Quantum Solid-State Physics

1. Introduction. General Properties of the Solid State of Matter.- 1.1 General Thermodynamic Description of the Solid State.- 1.2 Crystal Structure of Solids.- 1.3 Reciprocal Lattice.- 1.4 Examples of Simple Crystal Structures.- 1.5 Experimental Techniques for Determining the Periodic Atomic Structure of Solids.- 1.6 Qualitative Concepts of the Electronic and Nuclear Crystal Structure.- 1.7 Fundamental Concepts of the Chemical Bonding in Solids.- 1.7.1 Interaction Between Atoms (Ions) with Filled Electron Shells.- 1.7.2 Molecular Orbitals.- 1.7.3 The Heitler-London Method.- 1.7.4 Covalent Bond.- 1.7.5 Electrostatic Bonding Energy of Ionic Crystals.- 1.8 Types of Crystalline Solids.- 1.8.1 Ionic Crystals.- 1.8.2 Covalent Crystals and Semiconductors.- 1.8.3 Metals, Their Alloys, and Compounds.- 1.8.4 Molecular Crystals.- 1.8.5 Hydrogen-Bonded Crystals.- 1.8.6 Quasi-One-Dimensional and Quasi-Two-Dimensional Crystals.- 1.8.7 Quantum Crystals.- 1.9 Formulation of the General Quantum-Mechanical Problem of the Crystal.- 1.10 Properties of Disordered Condensed Systems.- 1.10.1 General Remarks.- 1.10.2 Metallic Glasses (Example of Amorphous Solids).- 2. Dynamic Properties of the Crystal Lattice.- 2.1 The Dynamics of the Ionic Lattice.- 2.1.1 A Linear Monatomic Array.- 2.1.2 A Linear Diatomic Array.- 2.1.3 The Three-Dimensional Crystal Case.- 2.1.4 Quantization of Ionic-Lattice Vibrations.- 2.2 The Specific-Heat Capacity of the Lattice.- 2.3 Allowance for Anharmonic Terms.- 2.3.1 Thermal Expansion of Crystals.- 2.3.2 Heat-Capacity Term Linear in Temperature.- 2.3.3 Thermal Conductivity of an Ionic Lattice.- 2.4 Localization of Phonons on Point Defects.- 2.5 Heat Capacity of Glasses at Low Temperatures.- 2.6 High-Frequency Permittivity of Ionic Crystals.- 2.7 Lattice Scattering and the Mössbauer Effect.- 2.7.1 Scattering Probability and the Correlation Function.- 2.7.2 Some Properties of the Phonon Operators and of the Averages Containing Them.- 2.7.3 Calculating the Dynamic Form Factor in the Harmonic Approximation.- 2.7.4 Elastic Scattering.- 2.7.5 Inelastic Scattering.- 2.7.6 The Mössbauer Effect.- 2.8 Conclusion.- 3. Simple Metals: The Free Electron-Gas Model.- 3.1 Types of Metals.- 3.2 Physical Properties of the Metallic State. Conduction Electrons.- 3.3 Classical Conduction-Electron Theory (Drude-Lorentz Theory).- 3.4 Itinerant Electron Theory According to Frenkel.- 3.5 Application of Fermi-Dirac Quantum Statistics to the Conduction-Electron Gas.- 3.5.1 The Case of T = 0 K.- 3.5.2 The Low-Temperature Case (T > 0 K, but T ??el).- 3.5.3 Atomic Volume, Compressibility, and Strength of Metals.- 3.5.4 Paramagnetism of a Degenerate Electron Gas.- 3.5.5 Diamagnetism of a Degenerate Electron Gas According to Landau.- 3.5.6 Oscillatory Effects in the Fermi Gas.- 3.5.7 Thermionic Emission (the Richardson Effect).- 3.6 Transport Phenomena.- 3.6.1 The Boltzmann Kinetic Equation.- 3.6.2 Electrical Conductivity.- 3.6.3 Thermal Conductivity and the Wiedemann-Franz Relation.- 3.6.4 Thermoelectric Phenomena.- 3.6.5 Galvanomagnetic Phenomena.- 3.7 High-Frequency Properties.- 3.7.1 Basic Equations.- 3.7.2 Skin Effect.