On the interaction between cubic anharmonicity and a second order electric moment in the optical absorption of crystals

Abstract

Abstract The infrared optical absorption coefficient of a crystal can be expressed as the Fourier transform of the crystal dipole moment correlation function. In this paper we study the contribution to the absorption coefficient from the second-order terms in the expansion of the dipole moment in powers of the atomic displacements, in the presence of cubic anharmonic terms in the crystal potential energy. The Fourier transform of the two-particle Green's function which appears in this discussion is expressed as the continuation to the real frequency axis of a vertex function defined only at discrete points in the complex frequency plane. The determination of the vertex function is carried out through the use of the temperature-dependent propagators of Luttinger and Ward. It is the solution of an integral equation in two complex variables. The required continuation is carried out by means of a double spectral representation of the vertex function. The absorption coefficient is given in terms of the spectral density, which is the solution of a real integral equation. The methods developed in this paper can also be used in the calculation of transport coefficients. In some cases the real integral equation becomes a Boltzmann equation.