±1 for majority and minority spin bands, and 0 is the electron DOS in the paramagnetic state. Inserting (7.19) into (7.18) one obtains Fm Fm d + 0 − d W = 0 + 2 2 F Fm Fm = 2 0 d − n + n = Wparamagnetic − m2 2 2 − 2 The last result is exactly the relation (7.13).

7.2.1 Static magnetic susceptibility The Curie temperature is the temperature above which spontaneous magnetization disappears. One can ﬁnd TC in terms of the Weiss ﬁeld F [see (7.3)]. Considering the paramagnetic phase of the system of itinerant electrons, an applied ﬁeld H will cause a ﬁnite magnetization, NmB , and this in turn will cause a ﬁnite Weiss ﬁeld, F m/2. If P is the paramagnetic static susceptibility, then F (7.20) NmB = P H + m 2 where N is deﬁned in Chapter 3 as the number of magnetic moments per unit volume. The identical equation has already been written in phenomenological language as (3.25). Let us assume, as in Chapter 3, that the paramagnetic susceptibility is given by the Curie law P = C/T , where C is the Curie constant. Substituting this equation in (7.20) one ﬁnds F NmB T = C H + m 2

(7.21)

130

Models of Itinerant Ordering in Crystals

or

=

NmB C = H T − TC

TC =

CF 2NB

(7.22)

The last equation known as Curie–Weiss law relates the Curie temperature to the Curie constant and the Weiss ﬁeld constant. Equation (7.22) for susceptibility is close to the experimental results, but it was derived in a phenomenological way, under the assumption that the specimen in the paramagnetic phase has the susceptibility expressed by the Curie law P = C/T . Expression (7.22) for the Curie temperature is the same as relation (7.12) and relation (3.28). Comparing deﬁnitions of Weiss ﬁeld in Section 3 [(3.25) and (3.18)], and in this section (7.21), one has F m = NB m 2

(7.23)

hence = F /2NB and from Chapter 3 [relations (3.26) and (3.28)] one obtains TC =

F nN2B F nB Fn = = 2NB kB 2 kB 2kB

(7.24)

as in (7.12). The experimental data for susceptibility (collected by Stanley [7.11]) are shown in Table 7.3. As T → TC from above, the susceptibility becomes proportional to T − TC − ; as T → TC from below, the magnetization NB m becomes proportional to TC − T . In the mean-ﬁeld approximation, = 1 and = 1/2. As we can see, the experimental data do not support the linear dependence of susceptibility on the inverse of T − TC . Detailed calculations for the itinerant model (see Section 7.7) predict that the susceptibility above the Curie temperature behaves as [3.2]

=

A A 1 ≈ 2TC T − TC T 2 − TC2

for T ≈ TC

Table 7.3 Critical point exponents for ferromagnets Element Fe Co Ni

1 33 ± 0 015 1 21 ± 0 04 1 35 ± 0 02

0 34 ± 0 04 _ 0 42 ± 0 07

(7.25)

Itinerant Ferromagnetism

131

Hirsch [6.15] has shown that

T =

2 T peff 3T − TC

(7.26)

where peff is the effective magnetic moment. Strong dependence of the effective moment on the temperature can hide the dependence of on temperatures different than 1/T − TC . Comparing Table 7.3 with (7.119) for susceptibility and (7.116) for magnetization in Section 7.7, one can see that there is still some work to be done on itinerant models to explain the critical exponents listed in Table 7.3. We will come back to this subject in Section 7.7.

7.3 STONER MODEL FOR FERROMAGNETISM For a simple magnetic material, the free energy per atom can be written as [3.2] = 0 + a 2 m 2 + a 4 m4 + · · ·

(7.27)

In the case of magnetization induced by the external ﬁeld, H, the magnetic ordering energy term can be expressed as a2 m2 = B mH =

2B m2 2 m2 ≈ B B m/H

(7.28)

with being the static magnetic susceptibility per atom. Finally one has = 0 + 2B m2 / + a4 m4 + · · ·

(7.29)

The essence of the Stoner model (see Section 7.2 and [7.12]) is that the energy of electrons with spins up and down is shifted with respect to each other to the amount of Fm/2, where F is equal to the exchange ﬁeld in the units of energy. With that assumption, the electron energy for both spins can be written as k = k −

Fm 2

= ±1

(7.30)

This equation is equivalent to (7.3). The total exchange ﬁeld, F , is now generalized to the sum of different on-site and inter-site interactions as in Appendix 6C.

132

Models of Itinerant Ordering in Crystals

The electron occupation for both spins is given by n± =

n±m 1 1 = 2 N k exp k − ∓ E/kB T + 1 =

D

−D

d exp k − ∓ E/kB T + 1

(7.31)

where for the total energy shift of the spin sub-band in the presence of an external ﬁeld, one has the expression (7.3). It should be pointed out that there is a connection between the Weiss theory for the itinerant band model expressed by (7.31) and (7.3) and the Weiss model for localized electrons. In order to transfer to the localized model, it is assumed in (7.31) that all band energies merge at the same limit of atomic level t0 . Moreover, the case of Boltzmann classical statistics is assumed (see Chapter 2), meaning that 1 ≈ exp − t0 /kB T exp±E/kB T = n± expt0 − ∓ E/kB T + 1 with E = Fm/2. Hence one can write

E m n − n− ≡ = tanh n n + n− kB T

(7.32)

Equation (7.32) is the same transcendental equation for magnetization, which was already discussed in Section 7.2. This equation is usually solved graphically or numerically (see [3.1] or [3.2]). However, approximate analytical solutions can be found around T = 0 and T = TC . Let us return now to the itinerant model. The magnetization is assumed to be a function of the external ﬁeld, m = mH, linear for small ﬁelds. After differentiating both sides of (7.31) (for + electrons) with respect to the external ﬁeld, one obtains E 1 dm D = m=0 PT d 2 dH H −D f 1 PT = − = f 2 exp − /kB T kB T

(7.33)

where f = exp − /kB T + 1−1 is the Fermi function. In the presence of the external ﬁeld, one has for + electrons E = F

m + B H 2

and

E F dm = + B H 2 dH

(7.34)

Itinerant Ferromagnetism

133

Inserting (7.34) into (7.33) and multiplying both sides of (7.33) by 2B one arrives at the expression for the static magnetic susceptibility per atom ( = mB /H:

= IT F + 22B

(7.35)

where IT =

D −D

m=0 PT d

(7.36)

hence

=

22B IT 1 − FIT

At T → 0 K, PT ⇒ − F , IT =

D

T = 0 K =

−D

(7.37)

− F d ⇒ F and

22B F 1 − F F

(7.38)

Using in (7.29) the expression (7.37) or (7.38), one arrives at a new form of Landau expansion, which is = 0 + m2

1 − FIT + a4 m 4 + · · · 2IT

(7.39)

The critical (minimum) value of the exchange ﬁeld for ferromagnetism, F = F cr , is obtained when the denominator of (7.37) [or (7.38) at T = 0 K] is zero. Hence, one arrives at the well-known Stoner criterion [3.2, 4.2] F cr IT ≥ 1

(7.40)

which at T = 0 K, when IT ⇒ F , becomes F cr F ≥ 1

(7.41)

Using the Stoner condition, F cr = 1/IT , in the expression (7.37) for susceptibility and in Landau expansion (7.39), one arrives at their corrected forms:

=

22B F cr − F

= 0 +

m2 cr F − F + a4 m4 + · · · 2

(7.42) (7.43)

134

Models of Itinerant Ordering in Crystals

Φ – Φ0 a b c

m

FIGURE 7.10 The dependence of Landau free energy on the magnetization m for various values of a2 = 1/2F cr − F. Curve (a): F < F cr , curve (b): F = F cr , curve (c): F > F cr . The arrows point to the stable minima for magnetization.

Since a4 > 0, the existence of the minimum in the above equation for m will depend on the sign of a2 in (7.27) or equivalently on the sign of the difference F cr − F in (7.43). As a result, the simple Stoner assumption (7.30) will lead to ferromagnetic instability, with the non-zero magnetic moment being given by the minimum of m curve in Fig. 7.10. This minimum appears only when F > F cr of a given material, where F cr is given by the condition (7.40) or (7.41) with the equality sign. It will be shown later (in Sections 7.5 and 7.6) that when correlation effects are included in the susceptibility, the form of energy expansion given by (7.43) remains unchanged. The correlation factors will change only the expression for F cr from the simple Stoner criterion, F cr = 1/IT (at T = 0 K), to the form given later in those sections.

7.4 STONER MODEL FOR RECTANGULAR AND PARABOLIC BAND 7.4.1 Rectangular band The ferromagnetic state in the case of constant DOS is depicted in Fig. 7.11a. For this DOS the general expression for the number of electrons in the band, (7.31), takes on the form n± =

n±m 1 D d = 2 2D −D exp − ∓ E/kB T + 1

(7.44)

135

Itinerant Ferromagnetism

ε

ε

εF εF +σ

+σ

–σ

–σ

2ΔE

2ΔE

ρ+σ

ρ–σ

ρ+σ

ρ–σ

(a)

(b)

FIGURE 7.11 Rectangular (a) and parabolic (b) DOS with the Stoner shift E = Fm/2. At zero temperature, electrons fill the states to the level of Fermi energy F .

At zero temperature, (7.44) gives the following relations n =

n+m 1 F = d 2 2D −D−E

n− =

n−m 1 F = d 2 2D −D+E

(7.45)

or Dn + m = F + D + E

Dn − m = F + D − E

(7.46)

By adding and subtracting these two equations, one arrives at F = Dn − 1 and

E = Dm

(7.47)

The last relation is puzzling. After inserting into it the value of energy shift, E = Fm/2, and using the Stoner condition F cr = 1/ F , one obtains F = 2D =

1 = F cr F

(7.48)

This means, that for any value of the Stoner ﬁeld (exchange ﬁeld), which fulﬁls the condition F ≥ F cr = 2D (even for F arbitrarily close to F cr , one will instantaneously have m = maximum. This effect is shown even better by directly solving two numerical equations (7.44) for n and m. The results, which were obtained for mT at, e.g. n = 1 1, when maximum mmax = 0 9, are depicted in Fig. 7.12. Looking at these curves one concludes, in agreement with (7.48), that at constant DOS, any Stoner ﬁeld, even slightly larger than the critical value of

136

Models of Itinerant Ordering in Crystals

1 mmax

0.8

m

0.6 0.4

F = 1.3 F = 1.15 F = 1.05

0.2 0

F = 1.015

0

0.1

0.2

0.3

0.4

0.5

kBT

FIGURE 7.12 Magnetization (in units of B versus temperature for the rectangular DOS. Electron occupation is n = 11. The parameters, at different curves, are the values of Stoner exchange field, F, in units of the total bandwidth 2D.

2D, will bring the same saturation magnetization. Later, these only slightly different ﬁelds will result in completely different Curie temperatures. For this reason, the constant DOS cannot describe the ferromagnetic elements within the Stoner model.

7.4.2 Parabolic nearly free electron band The parabolic DOS (see Fig. 7.11b) simulates the behaviour of nearly free electrons (see Section 4.1) and was originally used by Stoner [7.12] in his model. With this DOS (7.31) takes now the following form: n± =

n±m √ d = C 2 exp − ∓ E/kB T + 1 0

(7.49)

The formula derived in Chapter 4.1 for the parabolic DOS can be written as =

√ √ 4 2m3/2 = C 3 h

(7.50)

with constant C treated as the parameter, which will be adjusted to the width of the band from the condition n F = d (7.51) 2 0 which yields =

3 n √ 4 3/2 F

(7.52)

137

Itinerant Ferromagnetism

√ For the band with maximum capacity of n = 2 one has = 3/22D−3/2 . Assuming the value of bandwidth for transition ferromagnetic elements after [4.5], one can calculate the constant C = 3/22D−3/2 . With this constant, one arrives at the Fermi energy for a given 3d element by inserting it into (7.51) together with the electron occupation, n (see Table 7.4). For the parabolic DOS given by (7.52), the expressions (7.49) can be written as 3/2 √ 3 d 1 (7.53) n± = n 4 F exp − ∓ E/kB T + 1 0 From equations (7.53), one can obtain the magnetization m = n − n− (in units of B and electron occupation number n = n + n− : 3 kB T 3/2 m= n F1/2 + − F1/2 − 4 F

(7.54)

3 k T 3/2 n= n B F1/2 + + F1/2 − 4 F

where = /kB T = E/kB T and E = Fm/2, and the Fermi integral, Fp y, is deﬁned as xp dx (7.55) Fp y = 0 expx − y + 1 We will now try to ﬁt this general Stoner model with the parabolic DOS to the ferromagnetic 3d transition metals. To do this, ﬁrst the magnitude of the Stoner ﬁeld, F , will be ﬁtted to the value of the magnetic moment at zero temperature with the help of formulas (7.54) at T → 0. Next, using the same formulas with increasing temperature, the point at which mT → 0 will be found, which deﬁnes the Curie temperature, TC . The results are given in Table 7.4, together with the values of n, m and experimental TC taken after Kittel [3.1]. Table 7.4 Curie temperatures in the Stoner model for ferromagnetic 3d elements Element Fe Co Ni

exp

par

n

m

D (eV)

F eV

F (eV)

TC K

TC K

1 4 1 65 1 87

0 44 0 344 0 122

2 80 2 65 2 35

4 41 4 66 4 49

4 237 3 780 3 205

1043 1388 627

4820 3383 1021

n = 3d electron number, normalized to 2; m = magnetic moment at 0 K normalized to 2; F = Fermi exp energy in eV from (7.51) with C = 3/42/2D3/2 ; F = Stoner ﬁeld in eV; TC = experimental par Curie temperature in K; TC = Curie temperature from the Stoner model with parabolic DOS.

138

Models of Itinerant Ordering in Crystals

Comparison of the last two columns shows that in the Stoner model there is a large discrepancy between the value of exchange ﬁeld necessary to create the magnetic moment and the stability of that moment with temperature. This leads to the Curie temperatures from this model being much higher than the experimental data. In the original Stoner model the differences were even larger, since the constant C in expression (7.50) was smaller as it was assumed from the nearly free electron DOS model [relation (4.13)]. Such a result is not surprising since the parabolic DOS reﬂects only the nearly free electrons and not the tight-binding electron band responsible for the ferromagnetism. We will try to rectify this discrepancy in the following sections by introducing the DOS corresponding to the tight-binding approximation, and also the inter-site interactions in the modiﬁed H–F approximation. As was shown in Section 6.9.1, this approximation is changing the bandwidth of the band which is centred on the atomic level. For the parabolic nearly free electron band one cannot include the inter-site interactions, since there is no deﬁned atomic level and no rigorously deﬁned bandwidth, since the band extends to inﬁnite energies.

7.5 MODIFIED STONER MODEL Before proceeding further with the Stoner model, let us recall the general extended Hubbard Hamiltonian from Section 5.2: + t − tnˆ i− + nˆ j− + 2tex nˆ i− nˆ j− ci cj − 0 nˆ i − F n nˆ i H =−

+

i

i

+ + U V J + + nˆ i nˆ i− + nˆ i nˆ j + ci cj ci cj + J ci↑ ci↓ cj↓ cj↑ 2 i 2 2 (7.56)

In this section, it will be assumed temporarily that there is no Coulomb on-site correlation, U ≡ 0. The inﬂuence of Coulomb on-site correlation will be investigated in Section 7.6. For the inter-site interactions appearing in the above Hamiltonian, the modiﬁed H–F approximation has been developed in Section 6.9.1 and in Appendix 6C (following [6.15, 7.13]). In this approximation, the averages of two operators on the neighbouring lattice sites, in addition to the averages of two operators on the same site, which create the Stoner ﬁeld considered in the standard H–F approximation, have been retained. The inter-site averages are proportional to the average value of the kinetic energy. As a result, they contribute to the electron dispersion relation by changing the bandwidth (factor b ). This is in addition to the Stoner shift of both spin sub-bands coming from the averages of two operators on the same site. Final result is given by the expression below: k = k b − E

(7.57)

Itinerant Ferromagnetism

139

where the bandwidth change factor is b = 1 − 2

J −V t t J + J 2 n− + 2 ex n2− − I− − 2I I− − I − I t t t t − + I = ci cj

(7.58) (7.59)

F , The Stoner shift, E , depends on the Stoner ﬁeld for ferromagnetism, Ftot and the external magnetic ﬁeld H through the relation m F (7.60) + HB E = Ftot 2 F where the Stoner exchange ﬁeld, Ftot , is the sum of on-site exchange interaction, F , the inter-site exchange interaction, J , and the terms coming from kinetic interactions, t and tex (see Appendix 6C) M − − M 2 F = F + z J + 2tex I + I− − I − I− ntex − 2t (7.61) Ftot = m m

This total ﬁeld, composed of different interactions, will replace the simple exchange ﬁeld, F , from the previous sections of this chapter in all formulas for susceptibility, free energy, etc. Assuming for I− the lowest order approximation, I− = n− 1 − n− (this result is strict for the constant DOS at zero temperature), one arrives at the following relation: F Ftot = F + zJ + tex n2 − m2 + 2t1 − n

(7.62)

Expression (7.31) for the electron concentration is modiﬁed now and takes on the form n±m 1 1 = n± = ± 2 N k exp k b − − E± /kB T + 1 (7.63) D d = exp b± − − E± /kB T + 1 −D Assuming, as in Section 7.3, that the magnetization is a linear function of the applied external ﬁeld, m = mH and differentiating (7.63) for n+ with respect to H, one obtains F 1 dm D b dm Ftot dm PT b0 − (7.64) = + + B d 2 dH m dH 2 dH −D Multiplying both sides of the above equation by 2B , one arrives at the expression for the static magnetic susceptibility:

=

22B IT F 1 − Kij − Ftot IT

(7.65)

140

Models of Itinerant Ordering in Crystals

where IT was deﬁned previously by (7.36), and the inter-site correlation factor is given by D b m=0 PT b0 d (7.66) Kij = −2 m m=0 −D At zero temperature one has PT b0 ⇒ − F /b0 , with b0 = lim b m, m→0 which produces the following expression for the inter-site correlation factor at zero temperature: 2 F F b (7.67) Kij = − b0 m m=0 After equating the susceptibility denominator in (7.65) to zero one arrives at cr T = Ftot

1 − Kij T IT T

⇒

T→0 K

1 − Kij 0 F /b0

where IT →

T →0 K

F b0

(7.68)

cr We can see that the critical value of the Stoner ﬁeld, Ftot , is modiﬁed now by the inter-site interactions. Inserting (7.68) into (7.65) one has for the susceptibility

=

22B cr F Ftot − Ftot

(7.69)

It can be seen that the presence of the inter-site interactions described by the inter-site correlation factor does not change the expression for susceptibility, which is the same as the expression without correlation [relation (7.42)]. There will be also the same Landau expansion (7.43) for the free energy, but the cr = 1/IT → b0 / F , will be replaced now by (7.68). relation, Ftot T →0 K The parameter I entering susceptibility through the inter-site correlation factor (7.67) is, according to its deﬁnition (7.59), proportional to the average kinetic energy of electrons with spin : + + ci cj = −tz ci cj = −DI (7.70) K = −t ij

and therefore it can be written as D d − I± = D exp b± − − E± /kB T + 1 −D

(7.71)

The Curie temperature, at which magnetization vanishes, is calculated from the zero of the susceptibility denominator [(7.65) with the help of (7.36), (7.58), (7.62) and (7.67)] attained with temperature.

Itinerant Ferromagnetism

141

7.5.1 Modified Stoner Model for a semi-elliptic band In this section the model developed immediately above will be investigated numerically using semi-elliptic DOS. The reasons for using semi-elliptic DOS will now be summarized. In Section 7.4.1 the original Stoner model with constant DOS and without the inter-site interactions has been analysed, but it emerged that with this DOS there is no a unique relation between the strength of the Stoner exchange ﬁeld, F , and the magnetization. The other simple DOS used in connection with the Stoner model was the parabolic DOS in Section 7.4.2. When it was applied with the on-site Stoner ﬁeld to calculate experimentally the observed magnetic moments at zero temperature, it resulted in values of the Curie temperature that were much too high when compared with the experimental data for transition metals (see Table 7.4). Moreover, attempts to use it together with the inter-site interactions failed, since numerical values of m at T = 0 K versus the strength of inter-site interactions were decreasing with increasing interactions V and J . This non-physical feature does not allow for the use of parabolic DOS with the modiﬁed Stoner model. It may be not a coincidence that the parabolic DOS cannot describe ferromagnetic elements since this DOS merely describes the nearly free electrons. Ferromagnetism appears only in pure transition elements where the band showing ferromagnetism is of the tight-binding type (see Section 4.2). Such a band always centres on the atomic level and has a ﬁnite width. This width is modiﬁed by the inter-site interactions. Both the atomic level and the bandwidth cannot be deﬁned in the case of a parabolic DOS. For these reasons, the simple semi-elliptic DOS [5.4] will now be used as the DOS of the tight-binding type, with a ﬁnite width and centred on the atomic level. As opposed to constant DOS, it is proportional to the square root of the energy at the bottom and top of the band: 2 2 = 1− (7.72) D D This DOS at zero temperature from (7.63) gives the relation ⎡ ⎤ ± 2 ± 1 1 ⎣ ± F 1− F + arcsin F ⎦ n± = + 2 D D D

1 1 ± ± + sin ± F cos F + F 2 ±

with ± F = arcsin F /D . In agreement with (7.57) and (7.63) one has =

F =

F + E b

− F =

F + E− b−

(7.73)

(7.74)

142

Models of Itinerant Ordering in Crystals

or − F b − E = − − E− F b

(7.75)

The quantities b± and E are deﬁned by (7.58) and (7.60), respectively. The parameter I , given by (7.71), is proportional to the average kinetic energy K = −DI [see (7.70)] and therefore for semi-elliptic DOS at zero temperature takes on the following form: I± =

± F −D

± 2 3/2 2 2 2 2 1− − d = 1− F = cos3 ± F D D D 3 D 3 (7.76)

The set of equations (7.73)–(7.76) and (7.60) with the ignored external ﬁeld H = 0 can be solved numerically. One can ﬁnd the on-site Stoner ﬁeld necessary to create a given magnetization, Fm, for different values of inter-site interactions: V , J , J , t and tex . After ignoring in this section the on-site interaction, U, in the Hamiltonian (7.56), the critical temperature is inﬂuenced by (i) the inter-site interactions V , J and J and (ii) the kinetic interactions t = t − t1 and tex = t + t2 /2 − t1 . To limit the number of free parameters it will be assumed here that J = J [7.14] and V = 0. For the kinetic interactions it will be assumed that the ratio of hopping integrals with different occupation, t, t1 and t2 , is constant t1 /t = t2 /t1 = S, which produces t = t1 − S

tex = t

1 − S2 2

(7.77)

Under these simpliﬁcations, at the limit of T = 0 K and magnetization m → 0, one has from (7.62) and (7.77) a new equation for the critical on-site Stoner ﬁeld (the identity D = zt was used): 1 − S2 2 cr cr F = Ftot − zJ − D (7.78) n + 21 − S1 − n 2 cr where Ftot is calculated from (7.68). From this equation one can calculate the critical on-site Stoner ﬁeld, F cr , at a given electron concentration for various parameters J and S. The results are shown in Fig. 7.13. It can be seen from this ﬁgure that decreasing the factor S and increasing the inter-site exchange interaction J decrease the on-site Stoner ﬁeld required to create ferromagnetism, even to zero, at some electron concentrations. The presence of the hopping inhibiting factor S and inter-site exchange interaction J will also have a positive inﬂuence on the Curie temperature by decreasing it towards the experimental values.

143

Itinerant Ferromagnetism

3

Fcr[D]

2

1

0

0.5

0

1 n

1.5

2

FIGURE 7.13 The dependence of critical on-site Stoner exchange field, F cr , on the electron occupation, for different values of S and J; S = 1 and J = 0 (the Stoner model) – dot-dashed line; S = 06 and J = 05t – solid line; S = 1 and J = 05t – dashed line; S = 06 and J = 0 – dotted line. Table 7.5 Curie temperatures for ferromagnetic elements, modified stoner model Element

Fe Co Ni

n

m

No. 1 TC K S = 0 6, J = 0 5t

No. 2 TC K S = 0 6, J =0

No. 3 TC K S = 1, J = 0 5

No. 4 TC K S = 1, J ≡0

No. 5 exp TC K

1 4 1 65 1 87

0 44 0 344 0 122

2050 1690 620

3295 3300 870

3980 3880 1720

4290 4710 1960

1043 1388 627

n = 3d electron number, normalized to 2; m = magnetic moment at 0 K, normalized to 2. The other columns are nos. 1, 2, 3–TC S J = Curie temperature for different J , S, Fm is ﬁtted to m at 0 K; no. 4–TC S = 1 J = 0 = Curie temperature for the case of Stoner model, no inter-site interactions J = J = V = t = tex ≡ 0, Fm is ﬁtted to m at 0 K; exp no. 5–TC = experimental Curie temperature in K.

Now, using the same equations (7.73)–(7.76) and the electron occupation numbers corresponding to 3d ferromagnetic elements given in Table 7.5, we adjust the value of the Stoner on-site ﬁeld F [see (7.62)] at the given J and S in the limit of T → 0 K to the experimental magnetization at zero temperature mT → 0 K. The Curie temperature, at which magnetization vanishes, is calculated from the zero of the susceptibility denominator (7.65) attained with temperature with the help of (7.62), (7.66) and using the semi-elliptic DOS of (7.72). The same constants F , J and S were used, which were ﬁtted to the experimental magnetization at 0 K. The results are collected in Table 7.5.

144

Models of Itinerant Ordering in Crystals

It is very interesting to compare different results for TC , which are shown in this table. Column 4, which is the Stoner model for semi-elliptic DOS, and column 8 in Table 7.4 for Stoner model with parabolic DOS show that all theoretical results of the pure Stoner model are much higher than the experimental Curie temperatures (column, Table 7.5). This means that the Stoner model, which assumes that the on-site atomic ﬁeld creates ferromagnetism, overestimates, to a large extent, the Curie temperature. Perhaps ferromagnetism is the result of inter-site forces changing the bandwidth, since column 1 with strongest inter-site interactions is closest to the experimental results. The larger the component of on-site ﬁeld (i.e. S → 1, J → 0), the more the TC theoretical results exceed the experimental values. These simple calculations would conﬁrm the earlier attempts of understanding the itinerant ferromagnetism as the effect of ordering local moments, whose alignment disappears at Curie temperature. However, the moments themselves exist up to temperatures exceeding at least twice the Curie temperature [7.15]. One could conclude that the local moments were created by the on-site Stoner ﬁeld, but their ordering would be driven by the inter-site interactions, which are much weaker than the on-site ﬁeld. A conclusion of this type is similar to that of Hubbard [7.16]. His calculations “imply that two energy scales are operative in iron, one of the order of electron volts which is characteristic of the itinerant behaviour (e.g. the bandwidth and the exchange ﬁelds), and another of the order of one tenth of an electron volt characteristic of the localized behaviour [e.g. kB TC , the EV]”. In our model this larger ﬁeld would be the on-site ﬁeld creating local moments and the smaller ﬁeld would be the inter-site ﬁeld responsible for their ordering. The relatively weak inter-site interactions make the ordering of local moments less stable with temperature, which will encourage the creation of spin waves with increasing temperature, and will additionally decrease the magnetic moment. Their inﬂuence on TC will be analysed in Section 7.8.

7.6 BEYOND HARTREE–FOCK MODEL 7.6.1 General formalism In Section 6.5, the coherent potential approximation (CPA) for the simple Hubbard model with U = 0 has been developed. The results allow for a description of the process of deforming and ﬁnally splitting the band by the strong Coulomb repulsion. This change of band shape is something different to the shift of the entire band produced by the original H–F approximation. The modiﬁed H–F approximation was introduced in Section 6.9.1 for the intersite interactions, J , J , V , the kinetic interactions, t, tex , and it was used in Section 7.5. It allows for the expansion or contraction of the entire band with

Itinerant Ferromagnetism

145

respect to its atomic centre. However, it is not able to deform and split the band in the case of strong interactions. The many body results obtained in Section 6.9.2 and Appendix 6C will now be used to analyse the ferromagnetic ground state by combining the CPA for the on-site strong Coulomb correlation, U , with the modiﬁed H–F approximation for the weaker inter-site interactions. As was shown in Section 6.9.2 [see also (7.85)], the DOS per spin is a function of self-energy, U , describing the on-site Coulomb correlation, U , and the bandwidth factor, b , describing the inter-site interactions. The inter-site self-energy, 12 , was replaced by its ﬁrst-order approximation, which is the average inter-site energy [7.14]. For the number of electrons with spin ±, the equation is similar to those used previously [(7.31) and (7.63)] n± =

−

±

d exp − ∓ E± /kB T + 1

(7.79)

but now the spin-dependent DOS, ± , has replaced the previous paramagnetic DOS, . It has already been deformed by both on-site and inter-site correlations. The schematic depiction of the DOS deformed by the on-site Coulomb correlation is shown in Fig. 7.14.

ε

–

–

+σ

–σ

εF

+ +

ρ+σ

ρ–σ

FIGURE 7.14 Schematic DOS showing the influence of the strong on-site Coulomb correlation, U. The paramagnetic DOS for both spins, ± , are solid lines. When U is strong enough, the band is split into two sub-bands. Lower sub-bands have the capacity of 1 − n− for + electrons and 1 − n for − electrons; upper sub-bands have the capacity of n− for + electrons and n for − electrons. The changes in the spin electron densities integrated over energy up to the Fermi level are shown as the shaded areas in this figure and are equal to the correlation factor KU .

146

Models of Itinerant Ordering in Crystals

In the case of strong correlation, U D, the upper and lower sub-bands shown above can be described analytically for the semi-elliptic band (see Section 7.5) by the following equations: For the lower Hubbard sub-band 2 21 − n− = 1 − D D where −D ≤ ≤ D , D = 1 − n− Db ; For the upper Hubbard sub-band

2 2n− = 1 − D D √ where −D ≤ ≤ D , D = n− Db .

(7.80)

(7.81)

The modiﬁed H–F approximation was additionally introduced, which has changed the original half bandwidth, D, to the effective half bandwidth, D = Db , even in the case of the weak inter-site correlation (see Appendix 6C). The schematic depiction of the DOS deformed by the inter-site correlation is shown in Fig. 7.15.

ε

εF + –σ

+σ +

– –

ρ +σ

ρ –σ

FIGURE 7.15 Schematic DOS showing the influence of the inter-site interactions. The paramagnetic DOS for both spins, ± , are solid lines. The inter-site interactions [described by the b factors, see (7.58)] change the relative width of the bands with respect to each other. The Stoner field, which would displace the bands with respect to each other, is assumed to be non-zero. The shaded areas in this figure are the correlation factor Kij .

Itinerant Ferromagnetism

147

To calculate the susceptibility, one proceeds as previously and differentiates (7.79) with respect to the external ﬁeld H, obtaining d 1 dm dm = 2 dH m dH exp − /kB T + 1 − F Ftot dm + m=0 PT + B d 2 dH −

(7.82)

With this result, one arrives (see also Sections 7.3 and 7.5) at the following equation for the susceptibility

= B

dm 22B IT = F dH 1 − K − Ftot IT

(7.83)

where IT is given by (7.36) and K is the correlation factor deﬁned mathematically by the following equation: D − D d d − K= m exp − /kB T + 1 m exp − /kB T + 1 −D −D (7.84) D d =2 m exp − /kB T + 1 −D and is equal to the sum of all changes in the DOS over the energies below the Fermi level, which are depicted by the shaded areas in Figs 7.14 and 7.15. According to the many body theory (see Chapter 6), the density of states can be expressed as 1 = − ImF

(7.85)

where F is the Slater–Koster function [see [7.14] and (6.167)] depending on the self-energy responsible for the interaction, U , and on the factor b responsible for the inter-site interactions − 1 1 1 (7.86) F = = F N k − k b − b 0 b The derivative /m can be written as 1 F 1 F F b = − Im = − Im + m m m b m

(7.87)

which leads to the separation of the correlation factor into two correlation factors. One, KU , is responsible for the on-site Coulomb repulsion and another one, Kij , for all the inter-site interactions K = KU + Kij

(7.88)

148

Models of Itinerant Ordering in Crystals

where KU = −

2 D F d Im −D m exp − − E /kB T + 1

(7.89)

Kij = −

2 D F b d Im −D b m exp − − E /kB T + 1

(7.90)

Taking into account (7.88), the susceptibility (7.83) can be written as

=

22B IT F 1 − KU − Kij − Ftot IT

(7.91)

cr The critical value of the total Stoner ﬁeld, Ftot , can now be obtained. From the zero of the susceptibility denominator one has cr Ftot =

1 − KT 1 − KU T − Kij T = IT T IT T

→

T →0 K

1 − KU 0 − Kij 0

compare 7 68

(7.92)

F /b0

Using this formula back in (7.91), one again obtains the same expression for the susceptibility as in Sections 7.3 and 7.5.

=

22B cr F Ftot − Ftot

(7.93)

It is also the same equation (7.43) for the free energy, as was written for the case without correlation effects. But now, the critical value of the Stoner ﬁeld, which was given by (7.41), is replaced by the value from (7.92), which is lowered by the correlation effects.

7.6.2 Enhancement of magnetic susceptibility The susceptibility given by (7.91) or (7.93) is not divergent for most of the pure elements, as only few of them are magnetic. But for many pure elements the denominator of susceptibility is signiﬁcantly decreased, which produces the experimentally observed susceptibility enhancement. This phenomenon has been extensively studied in the past. When the H–F approximation is applied to the interaction UKU ≡ 0, and in the absence of the inter-site interactions F ≡ F + U , where the interaction U was added to the Kij ≡ 0, b0 = 1, one gets Ftot result of (7.61). Using these values and relation (7.92) in (7.91), one can write

22B 0 0F = P A

= (7.94) F 0 1 − Ftot 0F

Itinerant Ferromagnetism

149

where P = 22B 0 0F is the bare

Pauli susceptibility, which is enhanced by F 0 the factor A = 1/ 1 − Ftot 0F called the Stoner enhancement factor. In the case of non-zero correlations, this enhancement is given by F 0 A = 1/ 1 − KU − Kij b0 − Ftot 0F (7.95) F with Ftot given by (7.61). In the past, the CPA describing the on-site repulsion U , KU = 0, b0 = 1, was already used to calculate the susceptibility enhancement, see [6.4] for the general model of pure elements and [7.17–7.19] for susceptibility of disordered binary alloys. Experimental data and theoretical results from the local density functional method were collected more recently for pure transition elements by [4.2]. In this chapter, the factor A expressed by (7.95) is additionally increased by the inter-site interactions, Kij = 0, b0 < 1. This effect should also be included in the investigation of experimental data on susceptibility enhancement.

7.6.3 Critical values of interactions The results for the critical values of interactions in the higher order approximation will now be calculated. The inter-site interactions will still be treated in the modiﬁed H–F approximation (see Section 7.5) with b given by (7.58). The strong on-site Coulomb interaction will be described by the full CPA. For reasons mentioned in Section 7.5, the semi-elliptic DOS will be used in the numerical analysis [see (7.72)] for which the Slater–Koster (7.86) function has the following form: ⎡ ⎤ 2 2 − − ⎣ − − 1⎦ (7.96) F = Db Db Db This expression was derived from (6.99) and (6.96), with the additional change of D → Db . The above equation, together with the CPA expression for the on-site self-energy depending on the Coulomb correlation U in the Soven [6.2] form [see (6.91) and (6.92)] Un− − + U − F = 0

(7.97)

leads to the cubic equation for F

Db F 3 U − 2 Db F 2 +2 2 Db 2 4 − U Db F U − Un− − + +1 = 0 +2 2 Db 2 Db

(7.98)

150

Models of Itinerant Ordering in Crystals

The solution of (7.98) has three branches, two of which may be complex depending on the choice of parameters , U , D, b and n− . In further analysis, the complex solution, where the imaginary part is negative, will be most interesting because = −ImF . Equation (7.98) with eliminated selfenergy allows the correlation factor given by (7.89) to be written as 2 F n− d KU = − (7.99) Im − n− m exp − − E /kB T + 1

7.6.4 Numerical results Let us investigate, as in Section 7.5, how the critical temperature is inﬂuenced by (1) the inter-site interactions V , J and J (2) the kinetic interactions t = t − t1 and tex = t + t2 /2 − t1 . To these interactions it will be added now (3) the on-site Coulomb correlation. Figure 7.16 and 7.18 represent the dependence of intra-atomic critical ﬁeld, F cr n, calculated from (7.92) with the help of (7.78): F cr =

1 − KU 0 − Kij 0 F /b0

− zJ − D

1 − S2 2 n + 21 − S1 − n 2

(7.100)

where Kij is given by (7.66) and KU by (7.99). To obtain Kij , KU and F one has to solve (7.98) at a given U in the paramagnetic limit of m → 0 or n± → n/2. The strong on-site correlation, U , and zero (Fig. 7.16) or non-zero (Fig. 7.18) values of the inter-site interactions were assumed. In the case of non-zero inter-site 3

2

Fcr[D]

U=0

U = 3D

1 U=∞ 0

0

0.5

1 n

1.5

2

FIGURE 7.16 The dependence of the critical on-site Stoner field on electron occupation, F cr n, in the units of half band width. Figure shows the influence of the on-site correlation U alone treated in the CPA (it is assumed J = 0 S = 1).

Itinerant Ferromagnetism

151

interactions it was assumed as before that J = J , V = 0 and t tex are expressed by one parameter S, as in (7.77) and (7.100). Figure 7.16 shows the well-known result that the strong correlation in the CPA U D U = 3D decreases the values of the on-site Stoner ﬁeld creating ferromagnetism when compared to the results without correlation U = 0. The difference between the curves for U = 3D and U = is quite small. Therefore, from now on the case of U = will be assumed (although it seems to be unrealistic) just to simplify further calculations. For comparison we have shown in Fig. 7.17 results of the modiﬁed Hubbard III approximation in which we have included the effect of inter-site correlation + I− = ci− cl− . The self-energy was calculated from (6.103) with included T bandwidth correction, Bk , and bandshift correction, ST , which are both

+ proportional to the factor I− = ci− cl− . As can be seen from this ﬁgure, the inter-site correlation in the CPA causes, in the strong correlation limit U >> D, the rise of ferromagnetism at the whole range of electron occupations without the help of additional inter-site interactions. + Returning to the standard CPA (without the factor I− = ci− cl− , we show in Fig. 7.18 that the inter-site interactions, J and S, decrease the critical Stoner ﬁeld dramatically in the presence of the on-site correlation U . The increase of J causes the decrease of F cr for concentrations of a nearly half-ﬁlled band and also at small concentrations and concentrations close to a completely ﬁlled band. The decrease of S from 1 to 0 causes a drop of the critical Stoner

Fcr[D]

4

2

0

–2

0

0.5

1 n

1.5

2

FIGURE 7.17 The dependence of the critical on-site Stoner field on electron occupation,

+ F cr n, in the units of half band width. The inter-site kinetic correlation I = ci cj included in the Hubbard III approximation – bold line. Both the bandwidth factor BTk , and the

+ bandshift correction, ST , proportional to the correlation factor I− = ci− cl− , are present in relation (6.103) on which the calculations were based. Standard CPA – thin line. The curves are calculated in the strong correlation limit U D.

152

Models of Itinerant Ordering in Crystals

3

2

Fcr[D]

S = 1, J = 0

S = 0.6, J = 0

S = 1, J = 0.5t S = 0.6, J = 0.5t

1

0

0

0.5

1 n

1.5

2

FIGURE 7.18 The dependence of the critical on-site Stoner field on electron occupation, F cr n, in the units of half band width. Figure shows the influence of inter-site interactions, J = J , V = 0, and of the hopping interactions, t tex , represented by the hopping inhibiting factor S, in the presence of strong on-site correlation, U = , treated in the CPA. 0.5 0.4

m

0.3 0.2 0.1 0

0

1000

2000

3000

4000

5000

6000

T

FIGURE 7.19 Magnetization dependence on temperature, mT, without and with strong Coulomb correlation, n = 14 D = 28 eV. The curves are: U = 0, J = 0, S = 1 – solid line; U = , J = 0, S = 1 – dashed line; U = 0, J = 05t, S = 06 – dotted line; U = , J = 05t, S = 06 – dot-dashed line.

ﬁeld F cr , especially for small n < 0 5, and for 1 < n < 1 5, where both Hubbard sub-bands begin to ﬁll. The Curie temperature and the mT dependence are also inﬂuenced by the Coulomb correlation U . Fig. 7.19 shows the dependence mT with the strong U correlation and without it. In both cases, two pairs of inter-site interactions were assumed: zero J = 0, S = 1, or strong J = 0 5t, S = 0 6. The magnetization was calculated from coupled equations (7.79). For strong correlation U/D = , the analytical densities given by (7.80) and (7.81) have been used. In the case

Itinerant Ferromagnetism

153

of non-zero inter-site interactions J = 0 or S < 1, the parameter of bandwidth change b± = 1. It is expressed by (7.58), with the quantity I± given by (7.71) generalized to the case of spin-dependent DOS: I± =

d ± − ± D exp b± − − E± /kB T + 1 −D± D±

(7.101)

The densities given by (7.80) and (7.81) will be inserted here, but for sim+ cj will be assumed plicity, the zero temperature stochastic value of I = ci as the probability of electron hopping from the jth to ith lattice site. With these assumptions one arrives at the following approximate expressions [7.20]: For the lower Hubbard sub-band, I = n 1 − n/1 − n− ; For the upper Hubbard sub-band, I = n − 11 − n /n− . Inserting these stochastic values into (7.61) and neglecting the paramagnetic terms, one has in this case of strong on-site correlation the following Stoner exchange ﬁelds: For the lower Hubbard sub-band, F = F + zJ + 2z Ftot

1 − n 2t n2 − m2 + 4t1 − n 2 − n2 − m2 ex

(7.102)

For the upper Hubbard sub-band, F Ftot

n−1 = F + zJ + 2zn − 1 2tex − 4t 2 n − m2

(7.103)

The above Stoner ﬁelds were used in the numerical calculations of mT. The curves were calculated for parameters corresponding to Fe [n = 1 4 m0 = 0 44 and D = 2 8 eV]. Fig. 7.19 shows that the inter-site and kinetic interactions decrease the Curie temperature for both cases of U = 0 and U = , but the on-site correlation, U , does not decrease the Curie temperature. In fact it even slightly increases it, pushing it farther apart from the experimental value of 1043 K. The presence of on-site and inter-site correlations also lowers the values of the Stoner on-site ﬁeld, F , necessary to create the 0 44 B moment at zero temperature. This can be seen in Fig. 7.18 since the value of the Stoner ﬁeld creating m = 0 44B is only slightly larger than the critical values displayed there. Table 7.6 shows the Curie temperatures obtained for realistic values of inter-site interactions S = 0 6 J = 0 5 at U = . These values are in the range of experimental data especially for Co and Ni. It can also be seen that the inﬂuence of strong on-site correlation U = does not lower the theoretical Curie temperature when compared to the case without on-site correlation U = 0 shown in Table 7.5.

154

Models of Itinerant Ordering in Crystals

Table 7.6 Curie temperatures for ferromagnetic 3d elements, modified Stoner model with U = Element

Fe Co Ni

exp

n

m

TC K S = 0 6, J = 0 5t

TC K S = 0 6, J =0

TC K S = 1, J = 0 5t

TC K S = 1, J =0

TC K

1 4 1 65 1 87

0 44 0 344 0 122

2420 1850 630

4000 3270 910

3810 3990 1770

4960 5180 2020

1043 1388 627

n = 3d electron number, normalized to 2; m = magnetic moment at 0 K, exp normalized to 2; TC = experimental Curie temperatures, in K.

