Energy-transport models for charge carriers involving impact ionization in semiconductors

Energy-transport models taking into account impact ionization are derived from a kinetic framework by formal asymptotic methods. The derivation is based on a system of Boltzmann transport equations governing the distribution functions of conduction electrons and holes in a semiconductor. The charge carriers are assumed to obey a degenerate gas statistics. The energy exchanged during collisions between a charge carrier and a phonon is supposed to be weak. Ionization collisions and elastic phonon collisions are retained at leading order to carry out a rigorous diiusion limit of the system of Boltzmann equations. The resulting set of diiusion equations are balance laws for the total charge and for a shifted total kinetic energy, rather than for the densities and for the temperature of charge carriers. The diiusion limit is also investigated by a detour through an intermediate Spherical Harmonics Expansion model. Simpler expressions for the diiusion matrix are obtained leading to explicit formulas in some limiting cases.