Energy‐transport model for a nondegenerate semiconductor: Convergence of the Hilbert expansion in the linearized case

Abstract

This paper is concerned with the derivation of an energy‐transport model from the Boltzmann equation for a nondegenerate semiconductor with a parabolic band structure. Electron–electron collisions and elastic collisions (i.e., impurity scattering and the “elastic part” of phonon collisions) are retained as leading order terms. Then, a Hilbert expansion leads to an energy‐transport model, and a Chapman–Enskog expansion leads to the same energy‐transport model if the electric field vanishes. For the case of the Boltzmann equation linearized about a global equilibrium, the convergence of the Hilbert expansion is shown when the electric field vanishes.