The Physics of Phonons

Cover
Half Title
Title Page
Copyright Page
Dedication
Contents
Preface (Second Edition)
Acknowledgments (Second Edition)
Preface (First Edition)
Acknowledgments (First Edition)
Chapter 1: Elements of Crystal Symmetry
1.1. Direct lattice
1.2. Reciprocal lattice
1.3. The Brillouin zone
1.4. Crystal structure
1.5. Point groups
1.6. Space groups
1.7. Symmetry of the Brillouin zone
1.8. Jones zone
1.9. Surface Brillouin zone
1.10. Matrix representations of point groups
1.11. Effect of space group operations on plane waves
Chapter 2: Lattice Dynamics in the Harmonic Approximation – Semiclassical Treatment
2.1. Introduction
2.1.1. The phonon as an excitation in a crystal
2.1.2. Statement of the problem of lattice dynamics
2.1.3. Phonon statistics
2.2. Lattice dynamics of a linear chain
2.2.1. Monatomic linear chain
2.2.2. Diatomic linear chain
2.3. Lattice dynamics of monatomic two-dimensional mesh
2.3.1. Monatomic square mesh
2.3.2. Monatomic hexagonal close packed mesh
2.4. Lattice dynamics of three-dimensional crystals – phenomenological models
2.4.1. Models of interatomic forces
2.4.1.1. The Born model
2.4.1.2. The Born–von Kármán model
2.4.1.3. The valence force field model
2.4.1.4. Empirical many-body interatomic potentials
2.4.2. The rigid ion model
2.4.3. Dipole approximation models
2.4.3.1. The shell model
2.4.3.2. The bond charge model
2.4.4. Long-wavelength optical phonons in ionic crystals
2.4.5. Soft phonon modes
2.5. Density of normal modes
2.5.1. General expression for a three-dimensional crystal
2.5.2. Two-dimensional case
2.5.3. One-dimensional case
2.6. Numerical calculation of density of states
2.6.1. Root sampling method
2.6.2. Linear analytical approaches
2.6.2.1. The method of Gilat and Raubenheimer
2.6.2.2. The tetrahedron method
2.6.3. The isotropic continuum approximation
2.6.4. The Debye approximation
2.7. Lattice specific heat
2.7.1. Classical theory of Cv
2.7.2. Quantum theory of Cv
2.7.2.1. Einstein’s model
2.7.2.2. Debye’s model
2.8. Elastic waves in cubic crystals
Chapter 3: Lattice Dynamics in the Harmonic Approximation – Ab initio Treatment
3.1. Introduction
3.2. Total energy and forces
3.2.1. General consideration
3.2.2. Density functional theory
3.2.3. Momentum space formulation of electronic band structure
3.2.3.1. Kohn–Sham equations using plane wave pseudopotential method
3.2.3.2. Special k-points method for Brillouin zone summation
3.2.3.3. Self-consistent solutions of Kohn–Sham equations
3.2.3.4. Numerical solution of Kohn–Sham equation
3.2.4. Momentum space formulation of total energy
3.2.5. Hellmann–Feynman theorem for force calculation
3.3. Harmonic force constants and phonon eigensolutions
3.3.1. Direct methods
3.3.1.1. Frozen phonon method
3.3.1.2. Finite displacement method for zone-centre phonons
3.3.1.3. Planar force constant method
3.3.1.4. Real-space method for an arbitrary q in BZ
3.3.2. Linear response methods
3.3.2.1. Density functional perturbation theory (DFPT)
3.3.2.2. The dielectric matrix method
3.3.2.3. A simple perturbative method
Chapter 4: Anharmonicity
4.1. Introduction
4.2. Hamiltonian of a general three-dimensional crystal
4.3. Effect of anharmonicity on phonon states
4.4. Effects of the selection rules on three-phonon processes
4.5. Hamiltonian of an anharmonic elastic continuum
4.6. Evaluation of three-phonon scattering strengths
4.7. The quasi-harmonic approximation and Grüneisen’s constant
4.7.1. The equation of state in the quasi-harmonic approximation
4.7.2. Grüneisen’s constant and thermal expansion in the quasi-harmonic approximation
