Deterministic simulation of the Boltzmann-Poisson system in GaAs-based semiconductors

In this work we present a deterministic numerical scheme solving the one-dimensional Boltzmann–Poisson system for semiconductor materials with a complicated band structure such as GaAs. The model is based on a system of two kinetic equations for the electron population in each of the relevant valleys coupled through the Poisson equation to consider the self-consistent potential. Interaction of electrons with the crystal is taken into account by treating the most important inter- and intravalley scattering mechanisms in GaAs. The numerical scheme follows the main ideas in [J. A. Carrillo, I. M. Gamba, A. Majorana, and C.-W. Shu, J. Comput. Phys., 184 (2003), pp. 498–525] already developed for Si-based devices. The newest numerical features are the intervalley scattering, the singularity of the kernels of impurities, and optical polar phonon scattering. The charge conservation of the scheme is achieved by a suitable discretization of the collision operator within the accuracy of the middle-point integration rule. A detailed study of this numerical scheme is made by showing results on the bulk material, numerically validating the results by comparison to direct simulation Monte Carlo (DSMC) results, and testing our results for diodes and Gunn oscillations.