Quantum BGK model for electron transport in semiconductors V. D. Camiola and V.Romano Dipartimento di Matematica e Informatica, Universit`a di Catania, Viale A.Doria 6, I-95125 Catania, Italy romano@dmi.unict.it. November 10, 2009 Abstract A quantum BGK transport model is proposed. The Collision term is modelled as a perturbation of the classical one. The quantum correction to the 2 order is given as a relaxation term involving the equilibrium Wigner function in a thermal bath of phonons in the case of an arbitrary energy bands . Applications to the case of the Kane dispersion relation are shown. 1 Introduction In today semiconductor technology, the miniaturization of devices is more and more progress- ing. As a consequence, the simulation of the today nanoscale semiconductor devices requires advanced transport models that take into account also quantum effects. The full quantum approaches [1] based on the Sch¨odinger-Poisson or the Von Newmann- Poisson systems, look no practical on account of the computational complexity and the theoret- ical difficult in the formulation of the interaction Hamiltonian between electrons and phonons and the nonlocal character of the variables involved (wave function or density matrix). The quantum kinetic approach obtained with the Wigner transform [1, 2] should guarantee an ac- curate description, but again its numerical integration is a daunting computational task and the problem of modeling the collision term is still open [3]. 1