4.7.3. Grüneisen’s constant and phase transition
4.8. Analytic expressions for cubic and quartic anharmonic potential terms using a semi-ab initio scheme
4.9. Ab initio calculations of cubic and quartic anharmonic potential terms
4.9.1. Finite difference method
4.9.2. Third-order DFPT
Chapter 5: Theory of Lattice Thermal Conductivity
5.1. Introduction
5.1.1. The phonon Boltzmann equation
5.1.2. Expression for lattice thermal conductivity
5.2. Relaxation-time Methods
5.2.1. Single-mode relaxation-time method
5.2.2. Klemens’ model
5.2.3. Callaway’s model
5.2.4. Relaxing assumptions in original version of Callaway’s model
5.2.5. Srivastava’s model
5.2.6. Kinetic theory expression for K
5.2.7. An iterative approach
5.3. Variational methods
5.3.1. Structure of phonon collision operator and existence of a variational problem for the lattice thermal conductivity
5.3.2. Properties of the anharmonic phonon collision operator
5.3.3. Complementary variational principles using canonical Euler–Lagrange equations
5.3.4. Complementary variational principles using Schwarz’s inequality
5.3.5. Sequences of bounds and their convergence
5.3.5.1. Sequence of lower bounds
5.3.5.2. Sequence of upper bounds
5.3.6. Improvement of variational bounds by scaling and Ritz procedures
5.3.7. Thermodynamic interpretation of the variational principle
5.4. Green–Kubo linear-response theory
5.4.1. Evaluation of the correlation function G (t) by the Zwanzig–Mori projection operator method
5.4.1.1. Derivation of an integro-differential equation for G (t)
5.4.1.2. Evaluation of f (t) and g
5.4.1.3. Solution of Zwanzig’s equation in the weak-coupling limit
5.4.1.4. Conductivity expression
5.4.2. Evaluation of the correlation function G (t) by the double-time Green’s function method
5.5. Second sound and Poiseuille flow of phonons
Chapter 6: Phonon Scattering in Solids
6.1. Boundary scattering
6.2. Scattering by static imperfections
6.2.1. Mass difference scattering
6.2.2. Scattering by dislocations, stacking faults, and grain boundaries
6.2.2.1. Dislocations
6.2.2.2. Stacking faults
6.2.2.3. Grain boundaries
6.3. Phonon scattering in alloys
6.4. Anharmonic scattering
6.4.1. Three-phonon processes
6.4.1.1. Expressions for tqs and Pss' qq' from time-dependent perturbation theory
6.4.1.2. Expression for tqs from the projection operator method
6.4.1.3. Expressions for tqs and Δqs from the double-time Green’s function method
6.4.1.4. Acoustic–acoustic phonon interaction in the Debye model
6.4.1.5. Acoustic–optical phonon interaction
6.4.2. Four-phonon processes
6.5. Phonon–electron scattering in doped semiconductors
6.5.1. Phonon–electron scattering in moderately and heavily doped semiconductors
6.5.2. Phonon–electron scattering in lightly doped semiconductors
6.5.3. Phonon–hole scattering in lightly doped semiconductors
6.6. Phonon Scattering due to Magnetic Impurities in Semiconductors
6.7. Phonon scattering from tunnelling states of impurities
6.8. Phonon-spin interaction
6.9. Phonon–photon interaction
6.9.1. Infrared absorption
6.9.2. Raman scattering
6.10. Ab-initio evaluation of phonon-phonon interaction
6.10.1. Ab initio treatment of three-phonon processes
6.10.2. Ab initio treatment of four-phonon processes
6.10.3. Semi-Ab initio treatment of three-phonon processes
6.10.4. Semi-Ab initio treatment of four-phonon processes
6.11. Ab initio treatment of electron-phonon interaction
Chapter 7: Phonon Relaxation and Thermal Conductivity in Bulk Solids
7.1. Relaxation rate due to isotopic mass defects
