Quantum Theory of Solids

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Advances in Science and Technology in the USSR

Physics Series

Quantum Theory of Solids Edited by

I. M. Lifshits Mem. USSR Acad. Sc.

MIR Publishers Moscow

KBaHTOBaa Teopaa TBepfloro Teaa Ilod pedanifueu anad. H. M . Jlu0iauija Ha^aTejiLCTBO «Map» MocKBa

TO THE READER Mir Publishers would be grateful for your comments on the content, translation and de­ sign of this book. We would also be pleased to receive any other suggestions you may wish to make. Our address is: Mir Publishers, 2 Pervy Rizhsky Pereulok, 1-110, GSP, Moscow, 129820, USSR.

First published 1982 Ha amduucnoM &3biKe © HaftaTejibCTBO «Mnp», 1982 © English translation, Mir Publishers, 1982 Printed in the Union of Soviet Socialist Republics.

FOREWORD Academician ! •

M-• Lifshits

S. I. Vavilov Institute of Physical Problems, Academy of Sciences of the USSR

The basic problems of solid-state physics are connected with the quantum nature of solids. A number of new remarkable phenomena and properties of matter have been explained and predicted by developing the ideas and methods of the quantum theory of solid state. Although the scope of problems in this field is extremely broad, the inner logic of the development of the theory makes it possible to find unexpected links between what would seem to be rather distant phenomena and build on this basis a new approach to solving a wide spectrum of problems. This refers to most interesting problems con­ cerning new states of matter, to the fluctuation theory of phase transi­ tions, and to the problem of localization of quasiparticles in dis­ ordered systems. The present compilation includes four articles on various topics, each of which contains an exposition of the present state of art in the most interesting and timely problems of solid-state theory. The authors are physicists who have contributed significantly to this field of science. The paper by A. F. Andreev deals with a new state of matter, quantum crystals. In a certain sense this state stands between clas­ sical crystals and quantum liquids: the system possesses translation symmetry (an ordinary crystal) and has zero static shear moduli (a liquid). Mass transport in such systems is due to special quasipar­ ticles, delocalized imperfections (vacancies or impurities), and may be similar to a superfluid motion of an impuriton liquid. The peculiar properties of such systems (mainly solid 3He and solid 4He and their solutions) are at present intensely studied both theoretically and experimentally. These properties are the manifestations of quan­ tum laws of motion on the atomic-molecular level instead of the electronic level. The other three papers are devoted to the study of electronic prop­ erties of solids. The work of D. I. Khomskii deals with the connec­ tion between electronic phase transitions and the problem of mixed (nonintegral) valence in rare-earth compounds. Electronic phase transitions in such systems possess special properties, since the problem of mixed valence is related to other fundamental problems of the electron theory of solids, namely, the localization of electrons, the metal-insulator transition, or the magnetic-nonmagnetic tran­ sitions.

6

Foreword

The paper of B. L. Altshuler, A. G. Aronov, D. E. Khmelnitskii, and A. I. Larkin is devoted to coherent effects in disordered conductors. This problem is one of the most difficult and not yet completely solved problems in the theory of disordered systems, namely, the theory of localization of electrons and the behavior of transport characteristics of electrons near the mobility boundary (the boundary between localized and delocalized states). In a nar­ rower approach, coherent effects are quantum corrections due to the addition of amplitudes of multiply scattered particles in comparison to the classical theory of transport. These corrections grow with A/Z, where A is the electron wavelength, and I the mean free path. For example, in the electron-electron interaction the transition to the superconducting state constitutes a strongly coherent effect, while production of virtual Cooper pairs constitutes one of the mechanisms of coherent corrections in normal metals. The article analyzes from a unified position the role that coherent effects play in disordered metals. In addition, the same approach is used to examine the prob­ lem of electron localization in such systems, for one, localization in magnetic field and in a percolation structure. The last article, by P. B. Wiegmann, stands apart from the first three. It is devoted not to development of a new field but, rather, contains for the first time a full and exact solution to a problem that was formulated and studied in a great number of works but which did not as yet have any consistent solution. This is the solution of the so-called Kondo problem, which studies the effect of magnetic impurities on the transport and thermodynamic characteristics of nonmagnetic metals. The mathematical beauty of the method which enabled the author to solve a problem that did not yield to the efforts of many theoreticians, and the fact that the problem is inter­ esting in itself explains the reason for including this article. It goes without saying that the selection of material to illustrate the achievements in solid-state physics must inevitably be subjec­ tive. For one, the compilation does not hold works on superconductiv­ ity, a problem to which a separate book could be devoted. But we still believe that the articles should be of interest to readers who wish to know what is new in solid-state physics. Moscow February, 1982