Let us summarize now the results for the Curie temperature of 3d elements obtained in this chapter. First, the direct calculations within the original Stoner model (Section 7.4) have indicated that the Curie temperature, after ﬁtting the Stoner shift to the experimental magnetic moment at zero temperature, is much too high (see Table 7.4). Next, using the modiﬁed H–F approximation for the inter-site interactions (Section 7.5), the much lower Curie temperatures were arrived at, and considering the simplicity of the model, it was close enough to the experimental data (see Table 7.5). Apparently the inter-site interactions are softer and decrease faster with the temperature than the on-site Stoner ﬁeld used in the original Stoner model [7.16]. Adding the on-site strong correlation U in this section has decreased the critical ﬁeld and enhanced magnetic susceptibility, but did not improve the values of the theoretical Curie temperature (see Table 7.6). This fact can be understood better after examining Fig. 7.16, where the critical ﬁeld (initializing magnetization) has dropped at half-ﬁlling but not at the end of the band, where the 3d elements are located. As already established [7.20, 7.21] the electron correlation can help in creating antiferromagnetism (AF) at the half-ﬁlled band, where the antiferromagnetic 3d elements (Cr, Mn) are located, by dropping the critical ﬁeld for AF to zero. This type of ordering will be analysed in Chapter 8. As mentioned previously, the model is very simple; the details of the realistic DOS could be included as well as the magnetization decrease through the spin waves excitation, which would bring the theoretical results to complete agreement with the experimental data.

7.7 THE CRITICAL POINT EXPONENTS Within the Curie–Weiss law, the following relation for magnetic susceptibility was established earlier in this chapter

= with TC given by (7.24).

C T − TC

(7.104)

155

Itinerant Ferromagnetism

Comparing this result with Table 7.3, we can see that the critical point exponent in our model is = 1. To follow this model and obtain the exponent from Table 7.3, let us recall the free-energy expansion versus the ordering parameter = 0 + 2B m2 / + a4 m4 + · · ·

(7.105)

According to (7.104) one has (see also [3.2]) = 0 + 2B m2 T − TC /C + a4 m4 + · · ·

(7.106)

Hence for the minimum of free energy one gets 0=

d = 22B mT − TC /C + 4a4 m3 + · · · dm

from which it follows immediately that for T < TC ,

2B m= T − T 2a4 C C

1/2

which gives the mean-ﬁeld value of = 1/2 (compare with Table 7.3). In the itinerant model, the formula derived earlier in Section 7.6 for the static susceptibility including the on-site and inter-site correlations will be used

=

22B IT F 1 − K − Ftot IT

(7.107)

where K = KU + Kij

(7.108)

Inserting (7.107) into (7.105), expression (7.43) has been obtained = 0 +

m2 cr F + a4 m4 + · · · Ftot − Ftot 2

(7.109)

with the critical value of the interaction for magnetic ordering given by (7.92) cr Ftot T =

1 − KT 1 − KU T − Kij T = ≡ Fcr0 1 + T IT T IT T

(7.110)

cr where Fcr0 ≡ Ftot T = 0 K and

IT =

D −D

m=0 PT d

PT = −

f 1 = f 2 exp − /kB T kB T (7.111)

156

Models of Itinerant Ordering in Crystals

KU = 2

D −D

Kij = 2

D −D

d m U =0 exp − − E /kB T + 1 (7.112)

d m Vij =0 exp − − E /kB T + 1

where /mU =0 is the change in the DOS for an inﬁnitesimal magnetization at only on-site U = 0 (the inter-site interactions Vij = 0) and /mVij =0 is the change in the DOS for only Vij = 0 U = 0 (see Figs 7.14 and 7.15). Minimizing expansion (7.109) with respect to magnetization with the value cr T given by (7.110), one obtains of Ftot m2 =

F Ftot − Fcr0 1 + T b

b = 4a4

(7.113)

At T = TC the numerator in (7.113) will disappear, which gives

F Ftot = Fcr0 1 + TC

(7.114)

Returning this value into (7.113) one has

m = 2

Fcr0 TC − T

or

b

m=

Fcr0 TC − T b

(7.115)

At T = 0 K one has m = m0 and T = 0 in (7.115), hence m0 =

Fcr0 TC b0

and m=

Fcr0 TC 1 − T /TC b0

bT /b0

= m0

1 − T /TC bT /b0

≈ m0

1 − T /TC bTC /b0

(7.116)

where b0 K ≡ b0 , bT ≡ bT , and it was assumed that bT ≈ const around TC . From relation (7.116) one gets = 1/2 for the critical point exponent of magnetization in Table 7.3, when T ∼ T or even for T ∼ T 2 approximately in the area of T ≈ TC (see below) √ m 1 m = TC2 − T 2 ≈ 0 2TC − T1/2 m0 TC bTC /b0 TC bTC /b0

Itinerant Ferromagnetism

157

The assumption of T ∼ T 2 has been well justiﬁed for temperatures T TF ≡ F /kB where one has [7.22] 2 F F 2 T = − kB T2 ≡ aT 2 T > 0 − (7.117) 6 F F with F and F being the derivatives over energy at the Fermi level. Justiﬁcation of assumption T << TF ≡ F /kB can be seen from Table 4.1. The difference between ≈ 0 5 and 0 34 − 0 42 (Table 7.3) may be attributed to either ignoring the numerical result of (7.110) or to ignoring the spin waves in this approach [7.23]. For the susceptibility one has from (7.69) and (7.110) the expression 22B 22B = 0 (7.118) F F − Ftot Fcr 1 + T − Ftot

F At T = TC there is Fcr0 1 + TC = Ftot , which combined with (7.110) gives

=

cr Ftot

T =

Fcr0

22B

T − TC

(7.119)

Out of this expression one gets = 1 for the critical point exponent of susceptibility in Table 7.3, when T ∼ T or even for T ∼ T 2 [according to (7.117)] in the area of T ≈ TC (see below)

T =

22 2 1 22B

= B 2 0 B 2 0 0 F aT T − TC Fcr T − TC Fcr a T − TC C cr

(7.120)

As in the case of the exponent, the numerical result of (7.110) and the inﬂuence of spin waves were not taken into consideration.

7.8 SPIN WAVES IN FERROMAGNETISM The spin waves will be analysed within the frame of the classic Hubbard model given by expression (5.8). Transformed into the momentum space, it takes on the form U + + ck + ck+q↑ ck↑ ck+ −q↓ ck ↓ (7.121) H = k ck N k kk q The ground ferromagnetic state can be deﬁned as ! + F ≡ ck↑ 0 k < f

where 0 is the vacuum state.

(7.122)

158

Models of Itinerant Ordering in Crystals

Let us introduce the operators of magnetic excitation + + Skq = ck+q↑ ck↓

+ − Skq = ck+q↓ ck↑

(7.123)

and Sq± =

1 ± S N k kq

(7.124)

The state of one excited spin wave can be deﬁned as q = bq+ F

(7.125)

where 1 1 + + bq+ ≡ √ fq rj − ri eiq·ri ci↓ cj↑ = √ fq kck+q↓ ck↑ N ij N k

(7.126)

The dispersion relation for the spin-wave energy can be found from the relation " # $ % F bq H bq+ F − " % q = (7.127) F bq bq+ F Using in (7.127) the Hamiltonian (7.121) one obtains #

H bq+

$

1 1 U + =√ fq k k+q↓ − k↑ ck+q↓ ck↑ + √ fq k − N pr N k N k + + + + × cp+r↑ cp↑ ck+q−r↓ ck↑ − ck+q↓ ck−r↑ cp−r↓ cp↓

(7.128)

Let us now introduce the random phase approximation (RPA) to the last term above: " % " % + + + + + + cp+r↑ cp↑ ck+q−r↓ ck↑ ≈ cp+r↑ cp↑ ck+q−r↓ ck↑ + ck+q−r↓ ck↑ cp+r↑ cp↑ (7.129) " % " % + + + + − cp+r↑ ck↑ ck+q−r↓ cp↑ − ck+q−r↓ cp↑ cp+r↑ ck↑ Ignoring the averages over two operators with opposite spins in this expression, one has " % " % + + + + + + cp+r↑ cp↑ ck+q−r↓ ck↑ ≈ cp+r↑ cp↑ ck+q−r↓ ck↑ − cp+r↑ ck↑ ck+q−r↓ cp↑ (7.130)

159

Itinerant Ferromagnetism

For the second term in (7.128) one obtains % % " " + + + + + + ck−r↑ cp−r↓ cp↓ ≈ cp−r↓ cp↓ ck+q↓ ck−r↑ − ck+q↓ cp↓ cp−r↓ ck−r↑ ck+q↓

(7.131)

Inserting approximations (7.130) and (7.131) into (7.128) one arrives at $ # 1 + fq k k+q↓ − k↑ ck+q↓ ck↑ H bq+ = √ − N k 1 U " + % " + % + cp↑ cp↑ − cp↓ cp↓ ck+q↓ ck↑ +√ fq k N p N k

(7.132)

% " % 1 U " + + + +√ fq k ck↑ cp+q↓ cp↑ ck+q↓ ck+q↓ − ck↑ N N k p The second term of (7.132) is the Stoner shift equal to 2E in the H–F approximation (see Section 7.3) U " + % " + % cp↑ cp↑ − cp↓ cp↓ = 2E = Um (7.133) N p Using relation (7.132) one obtains the numerator of (7.127) in the form " # $ % 1 U F bq H bq+ F = nk k+q↓ − k↑ + Umfq k2 + fq k − N k N k % " % 1 " + + × ck+q↓ ck+q↓ − ck↑ ck↑ f pnp N p q + where fq pnp = cp+q↓ cp↑ and

nk =

(7.134)

1 k < f 0 k > f

The denominator of (7.127) can be written as % " 1 n f k2 F bq bq+ F = N k k q

(7.135)

Inserting (7.134) and (7.135) into (7.127), one obtains 1 1 2 q n f k = n − k↑ + Umfq k2 N k k q N k k k+q↓ +

k

% " % " + + U ck +q↓ ck +q↓ − ck ↑ ck ↑ fq k fq k N

(7.136)

160

Models of Itinerant Ordering in Crystals

or % " % " U fq k ck+ ↑ ck ↑ − ck+ +q↓ ck +q↓ N k fq k = k+q↓ − k↑ + Um − q

(7.137)

" % " % + + Multiplying this relation by ck↑ ck↑ − ck+q↓ ck+q↓ and summing over k, one has " % " % + + U ck↑ ck↑ − ck+q↓ ck+q↓ 1= N k k+q↓ − k↑ + Um − q

(7.138)

Equating the denominator to zero in (7.138) one obtains the dispersion relation for the excitation, which is the sum of the spin-wave excitation and the excitation ﬂipping the spin without gaining by electron momentum, called the Stoner excitation qk = k+q↓ − k↑ + Um

(7.139)

This expression will be illustrated in the example of nearly free electrons (see Section 4.1), where the electron dispersion relation k is given as k =

2 k2 2m∗

(7.140)

hence qk =

2 2 k + q2 − k2 + 2E = 2k · q + q2 + 2E ∗ 2m 2m∗

(7.141)

For q = 0, the excitation energy is equal to the Stoner shift 0k ≡ 2E, and for q = 0, this energy depends on the angle between the direction of electron k and the direction of excitation q and is given by values from an interval 2 ±2kF q + q 2 + 2E 2m∗

(7.142)

where kF is the Fermi sphere radius. The dependence of Stoner excitation on q is shown in Fig. 7.20.

161

hω k

F,q

Itinerant Ferromagnetism

2ΔE

0 0

q

FIGURE 7.20 The dependence of Stoner excitation on q.

7.8.1 Energy of spin-wave excitations The energy of spin-wave excitations can be calculated for small q, when energy q is much smaller than the Stoner shift 2E, by expanding the right hand side of (7.138) with respect to q : " % " % + + c c − c c k↑ k+q↓ k↑ k+q↓ U 1= N k k+q↓ − k↑ + Um1 − q / k+q↓ − k↑ + Um (7.143) " % " % + + U ck↑ ck↑ − ck+q↓ ck+q↓ ≈ 1 + q / k+q↓ − k↑ + Um N k k+q↓ − k↑ + Um Transforming this equation one obtains " % " %⎞ ⎡ " % " % ⎤−1 + + + + ck↑ ck↑ − ck+q↓ ck+q↓ ck↑ ck↑ − ck+q↓ ck+q↓ U U ⎦ ⎠⎣ q = ⎝1 − N k k+q↓ − k↑ + Um N k k+q↓ − k↑ + Um2 ⎛

(7.144)

7.8.2 Dynamic susceptibility of ferromagnets The dynamical transverse spin susceptibility in the real space is given by the two particle Green function:

ij+− t = Si+ t Sj− 0 where Si+ Si− is the spin-raising (lowering) operator at site i.

(7.145)

162

Models of Itinerant Ordering in Crystals

In the momentum–energy representation, this susceptibility is equal to −it −iq·r −r +− − i j

+− q = e e

ij tdt = Sq+ S−q (7.146) j

−

Using (7.124) one obtains from (7.146) the following relation: + −

+− q = Sq+ S−q = Skq Sk− −q

(7.147)

kk

+ Functions Skq Sk− −q can be calculated from the Green functions equation of motion (6.7), with the sign in the ﬁrst term on the right hand side changed to minus for bosons "# %% $ % ""# $ + + + Skq Sk− −q = Skq + Skq Sk− −q H Sk− −q −

−

(7.148) For the commutators in the ﬁrst term of (7.148) one has # $ + + Skq Sk− −q = k k+q ck↑ ck−q↑ − kk −q ck+ ↓ ck+q↓

(7.149)

−

and in the second term # $ U + + Skq H = k+q↓ − k↑ ck↑ ck+q↓ + − N pr + + + + × ck↑ ck+q+r↓ cp+r↑ cp↑ − cp−r↓ cp↓ ck+r↑ ck+q↓

(7.150)

Performing the same approximations as in (7.130) one obtains # $ + + + Skq H = k+q↓ − k↑ ck↑ ck+q↓ + Umck↑ ck+q↓ −

+

% " % U " + + + ck+q↓ ck+q↓ − ck↑ ck↑ cp↑ cp+q↓ N p

(7.151)

Inserting (7.149) and (7.151) into (7.148) one arrives at "" %% " % + + Skq Sk− −q = k k+q ck↑ ck−q↑ − kk −q ck+ ↓ ck+q↓

"" %% "" %% + + + k+q↓ − k↑ Skq Sk− −q + Um Skq Sk− −q

+

% " % "" %% U " + + + ck↑ Spq Sk− −q ck+q↓ ck+q↓ − ck↑ N p

(7.152)

163

Itinerant Ferromagnetism

From this equation the function

"" %% + Skq Sk− −q can be calculated

% " + "" %% k k+q ck↑ ck−q↑ − kk −q ck+ ↓ ck+q↓ + Skq Sk− −q = + k↑ − k+q↓ − Um +

U f k+q↓ − f k↑ N + k↑ − k+q↓ − Um

%% "" + Spq Sk− −q p

(7.153)

" % + where f k+q↓ = ck+q↓ ck+q↓ is the Fermi statistics [see (2.1)]. Using relation (7.147) one can calculate that

+− q =

%% "" + Skq Sk− −q = 0HF q − U 0HF q +− q

kk

(7.154)

where

0HF q =

1 f k↑ − f k+q↓ N k + k↑ − k+q↓ − Um

(7.155)

From (7.154) one obtains the relation for the dynamical transverse spin susceptibility

+− q =

0HF q 1 + U 0HF q

(7.156)

Equating the denominator above to zero one obtains at q = 0 and = 0 the Stoner condition for ferromagnetism (7.40). At q = 0 and = 0 expression (7.156) can have a pole on the real frequency axis outside the Stoner continuum region. The imaginary part of the dynamical susceptibility has a -function contribution 1 + U Re 0HF q = − q Re 0HF q U q

(7.157)

with 1 + U Re 0HF q = 0

(7.158)

This is a collective normal mode and corresponds to the spin-wave excitation given by (7.138).

164

Models of Itinerant Ordering in Crystals

7.8.3 Curie temperature The simpliﬁed method of estimating the Curie temperature is introduced here. One can assume that each spin-wave excitation reduces the magnetic moment by one, which can be written as mT = mcorr T − nq (7.159) q

where mcorr T is the magnetic moment at a given T already lowered by the correlations effects but without the spin waves and nq is the spin-wave occupation number at momentum q. The value of nq can be calculated from relation [7.24] " % − + S−q Sq = mcorr Tnq (7.160) which when combined with (7.159) gives mT = mcorr T −

1 mcorr T

"

% − + S−q Sq

(7.161)

q

The above average can be expressed by the Green functions as " % "" %% 1 − + − S−q Sq = − Im Sq+ S−q f d −

(7.162)

where f is the statistics for particles annihilated or created by the operators − S−q Sq+ . − Operators S−q Sq+ are of the bosonic type; therefore, the Bose–Einstein (B–E) statistics f = e /kB T − 1−1 will be used, obtaining the result "" %% + − Im S S q −q 1 1 mT = mcorr T + d mcorr T q − e /kB T − 1 (7.163) +− 1 1 Im q = mcorr T + d mcorr T q − e /kB T − 1 The B–E statistics for small energies and high temperatures T kB T can be approximated as 1 e /kB T

−1

≈

1 k T = B 1 + /kB T − 1

Inserting this equation into (7.163) one obtains kB T 1 Im +− q mT = mcorr T + d mcorr T q −

(7.164)

(7.165)

Itinerant Ferromagnetism

165

Let us now apply to this expression the Kramers–Kronig relations, which for the complex function B have the form Re B =

1 Im B 2 d − − 2

(7.166)

Using (7.166) in (7.165) one has mT = mcorr T +

kB T Re +− q 0 mcorr T q

(7.167)

This expression can be used for calculating the Curie temperature TC at which mTC = 0 obtaining the relation 1 1 =− 2 Re +− q 0 kB TC mcorr TC q

(7.168)

This relation is identical with the expressions from references [7.25] and [7.26]. Bruno [7.26] has calculated the Curie temperature within the RPA of formula (7.168), with the use of the renormalized exchange parameters, and he has shown that TC is comparable with the experimental values (for Fe Bruno obtained 1057 K and for Ni he obtained 634 K).

REFERENCES [7.1] [7.2] [7.3] [7.4] [7.5] [7.6] [7.7] [7.8] [7.9] [7.10] [7.11] [7.12] [7.13] [7.14] [7.15] [7.16] [7.17] [7.18] [7.19] [7.20]

J.F. Janak, A.R. Williams and V.L. Moruzzi, Phys. Rev. B 11, 1522 (1975). G.T. Rado and H. Suhl, Magnetism, Academic Press, New York (1963). J. Callaway and S.C. Wang, Phys. Rev. B 7, 1096 (1973). M.F. Collins and J.B. Forsyth, Phil. Mag. 8, 401 (1963). M. Liberati, G. Panaccione, F. Sirotti, P. Prieto and G. Rossi, Phys. Rev. B 59, 4201 (1999). R.M. Bozorth, Ferromagnetism, D. Van Nostrand Company, New York (1951). R.A. Tawil and J. Callaway, Phys. Rev. B 7, 4242 (1973). P. Weiss, J. Phys. 6, 661 (1907). P. Weiss and R. Forrer, Ann. Phys. (Paris) 5, 153 (1926). J.C. Slater, Phys. Rev. 49, 537 (1936). H.E. Stanley, Introduction to Phase Transitions and Critical Phenomena, Oxford University Press, New York (1971). E.C. Stoner, Proc. R. Soc. A 154, 656 (1936); E.C. Stoner, Proc. R. Soc. 165, 372 (1938). R. Micnas, J. Ranninger and S. Robaszkiewicz, Phys. Rev. B 39, 11653 (1989). G. Górski, J. Mizia and K. Kucab, Physica B 325, 106 (2003). J. Mizia, J. Phys. F: Met. Phys. 12, 3053 (1982). J. Hubbard, Phys. Rev. B 19, 2626 (1979). K. Levin, R. Bass and K.H. Bennemann, Phys. Rev. B 6, 1865 (1972). H. Fukuyama, Phys. Rev. B 5, 2872 (1972). H. Hasegawa and J. Kanamori, J. Phys. Soc. Jpn. 31, 382 (1971). G. Górski, J. Mizia and K. Kucab, Physica B 336, 308 (2003).

166 [7.21] [7.22] [7.23] [7.24] [7.25] [7.26]

Models of Itinerant Ordering in Crystals J. Mizia, Phys. Stat. Sol. (b) 84, 449 (1977). R. Kubo, Statistical Mechanics, North-Holland Publishing Co., Amsterdam (1995). T. Moriya, Spin Fluctuations in Itinerant Electron Magnetism, Springer, Berlin (1985). V. Yu Irkhin and M.I. Katsnelson, J. Phys. Condens. Matter 2, 7151 (1990). M.I. Katsnelson and A.I. Lichtenstein, J. Phys. Condens. Matter. 16, 7439 (2004). P. Bruno, Phys. Rev. Lett. 90, 087205 (2003).

CHAPTER

8 Itinerant Antiferromagnetism

Contents

8.1 8.2 8.3 8.4

Phenomenological Introduction Simple Model of Itinerant Antiferromagnetism Free Energy and the Magnetic Susceptibility Antiferromagnetism Induced by On-site and Inter-site Correlations 8.5 Free Energy and the Magnetic Susceptibility Including Correlation Effects 8.5.1 Longitudinal and transversal susceptibility 8.6 Onset of Antiferromagnetism 8.6.1 The case of zero Coulomb correlation: U = 0 8.6.2 The case of the strong correlation: U D 8.7 Numerical Results for Magnetization and Néel’s Temperature 8.8 Spin-Density Waves Appendix 8A: Antiferromagnetism in the Presence of On-site and Inter-site Coulomb Correlation References

167 168 174 175 180 182 186 187 189 191 193 199 202

8.1 PHENOMENOLOGICAL INTRODUCTION In the antiferromagnetic state the nearest-neighbour magnetic moments are opposite to each other. In the periodic table of elements there are only two elements, which are antiferromagnetic. These are chromium and manganese [3.1, 4.1, 8.1]. Their spin alignment is more complicated than in the case of simple antiferromagnetism (AF) [8.2], but for the purpose of simple models it can be represented by simple antiferromagnetic ordering. The small magnitude of the magnetic moments (Cr – 04 B per atom, Mn – 05 B per atom, at absolute zero temperature) and their appearance in the middle of the 3d row, where the critical curve for AF goes to zero (see Fig. 8.5 below), point to the itinerant nature of this ordering. 167

168

Models of Itinerant Ordering in Crystals

+

=

Sub-lattice α

Sub-lattice β

FIGURE 8.1 The simple model of antiferromagnetic ordering.

The crystal lattice of antiferromagnet can be divided into two interpenetrating sub-lattices with opposite magnetic moments. The new lattice constant, after transforming by which we return to the atom with the same magnetic moment, will be twice the original constant. The unit cell is doubled by this operation and correspondingly the Brillouin zone is reduced by one half. The simple examples of the lattice, which can be divided into two interpenetrating sub-lattices, are the simple cubic (sc) and body-centered cubic (bcc) lattices. In Fig. 8.1 the example of antiferromagnetic ordering for the two dimensional sc lattice is shown. The two interpenetrating sub-lattices will be called for the +m moment and for the −m moment. The average electron numbers for these sub-lattices are equal to n± =

n±m ≡ n± 2

n± =

n∓m ≡ n∓ 2

(8.1)

The magnetization per atom in Bohr’s magnetons is m = n − n− = n− − n ≡ n+ − n−

(8.2)

8.2 SIMPLE MODEL OF ITINERANT ANTIFERROMAGNETISM The simple model Hamiltonian enabling us to analyse itinerant (AF) is the basic Hubbard model (see Section 5.1), with the on-site Coulomb repulsion and the electron hopping limited only to the nearest neighbour sites tij ≡ t: H = −t

ij

+ ci cj − 0

i

nˆ i +

U nˆ nˆ 2 i i i−

(8.3)

+ ci is the creation (annihilation) operator for an electron of spin , where ci + nˆ i = ci ci is the electron number operator for electrons with spin , t is the

Itinerant Antiferromagnetism

169

hopping integral between the nearest-neighbour sites and 0 is the chemical potential. The zero of energy is located at the atomic level; therefore the term t0 i nˆ i ≡ 0 has been omitted. It should be pointed out here that the energy of atomic level, t0 , is the same as the energy a − in Chapter 4. After taking into account the existence of two interpenetrating sub-lattices the Hamiltonian (8.3) takes on the following form: H = −t

U + ˆ ˆ ˆ ˆ ˆ ˆ i j + + − n + + n n n + n n 0 i i i− j i i i i− 2 i i ij (8.4)

+ where + i i and i i are the creation (annihilation) operators for an electron of spin on the sub-lattice and , respectively and nˆ i = ˆ i = + + is the electron number operator for electrons with spin i i n i i on the sub-lattice . Applying the H–F approximation to the potential energy in Hamiltonian (8.4) one has

nˆ i nˆ i− nˆ i nˆ i− + nˆ i nˆ i− − nˆ i nˆ i− = n nˆ i− + n− nˆ i − n− n (8.5) and similarly for the sub-lattice . Inserting (8.1) into (8.5) one has for the sub-lattice nˆ i nˆ i−

n − m 2

nˆ i −

n 2 − m2 4

(8.6)

nˆ i −

n 2 − m2 4

(8.7)

and for the sub-lattice nˆ i nˆ i−

n + m 2

where = ±1 for spin up and down, respectively. Using (8.6) and (8.7) in the Hamiltonian (8.4) one arrives at the following effective Hamiltonian in the H–F approximation (the m2 terms are contributing the same energy for both spin directions and therefore they have been ignored): + ˆ ˆ i j + + − n + n nˆ i + nˆ i (8.8) H = −t i i j i − ij

i

i

i

where =U

m 2

and

n = 0 − U 2

(8.9)

In this simple model the on-site Coulomb interaction U plays the role of exchange interaction, since it is treated in the H–F approximation.

170

Models of Itinerant Ordering in Crystals

In order to ﬁnd the AF ground state, Hamiltonian (8.8) is transformed into the momentum space and used in the equations of motion for the Green functions in the energy representation [see (6.7)]: EA BE = A B+ + A H− BE

(8.10)

+ where A B ∈ + k k k k . As a result the following equations are obtained: G E k G E + + − k 1 0 E k = (8.11) 0 1 − k E + − G E k G E k + + with, e.g. G E k = k k , G E k = k k . Solving (8.11) one obtains the result E + −

k

k E + + G E k G E k = 2 E + − 2 − 2k G E k G E k

(8.12)

which is equivalent to the diagonalization of Plischke and Mattis [8.3], Brouers [8.4] and Mizia [8.21]. Choosing the energy scale at the chemical potential E + ⇒ E , one arrives at the following expressions for the Green functions: G E k =

E∓ E 2 − 2 − 2k

≡ G± E k

After expanding the simple fractions, this expression yields 1 E∓ ± G E k =

G k − G− k 2 E±

(8.13)

(8.14)

with

=

√

E 2 − 2

or

E = ± 2 + 2

(8.15)

and G k = 1/ − k being the unperturbed Green function. This Green function can be derived in a similar way to the derivation of (6.15) (in this and other chapters, following Chapter 6, the subscript + at the energy in the denominator of the Green function is dropped). Guided by the deﬁnition of the Slater–Koster function [see (6.85)] one deﬁnes such a function for the sub-lattice and calculates it using equations (8.14) and (8.15) F E =

1 1 E− G E k = ≡ F + E N k N k E 2 − 2 − 2k

(8.16)

Itinerant Antiferromagnetism

171

In a more general form one can write for both sub-lattices: 1 F E = 2

±

E∓

F − F0 − E± 0

(8.17)

with F0 = 1/N k − k −1 being the unperturbed Slater–Koster function. The above expression can be written as 1 F E = 2

±

E∓

F + F0 E± 0

where F0 = 1/N k + k −1 . Since in both sums appearing in F0 and in F0 , the dispersion relation k changes from −D to D; therefore one has F0 = F0 and as a result ±

F E =

E∓ F E± 0

(8.18)

From the relation between the Slater–Koster function and the DOS (6.17), one can write for the density of states on the sub-lattice : ⎡ ⎤ 1 1 E − E = − ImF + E = − Im ⎣ F ⎦ E+ 0 =

E− √ 2 E − 2 = E+ 0

(8.19) E− ≡ + E E+ 0

and for the sub-lattice E

1 1 = − ImF − E = − Im =

√

E+ E 2 − 2 = E− 0

E+ F E− 0

(8.20)

E+ ≡ − E E− 0

where the unperturbed DOS is given by relation (6.17): 0 = −1/ImF0 . The forms (8.19) and (8.20) show clearly that the AF DOS will vanish between energies − and + (the zero of energy was assumed at the atomic level which is located approximately in the center of the atomic band). Hence,

172

Models of Itinerant Ordering in Crystals

the quantity is the AF energy gap. The schematic shape of the DOS arising from the initial semi-elliptic DOS [see (7.72)] has the form ±

E = =

E∓ E± 0 2 E∓ 2 2 E∓ E∓ 2 − 1− = E ± D D D E ± D

(8.21)

and is shown in Fig. 8.2. The same solution was obtained in Appendix 8A as equation (8.A14). Using the general expression for electron occupation number (4.20), one obtains for the electron occupation numbers with spin on the sub-lattice : n ≡ n± =

−

fE ± E dE =

−

1 ± E dE exp E − /kB T + 1

(8.22)

where fE is the Fermi function.

ρα + σ

E D(1 + Δ2/D 2)0.5

E D(1 + Δ2/D 2)0.5

Δ

Δ

–Δ

ρα

ρα – σ

–D (1 + Δ2/D 2)0.5

Site α

ρβ + σ

–Δ

ρβ

ρβ – σ

–D(1 + Δ2/D 2)0.5

Site β

FIGURE 8.2 Schematic DOS in the antiferromagnetic state on sites and . The energy E has a gap from − to +, since for = 0 one has E = ± [relation (8.15)], and as a result the band is split into two sub-bands.

Itinerant Antiferromagnetism

173

Using relation (8.15) and (8.19) or (8.20) we can write (8.22) in the following form:

1 0 S ± 2 + 2 f 2 + 2 2 −

+ S ± − 2 + 2 f− 2 + 2 d ≡ n±

n± = n∓ =

(8.23)

or simply n± =

1 0 S ± E fE + S ± −E f−E d 2 −

(8.24)

where dE S E = d

±

E∓

= E± E

E∓ E±

(8.25)

√ The second term in the square bracket of (8.23) comes from E = − 2 + 2 in relation (8.15). Expression (8.24) for the electron numbers in the mean-ﬁeld approximation, n± , gives the following relation for the antiferromagnetic moment per atom (in Bohr’s magnetons): m = n + − n− = −

0 fE − f−E d

E −

(8.26)

= 0 tanh E − /2 − tanh −E − /2d 2E − from which, by using (8.9), we obtain 1 1 = 0 tanh E − /2 − tanh −E − /2d U 4E −

(8.27)

where = 1/kB T . The chemical potential, , is determined from the carrier concentration, n, on the basis of (8.24): n = n + + n− =

−

0 fE + f−E d

1 =1− 0 tanh E − /2 + tanh −E − /2d 2 −

(8.28)

174

Models of Itinerant Ordering in Crystals

8.3 FREE ENERGY AND THE MAGNETIC SUSCEPTIBILITY The expression for electron occupation in the antiferromagnetic state in the presence of an external ﬁeld, H, will be different from (7.63), which was suitable for ferromagnetism. Now the external ﬁeld acts differently on both spins ± in two different sub-lattices and . Therefore its energy has to be included in the energy of the internal exchange ﬁeld in the process of diagonalization. For the electron occupation on the sub-lattice one has to use (8.22) with the density of states ± E replaced by ± E x± , where x± = ± B H, obtaining the following result: 1 n± = (8.29) ± E x± dE − exp E − /kB T + 1 The static antiferromagnetic susceptibility is calculated from the following equation: n+ − n− = B (8.30) H H=0 This is the susceptibility of the sub-lattice . To ﬁnd its relation with the total experimental susceptibility one has to take into consideration the direction of the external ﬁeld, with respect to the easy magnetic axis [3.1, 8.1], and the expression for the longitudinal and transversal susceptibility given below. Using (8.29) in expression (8.30) for the susceptibility of the sub-lattice , one obtains =

22B Kx 1 − 2Kx m m→0

(8.31)

where the correlation factor Kx in the antiferromagnetic case can be written as 1 n+ n− Kx = − 2 m→0 1 = 2 − =

−

+ E − E dE − exp E − /kB T + 1 m→0

(8.32)

+ E fE dE m→0

From (8.9) one has /m = U/2, and on substitution into (8.31) one arrives at =

22B Kx 1 − Kx U

(8.33)

Itinerant Antiferromagnetism

175

The magnetic instability comes from the condition of the zero susceptibility denominator, which gives the critical value of the total ﬁeld U cr =

1 Kx n

(8.34)

After inserting this value into the sub-lattice susceptibility (8.33), one obtains =

22B cr U −U

(8.35)

Inserting (8.35) into the Landau expansion for free energy (7.29), one arrives at the same free energy expansion as for ferromagnetism (7.43), with the exchange interaction being replaced by the Coulomb on-site repulsion U . The critical ﬁeld, F cr , is also replaced by U cr . Thus the susceptibility of (8.35) and the generalized energy expansion of (7.43) are the same for F and AF. The difference between these two orderings lies in the different values of the critical (minimum) ﬁeld. In both cases, when existing in a given material F U > F cr U cr , the second-order term in the energy expansion versus magnetization becomes negative and there is a non-zero equilibrium value of the ordering parameter m, see Fig. 7.10.

8.4 ANTIFERROMAGNETISM INDUCED BY ON-SITE AND INTER-SITE CORRELATIONS The general model of itinerant antiferromagnetism is based on the extended Hubbard quasi-single-band Hamiltonian [6.15, 8.5], see (5.17): H =−

+ tij ci cj − 0

i

nˆ i − F

i

ni nˆ i + U

nˆ i↑ nˆ i↓ +

i

+ + J + + + ci cj ci cj + J ci↑ ci↓ cj↓ cj↑ + hc 2

V nˆ nˆ 2 i j (8.36)

In this Hamiltonian, the dominant on-site Coulomb correlation, U , and the on-site Hund’s ﬁeld, F , were also included. In addition, there are (as in the case of ferromagnetism) three explicit inter-site interactions [5.6, 6.15, 8.6]: the J – exchange interaction, the J – pair hopping interaction and the V – density–density interaction. The effective hopping interaction tij depends on the electron concentration (see Section 5.2) and is also of the inter-site type. The on-site Hund’s ﬁeld, F , can exist only as the interaction between different orbitals in a multi-orbital band, as is the case of the d-type band. Let us

176

Models of Itinerant Ordering in Crystals

assume the single band, composed from identical orbitals, which are fully degenerate, i.e. have the same DOS and the same electron occupation [5.5]. This is an obvious idealization but it allows our model to be simple. In such a band the effective exchange ﬁeld can be expressed as F = p − 1 Jin , where Jin is the exchange interaction between different orbitals within the same atomic site and p is the number of orbitals within the band. As a result the model is a quasi-single-band model. The intra-atomic Hund interaction in (8.36) is already expressed in the H–F approximation, which will be justiﬁed only for small values of this interaction. To avoid a large number of free parameters, it will be assumed later on in the numerical analysis that J = J V = 0 and also that t1 /t = S t2 /t1 = S. This will leave only two parameters: J representing the inter-site interactions and S representing the kinetic interactions [see (8.37) below]. For the kinetic interaction parameters, the previous relations (5.24) and (5.25) will be used: t = t − t1 = t1 − S

tex =

t + t2 t − t1 = 1 − S 2 2 2

(8.37)

In the Hamiltonian (8.36) the modiﬁed H–F approximation on all the intersite interactions t tex J J and V will be performed (see Appendix 6C). The set of these assumptions and approximations is the same as in the case of ferromagnetism, Chapter 7. In this case of AF it gives expressions (6.C22)–(6.C26) for bandwidth and molecular ﬁeld and the following simpliﬁed Hamiltonian: H = − teff

+ i j + + M − 0 nˆ i j i + ij

+

i

U M − 0 nˆ i + 2 i

nˆ i nˆ i−

(8.38)

i =

where teff = tbAF is the effective hopping integral and M is the spindependent modiﬁed molecular ﬁeld for electrons with spin on the sub-lattice . According to Appendix 6C in the case of AF the bandwidth change parameter is spin independent, b = b− ≡ bAF , and is equal to

b = b− ≡ bAF = 1 −

t t n + 2 ex t t

2J + J − V n 2 − m2 2 − − 3IAF IAF 4 t

(8.39)

After expressing the kinetic terms by a factor S [relation (8.37)], this expression takes on the following form: b

AF

= 1 − 1 − S n + 1 − S

2

n2 − m 2 2J + J − V 2 − 3IAF − IAF 4 t

(8.40)

Itinerant Antiferromagnetism

177

The generalized (modiﬁed) molecular ﬁeld depends on the spin and on the sub-lattice index or . From (6.C23) one has the following form for the sub-lattice : M = −Fn − zJ + 4tex IAF n− + zVn + 2ztIAF = M− ≡ M+

(8.41)

and for the sub-lattice M = −Fn− − zJ + 4tex IAF n + zVn + 2ztIAF = M− ≡ M−

(8.42)

with the effective total Stoner ﬁeld given by (6.C26) AF = Ftot

M − − M+ = F − zJ + 4tex IAF m

(8.43)

Equation (8.43) shows that the positive inter-site exchange interaction, J , and positive exchange-hopping interaction, tex , are opposing AF. It is contrary to the case of ferromagnetism [see (7.62)], where both these interactions (when positive) are aiding the ferromagnetism. + cj , is The parameter IAF , according to its deﬁnition I = I− = IAF ≡ ci proportional to the average kinetic energy (per spin) of electrons with spin , and in the AF state is independent of the spin index , and is given by K = −teff

+ + ci cj = −teff z ci cj = −Deff IAF

Deff = zteff

teff = tbAF

ij

(8.44)

Assuming for simplicity zero temperature and using the stochastic inter + ci , as the probability of electron hopping from the ith to pretation of IAF ≡ cj jth lattice sites and back, one arrives at expression (6.155) in the case of weak correlation [7.20]. For strong correlation one has to consider separately the lower and upper Hubbard band (see [7.20]) and as a result one obtains for the chemical potential located in the lower sub-band n ≤ 1 IAF =

2n − n2 − m2 1 − n 2 − n 2 − m2

(8.45)

for the chemical potential located in the upper sub-band n ≥ 1 IAF =

n − 1 2n − n2 − m2 n2 − m 2

(8.46)

178

Models of Itinerant Ordering in Crystals

In the present model, the effective total Stoner ﬁeld given by (8.43) replaces AF the Coulomb repulsion U from Section 8.2; therefore one has = Ftot m/2 instead of = Um/2, and the Hamiltonian (8.38) after taking into account (8.39), (8.41) and (8.42) will take on the following form: + H = −teff i j + + nˆ i − nˆ i + nˆ i j i − ij

i =

i

i

(8.47)

U + nˆ nˆ 2 i i i− =

where the modiﬁed chemical potential is now given by n = 0 − zVn − 2ztIAF + F + zJ + 4tex IAF 2

(8.48)

This Hamiltonian is merely the extension of Hamiltonian (8.8) with the change of t → teff = tbAF and the energy gap = Um/2 being replaced by AF = Ftot m/2. AF The extended H–F ﬁeld Ftot [see (8.43)] includes different on-site and intersite interactions, with the exception of interaction U , which is now singled out as the dominant interaction and will be treated in the CPA. For simplicity, in further analysis the interaction U will be treated in both the weak and the strong correlation limits. For weak correlation we use a ﬁrstorder approximation in the interaction constant over the bandwidth. This will reduce the U interaction again to the H–F approximation. The second case is the high correlation approximation, U >> D, within the CPA method. Later, the CPA approximation will be extended to the arbitrary strength of the on-site interaction, U . The main idea of the CPA formalism [5.4] is now used. The Hamiltonian (8.47) is divided into the homogeneous part analogous to the Hamiltonian (8.8) above: + H0 = −teff i j + + nˆ i + nˆ i + nˆ i + nˆ i (8.49) j i − ij

i

i

and the approximate stochastic part − nˆ + − nˆ HI = V V i i i

i

(8.50)

i

where are the self-energies on sites = for electrons with spin , and are the stochastic potentials given by V ± V P1 = 1 − n∓ ≡ P1± 1 = ∓ ≡ V1 (8.51) with probabilities V = = U ∓ ≡ V ± P2 = n∓ ≡ P2± V 2 2

Itinerant Antiferromagnetism

179

The potentials and probabilities (8.51) used in the CPA equation are described by only two self-energies = − ≡ + = 0 + 1

and

− = ≡ − = 0 − 1

(8.52)

for which we can write two CPA equations in the following form: 2

Pi

i=1

− V i =0 1 − Vi − F E

with = ±

(8.53)

where the Slater–Koster function, F ± E , is given in the form of (8.16) or (8.17). To proceed further, the Hamiltonian (8.49) with the self-energies calculated from (8.53) will be transformed into the momentum space and will be used in the equations of motion for the Green functions (8.10). As a result, the following equations are obtained: G E k G − k bAF 1 0 E + 1 − 1 E k = − k bAF E + 1 + 1 G 0 1 E k G E k

(8.54)

The magnetic self-energy, 1 , appearing in (8.54) is given from (8.52) as 1 = + − − /2. Equation (8.54) is the same as the simple set (8.11) after substituting ⇒ 1 = − 0 and ⇒ −1 . Solving this set of equations one obtains, as previously [see (8.14)], the following expressions for the Green functions: 1 G E k ≡ G± E k = 2

E + 1 ± 1

G k − G− k E + 1 ∓ 1

(8.55)

E + 1 2 − 21

(8.56)

with

=

and G k given as before by G k = 1/ − k bAF , with the modiﬁed dispersion relation k bAF . Using relation (6.17) generalized to the case of different sub-lattices 1 E = − ImF E

(8.57)

relations (8.18) and the expression G k =

1 1 1 1 = = G k bAF /bAF − k bAF 0 bAF

− k bAF

(8.58)

180

Models of Itinerant Ordering in Crystals

where G0 /bAF k is the unperturbed Green function [see (6.15)], one obtains 1 1 E = − ImF ± E = − Im ±

E + 1 ± 1 1 F E + 1 ∓ 1 bAF 0 bAF

(8.59)

When there are no inter-site interactions, bAF ≡ 1, and relation (8.59) is reduced to (8.19) and (8.20) from the simple case in Section 8.2. Shifting the Green functions in the energy scale to 1 E + 1 → E and using (8.22) we obtain the following expressions for the electron numbers: 1 1 ± 2 2 2 2

+ 1 f

+ 1 S n± = AF 0 bAF 2 − b ± 2 2 2 2 +S − + 1 f − + 1 d

where f

2 + 21 =

(8.60)

1 . exp

2 + 21 − 1 /kB T + 1

Treating the self-energies ± as the ﬁrst-order solutions of (8.53) with respect to the Coulomb on-site repulsion, U , one obtains using deﬁnition (8.52) the relations n 0 = U 2

m AF m 1 = − U + Ftot = − U + F − zJ + 4tex IAF ≡ − 2 2

(8.61)

The above expression for the antiferromagnetic energy gap, , is different from that discussed in connection with Hamiltonian (8.47) by adding to it the effective Coulomb repulsion, U , treated in the H–F approximation as all the other weak interactions in (8.43).

