## Physics of Transition Metal OxidesSadamichi Maekawa, Takami Tohyama, Stewart Edward Barnes, Sumio Ishihara, Wataru Koshibae, Giniyat Khaliullin The fact that magnetite (Fe304) was already known in the Greek era as a peculiar mineral is indicative of the long history of transition metal oxides as useful materials. The discovery of high-temperature superconductivity in 1986 has renewed interest in transition metal oxides. High-temperature su perconductors are all cuprates. Why is it? To answer to this question, we must understand the electronic states in the cuprates. Transition metal oxides are also familiar as magnets. They might be found stuck on the door of your kitchen refrigerator. Magnetic materials are valuable not only as magnets but as electronics materials. Manganites have received special attention recently because of their extremely large magnetoresistance, an effect so large that it is called colossal magnetoresistance (CMR). What is the difference between high-temperature superconducting cuprates and CMR manganites? Elements with incomplete d shells in the periodic table are called tran sition elements. Among them, the following eight elements with the atomic numbers from 22 to 29, i. e. , Ti, V, Cr, Mn, Fe, Co, Ni and Cu are the most im portant. These elements make compounds with oxygen and present a variety of properties. High-temperature superconductivity and CMR are examples. Most of the textbooks on magnetism discuss the magnetic properties of transition metal oxides. However, when one studies magnetism using tradi tional textbooks, one finds that the transport properties are not introduced in the initial stages. |

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### Contents

Introduction | 1 |

12 Crystal Structure and Physical Properties | 4 |

13 Exchange Interaction | 8 |

14 Orbital Degeneracy | 15 |

15 DoubleExchange Interaction | 19 |

16 Magnetic Anisotropy | 22 |

162 Anisotropic Exchange Interactions | 25 |

References | 34 |

412 Electronic Hamiltonian and Exchange Interaction | 170 |

413 JahnTeller Effect and Cooperative JahnTeller Effect | 178 |

414 Phase Diagram and Orbital Order | 182 |

415 Orbital Liquid State | 186 |

42 Manganite with Layered Structure | 190 |

422 Stability of Orbital and Magnetic Structure | 193 |

423 Experiments for Spin and Orbital Correlation | 194 |

43 Resonant Xray Scattering RXS | 197 |

Cuprates | 37 |

21 Underlying Electronic Structure of Cuprates | 38 |

212 Model Hamiltonian | 41 |

213 Superexchange Interaction CornerSharing Cuprates | 43 |

214 Cyclic FourSpin Interaction | 49 |

215 ZhangRice Singlet State | 50 |

216 Optical Excitations | 55 |

22 OneDimensional Cuprates | 58 |

222 Realization of SpinCharge Separation | 63 |

223 Charge Dynamics in Insulating Cuprates | 68 |

224 Nonlinear Optical Response | 74 |

225 Spin Dynamics in Insulating Cuprates | 79 |

23 TwoDimensional Cuprates | 80 |

231 Single Carrier in Mott Insulator | 81 |

232 Phase Diagram | 86 |

233 Optical Conductivity | 90 |

234 SingleParticle Spectral Function | 93 |

235 Chemical Potential | 94 |

24 Summary | 95 |

References | 96 |

Theory of Superconductivity | 101 |

31 The BCS Pairing Theory | 103 |

32 Phonons in Solids | 107 |

33 Phonons as Intermediate Bosons | 108 |

34 Theory of the Antiferromagnetic Parent Compounds | 111 |

35 The JordanWigner Transformation and Flux Tubes | 116 |

36 Coherent States Grassman Variables and Flux Tubes | 120 |

37 MeanField Approximations and Flux States | 122 |

38 Bogoliubov Theory for a Bose Superfluid | 124 |

39 Auxiliary Particle Methods | 126 |

310 Magnetic Exchange Interactions via Intermediate Bosons | 130 |

311 The MeanField RVB Slave Boson Theory | 131 |

312 The Gutzwiller Projection and a Ul Symmetry | 136 |

313 Auxiliary Particles and the Introduction of Flux Tubes | 137 |

314 Spin Pairing | 140 |

315 Fermionic Excitations in an Antiferromagnet | 141 |

316 SU3 Approach to Hole Coherent States | 143 |

317 The Effective Exchange for Coherent Doping | 147 |

318 SO5 Theory | 150 |

319 SO5 and SU3 Theories | 155 |

320 Gossamer Superconductivity | 158 |

321 Summary | 163 |

References | 165 |

Manganites | 167 |

431 Experiments | 198 |

432 Scattering Cross Section | 200 |

433 Azimuthal Angle Dependence | 203 |

434 Mechanism of RXS | 206 |

435 Microscopic Calculations of the RXS Intensity | 209 |

44 Orbital Excitation | 211 |

45 Other OrbitalRelated Topics | 216 |

46 Summary | 219 |

51 Introduction | 225 |

52 Orbital States | 226 |

522 Perovskite Vanadates | 229 |

53 Metal Insulator Transition | 230 |

54 Electronic State and Model Hamiltonian | 231 |

55 Summary | 238 |

Cobaltates | 241 |

62 Thermoelectric Materials and Cobalt Oxides | 245 |

63 Thermoelectric Effect | 246 |

64 Linear Response Theory for Thermoelectric Systems | 250 |

Approach from High Temperature Side | 252 |

66 Spin and Orbital States and the Thermopower | 254 |

67 Thermopower of the Degenerate Electron Gas | 257 |

References | 259 |

Quantum Effects in Orbitally Degenerate Systems | 261 |

71 Systems with eg Orbital Degeneracy | 262 |

712 OrbitalOnly Model | 269 |

713 OrbitalCharge Coupling Orbital Polarons | 272 |

714 Orbital Liquids Anomalous Transport | 279 |

72 Systems with t2g Orbital Degeneracy | 282 |

721 SpinOrbital Model | 283 |

722 OrbitalOnly Model | 291 |

73 High Spin Systems with t2g Orbital Degeneracy | 298 |

732 SpinOrbital Dimerization | 302 |

74 Summary | 306 |

A Optical Conductivity | 311 |

The Lanczos Method | 317 |

C Projection Method Memory Function Method Composite Operator Method | 321 |

DI Seebeck Effect | 323 |

D2 Peltier Effect | 325 |

D3 Thomson Effect | 326 |

Thomsons Considerations | 328 |

D5 The Third Law of Thermodynamics and Thermoelectric Response in Solids | 330 |

333 | |

335 | |

### Common terms and phrases

3d electrons anisotropy antiferromagnetic auxiliary particle axis bond bosons calculated carriers charge chemical potential condensate configuration correlation corresponds Coulomb interaction coupling cubic CuO2 plane cuprates defined degenerate degrees of freedom density diagonal dimensional discussed dispersion distortion doping dynamics effective eg orbitals electric energy exchange interaction experimental Fermi fermion ferromagnetic finite flux tube Hamiltonian Heisenberg hole holon Hubbard band Hubbard model implies insulating cuprates ions Jahn-Teller LaMnO3 lattice Lett linear Maekawa magnetic magnon manganites matrix elements mean-field momentum Mott insulator Neel obtained operators optical orbital degeneracy orbital degree orbital order orbital wave orbiton order parameter pairing perovskite phonon Phys physical pseudospin quantum reflects rotation scattering Sect shown in Fig singlet spin and orbital spin wave spin-orbital spinon sublattice superconductivity superexchange interaction symmetry t-J model temperature term theory thermopower Tokura transition metal transition metal oxides vector wave function zero