Numerical methods for a quantum drift-diffusion equation in semiconductor physics

We present the numerical methods and simulations used to solve a charge transport problem in semiconductor physics. The problem is described by a Wigner-Poisson kinetic system we have recently proposed and whose results are in good agreement with known experiments. In this model we consider doped semiconductor superlattices in which electrons are supposed to occupy the lowest miniband, exchange of lateral momentum is ignored, the electron-electron interaction is treated in the Hartree approximation and elastic and inelastic collisions are taken into account. Nonlocal drift-diffusion equations derived systematically elsewhere from the hyperbolic limit of a kinetic Wigner-Poisson are solved. The nonlocality of the original quantum kinetic model equations implies that the derived drift-diffusion equations contain spatial averages over one or more superlattice periods. Numerical methods are based upon prior knowledge on physical properties of the phenomenon and are shown to be effective enough in validating our formulation. Numerical solutions of the equations show self-sustained oscillations of the current through a voltage biased superlattice, in good agreement with known experiments.