Peter A. Markowich

A steady state potential flow model for semiconductors

SummaryWe present a three-dimensional steady state irrotational flow model for semiconductors which is based on the hydrodynamic equations. We prove existence and local uniqueness of smooth solutions under a smallness assumptions on the data. This assumption implies subsonic flow of electrons in the semiconductors device.

Asymptotic behaviour of solutions of the hydrodynamic model of semiconductors

Degond and Markowich discussed the existence and uniqueness of a steady-state solution in the subsonic case for the one-dimensional hydrodynamic model of semiconductors. In the present paper, we reconsider the existence and uniqueness of a globally smooth subsonic steady-state solution, and prove its stability for small perturbation. The proof method we adopt in this paper is based on elementary energy estimates.

On steady state Euler-Poisson models for semiconductors

This paper is concerned with an analysis of the Euler-Poisson model for unipolar semiconductor devices in the steady state isentropic case. In the two-dimensional case we prove the existence of smooth solutions under a smallness assumption on the prescribed outflow velocity (small boundary current) and, additionally, under a smallness assumption on the gradient of the velocity relaxation time. The latter assumption allows a control of the vorticity of the flow and the former guarantees subsonic flow. The main ingredient of the proof is a regularization of the equation for the vorticity.Also, in the irrotational two- and three-dimensional cases we show that the smallness assumption on the outflow velocity can be replaced by a smallness assumption on the (physical) parameter multiplying the drift-term in the velocity equation. Moreover, we show that solutions of the Euler-Poisson system converge to a solution of the drift-diffusion model as this parameter tends to zero.

Inverse-average-type finite element discretizations of selfadjoint second-order elliptic problems

Etude dune classe de discretisations a elements finis lineaires par morceaux des problemes aux valeurs limites elliptiques du second ordre autoadjoints

Eigenvalue problems on infinite intervals

Abstract : This paper is concerned with eigenvalue problems for boundary value problems of ordinary differential equations posed on an infinite interval. Problems of that kind occur for example in fluid mechanics when the stability of laminar flows is investigated. Characterizations of eigenvalues and spectral subspaces are given and the convergence of approximating problems which are derived by reducing the infinite interval to a finite but large one and by imposing additional boundary conditions at the far end is proved. Exponential convergence is shown for a large class of problems. (Author)

The Drift Diffusion Equations

The drift diffusion equations are the most widely used model to describe semiconductor devices today. The bulk of the literature on mathematical models for device simulation is concerned with this nonlinear system of partial differential equations and numerical software for its solution is commonplace at practically every research facility in the field. From an engineering point of view, the interest in the drift diffusion model is to replace as much laboratory testing as possible by numerical simulation in order to minimize costs. To this end, it is important that computations can be performed in a reasonable amount of time. This implies that the involved mathematical models cannot be too complicated, such as, for instance, the higher dimensional transport equations described in Chapter 1. For the current state of technology the drift diffusion equations seem to represent a reasonable compromise between computational efficiency and an accurate description of the underlying device physics. Therefore transport equations are used mainly to compute data for the model parameters in the drift diffusion equations in the engineering environment. It should be pointed out, however, that, with the increased miniaturization of semiconductor devices, one comes closer and closer to the limits of validity of the drift diffusion equations, even in an industrial environment.