8.5 FREE ENERGY AND THE MAGNETIC SUSCEPTIBILITY INCLUDING CORRELATION EFFECTS The expression for electron occupation in the antiferromagnetic state in the presence of external ﬁeld H will be the same as (8.29) in Section 8.3. To calculate the electron occupation on the sub-lattice , as before the relation (8.19) will be used, with quantity ± being replaced by x± = −± ± B H. The self-energies of the two sub-lattices are given by (8.52), with the paramagnetic self-energy 0 not depending on the ﬁrst power of magnetization: 0 /mm→0 = 0. Hence, the whole magnetic dependence of ± is contained in

181

Itinerant Antiferromagnetism

1 : /mm→0 = − − /mm→0 = 1 /mm→0 . Using this relation, the relation (8.29) for n± , and equation (8.30) for the susceptibility, one can write the following formula: =

22B Kx 1 1 + 2Kx m

(8.62)

m→0

where the correlation factor, Kx , in the antiferromagnetic case can be written as in (8.32). After calculating 1 in the H–F approximation [see (8.61)] one obtains the same susceptibility as given previously by (8.33): =

22B Kx AF 1 − Kx Ftot

(8.63)

with the replacement of repulsion U by the total ﬁeld, which is given by AF Ftot = U + F − zJ + 2tex IAF

(8.64)

IAF , according to (8.44), can be expressed as + 1 k + IAF = ci cj = − ck ck N k D 1 1 k 1 =− − ImG E k dE E/k N k D − e B T + 1 Using (8.54), we ﬁnd that ⎛

(8.65)

⎞

k 1 1 ⎜ ⎟ G − ⎝ ⎠ E k = 2 2k + 2 E + − 2k + 2 E + + 2k + 2

(8.66)

Recalling that energy E, as in all our Green functions, is really E+ = E + i0, and using identity (8.85), we ﬁnd that

k 2 2 2 2 Im G E k = − E + − k + + E + + k + 2 2k + 2 (8.67) Inserting this expression into (8.65) one obtains 1 k

k 2 2 2 2 IAF = − f

k + − f − k + (8.68) N k D 2 2 + 2 k

182

Models of Itinerant Ordering in Crystals

With the aid of identity 1 fEk − f−Ek = − tanh Ek − /2 − tanh −Ek − /2 2 where Ek = 2k + 2 , relation (8.68) is converted to

0

− tanh 2 + 2 − /2 IAF = − √ 4 D −D

2 + 2

− tanh − 2 + 2 − /2d

D

(8.69)

where 0 is the density of states in the paramagnetic state. From the condition that the antiferromagnetic instability takes place at the zero of the susceptibility denominator (as in Section 8.3), one arrives at the critical value of the total ﬁeld cr Ftot =

1 Kx n

(8.70)

After inserting this value into the sub-lattice susceptibility (8.63), we obtain =

22B cr AF Ftot − Ftot

(8.71)

Inserting this formula into the Landau free energy expansion (7.27), one has the same result as that for ferromagnetism given by (7.43): = 0 +

m2 cr AF + a4 m 4 + · · · Ftot − Ftot 2

(8.72)

cr but with a different critical ﬁeld Ftot which is given now by (8.70). The susceptibility expression of (8.71) and the modiﬁed energy expansion of (8.72) are the same for F and AF. The difference between these two orderings lies in the different values of the critical (minimum) ﬁeld. In both cases, when existing FAF cr in a given material Ftot > Ftot , the second-order term in energy expansion versus magnetization becomes negative and there is a non-zero equilibrium value of the ordering parameter m, see Fig. 7.10.

8.5.1 Longitudinal and transversal susceptibility The static magnetic susceptibility calculated throughout this book is the longitudinal susceptibility, i.e. the magnetic ﬁeld is applied in the direction parallel to the magnetization. Moreover, in the case of AF, it is the susceptibility of one sub-lattice (that with the magnetic moment parallel to the applied ﬁeld),

Itinerant Antiferromagnetism

183

and only at the zero point of magnetic curve, i.e. m H→0 → 0. In the real materials, usually one has the magnetic anisotropy, which creates the so-called easy magnetic axis. In the absence of an external ﬁeld, the magnetic moments already exist m H→0 = 0 . In different magnetic domains, moments fall onto the directions of their easy axes. On average two thirds of the easy directions are normal to the applied ﬁeld and one third is parallel. Therefore the total susceptibility of the real sample will be 1/3 + 2/3⊥ . The experimental parallel ﬁeld susceptibility tends to zero at zero temperature, while in our theoretical model the susceptibility is the probe of ordering transformation, and in the case of ordering it is divergent at T = 0 for the system which undergoes the para–ferro magnetic transformation. Let us derive now the expression for the net susceptibility in the case when magnetic ﬁeld is applied parallel to the direction of an easy axis at low temperature, when the system is already magnetized to the magnetization m0 on sub-lattice and −m0 on sub-lattice , with magnetization m0 being close to saturation. The Hamiltonian (8.8) written for the case of U = 0, with added external ﬁeld, now takes the following form: + ˆ ˆ H0 = − t i j + + − n + n nˆ i i j i i − ij

+

i

nˆ i

i

− B

i

H nˆ i + nˆ i

(8.73)

i

with the energy gap, , expressed by =

F zJ 1 1 dm m + m = F − zJ m0 + F + zJ H 2 2 2 2 dH

=

F zJ 1 1 dm m + m = − F − zJ m0 + F + zJ H 2 2 2 2 dH

(8.74)

This relation comes from (6.C12), where the paramagnetic terms, V and t, and also the magnetic term with exchange-hopping interaction, tex , were ignored. In (8.74), the provision was made for the general case of m = −m , since under the applied ﬁeld m = m0 + dm and m = −m0 + dm (m0 is the magnetic moment without the applied ﬁeld), and in consequence dm /dH = dm /dH ≡ dm/dH. To obtain (8.22) in an analogical way, assuming that the energy of the electron with spin on the sub-lattice is shifted to the amount of HB + HF + zJ /2dm/dH , one arrives at the following relations: F + zJ dm n = f E − H B + (8.75) E dE 2 dH − where as before it is assumed that = ±1.

184

Models of Itinerant Ordering in Crystals

For the total magnetization one has m =m + m = n − n− + n − n− F + zJ dm = f E − H B + + E dE 2 dH − F + zJ dm − − E dE f E + H B + 2 dH − F + zJ dm + f E − H B + − E dE 2 dH − F + zJ dm − f E + H B + + E dE 2 dH −

or

& F + zJ dm m + m = + E f E − H B + 2 dH − ' F + zJ dm −f E + H B + dE 2 dH & F + zJ dm + − E f E − H B + 2 dH − ' F + zJ dm −f E + H B + dE 2 dH

hence m + m =

1

E + − E 2 − + & F + zJ dm × − tanh /2 E − − H B + 2 dH ' F + zJ dm + tanh /2 E − + H B + dE 2 dH

dm + m 1 = dH 2kB T ×

−

F + zJ dm B + 2 dH

(8.76)

(8.77) −2

+ E + − E cosh E − /2dE

Itinerant Antiferromagnetism

185

and ﬁnally = with

2B S kB T − SF + zJ /2

1 S=

E + − E cosh−2 E − /2dE 2 − +

(8.78)

This formula is similar to the one introduced for parallel antiferromagnetic susceptibility by Morrish [8.1] based on the model for localized magnetic moments, providing that total effective ﬁeld F + zJ is negative. The sign in front of the inter-site exchange interaction J in the expression for above is different than in the case of initial susceptibility calculated at m H→0 → 0, later in Section 8.6. Assuming after many authors (Morrish [8.1], Yosida [8.3]) that the dominant is the negative inter-site exchange interaction, J < 0, one has a form consistent with the experimental results: =

C T +

(8.79)

with C = B S/kB and = SF + zJ /2kB . According to the formula (8.79) as the temperature drops to 0 K, the susceptibility also drops to zero, due to the saturation effect S → 0, because all the moments are parallel or anti-parallel to the applied ﬁeld and cosh−2 E − /2 → 0 in (8.78). With rising temperature the susceptibility increases until it reaches a maximum at the Néel’s temperature. For the magnetic ﬁeld perpendicular to the axis of the spins, one can calculate the susceptibility by elementary considerations. In the presence of a small applied ﬁeld, the energy density, with m = m = m , is F 2 U = N0 m · B m − N0 B H · m + m B 1 2 2 2 = −N0 Fm 1 − 2 − 2N0 HB m (8.80) 2 where 2 is the angle the spins make with each other (Fig. 8.3), N0 is the number of atoms in the unit volume. The energy is at the minimum when 0=

dU = 4N0 Fm2 − 2HB m d

=

H 2N0 Fm

(8.81)

so that 2N0 mB 2 = B (8.82) H F and transversal susceptibility is a constant independent of temperature. According to (8.79), the parallel susceptibility increases smoothly with temperature up to TN (see Fig. 8.4). ⊥ =

186

Models of Itinerant Ordering in Crystals

H mα

mβ 2φ

FIGURE 8.3 Calculation of a perpendicular susceptibility at 0 K.

χ

χ⊥

χ II

0

TN

T

FIGURE 8.4 The temperature dependence of the susceptibility of antiferromagnets.

8.6 ONSET OF ANTIFERROMAGNETISM Due to the competition between potential and kinetic energies, one has the critical (minimum) value of the interaction constant below which the antiferromagnetic ordering will not take place. The critical value of the total ﬁeld creating AF can be found from (8.70), with Kx from (8.32) expressed in the form E E fE dE = fE dE − x − 1 1 G E k =− fE Im dE N k − →0

Kx =

(8.83)

Differentiating (8.55) with 1 − over we can show that 1 1 1 G E k = − E + k E − k 2 k →0

(8.84)

Itinerant Antiferromagnetism

187

Using in (8.83) this expression and the well-known Green function identity 1 1 =P − iE ± k (8.85) E + ± k E+ ± k one obtains the following formula: 1 f k − f− k Kx = − N k 2 k 1 1 = tanh k − /2 − tanh − k − /2 N k 4 k

(8.86)

cr cr n can be calculated now from relation (8.70); Ftot = The critical ﬁeld Ftot 1/Kx . The above equation, although similar to the condition for critical interaction for superconductivity is different in two respects [8.7]. First, for antiferromagnetism the summation runs from k minimum to k maximum in the cr band, giving a positive Kx and positive Ftot n . Second, in the denominator of (8.86), in the case of AF one has the dispersion relation k , which is centred on the average energy of the band, while in the case of superconductivity it is centred around the chemical potential [8.8]. In this section, the model developed for AF will be investigated numerically, using semi-elliptic DOS given by (6.56) 2

2 = 1− (8.87) Deff Deff

and the H–F approximation for the self-energy [see (8.61)]. Since the inter-site interactions are included in the model the dispersion energy and the bandwidth in (8.87) are given by = 0 bAF and Deff = DbAF , where the bandwidth reduction parameter bAF is calculated from (8.39).

8.6.1 The case of zero Coulomb correlation: U = 0 From the condition of the zero of susceptibility denominator (8.70) and from (8.86) in the integral form, one obtains the condition for dependence of the critical total exchange ﬁeld on the electron concentration cr 1 = Ftot

Deff

−Deff

1 tanh − /2 − tanh − − /2d 4

(8.88)

This equation together with (8.28) for the chemical potential in the limit of cr → 0 gives the total critical ﬁeld as a function of occupation: Ftot n . Next, from (8.43) one ﬁnds the critical on-site exchange ﬁeld in the function of occupation cr n + zJ + 2tex IAF F cr n = Ftot

(8.89)

188

Models of Itinerant Ordering in Crystals

4

F cr[D ]

3 2 1 0

0

0.5

1 n

1.5

2

FIGURE 8.5 The dependence of the critical on-site exchange interaction, F cr , on the electron occupation for U = 0. The curves are for different values of S and J; S = 1 and J = 0 (no inter-site interactions) – dot-dashed line; S = 06 and J = 05t – solid line; S = 1 and J = 05t – dashed line; S = 06 and J = 0 – dotted line.

for different values of the inter-site and kinetic interactions, described by the parameters J and S (as in the case of ferromagnetism Section 7.5). This function is shown in Fig. 8.5. Analysing the curves in Fig. 8.5 we can see that the inter-site exchange interaction J increases values of the on-site exchange interaction, F cr , required for AF. The decrease in F cr can be achieved for J < 0. This effect is opposite in the case of ferromagnetism (considered in Section 7.5), where stabilization of ordering was obtained for J > 0. The fact that the inter-site exchange interaction inﬂuences F and AF differently can be roughly understood when one compares the expressions for total ﬁeld in the case of ferromagnetism F Ftot = F + z J + tex n2 − m2 + 2t1 − n

see 762

and antiferromagnetism AF Ftot = F − zJ + 4tex IAF

see 843

The word roughly is used, since in calculating the critical ﬁelds there is an additional factor of the bandwidth change, which does not appear explicitly in the effective ﬁeld. The result of the simpliﬁed approach [based on (7.62) and (8.43)], with respect to the inter-site exchange energy J , is in agreement with the Heisenberg term for the interaction energy of localized spins −J ij Si Sj , which points towards ferromagnetism when J > 0 and towards AF when J < 0. Including in our model the kinetic interactions t and tex has decreased the minimum of the on-site exchange interaction necessary for AF. The hopping interaction, t, decreases F cr by decreasing the bandwidth [7.20]. The positive exchange-hopping interaction, tex , also decreases the bandwidth but at the AF same time decreases the effective ﬁeld [see expression (8.43) for Ftot or (8.61)]. In effect, the inﬂuence of this interaction on AF is very small.

Itinerant Antiferromagnetism

189

8.6.2 The case of the strong correlation: U D In this case, the spin densities are calculated from the relation ± E± = −1/ImF ± E± , with F ± E± given by (8.A18) and (8.A19) for the lower subcr band. The total critical ﬁeld for n ≤ 1, Ftot , is calculated from (8.88), with the DOS in the paramagnetic state and U D: , which is the limit derived at m → 0 of ± E± from (8.A18) and (8.A19): 2 = Deff

1−

n

2 − 2 Deff

(8.90)

where, as before, one has Deff = DbAF , and bAF is given by relation (8.39). The chemical potential is calculated as before from the → 0 limit of (8.28), with the high correlation DOS, , given above: n=2

−

f d

(8.91)

cr , the on-site critical ﬁeld is calAfter calculating the total critical ﬁeld, Ftot culated from (8.89). The DOS in the U D limit is not usually analytical. It is analytical above only because the semi-elliptic DOS given by (8.87) was used as the initial DOS. Details of the method of calculating this DOS are described in Appendix 8A. Figure 8.6 shows the dependence of the on-site exchange interaction, F cr , on the electron concentration for different values of the inter-site and kinetic interactions, described by the parameters J and S in the case of strong Coulomb

4

F cr[D]

3 2 1 0

0

0.25

0.5 n

0.75

1

FIGURE 8.6 The dependence of the critical on-site exchange interaction, F cr , on the electron occupation in the case of the strong correlation U. The curves are different values of S and J; S = 1 and J = 0 (no inter-site interactions) – dot-dashed line; S = 06 and J = 05t – solid line; S = 1 and J = 05t – dashed line; S = 06 and J = 0 – dotted line. The double dotted-dashed line is for t = tt1 = 0 or S = 0) and tex ≡ 0 [see (8.96) below].

190

Models of Itinerant Ordering in Crystals

correlation U = . Minimum F cr is now shifted from n = 1 in the case of the weak correlation to n = 2/3. Figs 8.5 and 8.6 show us that including the inter-site interactions J J and V in the calculations does not shift the minimum of the critical on-site ﬁeld F cr n , which for the weak correlation is located at n = 1 and for the strong correlation at n = 2/3. The inter-site interactions (in the H–F approximation) do not change the character of the critical curves; they only lower them by a factor of bAF from (8.39) or (8.40). Recalling that = 0 bAF this can be proved analytically: Kx n = −

Deff −Deff

f − f− d

2

1 D 0 0 f 0 bAF − f− 0 bAF 0 1 = − AF d = AF Kx0 n 0 b b 2

−D

(8.92)

where Kx0 is the Kx factor without the inter-site interactions. cr Since Ftot = 1/Kx and F0cr = 1/Kx0 , one obtains that cr Ftot = F cr − zJ + 4tex IAF = bAF F0cr

(8.93)

where F cr and F0cr are the critical on-site exchange ﬁelds for AF with and without the inter-site interactions, respectively. Including the kinetic interactions, S < 1, leaves the minima of critical curves where they were, as it only changes the factor bAF in (8.93). In this chapter, the dependence of the hopping energy on the electron occupation number is expressed by the equation tij = t − tnˆ i− + nˆ j− + 2tex nˆ i− nˆ j−

(8.94)

This relation, together with the expressions for other inter-site interactions J J and V , in the generalized H–F approximation, brought relation (8.39) for bAF , with IAF in the case of U D given by (8.45) or (8.46). The only exception from the scaling rule for the critical on-site exchange interaction in the case of U D is the situation when bAF n = 1 = 0. The bandwidth factor falling to zero would cause the critical curve for AF to fall at n = 1 from inﬁnity to some ﬁnite value and would make AF possible at this concentration. In the itinerant band model, this is the situation when there is localization at half-ﬁlling. As a result one obtains at this concentration possible AF, and since the band will expand very rapidly with the occupation n departing from 1, there will be also a strong driving force towards SC [8.9]. In such circumstances, there will be the coexistence or rather competition at n = 1 between AF and SC. Unfortunately, since IAF n = 1 = 0, relation (8.39) gives bAF n = 1 = 0 unless t = t, and tex = 0, which is rather unrealistic. In the paper [7.20], the authors have obtained the

Itinerant Antiferromagnetism

191

zero bandwidth at n = 1, but they have neglected the tex term in (8.39) or (8.94). This is equivalent to assuming the Hirsch simpliﬁed linear approach [8.9–8.11] tij = t − tnˆ i− + nˆ j−

(8.95)

instead of (8.94). This would give for the bandwidth factor the relation bAF = 1 −

t 2J + J − V n− IAF t t

(8.96)

At t ≡ t1 − t = t or t1 = 0 (reminder: t1 is the hopping integral in the presence of one electron with opposite spin on one of the lattice sites involved in hopping), this equation gives bAF n = 1 = 0, and one has the possibility of AF ordering (see the double dotted-dashed line in Fig. 8.6). In the model with both t and tex present, as was mentioned above, AF can occur at n = 1 when t1 ≡ 0 (no hopping in the presence of other electrons with opposite spin), and additionally when tex ≡ 0, which leads to the strange condition that t2 = −t. Alternatively one can obtain relation (8.95) (which leads to AF at n = 1) from the basic equation for tij (5.21) in a ﬁrst approximation by ignoring all operator products of the type nˆ i− nˆ j− . At this point it is worthwhile making the connection with the Anderson model of semi-localization at n = 1. The occupationally dependent hopping according to relation (5.21) is given by the formula tij = t1 − nˆ i− 1 − nˆ j− + t1 nˆ i− 1 − nˆ j− + nˆ j− 1 − nˆ i− + t2 nˆ i− nˆ j− (8.97) where the ﬁrst and last terms represent hopping between unoccupied sites and doubly occupied sites, respectively. This hopping does not change the number of doubly occupied sites (doublons). The hopping term t1 nˆ i− 1 − nˆ j− + nˆ j− 1 − + nˆ i− when combined with the hopping operators ci cj in the Hamiltonian (8.3) will represent the energy change E = ±U corresponding to creating and annihilating doublons. This part of the Hamiltonian is eliminated in the diagonalization transformation [8.12], leading to what is called the t − J model. Here, it is assumed directly that this term, which at large U makes unfavourable energetic changes, has to be eliminated by assuming that t1 → 0 at U → .

8.7 NUMERICAL RESULTS FOR MAGNETIZATION AND NÉEL’S TEMPERATURE The analysis of (8.26) and (8.28) will give the magnetization dependence on the temperature for different values of the on-site and inter-site interactions. From these equations one ﬁnds numerically the sub-lattice magnetization and the Néel’s temperature, which is the temperature of dropping the sub-lattice magnetization to zero. The details of the method are described in Appendix 8A.

192

Models of Itinerant Ordering in Crystals

TN (K)

600

400

200

0

0.75

0.8

0.85

0.9

0.95

1

n

FIGURE 8.7 The dependence of the Néel’s temperature TN on the electron concentration for different values of S and J in the case of the weak Coulomb correlation; S = 1 and J = 0 (no inter-site interactions) – dot-dashed line; S = 07 and J = 02t – solid line; S = 1 and J = 02t – dashed line; S = 07 and J = 0 – dotted line. Parameters are D = 05 eV and F = 034 eV.

Fig. 8.7 shows the dependence of Néel’s temperature, TN , on electron concentration for different values of the inter-site and kinetic interactions in the case of zero Coulomb correlation U = 0. All the curves are for the same on-site exchange ﬁeld: F = 034 eV. The curves with non-zero inter-site exchange interaction J have a lower Néel’s temperature, because this interaction decreases AF AF : Ftot = F − zJ + 4tex IAF . This effect is the effective exchange interaction Ftot stronger than the increase of TN due to the decrease of the bandwidth with an increasing J [see (8.39)]. The assisted hopping interaction, t = t1 − S , and kinetic-exchange interaction, tex = t/21 − S 2 , increase the Néel’s temperature by decreasing the bandwidth [see (8.39)]. Figure 8.8 shows the magnetization versus temperature for electron occupation n = 096 and for various values of the inter-site and kinetic interactions. Again, all the curves are calculated for the same internal exchange ﬁeld F = 034 eV. Similarly, for the calculations of Néel’s temperature (see Fig. 8.7), the highest magnetization is obtained for parameter S < 1, which means that magnetization is enhanced when electron hopping is inhibited in the presence of other electrons. Table 8.1 shows the Néel’s temperature calculated for the 3d antiferromagnetic elements (Cr and Mn) in the case of the zero U correlation. First, the on-site exchange interaction, F , at a given assumed inter-site interactions J and AF ﬁtted to the experimental S was found from (8.43) after inserting total ﬁeld Ftot magnetic moment at T = 0. Next, using the same F , the Néel’s temperature was calculated. The results show that the inter-site, J , and kinetic, S, interactions decreased the Néel’s temperature (column no. 1). The difference between experimental and theoretical Néel’s temperature is even larger than in the case of ferromagnetism (confer Table 7.5). This large difference has to be attributed, as in the case of F, to the spin waves or spin ﬂuctuations [4.2, 7.23, 8.2].

193

Itinerant Antiferromagnetism

0.8

m/n

0.6

0.4

0.2 0

0

0.2

0.4

0.6

0.8

1

T/TN

FIGURE 8.8 Magnetization versus temperature for band filling n = 096, D = 05 eV, F = 034 eV and for different values of S and J; S = 1 and J = 0 (no inter-site interactions) – dot-dashed line; S = 07 and J = 02t – solid line; S = 1 and J = 02t – dashed line; S = 07 and J = 0 – dotted line. This is the case of the weak Coulomb correlation. Table 8.1 Néel’s temperatures and values of on-site Stoner field for antiferromagnetic elements, U = 0 Element

n

m

D (eV)

TN K , Fm (eV) No. 1: S = 06, J = 05t

No. 2: S = 06, J =0

No. 3: S = 1, J = 05t

No. 4: S = 1, J ≡0

No. 5, exp TN K

Cr

1.08 0.08

3.5

520, 2.37

1430, 1.47

1530, 3.17

2440, 2.27

311

Mn

1.24 0.48

2.8

2110, 2.41

5840, 2.65

7090, 4.47

10 820, 4.71

540

n = 3d electron number, normalized to 2; m = magnetic moment at 0 K, normalized to 2; D = 3d elements half bandwidth according to [4.5]. The next columns are nos 1, 2, 3, 4 - TN S J = Néel’s temperature and the on-site Stoner ﬁeld Fm (ﬁtted to m at 0 K) exp for different J and S; no. 5 – TN = experimental Néel’s temperature in K (from [8.2] for Cr and [4.2] for Mn).

8.8 SPIN-DENSITY WAVES In the preceding sections, the itinerant model of AF using the Green function method was developed. In this model, the nesting vector was Q = , and as a result the magnetic ordering was commensurate with the crystal lattice. Now the more complicated alignment will be considered of the spin-density wave (SDW), with the electron occupation in the sub-lattices and given by the following expression: ni = n + me−iQ·Ri /2

(8.98)

194

Models of Itinerant Ordering in Crystals

where, in agreement with (8.1), one has for the sub-lattice Ri = 2l a and for the sub-lattice : Ri = 2l + 1 a (a is the lattice constant, which will be assumed later on to be 1, a = 1, l − integer number). Vector Q is the reciprocal lattice vector in the presence of the AF ground state. For pure (commensurate) antiferromagnetism this vector is Q = . Thus exp−iQ · Ri = 1 for Ri in sub-lattice and exp−iQ · Ri = −1 for Ri in sub-lattice . The unit cell is doubled and in consequence the Brillouin zone is reduced to one half. Such a state is called commensurate antiferromagnetism (C-AF). At concentrations different from half-ﬁlling the vector Q = one may have a state of an incommensurate antiferromagnetism (IC-AF). In this state, the magnetic moment is modulated (see Fig. 8.9). Performing the H–F approximation on interaction U in the simple Hamiltonian (8.3), transforming it to the momentum space, and inserting the wave-like occupation values from (8.98), one obtains the following form: H=

1( k

2

) + + + + ck + k+Q − ck+Q ck+Q − ck+Q ck + ck ck+Q k − ck (8.99)

where for the 2D sc lattice one has the following dispersion relation (see Section 4.2)

k = −2tcos kx + cos ky

(8.100)

The above Hamiltonian can be diagonalized by the help of the Bogoliubov transformation + + = u k + ck k − k k + + ck+Q = k + k + uk k

(8.101)

FIGURE 8.9 The change of magnetic moment with distance along the x direction for the vertical incommensurate AF [Q = − , = 3/64 ].

195

Itinerant Antiferromagnetism

Taking the hermitian conjugate of this transformation one arrives at ck = uk k − k k ck+Q = k k + uk k

(8.102)

Functions uk and k are real and symmetric in the momentum space (u−k = uk and −k = k ), and they fulﬁl the normalization condition: u2k + k2 = 1

(8.103)

The transformation inverse to (8.101) has the following form: + + + k = uk ck + k ck+Q + + + k = −k ck + uk ck+Q

(8.104)

Using transformations (8.101) and (8.102) in the Hamiltonian (8.99) one obtains H=

1( k

+

2

1( k

+

) k − u2k + k+Q − k2 − 2k uk + k k

2

) k − k2 + k+Q − u2k + 2k uk + k k

(8.105)

1 ( ) + k+Q − k k uk + k2 − u2k + k k + k k 2 k

We need the non-diagonal last term above to disappear; therefore the following condition for functions uk and k will be imposed k+Q − k k uk = − k2 − u2k (8.106) Relations (8.103) and (8.106) allow to express the functions uk and k as

k − k+Q

k − k+Q 1 1 uk 2 = 1+ k 2 = (8.107) 1− 2 2Ek 2 2Ek 2uk k = −

Ek

where is given by (8.9) and Ek =

−

2 k k+Q + 2 2

(8.108)

196

Models of Itinerant Ordering in Crystals

Inserting relations (8.107) and (8.103) into Hamiltonian (8.105) the following diagonal form is obtained: H=

) 1 ( + + − Ek − + k k + Ek − k k 2 k

(8.109)

where the energies of quasi-particles and are given by Ek± =

k + k+Q ± Ek 2

(8.110)

and For the commensurate Q one has k+Q = − k , which will reduce (8.108)

(8.110) to the form (8.15) describing the simple C-AF: Ek± = ±Ek = ± 2k + 2 . The excitation spectra of quasi-particles and are shown in Fig. 8.10. Between them there is the energy gap of 2 = Um. In the SDW formalism, condition (8.2) for the magnetic moment can be written as 1 + m= ck ck+Q (8.111) N k Using transformations (8.101) and (8.102) one obtains + + ck ck+Q = uk + k − k k k k + uk k

( ) ) ( + = 2uk k f Ek+ − f Ek− = − f Ek − f Ek− Ek

(8.112)

Applying now the relation tanh /2 = 1 − 2f one arrives at the expression for magnetization: m=

( ) + 1 * tanh Ek+ − /2 − tanh Ek− − /2 2N k Ek

(8.113)

Ek+

Ek U/2

2Δ

–

Ek –π/2

0 kx = ky

π/2

FIGURE 8.10 Excitation spectrum of quasi-particles Ek+ and Ek− in the direction k x = ky .

Itinerant Antiferromagnetism

197

where the chemical potential satisﬁes the equation n−1 = −

( ) + 1 * tanh Ek+ − /2 + tanh Ek− − /2 2N k

(8.114)

In the case of the commensurate ordering Q = , the above equations reduce to m=

1 tanh Ek − /2 − tanh −Ek − /2 2N k Ek

(8.115)

and n−1 = −

1 tanh Ek − /2 + tanh −Ek − /2 2N k

(8.116)

This set is the same as relations (8.26) and (8.28) from Section 8.2. For SDW one can write the free energy [8.13] as F/N = U

) n 2 m2 1 * ( ln 2 cosh Ek+ − /2 − − kB T 4 4 N k

+ ln 2 cosh Ek− − /2

(8.117)

Solving (8.113) and (8.114) at m → 0 one obtains the dependence of Néel’s temperature on concentration for different modes. Fig. 8.11 shows the dependence TN n for C-AF, the diagonal IC-AF Q = − − and the vertical 800

TN (K)

600 400 200 0

0.8

0.85

0.9 n

0.95

1

FIGURE 8.11 The dependence of the Néel’s temperature TN on electron concentration for D = 1 eV, U = 055 eV and for commensurate AF [Q = ] – dotted line, the diagonal incommensurate AF Q = − − – dashed line, the vertical incommensurate AF [Q = − – solid line.

198

Models of Itinerant Ordering in Crystals

IC-AF Q = − . The incommensurability depends on the electron concentration and can be expressed as = 2 arcsin/2t [8.14]. The phase diagram for these types of AF ordering is shown in Fig. 8.12, where a given solution has been selected by searching for the minimum of the free energy given by (8.117). As can be seen the C-AF state at T = 0 is stable only at a half-ﬁlling. Such a result was found in Cr [8.2] and also in high TC cuprates [8.15]. At concentrations different from half-ﬁlling and at low temperatures one has the diagonal IC-AF. For smaller concentrations the dominant is the vertical IC-AF. The temperature dependence of magnetization for three different types of AF ordering is shown in Fig. 8.13 for an electron occupation n = 095. At low temperatures one has diagonal IC-AF and at high temperatures the C-AF.

800

T (K)

600 P

CAF

400 VICAF

200

DICAF 0 0.8

0.85

0.9 n

0.95

1

FIGURE 8.12 Phase diagram in the temperature-electron concentration plane for D = 1 eV, U = 055 eV, with commensurate (CAF) vertical incommensurate (VICAF) and diagonal incommensurate (DICAF) antiferromagnetic phases, and the paramagnetic (P) state.

0.4

m

0.3 0.2 0.1 0

0

200

400 T (K)

600

800

FIGURE 8.13 Temperature dependence of magnetization for n = 095, D = 1 eV, U = 055 eV. The notation is the same as in Fig. 8.11.

Itinerant Antiferromagnetism

199

APPENDIX 8A: ANTIFERROMAGNETISM IN THE PRESENCE OF ON-SITE AND INTER-SITE COULOMB CORRELATION The H–F approximation which was used above is the lowest order expansion of the self-energy in the difference of the two values of stochastic potential, which was U in the simple case of Section 8.2. Now the self-energy in a full CPA approach applied to the on-site Coulomb repulsion U will be calculated. All other interactions are treated in the H–F approximation, producing an average energy gap for electrons on sites (or − electrons on sites): − = −F AF tot m/2 = − F − zJ + 4tex IAF m/2 and an average energy gap for electrons on sites (or − electrons on sites): + = F AF tot m/2 = F − zJ + 4tex IAF m/2. The CPA equation (8.53) for the electrons on sites takes on the following form: 1 − n−

− − + U − − + + n = 0 − 1 − − − + F + E 1 − U − − + F + E

(8.A1)

and for the electrons on sites 1 − n+

− − U + − − + n = 0 + 1 − − − F − E 1 − U + − − F − E

(8.A2)

AF where = Ftot m/2. Solving these two equations self-consistently together with the conditions for electron concentration on different sub-lattices

n± = −

−

1 1 ImF ± E dE exp E − /kB T + 1

(8.A3)

the condition for sub-lattice magnetization m = n+ − n−

(8.A4)

and the condition for electron occupation n = n + + n−

(8.A5)

will give the self-energies: ± . They have a ﬁnite imaginary part as opposed to the self-energy coming from the H–F approximation and given by (8.61). For the initial semi-elliptic band given by (8.87) one has the following form of the Slater–Koster function [5.4]:

2 1 1 2

F0 = − ≡ −1 (8.A6) N k − 0k D D D

200

Models of Itinerant Ordering in Crystals

Equation (8.59) can be simpliﬁed to the form E − ∓ 1 F ± E ≡ F ± = F

= E − 0 2 − 21 0 E − ± bAF bAF

(8.A7)

with 0 = + + − /2 and 1 = + − − /2. Combining (8.A6) with (8.A7) one obtains 2 Deff F + 2 − 4E − − F + + 4

E − − = 0 E − +

2 Deff F − 2 − 4E − + F − + 4

E − + = 0 E − −

(8.A8) Deff = DbAF

(8.A9)

These two coupled equations are solved together with (8.A1) and (8.A2), leading to 4 Deff F + F − 2 D2 F + F − D2 F − + U − 2E eff + E 2 − EU F + + eff − E + 1 − n− U = 0 16 4 4 (8.A10) and 4 Deff F − F + 2 D2 F + F − D2 F + + U − 2E eff + E 2 − EU F − + eff − E + 1 − n+ U = 0 16 4 4 (8.A11)

The last two equations give the solution for the intermediate values of the Coulomb on-site repulsion. They have to be computed numerically together with (8.A4) and (8.A5) for m and n. The analytical solution for AF in the CPA with the semi-elliptic DOS can be obtained in two extreme cases of U D and of U = . In the case of weak correlation U 0 , the CPA equation reduces the H–F approximation with interaction U included in the Hartree ﬁeld. The self-energy is approximately equal to the average energy of electrons with a given spin on a given lattice site. Ignoring the paramagnetic part one has + = − = − − = =

m m AF = − U + F − zJ + 4tex IAF U + Ftot 2 2

m

U + F − zJ + 4tex IAF 2

(8.A12)

Solving the set of equations (8.A8) and (8.A9) with the above self-energies, one obtains the Slater–Koster functions in the following form: ⎡ ⎤ 2 E− 2 ⎣ E− E F + E = − − −⎦ Deff Deff Deff E+

Itinerant Antiferromagnetism

⎡ F − E =

2 ⎣ E+ − Deff Deff

E+ Deff

201

⎤

2 −

E+ ⎦ E−

(8.A13)

AF + U m/2 = E ± U + F − zJ + 4tex IAF m/2 and with E± = E ± ≡ E ± Ftot Deff = DbAF . Using relations ± E± = −1/ImF ± E± , one obtains the following DOS 2 E− E− 2 2 E+ E+ 2 + − E = − E = − Deff E + Deff Deff E − Deff (8.A14) which are the same as solutions (8.21) obtained in the simple model of AF. Now, the driving force for AF is not only the Coulomb repulsion U but all the on-site and inter-site exchange interactions, which contribute to the effective ﬁeld and to the energy gap: = U + F − zJ + 4tex IAF m/2. They also contribute to the bandwidth through the factor bAF [see (8.39)], which is a new way of including the correlation effects. In the case of strong correlation U = , the band in the paramagnetic state is split into two sub-bands (see section 6.5). For n < 1 the Fermi energy lies in the lower sub-band. For the energy in the lower sub-band, ≈ 0, the term U − − + F + 1 in (8.A1) and (8.A2). Therefore 1 can be ignored in the denominator of the second term of these relations, obtaining 1 − n−

− − + n − − = 0 1 − − − + F + F +

(8.A15)

− − n − + = 0 1 − − − F − F −

(8.A16)

and 1 − n+

Solving these two equations one arrives at + = − −

n− F+

− = −

n+ F−

(8.A17)

Now (8.A8) and (8.A9) are coupled with (8.A17). Substituting (8.A17) into (8.A8) and (8.A9) and performing rather laborious calculation, one arrives at ⎡ ⎤ 2 2 ⎣ E− E− − − f1 − n+ ⎦ (8.A18) F+ = Deff Deff Deff ⎡ F− =

2 ⎣ E+ − Deff Deff

E+ Deff

⎤

2 −

1 − n− ⎦ f

(8.A19)

202

Models of Itinerant Ordering in Crystals

with f being the complex quantity equal to E− 1 − n/2 m2 m m2 f= − 2 + 2 E+2 E−2 − E+ E− 1 − n/2 + E+ 1 − n+ 8E− 1 − n+ 2E+ 1 − n+ 16 (8.A20) where E ± = E± /Deff is the energy in the units of effective bandwidth. The spin densities, ± E± = −1/ImF ± E± , calculated from (8.A18) and (8.A19) for the lower sub-band show a similar behaviour to those in the U D case (see Fig. 8.2). But now the band is narrowed because the strong Coulomb repulsion pushes n electron states to the upper sub-band (the upper sub-band is not drawn in the ﬁgure). The magnetization of the system is the numerical solution of (8.A3)–(8.A5) with (8.A18) and (8.A19).

REFERENCES A.H. Morrish, The Physical Principles of Magnetism, Wiley New York (1965). E. Fawcett, Rev. Mod. Phys. 60, 209 (1988). M. Plischke and D. Mattis, Phys. Rev. B 7, 2430 (1973). F. Brouers, J. Giner and J. Van der Rest, J. Phys. F 4, 214 (1974). G. Górski and J. Mizia, Physica B 344, 231 (2004). D.K. Campbell, J.T. Gammel and E.Y. Loh Jr, Phys. Rev. B 38, 12043 (1988). J. Mizia and S.J. Romanowski, Phys. Stat. Sol. (b) 186, 225 (1994). J. Bardeen, L.N. Cooper and J.R. Schrieffer, Phys. Rev. 108, 1175 (1957). J.E. Hirsch and F. Marsiglio, Phys. Rev. B 39, 11515 (1989). J.E. Hirsch, Phys. Rev. B 59, 3327 (1993). J.E. Hirsch, Phys. Rev. B 62, 14487 (2000). P. Fazekas, Lecture Notes on Electron Correlation and Magnetism, Modern Condensed Matter Physics, Vol. 5 (1999). [8.13] R. Micnas, J. Ranninger and S. Robaszkiewicz, Rev. Mod. Phys. 62, 113 (1990). [8.14] H.J. Schulz, Phys. Rev. Lett. 64, 1445 (1990). [8.15] K. Machida and M. Ichioka, J. Phys. Soc. Jpn 68, 2168 (1999).

[8.1] [8.2] [8.3] [8.4] [8.5] [8.6] [8.7] [8.8] [8.9] [8.10] [8.11] [8.12]

CHAPTER

9 Alloys, Disordered Systems

Contents

9.1 Introduction 9.2 Order–Disorder Transformation and Bragg–Williams Approximation 9.2.1 Bragg–Williams approximation 9.3 Relation with the Band Model 9.4 Transition Metal Alloys 9.5 Different Types of Disorder in Bragg–Williams Approximation References

203 205 206 208 213 219 224

9.1 INTRODUCTION There are many crystals which are composed of two elements A and B that occupy at random the regular lattice sites of the structure, in proportion x and y = 1 − x for the composition Ax B1−x . In such a crystal, the translational symmetry is no longer perfect. The difference between the effective potentials of the components may be very weak in comparison to either potential alone. The consequences of alloying will be particularly small when both elements belong to the same column of the periodic table because the atomic cores will make a similar contribution to the effective potentials. Also the elements of the same d row within the periodic table frequently have a very small potential difference. This fact was used to set up a simple model of alloying using the coherent potential approximation (CPA) in chapter 6. The model will be applied to binary transition metal alloys in Section 9.4. One of the disorder measures is the residual electrical resistivity deﬁned as the low temperature limit of the resistivity. The residual resistivity increases with disorder [9.1, 9.2]. The effect is that the Cux Au1−x alloy cooled rapidly shows the residual resistivity depends on composition as x1 − x, see Fig. 9.1. 203

204

Models of Itinerant Ordering in Crystals 15

ρ (μΩ cm)

10

5

0

0

25

50 x (%)

75

100

FIGURE 9.1 Electrical resistivity for Cux Au1−x alloys as function of composition x, in the disordered state at 4.2 K (filled circles) and at 296 K (open circles) [9.1, 9.2]. The ordered alloy measured at 4.2 K (triangles) has been annealed, whereas the disordered alloy has been quenched rapidly. The composition of low residual resistivity corresponds to the ordered composition Cu3 Au and CuAu. Reprinted with permission from P.G. Huray, L.D. Roberts and J.O. Thomson, Phys. Rev. B 4, 2147 (1971). Copyright 2007 by the American Physical Society.

The same alloy when cooled slowly from the high temperature forms ordered structures of Cu3 Au and CuAu; these structures have a lower resistivity by virtue of their order, as in Fig. 9.1. Different phases of the binary alloys are depicted in the phase diagrams. This ﬁeld of research is quite extensively developed in textbooks. In this book, a good introduction to this subject given by Kubo [2.2] will be mentioned. In Section 9.2, a mathematical description of the order–disorder transformation in binary alloys will be developed. To prepare the reader for this section, an example of the phase diagram of the copper–zinc system is shown in Fig. 9.2, following [9.3]. In this phase diagram, the fcc structure ( phase) of pure copper (n = 1) persists on addition of zinc n = 2 until the electron concentration reaches 1.38. A bcc structure ( phase) occurs at a minimum electron concentration of about 1.48. The phase (complex cubic cell of 52 atoms) exists for the approximate range of n between 1.58 and 1.66. The dashed horizontal line in the phase (bcc) region in the phase diagram of the Cu–Zn system (at 454 and 468 C) represents the transition temperature between the ordered (low temperature) phase and the disordered (high temperature) phase of the alloy. The ordered phase of the 50–50% Cu–Zn alloy in the bcc structure can be visualized by dividing this lattice into the

Alloys, Disordered Systems

1000

α + liquid

Liquid

γ + liquid

834°C

800 Temperature (°C)

β + liquid

205

β 600

γ

β+γ

α+β

α

468°C 454°C

400

α + β′ 200 30

40

β′

β′ + γ

50 Weight percent zinc

60

FIGURE 9.2 Equilibrium phase diagram of Cu–Zn alloy [9.3]. The structures of phases are as follow: phase–fcc structure; phase–bcc structure; phase–complex structure. The phase is the ordered bcc structure (most of the Cu atoms are sitting on the sites belonging to one simple cubic sub-lattice, and most of the Zn atoms are sitting on the sites belonging to the second one; the second sub-lattice interpenetrates the first one) and the phase is the disordered bcc structure (the probability of occupying any lattice site is the same for both Cu and Zn atoms, independently of what kind of atom is in the neighbouring site). Reprinted with permission from H. Amar, K.H. Johnson and C.B. Sommers, Phys. Rev. 153, 655 (1967). Copyright 2007 by the American Physical Society.

two interpenetrating sub-lattices, each being occupied by only one sort of the atoms. To the ordering of atoms on such a lattice, an order–disorder theory developed in Section 9.2 will be applied.

9.2 ORDER–DISORDER TRANSFORMATION AND BRAGG–WILLIAMS APPROXIMATION This model is developed for a substitutional alloy in which the change of a given A atom for a B atom does not essentially inﬂuence the property of the neighbouring atoms. In such an alloy, the atoms A and B occupy equivalent sites in the structure at random. Hume-Rothery speciﬁed the empirical requirements for the stability of a solid solution of A and B as a single phase (see Hume-Rothery rules in [3.1]). More precisely, in band structure language, one can expect that the components of a substitutional alloy will have the same (or very similar) electronic

206

Models of Itinerant Ordering in Crystals

densities or equivalently the same electron dispersion relations, k . The difference between both components is characterized only by the difference of the average atomic level of both components, i = A or B , see (6.104).

