Quantum theory of materials

Contents
List of Figures
List of Tables
Preface
Acknowledgments
1 From Atoms to Solids
	1.1 Electronic Structure of Atoms
	1.2 Forming Bonds Between Atoms
		1.2.1 The Essence of Metallic Bonding: The Free-Electron Model
		1.2.2 The Essence of Covalent Bonding
		1.2.3 Other Types of Bonding in Solids
	1.3 The Architecture of Crystals
		1.3.1 Atoms with No Valence Electrons
		1.3.2 Atoms with s Valence Electrons
		1.3.3 Atoms with s and p Valence Electrons
		1.3.4 Atoms with s and d Valence Electrons
		1.3.5 Atoms with s, d, and f Valence Electrons
		1.3.6 Solids with Two Types of Atoms
		1.3.7 Hydrogen: A Special One-s-Valence-Electron Atom
		1.3.8 Solids with More Than Two Types of Atoms
	1.4 Bonding in Solids
	Further Reading
	Problems
2 Electrons in Crystals: Translational Periodicity
	2.1 Translational Periodicity: Bloch States
	2.2 Reciprocal Space: Brillouin Zones
		2.2.1 Nature of Wave-Vector k
		2.2.2 Brillouin Zones and Bragg Planes
		2.2.3 Periodicity in Reciprocal Space
		2.2.4 Symmetries Beyond Translational Periodicity
	2.3 The Free-Electron and Nearly Free-Electron Models
	2.4 Effective Mass, “k· p” Perturbation Theory
	2.5 The Tight-Binding Approximation
		2.5.1 Generalizations of the TBA
	2.6 General Band-Structure Methods
		2.6.1 Crystal Pseudopotentials
	2.7 Localized Wannier Functions
	2.8 Density of States
		2.8.1 Free-Electron Density of States
		2.8.2 Local Density of States
		2.8.3 Crystal DOS: Van Hove Singularities
	Further Reading
	Problems
3 Symmetries Beyond Translational Periodicity
	3.1 Time-Reversal Symmetry for Spinless Fermions
	3.2 Crystal Groups: Definitions
	3.3 Symmetries of 3D Crystals
	3.4 Symmetries of the Band Structure
	3.5 Application: Special k-Points
	3.6 Group Representations
	3.7 Application: The N-V-Center in Diamond
	Further Reading
	Problems
4 From Many Particles to the Single-Particle Picture
	4.1 The Hamiltonian of the Solid
		4.1.1 Born–Oppenheimer Approximation
	4.2 The Hydrogen Molecule
	4.3 The Hartree and Hartree–Fock Approximations
		4.3.1 The Hartree Approximation
		4.3.2 The Hartree–Fock Approximation
	4.4 Hartree–Fock Theory of Free Electrons
	4.5 Density Functional Theory
		4.5.1 Thomas–Fermi–Dirac Theory
		4.5.2 General Formulation of DFT
		4.5.3 Single-Particle Equations in DFT
		4.5.4 The Exchange–Correlation Term in DFT
		4.5.5 Time-Dependent DFT
	4.6 Quasiparticles and Collective Excitations
	4.7 Screening: The Thomas–Fermi Model
	4.8 Quasiparticle Energies: GW Approximation
	4.9 The Pseudopotential
	4.10 Energetics and Ion Dynamics
		4.10.1 The Total Energy
		4.10.2 Forces and Ion Dynamics
	Further Reading
	Problems
5 Electronic Properties of Crystals
	5.1 Band Structure of Idealized 1D Solids
		5.1.1 A Finite “1D Solid”: Benzene
		5.1.2 An Infinite “1D Solid”: Polyacetylene
	5.2 2D Solids: Graphene and Beyond
		5.2.1 Carbon Nanotubes
	5.3 3D Metallic Solids
	5.4 3D Ionic and Covalent Solids
	5.5 Doping of Ideal Crystals
		5.5.1 Envelope Function Approximation
		5.5.2 Effect of Doping in Semiconductors
		5.5.3 The p–n Junction
		5.5.4 Metal–Semiconductor Junction
	Further Reading
	Problems
6 Electronic Excitations
	6.1 Optical Excitations
