Existence of Weak Solutions to a Class of Degenerate Semiconductor Equations Modeling Avalanche Generation

Bin Wu

doi:10.1155/2009/873683

Abstract

We consider the drift‐diffusion model with avalanche generation for evolution in time of electron and hole densities n, p coupled with the electrostatic potential ψ in a semiconductor device. We also assume that the diffusion term is degenerate. The existence of local weak solutions to this Dirichlet‐Neumann mixed boundary value problem is obtained.

Highlights

In this paper, we consider the following degenerate semiconductor equations modeling avalanche generation:−∇ · ∇ψ p − n C x, 1.1 nt − ∇ · Jn R n, p g, Jn ∇ nγ − μ1n∇ψ, 1.2 pt ∇ · Jp R n, p g, −Jp ∇ pγ μ2p∇ψ1.3 with initial and boundary conditions ψ, n, p ψ, n, p, x, t ∈ ΣD ≡ ΓD × 0, T, ∂ψ ∂η ∂n ∂η ∂p ∂η0, 0, 0, x, t ∈ ΣN ≡ ΓN × 0, T, 1.5 n, p n0, p0, x ∈ Ω, t 0.Mathematical Problems in EngineeringHere the unknowns ψ, n, and p denote the electrostatic potential, the electron density, and the hole density, respectively
Function C x denotes the doping profile fixed charged background ions characterizing the semiconductor under consideration, while the term g α1 ∇ψ |Jn| α2 ∇ψ |Jp| models the effect of impact ionization avalanche generation of charged particles cf. 1, 2 for details
R n, p r n, p 1 − np is the net recombination-generation rate, where r characterizes the mechanism of particle transition

Summary

Introduction

We consider the following degenerate semiconductor equations modeling avalanche generation:. In this case, the parabolic equations 1.2 and 1.3 become of degenerate type, and existence of solutions does not follow from standard theory. Many authors 8–10 have studied the existence and uniqueness of weak solutions of this type of degenerate semiconductor equations without avalanche generation term. The existence and uniqueness results are shown under the assumption that the solution ψ of Poisson equation with Dirichlet-Neumann mixed boundary conditions had the regularity ψ ∈ W2,r Ω r > N , this amounts to a geometric condition on Ω, for example Ω ∈ C1,1 and ΓD ∩ ΓN ∅. When γ 1, that is, the diffusion term is not degenerate, the authors 14 obtained the existence of local weak solutions of problem 1.1 – 1.6. Under hypotheses (H1)–(H6), there exists at least one local weak solution to the problem 1.1 – 1.6

Approximate Problem

Findings

Proof of the Main Result