Quantum transport equations for high electric fields

We have studied quantum-mechanical transport equations for nondegenerate electrons in semiconductors in high electric fields. Our calculations use Kadanoff and Baym’s formalism based on Green functions, treat only fields that are constant in space and time, and are restricted to weak scattering. First, we derive an approximate solution to the equation of motion of the retarded Green function in an electric field, with a careful check of its validity. This is used in deducing the conditions under which the quantum-mechanical transport equation reduces to the Boltzmann equation. We find, in agreement with some previous studies, that the electric field causes a ‘‘broadening’’ of the δ function in semiclassical transition rates, a result of the ‘‘intracollisional field effect.’’ The Boltzmann equation fails when this broadening exceeds some characteristic energy scale (usually $k_B$T), which occurs at fields of a few MV $m^{\mathrm{-}1}$ in conventional semiconductors. These results are strongly dependent on the ansatz used to reduce the Green function to a distribution function. The scattering-out term is usually much less sensitive to the electric field than the scattering-in term. We exploit this to construct an integral transport equation, valid in high electric fields, which differs from the Boltzmann equation only in having a broadened function replacing the δ function in the scattering-in rates. It should be possible to solve this equation using standard numerical techniques and gain quantitative information on the intracollisional field effect.