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Metal-insulator transitions

Masatoshi Imada

Institute for Solid State Physics, University of Tokyo, Roppongi, Minato-ku, Tokyo, 106, Japan

Atsushi Fujimori

Department of Physics, University of Tokyo, Hongo, Bunkyo-ku, Tokyo, 113, Japan

Yoshinori Tokura

Department of Applied Physics, University of Tokyo, Hongo, Bunkyo-ku, Tokyo, 113, Japan

Metal-insulator transitions are accompanied by huge resistivity changes, even over tens of orders of magnitude, and are widely observed in condensed-matter systems. This article presents the observations and current understanding of the metal-insulator transition with a pedagogical introduction to the subject. Especially important are the transitions driven by correlation effects associated with the electron-electron interaction. The insulating phase caused by the correlation effects is categorized as the Mott Insulator. Near the transition point the metallic state shows fluctuations and orderings in the spin, charge, and orbital degrees of freedom. The properties of these metals are frequently quite different from those of ordinary metals, as measured by transport, optical, and magnetic probes. The review first describes theoretical approaches to the unusual metallic states and to the metal-insulator transition. The Fermi-liquid theory treats the correlations that can be adiabatically connected with the noninteracting picture. Strong-coupling models that do not require Fermi-liquid behavior have also been developed. Much work has also been done on the scaling theory of the transition. A central issue for this review is the evaluation of these approaches in simple theoretical systems such as the Hubbard model and t-J models. Another key issue is strong competition among various orderings as in the interplay of spin and orbital fluctuations.

Experimentally, the unusual properties of the metallic state near the insulating transition have been most extensively studied in d-electron systems. In particular, there is revived interest in transition-metal oxides, motivated by the epoch-making findings of high-temperature superconductivity in cuprates and colossal magnetoresistance in manganites. The article reviews the rich phenomena of anomalous metallicity, taking as examples Ti, V, Cr, Mn, Fe, Co, Ni, Cu, and Ru compounds. The diverse phenomena include strong spin and orbital fluctuations, mass renormalization effects, incoherence of charge dynamics, and phase transitions under control of key parameters such as band filling, bandwidth, and dimensionality. These parameters are experimentally varied by doping, pressure, chemical composition, and magnetic fields. Much of the observed behavior can be described by the current theory. Open questions and future problems are also extracted from comparison between experimental results and theoretical achievements. [S0034-6861(98)00103-2]

CONTENTS

I. Introduction 1040 A. General remarks 1040 B. Remarks on theoretical descriptions

(Introduction to Sec. II) 1044 C. Remarks on material systematics (Introduction

to Sec. III) 1046 D. Remarks on experimental results of anomalous

metals (Introduction to Sec. IV) 1047 II. Theoretical Description 1048

A. Theoretical models for correlated metals and Mott insulators in d-electron systems 1048 1. Electronic states of d-electron systems 1048 2. Lattice fermion models 1050

B. Variety of metal-insulator transitions and correlated metals 1053

C. Field-theoretical framework for interacting fermion systems 1057 1. Coherent states 1058 2. Grassmann algebra 1058 3. Functional integrals, path integrals, and

statistical mechanics of many-fermion systems 1059

Reviews of Modern Physics, Vol. 70, No. 4, October 1998 0034-6861/98/70

4. Green’s functions, self-energy, and spectral functions 1060

D. Fermi-liquid theory and various mean-field approaches: Single-particle descriptions of correlated metals and insulators 1062 1. Fermi-liquid description 1063 2. Hartree-Fock approximation and RPA for

phase transitions 1070 3. Local-density approximation and its

refinement 1074 4. Hubbard approximation 1078 5. Gutzwiller approximation 1079 6. Infinite-dimensional approach 1080 7. Slave-particle approximation 1084 8. Self-consistent renormalization

approximation and two-particle self- consistent approximation 1087

9. Renormalization-group study of magnetic transitions in metals 1089

E. Numerical studies of metal-insulator transitions for theoretical models 1092 1. Drude weight and transport properties 1093 2. Spectral function and density of states 1097 3. Charge response 1099 4. Magnetic correlations 1100

1039(4)/1039(225)/$60.00 © 1998 The American Physical Society

1040 Imada, Fujimori, and Tokura: Metal-insulator transitions

5. Approach from the insulator side 1103 6. Bosonic systems 1103

F. Scaling theory of metal-insulator transitions 1104 1. Hyperscaling scenario 1105 2. Metal-insulator transition of a noninteracting

system by disorder 1106 3. Drude weight and charge compressibility 1107 4. Scaling of physical quantities 1109 5. Filling-control transition 1109 6. Critical exponents of the filling-control

metal-insulator transition in one dimension 1110 7. Critical exponents of the filling-control

metal-insulator transition in two and three dimensions 1110

8. Two universality classes of the filling-control metal-insulator transition 1111

9. Suppression of coherence 1111 10. Scaling of spin correlations 1113 11. Possible realization of scaling 1114 12. Superfluid-insulator transition 1115

