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


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


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.


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


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


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


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 > 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


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


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).


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


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


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


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


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.


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


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


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


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


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


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


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


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


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) k0

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


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


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


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


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


+ + 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


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 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


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


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


(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


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)


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


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)


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


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


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


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.


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


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


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


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−


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


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


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


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)


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


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.


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


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.


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


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)


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

+


lmn

−FH0 C−

l

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

(6.118)

84


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


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


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)


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


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)


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


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


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


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.


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


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)


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


(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


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


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)


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


where =

til Wlmi tmi

(6.B20)

lm SS = −C− FH0

l=im


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


∓ + ± 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


∓ ∓ + ± ±

− ± ± − ∓ 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


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


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

(6.B42)


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


ST = FH C− − T SS + SB

= FH C− − T +

−C− FH0

l=im



(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


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


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


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


+ + 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


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)


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


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


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


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


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.


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


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.


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


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


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.


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


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.


0.3 hole

125

2.5 holes

εF

3d

4s

3d

0.8 electron

4.7 electrons

2.5 electrons

(b)


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


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)


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


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.


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


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)


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


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)


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


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


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


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


√ 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


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)


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


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.


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


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


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


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


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


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 .


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


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


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


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).


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


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


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


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


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


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


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 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


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


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.


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


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


(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


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)


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.


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


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)


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


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


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


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


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


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.


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


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)


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


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)


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


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.


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


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)


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


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)


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


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.


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


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)


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

331


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