Academician /. M. Lifshits

CONTENTS

Foreword 1 1.1 1.2 1.3 1.3.1 1.3.2 1.3.3 1.3.4 1.4. 1.4.1 1.4.2 1.4.3 1.5. 1.5.1 1.5.2 1.5.3 1.6 1.6.1 1.6.2 1.6.3 1.6.4 1.7

2

2.1 2.2 2.3 2.3.1

5

DEFECTS AND SURFACE PHENOMENA IN QUANTUM CRYS­ TALS BY A. F. ANDREEV (transl. by E . Yankovsky) 11 Introduction 11 Quantum Effects in Crystals 11 Impurity Quasiparticles: Impuritons 14 Diffusion in an Impuriton Gas 14 Diffusion of Strongly Interacting Impuritons 17 Phonon-Impuriton Interaction 20 One- and Two-Dimensional Impuritons 24 Vacancies 32 Vacancies in 4He Crystals 32 Zero-Point Vacancies 37 Vacancies in 23He Crystals 40 Surface Phenomena 42 The Equilibrium Shape of the Crystal-Liquid Interface 42 Crystallization and Melting 48 Crystallization Waves 51 Faceting Transitions in Crystals 55 The Role of Fluctuations 55 Thermodynamic Relations 58 A 6-fold Symmetry Axis 60 A 4-fold Symmetry Axis 61 Delocalization of Dislocations 64 References 66 ELECTRONIC PHASE TRANSITIONS AND THE PROBLEM OF MIXED VALENCE BY D. I. KIIOMSKII (transl. by E. Yan­ kovsky) 70 Introduction 70 Localization of Electrons and Insulator-Metal Transitions (MottHubbard Transitions) 72 Electronic Phase Transitions in Rare-Earth Compounds 75 The General Picture of Transitions 75

8

Contents

2.3.2

Theoretical Models for Describing Electronic Phase Transitions and MV States 77 Valence Transitions in the Falicov-Kimball Model 80 The Mean-Field Approximation 80 Beyond the Mean-Field Approximation. The Role of Local (Excitonic) Correlations 84 The Two-Level Model 87 The Periodic Model 89 The Electron-Lattice Interaction and Its Role in Transitions 91 Interaction with a Homogeneous Strain 91 The Change in the Width of the f Level and Its Role in Valence Transitions 97 Local (Polaron) Effects in the Electron-Lattice Interaction 99 Interaction via Short-Wavelength Phonons and Formation of Ordered Structures 102 Mixed-Valence States: The Basic Problems 105 Properties of Mixed-Valence States. The Experimental Situation and Statement of the Problem 106 The Anderson and Rondo Lattices 109 The Valence Transition and the Mott-Hubbard Transition 114 Excitonic Correlations in an MV Phase 117 Spatial Correlations in MV Systems 119 Systems with MV as a Model of Condensed Matter 121 Conclusion 123 References 126

2.4 2.4.1 2.4.2 2.4.3 2.4.4 2.5 2.5.1 2.5.2 2.5.3 2.5.4 2.6 2 . 6.1 2 . 6.2

2.6.3 2.6.4 2.6.5 2 . 6.6

2.7

3

3.1 3.2 3.2.1 3.2.2 3.2.3 3.2.4 3.3 3.3.1 3.3.2 3.4

COHERENT EFFECTS IN DISORDERED CONDUCTORS BY B. L. ALTSHULER, A. G. ARONOV, D. E. KHMELNITSKII AND A. I* LARKIN (transl. by E, Kaminskaya) 130 Introduction 130 Quantum Corrections to Conductivity of Noninteracting Electrons 131 Conductivity and Impurity Diagrammatic Technique 131 Gooperon and Quantum Correction to Conductivity 134 Effects of Spin Scattering 137 Properties of Samples of Finite Dimensions* Effective Space Dimensionality 142 Quantum Effects in High-Frequency Electromagnetic Field 144 Effects of High-Frequency Field on Quantum Corrections to Conductivity 144 Suppression of Coherent Effects by Electromagnetic Fluctuations 146 Electron-Electron Interaction in Disordered Metallic Systems 151

Contents 3.4.1 3.4.2 3.4.3 3.4.4 3.5 3.6 3.6.1 3.6.2 3.6.3 3.7 3.7.1 3.7.2 3.7.3 3.7.4 3.7.5 3.7.6 3.7.7 3.7.8 3.8 3.8.1 3.8.2 3.8.3 3.8.4 3.8.5 3.8.6 3.8.7 3.9 3.A 3.A.1 3.A.2 3.A.3 3.A.4

