Diffusion Approximation of a Scattering Matrix Model of a Semiconductor Superlattice

In this paper we derive a diffusion equation for electron transport in a superlattice. The starting model is a quantum scattering matrix model which relates the phase space density of each superlattice cell to that of the neighboring cells. Then, in the limit of a large number of cells, a diffusion equation for the particle number density in the position-energy space is obtained, which is of the "SHE" (spherical harmonics expansion) type. The diffusion constant retains the memory of the quantum scattering characteristics of the superlattice elementary cell (like, e.g., transmission resonances). An example is treated, for which the diffusion constants are analytically computed.