Redheffer Products and Numerical Approximation of Currents in One-Dimensional Semiconductor Kinetic Models

Abstract

When numerically simulating a kinetic model of an $n^+nn^+$ semiconductor device, obtaining a constant macroscopic current at steady state is still a challenging task. Part of the difficulty comes from the multiscale, discontinuous nature of both $p|n$ junctions, which create spikes in the electric field and enclose a channel where corresponding depletion layers glue together. The kinetic formalism furnishes a model holding inside the whole domain, but at the price of strongly varying parameters. By concentrating both the electric acceleration and the linear collision terms at each interface of a Cartesian computational grid, we can treat them by means of a Godunov scheme involving two types of scattering matrices. Combining both these mechanisms into a global S-matrix can be achieved thanks to “Redheffer's star-product.” Assuming that the resulting S-matrix is stochastic permits us to prove maximum principles under a mild CFL restriction. Numerical illustrations of collisional Landau damping and various $n^+nn^+$ devices are provided on coarse grids.