4 4.1 4.2 4.2.1

9

Electron-Electron Collision Time 151 Effects of Electron-Electron Correlations on the Density of OneParticle States 160 Conductivity of Interacting Electrons 170 Superconducting Fluctuations and Temperature Dependence of Conductivity 176 Temperature Dependence of Conductivity: Experiment 179 Hall Effect 187 Introduction 187 Hall Effect for Noninteracting Electrons 188 Hall Effect for Interacting Electrons 189 Anomalous Magnetoresistance 191 Magnetoresistance of Noninteracting Electrons 191 Magnetoresistance of Thin Films and Wires in Longitudinal Magnetic Field 194 Magnetoresistance and Scattering by Superconducting Fluctua tions 196 Magnetoresistance in Many-Valley Semiconductors 197 Effects of Spin-Orbit Scattering on Magnetoconductivity 199 Experiment 202 Aharonov-Bohm Effect in Disordered Metals 206 Effects of Electron-Electron Interaction in Magnetic Field 208 Electron Localization in Random Potential 213 Analogy Between Electron Properties in Disordered Systems and Statistical Mechanics of Ferromagnets 213 Q-Hamiltonian 214 Renormalization Group Equation 216 Conductivity in Two Dimensions 217 Localization in One and Three Dimensions 219 Localization in Magnetic Field 222 Localization in Percolating Structures 223 Conclusion 225 Appendices 226 Cooperon-Current Relation in the Space-Time Representation 226 Cooperon in External Electromagnetic Field 229 Calculation of Path Integrals 230 Expression for Conductivity of Interacting Electrons 232 References 235

AN EXACT SOLUTION OF THE RONDO PROBLEM BY P.B. WIEGMANN (transit by E. Yankovsky) 238 Introduction 238 The Basic Models 244 The Anderson Model 245

10

Contents

4.2.2

The Angular Dependence of Hybridization Amplitudes. A OueDimensional Hamiltonian 246 Hierarchy of Energies 247 Exchange Hamiltonians 248 The Simple Exchange Hamiltonians 250 Perturbation Theory 254 Bethe’s Method 257 General Survey. The Factorization Equations 257 An Effective Hamiltonian 263 The Bethe Ansatz for the s-d Exchange Model 264 Periodic Boundary Conditions 269 The Set of Commuting Operators 269 Diagonalizing T (a) 272 Discussion 275 The Thermodynamics of the s-d Exchange Model 277 Bethe’s Equations for the s-d Exchange Model. Going Over to the Continuous Limit 277 The Equilibrium Distribution 282 The Free Electron Gas (g ->■ 0 or S = 0) 284 Universality 287 The Limits of Strong and Weak Coupling 288 The Thermodynamics of an Impurity 290 Magnetic Susceptibility at T = 0 [30, 32, 33] 293 Solution of Equations (4.4.35) and (4.4.36) by an Iterative Method at T H [36, 38] and Perturbation Theory 297 Supplementary Discussion 300 Solution for the n-fold Degenerate Exchange Model 301 The Bethe Ansatz 301 Magnetic Susceptibility at T — 0 303 Thermodynamics 306 Discussion 308 Conclusion 308 References 311

4.2.3 4.2.4 4.2.5 4.2.6 4.3 4.3.1 4.3.2 4.3.3 4.3.4 4.3.5 4.3.6 4.3.7 4.4 4.4.1 4.4.2 4.4.3 4.4.4 4.4.5 4.4.6 4.4.7 4.4.8 4.4.9 4.5 4.5.1 4.5.2 4.5.3 4.5.4 4.6

D efects and Surface Phenomena in Quantum Crystals A . F . A n d r e e v , Corresponding Member of the USSR Academy of Sciences

S. I. Vavilov Institute of Physical Problems, Academy of Sciences of the USSR

1.1.

INTRODUCTION

The usual quantum theory of solids is based on the as­ sumption that the crystal lattice is a quasiclassical object. On the one hand, the quantum effects accounted for in this theory play an important role in the behavior of the phonons of the cyrstal, namely at temperatures below the Debye temperature. On the other hand, the particles that comprise the crystal are considered localized near definite equilibrium positions. The latter property is of a purely classical nature. Indeed, in this case the identical particles compris­ ing the crystal become distinguishable because they belong to different lattice sites. But quantum mechanics states that identical particles must be indistinguishable. Although due to this fact the quasiclassical picture of the crystal is an approximation, its accura­ cy is high for the majority of crystals and considerably exceeds the possibilities of experimental techniques. A quantitative criterion is the smallness of the ratio of the amplitude of zero-point vibrations in the majority of crystals to the lattice period. There is a small group of so-called quantum crystals (solid helium being the most striking example) in which the amplitude of the zero-point motion is exceptionally large and, in fact, is only several­ fold smaller than the lattice period. The anomalies in the properties of quantum crystals are the consequence of this fact because even at absolute zero the vibrational energy of the particles is of the order of the total energy of the crystal and the vibrations are markedly anharmonic. For the same reason the usual approach to calculating such properties of the quantum crystal as the ground-state energy, compressibility, and phonon spectrum is inapplicable. To describe these properties self-consistent methods of accounting for the zeropoint motion were developed (see the review article by Guyer [1]), and the results were in accord with the experimental data, at least qualitatively. However, there is a completely new effect associated with a new aspect in the motion of the constituent particles. Because of the large amplitude of zero-point vibrations in quantum crystals there is a high probability of the tunneling of the particles to neigh­ boring sites. In this way translational motion is imposed on the vibrational motion. As a result there emerges a picture of indistin-

