Diffusion Approximation and Homogenization of the Semiconductor Boltzmann Equation

Abstract

This paper deals with the diffusion approximation of the Boltzmann equation for semiconductors in the presence of spatially oscillating electrostatic potential. When the oscillation period is of the same order of magnitude as the mean free path, the asymptotics leads to the driftdiffusion equation with a homogenized electrostatic potential and a diffusion matrix involving the small-scale information. The convergence is proven rigorously for Boltzmann statistics, while it is incomplete for Fermi-Dirac statistics. Introduction.The drift-diffusion equation is a standard model of particle transport in many applications such as neutron transport [17,18], plasmas [14,22], semiconductors [25,26,33,34,41,42,40,44], gas discharges [39], etc. The driftdiffusion model, very well suited for numerical simulations, is designed to describe the macroscopic behavior of the device. It consists of a mass balance equation relying on the particle and current densities,

Homogenization and Hydrodynamic Limit for Fermi‐Dirac Statistics Coupled to a Poisson Equation

“…/ and the fact that r " is bounded in L 2 . Adding the fact that the potentialˆ" H converges two-scale strongly, we obtain the expression of the third term in the right-hand side of point (3). Finally, the convergence of h " 1 can be obtained using the properties ofˇdetailed in Lemma 3.12 and the compactness (in t and x) of the self-consistent part of the potentialˆ" P .…”

Homogenized Diffusion Limit of a Vlasov–Poisson–Fokker–Planck Model

“…Progressively, different fluid models have been obtained as a hydrodynamic limit of kinetic equations for prescribed macroscopic potential [25,26]. Nowadays, more attention is paid to the variations of the potential by coupling the distribution with the Poisson equation [4,5,11,15,16,18] and recently by adding a potential with microscopic variations in the same order of the mean free path of the diffusive operator, leading to phenomena of homogenization [10,17,19,22,30]. The concept of relative entropy dissipation is useful to approximate the solution of the kinetic model to its corresponding equilibrium state [6,7,12,21,24,28,29].…”

“…Various models are approximated using ideas like Hilbert expansion method and Chapman-Enskog development. When some coefficients have both macroscopic and microscopic variations, the multi-scale Hilbert development and the multi-scale convergence [1] are well adapted to extract the homogenized effects (see for example [10]). …”

Diffusion and Homogenization Approximation for Semiconductor Boltzmann–poisson System

“…We refer to [7] for the proof of proposition 2.2 and for the convergence (ε → 0) in the case of linear Boltzmann equation. In this paper, we are dealing with a more general situation since we have a coupling with Poisson and the major difficulty is to get enough compactness to pass to the limit in the nonlinear terms.…”

“…This generalizes the study done in [21] where the same problem is treated without the oscillating electrostatic potential. The study of the diffusion and homogenization of the linear Boltzmann system was also done in [7]. In the present paper, the major difficulty is the combination of the nonlinearity with the two scale limit which requires some compactness to pass to the limit.…”