Abstract
We present a new asymptotic-preserving scheme for the semiconductor Boltzmann equation with two-scale collisions --- a leading-order elastic collision together with a lower-order interparticle collision. When the mean free path is small, numerically solving this equation is prohibitively expensive due to the stiff collision terms. Furthermore, since the equilibrium solution is a (zero-momentum) Fermi-Dirac distribution resulting from joint action of both collisions, the simple BGK penalization designed for the one-scale collision [10] cannot capture the correct energy-transport limit. This problem was addressed in [13], where a thresholded BGK penalization was introduced. Here we propose an alternative based on a splitting approach. It has the advantage of treating the collisions at different scales separately, hence is free of choosing threshold and easier to implement. Formal asymptotic analysis and numerical results validate the efficiency and accuracy of the proposed scheme.
Highlights
The semiconductor Boltzmann equation describes the transport of charge carriers in semiconductor devices [20, 6, 18]
We propose an alternative based on a splitting approach
We study the asymptotic behavior of the numerical solution to the scheme (3.4–3.5)
Summary
The semiconductor Boltzmann equation describes the transport of charge carriers (electrons or holes) in semiconductor devices [20, 6, 18]. Since the delta function is involved in the collision operator, it is more realistic than many commonly used kinetic models that only deal with smoothed kernels [15, 8, 17, 9] It includes the electron-electron interaction which is usually neglected under a low-density assumption [4, 5] (not true in our case). While individual solvers for both regimes are available (or possible), it is desirable to have a unified scheme working for different α, as in practice α may not be uniformly small or large in the entire domain of interest To make it precise, we want a numerical scheme that is consistent to the kinetic equation (1.1), and when α approaches zero it automatically becomes a macroscopic solver for the limiting ET system, i.e., it is asymptotic-preserving (AP) [14].
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