G. Non-fermi-liquid description of metals 1116 1. Tomonaga-Luttinger liquid in one dimension 1116 2. Marginal Fermi liquid 1118

H. Orbital degeneracy and other complexities 1119 1. Jahn-Teller distortion, spin, and orbital

fluctuations and orderings 1119 2. Ferromagnetic metal near a Mott insulator 1122 3. Charge ordering 1123

III. Materials Systematics 1123 A. Local electronic structure of transition-metal

oxides 1123 1. Configuration-interaction cluster model 1123 2. Cluster-model analyses of photoemission

spectra 1125 3. Zaanen-Sawatzky-Allen classification scheme 1126 4. Multiplet effects 1128 5. Small or negative charge-transfer energies 1130

B. Systematics in model parameters and charge gaps 1131 1. Ionic crystal model 1131 2. Spectroscopic methods 1132 3. First-principles methods 1135

C. Control of model parameters in materials 1139 1. Bandwidth control 1139 2. Filling control 1141 3. Dimensionality control 1142

IV. Some Anomalous Metals 1144 A. Bandwidth-control metal-insulator transition

systems 1144 1. V2O3 1144 2. NiS22xSex 1151 3. RNiO3 1154 4. NiS 1157 5. Ca12xSrxVO3 1163

B. Filling-control metal-insulator transition systems 1165 1. R12xAxTiO3 1165 2. R12xAxVO3 1171

C. High-Tc cuprates 1173 1. La22xSrxCuO4 1173 2. Nd22xCexCuO4 1184 3. YBa2Cu3O72y 1187 4. Bi2Sr2CaCu2O81d 1195

D. Quasi-one-dimensional systems 1199 1. Cu-O chain and ladder compounds 1199 2. BaVS3 1205

E. Charge-ordering systems 1208

Rev. Mod. Phys., Vol. 70, No. 4, October 1998

1. Fe3O4 1208 2. La12xSrxFeO3 1211 3. La22xSrxNiO4 1212 4. La12xSr11xMnO4 1216

F. Double exchange systems 1218 1. R12xAxMnO3 1218 2. La222xSr112xMn2O7 1225

G. Systems with transitions between metal and nonmagnetic insulators 1228 1. FeSi 1228 2. VO2 1232 3. Ti2O3 1233 4. LaCoO3 1235 5. La1.172xAxVS3.17 1239

H. 4d systems 1241 1. Sr2RuO4 1241 2. Ca12xSrxRuO3 1243

V. Concluding Remarks 1245 Acknowledgments 1247 References 1247

I. INTRODUCTION

A. General remarks

The first successful theoretical description of metals, insulators, and transitions between them is based on noninteracting or weakly interacting electron systems. The theory makes a general distinction between metals and insulators at zero temperature based on the filling of the electronic bands: For insulators the highest filled band is completely filled; for metals, it is partially filled. In other words, the Fermi level lies in a band gap in insulators while the level is inside a band for metals. In the noninteracting electron theory, the formation of band structure is totally due to the periodic lattice struc- ture of atoms in crystals. This basic distinction between metals and insulators was proposed and established in the early years of quantum mechanics (Bethe, 1928; Sommerfeld, 1928; Bloch, 1929). By the early 1930s, it was recognized that insulators with a small energy gap between the highest filled band and lowest empty band would be semiconductors due to thermal excitation of the electrons (Wilson, 1931a, 1931b; Fowler, 1933a, 1933b). More than fifteen years later the transistor was invented by Shockley, Brattain, and Bardeen.

Although this band picture was successful in many re- spects, de Boer and Verwey (1937) reported that many transition-metal oxides with a partially filled d-electron band were nonetheless poor conductors and indeed of- ten insulators. A typical example in their report was NiO. Concerning their report, Peierls (1937) pointed out the importance of the electron-electron correlation: Strong Coulomb repulsion between electrons could be the origin of the insulating behavior. According to Mott (1937), Peierls noted

‘‘it is quite possible that the electrostatic interaction between the electrons prevents them from moving at all. At low temperatures the majority of the electrons are in their proper places in the ions. The minority which have happened to cross the potential barrier

1041Imada, Fujimori, and Tokura: Metal-insulator transitions

find therefore all the other atoms occupi