12

A. F. Arcdreez;

guishable (in the quantum mechanical sense) particles delocalized in the crystal lattice, similar to that in quantum liquids. The present paper is a review of properties of quantum crystals in which the quantum delocalization of the particles manifests itself. Recent years have seen results achieved by both theoretical and experimental investigations that undoubtedly indicate that here we are dealing with objects very different from ordinary crystals. 1.2

QUANTUM EFFECTS IN CRYSTALS

We will start by reviewing the quantitative role of quan­ tum effects in crystals and establishing the type of crystal in which the deviations from the quasiclassical theory are the greatest. As mentioned above, the relative magnitude of the quantum effects is determined by the dimensionless parameter A ~ w2/ 0, velocity u jumps whenever E, in varying its direction, passes through a plane that is perpendicular to one of the a„’s. The angular width of the transition region is equal to TleEa > O.jrThe presence of a small but finite tunneling probability for the vacancy to neighboring lattice sites leads to an energy band of a finite width Av. As long as Av is much smaller than E 0l we can use the method of strong binding to calculate the energy spectrum, and this

38

A. F. Andreev

method shows that the middle of the energy band coincides with the energy E 0 of the localized vacancy. This implies that as the tunneling probability (and, hence, the band width) increases, the minimal vacancy energy e0 corresponding to the bottom of the band decreases. For this reason in very pronounced quantum crystals such as solid helium there can be a situation in which s0 becomes negative. This means that the initial ground state of the crystal is in actual fact unstable towards vacancy production. The correct ground state corresponds to a state of the crystal in which there are some vacancies. If e0 is negative but small in absolute value, the vacancies with quasimomenta close to that corresponding to the bottom of the band have negative energies. The energy E (p) of the vacancies in this region near the energy minimum can be written as ^(p ) = E0+ - | i ,

(1.4.6)

where M is a certain effective mass, and p the quasimomentum reck­ oned from the minimal quasimomentum. In a Bose crystal of the 4He type, with which we will start our exposition, the vacancies are obviously bosons. To calculate their contribution (at low density) to the ground state energy of the crystal we can use (see [67]) the well-known formula for the ground state energy of a rarefied Bose gas with spectrum (1.4.6): E=

+

(1.4.7)

here E and N are the energy of vacancies and their number per unit volume, and / is the vacancy-vacancy scattering length. When / is positive, i.e. the vacancies repel each other, and s0 is small and nega­ tive, the quantity on the right-hand side of (1.4.7) has a minimum for small W’s, a fact that enables us to calculate the equilibrium den­ sity N 0 of zero-point vacancies in a Bose crystal: AT TW f 1so 1 (1.4.8) aV° 4ji/*2/ A crystal with zero-point vacancies should have many peculiar properties, for one, superfluidity, caused by the possibility of mass transfer via superfluid motion of the “vacancy liquid” through the crystal (see [31, 65, 66, 68]). The experimental data available at present, however, indicates that this possibility, in all likelihood, is not realized in solid 4He. The experimental data discussed in Sec. 1.4.1 on the diffusion of 3He impurity atoms induced by vacancies and on the mobility of charges indicates that the number of vacancies decreases exponential­ ly with temperature, which corresponds to a positive minimal ener­ gy of the vacancies. We must note, however, that this result is ob-

