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 $\psi$ 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 ${{\mathbf {J}}_n}$, ${{\mathbf {J}}_p}$ and the electric field strength $\left \| {\nabla \psi } \right \|$ 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.