9.2.1 Bragg–Williams approximation A very simple and interesting approach to the ordering of substitutional alloys exists, which bypasses the details of the electronic structure and its changes with ordering. It was applied not only to alloying but also to such processes as solidiﬁcation and to ferromagnetism and antiferromagnetism as will be shown later on in Section 9.5. First the atomic ordering in binary Ax B1−x alloys will be considered. We assume a bipartite crystal lattice (as in chapter 8) composed of two interpenetrating sub-lattices and . Let the probability of ﬁnding an atom A on sub-lattice be given by A, with similar notation for other probabilities. The nearest-neighbour energy bonds will be considered. The total bond energy per atom is U = nAA VAA + nBB VBB + nAB VAB

(9.1)

where nij is the number of nearest-neighbour ij bonds, and Vij is the energy of an ij bond. x1 + x1 − z 2 A A =z = x 1 − 2 2 2 4 y + x y − x z 2 B B nBB =z =z = y − x2 2 2 2 4 z A B B A nAB =z + = 2xy + x 2 4

nAA =z

(9.2)

where is the long-range order parameter. With the help of (9.2) the energy given by (9.1) becomes z U = U0 − 2 x2 Vex 2 z U0 = x2 VAA + y2 VBB + 2xyVAB 4 Vex =

VAA + VBB − VAB

2

(9.3)

Alloys, Disordered Systems

207

For the free energy and conﬁgurational entropy per atom one can write that = U − TS

S = −kB

4

Pi log Pi

(9.4)

i=1

where the four probabilities, Pi , are deﬁned as

A A B B and , and

they fulﬁl the relation x1 + x1 − y + x y − x + + + = 1

2 2 2 2 The meaning of the long-range order parameter, , can be understood from the following relations: A A − = x

B B − = x

(9.5)

Next, the equilibrium order is determined by the requirement that the free energy be a minimum with respect to the order parameter . Differentiating free energy with respect to one has 1+ y + x + ln = 0

−2z xVex + kB T ln 1− y − x

(9.6)

Near the transition ≈ 0, one may expand (9.6) to ﬁnd that TC =

zxyVex

kB

(9.7)

For a transition to occur, the effective exchange interaction Vex = VAA + VBB /2 − VAB must be positive. It means that the average interaction within the same sort of atoms should be weaker than between A and B atoms (all the interactions are negative). One can also see that for an alloy asymmetrical in concentration (x → 0 or x → 1) it is more difﬁcult to order, as at concentrations x → 0 or x → 1 one has TC → 0. The maximum TC is reached at concentration x = y = 1/2, where the fully ordered alloy forms the two interpenetrating sub-lattices of A and B atoms, and each atom of a given type will be surrounded by the atoms of the opposite type. The transcendental equation for , (9.6), may be solved easily and one ﬁnds the smoothly decreasing curve shown in Fig. 9.3 (x = y = 1/2 is set).

208

Models of Itinerant Ordering in Crystals

1.0

η

0.8 0.6 Second-order transformation 0.4 0.2 0

0

0.2

0.4

0.6 T/Tc

0.8

1.0

FIGURE 9.3 Long-range order parameter, , versus reduced temperature for an A05 B05 alloy. The transformation is of the second order.

9.3 RELATION WITH THE BAND MODEL The connection between the traditional Bragg–Williams theory of Section 9.2 and the band model (see [8.4] and Section 8.2 in here) is made in this section. The starting point of the band model is the simple tight-binding Hamiltonian, which in the case of two interpenetrating sub-lattices has the form already analysed in Section 8.2 for the case of antiferromagnetism: H=

i

+ i i +

+ i i − t

i

+ i j + h c

(9.8)

i =j

+ where + i i and i i are the creation (annihilation) operators for an electron of spin on the sub-lattices and , respectively. Using probabilities deﬁned in relation (9.2) the average atomic energies on sites and in the paramagnetic state can be found as

A B 2 = 2 A + B = + x

A B 2 = 2 A + B = − x (9.9)

where = xA + yB is the average electron energy in disordered alloy, factor 2 comes from the spin degeneration and = A − B is the energy difference of the atomic levels of A and B atoms. The Hamiltonian (9.8) with energies (9.9) has the form of the two sublattice Hamiltonian (8.8), which was diagonalized in Section 8.2 for the case of

Alloys, Disordered Systems

209

antiferromagnetism. Relations (8.15) and (8.21) will give the electron densities on sites in the case of an ordered alloy in the following form:

E =

E∓ 2 E± 0

(9.10)

√ where = E 2 − 2 , and the energy gap is now = x . The results of the atomic ordering will be illustrated using the same initial semi-elliptic band for both components of an alloy, since in (9.8) the same last term for both components was assumed. In this case one obtains

E =

2 E∓ 4 4 E∓ E∓ 2 1− =

− E ± D D D E ± D

(9.11)

These densities of states (DOS) depict the electron densities for the atomically ordered alloy. The degenerated spin densities are given by one half of (9.11) and they are shown (for an initial semi-elliptic DOS) in Fig. 9.4. They are analogous to the densities in Fig. 8.2 drawn for the antiferromagnetism. The critical curves for atomic ordering for any DOS will be calculated now in a way similar to the case of antiferromagnetism (see Section 8.2),

ρ+σ

E

E

Δ

Δ

−Δ

ρ−σ ρ+σ

−Δ

ρα

ρβ

Site α

Site β

ρ−σ

FIGURE 9.4 Schematic DOS of the ordered binary alloy on sites and . The energy gap extends from − , to +; = x. The number of electrons on sub-lattice n = n + x n0A − n0B , is smaller than on sub-lattice , n = n − x n0A − n0B , since n0A < n0B .

210

Models of Itinerant Ordering in Crystals

AF after replacing the antiferromagnetic energy gap = Ftot m/2 by = x . As AF a result, the exchange ﬁeld Ftot will be replaced by the components energy difference , which will eventually drive the atomic ordering transition. The cr critical value for atomic ordering, cr , will replace Ftot appearing in the case of antiferromagnetism. The precise condition for cr will now be derived. The number of electrons on sites and can be written as x1 + y − x A 0 0 B n = 2 nA + nB = n0A + n0B = n + x n0A − n0B 2 2 x1 − y + x A 0 0 B n = 2 nA + nB = n0A + n0B = n − x n0A − n0B (9.12) 2 2

with n = xn0A + yn0B being the average number of electrons per site. On the basis of (8.24), in full analogy with the derivation for antiferromagnetism (Section 8.2), one can write for the number of electrons on sites and :

n =

0 S ± EfE + S ± −Ef−Ed E −

=

0 S ± EfE + S ± −Ef−EdE −

where S ± E is given by the deﬁnitions from Chapter 8: S ± E = E ∓ /E ± . Using relation (9.12) for the left-hand side of this equation one can write that D n − n =2x n0A − n0B = fES + E + f−ES + −E 0 dE −D

− =

D

−D

+

D

−D

fES − E + f−ES − −E 0 dE

fES + E − S − E 0 dE D

−D

= − 2 = − 2

(9.13) f−ES + −E − S − −E 0 dE

D

−D

D

−D

fE

D

0

f−E 0 dE dE + 2 −D

0 fE − f−EdE

√ Since E = 2 + 2 , one has dE = dE/dd = /Ed. Inserting this formula and the relation = x into (9.13), one gets quite a simple form:

D

n = − fE − f−E 0 d n = n0A − n0B (9.14) E −D

Alloys, Disordered Systems

211

from which, in the limit of → 0, the critical (minimum) value of the driving force towards atomic ordering, cr , is obtained: n = − cr

D −D

f − f−

0 d

(9.15)

The electron occupation, n = xn0A + yn0B , is included in the last two expressions through the Fermi function which depends on the chemical potential (for T = 0). Equation (9.15) resembles the condition for antiferromagnetism [relations (8.34) and (8.86)]. Assuming (after Hasegawa and Kanamori [9.4]) that the Fermi level in both pure components is located at the same energy, one obtains for = A − B > 0 that n = n0A − n0B < 0. This would explain the negative signs in (9.14) and (9.15). The relations (9.14) and (9.15) are the ﬁrst of this type derived for the atomic ordering despite the extensive literature on the subject. Therefore, a short qualitative discussion of the consequences for atomic ordering coming from relation (9.15) will be included in here. For simplicity the constant DOS,

0 = 1/D, for −D < < D will be assumed. At constant DOS and at zero temperature, one has from relation (9.15)

F d D n 1 cr = − (9.16) with In = − = ln

0 In F −D The condition of equal Fermi levels for both pure components, in the case of constant DOS, gives n = − 0 , which allows us to write (9.16) as cr =

In

or

cr 1 =

In

(9.17)

For the constant DOS given above, the Fermi level is equal to F = Dn − 1, which after inserting into (9.16) gives the relation 1

In = ln (9.18) n−1 Figure 9.5 shows the dependence of cr / on a carrier concentration n given by relations (9.17) and (9.18). A look at this ﬁgure reveals that the alloy with purely metallic binding, where the cohesion electrons form one common band, will undergo ordering preferably close to a half-ﬁlled point. One can expect such an atomic ordering and > 0 in the concentration range between 0.63 and 1.37 electrons in the band. For these concentrations, cr is smaller than the real value of obtained from the relation = A − B . After calculating x from (9.14), one can calculate the internal energy from the formula

D U x = EfE E xdE (9.19) −D

212

Models of Itinerant Ordering in Crystals

5

4

δcr[δ ]

3

2

1

0

0

0.25

0.5

0.75

1 n

1.25

1.5

1.75

2

FIGURE 9.5 The dependence of the critical for atomic ordering energy difference cr = A − B cr , in units of the real existing difference = A − B , on the average electron concentration in the common A–B band.

where the total DOS can be written on the basis of (9.10) as ⎛ ⎞ √ E − E + 1 ⎠ 0 E 2 − 2

+

E x = E + E = ⎝ 2 E+ E−

(9.20)

Computational calculations of U x from expressions (9.19) and (9.20) show that the internal energy depends on the order parameter x in a quadratic way within 1% of accuracy [8.4]. Therefore one can write that U x = U0 − U0 − U x∗

x x∗

2

(9.21)

where U x∗ is the internal energy calculated at the point x∗ [for = x∗ ]. The similarity of this result to the result of the Bragg–Williams approximation (9.3) is not surprising, since both are molecular ﬁeld theories. The contact between the two theories, the Bragg–Williams and the band theory, is obtained by comparing (9.21) with (9.3) and identifying that U U0 − U x∗ z z ≡ = Vex = VAA + VBB − 2VAB

∗ 2 ∗ 2 x x 2 4

(9.22)

Alloys, Disordered Systems

213

Expression (9.22) together with (9.7) for the critical temperature leads to the relation for the critical temperature in terms of the energy difference calculated from the band model: TC =

2U / x∗ 2 xy

kB

(9.23)

Therefore one can explore the variation of TC with respect to the ﬁlling of the band and the difference of potentials, , within the band theory. The formalism described here is the formalism of atomic ordering, which takes place in the common cohesion band without the charge transfer from A to B atoms. Most of the ordered alloys are formed with a charge transfer, which leads to the forming of an electron compound. Their cohesion energy is larger than that of the atomic ordering, which results in a larger critical temperature. The transfer of charge will decrease the energy of those alloys, which by this transfer will ﬁll up the electron band of one component and empty (or almost empty) the electron band of another component. A good example here is the copper–zinc alloy for which the stability of the electron compound phase (with stochiometry CuZn) is of the order of 834 C (see phase diagram, Fig. 9.2). This is much higher than the temperature of its atomic ordering as the CuZn compound, at 454 or 468 C, below which the Cu and Zn atoms occupy sites on two different interpenetrating sub-lattices.

9.4 TRANSITION METAL ALLOYS Figure 7.6 shows the Slater–Pauling curve for the magnetization of different binary alloys of transition 3d metals and their compounds. This curve is taken from the book by Bozorth [7.6] and Kübler [4.2]. The Slater–Pauling curve has been reprinted in many other books and can be found, for example, in Kittel [3.1]. A close look at the Slater–Pauling curve reveals that the majority of magnetic moment versus concentration functions are linear. The explanation of these phenomena in the band language is quite simple. Linear behaviour takes place in these parts of the diagram, where the alloys are strong magnetics. They are strong magnetics for these concentrations, where the majority spin 3d band (called also the up spin band) is full: n = 5. Then one can write that − mAlloy = 5 − n− = 5 − xn− A + ynB

(9.24)

Since both components are strong magnetics, the alloying does not change the total number of 3d, − electrons. The assumption that the number of 4s electrons remains constant under alloying [4.2] which means that they do not transfer to the 3d band is made here. Hence − − − xn− A + ynB = xn0A + yn0B

(9.25)

214

Models of Itinerant Ordering in Crystals

− where n− 0A and n0B are the occupations of 3d band with − spin of pure A and B components before alloying. Inserting (9.25) into (9.24) one has − − − 0 0 mAlloy = 5 − xn− 0A + yn0B = x 5 − n0A + y 5 − n0B = xmA + ymB

(9.26)

where m0A and m0B are the magnetic moments of pure A and B components. This explains the linearity of magnetization dependence on concentration. In the alloy, the partial magnetic moments of its components may differ from the original values, i.e. mA = m0A and mB = m0B . The experimental results show for many alloys the almost constant behaviour of these moments on concentration. From the above derivation it can be concluded that the magnetization follows the major branch of the Slater– Pauling curve, when the majority spin band (up spin band) does not extend beyond the Fermi level. This condition requires strong enough exchange interactions and a very small value (with respect to half bandwidth) of the energy difference of the components A − B deﬁned in (9.28). Hasegawa and Kanamori [9.4] have applied the CPA of Section 6.5 to calculations of different properties of binary transition metals. They assumed that the components of the alloy differ only in their average band energies i H=

i nˆ i −

i

+ tij ci cj

(9.27)

ij

where i =

A = A + UA nA− B = B + UB nB−

with probabilities

Pi =

x

y

(9.28)

The Soven’s CPA equation (6.91) in this case will take on the following form: xA + yB − = A − B − F

(9.29)

with F deﬁned by (6.96), (6.85) and lemma (4.18) F = F0 − =

−

0 d − −

(9.30)

where 0 is the DOS per unit energy for the energy band determined by the second term in (9.27), i.e. it is the same for both pure components of an alloy. The average numbers of electrons on atom A or B are calculated from the expression ni =

−

i fd

i = A or B

(9.31)

Alloys, Disordered Systems

215

where f is the Fermi distribution function for a given temperature T f =

1 exp − /kB T + 1

(9.32)

and the partial DOS for i = A or B atoms [5.4] are 1 F

i = − Im

1 − i − F

(9.33)

In practical calculations, the authors have selected parameters A B UA and UB and for both components a common DOS function, 0 , as an input information. They determined six unknown quantities: ni (i = A or B and = ±) and , through six simultaneous, transcendental equations (9.29)– (9.31) with the help of (9.32) and (9.33). After solving these equations the authors obtained various physical quantities of the alloys. Relation (9.33) is a direct consequence of the general CPA equation [5.4] F = x

F F + y 1 − A − F 1 − B − F

(9.34)

which, after applying the operator −1/ Im to both sides, gives the DOS for electrons with spin :

= x A + y B

(9.35)

with AB given by (9.33). By integrating this equation over the energy one obtains the average number of electrons with spin per atom as n = xnA + ynB

(9.36)

The magnetic moment of A or B transition metal atom, in the unit of B , was obtained [9.4] as mi = 5ni − ni−

i = A or B

(9.37)

where the factor 5 arises from the ﬁve-fold degeneracy of the band. The average magnetic moment of the alloy, in the same units, was given by m = xmA + ymB

(9.38)

The authors [9.4] have assumed two different types of an initial unperturbed

0 . One for the bcc type of metals, see Fig. 9.6, and the second one for the fcc type of metals, see Fig. 9.7.

216

Models of Itinerant Ordering in Crystals

ρ (ε)

1.5/D

1/D

0.5/D

0

–D

0

ε

D

FIGURE 9.6 Model state densities function for bcc metals after [9.4].

ρ (ε)

1.5/D

1/D

0.5/D

0

–D

0

ε

D

FIGURE 9.7 Model state densities function for fcc metals after [9.4].

The DOS for bcc metals reﬂects the tight-binding DOS for this type of lattice (see the dashed line in Fig. 4.7), with the single maximum being doubled by the relatively strong Coulomb repulsion [although the same interaction U is treated according to these authors [9.4] as the source of the Hartree ﬁeld in (9.5)]. The DOS for fcc metals reﬂects closely the numerical results of tight-binding approximation for fcc type of the lattice (see Fig. 4.7, dotted-dashed line). As mentioned earlier, the numerical calculations based on (9.29), (9.30) and (9.31) allow us to ﬁnd the quantities ni (i = A or B and = ±) and from input parameters A B UA and UB and the assumed DOS function,

0 . This input information, together with the total number of 3d electrons normalized to 2, is shown in Fig. 9.8 for the bcc metals and in Fig. 9.9 for the fcc metals [9.4].

Alloys, Disordered Systems

2.0

U i [D ] 2.0

εi [D ]

1.5

–1

1.0

–2

0.5

–3

1.5 Ui

ni

ni

εi

1.0

0.5

217

0

0

0 Cr

Mn

Fe

Co

Ni

FIGURE 9.8 Assumed parameters for bcc metals [9.4]. The unit of energy is half of the 0 original bandwidth; ni is the number of electrons supplied by each atom. U i [D ] 2.0

1.5

Ui

εi [D]

2.0

0

1.5

–1

1.0

–2

0.5

–3

ni

ni

εi

1.0

0.5

0

0 Cr

Mn

Fe

Co

Ni

FIGURE 9.9 Assumed parameters for fcc metals [9.4].

The assumed parameters reﬂect the usual tendencies in the periodic table: (i) the average energy of the band, i , decreases with the increasing number of electrons in this band and (ii) the Coulomb correlation increases with the band occupation, as growing occupation makes screening more difﬁcult. Quantities ni (i = A or B and = ±) calculated numerically, from the assumed data (see Figs 9.8 and 9.9), allowed us to obtain the average moments in the alloy (9.38) as well as partial moments per A or B components (9.37). In many cases, partial moments in the binary transition metal alloys were measured by neutron diffraction. For these alloys Hasegawa and Kanamori [9.4] have obtained quite a good ﬁt between the theoretical model described in detail above and the experimental data.

218

Models of Itinerant Ordering in Crystals

2.0 Ni1–xCox

mCo

m[μB]

1.5 mav 1.0

0.5

0

mNi

0

0.2

0.4

0.6

0.8

1.0

x

FIGURE 9.10 The calculated magnetic moments of Ni and Co and the average magnetic moment of Ni–Co alloy together with the experimental data by neutron diffraction (by Cable et al. [9.5] and Collins and Wheeler [9.6]), after [9.7]. Reprinted with permission from N. Janke-Gilman, M. Hochstrasser and R.F. Willis, Phys. Rev. B 70, 184439 (2004). Copyright 2007 by the American Physical Society.

Figure 9.10 shows the results for Ni–Co alloy as one of the fcc alloys. As mentioned earlier, the average magnetic moment for this alloy is linear with concentration since it is a strong magnetic alloy, and the Fermi level does not cross the up spin (majority) band. One can see that the CPA theory is capable of reproducing partial moments of this alloy measured by neutron diffraction. Figure 9.11 shows the calculated magnetic moments of Fe and Co and the average magnetic moment per atom of the Fe–Co alloy as compared to experimental data. This is an example of the bcc alloy and a weak magnetic material. The calculated results are also in good agreement with the experimental data. The authors [9.4] also calculated other properties of these alloys using the CPA method, e.g. electronic speciﬁc heat, DOS at the Fermi level. In general, it can be said that the simple model given by (9.27), in which both components have the same band shape and the only difference between them is the difference of average energy of the band, is capable of describing many properties of binary alloys when treated beyond the Hartree–Fock approximation. To calculate the properties of the side branches of the Slater–Pauling curve (these are the branches which diverge from the main curve at cobalt or nickel), one has to assume that the energy level crosses the majority (up) spin band for some concentrations and to use much larger values of A − B [9.4]. Such large energy differences allow for the strong deformation of up and down spin bands, which were not possible in the rigid band model [4.2]. At present, there is a lot of new development in the explanation of Slater– Pauling curve, which can be found in the literature [4.2].

Alloys, Disordered Systems

219

4.0 Fe1–xCox mFe

3.0

m[μB]

mav 2.0 mCo

1.0

0

0

0.2

0.6

0.4

0.8

1.0

x

FIGURE 9.11 The solid lines express the magnetic moment of Fe, Co, and the average magnetic moment per atom calculated with the bcc DOS. The experimental points are the data of neutron diffraction by Collins and Forsyth [7.4] and Collins and Low [9.8], after [9.7]. Reprinted with permission from N. Janke-Gilman, M. Hochstrasser and R.F. Willis, Phys. Rev. B 70, 184439 (2004). Copyright 2007 by the American Physical Society.

9.5 DIFFERENT TYPES OF DISORDER IN BRAGG–WILLIAMS APPROXIMATION The Bragg–Williams approximation developed above has a wide use in physics. The simple examples, which will be described below, are the model of solid/liquid interface growth in pure metals, ferromagnetic and antiferromagnetic ordering of pure elements. The model of solid/liquid interface is based on the quasi-crystalline model for the liquid, i.e. the atoms in the liquid and contiguous solid phase are both located on the same crystal lattice, chosen as that of the solid. This is an obvious oversimpliﬁcation of the situation for the liquid. The second essential assumption is that the total energy can be expressed as the sum of pair interactions (usually between nearest neighbours). For the pure metal one has the following expression: U=

ij

V ij

⎧ ss ⎨V V ij = V ll ⎩ ls V

where indices s and l refer to the solid and liquid, respectively.

(9.39)

220

Models of Itinerant Ordering in Crystals

One can readily prove that the energy changes of an atom being ﬂipped, calculated from (9.39), are the same as the ones given by the so-called Ising model [9.9]: U =− where

Si =

Vex Si S j − L S i 2 ij i

+1 for solid atom −1 for liquid atom

Vex = V ls −

(9.40)

V ll + V ss 2

and the latent heat per atom is equal to L = V ll − V ss . The identity of (9.39) and (9.40) can be shown easily. For example, for the 2D diagram of Fig. 9.12 one can write down, following (9.39), that E = 4V ls − 4V ll = 4V ls − 2V ll − 2V ss − 2V ll − V ss = 4Vex − 2L = 8J − 2L where the Ising constant J = Vex /2. The last value can be readily obtained from (9.40). The transformation of (9.39) into (9.40) allows for the physical interpretation of the Ising model (9.40) on the basis of existing metal physics theories. These theories constitute what is known as the Jackson model [9.10]. In this model, the internal energy calculated for the single-layer interface from (9.39) is approximated by the Bragg–Williams (mean-ﬁeld) formulation. The assumption is made that the part of the single layer is in the solid state and 1 − in the liquid state. Counting as before the numbers of pairs one obtains for the energy of the interface: U = 2l1 − V ls + l2 V ss + l1 − 2 V ll + mV ls + lV ss + l1 − V ll = 2lVex 1 − + mV ls

(9.41)

where Vex = V ls − V ss + V ll /2 and l is the fraction of interactions in the plane of interface (l + 2m = 1, with m being the fraction of interactions above or below the plane). The last three terms were added to include the energy of the reference state. The term, mV ls , is the difference in out-of-plane interactions between the monolayer state and the reference state. To simulate the interface growth one needs, as before, the free energy = U − TS = 2lVex 1 − + mV ls + kB T ln + 1 − ln1 −

(9.42)

Let us introduce the normalized excess free energy of the monolayer interface: f =

w = 1 − + + ln + 1 − ln1 − = kB T kB T

(9.43)

Alloys, Disordered Systems

L L

S

L(S)

S(L)

L

S

ΔE = 8J + 2L

ΔE = 8J – 2L

L

S L

Flat interface

S

S(L)

S

L

ΔE = 4J + 2L

ΔE = 4J – 2L

L

S

L(S)

L

S

L

L(S)

L

Kink

S

S

S(L)

S

L

ΔE = 0 + 2L

ΔE = 0 – 2L

L S

S

S

L

S

L(S)

S

Flat interface

L

S(L)

S

ΔE = –4J – 2L

S

L

L(S)

L

L

ΔE = –4J + 2L

S

221

S

L

S(L)

S

L

ΔE = –8J + 2L

ΔE = –8J – 2L

L

FIGURE 9.12 Five possible transitions of the Ising model applied to the solid–liquid transformation on the square lattice. The change in internal energy is 8J ± 2L, 4J ± 2L, 0 ± 2L, −4J ± 2L and −8J ± 2L, respectively. The dashed line marks the position of the interface.

where =

2lVex kB T

=

mV ls kB T

w is the surface area/atom and is the surface tension. Figure 9.13 shows the free energy versus solid fraction of the interface for different values of the so-called Jackson’s coefﬁcient . For < 2 the curves show minima at = 1/2; these minima are responsible for the rough growth, since the new atoms may join the solid surface at any concentration. At = 2 these minima start to convex upwards, then the energy minima are found at

222

Models of Itinerant Ordering in Crystals

F (θ)

F(θ)

α>2

α>2

0

θ (a)

0

1

θ

α<2

α<2 1

(b)

FIGURE 9.13 “Out of crystal plane interaction” (b) pushes the free energy versus curve by = mV ls /kB T compared to the classical Jackson’s result (a). The minima of free energy are marked by the small crosses (after [9.11]).

0 and 1, and the argument is that the atoms may join the solid surface only at small concentrations or, equivalently, join the planes which already have been nucleated. This would result in an impeded or a smooth growth. The Bragg–Williams approximation, as applied to the solid/liquid interfaces, was developed much further than it can be reviewed here. There is a whole series of papers by Eustathopoulos et al. [9.12], extending the above approach to the interfaces in binary alloys. Later on, this approach was enriched by introducing to it the electronic properties of solid/liquid components [9.13]. Mathematics of the model was also developed further by applying the Monte-Carlo simulations [9.9, 9.14], but the basic physical features sketched in here have not been changed. The Bragg–Williams approximation will be applied now to ferromagnetic and antiferromagnetic ordering of pure elements. In this application, the probabilities and interactions are redeﬁned in a simple way. Let the energies of two electrons with parallel spins on the same atom be −J ≡ −J and −J−− ≡ −J− and that of the antiparallel spins be −J− , where all J±± > 0. Then, in the case of ferromagnetism, one can write for the interaction energy per atom: U = −J n2 − J− n2− − 2J− n n− 1 1 = U0 − J − J− mn − Jex m2 2 2

U0 = − with

J + J− + 2J− 4

J + J− Jex = − J−

2

(9.44)

Comparing this equation with (9.3) one can see that the role of the atomic order parameter, , is resumed now by x = m. To remove the linear term in energy, the additional assumption J = J− was made. To include in the model the neighbouring atoms, the numbers of electrons with parallel and antiparallel spins on these atoms will be also counted. These

Alloys, Disordered Systems

223

numbers will be multiplied by the constants of inter-site interaction, −Jinter , inter inter −J− and −J− . As a result one obtains for the interaction energy 1 1 inter U = U0 − m2 Jex − z m2 Jex 2 2 1 inter = U0 − m2 Jex + zJex 2

with

inter Jex =

inter Jinter + J− inter − J−

2

(9.45)

Next, the conﬁgurational entropy per atom is added to obtain the free energy:

n n n n S = −kB n log + 1 − log 1 − n n n n (9.46) n

n

n n = −2kB n log + 1− log 1 −

n n n n Minimizing free energy with respect to m one obtains for the long-range order parameter inter inter nP Jex + zJex n Jex + zJex m P = = tanh and TC =

(9.47) n 2kB T 2kB The above relation for P is in full analogy to given by (9.6) for binary alloys, shown in Fig. 9.3, and the relation for TC to the relation for TC given by (9.7). Moreover, the results for ferromagnetism coming out from this Bragg– Williams approximation are the same as the results of the simple Weiss model given by relations (7.9) and (7.12). In turn, it was shown in Section 7.3 that the Weiss model is the result of the Hartree–Fock approximation with the suppression of all the details of electronic structure, k → a (atomic level). For antiferromagnetism the following probabilities for ﬁnding a + or − electron on sites or are deﬁned: 1

m 1

m = 1+ = 1− 4 n 4 n 1

m 1

m − − = 1− = 1+

(9.48) 4 n 4 n Comparing this relation for AF with relation (9.2) for disordered binary alloys one can see that now x = y = 1/2 and ⇒ P = m/n (in the case of on-site interaction, Jex , we have z = 1). We assume again that the interactions leading to AF are the on-site and inter-site exchange interactions; the same as in the above case of ferromagnetism. Counting the numbers of electrons with parallel

224

Models of Itinerant Ordering in Crystals

spins on the same atoms or and the numbers of electrons with parallel spins between neighbouring atoms and , one has U = U0 −

1 m 2 1 m 2 inter 1 m 2 inter Jex + z Jex = U0 − Jex − zJex

8 n 8 n 8 n

(9.49)

Adding the conﬁgurational entropy to this expression and minimizing the free energy, one obtains the same results for AF as given by expressions (9.47) for ferromagnetism. The difference lies in the value of the exchange constant for F inter = Jex + zJex , and both these orderings. For ferromagnetism this constant is Ftot AF inter for antiferromagnetism Ftot = Jex − zJex , as for AF the inter-site interactions have increased the internal energy and opposed the ordering. The same exchange ﬁeld was derived for ferromagnetism, with the additional account for the kinetic interactions: t, tex [see (7.62)]. n2 − m 2 F Ftot = F + z J + tex + 2t1 − n 2

(9.50)

and for the antiferromagnetism as [see (8.43)] AF Ftot = F − zJ + 2tex IAF

(9.51)

inter and tex = t = 0; therefore under this In here, one has F ≡ Jex J ≡ Jex substitution (9.50) and (9.51) will become identical with the results quoted above: F inter Ftot = Jex + zJex

AF inter Ftot = Jex − zJex

All these results of Bragg–Williams approximation lack any indication of the competition between the kinetic and potential energy, which is present only in the itinerant band model for electrons. Therefore for atomic ordering, ferromagnetism and antiferromagnetism, they give us only the effective molecular ﬁeld model without the critical values of interaction for these orderings.

REFERENCES [9.1] [9.2] [9.3] [9.4] [9.5] [9.6] [9.7] [9.8]

C.H. Johansson and J.O. Linde, Ann. Physik 25, 17 (1936). P.G. Huray, L.D. Roberts and J.O. Thomson, Phys. Rev. B 4, 2147 (1971). H. Amar, K.H. Johnson and C.B. Sommers, Phys. Rev. 153, 655 (1967). H. Hasegawa and J. Kanamori, J. Phys. Soc. Jpn 31, 382 (1971); H. Hasegawa and J. Kanamori, J. Phys. Soc. Jpn 33, 1599, 1607 (1972). J.W. Cable, E.O. Wollan and W.C. Koehler, Phys. Rev. 138, A755 (1965). M.F. Collins and D.A. Wheeler, Proc. Phys. Soc. 82, 633 (1963). N. Janke-Gilman, M. Hochstrasser and R.F. Willis, Phys. Rev. B 70, 184439 (2004). M.F. Collins and G.G. Low, Proc. Phys. Soc. (GB) 86, 535 (1965).

Alloys, Disordered Systems [9.9] [9.10] [9.11] [9.12] [9.13] [9.14]

225

R. Harris and M. Grant, Phys. Rev. B 38, 9323 (1988). D.P. Woodruff, The Solid–Liquid Interface, Cambridge University Press, Cambridge (1973). J. Mizia and R.W. Smith, J. Crystal Growth 98, 659 (1989). D. Camel, G. Lesoult and N. Eustathopoulos, J. Crystal Growth 53, 327 (1981). P. Hicter, D. Chatain, A. Pasturel and N. Eustathopoulos, J. Chim. Phys. 85, 941 (1988). H. Gould and J. Tobochnik, An Introduction to Computer Simulation Methods, Application to Physical Systems, Addison-Wesley, Reading, MA (1996).

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CHAPTER

10 Itinerant Superconductivity

Contents

10.1 Phenomenological Introduction and Historical Background 10.2 Physical Properties of the High-Temperature Superconductors 10.2.1 General properties 10.2.2 Crystal structure of the HTS 10.2.3 Symmetry of the energy gap 10.2.4 Dependence of the critical temperature on concentration 10.2.5 Phase diagrams of the ordering 10.3 Classic (BCS) Model for Superconductivity 10.4 Electron–Electron Interaction as a Source of Superconductivity 10.4.1 Introduction 10.4.2 Single-band model 10.4.2.1 Model Hamiltonian 10.4.2.2 Moments method for the superconductivity equation 10.4.2.3 Analysis of the solution: critical temperature dependence on concentration 10.4.2.4 Effect of internal pressure on superconductivity 10.4.2.5 Symmetry of the energy gap 10.4.3 Three-band model 10.4.3.1 Introduction 10.4.3.2 The Model Hamiltonian 10.4.3.3 Hamiltonian diagonalization and the pairing interaction 10.4.3.4 Results Appendix 10A: Transformation of Superconductivity Hamiltonian to Momentum Space

228 230 230 231 232 235 236 237 240 240 243 243 245

248 252 257 259 259 259 263 266 269

227

228

Models of Itinerant Ordering in Crystals

Appendix 10B: Green Function Equations for Singlet Superconductivity Appendix 10C: Effective Pairing Potential in the Single-Band Model Appendix 10D: Bogoliubov Transformation References

10.1 PHENOMENOLOGICAL INTRODUCTION AND HISTORICAL BACKGROUND Superconductivity (SC) had already been discovered in 1911, but new resurgence of research in this ﬁeld started after 1986. In 1986, Bednorz and Müller [10.1] discovered high-temperature SC in the La5−x Bax Cu5 O53−y compound at a temperature of ∼ 30 K. This discovery was immediately followed by other researchers and already by 1987 Wu et al. [10.2] reached the critical temperature, TSC , of the order of 90 K in YBa2 Cu3 O7− (Y123). Even higher critical temperatures were achieved in compounds Bi2 Sr2 Ca2 Cu3 O10 TSC = 110 K [10.3] and Tl2 Ba2 Ca2 Cu3 O10 TSC = 125 K [10.4]. The highest critical temperature, TSC = 133 K, was obtained for those compounds (called cuprates) for HgBa2 Ca2 Cu3 O8+x [10.5] and for Hg08 Pb02 Ba2 Ca2 Cu3 O8+x [10.6] in 1993. To further increase this critical temperature one can apply high external pressure. HgBa2 Ca2 Cu3 O8+x is particularly sensitive to this pressure, for which the temperature TSC = 155 K was achieved under the pressure of 25 GPa and TSC = 164 K under 30 GPa [10.7]. Progress in the critical temperature of superconductors from its discovery to the present day is shown in Fig. 10.1. The compound HgBa2 Ca2 Cu3 O8+x has the highest critical temperature among the cuprates. Further investigations after 1994 did not bring any increase of the critical temperature, but new interesting materials were discovered. Superconducting strontium ruthenate Sr2 RuO4 discovered in 1994 [10.8] has a similar structure as the cuprate La2−x Bax CuO4 , which is the spin singlet d-wave superconductor, but Sr2 RuO4 is the spin triplet p-wave or f-wave superconductor with a very low critical temperature, TSC ∼ 15 K. Spin triplet SC was also discovered in the UGe2 compound [10.9] in 2000. The characteristic feature of this compound and of compounds ZrZn2 and URhGe is the coexistence of the ferromagnetic phase with a superconducting state. In superconducting cuprates and heavy-fermion materials discovered up to that time, there was coexistence or rather competition between the superconducting and antiferromagnetic phases. Such coexistence also takes place in borocarbide RM2 B2 C, discovered in 1994, where M is usually Ni and R is a rare earth element. In those compounds, the maximum of the critical temperature

274 277 280 282

229

Itinerant Superconductivity

175 HgBa2Ca2Cu3O8 + δ (30 GPa)

150 High-temperature superconductors

125

HgBa2Ca2Cu3O8+δ Tl2Ba2Ca2Cu3O10

TSC (K)

Bi2Sr2Ca2Cu3O10

100

YBa2Cu3O7–δ Liquid nitrogen

75 Cs3C60

50 Classical superconductors

25 Hg Pb

1920

Nb

NbN0.96

BaxLa5–xCu5Oy Nb3Ge

Nb3Sn

MgB2

YPd2B2C Sr2RuO4 UGe2(0.8 K)

1940

1960 Discovery date

1980

2000

FIGURE 10.1 Changes in the critical temperature since the discovery of superconductivity.

was found in YPd2 B2 CTSC ∼ 23 K. The borocarbides are the ﬁrst quaternary intermetallic systems that are superconducting. These materials are 3D in their behaviour, and thus they are in fact quite different than the layered cuprates. A separate group of superconductors was discovered, buckminsterfullerenes or fullerenes for short, in 1991. They are compounds of alkali metals with C60 molecules. The compound C60 is not a superconductor, but when alkali metals are added it becomes superconducting. Within this group the maximum of critical temperature is observed in Cs3 C60 TSC ∼ 47 K. In 2001, SC was discovered in MgB2 for which the critical temperature is TSC ∼ 40 K. This compound has a simple crystal structure, large coherence lengths, high critical current densities and ﬁelds. Therefore its application prospects are higher than for cuprates. It was established [10.10] for all groups of superconductors that with the increase of critical temperature the isotope effect decreases. The isotope effect was a strong conﬁrmation of the BCS theory [8.8] in the case of classic superconductors. This effect is the change of the critical temperature with the isotope mass of superconductor. According to the BCS model, after assuming that atoms in the lattice are the harmonic oscillators, the critical temperature is proportional to the atomic oscillator angular frequency: TSC ∼ =

k M

1/2

or TSC ∼ M −

where = 1/2

(10.1)

230

Models of Itinerant Ordering in Crystals

A majority of classical superconductors have the coefﬁcient close to the above value. In the case of superconducting cuprates, the isotope effect is caused by the mass change of vibrating oxygen atoms in the CuO2 plane. The O16 atoms can be replaced by O18 . Experimental data have shown the decrease of coefﬁcient with increasing critical temperature. For La2−x Srx CuO4 (La124) with TSC = 35 Kx = 015 coefﬁcient is 0.1 [10.11], but for Y123 with TSC = 92 K coefﬁcient is only 0.02 [10.12]. It decreases to zero with the increase of critical temperature to 100 K [10.10]. Decreasing is the evidence of a decreasing electron–phonon mechanism in creating high critical superconducting temperatures. Also the share magnitude of the critical temperature eliminates possible phonon effect as the SC mechanism. After BCS [8.8] one can assume that the critical temperature is given by TSC = exp−1/

(10.2)

where is the Debye temperature and is the electron–phonon coupling constant. For TSC of the order of 100 K one needs ∼ 04 even at a very high Debye temperature = 1250 K [10.13]. The values of the Debye temperature and electron–phonon coupling constant for the cuprates (e.g. for La124 = 01 [10.14] and = 360 K [10.15]; for Y123 = 02 [10.14] and = 410 K [10.15, 10.16]) are not high enough to reach their high critical temperature. In conclusion, the new SC cannot be explained by the classic phonon effect described in the BCS theory [8.8], but it will require the electron–electron coupling which has much higher binding energies.

10.2 PHYSICAL PROPERTIES OF THE HIGH-TEMPERATURE SUPERCONDUCTORS 10.2.1 General properties The high-temperature superconductors (HTS) have many interesting properties. A majority of them have planes of CuO2 which will be described below. Apparently these planes are important for creating the SC state. The next feature of HTS is the metallic character of their conductivity in the CuO2 planes, which is of the same order as the conductivity of some disordered metallic alloys. The HTS crystals show different defects such as crystal grains, grain boundaries, twinning, oxygen vacancies and porosity, which have a negative inﬂuence on the maximum critical currents [10.17]. In the direction perpendicular to the CuO2 plane, they show smaller conductivity, which increases with temperature as in the semiconductors. The HTS are characterized by their small coherence length in the CuO2 plane, in the order of 15 Å for Y123, which is comparable with the size of the elementary unit cell. The coherence length depends strongly on the direction

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231

in the unit cell. Along the c axis it is only 4 Å for Y123 and is close to the distance between two adjacent CuO2 planes. This suggests that the HTS can be described by a 2D model of the CuO2 planes, which are only weakly coupled [10.17]. Due to a very small coherence length, the cuprates are type II superconductors with a large upper critical ﬁeld Hc2 , but with a small value for the critical current. The Hall effect measurements on HTS show that the majority of them are of the positive p type; the exception here is R2−x Cex CuO4 (R = Nd, Sm, Eu and Pr) [10.18] with a negative Hall constant. The carrier concentration from Hall’s measurements is of the order of 7 × 1021 cm−3 for Y123 [10.17]. Considering carriers as coming from CuO2 planes, their concentration is close to the carrier concentration of copper.

10.2.2 Crystal structure of the HTS The main feature of the HTS is their layered structure. A schematic structure of the HTS will be shown for the example of YBa2 Cu3 O7− . It is a superconductor for 0 < < 06 and insulator for 06 < < 10. The general structure of this compound is shown in Fig. 10.2. A simple layer model of this compound is shown in Fig. 10.3. The CuO2 quasi-2D planes are responsible for the SC. In the compound with too small carrier concentration (underdoped, = 1), Cu2+ ions in the CuO2 plane have an electron conﬁguration d9 and are coupled antiferromagnetically with their neighbours. Such a plane will behave as an insulator.

Y

CuO chains

Ba BaO plane

Cu O

CuO2 plane

Y plane

CuO2 plane

BaO plane

CuO chains

c B

b

a

FIGURE 10.2 Crystal structure of YBa2 Cu3 O7− .

232

Models of Itinerant Ordering in Crystals

CuO chains BaO CuO2 plane Y CuO2 plane BaO CuO chains

FIGURE 10.3 Simplified layer structure of YBa2 Cu3 O7− .

By doping we increase the amount of oxygen in YBa2 Cu3 O6 and change the carrier (holes) concentration. In the doping process, oxygen enters the compound as O2− ion on the B sites in Fig. 10.2 creating together with Cu ions the CuO chains. To keep the neutrality of charge the electrons are removed from the CuO2 planes, creating holes with high mobility, which can conduct the current. These holes in the copper-oxide planes below the critical temperature form the Cooper pairs, which are responsible for the SC. The conductivity of the CuO2 planes increases with carrier concentration, but TSC increases only initially, reaching a maximum and later dropping. In the case of Y123, the SC disappears at the hole concentration of 0.3 per copper atom [10.17]. The chains Cu–O can be treated as a reservoir of charge, which can be transferred to and from the CuO2 planes (Fig. 10.4).

10.2.3 Symmetry of the energy gap There are many papers on the symmetry of the energy gap in HTS. Most of the authors found that the energy gap of the hole superconductors have a symmetry of d dx2 −y2 -wave. The d-wave symmetry of the energy gap

Charge storage CuO2 plane Y CuO2 plane Charge storage

FIGURE 10.4 Layered model of YBa2 Cu3 O7− .

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233

was supported by angle-resolved photoemission spectroscopy (ARPES) experiments for Bi2 Sr2 Ca2 Cu3 O10 [10.19], superconductor quantum interference device (SQUID) [10.20] and nuclear magnetic resonance (NMR) [10.21] for Y123. In compounds with a negative Hall’s constant, e.g. R2−x Cex CuO4 (R = Nd and Pr), the isotropic s-wave symmetry energy gap was observed [10.22]. Recapitulating, in compounds with a positive Hall constant one has the hole SC with d-wave symmetry and in compounds with a negative Hall constant one has the electron SC, generally with s-wave symmetry. For the d-wave symmetry one can write for the energy gap [10.23] k = T cos kx − cos ky

(10.3)

where

TSC T = 0 tanh −1 T

(10.4)

Parameters 0 and can be ﬁtted to the experimental data. For Y123 Pines and co-workers [10.23] estimated, on the basis of NMR and Knight shift experiments [10.24], that 0 = 3kB TSC and = 22. There are also interesting results from Shen’s group. They drew the phase diagram with the dependence of energy gap width and critical temperature on hole concentration for Bi2 Sr2 CaCu2 O8+ (Bi2212) as obtained by the ARPES method [10.25, 10.26]. This dependence closely agrees with the model assuming that the transition temperature from normal to SC state in the underdoped region is not the temperature of forming pairs but rather the temperature at which pairs already existing become coherent in phase. A decrease of the critical temperature with decreasing doping in the underdoped region is caused by decreasing carrier concentration, while the decrease of critical temperature with increasing doping in the overdoped region is caused by decrease of the pairing potential with increasing carrier concentration [10.26]. Schematic dependence of the energy gap on concentration for Bi2 Sr2 Ca1−x Dyx Cu2 O8+ is shown in Fig. 10.5. In the review paper [10.27], the authors described many experiments on the symmetry of the energy gap for compounds Bi2212 and Y123. These experiments supported the existence of the d-wave gap in these compounds. Figure 10.6 shows the angular measurements of the energy gap made by Ding et al. [10.28] using ARPES for Bi2212, which supports very clearly the d-wave character of the gap. Theoretical papers also predict the existence of the d-wave superconducting gap which may coexist with the s-wave gap [10.29–10.31].

