A Finite Difference Method for the Basic Stationary Semiconductor Device Equations

In this paper we analyse a special-purpose finite difference scheme for the basic stationary semiconductor device equations in one space dimension. These equations model potential distribution, carrier concentration and current flow in an arbitrary one-dimensional semiconductor device and they consist of three second order ordinary differential equations subject to boundary conditions. A small parameter appears as multiplier of the second derivative of the potential, thus the problem is singularly perturbed. We demonstrate the occurence of internal layers at so called device-junctions, which are jump-discontinuities of the data, and present a finite difference scheme which allows for the resolution of these internal layers without employing an exceedingly large number of grid-points. We establish the relation of this scheme to exponentially fitted schemes and give a convergence proof. Moreover the construction of efficient grids is discussed.