1. Defects and Surface Phenomena in Quantum Crystals

39

tained for conditions with constant pressure. Measurements of the charge-mobility temperature dependence along the crystal-liquid equilibrium curve show that the mobility is approximately constant (see [64]) down to the lowest temperatures studied, of the order of 1 K. In 3He crystals the quantum effects are more pronounced than in 4He thanks to the smaller mass of the atoms. But here too there is experimental data on x-ray scattering [69] that shows that the number of vacancies at temperatures of the order of 1 K decreases with tem­ perature more or less exponentially. But this does not mean that a 3He crystal has no vacancies at absolute zero. The point is that the vacancies in 3He crystals behave in a peculiar way because 3He atoms possess nuclear spins. If all the nuclear spins are parallel, the 3He crystal, just as the 4He crystal, is periodic. Then due to the tunneling effect the vacancies become delocalized quasiparticles with a band width of the order of 10 K. Actually, however, down to temperatures of the order of 10~3 K the nuclear spins in 3He are not aligned and for this reason the crystal is not periodic. The vacancies in such a crystal are not quasiparticles with definite quasimomenta. Leaving the discussion of this aspect to the next section, we will nevertheless consider the case of zero-point vacancies in a crystal with all nuclear spins parallel. This state can in fact be achieved by applying a strong magnetic field to the sample. Since in the case under consideration the crystal has a periodic structure, the vacancy energy spectrum with a small negative ener­ gy e0 corresponding to the bottom of the band can still be described by formula (1.4.6). In the Fermi crystal of solid 3He the vacancies are fermions, however, since for vacancy production one fermion, the atom of 3He, must be annihilated. The ground state for fermions with the spectrum given by (1.4.6) corresponds, obviously, to a complete occupation of all negative-energy states, i.e. states with p i?0, with P

2P 0

PstfSfX •

(1.5.3)

The situation here is similar to that discussed by Lifshits and Kagan [89] and Iordanskii and Finkel’shtein [90] concerning nucleus production in a first-order phase transition from a metastable state and to that considered by Petukhov and Pokrovskii [91] on moving dislocations. Crystal growth at sufficiently low temperatures is the result of quantum subbarrier production of nuclei with R = R 0i which then rapidly grow in size. Let us calculate the probability w of the nucleus of a new atomic row being created. We assume that on the given surface the step is quantum-rough. This case is favorable for crystal growth, since we saw in Sec. 1.5.1 that such steps have station­ ary states corresponding to a continuous motion of the step and as

1. Defects and Surface Phenomena in Quantum Crystals

49

close to the ground state as desired. In other words, in this case the step can move freely. But if the step is smooth, its motion is hindered by the fact that it is the result of subbarrier creation of pairs of kinks of opposite signs. The energy U (R) defined in (1.5.2) plays the role of the nucleus potential energy. To calculate the kinetic energy K associated with the growth of the nucleus at rate R we note that the step’s movement at rate R is accompanied, due to the difference of the densities ps and p! of the solid and liquid, respectively, by a flow of a mass of liquid to the solid (per unit length of step per unit time) equal to (Ps — Pi)&fl- It is easy to find the speed of liquid flow at distances r from the step, where a 0, 0 < c/3 < 6. By finding the derivatives of / with respect to the rj^sw e arrive at the following expressions for the angular variables: K = ’ll (at + br\{ + ct]-), h2 = r]2 {at + br\l + cr]f).

(1.6.22)

To find the specific structure of the crystal faceting at t < 0 we note the following. For a fixed direction of vector r] ^ the potential f is an increasing function of r\ for small rj’s and a decreasing function for large values. As to the dependence of / on the direction of it is different for b > c and 6 < c . Suppose that b > c. Then / as a function of the direction of r]^ at fixed r] has minima along the coordinate axes = 0 and r]2 = 0. Since sections with zero curva­ ture must appear where the initial potential f (and coordinate z in view of (1.6.13)) is minimal, such sections at b >> c must lie along the coordinate axes. The first formula in (1.6.22) shows that for a fixed value of tj2 (t]| < a\ t \tc) hx vanishes at two values of % (differing only in sign):

T]^-2)= ± (-a|tJ-~ c.ni-)1/2, At these points the potential / (and, hence, coordinate z)7 which depends only on r]*, assumes the same value. Whence it is clear that the entire segment r\[2) < r\x < r\[i} corresponds, in fact, to a con­ stant (zero) value of the angular variable hv Similarly, for a fixed value of ill (ti? < a\ t \!c) h 2 vanishes in the segment < ri2 lskii,Cand.

Sc. (Physics and Mathematics)

