Semi-classical limits in a crystal with exterior potentials and effective mass theorems

We study the semi-classical limit of the dynamics of electrons in a crystal under the influence of an external electric field. Two small parameters e and h are introduced. They are respectively related to the scaled lattice length and the scaled Planck constant. When ε/h and h go to zero, we prove that in the limit process the dynamics of electrons is described by a Vlasov equation. When h is fixed and ε tends to zero, the dynamics is governed by an effective Schrödinger equation. In both cases, the mass of electrons is replaced by a matrix, the so-called effective mass.