	6.2 Conductivity and Dielectric Function
		6.2.1 General Formulation
		6.2.2 Drude and Lorentz Models
		6.2.3 Connection to Microscopic Features
		6.2.4 Implications for Crystals
		6.2.5 Application: Optical Properties of Metals and Semiconductors
	6.3 Excitons
		6.3.1 General Considerations
		6.3.2 Strongly Bound (Frenkel) Excitons
		6.3.3 Weakly Bound (Wannier) Excitons
	Further Reading
	Problems
7 Lattice Vibrations and Deformations
	7.1 Lattice Vibrations: Phonon Modes
	7.2 The Born Force-Constant Model
	7.3 Applications of the Force-Constant Model
	7.4 Phonons as Harmonic Oscillators
	7.5 Application: Specific Heat of Crystals
		7.5.1 The Classical Picture
		7.5.2 The Quantum-Mechanical Picture
		7.5.3 The Debye Model
		7.5.4 Thermal Expansion Coefficient
	7.6 Application: Mo¨ssbauer Effect
	7.7 Elastic Deformations of Solids
		7.7.1 Phenomenological Models of Solid Deformation
		7.7.2 Elasticity Theory: The Strain and Stress Tensors
		7.7.3 Strain Energy Density
		7.7.4 Isotropic Solid
		7.7.5 Solid with Cubic Symmetry
		7.7.6 Thin Plate Equilibrium
	7.8 Application: Phonons of Graphene
	Further Reading
	Problems
8 Phonon Interactions
	8.1 Phonon Scattering Processes
		8.1.1 Scattering Formalism
	8.2 Application: The Debye–Waller Factor
	8.3 Phonon–Photon Interactions
		8.3.1 Infrared Absorption
		8.3.2 Raman Scattering
	8.4 Phonon–Electron Interactions: Superconductivity
		8.4.1 BCS Theory of Superconductivity
		8.4.2 The McMillan Formula for Tc
		8.4.3 High-Temperature Superconductors
	Further Reading
	Problems
9 Dynamics and Topological Constraints
	9.1 Electrons in External Electromagnetic Fields
		9.1.1 Classical Hall Effect
		9.1.2 Landau Levels
		9.1.3 Quantum Hall Effect
		9.1.4 de Haas–van Alphen Effect
	9.2 Dynamics of Crystal Electrons: Single-Band Picture
	9.3 Time-Reversal Invariance
		9.3.1 Kramers Degeneracy
	9.4 Berry’s Phase
		9.4.1 General Formulation
		9.4.2 Berry’s Phase for Electrons in Crystals
	9.5 Applications of Berry’s Phase
		9.5.1 Aharonov–Bohm Effect
		9.5.2 Polarization of Crystals
		9.5.3 Crystal Electrons in Uniform Electric Field
	9.6 Chern Numbers
	9.7 Broken Symmetry and Edge States
		9.7.1 Broken Symmetry in Honeycomb Lattice
		9.7.2 Edge States of Honeycomb Lattice
	9.8 Topological Constraints
	Further Reading
	Problems
10 Magnetic Behavior of Solids
	10.1 Overview of Magnetic Behavior of Insulators
	10.2 Overview of Magnetic Behavior of Metals
		10.2.1 Free Fermions in Magnetic Field: Pauli Paramagnetism
		10.2.2 Magnetization in Hartree–Fock Free-Electron Model
		10.2.3 Magnetization of Band Electrons
	10.3 Classical Spins: Simple Models on a Lattice
		10.3.1 Non-interacting Spins on a Lattice: Negative Temperature
		10.3.2 Interacting Spins on a Lattice: Ising Model
	10.4 Quantum Spins: Heisenberg Model
		10.4.1 Motivation of the Heisenberg Model
		10.4.2 Ground State of Heisenberg Ferromagnet
		10.4.3 Spin Waves in Heisenberg Ferromagnet
		10.4.4 Heisenberg Antiferromagnetic Spin Model
	10.5 Magnetic Domains
	Further Reading
	Problems
Appendices
Appendix A Mathematical Tools
Appendix B Classical Electrodynamics
Appendix C Quantum Mechanics
Appendix D Thermodynamics and Statistical Mechanics
Index