P. N. Lebedev Institute of Physics, Academy of Sciences of the USSR

2.1

INTRODUCTION

Electronic phase transitions (EPT) are usually understood to be the phase transitions in a solid that basically involve the electron subsystem. Such, for instance, are the Mott metal-insulator transitions [1-3] and valence-change transitions in rare-earth metals and compounds [4-8], Another example is the transition from localized to delocalized electrons (the Anderson transition) [9, 10]. It is natural that the change in the state of the outermost (valence) electrons affects other properties of solids, namely, those associated with the crystal lattice. We can also say that the interaction of electrons with the lattice influences the parameters of EPT. From this viewpoint singling out EPT is a somewhat artificial procedure, since in the final analysis all phase transitions in solids (e.g. structural transitions) are “electronic’’ in origin because the types of chemical bond and lattice structure depend on the configuration of the valence electrons. However, this basic fact often remains concealed, and many tran­ sitions are described without such analysis. For instance, the usual way of interpreting ferroelectric phase transitions is to resort to the anharmonicity of lattice vibrations as the basic mechanism [11]. At the same time, the main features of phenomena like the Mott transition or a transition with valence change can be interpreted mainly by using the language of valence electrons, which justifies our singling out such phenomena and employing the term EPT. The present review focuses on transitions in rare-earth compounds associated with a change in the electron configuration of the rareearth ions but not with a change in lattice symmetry (so-called isostructural phase transitions). It also describes the electronic states that occur in such transitions, mixed valence (MV) states. Only recently have such phenomena been thoroughly studied (e.g. see [4-8]), although the first experimental works appeared in the early sixties [12, 13]. The class of substances is fairly large in which, in contrast to typical rare-earth compounds, the 4f shell loses its stability while in many ways retaining its atomlike character. At present more than a hundred such compounds have been discovered. In such substances states having different numbers of f-electrons per site

2. Electronic Phase Transitions and the MV Problem

71

(e.g. states 4fn and (4fn-1 + one electron in the conduction band)) prove to lie close in energy. Due to this resonance transitions may occur between these configurations, the f-electrons acquire a par­ tially band nature, the mean number of f-electrons per center (the ion’s valence) becomes nonintegral, etc. Such compounds have come to be known as mixed-valence compounds. States with mixed valence have a number of unique properties, which are of interest in themselves but can also contribute to our understanding of the behaviour of electrons in a solid. Under varia­ tion of external conditions (temperature, pressure, composition) compounds of this type often undergo phase transitions involving variation in the filling of electron levels. In some cases these tran­ sitions are of the insulator-metal type. Also there is often a change in magnetic properties (localized magnetic moments vanish)* i.e. these transitions are of the magnetic-nonmagnetic state type. Systems with mixed-valence states show marked anomalies in practically all experimentally measured characteristics: in lattice properties (anomalously high compressibility), in specific heat (anomalously high linear specific heat at low temperatures), in magnetic susceptibility, and transport characteristics (especially in electric conductivity). The fundamental problem that we can hope to solve by studying such substances is the relationship between the pictures of localized and collective states in describing the behavior of the electrons in solids, i.e. when does one or the other language fit, how do tran­ sitions between these states occur, and what peculiar features in the behavior of the electron (and lattice) system accompany such tran­ sitions?. Since the f-electrons do to a great extent retain their atom­ like character, there is the hope that at least in certain respects the corresponding substances may be simpler than compounds of d metals and actinides, for which all characteristic parameters are of the order of unity. We may assume, therefore, that an understanding of the situation with mixed valence will also help clarify the properties of transition metals and their compounds. The physics of mixed valence is, in fact, closely related to a num­ ber of problems and phenomena in solids. These include the question of tlie nature of magnetism, the Kondo effect, insulator-metal tran­ sitions, spatial ordering and Wigner crystallization; one can also add superconductivity [95]. We must also note that EPT in such com­ pounds resemble phenomena occurring in superdense matter, a topic that is important for astrophysics [96]. To a certain extent systems with mixed valence may serve as a general model of condensed matter (see Chap. 2.6). All this together with the intrinsic importance of such compounds explains the growing interest in this class of physical objects, which in many respects are unique. This review is concerned mainly with the theoretical aspects of

D. /. Khomskii

72

the MY problem. Experimental results are brought in largely to illustrate and justify the theoretical discussion. Although solid-state theoreticians in many countries are working in this field and our aim is to give a comprehensive picture of the present state of this field, we will dwell chiefly on the works of Soviet authors. 2.2

LOCALIZATION OF ELECTRONS AND INSULATOR-METAL TRANSITIONS (MOTT-HUBBARD TRANSITIONS)

We will start with a brief survey of a problem that is important not only to the physics of rare-earth compounds and the MV problem but has general significance. This problem is the rela­ tionship between the localized and collective states of electrons in a solid. As is known, there are two limiting pictures of the behavior of electrons in a solid. The usual band scheme, which works well for normal metals and semiconductors, deals with almost free electrons moving in the periodic lattice potential. The corresponding states (Bloch waves) prove to be delocalized; the electron-electron interac­ tion is usually taken into account by the Hartree-Fock approxima­ tion and does not violate the one-electron picture. On the other hand, in some cases a more appropriate way to de­ scribe the behavior of electrons is to use the language of localized states (in the theory of molecules this is the Heitler-London approxi­ mation). This approach is justified for strong interelectronic corre­ lation; in reality it is valid for the electrons of inner shells. For filled shells both approaches are completely equivalent, but for partially filled shells this is not so. These are the partially filled d and f states in transition and rare-earth metals and actinides and their compounds. Since the electrons are localized, such substances are magnetic, as a rule. It is this localization that explains the fact that, in contrast to the predictions of the band theory, many such sub­ stances are insulators (Mott insulators). The criterion that enables us to choose between the two types of states involves the mean Coulomb interaction energy between two electrons, U ~ e2/r, and the characteristic kinetic energy, E%\n ~~ h 2lmr*: at 2?kin > U the electrons are delocalized, while the opposite inequality enables us to use the picture of localized electrons. We see that the second case is realized in a rarefied system: ft* __

r > me? ~ a° ’

