Multiphase semiclassical approximation of an electron in a one-dimensional crystalline lattice ⋆ II. Impurities and Bloch oscillations

We present a computational approach for the WKB approximation of the wave function of an electron moving in a periodic one-dimensional crystal lattice by means of a nonstrictly hyperbolic system whose flux function stems from the Bloch spectrum of the Schrödinger operator. This second part focuses on the handling of the source terms which originate from slowly-varying exterior potentials. Physically relevant examples are the occurrence of Bloch oscillations in case this exterior potential is linear and the harmonic Coulomb term resulting from the inclusion of a “donor impurity” inside an otherwise perfectly homogeneous lattice.