Wigner-Poisson and nonlocal drift-diffusion model equations for semiconductor superlattices

TL;DR

This paper introduces a quantum kinetic Wigner-Poisson model for charge transport in semiconductor superlattices, deriving nonlocal drift-diffusion equations and demonstrating their ability to reproduce experimentally observed current oscillations.

Contribution

It develops a systematic derivation of nonlocal drift-diffusion equations from a quantum kinetic model using the Chapman-Enskog method, incorporating quantum effects and energy dissipation.

Findings

Derived drift-diffusion equations include spatial averages over superlattice periods.

Numerical solutions show self-sustained current oscillations.

Results agree with experimental observations.

Abstract

A Wigner-Poisson kinetic equation describing charge transport in doped semiconductor superlattices is proposed. Electrons are supposed to occupy the lowest miniband, exchange of lateral momentum is ignored and the electron-electron interaction is treated in the Hartree approximation. There are elastic collisions with impurities and inelastic collisions with phonons, imperfections, etc. The latter are described by a modified BGK (Bhatnagar-Gross-Krook) collision model that allows for energy dissipation while yielding charge continuity. In the hyperbolic limit, nonlocal drift-diffusion equations are derived systematically from the kinetic Wigner-Poisson-BGK system by means of the Chapman-Enskog method. 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 solutions of the latter equations show self-sustained oscillations of the current through a voltage biased superlattice, in agreement with known experiments.