(2.2.1)

i.e. when the mean distance between electrons r is greater than the Bohr radius a0. In other words, the overlap of the wave functions of

2. Electronic Phase Transitions and the MV Problem

73

neighboring centers must be small, a situation realized for d- and especially f-electrons * The fact that in a system with low density the electrons are localized and because of this lose their “metallic” nature was first pointed out for an electron gas by Wigner [14] (the jellium model) and for real solids by Landau and ZeFdovich [1]. The latter work, which unfortunately remained largely unnoticed, contains practically all the ideology of insulator-metal transitions and the argument (related to R. Peierls) showing that such transitions must be of the first order [1]: A dielectric differs from a metal in that it has an energy gap in its electronic spectrum. But does this gap tend to zero when we approach the point of transition from dielectric to metal? If this was the case, we should have a transition without latent heat, without jumps in volume, and other properties. Peierls has shown that a transition continuous in this sense is impos­ sible. Suppose we are studying an excited state of a dielectric, a state in which it conducts electricity. An electron leaves its place and is moving in the lattice, leaving a positive charge in another place in the lattice. When the electron has traveled far from the positive charge, it is attracted by a Coulomb force that tends to return the electron to its site. In an attractive Coulomb field there are always discrete levels of negative energy, cor­ responding to an electron binding. Therefore, the excited and conducting states of a dielectric are always separated from the ground state, in which the electrons are bound, by a gap of finite width. Later Mott [16] expounded these ideas independently. By esti­ mating the screening of the Coulomb potential as the number of excited electrons grows and using the condition for a bound electron state to disappear, he found the criterion for an insulator-metal transition: F~

(2.2.2)

(n is the electron concentration). The phenomenon associated with this transition is often called a Mott transition. The simplest model employed in studying Mott transitions was proposed by Hubbard [17], In this model the electrons in a non­ degenerate band interact via a repulsive force at one site: H =

2 tij&ioG'ja H 2~ 2 ijt a io

-a .

(2.2.3)

The tunneling integral t is the measure of kinetic energy, and criterion (2.2.1) here has the form t h In essence the difficulty in building the theory of Mott transitions is due to the fact that to describe the transition we must change the mode of description: from electrons localized at The centers with a “single” filling of each level we must switch to the usual band "double” filling. When this is done even the statistics of the electrons changes [89, 90]: the “atomic” statistics becomes the usual Fermi statistics* In teality the systems closest to the Hubbard model appear to be the transition metals and their compounds [of course many impor­ tant details are ignored in the Hamiltonian (2.2.1)]. In such substances the width of the d band, t, is of the order of the Coulomb interaction, U-, and, therefore, these substances constitute, ms Varma noticed in. [5], a “theorist’s nightmare”. With this fact are associated the wellknown; difficulties in describing magnetism in substances of. the type of iron [22, 23] and insulator-metal transitions often observed in such compounds as vanadium oxides [2, 24]. * Because of this the insulator-metal transition in the “pure” Hubbard model may not be a real phase transition in the usual sense and becomes one only after an extra degree of freedom has been added (a change in the lattice constant, for one).

2. Electronic Phase Transitions and the MV Problem

2.3

ELECTRONIC PHASE TRANSITIONS IN RARE-EARTH COMPOUNDS

2.3.1

The General Picture of Transitions

75

At first glance the situation with rare-earth compounds seems simpler than with transition metals. In typical rare-earth compounds there are two groups of electrons: the outer valence electrons, which form chemical bonds or form the metallic band (they are described by usual Bloch waves), and the electrons of the inner partially filled 4f shells.* Since the radius of the 4f wave func­ tion is exceptionally small (~ 0 .5 A), we can neglect the overlap of the .wave functions . of neighboring centers. At the same time the Coulomb interaction of f-electrons at the center is strong (Utt ~ ~ 5-7 eV), and we can assume that the f-electrons are far on the dielectric “side” of the Mott-Hubbard transition (tft )5 whence Gu(o>) with

{(fmlfm))

(2.4.3) (2.4.4)

