A Scattering Matrix Model of Semiconductor Superlattices in Multidimensional Wave-Vector Space and Its Diffusion Limit

We first establish a quantum microscopic scattering matrix model in multidimensional wave-vector space, which relates the phase space density of each superlattice cell with that of the neighbouring cells. Then, in the limit of a large number of cells, a SHE (Spherical Harmonics Expansion)-type model of diffusion equations for the particle number density in the position-energy space is obtained. The crucial features of diffusion constants on retaining the memory of the quantum scattering characteristics of the superlattice elementary cell (like e.g. transmission resonances) are shown in order. Two examples are treated with the analytically computation of the diffusion constants.