  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.