2ji E ki i*e- the magnetic state proves preferable. The possi­ bility of a magnetic solution in the Kondo lattice at J ^ 0.3 W was first obtained for the one-dimensional model [74]; further calcula­ tions (see [84]) have not corroborated the existence of two different regimes in one dimension., but in three dimensions the situation is, apparently, what we have described. The deep f levels, in agreement with (2.3.3), give a small exchange interaction / , and for these cen­ ters the magnetic regime is realized; in fact such are the typical magnetic compounds of rare-earth metals. On the other hand, as Et ->• eF (when we approach a state with MV), | J | grows and a tran­ sition is possible to the Kondo regime, in which at N e = N s a gap appears at the Fermi level. We note that from what has been said a change from one regime to the other will take place at ( / | ex 0.3W = 0.3p~\ i.e., in ac­ cordance with (2.3.3), at eF — Ef ~ 3pV2 ~ T, which is exactly in an MV phase. In this case, however, the transition via SchriefferWolff transformation from the Anderson lattice (2.3.4) to the Kondo lattice (2.5.3) cannot be achieved, and one must resort to the orig­ inal model (2.3.4). Therefore, we see that the singlet regime with a gap realized in a three-dimensional Kondo lattice for strong coupl­ ing, p / ^ 1, apparently can serve only as an indication that a sim­ ilar situation is present in the mixed-valence case in the realistic model of the Anderson lattice as well. Although the studies of the Kondo lattice are far from complete, the situation appears to be such as discussed above; possibly, the appropriate one-dimensional model can even be solved exactly (in a way similar to the solution of the Kondo problem [29] or the one­ dimensional Hubbard model [86]). However, as yet there is no reliable data on the Anderson lattice. Among the exact results we can recall only the interesting work of Jullien and Martin [88], who made a nu­ merical calculation for a one-dimensional model with a finite number of centers and, by applying an extrapolation technique, established that the gap at N c = A s for deep-lying levels (the Kondo lattice) is present here for any position of the f level (including the mixedvalence case) and the chemical potential lies inside the gap. The authors study the case where there are two electrons per site and in the initial state, at E t = 0, (U (r) U (r')> = 6 (r - r'), (3.2.9) where the angle brackets denote averaging over the impurity con­ figuration. Equation (3.2.9) corresponds to the Born approxima-

3.

Coherent Effects in Disordered Conductors

133

tion for the interaction with short range impurities, and ( U2)= = cim ( j V (r) d r)2, where cim is the impurity concentration and V (r) is the potential of a single impurity. The sum of diagrams with non-intersecting dashed lines defines the Green function and Ke(P y(o)

Fig. 3.1.

conductivity to the leading approximation in In this ap­ proximation the averaged Green function is given by the formula [4] (r, r')>= j (dp) exp {ip -(r — r')}Gg(p),

where (dp) = dp/(2jt)d, d

is

the space

dimensionality and

Here energy e is counted from the chemical potential jx, 1 / t = = jxv (U2) is the frequency of elastic collisions and v is the elec­ tronic density of states. If scattering is isotropic, then in calculat­ ing the conductivity to the leading approximation in 1/fxx we can ignore the graphs with vertical dashed lines. These graphs give zero contribution after integration over angles of p due to the vectorial character of the current vertex. The first term in (3.2.6) gives K u (e, to)

e2w i Gq +r ____________ d|_____________ 2nd

-‘- ( - E + ^ + e)

+

i7j.ve2T \ (3.2.11) d l + icox ’ p2 where an energy variable £ = • — jx is introduced. It is seen from (3.2.11) that K (e, co) does not depend on e, so we can set e to zero. In the integral over £ the main contribution comes from

B. L. Altshuler et ah

134

a narrow range 1/t . Therefore we can integrate between infinite limits. In each of the last two terms in Eq. (3.2.6) the poles of the Green functions lie in the same half-plane of the complex variable hence the integrals over £ between infinite limits are equal to zero, and the contribution of these terms is small in the parameter (p'r)*’1 and independent of co. Neglecting this contribution, we obtain Eq. (3.2.7). Substituting (3.2.11) into (3.2.7) and (3.2.4), we obtain the Drude formula (3.2.1). The density correlator &C (co, q) = j dz [n (e -f co) —n (e)]

(co, q)

can also be calculated by means of the impurity technique: (co, q) = (

j

G&% (p + q) G$ (p) (dp)') .

'

(3.2.12)

In the leading approximation &£ 6 is given by the sum of laddertype diagrams (Fig. 3.1): &£ z (w, q) = n vt t, (q, co) D (q, co), (3.2.13) where < . £ (q. «) = — -

j

(dp)

(e + CO, p + q) Ga (e, p).

In the limit ql