- 3.7.3 Cyclotron Resonance.- 3.7.4 Electromagnetic Waves in Metals.- 3.8 Conclusions.- 4. Band Theory.- 4.1 Preliminary Observations and the One-Dimensional Model.- 4.1.1 Electron Waves in a Crystal.- 4.1.2 The Array of Rectangular Potential Barriers.- 4.1.3 Linear Atomic Array.- 4.1.4 Rigorous Theory of Electron Motion in a One-Dimensional Array.- 4.2 General Theory of the Electron Motion in a Three-Dimensional Crystal.- 4.2.1 Bloch’s Theorem.- 4.2.2 Brillouin Zones.- 4.2.3 Electron Energy Spectrum.- 4.2.4 The Properties of Constant Energy Surfaces.- 4.2.5 Density of Electron States in Energy Bands. Topological Electronic Transitions.- 4.3 Nearly-Free-Electron Approximation.- 4.3.1 Statement of the Problem.- 4.3.2 Empty-Lattice Model.- 4.3.3 Allowance for a Weak Periodic Field.- 4.4 Effect of an Electric Field on Electronic States.- 4.4.1 Acceleration and Effective Electron Mass.- 4.4.2 Zener Breakdown.- 4.4.3 Quantum Theory of the Electric Inertia Effect.- 4.5 The Metal-Semiconductor Criterion.- 4.5.1 The Metal-Nonmetal Criterion in Band Theory.- 4.5.2 The Peierls Transition.- 4.5.3 The Mott Transition.- 4.5.4 Disordered Systems.- 4.6 Computing the Electron Energy Spectrum of Crystals.- 4.6.1 Self-Consistent Field Approximation.- 4.6.2 Solving the Schrödinger Equation. Formulation of the Problem and the Cellular Method.- 4.6.3 The LCAO Method and Tight-Binding Approximation.- 4.6.4 The Orthogonalized Plane Waves (OPW) Method. Pseudopotential.- 4.6.5 The Augmented Plane Waves (APW) Method.- 4.6.6 K?P Perturbation Theory.- 4.6.7 Fermi Surfaces in Real Metals.- 4.7 Band Electrons in a Magnetic Field.- 4.7.1 The Effective Hamiltonian.- 4.7.2 Classical Paths.- 4.7.3 Quasi Classical Energy Levels. Oscillatory Effects.- 4.8 Impurity States.- 4.8.1 A Simple Model.- 4.8.2 Green’s Functions and the Density of States.- 4.8.3 Priedel Oscillations.- 4.9 The Electronic Structure of Disordered Systems.- 4.9.1 The Average Green’s Function in the Diagonal Disorder Model.- 4.9.2 Approximate Methods of Computing the Average Green’s Function in the Binary Alloy Model.- 4.9.3 Anderson Localization.- 4.10 Conclusion. The Role of Many-Particle Effects.- 5. Many-Particle Effects.- 5.1 Plasma Phenomena. Screening.- 5.1.1 A Discussion of the Model.- 5.1.2 The Equation for a Self-Consistent Plasma Potential.- 5.1.3 Static Screening.- 5.1.4 Plasmons.- 5.1.5 Phonons in the Plasma Model.- 5.1.6 Fluctuation-Dissipation Theorem.- 5.2 The Fermi-Liquid Theory.- 5.2.1 Major Postulates of the Landau Theory.- 5.2.2 Thermodynamic Properties.- 5.2.3 Kinetic Equation for Quasiparticles.- 5.3 Electron-Phonon Interaction.- 5.3.1 Formulation of the Problem.- 5.3.2 Conditions for the Applicability of the Adiabatic Approximation.- 5.3.3 Temperature Dependence of the Electrical Conductivity in Metals.- 5.3.4 Polarons.- 5.3.5 The Cooper Phenomenon.- 5.4 Superconductivity.- 5.5 Excitons.- 5.6 Transition Metals and Their Compounds.- 5.6.1 Properties of d and ƒ States.- 5.6.2 The Heisenberg Model.- 5.6.3 d Metals.- 5.6.4 Magnetism in the 4 ƒ Metals.- 5.7 Anderson’s Orthogonality Catastrophe.- 5.8 Conclusion.- Addenda (Recent Developments).- References.