On a hierarchy of macroscopic models for semiconductors

Abstract

This paper shows that various models of electron transport in semiconductors that have been previously proposed in the literature can be connected one with each other by the diffusion approximation methodology. We first investigate the diffusion limit of the semiconductor Boltzmann equation towards the so-called ‘‘spherical harmonic expansion model,’’ under the assumption of dominant elastic scattering. Then, this model is again connected, either to the energy-transport model or to a ‘‘periodic spherical harmonic expansion model’’ through a diffusion approximation, respectively making electron–electron or phonon scattering large. We provide the mathematical background which makes the Hilbert expansions associated with these various diffusion limits rigorous.

Quantum Energy-Transport and Drift-Diffusion Models

“…The Classical Energy-Transport model appears in the early work [70]. Its first derivation from the semiconductor Boltzmann equation is due to [15] and [13] (see also the [23]). It has been analyzed in [25], [26].…”

A Boltzmann Model for Trapped Particles in a Surface Potential

“…It is possible when the physical conditions allow one to do it, for example when particles are subject to many collisions. The resulting model depends on the considered physical process (see [6] and the references given there).…”

“…It is possible when the physical conditions allow one to do it, for example when particles are subject to many collisions. The resulting model depends on the considered physical process (see [6] and the references given there).Here we want to establish an asymptotic model when the applied potential confines the particles close to the surface, increasing the number of collisions with it. Furthermore, we assume that the dominant surface collision process is specular; the other collisions are supposed to be only a perturbation.…”

Diffusion Limit of a Semiconductor Boltzmann–Poisson System

“…The linear operator Q describes physical conservation properties during collisions. Here, we only assume that the charge is conserved during the collision [2,27]. A typical model for such situation is the linear approximation of the electron-phonon interaction, given by…”