Macroscopic Models for Almost Elastic Nonlinear Electron-Phonon Interaction in Semiconductors

A semiclassical transport equation for electrons in a semiconductor crystal is considered. It includes a nonlinear scattering operator modelling electron-phonon interaction. A formal macroscopic limit under the assumption of a small phonon energy is carried out, leading to the so called spherical harmonics expansion (SHE-) model for an energy-dependent distribution function with a scattering operator of Fokker-Planck type. By a further macroscopic limit for electric fields balancing the inelastic scattering operator, a nonlinear, hyperbolic conservation law for the macroscopic density is derived. The flux depends nonlinearly on the electric field. The nonlinear structure of the conservation law and the behaviour for large fields are studied.