A solution to the simplified multi-dimensional energy-transport model with a general conductivity for semiconductors

Abstract

The Dirichlet–Neumann mixed boundary value problem of a simplified multi-dimensional energy-transport model for semiconductors with the conductivity κ(n,θ)=nθ is studied. It consists of a drift–diffusion equation for carrier’s density, involving temperature gradients, a nonlinear heat equation for the carrier’s temperature, and the Poisson equation for the electric potential. The existence of weak solution and zero energy relaxation time limit for the problem are proved. For a small variation of the lattice temperature, the resulting limit for the problem is well-known drift–diffusion model. The proofs are based on the analysis of a time-discrete approximate system for the transient problem with the Stampacchia’s truncation approach and some careful calculations concerning the stability estimates needed for convergence of the scheme. Under some regularity assumption of the solution, the uniqueness of solution is shown.