The Physics of Phonons

A.A. Maksimov

Annotation

In the first part of the course “The Physics of Phonons” the dynamics of atoms in one- and three- dimensional crystalline lattices is considered in detail. The quasiparticle conception in solids, taking phonons as an example, is discussed; creation and annihilation operators of phonons, main properties of acoustic and optical phonons dispersion laws, density of phonon states. In the second part the phonon contribution into the thermodynamics of solids is discussed, heat capacity of dielectrics at different temperatures, thermal expansion of solids, Gruneisen relation, instability of one-dimensional and two-dimensional crystals. The main third part of the course is devoted to the study of different kinetic processes in phonon system of solids. Different phonon propagation modes are considered: ballistic regime, heat conductivity of dielectrics at low and high temperatures, independent diffusion mode, second sound. Relaxation of high-frequency nonequilibrium acoustic phonons, the model of generations, quasiballistic and quasidiffusion phonon propagation regimes. Some problems of electron-phonon interaction and phonon interactions with other excitations in solids are discussed as well.

1. Crystalline state. Translation symmetry. Bravais lattice, Wigner-Seitz cell. Periodic functions in three dimensions. Main properties of a reciprocal lattice, Brillouin zone. Adiabatic approximation.
2. Dynamics of the linear chain. Harmonic approximation. Periodic boundary conditions. Geometrical interpretation in one-dimensional case. Normal coordinates. Independent harmonic oscillators, phonons. Phase and group velocity of phonons. Creation and annihilation operators of phonons. Complex one-dimensional lattice. Acoustic and optical phonons. Movement of atoms in a long-wave and short-wave limit. Dynamics of a three-dimensional crystalline lattice.
3. The general case of the three-dimensional lattice with the basis. Tensor of the force constants, its main properties. Born-Karman boundary conditions. Bloch theorem. Dynamic matrix, general properties of normal oscillations of a three-dimensional crystalline lattice. Polarization vectors. Property of long-wave acoustic and optical phonons in a three-dimensional case. Relationship between displacement operator and creation and annihilation operators of phonons. Density of phonon states. Van Hove singularities in the crystals of different dimensionalities.
4. Statistics of phonons. Planck distribution. Amplitude of the thermal oscillations of the atoms, applicability of the phonon description. Instability of one-dimensional and two-dimensional crystals. Contribution of phonons to thermodynamics. Internal energy and a heat capacity of dielectrics at high and low temperatures. Debye and Einstein models.
5. Anharmonicity. Interaction forces of atoms in a crystal. Gruneisen parameter. Equation of state of a solid. Gruneisen relation. Thermal expansion of solids. Phonon-phonon interaction and the lattice anharmonicity. Conservation law of quasi-momentum. Probabilities of three-phonon processes. Selection rules for scattering of acoustic phonons.
6. Thermal conduction in solids. Phonon kinetic equation. Normal processes and umklapp processes, their temperature dependencies. Planck distribution in moving coordinate system. Relaxation method. Cowley and Guyer-Krumhansl models. Heat conductivity of dielectrics at low and high temperatures. Contribution of scattering on sample boundaries and defects. Knudsen and Poiseuille modes of phonon propagation.
7. Nonequilibrium phonons. Establishing of equilibrium at different initial phonon distributions. Model of generations for the relaxation of nonequilibrium acoustic phonons. Estimation of equilibration times. Spatially non-uniform distribution of nonequilibrium phonons. Ballistic regime of phonon propagation, phonon focusing. Diffusion modes: heat conductivity and independent diffusion mode. Hydrodynamic regime, second sound. Quasiballistic and quasidiffusion phonon propagation regimes in the case of strongly nonequilibrium distribution.
8. Electron-phonon interaction. Matrix element of one-phonon process. Macrofield and microfield. Electron scattering kinematics by acoustic phonons. Electron-hole droplets and phonon wind. Polarons.
9. Interaction of phonons with other excitations. Infrared photon absorption. Polaritons. Mandelstam-Brillouin and Raman scattering of light. "Folded" acoustic phonons in semiconductor superlattices. Scattering of X-rays and gamma-quanta. Scattering of thermal neutrons on phonons

Literature

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