Semi-classical limit in a semiconductor superlattice

In this paper, we perform a mathematical study of a semiconductor superlattice. Since the thickness of the layers is very small, quantization plays an important role. The modelling is therefore given by a Schrodinger equation with a periodic potential. The scaled lattice thickness is denoted by a small parameter epsilon which is of the same order of magnitude as the Planck constant. When this parameter tends to zero, i.e., the semi-classical limit, we obtain classical transport of the charge carriers described by a Vlasov equation.