234

Models of Itinerant Ordering in Crystals

ΔSC (meV)

30

20

10 2.14 kBTSC

0

0.1

0.05

Bi2Sr2Ca1–xDyxCu2O8+δ T = 13 K

0.15 p

0.2

0.25

FIGURE 10.5 The dependence of the energy gap (open circles) on concentration for compound Bi2 Sr2 Ca1−x Dyx Cu2 O8+ [10.25]. For comparison the BCS relation SC = 214kB TSC is drawn as a dashed line. The energy relation, 214kB TSC , shows very different behaviour from the linear fit to the SC data points (straight line) for underdoped samples. The gap values for overdoped samples decrease in the conventional way. Reprinted with permission from J.M. Harris, Z.X. Shen, P.J. White, D.S. Marshall, M.C. Schabel, J.N. Eckstein and I. Bozovic, Phys. Rev. B 54, R15665 (1996). Copyright 2007 by the American Physical Society. M

Y

E

40

1 15 15

Δ (meV)

30 Γ

M 1

20

10

0

0

20

40 60 FS angle

80

FIGURE 10.6 The superconducting gap at 78 K in overdoped Bi2212, extracted from fits, versus angle on the Fermi surface (filled circles) compare to a d-wave gap (solid line). Locations of measured points and the Fermi surface are shown in the inset [10.28]. Reprinted with permission from H. Ding et al., Phys. Rev. B 54, R9678 (1996). Copyright 2007 by the American Physical Society.

235

Itinerant Superconductivity

10.2.4 Dependence of the critical temperature on concentration Pure cuprates compounds (e.g. La2 CuO4 and YBa2 Cu3 O6 ) have a half-ﬁlled band [10.32]. For La2 CuO4 the number of holes per Cu site is equal to 1 ± , where is determined by doping, oxygen defects and the states of atoms outside the CuO2 planes. Thinking along the lines of the Hubbard model, one can expect that a large on-site Coulomb repulsion U splits the conduction band into two bands (upper and lower Mott–Hubbard bands). For undoped (pure) compounds the lower band is ﬁlled and the upper band is empty. Experimental dependence of TSC on concentration for Y123 was presented in [10.33–10.36], for the family of Bi2212 in papers [10.25, 10.37] and for the family of La124 in papers [10.34, 10.37]. Despite a different TSC it was found [10.34– 10.37] that for these three groups the ratio TSC /TSCmax in function of holes concentration in the lower Hubbard band, p, is identical (see Fig. 10.7). To describe the experimental results Tallon and co-workers [10.36] introduced an empirical formula for the TSC dependence on hole concentration p on copper atoms in the CuO2 plane: TSC p = 1 − 826p − 0162 TSCmax

(10.5)

Derivation of the theoretical formula for critical temperature versus concentration is the goal of many papers, e.g. the results for compounds Bi2212 [10.38] were ﬁtted in [10.39] to the theoretical model. Schematic dependence of the critical temperature on concentration is presented in Fig. 10.7. In addition to critical temperature and the energy gap, the superconductors are also characterized by the ratio 2 /kB TSC . A comparison of theoretical with

TSC / TSC,max

1.2 123 214 2126 Ca-123: δ=0 δ=1 x = 0.1

0.8

0.4

0

0

0.1

0.2

0.3

p

FIGURE 10.7 The dependence TSC /TSCmax on hole concentration p for YBa2 Cu3 O7− (123), La2−x Srx CuO4 (214), La2−x Srx CaCu2 O6 (2126) and Y1−x Cax Ba2 Cu3 O7− (Ca-123) [10.35]. Reprinted with permission from J.L. Tallon, C. Bernhard, H. Shaked, R.L. Hitterman and J.D. Jorgensen, Phys. Rev. B 51, 12911 (1995). Copyright 2007 by the American Physical Society.

236

Models of Itinerant Ordering in Crystals

the experimental value of this parameter is a usual test for theoretical models. In the BCS model, this ratio was 3.5 and was close to the experimental values for the classic phonon-mediated superconductors. In the HTS, this parameter is larger and depends on carrier concentration. At the optimum doping (maximum critical temperature) it is 4.3 for the group La124 [10.37] and Bi2212 [10.40, 10.41] and 4.0 for Y123 [10.12]. When the concentration decreases from its optimum value the ratio 2 /kB TSC increases since the energy gap increases and the critical temperature decreases [10.25, 10.37, 10.40] (see Fig. 10.5). In the HTS, there is a strong interdependence between crystal structure and the critical temperature. In the compound Y123, a large inﬂuence on the critical temperature has the atomic ratio Ba/Y in the unit cell and also the length of the c axis. Data obtained by the metal organic chemical vapour deposition (MOCVD) technique [10.34, 10.42, 10.43] for Y123 have shown that the optimum ratio of Ba/Y is between 1.6 at TSC = 89 K [10.42] and 2.2 at TSC = 93 K [10.43]. The maximum critical temperature for Y123 was reached for c = 11685 Å [10.34, 10.44]. The inﬂuence of the Ba/Y ratio and the c axis length on the critical temperature is related to the pressure exerted on the CuO2 plane in the compound. This pressure can be one of the driving forces towards SC.

10.2.5 Phase diagrams of the ordering As mentioned above many experimental papers support the common phase diagram for the cuprates despite their complicated structures and a variety of chemical methods introducing holes into the CuO2 planes. This phase diagram is presented in Fig. 10.8.

200

0

Overdoping

Underdoping Antiferromagnetic

T (K)

400

0

Optimal doping

600

Normal Pseudogap

Superconductor 1 Doping (p/poptimal)

FIGURE 10.8 Schematic phase diagram for cuprates.

Itinerant Superconductivity

237

The terms underdoping (too small doping), overdoping (too large doping) and optimal doping describe the hole doping, which is the key parameter for electronic properties of the HTS. Figure 10.8 shows the two main states of the layered cuprates, where p = 0 corresponds to half-ﬁlled band and the optimal doping to poptimal ∼ 018. In the insulating phase at concentration p < 003, the layered cuprates show long-range antiferromagnetic order. For concentrations in the range p = 005–03 the cuprates are in the superconducting state. Outside these two major regions there are regions with antiferromagnetic correlation and with the pseudogap (region of Cooper pairs which are incoherent). New regions in the phase diagram are accessed due to the high-intensity magnetic pulse ﬁelds of the order of 60 T. These pulses are strong enough to destroy SC below the critical temperature (without the presence of constant ﬁeld) [10.45]. The phase diagrams show the importance of electron–electron correlations. In the insulating phase, due to hybridization of orbital 3dx2 −y2 and orbital 2p in the CuO2 planes, the cuprates form the band which is half ﬁlled. We then observe antiferromagnetic ordering. With the increase of doping by holes in the CuO2 planes the Néel temperature drops rapidly to zero, but the dynamic antiferromagnetic correlations are still present [10.46].

10.3 CLASSIC (BCS) MODEL FOR SUPERCONDUCTIVITY First, one needs a word of explanation. It is the quantum model; the word “classic” is used here in the colloquial sense to underline its usefulness, even today, in explaining experimental phenomena. Formalism of this model is also used at present to describe the HTS. What has changed it is the mechanism of raising the negative pairing potential: from the electron–phonon to the electron–electron interaction. In 1957, Bardeen, Cooper and Schrieffer (BCS) [8.8] introduced the mechanism for creating Cooper pairs, which are pairs of electrons bound by the virtual phonons. As a result of phonon exchange between two electrons with momentum k1 k2 , they scatter to the state with momentum k1 − q k2 + q (q – phonon momentum). A phonon is virtual; therefore its emission or absorption does not have to obey the law of conserving energy. Phonon energy larger than the sum of energies of both electrons will create attractive interaction. Electrons which are not further from the Fermi energy than the maximum phonon’s energy D (here we assume that the maximum energy is the Debye’s energy) will feel the attractive potential. When this attraction overpowers the Coulomb repulsion then at a sufﬁciently low temperature a superconducting state can exist. The highest decrease in energy takes place when the moments of electrons are −k1 = k2 ≡ k, or scattering takes place from the state −k k to state

238

Models of Itinerant Ordering in Crystals

−k + q k + q. To decrease the exchange energy the spins of electrons in the pair have to be opposite. As a result the superconducting Cooper pair is the bound state of two electrons with opposite momenta and spins (the singlet pair). Due to the existence of these pairs the excited electron states are separated from the ground state by the energy gap equal to the binding energy of the Cooper pair. We will now introduce the major results of the BCS theory, which after changing the pairing potential can be applied to HTS also. The interaction Vkk is responsible for scattering Cooper pair from state k ↑ −k ↓ to k ↑ −k ↓. The total internal energy at T = 0 K is the sum of the kinetic and potential energies. The chemical potential is assumed as the zero energy level. For the free energy one obtains =

2 − fk + 1 − 2fk hk N k k (10.6)

1 + 2 V h 1 − hk hk 1 − hk 1/2 1 − 2fk 1 − 2fk − TS N kk kk k

where hk is the probability of ﬁnding pair in the state k ↑ −k ↓; fk = fEk = expEk + 1−1 is the probability of ﬁnding an electron in the state k k − is the electron dispersion relation centred around the chemical potential; T is the temperature; Ek is the electron dispersion energy in the superconducting state and S is the conﬁgurational entropy expressed as S = −kB

1 f ln fk + 1 − fk ln1 − fk N k k

(10.7)

After minimizing the free energy over hk = 0 hk

(10.8)

one has 2k − +

1 − 2hk 1 V h 1 − hk 1/2 1 − 2fk = 0 1/2 hk 1 − hk N k kk k

(10.9)

and after some simple calculations the following expression is obtained: k = −

1 1 Vkk k 1 − 2fEk = − Vkk k tanhEk /2 N k 2Ek N k 2Ek

(10.10)

where Ek is given by Ek =

k − 2 + 2k

(10.11)

Itinerant Superconductivity

239

and k is the energy gap depending on the momentum k. The relation between probability hk and the energy Ek has the form 1 − (10.12) 1− k hk = 2 Ek Equation (10.10) is the main relation describing the superconducting state. Let us introduce a simpliﬁed potential which is constant around the chemical potential inside a shell of thickness Vcoff coff = cut off: V0 < 0 for k − < Vcoff and k − < Vcoff (10.13) Vkk = 0 otherwise and the constant density of states (DOS), 0 . Using this in (10.10) the following expression for the critical temperature is arrived at 1 (10.14) kB TSC = 114Vcoff exp 0 V0 The negative potential, V0 , in the expressions above was cut off at the energy distance, Vcoff , from the chemical potential. At zero temperature T = 0 and for the constant effective potential (10.13), one obtains from (10.10) the following formula: 1 (10.15) 2 0 = 4Vcoff exp 0 V0 which gives the energy gap independent of the wave vector k ≡ 0 . Comparing the last two expressions one arrives at the formula 2 0 = 35 kB TSC

(10.16)

This ratio is very important in the theory of SC, since it compares very well with the experimental results. Agreement between this theoretical value and the experimental data for low-temperature superconductors was one of the major successes of the BCS theory. As it turned out after many years [8.7] formalism BCS is also capable of describing itinerant antiferromagnetism. One has only to change some basic deﬁnitions. In the BCS model,

as a result of attractive pairing potential one obtains two bands: EkSC = ± k − 2 + 20 , separated by the SCenergy SC gap 2 0 SC ≡ 1 − 2f E Ek used in located on the Fermi level. The function F E k k (10.10) is symmetric, i.e. F −EkSC = F EkSC . As a result the summation in (10.10) over the ﬁrst band gives the same result as summation over the second band, therefore in (10.10) there is summation only over one band. In the case of

240

Models of Itinerant Ordering in Crystals

antiferromagnetism, the energy gap is centred around the atomic level t0 and one has two bands given by EkAF = ± k − t0 2 + 2AF . The expression for the probability of ﬁnding electron in the state k will now have to be modiﬁed: 2 2 f ± k − t0 + AF =

1

exp ± k − t0 2 + 2AF − + 1

(10.17)

Fermi function when used in the relation, F EkAF ≡

This modiﬁed 1 − 2f EkAF /EkAF , makes it non-symmetrical, F −EkAF = F EkAF ; therefore in the equation describing antiferromagnetism, see (10.18) below, this function has to be summed up over both bands. In addition, the antiferromagnetic potential VAF is positive and usually assumed to be constant for all energies in the band. Under these interchanges one obtains from the BCS equation the following relation for the antiferromagnetism: ⎤ 2 2 2 2 1 − 2f k + AF 1 − 2f − k + AF ⎥ V AF ⎢ ⎢ ⎥ AF = AF + ⎣ ⎦ N k 2 2 2k + 2AF −2 2k + 2AF ⎡

(10.18)

Equation (10.18) is identical with relation (8.26) after inserting AF = VAF m/2. It allows us to calculate the Néel’s temperature [TN , when AF ⇒ 0 in (10.18)] and also the value of AF at a given temperature below Néel’s temperature. The BCS formalism survived to today as the formalism fully capable of describing high-temperature SC after replacing the negative pairing potential of the electron–phonon origin by the negative electron–electron (hole–hole) potential. Moreover, as was explained above after a simple change of deﬁnitions, it also describes the band antiferromagnetism which is in agreement with the later developed model [8.7, 10.47].

10.4 ELECTRON–ELECTRON INTERACTION AS A SOURCE OF SUPERCONDUCTIVITY 10.4.1 Introduction In Section 10.1, we have described two effects (the isotope effect and the size of the critical temperature), the reasons justifying the new non-phonon mechanisms of the HTS. Several unconventional new mechanisms have been proposed. Among them are the plasmon mechanism [10.48], solitons [10.49], excitons and the RVB (resonating valence bond) theory [10.50] introduced by Anderson. In general, the electron interaction models are important, since their

Itinerant Superconductivity

241

energy scale is much higher than that of the phonon models. In this textbook, we focus on the electron models. To describe SC we will use the Hubbard model of Chapter 5. The Hamiltonian of this single-band model, which will be used here, includes the onsite and inter-site interactions given by (5.22) with the exception of inter-site exchange and pair hopping interactions, J , J : + t − tnˆ i− + nˆ j− + 2tex nˆ i− nˆ j− ci cj − 0 nˆ i H =−

i

(10.19)

U V + nˆ i nˆ i− + nˆ nˆ 2 i 2 i j

The magnetic term, −F i n nˆ i , is omitted now since the possibility of simultaneous magnetic and superconducting ordering is not considered in this chapter. The Coulomb interaction parameters can be estimated using Wannier functions or approximately atomic orbitals. They usually fulﬁl the relation U > V > t but in the case of strong screening one may have [10.51] U > t > V The Coulomb interactions U , V and t, without screening are repulsive for the electrons and they do not contribute to SC. The situation is different + for holes. Replacing the creation ci (annihilation ci ) operator for electrons by + annihilation ai (creation ai ) operator for holes, one has the relation for hole number operator: + ˆ i nˆ hi = a+ i ai = ci ci = 1 − n

(10.20)

Introducing the hole operators into (10.19) and ignoring constant terms in energy one obtains the following Hamiltonian: h h H =− t2 + th nˆ hi− + nˆ hj− + 2tex nˆ hi− nˆ hj− a+ nˆ i i aj − 0

i

U h h V h h + nˆ i nˆ i− + nˆ nˆ 2 i 2 i j

(10.21)

where the exchange-hopping interaction is deﬁned the same way as in Chapter 5: tex = t + t2 /2 − t1 , but a hopping interaction for holes has the new form th = t1 − t2

(10.22)

242

Models of Itinerant Ordering in Crystals

In the initial Hamiltonian (10.19) the hopping interaction for electrons t was repulsive, but in (10.21) the hopping interaction for holes − th is attractive. The idea of attractive hopping interaction for holes was extensively studied as the main mechanism for SC in the papers by Marsiglio and Hirsch [8.9, 10.52] and others [7.13, 10.53, 10.54]. The advantageous feature of this interaction is its naturally negative sign and purely electronic character. On the negative side is the s-wave symmetry of the energy gap produced by hopping interaction, while experimental data show that the dominant contribution is of the d-wave symmetry gap (see Fig. 10.6). Therefore the hopping interaction may play only a supplementary role for d-wave superconductors, but it can describe the superconductors with mixed s + d-wave symmetry gap. The d-wave symmetry SC is created by the diagonal charge–charge V interaction. The effective potential coming from this interaction contributes to all: the s-wave, d-wave and p-wave (for triplet states) SC. The condition for obtaining the binding potential is the negative sign of this interaction. In the single-band model, the bare interaction V is positive and it may be converted to the attractive interaction by phonon screening or by Weber’s mechanism described in Section 10.4.3. In the single-band models, it is possible to introduce the negative interaction U (“negative centres U model”) [8.13, 10.55]. Such a model has the isotropic energy gap of s0 -wave symmetry, which is independent of wave vector k, but the negative sign of interaction U is difﬁcult to be justiﬁed physically. As was found in the preceeding section the BCS formalism does not depend on the mechanism creating the negative (attractive) pairing potential. Therefore it can be still used after exchanging the weak electron–phonon interaction by the stronger electron–electron potential originated by the electron correlations. This will be the subject of further considerations now. The models used most frequently are the single-band or three-band models. Single-band models in order to describe the HTS need a large number of simpliﬁcations (e.g. ignoring two kinds of atoms (Cu and O) in the superconducting planes CuO2 . On the other side the three-band and multi-band models have a very complicated numerical analysis which causes the loss of typical simple relation for models between the driving force and resulting physical properties. The intermediate two-band model was used since the discovery of HTS [10.56, 10.57] but initially it was based on the t–J model. Later on, the two-band models based on the completely itinerant picture were also introduced [10.58–10.62]. In this chapter, we will review the single-band model describing the interaction between oxygen holes in the CuO2 plane and the three-band model describing the interaction between two degenerated copper orbitals (3dx2 −y2 and 3d3z2 −r 2 ) and the degenerated oxygen 2px = 2py ≡ 2p orbital. The single-band Hamiltonian is given by (10.21). The driving forces for the SC in this model are the hopping and exchange-hopping interactions

Itinerant Superconductivity

243

( th and tex ) and the effective negative charge–charge interaction V . To include the pressure effect as another driving force the local deformation parameter, ui , will be added. The driving force for SC in the three-band model will be the inter-band charge–charge interaction V between the two copper orbitals (3dx2 −y2 and 3d3z2 −r 2 ) and the oxygen orbital 2p. The similarity of the SC formalism developed in this chapter to the magnetic models will allow us to explain in Chapter 11 the competition and coexistence between SC on one side and AF or F on the other side.

10.4.2 Single-band model 10.4.2.1

Model Hamiltonian

This is the most frequently used model for SC. It is relatively simple and is based on representing the CuO2 plane by the lattice of atoms with hybridized quasi-particles. In the early single-band models [7.13, 8.9, 10.32], the authors have assumed that the SC state is created by the holes on the oxygen atoms. The experimental data revealed the d-wave symmetry of the energy gap cause considering the copper atoms in the CuO2 plane [10.63] as a source of SC state. To take into account in the single-band model both oxygen and copper holes we assume that the quasi-particles created in the hybridization process of copper and oxygen orbitals are responsible for SC. The Hamiltonian (10.21) is employed and analysed in the Hartree–Fock (H–F) approximation. This Hamiltonian is written for holes. In Appendix 6C, the H–F approximation is applied to the Hamiltonian (10.19) in electron representation. The only difference in applying this approximation now to holes will be the change of sign at the hopping interaction from positive for electrons in (10.19) to negative for holes in (10.21). We split the Hamiltonian (10.21) for holes and use the modiﬁed H–F approximation as in Appendix 6C, obtaining H = H0 + HS + HT

(10.23)

where H0 is the kinetic term, HS is the singlet superconducting term and HT is the triplet equal spin superconducting term. Such a choice of ordering terms is dictated by the experimental evidence showing a singlet SC in all cases of HTS with the exception of Sr2 RuO4 , heavy-fermion compounds and organic superconductors, which are the spin triplet SC. The Hamiltonian H0 has the form H0 = −

h + teff ai aj −

i

nˆ hi

(10.24)

244

Models of Itinerant Ordering in Crystals

h where the effective hopping integral is teff = t2 bh , with 2 nh 1 h h h h 2 2 2 2 h t n + 2tex − 3I − 0 − ij + − + VI b = 1+ t2 2

(10.25) the average hole occupation number is nh = a+ i ai and the Fock’s parameter proportional to the kinetic energy is I h = a+ a i j (at the moment only the h h paramagnetic state is considered, thus I = I− ≡ I h ), the modiﬁed chemical potential is = h0 − U + 2zV

nh + 2z th I h + 2ztex I h nh + ij ∗0 + ∗ij 0 2

(10.26)

The superconducting parameters 0 , ij and , appearing in (10.25) and (10.26), are deﬁned as 0 =

1 ai− ai 2

ij =

1 aj− ai 2

= aj ai

(10.27)

where 0 is the single-site singlet superconducting parameter, ij is the intersite singlet superconducting parameter, is the inter-site triplet superconducting parameter, = ±1 for spins ↑ ↓. The singlet superconducting part of Hamiltonian (10.23) will have the form + HS = U 0 − 2z th ij − 2ztex nh ij − 2I h 0 a+ i↑ ai↓ + hc i (10.28) ! 1 + + + − th 0 + V ij − tex nh 0 − 2I h ij a+ + i↑ aj↓ − ai↓ aj↑ + hc 2 and the triplet part is expressed by 1 + h HT = a+ V − 2tex I i aj + hc 2 ij

(10.29)

In the ﬁrst term of (10.28) and in the modiﬁed chemical potential (10.26), the Coulomb on-site repulsion in the H–F approximation was included, in addition to the effective Hamiltonian from Appendix 6C. Since the expectation values in (10.27) have to be real, the hermitian conjugate terms were added to the operators in (10.28) and (10.29). Transforming Hamiltonian (10.23) into the momentum space (see Appendix 10A) one obtains h ↑↓ + + + H= k − nˆ hk − kT a+ kS ak↑ a−k↓ + hc − (10.30) k a−k + hc k

k

k

Itinerant Superconductivity

245

where hk = k bh (bh ≡ bijh ), given by (10.A19), is the modiﬁed dispersion relation of the original band described by dispersion relation for holes k . A majority of the HTS show the 2D character of SC (in all cuprates superconducting current is carried by the CuO2 plane); therefore 2D unperturbed dispersion relation k for the sc lattice in the tight-binding approximation [see relation (4.34) with a ≡ 1] will be assumed: k = −2t2 cos kx + cos ky

(10.31)

↑↓

The parameter kS describes the singlet SC. The triplet superconducting ordering parameter for parallel spins is denoted by kT . ↑↓ For the dispersion relation (10.31) the parameter kS is the sum of the swave and d-wave terms: ↑↓

kS = d0 + ds k + dd k

(10.32)

where for the 2D lattice one has k = cos kx + cos ky

k = cos kx − cos ky

(10.33)

Parameters d0 and ds are components of the isotropic energy gap and the anisotropic energy gap with the symmetry sx2 +y2 , respectively. They are given by (10.A29) and (10.A30) in Appendix 10A. Parameter dd describes the d-wave SC with the wave function characterized by symmetry dx2 −y2 and is given by (10.A31). The parameter of the triplet SC is y

kT = dx kx + dy k

with dxy = 4tex I− − V

1 xy k a−k ak N k

(10.34)

where xy

k

= sin kxy

(10.35)

These rather complicated equations for the SC developed in this section will be solved by the moments method described below.

10.4.2.2

Moments method for the superconductivity equation

Many of the HTS have the singlet superconducting phase; therefore the properties of the s-wave and the d-wave symmetry singlet superconductors will be described now, using the Hamiltonian (10.30) and the Green functions method. From the equation of motion for the Green functions, (6.7), with this Hamiltonian one obtains (see Appendix 10B) ⎤ ⎡ "" ⎡ ## ↑↓ ⎤ kS − hk + ak↑ a+ a a k↑ −k↓ k↑ ⎥⎣ ⎢ ∗ ˆ ## "" ##⎦ = 1 (10.36) ⎦ "" ⎣ ↑↓ h + + kS + k − a−k↓ a+ a a −k↓ k↑ −k↓

246

Models of Itinerant Ordering in Crystals

where 1ˆ is the unit matrix. Solutions of (10.36) are the Green functions ↑↓

ak↑ a−k↓ =

kS − Ek − − Ek

(10.37)

## "" ak↑ a+ k↑ =

+ hk − − Ek − − Ek

(10.38)

where

Ek =

h 2 ↑↓ 2 k − + kS

Using the above Green functions and the Zubarev [6.1] relation 1$ AB = − fImB A d

(10.39)

(10.40)

one can calculate superconducting parameters 0 , S and D 0 =

1 1 ai− ai = a−k− ak 2 2N k

1 1 $ =− fImak↑ a−k↓ d N k 2 S =

1 1 ai+x− ai + ai+y− ai = k a−k− ak 2 2N k

1 1 $ =− fImak↑ a−k↓ d N k k 2 D =

(10.41)

(10.42)

1 1 ai+x− ai − ai+y− ai = k a−k− ak 2 2N k

1 1 $ =− fImak↑ a−k↓ d N k k

(10.43)

Inserting into (10.41)–(10.43) expressions (10.37) and (10.32) one obtains the following relations for these parameters: 0 = d0 J0 + ds J1

(10.44)

2 S = d0 J1 + ds J2

(10.45)

2 D = dd L2

(10.46)

247

Itinerant Superconductivity

where the moments Jn and Ln are deﬁned by the following expressions: Jn = −

1 1 n$ fImG1 k G1 k −d N k k

(10.47)

Ln = −

1 1 n$ fImG1 k G1 k −d N k k

(10.48)

with G1 k =

1 − Ek

(10.49)

In equations (10.44)–(10.46), the moments proportional to the odd powers of k are ignored, since they are equal to zero for the dispersion relation given by (10.31). Using (10.44)–(10.46) in (10.A29)–(10.A31) one obtains the following set of equations for the s-wave SC:

−U + 4ztex I h ztex nh + th d0 J0 + ds J1 z th + tex nh −V + 4tex I h

d0 J1 + ds J2

=

d0 ds

(10.50)

and the following equation for the d-wave SC: 1 = −V + 4tex I h L2 For the hole occupation nh = nh =

(10.51)

+ ai ai , the Zubarev relation (10.40) gives

# " # # " # 1 " + 1 " + + ak↑ ak↑ + a+ a a = 1 + a − a a k↓ k↑ k↓ k↓ k↑ k↓ N k N k

## "" ## "" 1 1 $ + = 1− f Im ak↑ a+ d k↑ − ak↓ ak↓ N k

(10.52)

Similarly the Fock’s parameter for holes in the paramagnetic state is given by 1 I h = a+ expik · h a+ i aj = k ak N kh $ 1 1 =− expik · h f Im ak a+ k d N kh where h = ri − rj is the nearest-neighbour distance.

(10.53)

248

Models of Itinerant Ordering in Crystals

The moments Jn and Ln appearing in (10.50) and (10.51) can be calculated with the help of identity 1 1 − i − 1 =P (10.54) − 1 + i0+ − 1 where P is the principal value of the integral. As a result one obtains Jn =

1 kn tanhEk /2 N k 2Ek

(10.55)

Ln =

1 kn tanhEk /2 N k 2Ek

(10.56)

1 hk − tanhEk /2 N k Ek

(10.57)

and nh = 1 −

Ih = −

1 hk tanhEk /2 N k k 2Ek

(10.58)

Using (10.50) and (10.51), moments deﬁnitions (10.55), (10.56) and the above relations for the carrier concentration nh and for Fock’s parameter in the paramagnetic state, I h , we calculate the critical SC temperature versus concentration for singlet superconductors. After setting in (10.50) and (10.51) the inter-site interactions to zero, (10.51) disappears and (10.50) takes on the well-known form 1 = −UJ0

hence 1 = −U

1 1 tanhEk /2 N k 2Ek

(10.59)

which is the BCS equation for SC, (10.10), with the pairing potential, Vkk , replaced by a constant potential U < 0 (negative U model).

10.4.2.3

Analysis of the solution: critical temperature dependence on concentration

An important characteristic of the HTS is their critical temperature dependence on carrier concentration. This dependence will be calculated separately for swave and d-wave superconductors, since equations describing them, (10.50) and (10.51) are independent. For the s-wave SC one solves (10.50), (10.57) and (10.58) with the relation (10.55) deﬁning moments Jn . From (10.50) one can see that the superconducting state may depend on up to four interaction constants: Coulomb repulsion U , charge–charge interaction V and two kinetic interactions th and tex . The inﬂuence of the correlated hopping interaction, th , on

Itinerant Superconductivity

249

SC was extensively analysed by Marsiglio and Hirsch [8.9, 10.52] and other authors [7.13, 10.53, 10.54]. These papers have shown that the driving force for the extended s-wave SC with the energy gap of the type k = ds k is the hopping interaction, th . To ﬁnd the SC one needs th > 0, which according to (10.22) means that t1 > t2 (see Fig. 10.9). In the majority of compounds, there is the relation: t > t1 > t2 , thus one has the positive hopping interaction, although its size is questionable, i.e. if it would be sufﬁcient for obtaining the superconducting state. The interactions V and tex also assist the s-wave SC of the extended s-wave type [see (10.50)]. To drive the SC the exchange-hopping interaction tex has to be negative, which means that the relation t + t2 − 2t1 < 0 has to hold, therefore the hopping integrals, t, t1 , t2 , cannot depend linearly on the concentration. Such a nonlinear dependence of t, t1 , t2 in the CuO2 plane was shown by Feiner et al. [10.64] in the effective single-band model, which came about by reducing the three-band extended Hubbard model representing the CuO2 planes in the highTSC cuprates (see Fig. 10.9). The hopping integrals obtained in that paper are in the range 0.1–0.5 eV, depending on the parameters used in the three-band model. The hopping exchange interaction tex calculated from these parameters 0.5

Hopping integrals (eV)

0.4 t2

t1

0.3

0.2 t 0.1

0

0

2

4

6

8

10

ε (eV) FIGURE 10.9 The effective hopping integrals for electrons t (dashed lines), for holes t2 (dotted line) and for electron–hole transition t1 (solid line) in the CuO2 plane as a function of the effective charge transfer energy of the oxygen band = p − d − 14536tpp for Ud = . The upper part of this figure corresponds to tpp = 05 eV and the lower part to tpp = 0. Reprinted with permission from L.F. Feiner, J.H. Jefferson and R. Raimondi, Phys. Rev. B 53, 8751 (1996). Copyright 2007 by the American Physical Society.

250

Models of Itinerant Ordering in Crystals 125

TSC (K)

100 75 50 25 0

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

nh

FIGURE 10.10 The dependence of the critical temperature TSC for the s-wave SC on carrier concentration nh for t2 = 04 eV, U = 2 eV and for different values of interactions V, t h and tex : solid line (V = 0, t h = 0225 eV and tex = 0); dotted line (V = −007 eV, t h = 02 eV and tex = −026 eV); dashed line (V = 0,t h = 02 eV and tex = −04 eV).

(t, t1 , t2 ) is always negative. In their paper, the authors also estimated the effective Coulomb repulsion: U ≈ 2–15 eV. In Fig. 10.10, the dependence of critical temperature for the s-wave SC on carrier concentration is shown for different values of driving parameters: th , V and tex . This dependence was obtained from (10.50), (10.57) and (10.58) with the relation (10.55) deﬁning moments Jn . Values of different interactions were adjusted to a maximum critical temperature of approximately 100 K at only th = 0. The half bandwidth of the original undeformed band D = 4t2 is equal to 1.6 eV. The maximum of the critical temperature at these interactions is obtained for carrier concentrations in the range of nh = 0–035. For nh = 0 SC vanishes for lack of carriers, but for nh > 035 it is destroyed by decreasing attractive potential which cannot offset the Coulomb repulsion. In all the above curves of TSC nh , the main driving force for the s-wave SC was the hopping interaction. The dependence of the critical temperature for the d-wave SC on carrier concentration is calculated from (10.51) [with moments Ln deﬁned by (10.56)] and the condition for carrier concentration (10.57). The critical temperature for d-wave state depends on charge–charge interaction V and hopping exchange interaction, tex . There is also a weak inﬂuence of hopping interaction th on the d-wave SC which comes from the bandwidth dependence on occupation. In the H–F approximation, under which these relations were derived, the positive Coulomb interaction U does not inhibit the d-wave SC. The effect of hopping exchange interaction tex on SC was analysed by Aligia and co-workers [6.16, 6.18]. Their papers have shown that the tex interaction affects both singlet (s-wave and d-wave) SC [6.16] and triplet (p-wave) SC [6.18]. Figure 10.11 shows the dependence of critical temperature for the d-wave SC on the carrier concentration for different charge–charge and

Itinerant Superconductivity

251

100

TSC (K)

80 60 40 20 0

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

1.4

h

n

FIGURE 10.11 The dependence of the critical temperature TSC for the d-wave SC on carrier concentration nh for different values of V, tex and t h : solid line V = −0154 eV, tex = 0, t h = 0; dashed line: V = 0, tex = −04 eV, t h = 02 eV; dotted line V = −007 eV, tex = −026 eV, t h = 02 eV. The remaining parameters are: t2 = 04 eV and U = 0.

exchange-hopping interactions. Values of interactions were chosen to obtain a maximum of the critical temperature for d-wave SC of about 100 K. Values of the interactions V , tex and th for the dotted and dashed curves in this ﬁgure are the same as values used for the s-wave SC in Fig. 10.10. The interaction U is not relevant in this case. The maximum critical temperature for the d-wave SC is obtained at half-ﬁlled band. For V = −0154 eV (solid line in Fig. 10.11) there is no TSC nh curve of the s-wave SC (Fig. 10.10) since this V is too weak to overcome the repulsive Coulomb potential U . Similarly for th = 0225 eV (solid line in Fig. 10.10) there is no TSC nh curve of the d-wave SC since this th is not the driving force for the d-wave SC. In the H–F approximation, the Coulomb repulsion U does not inﬂuence the d-wave SC since in the pairing potential it is present only in the s0 -wave symmetry term, and it does not deform the DOS. Domanski and Wysokinski ´ ´ [10.65] calculated the inﬂuence of repulsion U in the CPA approximation and Hubbard I approximation on the critical temperature. Both these approximations reveal the dependence of critical temperature for the d-wave SC on Coulomb repulsion U . The Hubbard I approximation limits the existence of d-wave SC to nh < 1. At the half-ﬁlled band the Hubbard I approximation always splits the band for any U , therefore at half ﬁlling, the lower sub-band is always full at any U = 0 and there are no free carriers for SC. The SC state at 04 < nh < 08 is not destroyed even when U = . In the CPA approximation, the d-wave SC exists for 02 < nh < 08, at an arbitrarily strong Coulomb repulsion, if the charge–charge interaction V > Ucr ≈ D. For smaller values of charge–charge interaction the SC state is destroyed by repulsion U , if strong enough.

252

10.4.2.4

Models of Itinerant Ordering in Crystals

Effect of internal pressure on superconductivity

The majority of superconducting cuprates have strong dependence of the critical temperature on the external pressure. At small pressure one usually has a positive and large dTSC /dp, e.g. for La1875 Ba0125 CuO4 dTSC /dp = 063 K/kbar [10.66]. The increase of critical temperature with pressure is limited, since at higher pressure dTSC /dp decreases and eventually becomes negative [10.67, 10.68]. A similar dependence of critical temperature on pressure was observed in heavy-fermion superconductors [10.69] and in ferromagnetic superconductors (e.g. UGe2 ) which will be described in Chapter 11. The ratio dTSC /dp depends on doping. For most compounds in the underdoped (overdoped) region dTSC /dp is positive (negative), while at optimal doping dTSC /dp is zero. A good example of such a compound is Y123 [10.70]. Tallon and co-workers [10.68] have explained this effect by assuming that charge reservoir planes (such as BaO plains in Y123, see Figs 10.3 and 10.4) under growing pressure transfer charge to the CuO2 layers. Thus pressure raises the hole concentration on the CuO2 layers, which explains the similarity between pressure and doping experiments. The inﬂuence of the lattice defects on the SC in the HTS is a very interesting effect. In the classic low-temperature superconductors, the decreasing number of lattice defects increases the critical temperature of SC. In the cuprates, one observes the opposite effect. Undoped compounds (e.g. La2 CuO4 or YBa2 Cu3 O6 ), which have regular ordered structure, do not show superconducting properties. Only after introducing defects, which generate internal pressures, is there a superconducting state. The size of this effect depends on the kind of dopant. For the family of La(Ba,Sr)CuO the critical temperature increases with substituting Ba or Sr atoms in place of La atoms; for compounds Y123, Tl2 Ba2 Ca2 Cu3 O10 and HgBa2 Ca2 Cu3 O8+x (HgBaCaCuO), the critical temperature increases with the addition of O atoms. It is important to note that at the optimum concentration of Ba atoms in La2−x Bax CuO4 , replacement of Ba by Sr, which has a very small ionic radius, causes further increase in critical temperature [10.71]. The same substitution of Ba by Sr in Y123 and HgBaCaCuO causes a decrease of the critical temperature [10.72, 10.73], because in these compounds it reduces the strain of the Ba/Sr layers that is needed for optimizing the charge transfer between the chains and planar Cu cations. We will now try to explain how the internal pressure affects the critical temperature in the simple single-band model of the Hamiltonian (10.21) in which in addition to the Coulomb repulsion U and the nearest-neighbour attractive interaction V , there is also a local deformation of crystal lattice responsible for the internal pressure in the CuO2 plains. The quantum expression for the local deformation has the parameter of the inter-site volume–charge coupling, which produces the negative contribution to the nearest-neighbour interaction. This allows for some choice of parameters to obtain the negative effective nearest-neighbour interaction Veff < 0, which can create the d-wave SC. The effective Hamiltonian is transformed by the Bogoliubov method into the

Itinerant Superconductivity

253

momentum space. In this process, the hole creation and annihilation operators are replaced by the new operators characteristic for the superconducting state. This method is an alternative method to the Green function method described in Section 10.4.2.2 for obtaining the equation of the energy gap. Next we solve the standard BCS SC equation obtained in the diagonalization process pairing potential using the generalized moments method presented above. The main idea in introducing the effect of internal pressure is to show how the local deformation of the unit cell ui = V0i /V0 (V0 is the initial unit-cell volume) inﬂuences the electronic properties of the crystal and in particular its superconducting state. It is assumed that the local deformation changes the electronic structure on the ith lattice site and also on neighbouring lattice sites. The inﬂuence of the local deformation ui on the electronic structure is described by three electron-deformation coupling constants: s, s1 and . The ﬁrst constant, s, represents the coupling between the on-site deformation ui and the onsite electron density. Constant, s1 , represents the coupling between the on-site deformation and the nearest-neighbour electron density. The third constant, , describes the coupling between the on-site deformation and the electron kinetic energy. This term is similar to Hirsch’s [8.9] hopping correlation term th . Summing up, in the Hamiltonian (10.21) the dependence of hopping integral on occupation was ignored by assuming th = tex = 0; the pressure effects described by the electron-deformation coupling constants s, s1 and were included. As a result the following effective Hamiltonian is obtained: H = − t2

a+ i aj +

U h h V h h nˆ i nˆ i− + nˆ i nˆ j + s ui nˆ hi 2 i 2 i

h V0 1 h + s1 ui nˆ hj − u i a+ nˆ i + V0 Kel u2i i aj − 0 8 2 i i

(10.60)

The last term in (10.60) represents the energy of elastic deformation, where Kel is the elastic module of the material. The parameter of the local deformation of the unit cell, ui introduced here, depends on the band occupation. It is calculated from the condition for the minimum of energy: H = 0 ui which when applied to (10.60) gives the following quantum expression for this deformation: h h

+ ui = − s nˆ i + s1 (10.61) nˆ j + a a 8 i l <j−i>

254

Models of Itinerant Ordering in Crystals

where = −V0 is the parameter of bandwidth increase with the increase of the unit cell volume, = 1/Kel V0 is the quantity representing lattice vibrations. Typically ≈ 0007–0013 eV−1 for the superconducting ceramic of Y123 [10.74]. After inserting (10.61) back into (10.60) one obtains the following Hamiltonian H = − t2

h a+ i aj − 0

nˆ hi + Ueff

i

nˆ hi↑ nˆ hi↓ +

i

Veff 2

nˆ hi nˆ hj (10.62)

A K + + ai aj nˆ hi− + nˆ hj− + a + a a+ a 4 4 ijl i j i− l− where Ueff = U − s2 + s12

Veff = V − ss1

K=−

s + s1 2

A=−

2 (10.63) 32

Transforming Hamiltonian (10.62) into the momentum space one arrives at the BCS reduced Hamiltonian (see Appendix 10C): Hred =

h 1 + k − a+ Vkk a+ k ak + k↑ a−k↓ a−k ↓ ak ↑ N k kk

where h hk = −2teff k

K V −A h h teff = t2 1 − nh + eff I t2 t2

= 0 − Ueff

nh − Veff nh − KI h 2

(10.64)

(10.65)

(10.66)

and the effective pairing potential is given by Vkk = Ueff + Kk + k + Ak + k 2 + Veff k k + k k

(10.67)

with functions k and k given by (10.33). The ﬁrst term in the above pairing potential is the reduced Coulomb repulsion [see (10.63)], the second term comes from the volume–charge coupling. It has the form of the correlated hopping interaction [8.9]. The explicit correlated hopping interaction term is ignored in here as it is smaller than the nearest-neighbour interaction V , and it eventually contributes only to the swave SC, which was not supported experimentally. The third term in the potential (10.67) describes the volume–charge effect related to the inter-site hopping [10.74], which contributes only to the s-wave SC. The last term, when negative, contributes to the effective nearest-neighbour interaction and, by decreasing it due to the volume–charge coupling (see (10.63)), is in favour of

Itinerant Superconductivity

255

d-wave SC. The d-wave SC is additionally augmented by the reduction of the kinetic energy [see (10.65)]. After applying the Bogoliubov transformation (see Appendix 10D) one obtains the standard BCS equation for the energy gap: k = −

1 1 − 2fEk V N k kk k 2Ek

(10.68)

where the dispersion relation in the superconducting state is given by Ek =

h 2 k − + 2k

(10.69)

and the energy gap is of the mixed s + d-wave symmetry: k = ds k2 + ak + b + dd k

(10.70)

Equation (10.68) is solved by the moments method. By inserting into (10.68) relation (10.67) for the effective potential, relation (10.70) for the energy gap and then comparing coefﬁcients at the same powers of function k and k on both sides of this equation one obtains the desired relations. Eliminating from these relations constants a and b one arrives at the ﬁnal result, which for the s-wave SC has the following form: K2 2 1 − XJ2 1 + AJ2 2 = A2 J4 − U − XJ1 + I0 1 − XJ2 X K − + AJ3 −KJ0 + 2XJ1 − AXJ0 J3 − 2J1 J2 X

(10.71)

where X = −2A − Veff and the moments Jn are deﬁned by (10.55). Equation for the d-wave SC has the form 1 = −Veff L2

(10.72)

where the moments Ln are deﬁned by (10.56). As can be seen from (10.72) the d-wave SC is stimulated by the reduced nearest-neighbour interaction, Veff , which has to be negative The other paramh eter which is hidden in this equation is the bandwidth, D = 4teff . The s-wave SC depends on many parameters of the model, see (10.71). The numerical analysis of the model will be concentrated on the d-wave SC, thus it will be assumed that = −V0 = 0, since the coupling between the on-site deformation and the electron kinetic energy, is not driving the d-wave SC. The pressure effect will be described by the electron-deformation coupling

256

Models of Itinerant Ordering in Crystals

constants, s and s1 . Strong and positive on-site Coulomb repulsion is opposing the s-wave SC, but in the mean-ﬁeld approximation it does not inﬂuence the dwave SC, see (10.72). As can be seen from this equation the d-wave SC is caused by the effective nearest-neighbour interaction Veff = V − ss1 , if it is negative. The dependence of the critical temperature on the carrier concentration, nh , d for the d-wave SC, TSC n, was obtained from the self-consistent solution of (10.72) and the equations for carrier concentration nh [relation (10.57)] and Fock’s parameter I h (10.58). ¯ = −ss1 The strain contribution to the nearest-neighbour interaction Veff u was calculated for the average deformation of the unit cell depending on the ¯ carrier concentration un. This dependence was estimated from the experimental data ([10.35, 10.75]) for compound Y123 and is shown in Fig. 10.12. The broken curve is the approximate linear dependence in the range of n = ¯ 08–1un −008461 − n. ¯ ¯ , s and s1 , also The dependence un causes the parameters depending on u: to change with the concentration. They can be calculated from (10.61), which for the average values of occupation take on the form u¯ = −snh + zs1 nh + z I h /4, ¯ h one can obtain where z is the number of nearest neighbours of site i. Using un d h varying concentration dependence of TSC n , which is shown in Fig. 10.13. The nearest-neighbour interaction Veff = V − ss1 is a function of nh and is ¯ estimated from the expression for un at = 0 and s1 = 025 s. Initial value (without stress effect) of the nearest-neighbour interaction, V = −00453 eV, is chosen to obtain TSC ≈ 100 K at concentration n = 08 (see Fig. 10.13). The decrease of Veff n at n ≈ 1 leads to the decrease of the critical temperature at this concentration.

