Green's function approach to electrical conductivity and phonon drag

Abstract

The Ku bo expression for the frequency dependent electrical conductivity σ(ω) of a system of electrons interacting with phonons by the Fröhlich Hamiltonian, is studied by many body green's function theory. The contributions of lowest order in the electron-phonon coupling to σ(ω) are obtained by deriving equations of motion for Green's functions and applying decoupling techniques. They have to be calculated from the solution of a set of two coupled kinetic equations for both the electron and the phonon distribution functions in which the so called phonon drag appears in a natural way. In the particular case of a static electrical field ( ω = 0), the set of equations reduces to two linearized Boltzmann equations. As a consequence of the phonon drag terms (being of lowest order in the coupling), this set has no finite solution, “Umklapprozesse” being neglected. The static electrical conductivity tensor then becomes infinite. Furthermore it is stressed that the phonon drag occurs also in the case of the low electron density limit. If one considers Umklapprozesse, supplementary terms appear in the kinetic equations, which prevent the electric conductivity to become infinite for zero frequency.