A new approach to the theory of polaritons in semiconductors at finite temperatures: local-field effects and crystal optics approximation

Abstract

The Green function approach to the theory of the interacting photon-electron-phonon system is presented. The method of Legendre transforms is used to derive the Dyson equations for the photon and phonon Green functions, as well as the Bethe-Salpeter equation for the two-particle electron-hole Green function. Knowledge of those Green functions provide the excitation energies of the quasiparticles (polaritons) formed by coupling of excitons with photons and phonons. Examination of the poles of the two-particle electron-hole Green function leads to an equation for the Bethe-Salpeter amplitudes, as well as to a microscopic derivation of the Maxwell equations taking account of the local-field effects. A normalisation condition for the Bethe-Salpeter amplitudes is derived. In the limit of an instantaneous electron-phonon interaction the equations for determining the renormalised phonon frequencies are obtained, as well as the sum rule for them, similar to the Lyddane-Sachs-Teller relation. In order to define a dielectric function which does not depend on reciprocal lattice vectors (crystal optics approximation), the special eigenvalue problem is treated by successive integrations, first over rapidly varying photon fields and then over slowly varying photon variables.