Finite element solution of the fundamental equations of semiconductor devices. I

Abstract

We investigate the nonstationary equations of the semiconductor device theory consisting of a Poisson equation for the electric potential ψ \psi and of two highly nonlinear continuity equations for carrier densities n and p. We use simplicial elements with linear polynomials and four-node two-dimensional and eight-node three-dimensional isoparametric elements. There are constructed finite element solutions such that the current densities J n {{\mathbf {J}}_n} , J p {{\mathbf {J}}_p} and the electric field strength ‖ ∇ ψ ‖ \left \| {\nabla \psi } \right \| are constant on each element. Two schemes are proposed: one is nonlinear, the other is partly linear. The schemes preserve the property of the exact solution (corresponding to the physical meaning) that the carrier densities n and p are positive. Existence of the solution is proved in both cases, unicity in the second case. A subsequent paper II will be devoted to problems of stability and convergence.