1–n h 0

0

0.05

0.1

0.15

0.2

u

–0.004

–0.008

–0.012

–0.016

FIGURE 10.12 The dependence of the average value of unit cell deformation u for compound YBa2 Cu3 O6+x on carrier concentration nh . Dashed curve is the linear eye guide approximation to the experimental data after [10.35, 10.75].

Itinerant Superconductivity

257

100

TSC (K)

80 60 40 20 0

0.8

0.85

0.9

0.95

1

nh

FIGURE 10.13 The dependence of the critical temperature TSC on holes concentrations nh for the d-wave superconductivity with unit cell average deformation u depending on concentration nh as shown in Fig. 10.12, for V = −00453 eV and Veff estimated from u at = 0007 eV−1 , = 0 and s1 = 025 s.

10.4.2.5

Symmetry of the energy gap

The single-band model analysed here has the mixed symmetry of the energy gap. Symmetry of the s-wave type is the sum of the following symmetries: s0 → 1, sx2 +y2 → k = cos kx + cos ky , sxy → cos kx cos ky – second neighbour symmetry and sxIII2 +y2 → 2k = cos 2kx + cos 2ky , which is the sx2 +y2 symmetry of third neighbours. Symmetry of the d-wave has one component: dx2 −y2 → k = cos kx − cos ky For the s-wave SC the ratio of contributions from different symmetries can be calculated from (10.70) as 1 s0 sx2 +y2 sxy sxIII2 +y2 = 1 + b a 2 2 and it depends on coefﬁcients a and b. Coefﬁcients a and b characterize symmetries sx2 +y2 and s0 . They are calculated as the solution of BCS equation (10.68) with the effective potential (10.67) and the energy gap (10.70) by the moments method (see Section 10.4.2.2) in which the coefﬁcients at the same powers of functions k and k are compared on both sides of the BCS equation. For temperature T < TSC coefﬁcients a and b are present in the moments Jn , Ln , and thus they will depend on all model parameters (U , A, V , K) and on the

258

Models of Itinerant Ordering in Crystals

carrier concentration nh (moments Jn and Ln depend on nh ). However, in the s , moments Jn and Ln do critical temperature for disappearance of s phase, TSC not depend on a and b, therefore these coefﬁcients can be expressed as KJ0 + XJ1 + XAJ3 J0 − J1 J2

A XJ12 + J0 1 − XJ2

(10.73)

1 − XJ2 1 − AJ2 − J1 XAJ3 + K

A XJ12 + J0 1 − XJ2

(10.74)

a= and b=

At zero temperature, inserting into these expressions, nh = 006, U = 2 eV, V = −0095 eV, K = 025 eV, A = 0044 eV and D = 04 eV, one obtains a = 219 and b = −184, which gives the following ratios for different symmetries: s0 sx2 +y2 sxy sxIII2 +y2 = −174 219 2 05 From these values one can conclude that for the s-wave SC the dominant are the isotropic s0 -wave SC and anisotropic sx2 +y2 -wave (s-extended) SC. The other two components are very small. Another important parameter characterizing SC is the experimentally measured ratio: 2 0 /kB TSC . Quantity 0 is calculated as the value of the energy gap of a given symmetry (of s-wave or d-wave type) on the Fermi level at zero temperature. For the symmetry of s-wave type the energy gap on the Fermi level s0 has a constant value which can be calculated from (10.70) as −2F 2 −2F s + a+b (10.75) 0 = s D D For the d-wave SC the value of the energy gap on a Fermi level changes, depending on the direction of wave vector k. The maximum value of the energy gap for the d-wave SC d0 can be obtained from (10.70) as 2F d 0 = d 2 − (10.76) D The numerical calculations are based on (10.50) and (10.51). At T = 0 K one sd obtains s F a b and d . Assuming s = 0 and d = 0 the values of TSC are calculated. Next using (10.75) and (10.76) the following maximum values for sd sd the ratio 2 0 /kB TSC are obtained: • for the s-wave symmetry, at nh = 006, U = 2 eV, V = −0095 eV, K = 025 eV, A = 0044 eV and D = 04 eV 2 s0 = 357 s kB TSC

Itinerant Superconductivity

259

• for the d-wave symmetry, at n = 1, V = −00765 eV and D = 04 eV 2 d0 = 402 d kB TSC sd

sd

As one can see, the results obtained for 2 0 /kB TSC are different for different symmetries. For the s-wave symmetry this ratio is equal approximately to the BCS result of 3.5. The result obtained for the d-wave symmetry is close to the experimental results of 4.0 in 0 direction for the Y123 compound [10.12].

10.4.3 Three-band model 10.4.3.1

Introduction

As was mentioned in the beginning of this chapter, all the HTS cuprates have the CuO2 planes separated by other planes composed of different ions like La, Y, Bi, Th, Hg, etc. These CuO2 planes play the major role in creating the superconducting state. The experimental data for the HTS point to the ﬂow of superconducting current in the CuO2 planes. In the normal direction, this current is several times smaller [10.76]. Therefore in the ﬁrst approximation one can consider the effect of SC as taking place in the CuO2 planes. The critical temperature depends on the number of CuO2 planes in the unit cell. An increase of that number causes an increase in TSC but at n = 4 there is already a decrease in TSC , e.g. for HgBa2 Can−1 Cun O2n+2 at n = 1–6 one has TSC equal to 96, 128, 135, 127, 110, 107 K, respectively [10.77]. Similar trends are observed for Bi2 Sr2 Can−1 Cun O2n+4 and Tl2 Ba2 Can−1 Cun O2n+4 . The CuO2 planes are built as follows. Each copper atom in the plane is surrounded by four oxygen atoms. A schematic picture of the CuO2 plane is shown in Fig. 10.14. Each copper atom has ﬁve orbitals in the 3d band (3dx2 −y2 , 3d3z2 −r 2 , 3dxy , 3dxz , 3dyz ) shown in Fig. 10.15. Experimental investigations have shown that the major role in creating the superconducting state is played by the orbital 3dx2 −y2 , but one also has to consider the orbital 3d3z2 −r 2 [10.78, 10.79]. In the Hamiltonian, we include two copper orbitals (3dx2 −y2 further called 3dx , and 3d3z2 −r 2 further called 3dz ) and one oxygen orbital 2p (2px or 2py orbitals are denoted as 2p orbital).

10.4.3.2

The Model Hamiltonian

The Model Hamiltonian describing interaction of two copper orbitals with one degenerated oxygen orbital has the form H = H 0 + HI

(10.77)

260

Models of Itinerant Ordering in Crystals

tpx

tpz

3d3z 2–r 2 O2p Cu3d

3dx 2–y 2

FIGURE 10.14 Major copper and oxygen orbitals in the CuO2 plane of the three state model. Each copper contributes a 3dx2 −y2 and 3d3z2 −r2 orbital and each oxygen contributes a 2px or 2py orbital (denoted as 2p), as shown. The clear and shadowed areas indicate the opposite signs of the wave functions. The integrals tpx and tpz are the hopping integrals between 2p orbital and 3dx2 −y2 , and 3d3z2 −r2 orbitals, respectively. z

3d3z2–r 2

z

3dx2–y 2

+ –

+

–

–

y

y

+

+ x

3dxy

–

x

3dxz

y –

+

+

–

3dyz

z –

+

+

–

x

z –

+

+

–

x

y

FIGURE 10.15 Spatial distribution of electron density for the five 3d orbitals.

261

Itinerant Superconductivity

The kinetic part of the Hamiltonian (H0 ) for two copper orbitals (3dx and 3dz ) and the oxygen orbital 2p has the form H0 = p −

j

−tpx

nˆ pj + x −

nˆ xi + z −

i

nˆ zi

i

+ + dxi pj + hc − tpz dzi pj + hc

(10.78)

+ + and dzi are operators creating hole with spin on copper levels where dxi 3dx and 3dz on lattice site i; pj is the operator annihilating hole with spin on the oxygen level 2p and the lattice site j; x , z and p are energies of the levels 3dx , 3dz and 2p; nˆ xi , nzi and npi are operators of hole numbers with spin on sites i and the level 3dx , 3dz and 2p, respectively. Quantity tpx is the inter-band hopping integral between 3dx and 2p level, tpz is the inter-band hopping integral between 3dz and 2p level. Between integrals tpx and tpz there √ is relation tpx = 3tpz [10.80]. In further analysis, it will be assumed that tpx = tpd √ and tpz = tpd / 3 with tpd ≈ 15 eV. In the kinetic part of the Hamiltonian, copper and oxygen bands are treated as levels meaning that the intra-band hopping integrals (tpp = tdd = 0) are ignored. The potential energy HI has four interactions: Coulomb repulsion on oxygen (Up ), Coulomb repulsion on copper (Ud ), and the inter–band charge– charge interaction between holes on the oxygen and on the copper levels 3dx or 3dz (Vx and Vz ). It has the form

HI =

Up V Ud nˆ i nˆ i 1 − + nˆ pj nˆ pj− + nˆ nˆ 2 i 2 j 2 i pj =xz

=xz

(10.79) where Vz is the inter-band charge–charge repulsion between a hole on oxygen 2p orbital on site j and a hole on 3d3z2 −r 2 copper orbital on site i, Vx is the inter-band charge–charge repulsion between a hole on oxygen 2p orbital on site j and a hole on 3dx2 −y2 copper orbital on site i. Sites i and j are the nearest neighbours. Coulomb repulsion on copper is characterized by its highest value of energy in copper oxides (Ud ≈ 8−12 eV), much larger than the width of copper orbitals (2D ∼ 015–06 eV, [10.81]), hence one can assume that on copper there is a strong correlation: Ud /2D → . Such a correlation will limit, for energy reasons, the number of holes on both copper orbitals to one: nˆ di = nˆ xi + nˆ zi ≤ 1

(10.80)

The single occupation on copper can be introduced in an alternative way by modifying the inter-band hopping integral [10.82]. This integral depends

262

Models of Itinerant Ordering in Crystals

strongly on carrier concentration on copper. The modiﬁed hopping integral t˜ is given by the relation √ t˜ = tpd Z (10.81) where the correlation factor Z responsible for correlation on copper in the mean-ﬁeld approximation is equal to Z = 1 − nd

(10.82)

In the Gutzwiller approximation [10.83], this factor is Z=

1 − nd 1 − nd /2

(10.83)

After taking into account the strong correlation on copper through the relation (10.81) the kinetic part of the Hamiltonian takes on the form H0 =˜p − nˆ pj + x − nˆ xi + z − nˆ zi j

− t˜

i

i

+ t˜ + dxi pj + hc − √ dzi pj + hc 3

(10.84)

where ˜ p = p + Vx is the modiﬁed energy of an O orbital 2p. As stated above the strong correlation limits the total number of holes on copper to one. Including this condition in relation (10.80) gives the result nˆ xi = 1 − nˆ zi which after substituting into the potential part brings the expression HI = Up nˆ pj nˆ pj− + V nˆ zi nˆ pj j

(10.85)

where V = Vz − Vx The inter-band repulsion between orbitals 3dx and 2p Vx is larger than the repulsion between orbitals 3dz and 2p Vz since the overlap of orbital 3dx with oxygen orbital 2p is stronger than the overlap of 3dz with 2p (see Fig. 10.16). Therefore the difference of these two interactions is negative: V = Vz − Vx < 0 Weber [10.80] has estimated this difference as V ≈ −03–05 eV.

(10.86)

263

Itinerant Superconductivity

z

z

+

+ +

–

– +

–

–

2p

3d z

x

–

y

3d x

+ x

FIGURE 10.16 Two copper orbitals and oxygen y direction orbital py in the CuO2 plane.

The negative V interaction is the negative charge–charge interaction which can create the d-wave type SC.

10.4.3.3

Hamiltonian diagonalization and the pairing interaction

After transforming Hamiltonian H0 , (10.84), to the momentum space, in a similar way to transformation in Appendix 10A, one obtains the form (see [10.84]) H0 = ˜p −

+ pk pk + x −

k

−

+ dxk dxk + z −

k

+ dzk dzk

k

+ + itxk dxk itzk dzk pk + hc + pk + hc k

(10.87)

k

where ky kx ˜ txk = 2t sin − sin 2 2

ky 2t˜ kx tzk = √ sin + sin 2 2 3

(10.88)

and the lattice constant a ≡ 1 is the distance between two nearest copper atoms in the CuO2 plane. Interaction Hamiltonian HI , (10.85), in the momentum representation has a form HI = Up

kk

+ + pk p−k− p−k − pk +

V ik−k ·h + + e dzk p−k− p−k − dzk (10.89) N kk h

To simplify the model both energies x and z will be assumed equal to each other and equal to zero (x = z = 0). Energy of the oxygen level ˜ p with respect to the copper level, x = z ≡ 0, will be denoted by .

264

Models of Itinerant Ordering in Crystals

To analyse this model one has to diagonalize the Hamiltonian H0 with the help of the following transformation (see [10.84]): ⎞ ⎛ Ek+ Ek− 0 ⎟ ⎜ 2 − 2 ⎜ E + + 2 Ek + 2k ⎟ ⎟ ⎜ k k ⎟⎛ ⎞ ⎛ ⎞ ⎜ ⎟ k ⎜ x z x pk tk −itk ⎟ ⎜ −itk ⎟ ⎝ k ⎠ ⎝dxk ⎠ = ⎜ 2 (10.90) ⎜ + 2 k − 2 2⎟ E + E + ⎜ k k k k ⎟ k dzk ⎟ ⎜ ⎟ ⎜ itkz tkx itkz ⎟ ⎜ ⎠ ⎝ + 2 2 k − 2 2 Ek + k Ek + k As a result the kinetic energy of the three orbitals is transformed into three new quasi-particle states given by H0 =

+ 0 + Ek − + Ek − k k + Ek− − + k k + k k k

k

(10.91)

k

with the dispersion relations of new bands given by Ek± = 1/2 ± 2 + 42k and Ek0 = 0

(10.92)

where 2k = txk 2 + tzk 2

(10.93)

Further analysis of this model will be carried for the lower hybridized band k . The constant energy surfaces for the lower hybridized band and the dispersion relation are shown in Figs 10.17 and 10.18. Using transformation (10.90) for the interaction Hamiltonian (10.89) one obtains [10.84] the reduced effective potential of pair interaction represented by operators k in the form Ek− Ek− eik−k ·h tzk tzk (10.94) Vkk = Up Ek− Ek− + V 2 2 2k + Ek− 2k + Ek− h Potential (10.94) can be separated into a singlet part having symmetries of the types k = cos kx + cos ky and k = cos kx − cos ky and a triplet part with antisymmetric functions, e.g. sin kx + sin ky . The reduced pairing potential for the singlet part has the form + !, 4 2 1 1 eff − − ˜ Vkk = Up Ek Ek + V t 1 − k + k + k k + k k Fk Fk (10.95) 3 2 4

265

Itinerant Superconductivity

π

Y

0

–π

Γ

–π

π

0

X

FIGURE 10.17 Constant energy surfaces for the lower hybridized band in the threeband model. The symmetry points are: = 0 0, X = /a − /a, Y = /a /a.

Ek+

εp Ek0

εx = εz

Ek– Γ

X

Y

FIGURE 10.18 Dispersion relation for the lower Ek− and upper Ek+ hybridized bands in a three-band model. The symmetry points are defined as in Fig. 10.17.

where Fk =

Ek− 2 2k + Ek−

(10.96)

The potential (10.95) has two terms. The ﬁrst term describes the screened Coulomb repulsion, which opposes SC. The second term is generated by the attractive interaction V < 0, and it is the driving force for SC. The potential

266

Models of Itinerant Ordering in Crystals

has two types of symmetry: the s-wave for functions Ek− , k and const and the d-wave for function k .

10.4.3.4

Results

The energy gap, k , which corresponds to the symmetry of potential (10.95) has the following form: k = S Fk Ek− + ak + c + d Fk k

(10.97)

where the parameters S , a, c describe the s-wave gap and have the dimension of energy; parameter d describes the d-wave gap and has the dimension of the square of energy. The expressions for the energy gap (10.97) and the effective potential (10.95) are inserted into (10.68). The resulting expression is solved by the moments method, as in Section 10.4.2.2. Obtained in this way the critical temperature for the d-wave state depends only on V , the critical temperature for the s-wave state depends on V and the Coulomb repulsion on oxygen Up . According to (10.95) the effective poteneff tial Vkk is reduced by function Fk , hence to reach the critical temperature of ∼ 100 K for the d-wave SC one needs larger negative values of interaction V . This value can be reduced by decreasing the difference of copper and oxygen energy levels, = p − d and by replacing oxygen level p by oxygen band with the tight-binding dispersion relation: k = +4tpp sinkx a/2 sinky a/2. For example, for tpp = 0 and = 3tpd one has V = −17tpd ; for tpp = 0, = 2tpd one has V = −1104tpd ; and for = 2tpd and tpp = −05tpd one has V = −802tpd . The critical temperature ∼ 100 K for the s-wave state is reached at smaller values of interaction V . The reason is that in the potential (10.95) there are more terms creating the s-wave SC than the d-wave SC. Maximum critical temperature for both s-wave and d-wave state is reached at high concentration of carriers on copper nd = 07–09. Figure 10.20 shows the angular dependence of the d-wave symmetry parameter dk = d Fk k , normalized to its maximum value dMax . Angle is deﬁned in Fig. 10.19 showing the surfaces of the constant energy in the three-band model. This ﬁgure is an insert from Fig. 10.17 showing the reduced Brillouin zone. To calculate equipotential surfaces the expression (10.92) was used with the parameters: tpd = 1 eV, nd = 085 and = 3tpd . The angular dependence shown in Fig. 10.20 is typical for the d-wave superconductors. Figure 10.21shows the angular dependence of the s-wave ordering param eter Sk = S Fk Ek− + ak + c , normalized to its maximum value sMax . The minimum value of Sk at the angle = 90 is about 10% of its maximum value sMax at the angle = 45 . Those values would suggest that the s-wave energy gap has the dispersion relation close to Ek− . For parameters tpd = 1 eV, nd = 085, = 3tpd and V = −38tpd , at the critical temperature, the constants a and c in the expression for Sk are a = −012 and c = 024.

Itinerant Superconductivity

267

nd = 0.85 π

φ

0

–π

–π

π

0

FIGURE 10.19 The equipotential surfaces in the three-band model for parameters: tpd = 1 eV, nd = 085 and = 3tpd .

1

Δd / ΔdMax

0.5 0 –0.5 –1

0

20

40

60

80

φ FIGURE 10.20 Angular dependence of the d-wave type ordering parameter dk = d Fk k , normalized to its maximum value dMax . Angle is defined in Fig. 10.19.

In (10.97), there are energy gaps of the s-wave symmetry, of the s0 -wave symmetry (constant value) and of the k symmetry (sx2 +y2 -extended). The relative strength of these terms can be found by the projection method based on the orthogonality of functions k , k and 1k . For example, for the sx2 +y2 extended symmetry one has S−extended = k k k / k k2 , and for the s0 -wave symmetry one has S0 = k k · 1k / k 1k .

268

Models of Itinerant Ordering in Crystals

Δs / ΔsMax

1

0.95

0.9

0

20

40

60

80

φ FIGURE 10.21 Angular dependence of the s-wave type ordering parameter Sk = S Fk − s Ek + ak + c , normalized to its maximum value sMax . Calculations performed for T = TSC .

Calculated in this way ratio of s0 state (with constant symmetry) to sx2 +y2 state is 3.6, which means that the s0 -wave SC dominates. Since the pure s0 -wave SC has a maximum of TSC at half ﬁlling and the sx2 +y2 SC has a maximum of TSC for carriers concentrations close to zero, the maximum of total TSC appears at large values of nd . The results of the three-band model will now be compared with the results of the single-band model of Section 10.4.2. The charge–charge interaction in the single-band model with the d-wave symmetry has the form d d Vkk = V 0 k k

(10.98)

where the ratio V0d /D in the single-band model for the critical temperature TSC = 100 K and concentration n = 1 (corresponding to maximum of TSC n) depends on D. For D = 045 eV one has V0d /D ≈ −018; for D = 145 eV (the same as in Fig. 10.11) one has V0d /D ≈ −011. Increase of D at a constant maximum of critical temperature TSC = 100 K causes a decrease of V0d /D necessary to reach it. To compare the three-band model with the single-band model, the projection method is used for the effective potential (10.95), which gives for the interaction V0d the following formula: V0d =

kk

- eff Vkk k k

2 k2

(10.99)

k

Using in (10.99) the parameters of the three-band model for the maximum of the critical d-wave temperature, tpd = 1 eV, nd = 085, = 3tpd and V = −17tpd , one obtains a projected value of the charge–charge interaction in the singleband model V0d /D− = −02054, where D− is the half bandwidth of the lower

269

Itinerant Superconductivity

− − hybridized band 2D− = E00 − E− , D− = 045 eV. For nd = 075, = 2tpd , tpp = −05tpd and V = −802tpd one obtains D− = 115 eV and V0d /D− = −014. This values of V0d /D− are comparable with those for the single-band model counterpart. Recapping, the single-band model can be treated as the effective simpliﬁed three-band model describing properties of the CuO2 planes.

APPENDIX 10A: TRANSFORMATION OF SUPERCONDUCTIVITY HAMILTONIAN TO MOMENTUM SPACE The starting point is the Hamiltonian (10.23), which will be transformed from the real space to the momentum space H = H 0 + H S + HT

(10.A1)

where H0 = −

h + teff ai aj −

HS =

nˆ hi

+ + + + a1 a+ a2 0 + a3 ij a+ i↑ ai↓ + hc + j↑ ai↓ − aj↓ ai↑ + hc

i

(10.A2)

i

(10.A3)

and a1 = −2 th

j

ij + U 0 − 2tex nh j ij − 2zI h 0

(10.A4)

a2 = − th − tex nh

(10.A5)

1 a3 = V + 4tex I h 2

(10.A6)

where 0 = ai↓ ai↑

and

ij =

1 aj− ai 2

(10.A7)

The symbol j stands for the summation over nearest neighbours of site i. The triplet part has the form 1 + h HT = −2tex I + V a+ a + hc (10.A8) i j 2 ij Performing in (10.A4) the summation over nearest neighbours one obtains ij = ii+x + ii−x + ii+y + ii−y (10.A9) j

270

Models of Itinerant Ordering in Crystals

where x y = a, and a is the lattice constant. For the singlet hole pairs one has the relation ii−x = ii+x , therefore (10.A9) can be written in the form j

ij = 2 ii+x + ii+y = 4 S = z S

(10.A10)

The third term in (10.A3) can be written as + + + a3 ij a+ a − a a + hc j↑ i↓ j↓ i↑

= a3

(10.A11) ˆ ii+x + ii−x ˆ ii−x + ii+y ˆ ii+y + ii−y ˆ ii−y + hc ii+x

i

where the singlet pair operator has been introduced: ˆ ii+x = a+ a+ − a+ a+ i+x↑ i↓ i+x↓ i↑

(10.A12)

Transforming (10.A11) one arrives at + + + a3 ij a+ j↑ ai↓ − aj↓ ai↑ + hc

= a3

ˆ ii+x + ii−x ˆ ii−x + ii+y ˆ ii+y + ii−y ˆ ii−y + hc ii+x

i

1 ˆ ii+x + ˆ ii−x + ˆ ii+y + ˆ ii−y = a3 + ii+y 2 ii+x i

(10.A13)

! 1 ˆ ii+x + ˆ ii−x − ˆ ii+y − ˆ ii−y + hc + ii+x − ii+y 2 which after introducing S = 12 ii+x + ii+y and D = 12 ii+x − ii+y becomes a3

+ + + ij a+ a − a a + hc j↑ i↓ j↓ i↑

= a3

(10.A14) ˆ ii+x + ˆ ii−x + ˆ ii+y + ˆ ii−y + D ˆ ii+x + ˆ ii−x − ˆ ii+y − ˆ ii−y + hc S

i

For transformation from real to momentum space the following relations will be applied 1 ik·ri + a+ e ak i = √ N k

1 −ik·ri ai = √ e ak N k

where N is the number of lattice sites.

(10.A15)

Itinerant Superconductivity

271

Transforming the kinetic part H0 of the Hamiltonian (10.A2) by (10.A15) one has 1 h ik·ri −ik ·rj + 1 ik·ri −ik ri + H0 = − t e e ak ak − e e · ak ak N kk eff N i kk (10.A16) 1 h ik−k ·ri ik ·ri −rj + 1 ik−k ·ri + =− t e e ak ak − e ak ak N kk eff N i kk which after taking into account that 1 ik−k ·ri e = kk N i

(10.A17)

will have the form h teff kk eik ·h a+ kk a+ H0 = − k ak − k ak kk h

(10.A18)

kk

where h = ri − rj is the vector pointing to the nearest neighbours. Using property of the Kronecker’s delta function and deﬁning the dispersion relation as h ik·h hk = − teff e = −bh t2 eik·h = bh k (10.A19) h

h

where bh is given by (10.25), one can write (10.A18) as h k − nˆ hk H0 =

(10.A20)

k

with the hole number operator deﬁned as nˆ hk = a+ k ak . The Hamiltonian HS , (10.A3), is transformed in a similar way obtaining + + 1 ik·ri ik ·ri + + a1 ai↑ ai↓ + hc = a1 e e ak↑ ak ↓ + hc N i kk i (10.A21) + + + = a1 k−k a+ a + hc = a a a + hc 1 k↑ k ↓ k↑ −k↓ kk

k

1 + + + + + + a2 0 a+ a2 0 eik·rj eik ·ri a+ j↑ ai↓ −aj↓ ai↑ +hc = k↑ ak ↓ −ak↓ ak ↑ +hc N kk

ik·h ik+k ·r + + 1 + + i = a e e ak↑ ak ↓ −ak↓ ak ↑ +hc N kk 2 0 h i =

k

+ + + a2 0 2k a+ k↑ a−k↓ −ak↓ a−k↑ +hc

(10.A22)

272

Models of Itinerant Ordering in Crystals

where as before k = cos kx + cos ky

(10.A23)

The ﬁrst term in (10.A14) after transformation will take the form ˆ ii+x + ˆ ii−x + ˆ ii+y + ˆ ii−y a3 S i

1 ik·x ik·y ik+k ·ri + + + = a3 S ak↑ ak ↓ − a+ e + e e k↓ ak ↑ N kk x y i = a3 S

(10.A24)

+ + + 2k a+ k↑ a−k↓ − ak↓ a−k↑

k

and the second term will be ˆ ii+x + ˆ ii−x − ˆ ii+y − ˆ ii−y a 3 D i

1 ik·x ik·y ik+k ·ri + + + e − e e ak↑ ak ↓ − a+ = a3 D a k↓ k ↑ N kk x y i = a3 D

(10.A25)

+ + + 2k a+ k↑ a−k↓ − ak↓ a−k↑

k

where k = cos kx − cos ky . Collecting together all terms of the Hamiltonian HS one obtains + + + + + HS = a1 ak↑ a−k↓ + hc + a2 0 + a3 S 2k a+ + hc a − a a k↑ −k↓ k↓ −k↑ k

+

k

a3 D

+ + + 2k a+ a − a a + hc k↑ −k↓ k↓ −k↑

(10.A26)

k

or HS = −

↑↓ + + kS ak↑ a−k↓ + hc

(10.A27)

k

where ↑↓

kS = d0 + ds k + dd k

(10.A28)

Parameters d0 and ds are the components of the isotropic energy gap and energy gap with the symmetry sx2 +y2 , respectively; parameter dd describes the

273

Itinerant Superconductivity

d-wave SC with the wave function characterized by the symmetry dx2 −y2 . They are equal to d0 = −U + 4ztex I h 0 + 2z th + tex nh S

(10.A29)

ds = z th + tex nh 0 − 2V + 4tex I h S

(10.A30)

dd = −2V + 4tex I h D

(10.A31)

Now the term in Hamiltonian (10.A1) describing the triplet SC will be considered. Using transformation (10.A15) one obtains 1 1 + V − 2tex I− e−ik·ri ak e−ik ·rj ak eik ·ri eik ·rj a+ HT = 2 k ak + hc N kk 2 k k (10.A32) 1 1 = 2 V − 2tex I− eik −k·ri −rj ei−k+k −k +k ·rj ak ak N kk 2 k k

+ + ak ak + hc which after some algebra will give us 1 ik −k·h 1 + e V − 2tex I− ak a−k a+ HT = k a−k + hc N kk h 2 In the case of 2D crystal lattices the sum h

(10.A33)

eik −k·h can be written as

h

eik −k·h = 2 cos kx − kx + cos ky − ky = 2 cos kx cos kx + sin kx sin kx + cos ky cos ky + sin ky sin ky y y = k k + k k + 2 kx kx + k k

(10.A34)

where xy

k

= sin kxy

(10.A35)

Inserting expansion (10.A34) into (10.A33) and ignoring, for triplet SC, the xy even-parity functions k and k (since the triplet SC function, k = sin kxy , has the odd parity) one arrives at 2 1 y y + V − 2tex I− ak a−k kx kx + k k a+ HT = k a−k + hc N kk 2 (10.A36)

274

Models of Itinerant Ordering in Crystals

or ﬁnally HT = −

+ + kT ak a−k + hc

(10.A37)

k

where y

kT = dx kx + dy k

(10.A38)

and dxy = 4tex I− − V

1 xy ak a−k N k k

(10.A39)

Combining results (10.A20), (10.A27) and (10.A37) one obtains the following Hamiltonian h ↑↓ + + + H= k − nˆ hk − kS ak↑ a−k↓ + hc − kT a+ k a−k + hc (10.A40) k

k

k

APPENDIX 10B: GREEN FUNCTION EQUATIONS FOR SINGLET SUPERCONDUCTIVITY In this appendix the equations for the singlet SC will be derived. The Hamiltonian (10.30) without the parameter of triplet SC kT = 0 will be used: H=

↑↓ + + h k − nˆ hk − kS ak↑ a−k↓ + hc k

(10.B1)

k

This Hamiltonian relation ## (6.7) to calculate the Green ## will be inserted into "" "" + + functions: ak↑ ak↑ and ak↑ a−k↓ . For ak↑ ak↑ one obtains the following equation of motion: ""

ak↑ a+ k↑

##

" # = ak↑ a+ + k↑ +

.. ak↑

h k − nˆ hk k

// ↑↓ + + + − k S ak ↑ a−k ↓ + hc ak↑ k

(10.B2)

−

To the ﬁrst term on the right-hand side the fermions anticommutation relations is applied: ak↑ a+ = k↑k↑ = 1 k↑ +

(10.B3)

275

Itinerant Superconductivity

The second term is the sum of two components. For the ﬁrst component one has

h k − nˆ hk ak↑ k

=

h + k − ak↑ a+ k ak − ak ak ak↑ k

−

=

h + k − k↑k − a+ k ak↑ ak − ak ak ak↑ k

=

h + k − k↑k ak − a+ k −ak ak↑ − ak ak ak↑ k

= hk − ak↑ (10.B4) thus ..

h k − nˆ hk ak↑

//

a+ k↑

k

## "" = hk − ak↑ a+ k↑

(10.B5)

−

For the second component one has ↑↓ + + ↑↓ ∗ k S ak ↑ a−k ↓ + k S a−k ↓ ak ↑ ak↑ −

k

=−

k

=−

k

=−

k

=−

k

− ↑↓

+ k S ak↑ a+ k ↑ a−k ↓ −

k

↑↓ ∗

k S ak↑ a−k ↓ ak ↑ −

↑↓ ↑↓ ∗ + + + k S ak↑ a+ k S ak↑ a−k ↓ ak ↑ − a−k ↓ ak ↑ ak↑ k ↑ a−k ↓ − ak ↑ a−k ↓ ak↑ − k

↑↓ k S

k↑k ↑ − a+ k ↑ ak↑

+ + a+ −k ↓ − ak ↑ a−k ↓ ak↑

(10.B6) −0

↑↓ ↑↓ + + + + + k S k↑k ↑ a+ −k ↓ − ak ↑ k↑−k ↓ − a−k ↓ ak↑ − ak ↑ a−k ↓ ak↑ = − kS a−k↓

which gives ..

// "" ## ↑↓ + + ↑↓ ∗ ↑↓ + + k S ak ↑ a−k ↓ + k S a−k ↓ ak ↑ ak↑ = − kS a+ ak↑ − −k↓ ak↑ k

−

(10.B7)

276

Models of Itinerant Ordering in Crystals

"" ## Collecting together (10.B3), (10.B5) and (10.B7) we obtain for ak↑ a+ the k↑ following relation: "" ## ## "" ## h "" ↑↓ + ak↑ a+ ak↑ a+ a+ k↑ = 1 + k − k↑ − kS −k↓ ak↑

(10.B8)

Now the equation of motion (6.7) will be applied to the function ak↑ a−k↓ : .. ak↑ a−k↓ =ak↑ a−k↓ + +

ak↑

h k − nˆ hk k

// ↑↓ + + k S ak ↑ a−k ↓ + hc − a−k↓ k

(10.B9)

−

The ﬁrst term on the right is equal to 0 on the base of fermion anticommutation rules. Using for the second term the results (10.B4) and (10.B6) we arrive at "" ## ↑↓ ak↑ a−k↓ = hk − ak↑ a−k↓ − kS a+ −k↓ a−k↓

(10.B10)

"" ## "" ## + The functions a+ and a+ are calculated in a similar way, −k↓ a−k↓ −k↓ ak↑ obtaining "" ## ## ∗ h "" + ↑↓ a+ a = 1 − − a a − ak↑ a−k↓ −k↓ −k↓ −k −k↓ −k↓ −kS "" ## ## ## ∗ "" h "" + ↑↓ + a+ a−k↓ a+ ak↑ a+ −k↓ ak↑ = − −k − k↑ − kS k↑

(10.B11)

(10.B12)

Relations (10.B8), (10.B10)–(10.B12) can be written together in the matrix form: ⎡

− hk +

⎢ ⎣ ∗ ↑↓ kS

## ⎤ ⎤ ⎡ "" ak↑ a+ ak↑ a−k↓ k↑ 10 ⎥ ⎥⎢ ˆ = 1 ⎦ ⎣"" ## "" ##⎦ = 01 + + + hk − a+ a a a −k↓ −k↓ k↑ −k↓ ↑↓

kS

The above matrix relation is (10.36) from the main text.

(10.B13)

Itinerant Superconductivity

277

APPENDIX 10C: EFFECTIVE PAIRING POTENTIAL IN THE SINGLE-BAND MODEL We start from the Hamiltonian (10.62):

H = −t2

h a+ i aj − 0

nˆ hi + Ueff

i

nˆ hi↑ nˆ hi↓ +

i

Veff 2

nˆ hi nˆ hj

A K + + ai aj nˆ hi− + nˆ hj− + a+ a a+ a 4 4 ijl i j i− l−

(10.C1)

and transform it to the momentum space. Transformation of the kinetic energy term has been already performed in Section 5.1. The Coulomb repulsion term, Ueff , can be written as Ueff

nˆ hi↑ nˆ hi↓ = Ueff

i

+ a+ i↑ ai↑ ai↓ ai↓

(10.C2)

i

The holes creation and annihilation operators are transformed from real to momentum space taking into account (10.A15). As a result one obtains Ueff

nˆ hi↑ nˆ hi↓ = Ueff

i

1 ik−k +k −k ·ri + + e ak↑ ak ↓ ak ↓ ak ↑ N 2 i kk k k

(10.C3)

Summing up over all lattice sites i one obtains 1 ik−k +k −k ·ri 1 for k − k + k − k = 0 e = N i 0 otherwise

(10.C4)

The condition k − k + k − k = 0 can be written as k + k = k + k = q, which means that (10.C3) is annihilating and creating the opposite spins pair with momentum q. Imposing the additional physical condition that the total momentum of the pair before and after scattering is zero (see [10.85]) one obtains k = −k

and

k = −k

hence Ueff

i

nˆ hi↑ nˆ hi↓ Ueff

1 + + 1 + ak↑ a−k↓ a−k ↓ ak ↑ = Ueff b b N kk N kk k k

(10.C5)

+ with operators bk+ = a+ k↑ a−k↓ and bk = a−k ↓ ak ↑ , creating and annihilating the pair of holes, respectively.

278

Models of Itinerant Ordering in Crystals

In the term of inter-site charge–charge interaction, Veff , in the case of the singlet SC only the component = − will be kept: V + Veff h h nˆ i nˆ j− = eff a a a+ a 2 2 i i j− j−

(10.C6)

Using transformations (10.A15) one obtains V 1 ik−k ·ri +k −k ·rj + Veff h h nˆ i nˆ j− = eff e ak ak a+ k − ak − 2 2 N 2 kk k k =

Veff 1 ik −k ·h ik−k +k −k ·ri + + e e ak ak − ak − ak 2 N 2 kk h i k k

(10.C7) with h = rj − ri . The summation over i gives, like in the previous case, the condition k − k + k − k = 0, which together with the physical condition that the total momentum of created and annihilated superconducting pair is zero produces the relation k + k = k + k = q = 0, which in turn gives a result similar (but not identical) to the one before: 1 kk + Veff h h nˆ nˆ = V b b 2 i j− N kk eff k k

kk with Veff = Veff

ei−k−k ·h

(10.C8)

h

Summing up over all nearest-neighbour lattice vectors h in the 2D simple cubic lattice one obtains kk Veff = Veff ei−k−k ·h = 2Veff cos kx − kx + cos ky − ky = Veff k k +k k h

(10.C9)

where (as before) one has k = cos kx +cos ky and k = cos kx −cos ky . In writing (10.C9) the function sinus has been ignored, since it corresponds to the triplet SC. Fourier transform of the hopping interaction term K proceeds as follows: K K + + + + a+ ai aj nˆ hi− + nˆ hj− = i aj ai− ai− + ai aj aj− aj− (10.C10) 4 4 Using transformations (10.A15) one obtains K + ai aj nˆ hi− + nˆ hj− 4 =

K 1 ik+k −k ·ri −k ·rj + e + eik·ri +k −k −k ·rj a+ k ak ak − ak − 4 N 2 kk k k (10.C11)

Itinerant Superconductivity

279

Changing this expression in analogous way to the interaction term Veff one has K + K −ik ·h 1 ik−k +k −k ·ri h h a a nˆ + nˆ j− = e e 4 i j i− 4N kk N i h k k

+

h

1 + e−ik·h eik−k +k −k ·rj a+ k ak − ak − ak N j (10.C12)

Summing up above over h [as in (10.C9)] one obtains K + K K ai aj nˆ hi− + nˆ hj− = 4k + k bk+ bk = + k bk+ bk 4 4N kk N kk k (10.C13) In a similar way, one can write for the pressure term A + a a + hc a+ i− al− + hc 4 ijl i j =

A + + + + + + + ai aj a+ i− al− + aj ai ai− al− + ai aj al− ai− + aj ai al− ai− 4 ijl (10.C14)

Transforming it to the momentum space one has ⎛ −ik ·h A A + li ⎝ e−ik ·hji a+ e i aj + hc ai− al− + hc = 4 ijl 4N kk hji hli k k

+

eik·hji

hji

× =

hli

e

−ik ·hli

+

e

−ik ·hji

hji

e

ik ·hli

hli

+

hji

eik·hji

⎞ e

ik ·hli

⎠

hli

1 ik−k +k −k ·ri + + e ak ak − ak − ak N i

A 1 ik−k +k −k ·ri + + k k + k k + k k + k k e ak ak − ak − ak N kk N i k k

where hji = rj − ri hli = rl − ri .

280

Models of Itinerant Ordering in Crystals

Taking into account the condition for zero momentum of the superconducting pairs: k = −k and k = −k , one obtains A A + ai aj + hc a+ + k 2 bk+ bk i− al− + hc = 4 ijl N kk k

(10.C15)

Collecting together the transformations (10.C5), (10.C8), (10.C13) and (10.C15) in (10.C1) one arrives at the ﬁnal Hamiltonian in the momentum representation: H=

h 1 + k − a+ Vkk a+ k ak + k↑ a−k↓ a−k ↓ ak ↑ N k kk

(10.C16)

with the effective pairing potential given by Vkk = Ueff + Kk + k + Ak + k 2 + Veff k k + k k

(10.C17)

APPENDIX 10D: BOGOLIUBOV TRANSFORMATION The reduced Hamiltonian given by (10.64), describing scattering of electron pair from the state k ↑ −k ↓ to state k ↑ −k ↓, can be written in the form Hred =

1 h + k − a+ + a + a a V a+ a+ a a k↑ −k↓ k↑ −k↓ N kk kk k↑ −k↓ −k ↓ k ↑ k

(10.D1)

where the effective pairing potential can take any form. In our case, it is given by the expression Vkk = Ueff + Kk + k + Ak + k 2 + Veff k k + k k

(10.D2)

The above Hamiltonian will be transformed by the Bogoliubov transformation ak↑ = uk k + vk + −k a−k↓ = uk −k − vk + k

+ a+ k↑ = vk −k + uk k

and

+ a+ −k↓ = −vk k + uk −k

(10.D3)

where uk and vk are the real functions, symmetric in the momentum representation (u−k = uk and v−k = vk ). These functions fulﬁl the normalization condition u2k + vk2 = 1 hence the new operators obey the fermion commutation rules.

(10.D4)

281

Itinerant Superconductivity

As a result of transformation one obtains h + + + Hred = k − 2vk2 + u2k − vk2 + k k + −k −k + 2uk vk k −k + −k k k

+

1 + + 1 − + Vkk uk vk uk vk 1 − + −k −k − k k −k −k − k k N kk

2 + + + + uk vk 1 − + −vk k −k + u2k −k k −k −k − k k

(10.D5)

2 + + + + uk vk 1 − + uk k −k − vk2 −k k −k −k − k k 2 + + + 2 + u2k + −vk k −k + u2k −k k k −k − vk −k k The reduced Hamiltonian can be written as 0 0 1 2 Hred = Ered + Hred + Hred + Hred

(10.D6)

where the constant factor is 0 Ered =

k

1 2vk2 hk − + V u v u v N kk kk k k k k

(10.D7)

0 1 and Hred contain only the products of two operators. Hamiltonians Hred 0 Diagonal term Hred has the form h 2 1 + 0 2 Hred = k − uk − vk + 2uk vk Vkk uk vk + (10.D8) k k −k −k N k k 1 and the non-diagonal term Hred is h 2 + 1 2 1 k − 2uk vk + uk − vk Hred = Vkk uk vk + k −k + −k k (10.D9) N k k 2 Hamiltonian Hred contains the products of four operators: 2 Hred =

+ 1 −k −k + + V u v u v + + + k k k k N kk kk k k k k −k −k + + + + uk vk vk2 − u2k + k −k + −k k −k −k + k k

(10.D10)

2 + + + 2 + u2k + −vk k −k + u2k −k k k −k − vk −k k 2 For the excited low-energy states the term Hred is much smaller than the two operator terms and it will be ignored now.

