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Metal-Insulator Transitions 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 4. Green’s functions, self-energy, and spectral functions 1060 D. Fermi-liquid theory and various mean-field I. Introduction 1040 approaches: Single-particle descriptions of A. General remarks 1040 correlated metals and insulators 1062 B. Remarks on theoretical descriptions 1. Fermi-liquid description 1063 (Introduction to Sec. II) 1044 2. Hartree-Fock approximation and RPA for C. Remarks on material systematics (Introduction phase transitions 1070 to Sec. III) 1046 3. Local-density approximation and its D. Remarks on experimental results of anomalous refinement 1074 metals (Introduction to Sec. IV) 1047 4. Hubbard approximation 1078 II. Theoretical Description 1048 5. Gutzwiller approximation 1079 A. Theoretical models for correlated metals and 6. Infinite-dimensional approach 1080 Mott insulators in d-electron systems 1048 7. Slave-particle approximation 1084 1. Electronic states of d-electron systems 1048 8. Self-consistent renormalization 2. Lattice fermion models 1050 approximation and two-particle self- B. Variety of metal-insulator transitions and consistent approximation 1087 correlated metals 1053 9. Renormalization-group study of magnetic C. Field-theoretical framework for interacting transitions in metals 1089 fermion systems 1057 E. Numerical studies of metal-insulator transitions 1. Coherent states 1058 for theoretical models 1092 2. Grassmann algebra 1058 1. Drude weight and transport properties 1093 3. Functional integrals, path integrals, and 2. Spectral function and density of states 1097 statistical mechanics of many-fermion 3. Charge response 1099 systems 1059 4. Magnetic correlations 1100 Reviews of Modern Physics, Vol. 70, No. 4, October 1998 0034-6861/98/70(4)/1039(225)/$60.00 © 1998 The American Physical Society 1039 1040 Imada, Fujimori, and Tokura: Metal-insulator transitions 5. Approach from the insulator side 1103 1. Fe3O4 1208 6. Bosonic systems 1103 2. La12xSrxFeO3 1211 F. Scaling theory of metal-insulator transitions 1104 3. La22xSrxNiO4 1212 1. Hyperscaling scenario 1105 4. La12xSr11xMnO4 1216 2. Metal-insulator transition of a noninteracting F. Double exchange systems 1218 system by disorder 1106 1. R12xAxMnO3 1218 3. Drude weight and charge compressibility 1107 2. La222xSr112xMn2O7 1225 4. Scaling of physical quantities 1109 G. Systems with transitions between metal and 5. Filling-control transition 1109 nonmagnetic insulators 1228 6. Critical exponents of the filling-control 1. FeSi 1228 metal-insulator transition in one dimension 1110 2. VO2 1232 7. Critical exponents of the filling-control 3. Ti2O3 1233 metal-insulator transition in two and three 4. LaCoO3 1235 dimensions 1110 5. La1.172xAxVS3.17 1239 8. Two universality classes of the filling-control H. 4d systems 1241 metal-insulator transition 1111 1. Sr2RuO4 1241 9. Suppression of coherence 1111 2. Ca12xSrxRuO3 1243 10. Scaling of spin correlations 1113 V. Concluding Remarks 1245 11. Possible realization of scaling 1114 Acknowledgments 1247 12. Superfluid-insulator transition 1115 References 1247 G. Non-fermi-liquid description of metals 1116 1. Tomonaga-Luttinger liquid in one dimension 1116 2. Marginal Fermi liquid 1118 I. INTRODUCTION H. Orbital degeneracy and other complexities 1119 1. Jahn-Teller distortion, spin, and orbital A. General remarks fluctuations and orderings 1119 2. Ferromagnetic metal near a Mott insulator 1122 The first successful theoretical description of metals, 3. Charge ordering 1123 insulators, and transitions between them is based on III. Materials Systematics 1123 noninteracting or weakly interacting electron systems. A. Local electronic structure of transition-metal The theory makes a general distinction between metals oxides 1123 1. Configuration-interaction cluster model 1123 and insulators at zero temperature based on the filling of 2. Cluster-model analyses of photoemission the electronic bands: For insulators the highest filled spectra 1125 band is completely filled; for metals, it is partially filled. 3. Zaanen-Sawatzky-Allen classification scheme 1126 In other words, the Fermi level lies in a band gap in 4. Multiplet effects 1128 insulators while the level is inside a band for metals. In 5. Small or negative charge-transfer energies 1130 the noninteracting electron theory, the formation of B. Systematics in model parameters and charge band structure is totally due to the periodic lattice struc- gaps 1131 ture of atoms in crystals. This basic distinction between 1. Ionic crystal model 1131 metals and insulators was proposed and established in 2. Spectroscopic methods 1132 the early years of quantum mechanics (Bethe, 1928; 3. First-principles methods 1135 C. Control of model parameters in materials 1139 Sommerfeld, 1928; Bloch, 1929). By the early 1930s, it 1. Bandwidth control 1139 was recognized that insulators with a small energy gap 2. Filling control 1141 between the highest filled band and lowest empty band 3. Dimensionality control 1142 would be semiconductors due to thermal excitation of IV. Some Anomalous Metals 1144 the electrons (Wilson, 1931a, 1931b; Fowler, 1933a, A. Bandwidth-control metal-insulator transition 1933b). More than fifteen years later the transistor was systems 1144 invented by Shockley, Brattain, and Bardeen. 1. V2O3 1144 Although this band picture was successful in many re- 2. NiS22xSex 1151 spects, de Boer and Verwey (1937) reported that many 3. RNiO3 1154 transition-metal oxides with a partially filled d-electron 4. NiS 1157 band were nonetheless poor conductors and indeed of- 5. Ca12xSrxVO3 1163 B. Filling-control metal-insulator transition systems 1165 ten insulators. A typical example in their report was NiO. Concerning their report, Peierls (1937) pointed out 1. R12xAxTiO3 1165 2. R12xAxVO3 1171 the importance of the electron-electron correlation: C. High-Tc cuprates 1173 Strong Coulomb repulsion between electrons could be 1. La22xSrxCuO4 1173 the origin of the insulating behavior. According to Mott 2. Nd22xCexCuO4 1184 (1937), Peierls noted 3. YBa Cu O 1187 2 3 72y ‘‘it is quite possible that the electrostatic interaction 4. Bi2Sr2CaCu2O81d 1195 D. Quasi-one-dimensional systems 1199 between the electrons prevents them from moving at 1. Cu-O chain and ladder compounds 1199 all. At low temperatures the majority of the electrons 2. BaVS3 1205 are in their proper places in the ions. The minority E. Charge-ordering systems 1208 which have happened to cross the potential barrier Rev. Mod. Phys., Vol. 70, No. 4, October 1998 Imada, Fujimori, and Tokura: Metal-insulator transitions 1041 Mott, 1990), in his original formulation Mott argued that the existence of the insulator did not depend on whether the system was magnetic or not. Slater (1951), on the other hand, ascribed the origin
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