7.2. Spectrum of three-phonon relaxation times – ab initio results
7.3. Anharmonic decay of phonons
7.3.1. High-frequency acoustic phonons
7.3.2. Long wavelength optical phonons
7.4. Lattice thermal conductivity of undoped semiconductors and insulators
7.4.1. Relaxation time results
7.4.1.1. Simple phenomenological approach
7.4.1.2. Detailed phenomenological approach
7.4.1.3. Non-phenonomenological isotropic continuum approach
7.4.1.4. Semi-ab initio approach
7.4.1.5. Fully-ab initio approach
7.4.1.6. High-temperature results: role of thermal expansion and four-phonon processes
7.4.2. Results from variational principles
7.4.2.1. Two-sided, or complementary, variational results
7.4.2.2. Improvement of the lowest variational bound by the Rayleigh and Ritz procedure
7.4.2.3. Low-temperature results
7.4.2.4. Contributions from different scattering processes
7.5. Non-Metallic crystals with high thermal conductivity
7.6. Thermal conductivoity of complex crystals
7.7. Low-temperature thermal conductivity of doped semiconductors
7.7.1. Heavily doped semiconductors
7.7.2. Lightly doped semiconductors
7.7.3. Semiconductors with magnetic impurities
7.7.4. Thermal conductivity of doped alkali halides
7.8. Thermal conductivity of different forms of diamond
Chapter 8: Phonons on Crystal Surfaces and Layered Crystals
8.1. Introduction
8.2. Continuum theory
8.3. Lattice dynamical theory
8.3.1. Monatomic linear chain
8.3.2. Diatomic linear chain
8.3.3. A crystal with a surface
8.4. Phonons on semiconductor surfaces
8.4.1. Unreconstructed III-V(110) surfaces
8.4.2. Reconstructed silicon surfaces
8.4.2.1. Si(001)-(2x1)
8.4.2.2. Si(111)-(2x1)
8.4.2.3. Si(111)-(7x7)
8.4.3. Molecular adsorption on surfaces
8.5. Phonons on monolayer transition metal dichalcogenides
8.6. Surface specific heat
8.7. Attenuation of surface phonons
8.7.1. Interaction with impurities and imperfections
8.7.2. Interaction with bulk phonons
Chapter 9: Phonons and Thermal Transport in Nanocomposites
9.1. Introduction
9.2. Continuum theory of phonons in planar superlattices
9.3. One-dimensional approach for lattice dynamical theory of phonons in planar superlattices
9.4. Raman scattering studies of planar superlattice phonons
9.4.0.1. Folded LA phonons
9.4.0.2. Confined LO phonons
9.5. Three-dimensional treatment of phonon dispersion relations in periodic nanocomposite structures
9.6. Thermal conductivity of periodic nanocomposite structures
9.6.1. Introduction
9.6.2. Direct calculation for ultrathin nanocomposites
9.6.3. Effective medium theory for nano- and micro-composites
9.6.4. Results for composites with unit cell size in the range low-nm to microns
9.7. Dimensionality dependence of thermal conductivity
9.8. Nanostructuring for enhanced thermoelectric properties
9.9. Breakdown of Fourier’s law
9.10. Phonon interaction with a two-dimensional electron gas
9.10.1. Acoustic phonon interaction with 2DEG
9.10.1.1. Phonon emission
9.10.1.2. Phonon absorption
9.10.1.3. Measurements and theoretical results
9.10.2. Emission of acoustic phonons by a 2DEG in a quantising magnetic field
9.10.3. Magnetophonon effect in 2D systems
Chapter 10: Topological Nanophononics and Chiralphononics
10.1. Introduction
10.2. Topological solids
10.3. Topological phonons
10.3.1. Topological phonons in 1D systems
10.3.2. Topological phonons in 2D systems
10.3.2.1. Lattice dynamics of graphene
10.3.2.2. Topological phonons on graphene
10.3.2.3. Topological phonons on graphene-like structure with broken inversion symmetry
10.3.3. Topological phonons in 3D systems
10.3.3.1. Dirac phonons in body-centred Si