282

Models of Itinerant Ordering in Crystals

1 The functions uk and vk are speciﬁed from the condition that Hred vanishes. This gives the following relation:

1 V u v 2uk vk hk − = − u2k − vk2 N k kk k k

(10.D11)

Introducing the notation k = −

1 V u v N k kk k k

and using (10.D11), (10.D4), one obtains ⎞ ⎛ u2k =

hk −

1⎜ ⎟ ⎠ ⎝1 + 2 2 h 2 k − + k

vk2 =

(10.D12)

⎞

⎛ hk −

1⎜ ⎟ ⎠ ⎝1 − 2 2 h 2 k − + k

k 2uk vk = 2 hk − + 2k

(10.D13)

(10.D14)

Inserting (10.D14) to (10.D12) one arrives at the equation for the energy gap, k , k = −

1 k V 2 N k kk 2 hk − + 2k

(10.D15)

0 , one Using (10.D12)–(10.D14) for the diagonal part of the Hamiltonian, Hred has the form h 2 + + 0 Hred = k − + 2k + Ek k k + + k k + −k −k = −k −k (10.D16) k

k

where

Ek =

h 2 k − + 2k

(10.D17)

is the elementary excited energy of the superconducting state.

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Models of Itinerant Ordering in Crystals

[10.40] M. Oda, K. Hoya, R. Kubota, C. Manabe, N. Momono, T. Nakano and M. Ido, Physica C 281, 135 (1997). [10.41] H. Won and K. Maki, Phys. Rev. B 49, 1397 (1994). [10.42] J. Zhao and P. Norris, Thin Solid Films 206, 122 (1991). [10.43] N.G. Chew, S.W. Goodyear, J.A. Edwards, J.S. Satchell, S.E. Blenkinsop and R.G. Humphreys, Appl. Phys. Lett. 57, 2016 (1990). [10.44] Y.Q. Li, J. Zhao, C.S. Chern, P. Lu, B. Gallois, P. Norris, B. Kear and F. Cosandey, Physica C 195, 161 (1992). [10.45] B. Batlogg, Physica C 282–287, XXIV–XXX (1997). [10.46] J.M. Tranquada, High-Temperature Superconductivity (ed. J. Ashkenazi), 629, Plenum Press, New York (1991). [10.47] J. Mizia, Phys. Stat. Sol. (b) 84, 449 (1977). [10.48] J. Ruvalds, Phys. Rev. B 35, 8869 (1987). [10.49] A.S. Davydov, Preprint ITP-88-55E, Institute for Theoretical Physics, Kiev, USSR (1988). [10.50] P.W. Anderson, Science 235, 1196 (1987). [10.51] D.K. Campbell, J. Tinka Gammel and E.Y. Loh Jr, Phys. Rev. B 42, 475 (1990). [10.52] F. Marsiglio and J.E. Hirsch, Phys. Rev. B 41, 6435 (1990). [10.53] T. Domanski, G. Litak and K.I. Wysokinski, Phys. Rev. B 49, 12182 (1994). ´ ´ [10.54] S. Das and N.C. Das, Phys. Rev. B 46, 6451 (1992). [10.55] G. Litak, B.L. Györffy and K.I. Wysokinski, Physica C 308, 132 (1998). ´ [10.56] J. Spałek, Phys. Rev. B 38, 208 (1988). [10.57] J. Spałek and P. Gopalan, Physica B 163, 151 (1990). [10.58] K. Penc and A. Zawadowski, Phys. Rev. B 50, 10578 (1994). [10.59] W.A. Atkinson and J.P. Carbotte, Phys. Rev. B 51, 16371 (1995). [10.60] F. Barriquand and G.A. Sawatzky, Phys. Rev. B 50, 16649 (1994). [10.61] G. Górski and J. Mizia, Physica C 309, 138 (1998). [10.62] J.E. Hirsch and F. Marsiglio, Phys. Rev. B 43, 424 (1991). [10.63] A. Nazarenko and E. Dagotto, Phys. Rev. B 53, R2987 (1996). [10.64] L.F. Feiner, J.H. Jefferson and R. Raimondi, Phys. Rev. B 53, 8751 (1996). [10.65] T. Domanski and K.I. Wysokinski, Phys. Rev. B 59, 173 (1999). ´ ´ [10.66] W.J. Liverman, J.G. Huber, A.R. Moodenbaugh and Y. Xu, Phys. Rev. B 45, 4897 (1992). [10.67] L. Gao, Y.Y. Xue, F. Chen, Q. Xiong, R.L. Meng, D. Ramirez, C.W. Chu, J.H. Eggert and H.K. Mao, Phys. Rev. B 50, 4260 (1994). [10.68] D. Tristan Jover, R.J. Wijngaarden, R. Griessen, E.M. Haines, J.L. Tallon and R.S. Liu, Phys. Rev. B 54, 10175 (1996). [10.69] C. Varma, Phys. World 11, 22 (1998). [10.70] C.C. Almasan et al., Phys. Rev. Lett. 69, 680 (1992). [10.71] J.P. Locquet, J. Perret, J. Fompeyrine, E. Maächler, J.W. Seo and G. Van Tendeloo, Nature 394, 453 (1998). [10.72] M.A. Subramanian and M.H. Whangbo, J. Solid State Chem. 109, 410 (1994). [10.73] M. Marezio, F. Licci and A. Gauzzi, Physica C 337, 195 (2000). [10.74] G. Górski and J. Mizia, Phys. Stat. Sol. (b) 215, 1083 (1999). [10.75] J.D. Jorgensen, B.W. Veal, A.P. Paulikas, L.J. Nowicki, G.W. Crabtree, H. Claus and W.K. Kwok, Phys. Rev. B 41, 1863 (1990). [10.76] S.W. Tozer, A.W. Kleinsasser, T. Penney, D. Kaiser and F. Holtzberg Phys. Rev. Lett. 59, 1768 (1987). [10.77] T. Thonhauser, H. Auer, E.Ya. Sherman and C. Ambrosch-Draxl, Phys. Rev. B 69, 104508 (2004). [10.78] Y. Ohta, T. Tohyama and S. Maekawa, Phys. Rev. B 43, 2968 (1991). [10.79] N. Nücker, H. Romberg, X.X. Xi, J. Fink, B. Gegenheimer and Z.X. Zhao, Phys. Rev. B 39, 6619 (1989).

Itinerant Superconductivity [10.80] [10.81] [10.82] [10.83] [10.84] [10.85]

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CHAPTER

11 The Coexistence Between Magnetic Ordering and Itinerant Electron Superconductivity

Contents

11.1 Experimental Evidence of Coexistence Between Magnetic Ordering and Superconductivity 11.2 Coexistence of Ferromagnetism and High-Temperature Superconductivity 11.2.1 Model Hamiltonian 11.2.2 Green function solutions 11.2.2.1 General equations 11.2.2.2 Ferromagnetism coexisting with singlet superconductivity 11.2.2.3 Ferromagnetism coexisting with triplet opposite spins pairing superconductivity 11.2.2.4 Ferromagnetism coexisting with triplet parallel (equal) spins pairing superconductivity 11.2.3 Comparison with experimental results (for UGe2 , ZrZn2 , URhGe) 11.3 Coexistence of Antiferromagnetism and High-Temperature Superconductivity 11.3.1 Model Hamiltonian 11.3.2 Formalism of the model 11.3.3 Numerical examples 11.4 Superconducting Gap in Stripe States Appendix 11A: Coexistence of Ferromagnetism and Singlet Superconductivity Appendix 11B: Coexistence of Antiferromagnetism and Singlet Superconductivity References

288 297 297 300 300 301 304

304 306 309 310 312 313 315 318 321 323

287

288

Models of Itinerant Ordering in Crystals

11.1 EXPERIMENTAL EVIDENCE OF COEXISTENCE BETWEEN MAGNETIC ORDERING AND SUPERCONDUCTIVITY New superconducting compounds and materials quite frequently possess magnetic alignments. There is a problem of coexistence between these alignments and superconductivity (SC), especially in the high-temperature superconductors. In a majority of perovskite compounds, there are two phases: SC and AF, the presence of which depends on doping. Between the phase of insulating AF and metallic SC there is sometimes also the phase of spin glass, which at present is identiﬁed rather as the diagonal stripe phase [11.1]. The concentration range and properties of this intermediate phase depend on the kind of compound. During the last few years new compounds have been discovered, which have the superconducting phase coexisting with ferromagnetism. The experimental evidence of both types of these compounds is presented below. The coexistence between magnetic phase and SC is particularly frequent in high-temperature superconductors, where the SC most likely has an electronic origin. The most popular superconducting cuprates, such as YBa2 Cu3 O6 + x and La2−x Srx CuO4−y , were extensively studied for the occurrence of the antiferromagnetic (AF) order. The schematic dependence of SC critical temperature and Néel temperature on the concentration of oxygen in YBa2 Cu3 O6 + x is shown in Fig. 11.1 (based on [11.2]). The AF phase is observed at x< 04. The maximum of the Néel temperature is TN ∼ 500 K, and the average magnetic moment of YBa2Cu3O6+x 600

Temperature (K)

500 400

TN TSC

300 200

AF

100 SC 0

0.0

0.2

0.4

0.6

0.8

1.0

x

FIGURE 11.1 Schematic phase diagram for YBa2 Cu3 O6+x showing Néel temperature, TN , and superconducting temperature, TSC , versus concentration of oxygen (after [11.2]). Reprinted with permission from J.M. Tranquada et al., Phys. Rev. B 38, 2477 (1988). Copyright 2007 by the American Physical Society.

Coexistence: Magnetic Ordering and Itinerant Electron SC

289

066 ± 007 B per magnetic Cu atom is reached at x = 0 [11.2]. The maximum of the superconducting temperature ∼ 90 K is obtained for 09 ≤ x ≤ 1. The coexistence of AF and SC ordering takes place in the underdoped region 035 ≤ x ≤ 06 (see Fig. 11.2). In this region, the compound YBa2 Cu3 O6+x is superconducting at temperatures of the order 25–60 K [11.3–11.5]. Using measurements of inelastic neutron scattering [11.5–11.7], nuclear magnetic resonance [11.8] and muon spin relaxation [11.4] the magnetic order was discovered at doping speciﬁc for the superconducting phase (e.g. for x = 04 the magnetic order existed below T = 5 K [11.4]). This magnetic order in the underdoped cuprates was deﬁned as dynamical magnetic correlation [11.9]. In the underdoped cuprates, the increasing doping increases the carrier concentration and as a result the Néel static AF order disappears, but dynamic AF spin correlations survive and coexist with the induced SC [11.10]. Similar coexistence of dynamic magnetic correlations and the hightemperature SC is observed in the underdoped regime also in the other cuprates and especially in La2−x Srx CuO4−y [11.9, 11.11] (see left side of Fig. 11.3). In these compounds, the long-range AF order depends strongly on

150

T (K)

TN 100

TSC 50

0

0.35

0.45 x

0.55

FIGURE 11.2 Phase diagram of YBa2 Cu3 O6+x near the magnetic-superconducting boundary x = 04 (description of the experimental points is given in [11.6]). Reprinted with permission from G. Shirane, J. Als-Nielsen, M. Nielsen, J.M. Tranquada, H. Chou, S. Shamoto and M. Sato, Phys. Rev. B 41, 6547 (1990). Copyright 2007 by the American Physical Society.

290

Models of Itinerant Ordering in Crystals

300 Nd2–xCexCuO4

La2–xSrxCuO4

250

T (K)

200

TN TN

150

100

AF

50

Tsc SC

0

AF

0.2

Tsc

SG 0.1

SC 0

0.1

0.2

Dopant concentration (x )

FIGURE 11.3 Schematic phase diagram for both electron Nd2−x Cex CuO4−y and hole La2−x Srx CuO4−y superconductors, showing antiferromagnetic (AF), spin glass (SG) and superconducting (SC) phases. Reprinted with permission from G.M. Luke et al., Phys. Rev. B 42, 7981 (1990). Copyright 2007 by the American Physical Society.

doping concentration and disappears at x ≈ 002. The domain of SC is relatively broader 005 < x < 02 with the maximum critical temperature of 37 K at x = 015. From the doping concentration x, one can calculate the electron occupation from the following formula: n = 1 − x (pure undoped La2 CuO4 compound has a half-ﬁlled band [10.72]). The superconducting family of La2−x Srx CuO4−y has the same structure as Nd2−x Cex CuO4−y compounds, but their properties are different. Compound La2−x Srx CuO4−y is the hole-doped superconductor, while Nd2−x Cex CuO4−y is of the electron type. Both the Néel and the superconducting critical temperatures for Nd2−x Cex CuO4−y are smaller than for La2−x Srx CuO4−y , and they are 252, 320 and 22, 40 K, respectively. The doping dependence of these temperatures is interesting. As already mentioned, in the case of La2−x Srx CuO4−y the Néel temperature decreases rapidly with doping, and at 2% the long-range magnetic ordering disappears, being replaced by the diagonal stripe phase. For Nd2−x Cex CuO4−y the dependence of the AF order on electron doping is much weaker; doping destroys the AF long range only for dopant concentrations of the order of 15% [11.12]. SC is observed in the narrow range of doping 015 < x < 018. Between the long-range AF phase and SC phase there is no spin glass phase (diagonal stripe phase), and the transition is abrupt, rather characteristic for heavy-fermion systems. The schematic comparison of TSC and TN for the hole- and electron-doped compounds is shown in Fig. 11.3 [11.12].

Coexistence: Magnetic Ordering and Itinerant Electron SC

291

In order to describe doped holes in the hole superconductors (YBa2 Cu3 O6+x and La2−x Srx CuO4−y ), the stripe picture has been proposed [11.13, 11.14], where the 1D spin and charge arrangement is characterized by incommensurate wave vector Q. A variety of experiments now support the stripe picture [11.15]. Investigations of La2−x Srx CuO4−y by the elastic neutron scattering [11.16] have shown that in the metallic superconducting phase x > 005 there is a vertical stripe structure with Q = 2 1/2 ± 1/2 or Q = 2 1/2 1/2 ± , which changes to the diagonal stripes with Q = 2 1/2 ± 1/2 ± or Q = 2 1/2 ± 1/2 ∓ in the insulating phase x < 005. Between carrier concentration x and the parameter there is linear dependence ∼ x/2 characteristic of an insulating phase [11.9]. The coexistence of AF and SC was also found experimentally in the layered organic superconductors [11.17], which have a lower superconducting temperature, e.g. salts (TMTSF)X, where X = PF6 , compounds AsF6 ClO4 [11.18, 11.19] and -BEDT − TTF2 X (see Fig. 11.4) with X = CuNCS2 , Cu NCN2 Br and Cu NCN2 Cl, with Néel temperature of the order of 27 K and SC critical

100

“Ethylene-liquid” state T g

T (K)

“Glassy” state

T∗

TN 10 Tsc

AFI

PM

SC 1 X = Cu[N(CN)2]Cl

D8

Phydr

H8

Cu[N(CN)2]Br

Cu(NCS)2

1 kbar

FIGURE 11.4 Temperature/hydrostatic pressure phase diagram for the -BEDT-TTF2 X compounds. Arrows indicate the location of different compounds at ambient pressure. Solid lines represent the hydrostatic pressure dependences of TN and TSC . Circles denote -ET2 CuNCN2 Cl, down and up triangles denote deuterated and hydrogenated -ET2 CuNCN2 Br, respectively, and squares stand for -ET2 CuNCS2 . The superconducting and antiferromagnetic transitions are represented by solid and open symbols, respectively. Diamonds stand for the glasslike transitions and crosses for the maxima of the anomalous expansivity contributions at intermediate temperatures 30–50 K (after [11.20]). Reprinted with permission from J. Müller, M. Lang, F. Steglich, J.A. Schlueter, A.M. Kini and T. Sasaki, Phys. Rev. B 65, 144521 (2002). Copyright 2007 by the American Physical Society.

292

Models of Itinerant Ordering in Crystals

Tsc (K)

0.4 10

0

2.5 3 p (GPa)

T (K)

TN

5

2Tsc 0

0

1

2

3

p (GPa)

FIGURE 11.5 Pressure–temperature phase diagram of CePd2 Si2 . Magnetic TN and superconducting TSC transition temperatures have been determined from the mid points of d /dT and , respectively. The inset shows the superconducting part of the phase diagram in more detail (after [11.26]).

temperature ∼ 12 K [11.20]. The coexistence of SC and AF at high pressures was also found in uranium (U)-and cerium (Ce)-based heavy-fermion systems (e.g. UPt3 , URu2 Si2 [11.21], UPd2 Al3 [11.22], CePd2 Si2 (see Fig. 11.5) CeRhIn5 [11.23]), where maximum Néel temperature is 14.3 K for UPd2 Al3 , and the critical SC temperature is approximately 2 K. This coexistence also takes place in the family of rare-earth nickel boride carbides RNi2 B2 C (R = Y, Lu, Ho, Tm, Er) [11.24, 11.25]. The coexistence of AF and SC in the low-temperature superconductors mentioned above has been established in materials in which (i) different groups of electrons have been responsible for both types of ordering, (ii) the superconducting coherence length is extended over many elementary cells of the AF order. In recent years new materials have been found where the ferromagnetism coexists with weak SC. These two cooperative phenomena are mutually antagonistic because SC is associated with the pairing of electron states related to time reversal, while in the magnetic states, the time-reversal symmetry is lost and therefore there is strong competition between them. Ginzburg [11.27] has already pointed out the possibility of this coexistence under the condition that the magnetization is smaller than the thermodynamic critical ﬁeld

Coexistence: Magnetic Ordering and Itinerant Electron SC

293

multiplied by the susceptibility of a given material. Matthias and co-workers [11.28] demonstrated that a very small concentration of magnetic impurities was enough to completely destroy SC, when ferromagnetic ordering was present. The theoretical possibility of ferromagnetism coexisting with the triplet parallel spins SC was suggested by Fay and Appel for ZrZn2 [11.29], while the model for coexisting of F with singlet SC was theoretically developed by Fulde and Ferrell [11.30] and Larkin and Ovchinnikov [11.31]. Further theoretical development took place after ﬁnding experimental evidence for coexistence of triplet SC with F [11.32, 11.33] or singlet SC with F [11.32, 11.34, 11.35]. More impulse to this research has come from the discovery of SC in Sr2 RuO4 [10.8], which has low TSC ∼ 15 K, but as opposed to cuprates, the SC energy gap has the odd-parity (spin triplet) character with a p-wave symmetry. Other compounds of this type have also ferromagnetic properties; for example, SrRuO3 has a Curie temperature of ∼ 165 K. For multilayer ruthenates (Srn+1 Run O3n+1 , with n being the number of RuO2 planes per unit cell), the Curie temperature depends on the number of RuO2 planes [11.36]. For n = 3 it was found that TC ≈ 148 K and for n = 2 TC ≈ 104 K. This demonstrates the tendency that with decreasing layer number, TC is reduced and ﬁnally vanishes. The number of RuO2 planes is the parameter which determines the quantum phase transition between SC and F phases. It is only recently that the so-called ferromagnetic superconductors have been discovered, which at some high pressures exhibit the ferromagnetic and a spin triplet superconducting phases at the same time. At present, UGe2 [10.9, 11.37, 11.38], URhGe [11.39] and ZrZn2 [11.40] belong to the ferromagnetic superconductors. The schematic structure of UGe2 is shown in Fig. 11.6 [11.41]. In this compound, the superconducting and ferromagnetic ordering is created by 5f electrons of uranium atoms. Schematic dependence of SC critical temperature

b a

c

U Ge

FIGURE 11.6 The crystal structure of UGe2 .

294

Models of Itinerant Ordering in Crystals

60 Paramagnetism

T (K)

40 Ferromagnetism

Tc

20 Tx 0

10Tsc SC

0

5

10 p (kbar)

15

20

FIGURE 11.7 The dependence of Curie temperature TC and superconducting critical temperature TSC on pressure for the compound UGe2 . Reprinted with permission from C. Pfleiderer and A.D. Huxley, Phys. Rev. Lett. 89, 147005 (2002). Copyright 2007 by the American Physical Society.

and the magnetic Curie temperature on the pressure for UGe2 compound is shown in Fig. 11.7 (based on the paper by Pﬂeiderer and Huxley [11.37]). At ambient pressure, UGe2 is an itinerant ferromagnet below the Curie temperature of 52 K, with low-temperature ordered moment of m = 14B per U atom (see Fig. 11.8). Uranium compounds are the heavy-fermion systems, and the 5f electrons are expected to be strongly correlated electron states. However, the speciﬁc heat measurements show that the coefﬁcient = CT /T is about 10 times smaller than in conventional heavy-fermion U-compounds [11.41], which suggests that these electrons behave more like the 3d electrons in the traditional itinerant ferromagnets such as Fe, Co and Ni.

2

m (μB/f.u.)

1.5 1

ms(H = 0) mx(H → 0 from H > Hx )

0.5

T = 2.3 K

0

0

5

10

15

20

p (kbar)

FIGURE 11.8 Pressure dependence of the dimensionless magnetization per lattice site.

Coexistence: Magnetic Ordering and Itinerant Electron SC

295

It is plausible that increasing the pressure on UGe2 changes the anisotropy, which in turn shifts the system from itinerant behaviour to a high-pressure phase which is dominated by localized spins. The superconducting phase is detected at pressures of 1.0–1.6 GPa. Maximum critical temperature TSC = 08 K is reached at 1.2 GPa, where the ferromagnetic state is still stable with TC =32 K. A characteristic feature of ferromagnetic superconductors is the existence of SC only in the domain of ferromagnetic ordering [11.38]. Transition to the paramagnetic state causes the disappearance of SC, see Figs. 11.7 and 11.9. The ferromagnets ZrZn2 and URhGe are superconducting at ambient pressure with superconducting critical temperatures TSC = 029 and 0.25 K, respectively. ZrZn2 is ferromagnetic below the Curie temperature TC = 285 K, with the low-temperature magnetic moment of m = 017 B per unit cell. Dependence of superconducting critical temperature and Curie temperature on the pressure for ZrZn2 is shown in Fig. 11.9 [11.42]. Fig. 11.10 shows the dependence of the magnetic moment on pressure for the same compound. Both the magnetic moment and the Curie temperature of ZrZn2 drop discontinuously at the pressure p = 165 kbar, indicating the ﬁrst-order transition from the ferromagnetic to the paramagnetic phase. Both superconducting and ferromagnetic phases originate in 4d electrons of Zr. The compound URhGe at p = 0 pressure has the Curie temperature TC = 95 K and magnetization m = 042 B at T = 0. Fig. 11.11 shows the dependence of Curie temperature and superconducting critical temperature on pressure for this compound [11.44]. The Curie temperature increases with pressure. This effect is opposite to the dependence TC p observed in UGe2 and ZrZn2 . Both the ferromagnetic and the superconducting states are originated by 5f electrons of U atoms.

30 Paramagnetism

T (K)

20 Ferromagnetism

Tc

10 10Tsc

0

0

5

10

15

20

25

p (kbar)

FIGURE 11.9 The dependence of Curie temperature TC and superconducting critical temperature TSC on pressure for the ZrZn2 compound.

296

Models of Itinerant Ordering in Crystals

0.2

m (μB/f.u.)

0.15

0.1

0.05

0 0

5

10 p (kbar)

15

20

FIGURE 11.10 Pressure dependence of the magnetization per lattice site for the ZrZn2 compound (after [11.43]). 20 Paramagnetism

T (K)

15

Tc

10

5

Ferromagnetism

20Tsc SC

0 0

20

40

60

80

100

120

140

p (kbar)

FIGURE 11.11 The dependence of Curie temperature TC and superconducting critical temperature TSC on pressure for the compound URhGe (after [11.44]).

Coexistence of SC and F is observed also in other systems, e.g. ErRh4 B4 HoMo6 S8 and ErNi2 B2 C compounds, which have TC < TSC . For ErRh4 B4 , one has TC = 12 K and TSC = 87 K. These compounds differ from uranium compounds by originating ferromagnetism and SC on different atoms, e.g. in ErRh4 B4 the ferromagnetic state is created by 4f electrons of Er atoms and SC by 4d electrons of Rh atoms. There are also results showing the coexistence of magnetic ordering and SC in a new family of hybrid ruthenate–cuprate compounds. In the compound RuSr2 GdCu2 O8 , there is a coexistence of SC and weak ferromagnetism. This compound exhibits ferromagnetic order at a rather high Curie temperature TC = 133–136 K and becomes superconducting at a signiﬁcantly lower critical temperature TSC = 15–40 K [11.45]. The other compound R14 Ce06 RuSr2 Cu2 O10−

297

Coexistence: Magnetic Ordering and Itinerant Electron SC

(R = Gd and Eu) exhibits coexistence of bulk SC (TSC =32 and 42 K for R = Gd and Eu, respectively) with the AF state (TN = 122 and 180 K for R = Gd and Eu, respectively) [11.46].

11.2 COEXISTENCE OF FERROMAGNETISM AND HIGH-TEMPERATURE SUPERCONDUCTIVITY 11.2.1 Model Hamiltonian The experimental results for ZrZn2 , URhGe and in some pressure ranges also for UGe2 have shown that the ferromagnetic superconductors are weak itinerant ferromagnets. This would allow us to describe them by the extended Stoner model (Section 7.5), which includes the on-site exchange ﬁeld and the inter-site charge–charge interaction (see Section 5.2): H = −t

+ ci cj − 0

nˆ i − F

i

n nˆ i +

i

V nˆ nˆ 2 i j

(11.1)

The model will be treated in the modiﬁed Hartree–Fock (H-F) approximation (see Section 6.9 and Appendix 6C). The inter-site interaction will modify the bandwidth (see Section 6.9). In the superconducting state, the experimentally observed spin triplet SC and spin singlet SC will be included (e.g. cj ci and cj− ci ). As a result the itinerant Stoner model for ferromagnetism (see Section 7.5), which interacts with different types of SC, is obtained. In this model, the Stoner F ﬁeld, Ftot , is given by F Ftot m = Mi − Mi−

(11.2)

where the spin-dependent modiﬁed molecular ﬁeld is the same on site i or j, Mi = Mj ≡ M , and is given by relation (11.6). This model can describe such compounds as ZrZn2 and URhGe, which are weak itinerant ferromagnets. For the UGe2 compound, which has itinerant as well as localized moments, one should also include in the model Hamiltonian the term describing interaction between localized and itinerant moments [11.35]. Using (11.1) one can write the following simpliﬁed Hamiltonian: H = H0 + HOSP + HESP

(11.3)

Hamiltonian H0 is the kinetic energy plus the coherent molecular ﬁeld: H0 = −

+ teff ci cj −

i

0 − M nˆ i

(11.4)

298

Models of Itinerant Ordering in Crystals

where the effective hopping interaction is teff = tb , with b given by relation (6.C11) adopted to Hamiltonian (11.1)

b = 1 +

V I t

(11.5)

In this case, the spin-dependent modiﬁed molecular ﬁeld (see Appendix 6C) is given by M = −Fn + zVn

(11.6)

The next two terms of the Hamiltonian (11.3) (HOSP and HESP ) are related to the opposite spin pairing (OSP) and equal (parallel) spin pairing (ESP). The OSP term describes the singlet SC with the total spin being 0 and the triplet SC with total spin 1 and its projection 0 and is given by HOSP =

V V S + + + + + + + + ij ci↑ cj↓ − ci↓ cj↑ + hc + Tij ci↑ cj↓ + ci↓ cj↑ + hc (11.7) 2 2

where the inter-site singlet superconducting parameter is Sij =

1 cj− ci 2

(11.8)

and the inter-site opposite spin triplet superconducting parameter is Tij =

1 c c 2 j− i

(11.9)

The ESP term is given by HESP =

V + + ci cj + hc 2

(11.10)

with the ESP parameter, , given by = cj ci

(11.11)

Transforming Hamiltonian (11.3) into the momentum space by the method as in Appendix 10A, one arrives at H=

k

k − 0 + nˆ k −

k

↑↓ ↑↓ + + + + kS + kT ck↑ k ck c−k↓ + hc − c−k + hc k

(11.12)

Coexistence: Magnetic Ordering and Itinerant Electron SC

299

where k = k b is the spin-dependent modiﬁed dispersion relation of the original band described by dispersion relation, k , which in the case of the 2D simple cubic lattice has the following form: k = −2tcos kx + cos ky = −2t k

(11.13)

In the model presented here, the self-energy is equal to the modiﬁed molecular ﬁeld. It describes the on-site exchange ﬁeld and the inter-site charge– charge interactions in the H–F approximation and is given by the expression: ≡ M = −Fn + zVn

(11.14)

Introducing the self-energy would allow, in the next step, the calculation of SC in higher order approximations rather than in the H–F approximation (e.g. CPA, Hubbard I). ↑↓ The singlet SC parameter, kS , is the sum of the s-wave and d-wave ↑↓ symmetry terms (which are characterized by the symmetric energy gap, −kS = ↑↓ ↑↓ kS ). The opposite spin triplet term, kT , is characterized by the antisymmetric ↑↓ ↑↓ energy gap, −kT = −kT . The triplet superconducting ordering parameter for parallel spins is denoted by k . ↑↓ The singlet part of kS is the sum of the s-wave and d-wave terms: ↑↓

kS = ds k + dd k

(11.15)

where the functions k and k , describing s-wave and d-wave SC, are expressed (for the 2D lattice) by (10.33). The parameter ds for the s-wave SC, obtained in the process of transforming (11.3) to the momentum space, is given by ds = −2VS

(11.16)

where S is deﬁned as S =

Sii+x + Sii+y 2

(11.17)

and Sij is given by (11.8). Index i + x is the nearest neighbours of atom i in the x direction, and i + y is the nearest neighbours of atom i in the y direction. Parameter dd for the d-wave SC is given by dd = −2VD

(11.18)

where D is equal to D =

Sii+x − Sii+y 2

(11.19)

300

Models of Itinerant Ordering in Crystals ↑↓

The triplet part of kT , which corresponds to antiparallel spin alignment, has the form ↑↓

↑↓

↑↓

y

kT = dTx kx + dTy k xy

where k

(11.20)

↑↓

and the parameter dTxy are given by xy

k

= sin kxy

(11.21)

↑↓

dTxy = −VTii+xy

(11.22)

The parameter of the triplet SC with parallel spins (ESP) is equal to y

k = dx kx + dy k

(11.23)

dxy = −Vxy = −V ci+xy ci

(11.24)

11.2.2 Green function solutions 11.2.2.1

General equations

The Hamiltonian (11.12) will be analysed using Green functions. From the equation of motion for the Green functions, (6.7), with this Hamiltonian, one obtains ⎛ ↑ ↑ 0 2k − k + 0 − ↑ ⎜ ↓ ↑↓ ↑↓ ⎜ 0 − k + 0 − ↓ −kS + kT ⎜ ∗ ∗ ∗ ↑ ↑↓ ↑↓ ↑ ⎜ 2k −kS + kT + −k − 0 + ↑ ⎝ ↑↓ ∗ ↑↓ ∗ kS + kT

↓∗ 2k

0

↑↓

⎞

↑↓

kS + kT

⎟ ⎟ ⎟ Gk ˆ ˆ = 1 ⎟ ⎠

↓

2k 0

↓ + −k − 0 + ↓

(11.25)

where 1ˆ is the identity matrix, and the Green function matrix has the form ⎛

+ ck↑ ck↑

⎜ ⎜ + ⎜ ck↓ ck↑ ⎜ ˆ Gk = ⎜ ⎜ c+ c+ ⎜ −k↑ k↑ ⎝ + + c−k↓ ck↑

⎞ + ck↑ ck↓ ck↑ c−k↑ ck↑ c−k↓ ⎟ ⎟ + ck↓ ck↓ ck↓ c−k↑ ck↓ c−k↓ ⎟ ⎟ ⎟ + + + + c−k↑ ck↓ c−k↑ c−k↑ c−k↑ c−k↓ ⎟ ⎟ ⎠ + + + + c−k↓ c−k↓ c−k↓ ck↓ c−k↑ c−k↓

(11.26)

In further analysis based on these general equations, the coexistence of ferromagnetism with the singlet and two kinds of triplet SC (OSP and ESP) will be considered separately.

Coexistence: Magnetic Ordering and Itinerant Electron SC

11.2.2.2

Ferromagnetism coexisting with singlet superconductivity ↑↓

↑↓

301

↑↓

In this case, the energy gaps in (11.25) are k = 0, kT = 0 and kS = 0, where kS is given by (11.15). To calculate the inter-site singlet superconducting parameters deﬁned in (11.8), the Green functions method and the Zubarev relation [5.1, 6.1] will be used for the average of the operators product: BA = −

1 fIm A B

d

(11.27)

This relation when applied to the averages appearing in the parameters S and D [relations (11.17) and (11.19)] gives the following result: 1 1 ci+x− ci + ci+y− ci = c−k− ck

2 2N k k 1 1 fIm ck↑ c−k↓

− ck↓ c−k↑

d =− 2N k k

(11.28)

1 1 ci+x− ci − ci+y− ci = c−k− ck

2 2N k k 1 1 =− fIm ck↑ c−k↓

− ck↓ c−k↑

d 2N k k

(11.29)

2S =

2D =

The solution of these equations in the H-F approximation for the singlet SC coexisting with ferromagnetism is given in Appendix 11A. The superconducting state according to (11.15) is described by two parameters, for which the following equations (Appendix 11A) are valid in different concentration ranges: for the s-wave SC 1 = −VJ2

(11.30)

1 = −VL2

(11.31)

for the d-wave SC

The solutions of above equations depend on carrier concentration and temperature. The moments Jn and Ln appearing above are expressed by (11.A13) and (11.A14) and they depend on the self-energies 0 and 1 , which can be calculated

302

Models of Itinerant Ordering in Crystals

in any approximation. In the H–F approximation, the self-energies, according to (11.14) and (11.A6), are real and equal to F 0 = − + zV n (11.32) 2 F 1 = − m 2

(11.33)

The moments Jn and Ln can be calculated using these self-energies and the identity 1 1 =P − i − 1 (11.34) − 1 + i0+ − 1 where P is the principal value of the integral. As a result one obtains Jn =

1 kn tanh Ek + k1 + 1 /2 + tanh Ek − k1 − 1 /2 2N k 2Ek

(11.35)

Ln =

1 kn tanh Ek + k1 + 1 /2 + tanh Ek − k1 − 1 /2 2N k 2Ek

(11.36)

with

Ek = k0 =

2 ↑↓ k0 + 0 − 0 2 + kS

k + − k 2

and

k1 =

(11.37)

k − − k 2

(11.38)

where k0 (k1 ) is the paramagnetic (ferromagnetic) part of the dispersion relation k = k b , respectively. The above equations, together with the equations for the electron occupation (11.A17) and magnetization (11.A18) obtained in the H–F approximation ↑ 1 k − 0 + 0 + 1 n =1− tanh Ek + k1 + 1 /2 2N k Ek

+

m=

↓ k − 0 + 0 − 1

Ek

(11.39)

tanh Ek − k1 − 1 /2

1 tanh Ek + k1 + 1 /2 − tanh Ek − k1 − 1 /2 2N k

and the equation for Fock’s parameter

(11.40)

Coexistence: Magnetic Ordering and Itinerant Electron SC

I = −

1 k − + 0 + Ek tanh Ek + k1 + 1 /2 2N k 2Ek k0

303

(11.41)

+ k0 − + 0 − Ek tanh Ek − k1 − 1 /2 constitute a set of self-consistent equations for parameters of ferromagnetic and superconducting states. Using equations (11.30) and (11.31), moments deﬁnitions (11.35) and (11.36) and relations (11.39)–(11.41), the critical SC temperature and magnetic Curie temperature versus concentration for singlet superconductors were calculated. The maximum of ferromagnetic ordering takes place at the half-ﬁlled point. At this point there is also the maximum of the singlet SC of the s0 -wave and d-wave type; the s-extended SC sx2 +y2 has its maximum at the almost empty or almost full band. Therefore, the ferromagnetism will coexist with the SC of the isotropic s0 -wave type and anisotropic d-wave type. Dependence of the critical temperature for s0 -wave SC, TSC , and the Curie temperature, TC , on the carrier concentration is shown in Fig. 11.12. This expression was obtained from (11.30) after substituting the moment J2 by J0 . This substitution corresponds to changing the expression V k2 2 appearing in (11.30) by effective mean value Veff = V k = V . Curves of critical temperatures presented above show that even a very small ferromagnetic moment destroys the singlet SC, assuming that they take place within the same band. The coexistence can take place in the systems, where magnetism and SC are created by electrons belonging to different bands or atoms. Fulde and Ferrell [11.30] and Larkin and Ovchinnikov [11.31] have studied systems with magnetic ordering coming from localized impurities and the Cooper pairs formed by the itinerant electrons.

0.08

Tcr[D0]

0.03 0.02

+F

0.06

SC

0.01 0

0.04

0.85

F

SC

Tcr[D0]

0.04

0.86

0.87

0.88

0.89

n

0.02

0

F

SC

0

0.2

0.4

0.6

0.8

1

n

FIGURE 11.12 The dependence of the critical temperature on the carrier concentration n. Critical temperature TSC for the s0 -wave SC without ferromagnetism – dashed line; ferromagnetic Curie temperature without SC – solid line; critical temperature for the s0 -wave SC coexisting with ferromagnetism – dotted line. Interaction constants are V ≈ −013D0 and F = 102D0 . Inset: magnified region of coexistence between F and SC.

304

Models of Itinerant Ordering in Crystals

11.2.2.3

Ferromagnetism coexisting with triplet opposite spins pairing superconductivity ↑↓

In the case of triplet OSP SC, the following energy gaps: k = 0 kS = 0 and ↑↓ ↑↓ kT = 0 [where kT is expressed by (11.20)] were inserted into (11.25). Proceeding in a similar way to the case of singlet SC and using the H–F approximation, one obtains the following equations for the OSP SC: 2 xy 1 k 1=− V tanh Ek + k1 + 1 /2 + tanh Ek − k1 − 1 /2 4N Ek k (11.42) where now the dispersion relation in the superconducting state is given by 2 ↑↓ (11.43) Ek = k0 + 0 − 0 2 + kT The equation describing the triplet SC with opposite spins (11.42) has a similar structure to the equations for singlet SC (11.30) and (11.31). One can also compare the energy of the SC state for singlet SC (11.37) and that for OSP triplet SC (11.43). Overall it is observed that the inﬂuence of ferromagnetism in these two cases is qualitatively similar.

11.2.2.4

Ferromagnetism coexisting with triplet parallel (equal) spins pairing superconductivity ↑↓

↑↓

In this case, the opposite spins parameters in (11.25) are zero (kS = 0 and kT = 0, and there is only the parallel spins energy gap k = 0 expressed by (11.23). Using deﬁnitions (11.11), (11.24) and the Zubarev’s relation (11.27), one can write the expression for the triplet SC parameter in the x direction x : = ci+x ci =

1 x c c

N k k −k k

1 1 x fIm ck c−k

d =− N k k

(11.44)

A similar expression can be written for the y direction by replacing in (11.44) y the function kx by k [these functions are given by relation (11.21)]. Proceeding with (11.44) in a similar way to the singlet SC and using the H–F approximation, one obtains the SC equations in the following form: 1=−

2 1 kx V tanh Ek /2 for the x direction N E k k

(11.45)

1=−

y 2 1 k tanh Ek /2 for the y direction V Ek N k

(11.46)

305

Coexistence: Magnetic Ordering and Itinerant Electron SC

where

Ek =

k − 0 +

2

2 + 2k

(11.47)

Relations for carrier concentration n and for I have the form n =

1 1 k − + − tanh Ek /2 2 2N k Ek

(11.48)

I =

− + 1 k 1 − k tanh /2 E k 2N k Ek

(11.49)

Equations (11.45) and (11.46) together with relation for carrier concentration (11.48) and relation (11.49) are the set of self-consistent complex equations. From this set the critical temperature versus concentration can be found. The results are shown in Fig. 11.13 for the A1 phase with pairing parallel to magnetization, ↑↑, and for the A2 phase with pairing opposite to magnetization, ↓↓. These curves are shown in Figs 11.13 and 11.14 by dotted and dashed lines, respectively. One can see that the critical temperatures for the A1 phase (dotted line) and A2 phase (dashed line) are different at concentrations at which magnetization appears. To explain this effect, the simple Stoner model (see Section 7.3) represented by the Hamiltonian (11.1) is used, with the on-site exchange ﬁeld F generating ferromagnetism and addition of the negative inter-site charge–charge interaction V promoting SC. Within this model the ferromagnetism depends only on the mutual shift of spin sub-bands. As a result, for concentrations n < 1 at non-zero magnetization, the critical SC temperature in the A1 phase is higher than in the

Tsc[D0]

0.01

0.01

0.008 0.007 0.8

0.008 Tsc[D0]

0.009

0.9

1 n

1.1

1.2

0.006 0.004 0.002 0 0

0.25 0.5 0.75

1 n

1.25 1.5 1.75

2

FIGURE 11.13 The dependence of critical temperature for triplet parallel spins superconductivity TSC on carrier concentration n in the presence of ferromagnetism. The interactions are V = −024D0 and F = 0828D0 . Solid line – SC without F; dotted line – the A1 phase (only ↑↑ pairing); dashed line – the A2 phase (only ↓↓ pairing).

306

Models of Itinerant Ordering in Crystals

Tsc[D0]

0.0095

0.0085

0.0075 0.82

0.84

Fcr

F [D0]

0.86

FIGURE 11.14 The dependence of critical temperature for triplet parallel spins superconductivity TSC on the on-site Stoner field F. The parameters are V = −024D0 and n = 09. Dotted line – the A1 phase; dashed line – the A2 phase.

A2 phase. At half-ﬁlling, both these temperatures become equal, and at n > 1 the critical temperature for the A1 phase is lower than for the A2 phase. The size of this difference depends on the magnitude of the Stoner on-site exchange ﬁeld (see Fig. 11.14). For the exchange interaction smaller than the critical value, F < Fcr , there is no ferromagnetic order (at given occupation n) and as a result there is no temperature split between the A1 and A2 phases. Increase of F for F > Fcr causes an increase of the magnetic moment, which results in an increase of the temperature difference between the A1 and A2 phases. The simple explanation of the difference in critical temperature between the two phases is based on dependence of the critical temperature on the concentration in the paramagnetic state. For carrier concentrations less than half-ﬁlling, n < 1, one has TSC /n > 0. The A1 phase has larger electron concentration than the A2 phase; hence it has a larger critical temperature. For n > 1 one has TSC /n < 0; therefore the A1 phase has a lower critical temperature than the A2 phase. Modiﬁcation of the density of states by the inter-site interactions will increase this temperature difference even further.

11.2.3 Comparison with experimental results (for UGe2 ZrZn2 , URhGe) As mentioned earlier, in the ferromagnetic ZrZn2 compound, the ambient pressure strength affects the Curie temperature. This compound has the quasi-linear dependence of both magnetic moment and Curie temperature on pressure (see Figs 11.9 and 11.10). Additionally, at pc = 165 kbar there is a rapid drop of the magnetic moment, which seems to be the ﬁrst-order phase transition. This is caused by the coupling between long-range itinerant magnetization modes and weak particle-hole

Coexistence: Magnetic Ordering and Itinerant Electron SC

307

excitations. The coupling causes the appearance of the non-analytical terms in the free energy near the phase transition point [11.47]. For the ZrZn2 compound the superconducting phase coexists with ferromagnetism. The superconducting critical temperature is about 100 times smaller than the magnetic Curie temperature for small pressures. The experimental data are shown in Fig. 11.9 (after [11.42]). The constants t and V will be assumed to be dependant on the external pressure. The results of [11.48] give us the following dependence of effective mass on pressure: 1 m∗ 0

m∗ p = −0017 ± 0004 kbar−1 p

(11.50)

From this relation one obtains tp =

At A − Bp

(11.51)

where A ≈ 147776 and B ≈ 025122 kbar−1 . In the calculation, the lattice constant ´ [11.49] was used. for ZrZn2 : R0 = 7393 Å The pressure effect on V is assumed as in [10.52] and including the Thomas– Fermi screening correction [5.7, 11.50] V˜ R = VRe−R

(11.52)

where V˜ R is the effective screened interaction. The screening parameter is given by 2 = 4e2 F

(11.53)

where e is the electron charge and F is the density of states on the Fermi level. Knowing the dependence of inter-atomic distance on pressure R = Rp and the density of states on the Fermi level for different pressures [11.49], one can write the following expression for inter-atomic interaction V versus pressure including the Thomas–Fermi screening effect: V˜ p = V0 D exp C · pp − p1 p − p2

(11.54)

where D ≈ 0143721, C ≈ −305466 × 10−6 kbar−3 and p12 ≈ 365462 ± 325989 i kbar. This approximate expression gives us the results with error less than 0.1% for maximum applied pressure. The complex values of p12 are strange, but a close look at the expression (11.54) shows that the interaction V˜ p is always real. From now, the symbols t and V will mean the modiﬁed (by pressure) value of hopping integral and inter-atomic interaction deﬁned by (11.51) and (11.54). The papers [11.49, 11.51, 11.52] show that ZrZn2 has a triplet parallel spin SC. Therefore the calculations will be performed for the coexistence of F with triplet SC and

308

Models of Itinerant Ordering in Crystals

will be based on the equations for the triplet parallel spin SC, (11.45) and (11.46), and the equation for carrier concentration (11.48). At the critical temperature, the assumption of k = 0 will be made in these equations. Figure 11.15 shows the dependence of the Curie TC and superconducting critical TSC temperatures on ambient pressure, compared with experimental results obtained by [11.42]. The parameters used here are chosen to ﬁt to experimental data of the Curie temperature and magnetic moment at p = 0 (see Fig. 11.16). Solving self-consistently (11.45), (11.46), (11.48) and (11.49), one obtains the dependence of magnetization versus pressure. The results are shown in Fig. 11.16.

