Semi‐classical limits in a crystal with a Coulombian self‐consistent potential: Effective mass theorems

Abstract

We study the semi‐classical limit of the dynamics of electrons in a crystal under the influence of a self‐consistent Coulombian potential. To do this, we introduce two small parameters: $\varepsilon$ , the typical length of the crystal, and $h$ , the macroscopic Planck constant. When $\varepsilon$ and $h$ tend to zero, under special conditions, the electrons distribution limit follows a Vlasov equation where the electrons mass is replaced by an effective mass.