Quantum Theory of Solids

Lectures: 64 hours

Practical studies: ---

Self-instruction:13 hours

4 course (7 term)
Academics: 
Course program: 

Geometric structure of crystals.

Translation symmetry of crystals. Basic vectors. Unit cell. Bravais lattices and lattices with a basis. Primitive, base-centered, face-centered та body-centered cells. Elements of lattice point symmetry. Crystal systems,  spatial groups. Examples of crystal structures. Reciprocal lattice. Brillouin zones. 

Nature of interatom forces and types of crystals.

General Hamiltonian of crystal. Adiabatic approximation. Interatomic interaction and types of crystals: ionic, covalent, molecular and metallic crystals.

Atom vibrations in crystals.

Equation of motion of atoms in crystal. Harmonic approximation, force matrix. Atomic oscillations of one-dimensional  lattices. Acoustic and optical vibrations. Born-Karman cyclic conditions. Spectral density of vibrations. Atomic oscillations of three-dimensional lattices. Dynamic matrix. Normal vibrations. Quantum theory of lattice vibrations. Phonons. Vibrations of lattice with defects. Local and quasilocal vibrations. Lattice heat capacity of crystals. Infrared dispersion  of permittivity of crystals. Macroscopic theory of longwave optical vibrations. Polaritons. Anharmonicity and  thermal expansion of lattice. Phonon-phonon interaction.Thermal phonon conductivity of lattice. Normal and umklapp processes. 

Band theory of electronic states.

Approximation of self-consistent field: Hartree-Fock method, single-electron states, Sleter determinant, crystal potential. Bloch theorem. General electron properties in periodic field. Nearly-free-electron approximation. Band structure. Tight-binding-electron approximation. Electron effective mass. Electron motion in external field. Method of effective mass. Methods of calculation of energetic bands: cell method, plane wave method, orthogonalized plane wave method, augmented plane wave method, pseudopotential method. Impurity electronic levels in crystals.

Equilibrium statistics of electrons in crystals.

Fermi statistics for electrons. Fermi surface. Charge carrier statistics in semiconductors. Electron heat capacity.

Kinetic effects in crystals.

Boltzmann kinetic equation. Relaxation time approximation. Calculation of relaxation time. Electron-phonon interaction. Interaction with acoustic vibrations, deformation potential method. Interaction with optical vibrations in ionic crystals. Polarons.

Interaction between electrons.

Electron gas permittivity. Lindhard formula. Plasma oscillations, plasmons. Screening of static field in metals and semiconductors. Correlation effects. Mott transition.

Optical properties of crystals.

Coefficient of light absorbtion. Kubo-Greenwood formula. Light absorbtion by charge free carriers. Interband light absorbtion. Allowed and forbidden direct transitions. Shape of absorbtion band long-wave edge. Van Hove singularities. Indirect transitions. Exciton states. Wannier-Mott excitons and Frenkel excitons. Davydov splitting of spectral bands.

Magnetic properties of crystals.

Paramagnetism of atoms and free electrons in metals and semi-conductors. Diamagnetism of atoms and  conduction electrons. Landau levels. Experimental methods of Fermi surface investigation. De Haas - van Alphen effect. Diamagnetic and cyclotron resonances. Ferromagnetism of crystals. Exchange interaction. Heisenberg Hamiltonian. Ferromagnetics, antiferromagnetics, ferrites. Spin waves. Magnons. Magnon heat capacity. Dependence of spontaneous magnetization on temperature. Magnon absopbtion of electromagnetic radiation. 

Knowledge tests: 

Exam

Literature: 

Basic

1. І.П. Пінкевич,  В.Й. Сугаков. Теорія твердого тіла. – К., 2006.

2.  J. M. Ziman.  Princeples of the theory of solids. - Cambridge University Press., 1995. 

3.  А.И. Ансельм.  Введение в теорию полупроводников.- М., 1972. 

 Optional

1. Ch. Кittel. Introduction to solid state physics. - 2005 

2  N. W. Аshcroft, N. D. Меrmin. Solid state physics. - 1976. 

3.  А.С. Давыдов. Теория твердого тела. - М., 1977.

4.  W. A.. Harrison. Solid state theory. -  1980. 

5  А. O. Аnimalu. Intermediate quantum theory of crystalline solids. - 1991.