10.3.3.2. Weyl phonons in zinc-blende CdTe
10.3.3.3. Weyl phonons in wurtzite ZnSe
10.3.3.4. Experimental investigation of DPs and WPs
10.4. Phononic systems
10.5. Nanophononic topology
10.6. Phonon chirality
10.6.1. Phonon angular momentum in crystals
10.6.2. Phonon pseudo angular momentum in monolayer thin crystals
10.6.3. Phonon chirality in monolayer thin crystals
10.6.4. Topologically chiral phonons
Chapter 11: Phonons and Thermal Transport in Impure and Mixed Crystals
11.1. Introduction
11.2. Localised vibrational modes in semiconductors
11.2.1. Infrared absorption measurements
11.2.2. Theory of localised modes
11.3. Experimental studies of long-wavelength optical phonons in mixed crystals
11.3.1. Infrared reflectance measurements
11.3.2. Raman scattering measurements
11.4. Theoretical models for long-wavelength optical phonons in mixed crystals
11.4.1. Linear chain model
11.4.2. Random element isodisplacement (REI) model and its modifications
11.4.3. Application of the MREI model to mixed crystals
11.5. Phonon conductivity of mixed crystals
11.5.1. Thermal resistivity of undoped single crystal alloys
11.5.2. Low-temperature thermal conductivity of doped semiconductor alloys
11.5.3. Thermal conductivity of sintered semiconductor alloys
11.5.4. Role of sintering additives in increasing thermal conductivity of ceramics
Chapter 12: Phonons in Quasi-Crystalline and Amorphous Solids
12.1. Introduction
12.2. Phonons in quasi-crystals
12.2.1. Structure of quasi-crystals
12.2.2. Vibrational properties of the Fibonacci monatomic linear chain
12.2.3. Fibonacci superlattices
12.3. Structure and vibrational excitations of amorphous solids
12.3.1. Structure
12.3.2. Phonons
12.3.3. Tunnelling states
12.3.4. Fractal structure and fractons
12.4. Vibrational properties of amorphous solids
12.4.1. Numerical methods
12.4.2. Analytical methods and other ideas
12.4.3. Comparison with crystalline density of states
12.5. Low-temperature properties of amorphous solids
12.5.1. Specific heat
12.5.2. Thermal conductivity
12.5.2.1. Below the plateau temperature (< 1 K, or T=/OD ≤ 10-2)
12.5.2.2. The plateau region (10-2 ≤ T/OD ≤ 10-1)
12.5.2.3. Above the plateau temperature (T/OD ≥ 10-1)
12.5.3. Acoustic and dielectric properties
12.5.4. An interactive defect model
Chapter 13: Phonon Spectroscopy
13.1. Introduction
13.2. Heat pulse techniques
13.3. Superconducting tunnel junction technique
13.4. Optical techniques
13.5. Phonons from Landau levels in 2DEG
13.6. Phonon focusing and imaging
13.7. Frequency crossing phonon spectroscopy
13.8. Phonon echoes
13.9. Time- and frequency-resolved phonon spectroscopies
13.9.1. Time-resolved coherent anti-Stokes Raman scattering
13.9.2. Ultrafast pump-probe techniques
Chapter 14: Phonons in Liquid Helium
14.1. Introduction
14.2. Dispersion curve and elementary excitations
14.3. Specific heat
14.4. Interactions between the excitations
14.4.1. Anharmonic interaction below about 600 mK
14.4.2. Anharmonic interaction above about 600 mK
14.4.2.1. Four-phonon processes
14.4.2.2. Roton–roton scattering
14.4.2.3. Phonon–roton scattering
14.4.3. Second sound
14.5. Kapitza resistance
14.6. Quantum evaporation
Appendix A: Density Functional Formalism
Appendix B: The Pseudopotential Method
Appendix C: The 2n+1 Theorem of Perturbation Theory
Appendix D: Derivation of Tensor Expression for Thermal Conductivity
Appendix E: Evaluation of Integrals in Section 6.4.1.4
Appendix F: Negative-definiteness of the Phonon Off-diagonal Operator Λ
Appendix G: Geometry-dependent Depolarization Tensor
References
Index