30 TC TCexp 10 ⋅TSC

20

exp

T (K)

10 ⋅TSC

10

0

4

0

8

12

16

20

p (kbar)

FIGURE 11.15 The Curie TC and superconducting TSC critical temperatures versus ˜ pressure including relation (11.63) for the inter-site charge–charge interaction V = Vp. Assumed constants are F ≈ 088D0 and V0 ≈ −0144D0 . The carrier concentration n = 1. Experimental results are also shown after [11.42].

m (μ B)

0.15

0.1

0.05

0

0

4

8

12

16

20

p (kbar)

FIGURE 11.16 Magnetic moment versus pressure; for constant V0 , Vp ≡ V0 = V0 – solid line; for V = Vp – dashed line. Circles are the experimental points from [11.43]. Results were obtained for T = 23 K, F = 088D0 , V0 = −0144D and n = 1.

Coexistence: Magnetic Ordering and Itinerant Electron SC

309

m (μ B)

0.15

0.1

0.05

0

0

5

10

15

20

25

30

T (K)

FIGURE 11.17 Magnetic moment versus temperature: F ≈ 088D0 , V0 ≈ −0144D0 , external pressure p = 0 kbar and n = 1. Both theoretical (solid line) and experimental (dashed line after [11.43]) curves overlap approximately.

The theoretical dashed curve shown in Fig. 11.16 does not closely match the experimental data, but the character of the curve is preserved. The differences are caused by the values of parameters used here, which were ﬁt to the magnetic moment and Curie temperature at zero pressure. From our theoretical equations the rapid drop in magnetization (and also in the Curie temperature) for the pressure value of p∼165 kbar cannot be obtained numerically. This is because the drop is caused by a change in the internal structure (free energy) of ZrZn2 at this pressure, which is not described by our Hamiltonian. The magnetization dependence (in Bohr magnetons) on temperature is shown in Fig. 11.17. Our theoretical data (solid line) coincides very closely with the experimental values (dashed line). As one can see, at zero pressure the magnetic moment falls to zero at T ≡ TC ≈ 285 K, which is in close agreement with the Curie temperature presented in Fig. 11.15 for p = 0 kbar.

11.3 COEXISTENCE OF ANTIFERROMAGNETISM AND HIGH-TEMPERATURE SUPERCONDUCTIVITY In the majority of cuprates, the superconducting state is the spin singlet state of anisotropic d-wave symmetry [11.53]. The other superconductors, which exhibit the AF phase, are of the spin singlet type. Therefore only the spin singlet superconducting phase will be analysed in this chapter. For simplicity, a model will be assumed, in which the SC phase coexists with the commensurate AF Q = ordering. A more detailed approach would require assuming incommensurate AF ordering Q = coexisting with the striped superconductor [11.9, 11.54–11.56]

310

Models of Itinerant Ordering in Crystals

Another important simpliﬁcation of the model is assuming the mean-ﬁeld approximation in the model Hamiltonian describing the interplay between AF and SC. The analysis will be performed using the Green function formalism.

11.3.1 Model Hamiltonian Experimental results described in the introduction point to the coexistence of high-temperature singlet SC with AF. To analyse this coexistence, the extended Hubbard Hamiltonian (11.1) will be employed, and additionally two interpenetrating sub-lattices and will be included (as in Chapter 8). In the case of a simple commensurate AF, the average number of electrons on these sub-lattices is equal to n± = n ± m/2

n± = n ∓ m/2

(11.55)

where the AF moment per atom in Bohr’s magnetons is − m = n − n− = n − n

Using in the Hamiltonian (11.1) the H–F approximation and the electron occupations from (11.55), one obtains a Hamiltonian, which is the sum of the unperturbed, H0 , and superconducting, HSC , part: H = H0 + HSC

(11.56)

The unperturbed part is given by H0 = −

teff + nˆ i + nˆ i + nˆ i i j + hc − 0

i

i

(11.57)

i

+ where + i i and i i are the creation (annihilation) operators for an electron + of spin on the sub-lattice and , respectively, nˆ i = i i is the electron number operator for electrons with spin on the sub-lattice = 0 is the chemical potential and teff is the spin-independent effective hopping integral (see Chapter 8) given by teff = tbAF , where

bAF = 1 +

V I t AF

(11.58)

The self-energy, , in the H–F approximation can be written as ± = 0 ± 1

and

± = 0 ∓ 1

(11.59)

where the paramagnetic part, 0 , is given by 0 = F

n + zVn 2

(11.60)

Coexistence: Magnetic Ordering and Itinerant Electron SC

311

and the magnetic part, 1 , has the form 1 = −F

m 2

(11.61)

The superconducting part of the Hamiltonian (11.56) is equal to HSC =

V + + + hc i j− 2 ij

(11.62)

where the average is given by =

1 j− i 2

(11.63)

After transforming Hamiltonian (11.56) into momentum space, the spin-density wave (SDW) model developed in Section 8.8 will be employed. According to this model, the electron occupation on the sub-lattices and can be expressed by the following expression: ni = n + me−iQ·ri /2

(11.64)

where for sub-lattice one has ri = 2la and for sub-lattice one has ri = 2l + 1a (a is the lattice constant, which will be assumed equal to one: a = 1; l is an integer number). The vector Q is the reciprocal lattice vector in the presence of the AF ground state. For the pure (commensurate) AF this vector is Q = . Thus exp −iQ · ri = 1 for ri on sub-lattice and exp −iQ · ri = −1 for ri on sub-lattice and (11.64) reduces to (11.55). Inserting (11.64) into the model Hamiltonian (11.56) and transforming it into the momentum space, one has the following expression: + + + HMF = k − eff nˆ k + 1 ck ck+Q − k ck↑ c−k↓ + hc (11.65) k

k

k

with the effective dispersion relation for the simple cubic 2D lattice in the form: k = −2t k bAF . Quantity eff appearing in (11.65) is the effective chemical potential deﬁned as eff = 0 − 0

(11.66)

The energy gap, k , in the superconducting state is expressed as k = ds k + dd k

(11.67)

where ds is the s symmetry parameter and dd is the d symmetry parameter. The parameters ds and dd are given by (11.16) and (11.18), where the following notation is used S =

+x + +y 2

D =

+x − +y 2

(11.68)

312

Models of Itinerant Ordering in Crystals

with + xy being the nearest neighbour of atom on sub-lattice , in the direction xy, located on sub-lattice . With the help of (11.63) and expression (11.68), one can calculate the averages S and D as 2S =

1 1 i+x i + i+y− i = k c−k− ck 2 2N k

(11.69)

2D =

1 1 i+x− i − i+y− i = k c−k− ck 2 2N k

(11.70)

According to the SDW formalism introduced in Section 8.8, the magnetic moment m is calculated from the expression m=

1 + ck ck+Q N k

(11.71)

The electron concentration, n, and the Fock’s parameter, IAF , can be expressed as n=

IAF =

1 + + ck ck + ck+Q ck+Q 2N k

(11.72)

1 ik·h + + e ck ck − ck+Q ck+Q N kh

(11.73)

where h = ri − rj .

11.3.2 Formalism of the model Using the Green’s function method developed in Appendix 11B for the coexistence of AF and SC, one can obtain from relations (11.16), (11.18), (11.69) and (11.70) the following formula for the s-wave SC: V n tanhE˜ k1 /2 tanhE˜ k2 /2 1=− + 4N k k E˜ k1 E˜ k2

(11.74)

and the formula for the d-wave SC V n tanhE˜ k1 /2 tanhE˜ k2 /2 1=− + 4N k k E˜ k1 E˜ k2

(11.75)

Coexistence: Magnetic Ordering and Itinerant Electron SC

where E˜ k1 = E˜ k2 = EkAF =

1/2 2 −EkAF − eff + 2k

1/2 2 EkAF − eff + 2k

313

(11.76)

2k + 21

Equations (11.74) and (11.75) together with the equation for electron concentration and magnetization allow to calculate the energy gap parameters ds dd at a given temperature or the critical temperature at k → 0. Conditions for the electron concentration and magnetization are obtained in a similar way to equations for the energy gaps and are given by EkAF + eff 1 EkAF − eff ˜ ˜ n = 1− tanhEk2 /2 − tanhEk1 /2 (11.77) 2N k E˜ k2 E˜ k1 and m=−

1 1 + f Im ck ck+Q

d N k

=

1

4N

k

eff 1 + AF Ek

tanhE˜ k1 /2 eff tanhE˜ k2 /2 + 1 − AF Ek E˜ k1 E˜ k2

(11.78)

11.3.3 Numerical examples Within the framework of the mean-ﬁeld approximation, the coexistence of SC and AF is possible only for the SC of the s0 -wave and d-wave type because at the weak Coulomb correlation the critical temperature of the d-wave and s0 -wave superconductivity has its maximum at the half-ﬁlled band, where AF exists. The critical temperature for the s-extended SC sx2 +y2 exists in a different carrier concentration range than the Néel temperature [8.13]. The numerical analysis presented below for zero and non-zero temperatures shows that the AF state will suppress the SC state, causing the critical temperature for SC to drop to zero at the half-ﬁlled band, in agreement with the experimental data. From (11.61), (11.75), (11.77) and (11.78) we can calculate the dependence of d-wave SC ordering parameter D and the magnetic part of the self-energy 1 on carrier concentration at a given temperature and also the dependence of the critical temperature for SC ordering, TSC , and the Néel temperature, TN , on the carrier concentration n. Figure 11.18 shows the dependence of D and 1 on n at T = 0 K. This dependence was calculated for the negative inter-site density–density interaction V which created the d-wave type of SC and for the positive on-site exchange interaction F which created the AF state at n>0917. For the half-ﬁlled band n = 1 the superconducting

314

Models of Itinerant Ordering in Crystals

0.08

ΔD(Σ1)[2D ]

0.06 AF 0.04

0.02 SC 0

0.6

0.7

0.8 n

0.9

1

FIGURE 11.18 The dependence of d-wave SC ordering parameter D and the magnetic part of the self-energy 1 on the carrier concentration n for interaction constants: V = −0188D and F = 064D. The solid line is for AF ordering in the presence of SC and the dashed line is for d-wave superconducting ordering in the presence of AF. In addition, the dotted line shows pure AF ordering (without SC) and the dotted-dashed line shows pure SC ordering (without AF).

ordering parameter is equal to zero, but the AF ordering parameter has its maximum value. To illustrate better the mutual competition between AF and SC, the ordering parameter for the SC state without AF at zero temperature (dotted-dashed line) and the ordering parameter for AF without SC (dotted line) are also shown. Qualitatively similar behaviour is obtained for the dependence of the critical temperature of the d-wave superconductivity, TSC , and the Néel temperature, TN , on carrier concentration n, which is shown in Fig. 11.19. Analysing this ﬁgure one can see, that from the moment of creating the AF state, the critical temperature for

0.04

T [2D ]

0.03 AF 0.02 0.01 0

SC 0.6

0.7

0.8 n

AF + SC 0.9

1

FIGURE 11.19 The dependence of the critical temperature for d-wave SC TSC and the Néel temperature TN on the carrier concentration for V = −0188D and F = 064D. The solid line is for the Néel temperature in the presence of SC; dashed line is for the critical d-wave SC temperature in the presence of AF. Additional notation is the same as in Fig. 11.18.

Coexistence: Magnetic Ordering and Itinerant Electron SC

315

V/D 1 CDW

CDW

SDW –2

1

–1 SS (on-site)

d-wave

U/D

2

SDW

–1

–2

FIGURE 11.20 Schematic phase diagram of the Hubbard model in two dimensions on a square lattice for the half-filled band: CDW, charge density wave; SDW, spin-density wave; SS, on-site SC singlet pairing; d-wave, d-wave SC pairing (after [8.13]). Reprinted with permission from R. Micnas, J. Ranninger and S. Robaszkiewicz, Rev. Mod. Phys. 62, 113 (1990). Copyright 2007 by the American Physical Society.

SC rapidly decreases to zero. This is the result of destroying SC by the AF correlations. For some carrier concentrations, one has two values of the Néel temperature, which are the result of the model with commensurate AF ordering, Q = . At these concentrations, pure AF ordering (commensurate AF) does not yield the absolute minimum of the total magnetic energy. Taking into consideration the stripe model, in which Q = , removes this ambiguity (see Section 8.8). In Figs 11.18 and 11.19, the AF destroys SC only at n = 1, but in some ranges of concentration both these orderings exist together. This is contrary to the experimental results (see Fig. 11.1). A better ﬁt to the experimental results can be achieved by including the Coulomb correlation U in the CPA, since the strong Coulomb correlation will itself destroy SC at the half-ﬁlling point [10.53]. In addition, the ground-state phase diagram for the half-ﬁlled band n = 1 is shown here (see Fig. 11.20) after Micnas and co-workers [8.13], based on a comparison of critical temperatures for pure SC, SDW(AF) and charge density wave (CDW) phases. For U > 0 and V > 0 there are only the CDW and the SDW phases. For U > 0 and V < 0 the SDW and d-wave SC phases are possible, while for U < 0 and V > 0 one has the single CDW phase. For U < 0 and V < 0 one has the on-site singlet SC or d-wave pairing.

11.4 SUPERCONDUCTING GAP IN STRIPE STATES As mentioned in the introduction, between the insulating AF phase and the metallic SC phase, there is the diagonal stripe phase on the phase diagram, in which

316

Models of Itinerant Ordering in Crystals

one observes correlations of SC and AF. The experimental results show that the superconducting compounds La2−x Srx CuO4 have the vertical stripe state with ordering vectors Q = 21/2 ± 1/2 or Q = 21/21/2 ± in the superconducting phase x > 005, which is changing to the diagonal stripe state with ordering vectors Q = 21/2 ± 1/2 ± or Q = 21/2 ∓ 1/2 ± in the insulating phase x < 005. Parameter depends strongly on the doping x. The ratio /x changes at the superconductor–insulator transition. In the superconductor phase ∼ x, which means that the hole density is 0.5 per stripe, suggesting metallic behaviour. In the insulator phase, ∼ x/2, which means that the hole density is 1 per stripe, suggesting insulating behaviour with the fully ﬁlled hole band. The stripe instability for doped antiferromagnets was predicted theoretically by Zaanen and Gunnarsson [11.13] within the H–F approximation applied to the extended Hubbard model and conﬁrmed by a number of subsequent investigations [8.14, 11.1, 11.14, 11.54, 11.56–11.59]. To describe the stripe phase, the conventional self-consistent mean-ﬁeld approximation of the Hubbard model will be used. The 2D square lattice will be assumed. The Hamiltonian of this model is expressed by the simpliﬁed version of (11.1), in which the dependence of hopping integrals on the site occupation is ignored (t = t1 = t2 , hence t = tex = 0, and the only non-zero interactions are the Coulomb repulsion U and the negative inter-site density–density interaction V . As was shown in Chapter 10, such interaction V creates the d-wave SC with the pairing function of the symmetry: cos kx − cos ky . To obtain superconducting state, the Bogoliubov-de Gennes method will be used. In the H–F approximation, this simpliﬁed Hamiltonian is given by + ∗ + + + H = −t ci cj + U ni− ci ji cj↓ ci↑ + ji ci↑ (11.79) ci + V cj↓

i

+ where ni = ci ci is the average occupation of electrons on the i site, which can vary from site to site, e.g. ni↑ = ni↓ and ni = ni+ . The inter-site singlet superconducting parameter, ji , is deﬁned as

ji = cj↓ ci↑

(11.80)

In this model, one can write the following expressions for ni and ji : ilQ·r ni = e i nlQ (11.81) 0≤l≤NS

xj =

eilQ·rj xlQ

yj =

0≤l≤NS

eilQ·rj ylQ

(11.82)

0≤l≤NS

To have the NS -site periodicity in the y direction, the nesting vector is assumed to be Q = 21/2 1/2 − , = 1/NS (NS – the distance between stripes in units of the lattice constant). After the Bogoliubov transformations ∗ +

∗ + cj↓ = vj + u j + and cj↑ = uj + vj (11.83)

Coexistence: Magnetic Ordering and Itinerant Electron SC

317

where is the label of the eigenstates, one obtains the energy of eigen state, E , and the wave functions ui and vi at i site given by the Bogoliubov-de Gennes equation in the form Kij↑ Dij uj ui = E (11.84) ∗ ∗ D −K v vi j ij ij↓ j where the kinetic part is Kij = −tij + ij Uni− −

(11.85)

and the potential part driving the SC is given by Dij = Vij ji+x + ij ji+y

(11.86)

From (11.83), (11.81) and (11.82) one obtains the following self-consistent conditions for the pairing potential and the carrier density: ∗ fE (11.87) ij = cj↓ ci↑ = uai vj

+ ci↑ = ni↑ = ci↑

uai 2 fE

(11.88)

+ ci↓ = vai 2 1 − fE ni↓ = ci↓

(11.89)

The charge density is ni = ni↑ + ni↓ , and the spin density has the form Szi = 1/2ni↑ − ni↓ . Schematic distribution of charge density (circles) and spin density (arrows) for the vertical stripe phases characterized by = 1/12 is shown in Fig. 11.21. The density of holes is proportional to the diameter of the circles, and similarly the length of the arrows gives the expectation value of magnetic moment Sz . The ﬁgure shows that the holes form 1D stripes, with magnetic domains between these stripes. Pairs of holes from the stripes separated by the magnetic domain can obtain pairing correlations by virtually hopping into the magnetic domains. Below some temperatures, the stripes phases are coherently locked via the Josephson coupling, leading to a long-range (3D) superconducting order. Figure 11.22 shows the schematic phase diagram of the transition temperatures versus doping (after [11.56]). A pure AF phase can be considered as the stripe phase with an inﬁnite stripe period modulation. At increasing temperatures the doping range of existing commensurate AF increases, thus incommensuration is a decreasing function of temperature. Martin and co-workers [11.56] have found that the SC does not disappear in the region of the AF stripes, but rather becomes striped. Due to meandering of the stripes and their break-up into ﬁnite segments [11.15] the state is likely to be highly inhomogeneous and neither an insulator nor a superconductor, but also not a simple metal. In agreement with the experimental attribution, the authors refer to this region as a strange metal (SM).

318

Models of Itinerant Ordering in Crystals

T

FIGURE 11.21 Schematic vertical stripe phases for = 1/12 ( = 1/NS is the incommensurate parameter, here NS = 12) showing charge (circles) and magnetization (arrows) density (after [11.57]). Reprinted with permission from M. Fleck, A.I. Lichtenstein, E. Pavarini and A.M. Ole´s, Phys. Rev. Lett. 84, 4962 (2000). Copyright 2007 by the American Physical Society.

AF

ICAF

SM

SC

SSC x

FIGURE 11.22 Schematic phase diagram in the temperature versus doping plane showing commensurate (AF) and striped (incommensurate) (ICAF) antiferromagnetism, d-wave superconducting (SC), stripe superconducting (SSC) and non-superconducting strange metal (SM) phases (after [11.56]).

APPENDIX 11A: COEXISTENCE OF FERROMAGNETISM AND SINGLET SUPERCONDUCTIVITY The formalism describing coexistence of the singlet SC and ferromagnetism will be developed here. For the singlet SC the energy gaps in (11.25) are k = 0 and ↑↓ ↑↓ k ≡ kS , which gives the following relation: ⎛ ⎜ ⎜ ⎜ ⎜ ⎝

↑

↑↓

− k + 0 − ↑

0

0

0

− k + 0 − ↓

−kS

0

−kS

+ −k − 0 + ↑

0

0

0

+ −k − 0 + ↓

↑↓ ∗

kS

↓

↑↓ ∗

kS ↑↓

0

↑

⎞ ⎟ ⎟ ⎟ Gk ˆ ˆ = 1 ⎟ ⎠

↓

(11.A1)

Coexistence: Magnetic Ordering and Itinerant Electron SC

319

↑↓

with kS given by (11.15). Equation (11.A1) with Green functions deﬁned by (11.26) can be split into two equations: ↑ ↑↓ + ck↑ ck↑

ck↑ c−k↓

kS − k + 0 − ↑ ˆ (11.A2) = 1 + + + ↑↓∗ ↓ c c kS + −k − 0 + ↓ −k↓ k↑

c−k↓ c−k↓

and ↓ − k + 0 − ↓ ↑↓∗ −kS

↑↓

−kS

↑ + −k − 0 + ↑

+ ck↓ ck↓

ck↓ c−k↑

+ + c−k↑ ck↓

+ c−k↑ c−k↑

ˆ (11.A3) = 1

which will be used to ﬁnd functions ck↑ c−k↓

and ck↓ c−k↑

. From the set (11.A2) one has ck↑ c−k↓

= −

↑↓ kS ↑ ↓ ↑↓ − k + 0 − ↑ + −k − 0 + ↓ − kS 2

(11.A4)

and from (11.A3) ↑↓

ck↓ c−k↑

=

kS

2 ↓ ↑ ↑↓ − k + 0 − ↑ + −k − 0 + ↓ − kS

(11.A5)

Using the relation −k = k and introducing the notation for the paramagnetic and ferromagnetic self-energies and dispersion relations 0 =

↑ + ↓ 2

k0 =

k + k 2

↑

↑ − ↓ 2

1 =

↓

↑

and

k1 =

(11.A6) ↓

k − k 2

(11.A7)

one obtains from (11.A4) and (11.A5) ↑↓

ck↑ c−k↓

=

kS − 1 − k1 − Ek − − k1 − 1 − Ek

(11.A8)

↑↓

ck↓ c−k↑

= −

kS + 1 + k1 − Ek − + 1 + k1 − Ek

where

Ek =

2 ↑↓ k0 + 0 − 0 2 + kS

(11.A9)

(11.A10)

Using the moments method [8.9, 11.60], together with the deﬁnition of the singlet superconducting parameter (11.15) and Green functions (11.A8) and (11.A9),

320

Models of Itinerant Ordering in Crystals

one can write relations (11.28) and (11.29) for parameters S and D in the following form: 2S = ds J2

(11.A11)

2D = dd L2

(11.A12)

with the moments Jn and Ln deﬁned by the expressions 1 1 n fIm G1 k − k1 − 1 G1 k − + k1 + 1 2N k k + G1 k + k1 + 1 G1 k − − k1 − 1 d

(11.A13)

1 1 n fIm G1 k − k1 − 1 G1 k − + k1 + 1 2N k k + G1 k + k1 + 1 G1 k − − k1 − 1 d

(11.A14)

Jn = −

Ln = −

where G1 k = 1/ − Ek . Using (11.A13) and (11.A14) in (11.16) and (11.18) one obtains ds = −Vds J2

(11.A15)

dd = −Vdd L2

(11.A16)

The above equations, together with equations for electron concentration and magnetization n =1+

1 1 + + fIm ck↑ ck↑

+ ck↓ ck↓

2N k

(11.A17)

+ + − c−k↑ c−k↑

− c−k↓ c−k↓

d

m=−

1 1 + + fIm c−k↑ c−k↑

− c−k↓ c−k↓

d N k

(11.A18)

and with the equation for Fock’s parameter I = −

1 1 + + k fIm ck↑ ck↑

− c−k↑ c−k↑

d 2N k

(11.A19)

constitute a set of self-consistent equations for parameters of ferromagnetic and superconducting states.

Coexistence: Magnetic Ordering and Itinerant Electron SC

321

APPENDIX 11B: COEXISTENCE OF ANTIFERROMAGNETISM AND SINGLET SUPERCONDUCTIVITY The superconducting parameters S and D deﬁned by (11.69) and (11.70) can be calculated by the Green function method using the equations of motion for the Green functions (6.7) with the Hamiltonian (11.65). As a result one obtains ⎡ − k + eff k 1 ∗ ⎢ + − 0 k eff ⎢ k ⎢ ⎣ 1 0 − k+Q + eff 0

∗k+Q

1

0 1 k+Q

⎤ ⎥ ⎥ ˆ ˆ ⎥ · Gk = 1 ⎦

(11.B1)

+ k+Q − eff

where 1ˆ is the identity matrix, and the Green functions matrix in the momentum representation has the form ⎛

+

ck↑ ck↑

ck↑ c−k↓

+ ck↑ ck+Q↑

ck↑ c−k−Q↓

⎞

⎜ ⎟ + + + + + + ⎜ c−k↓ ck↑

c−k↓ c−k↓

c−k↓ ck+Q↑

c−k↓ c−k−Q↓

⎟ ⎟ ˆ Gk = ⎜ + + ⎜ c ck+Q↑ c−k↓

ck+Q↑ ck+Q↑

ck+Q↑ c−k−Q↓

⎟ k+Q↑ ck↑

⎝ ⎠ + + + + + + c−k−Q↓ ck↑

c−k−Q↓ c−k↓

c−k−Q↓ ck+Q↑

c−k−Q↓ c−k−Q↓

(11.B2) It is worth recalling here that to obtain the set of equations (11.B1) one has to break the chain of equations for the Green functions by projecting higher order Green functions into lower ones as follows: + + + + c−k ↓ ck ↑ c−k↓ ck↑

≈ c−k ↓ ck ↑ · c−k↓ ck↑

(11.B3)

The Green functions calculated from (11.B1) will allow us to ﬁnd the averages (11.69) and (11.70) with the aid of relation (11.27). The average c−k↓ ck↑ used in expressions (11.69) and (11.70) for SC will have the following form: c−k↓ ck↑ = −

1 f Im ck↑ c−k↓

d

(11.B4)

The function ck↑ c−k↓

is found from (11.B1) as ' ( )* k 2 − k + eff 2 + 2k + 21 ck↑ c−k↓

= − 2 2 2 − E˜ k1 2 − E˜ k2

(11.B5)

322

Models of Itinerant Ordering in Crystals

with E˜ k1 = E˜ k2 = EkAF =

1/2 2 −EkAF − eff + 2k

1/2 2 EkAF − eff + 2k

(11.B6)

2k + 21

Using the Green function given by (11.B5) one can write the average c−k↓ ck↑

in the following form: 1 c−k↓ ck↑ = − f Im ck↑ c−k↓

d = k 4

tanhE˜ k1 /2 tanhE˜ k2 /2 + E˜ k1 E˜ k2 (11.B7)

With the aid of this equation and the deﬁnitions (11.67) one can write (11.69) and (11.70) as 1 2 tanhE˜ k1 /2 tanhE˜ k2 /2 2S = ds + 4N k k E˜ k1 E˜ k2

(11.B8)

1 2 tanhE˜ k1 /2 tanhE˜ k2 /2 + 2D = dd 4N k k E˜ k1 E˜ k2

(11.B9)

After inserting (11.B8) and (11.B9) into (11.16) and (11.18) one obtains the set of two self-consistent equations, which together with the equation for electron concentration and magnetization, will allow us to calculate the energy gap parameters ds dd at a given temperature or the critical temperature at k → 0. Conditions for the electron concentration and magnetization are obtained in a similar way to the equations for the energy gaps: EkAF + eff 1 EkAF − eff ˜ ˜ n = 1− tanhEk2 /2 − tanhEk1 /2 2N k E˜ k2 E˜ k1

(11.B10)

and m=−

1 1 + f Im ck ck+Q

d N k

=

1

4N

k

eff 1 + AF Ek

tanhE˜ k1 /2 eff tanhE˜ k2 /2 + 1 − AF Ek E˜ k1 E˜ k2

(11.B11)

Coexistence: Magnetic Ordering and Itinerant Electron SC

323

Relations for Fock’s parameter can be expressed as

IAF

1 = 2N k k k

eff 1 + AF Ek

tanhE˜ k1 /2 eff tanhE˜ k2 /2 (11.B12) + 1 − AF Ek E˜ k1 E˜ k2

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SUBJECT INDEX

3d metals, 119 3d orbitals, 260 3d row, 116 phase, 204 A1 phase, 305 A2 phase, 305 Alkali metals, 229 Alloys, binary, 204, 206, 223 Bragg–Williams approximation, 205–7 disorder, 203, 208, 214 ordered, 207 substitutional, 205 transition metal, 213–19 Angle-resolved photoemission spectroscopy (ARPES), 233 Anticommutation rules, 53 Antiferromagnetism, commensurate, 194–8 with correlation effects, 187–91 incommensurate, 194, 198 diagonal, 197, 198 vertical, 197, 198 magnetic susceptibility see Magnetic susceptibility phenomenological, 167–8 spin conﬁguration, 168 Atomic, distance, 38, 42, 43 orbital, 36 overlap, 37 ordering, 206, 209–12 critical value, 210 phase, 204, 213 ’ phase, 204

Band model, single band, 55–6, 243–59 three-band, 259–66 Band split, 69 Bandwidth, 38–40 Bandwidth modiﬁcation factor, 93, 110 antiferromagnetism, 112, 175, 188 ferromagnetism, 112, 139 superconductivity, 187 Bardeen, J., 237 BCS theory see Superconductivity, BCS model Bi2 Sr2 Ca2 Cu3 O10 , 228, 233 Bloch function, 43–4 Bloch theorem, 43–4 Bogoliubov-de Gennes equation, 317 Bogoliubov diagonalization, 194, 255, 280–2, 316 Bohr magneton, 20 Boltzmann, constant, 14 statistics, 12 Born–von Karman boundary condition, 29 Borocarbide, 229 Bose–Einstein (B–E) statistics, 11–13, 16 Bosons, 11, 16 Bragg law, 5, 7, 8 Bragg, W.L., 5 Bragg–Williams approximation, 205–7, 219–24 Bragg’s reﬂection, 28 Brillouin function, 21–4 Brillouin zone, 8–10 boundary, 10 325

326

Subject Index

Chain equation, 60–3 Charge–charge (density–density) interaction, 56 Charge transfer, 213 Chemical potential, 12 Chromium (Cr), 167, 193 Classical distribution see Boltzmann, constant, statistics Cobalt (Co), 121–3, 143, 154 Coexistence of, antiferromagnetism and superconductivity, 288, 289, 309–18, 321 ferromagnetism and superconductivity, 297–309 Coherence length, 229, 230 Coherent potential, 72, 79 Coherent potential approximation (CPA), 72–81 alloys, 214, 215 antiferromagnetism, 177, 179, 199, 200 extended Hubbard model, 94 ferromagnetism, 144, 145, 149 Cohesion energy, 40–3 Cooper, L.N., 237 Cooper pair, 232, 237 Copper (Cu), 119 Correlation factor, AF, 174, 181 inter-site ferromagnetism, 146–8 on-site ferromagnetism, 145–8 Coulomb integral, 51 Coulomb repulsion, 52 Creation and annihilation operators, electrons, 51, 52 holes, 241 Critical exchange ﬁeld, ferromagnetism, 133 on-site, 142, 150 total, 139, 140, 148 antiferromagnetism, 175, 182, 187

Critical point exponents, 130, 154–7 Critical temperature, superconductivity, 228–36, 248–57 Crystal structures, 4 Cux Au1−x alloy, 203, 204 Cubic structure body-centred cubic (bcc), 4, 6 face-centred cubic (fcc), 4, 6 simple cubic (sc), 4, 6 CuO chain, 231–2 CuO2 plane, 230–7, 242–5, 249, 252, 259 Curie, constant, 20, 25, 129, 130 law, 20, 25, 129, 130 point, 24 Curie temperature, Stoner model, 127–31, 140, 142–4 Stoner model with correlations, 140–4 Curie–Weiss law, 25, 130, 154 Cu–Zn alloy, 204, 205 de Broglie relation, 28 de Broglie wavelength, 13 Debye, energy, 237 temperature, 230 Defects, 230, 235, 252 Degenerate band, 54–6 Density of states, for antiferromagnetism, 171 Gaussian, 89 general result, 31, 32, 61 nearly free electrons, 33, 34 parabolic, 135–8 rectangular, 41, 92, 134–6 semi-elliptic, 68, 69, 78, 79, 141–6, 149 Diagonalization, 263 Diffraction, 5 Dispersion relation, 2D sc lattice, 39 d-dimensional sc lattice, 89 nearly free electrons, 28–33

Subject Index

327

spin wave, 158 tight binding, 38–40 Double hopping, 83 Doublons, 65 Dynamical mean-ﬁeld theory (DMFT), 88–90 Dynamical transverse spin susceptibility, 161–3 Dyson equation, 73, 74, 90 Dyson’s time-ordering operator, 95

Exchange, inter-site interaction, 56 ﬁeld, 125, 126, 128, 131, 135–8, 143, 144, 153 hopping interaction, 56 on-site interaction, 56, 126, 128, 131, 138 Excitons, 240 Extended Hubbard Hamiltonian see Hubbard model, simple External ﬁeld, 131, 132, 139, 142, 147, 174, 180, 183–6

Effective, potential, 239, 254 interaction, 243, 252 Einstein identity, 302 Elastic module, 253 Electron-deformation coupling constant, 253, 255 Electron–electron, correlations, 237 coupling, 230 Electron moment, orbital, 126 spin, 126 total, 126 Electron–phonon, coupling constant, 230 interaction, 230, 237, 240, 242 mechanism, 230 Energy barrier, 126 Energy gap, alloy, 209 antiferromagnets, 172, 178, 180, 183, 196, 201 superconductors, 232–4, 239 d-wave symmetry, 232–4, 245, 247, 255, 257–9, 266, 268 s-wave symmetry, 245, 247, 255, 257, 258, 266–8 s + d-wave symmetry, 255 Energy shift, 129, 132, 135 Entropy, 14 binary alloys, 207 ferromagnets, 223 superconductors, 239 Equation of motion, 60–3, 65, 70, 95–7, 101

Fe–Co alloy, 218 Fermi energy, 12, 33 Fermi liquid, 118 Fermi surface, 122 Fermi temperature, 35 Fermi–Dirac (F–D) statistics, 11, 12, 15 Fermions, 11 Ferromagnetism, itinerant model, 125–57 strong, 118–22 weak, 118, 122, 124 Feynman–Dyson perturbation series, 73 Field, molecular see Molecular ﬁeld total, antiferromagnetism, 175, 177, 181, 182 ferromagnetism, 139, 142, 148 Fock’s parameter (inter-site average), 92, 320 antiferromagnetism, 175, 181, 182, 309 ferromagnetism, 141, 144, 303 superconductivity, 248 Fourier series, 45 Free energy, alloy, 207 antiferromagnets, 174, 175, 182, 198 ferromagnets, 131, 133, 134, 157 Landau expansion, 133 solid/liquid, 220 superconductors, 239 Fullerens, 229

328

Subject Index

phase, 204 -point, 38 Gas constant, 35 gJ factor, 20 Green function, advanced, 95 antiferromagnetism, 170, 179, 181 deﬁnitions, 60–76, 95, 97 retarded, 95, 96 unperturbed, 74, 76 Group velocity, 28, 32, 38 Gyromagnetic ratio, 20 Hall, constant, 231, 233 effect, 231 Hartree–Fock approximation, classic, 63–4 ferromagnetism, 138–44 modiﬁed, 90–4 Heat capacity, electron gas, 34 Heavy-fermion, 228, 243, 252, 290, 292, 294 Heisenberg representation, 95, 96 Heisenberg term, 188 Helical types of order, 40 Hexagonal close packed structure, 4 HgBa2 Ca2 Cu3 O8+x , 228, 252, 259 Hole operators, 241 Hole superconductivity, 232–7, 243–59 Hopping integral, 37 inter-band, 261 Hopping interaction, 56 Hubbard I approximation atomic limit, 64–6 ﬁnite bandwidth limit, 66–9 Hubbard III approximation, 69–72, 99–105 resonance broadening effect, 70, 100–4 scattering effect, 97–100 Hubbard model, simple, 51–4 extended by inter-site interactions, 54–7

Hubbard sub-band, lower, 68, 146, 177, 201, 202 upper, 68, 145, 177, 202 Hume-Rothery rules, 205 Hund’s interaction, 56, 125 Hybridization, 237, 243 Hydrogen molecule, 57 Impurity effective action, 90 Impurity problem, 89 Insulator, 231 Inter-band interactions, 261 Inter-site average see Fock’s parameter (inter-site average) Inter-site interactions (two-site interactions), 55–7 Intermetallic systems, 229 Internal ﬁeld, 125, 126 Iron (Fe), 122, 124, 143, 153 Ising model, 220, 221 Isotope effect, superconductors, 229, 230 coefﬁcient , 230 Itinerant electron model, 51–7 Itinerant moments, 25 Jackson, coefﬁcient, 221 model, 220 Kinetic energy, 51–7 Knight shift experiments, 233 Kramers–Kronig relations, 165 La2-x (Ba,Sr)x CuO4 (La124), 228, 230, 235, 236, 288–91 La5-x Bax Cu5 O53−y , 228 Landau free energy see Free energy, alloy Landé equation, 20 Langevin function, 24 Lattice vibrations, 254 Lattices, cubic see Cubic structure hexagonal see Hexagonal close packed structure

Subject Index

reciprocal, 4, 7, 8 axis vectors, 7 Layer model, 231 Lifetime, 69 Local deformation, 243, 252, 253 Local effective action, 89 Localized model, 132 Localized moments, 25, 26 Long range order parameter, 206–8, 223 Lorentzian function, 77 Magnetic dipole, 125 Magnetic excitation operators, 158 Magnetic moment, 19–25, 118–43, 152–6 Magnetic susceptibility, 174, 180 Magnetization, 19–25, 118–43, 152–6 Magnon, 15, 16 see also Spin-wave theory Majority spins, 118–24, 129 Manganese (Mn), 167, 192, 193 Mass, effective, 28, 29 dependence on pressure, 307 free electron, 29 Maxwell statistics, 12 Mean ﬁeld approximation see Hartree–Fock approximation, classic Metal organic chemical vapour deposition (MOCVD), 236 MgB2 , 229 Minority spins, 119–24, 129 Modiﬁed alloy analogy approximation (MAA), 86–8 self-energy, 87 Molecular ﬁeld, 93, 94, 111, 112, 176, 177 Moments method for superconductivity, 245–8, 255, 258 Momentum representation, 60, 63, 74, 105 Monte-Carlo simulations, 222

329

Mott–Hubbard bands, 235 Mott transition, 90 Multi-orbital single-band model, 56 Nd2−x Cex CuO4−y , 290 Nearly free electron model, 27–35 Néel temperature, 185, 191–3, 197, 237, 240 Negative centers U model, 242 Nesting vector, 316 Nickel (Ni), 120, 121, 143, 154 Ni–Co alloy, 218 Nuclear magnetic resonance (NMR), 233 Number of particles, 12 Operator, annihilation, 51, 52 creation, 51, 52 electron number, 52 Orbital moment see Electron moment, orbital Order parameter, 206–8, 212, 222, 223 Order–disorder transformation, 204, 205 Organic superconductors, 291 Orthogonality of functions, 267 Overdoping, 236, 237 Overlapping bands, 118 Pair hopping interaction, 56 Paramagnetic static susceptibility, 129 Paramagnetism, 19–25 Pauli exclusion principle, 11, 52, 116 Pauli susceptibility, 149 Periodic boundary conditions, 30 Periodic table of elements, 6, 115–17 Perturbation term, 54 Phonon, 15, 16 effect, 230 energy, 237 Planck distribution, 15, 16 Plane waves, 27, 29, 35, 44

330

Subject Index

Plasmon mechanism, 240 Positrons, 11 Potential energy, 51–3 Pressure external, 228, 252 internal, 252 Primitive axes, 3 basis vectors, 3, 4 cell, (elementary cell), 3, 4, 8 Wigner–Seitz, 4, 8 Projection method, 267, 268 Protons, 11 Pseudogap, 237 Quantum Monte-Carlo simulations, 90 Quantum number, magnetic, 116 orbital, 116 principal, 116 spin, 116 R2−x Cex CuO4 , 231, 233 R1 4 Ce0 6 RuSr2 Cu2 O10− , 296 Random phase approximation (RPA), 158, 165 Random potential, 54 Resonating valence bond (RVB) model, 240 RNi2 B2 C, 292, 296 Roth’s two-pole approximation, 82 RuSr2 GdCu2 O8 , 296 Saturation magnetization, 21, 127, 136 Schrieffer, J.R., 237 Schrödinger equation, 36, 45 Screening parameter, 307 Self-energy, 66, 68, 74–8, 81, 85, 87, 89 antiferromagnetism, 179, 180, 199 Single site approximation, 75 Single-site problem, 88, 90

Singlet superconductivity, 240–59, 274–6 Slater–Koster function, 62, 75–8, 147 antiferromagnetism, 170, 171, 200 modiﬁed CPA, 94 semi-elliptic band, 149 Slater–Pauling curve, 123, 213, 214, 218 Solid/liquid, model, 219–21 interfaces, 220 Solitons, 240 Speciﬁc heat, electronic, 35 Spectral density, 82, 85 Spectral density approach (SDA), 81–6 self-energy, 85 Spectral weights, 82, 84 Spin-density waves, 193–8 Spin exchange, 83 Spin glass phase, 290 Spin moment, 126 Spin-wave theory, 128, 157–64 see also Magnon Spin-wave excitations, 161 Standing waves, 29, 38, 47 velocity, 29 Stochastic potential, 72, 76, 81, 94 Stoner criterion, ferromagnetism, 133, 134 Stoner enhancement factor, 149 Stoner excitation, 160, 161 Stoner ﬁeld, 112, 135–42 Stoner gap, 119 Stoner model, classic, 125–38 modiﬁed, 138–44 Stripe states, 315–18 diagonal, 288–91, 316 vertical, 291, 316 Strong-correlation regime, 87 Strontium ruthenate (Sr2 RuO4 , 228, 243 Sub-band interaction, 55

Subject Index

Sub-lattices, antiferromagnetic, 168–83 interpenetrating, 205–8, 213 magnetization, 191 Superconductivity, BCS model, 229, 230, 237–40 high temperature, 228–69 historical background, 228–30 low temperature, 230, 237–40 phenomenological introduction, 228–30 under pressure, 252 Superconductor quantum interference device (SQUID), 233 Superparamagnetic particles, 24 Susceptibility, ferromagnetic, 129–31, 133 longitudinal and transversal, 182–6 static antiferromagnetic, 174, 175 Temperature fundamental, 14 Thermodynamics law, ﬁrst, 14 second, 14 Thomas–Fermi screening effect, 307 Tight binding approximation, 35–40 t−J model, 191, 242 Tl2 Ba2 Ca2 Cu3 O10 , 228, 252 Total ﬁeld see Field, molecular Triplet superconductivity, 228, 242–5, 250, 269, 273, 274 equal spins pairing (ESP), 298, 304–6

opposite spins pairing (OSP), 304 Two-plane waves model, 29, 44–7 Two-site interaction, 55 UGe2 , 228, 252, 293–5, 306 Underdoping, 231–7, 252 Unperturbed part, 54 Upper critical ﬁeld for SC, 231 URhGe, 228, 293–5 Van Hove singularity, 40 Virtual phonons, 237 Volume–charge coupling, 252, 254 Wannier functions, 51 Wave function, 29, 36, 43–5 Wave packet, 28 Wave vector, 28, 30, 44, 46 Weber’s mechanism, 242, 262 Weiss, ferromagnetism, 25, 26 ﬁeld, 25, 125, 129, 130 function, 90 model, 25, 26, 132 YBa2 Cu3 O7– (Y123), 228–36, 252, 254, 256, 259, 288 ZrZn2 , 228, 293, 295, 297, 306–9 Zubarev relation, 246, 247, 301

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