Introduction

Abstract

Semiconductor device modeling started in the early fifties just after Van Roosbroeck had formulated the so-called fundamental semiconductor device equations, a nonlinear system of partial differential equations, which describes potential distribution, carrier concentrations and current flow in arbitrary semiconductor devices (see [1.32]). In the early stages highly simplified one-dimensional models accessible to direct analytic treatment were used in order to understand device characteristics and to improve device design (see [1.28], [1.29]). The trend towards miniaturisation in VLSI and device design, mainly caused by the increasing demand for fast computers with large storage, rendered the simplified models and consequently the fully analytic approach obsolete. Instead, the emphasis shifted towards numerical simulation techniques, i.e. the computational solution of the semiconductor device equations based on numerical discretisation methods. This approach was suggested by Gummel [1.9] for the bipolar transistor. De Mari [1.3], [1.4] applied the fully computational approach to pn-junction diodes. It became clear very soon that standard methods and theories of discretisation techniques are inappropriate because they require an enormous amount of computer resources in order to give reasonably accurate results when modeling practically relevant devices. The main reason for this is that the equations are stiff and allow for solutions of locally different behaviour. The stiffness problem was — to a certain extent — overcome by the ingenuity of Scharfetter and Gummel, who developed a nonstandard, special purpose discretisation method, which — sometimes in a modified and extended way — is being used up to now (see [1.25]).