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We present a qualitative analysis of the fundamental static semiconductor device equations which is based on singular perturbation theory. By appropriate scaling the semiconductor device equations are reformulated as singularly perturbed elliptic system (the Laplacian in Poisson's equation is multiplied by a small parameter ?2, the so‐called singular perturbation parameter). Physically the singular perturbation parameter is identified with the square of the normed minimal Debye length of the device under consideration. Using matched asymptotic expansions for small A we characterize the behaviour of the solutions locally at pn junctions, Schottky contacts and oxide‐semiconductor interfaces and demonstrate the occurrence of exponential internal/boundary layers at these surfaces. The derivatives of the solutions blow up within these layer regions (as ?2 decreases) and they remain bounded away from the layers. We demonstrate that the solutions of the ‘zero‐space charge approximation’ are close to the solutions of the ‘full’ semiconductor problem (when ? is small) away from layer regions and derive a second‐order ordinary differential equation which (when subjected to appropriate boundary/interface conditions) ‘describes’ the solutions within layer regions.

In this paper we carry out a conditioning analysis for the steady state semiconductor device problem. We consider various quasilinearizations as well as Gummel‐type iterations and obtain stability bounds which may allow ill‐conditioning in general. These bounds are exponential in the potential variation, and are sharp e.g. for a thyristor. But for devices where each smooth subdomain has an Ohmic contact, e.g. a pn‐diode, moderate bounds guaranteeing well‐conditioning are obtained. Moreover, the analysis suggests how various row and column scalings should be applied in order for the measured condition numbers of the linearized discrete problem to correspond more realistically to the true loss of significant digits in the calculations.

The singular perturbation character of the semiconductor device problem is well known by now. Various results on the structure of solutions (i.e. existence of spatial and temporal layers) have been obtained by means of singular perturbation theory. We use a rescaled form of the equations, which describes the evolution on the fast time scale, and discuss the asymptotic behavior of this system, i.e. its relationship to the initial layer problem, to the corresponding stationary problem and to the reduced problem. We show that the transient semiconductor problem fits in the framework of ‘fast reaction‐slow diffusion’ type equations, which are known from the analysis of chemical reacting systems. We use a multiple time scale expansion to give a new ‘dynamical’ derivation of the reduced problem.

State of the art programs for the solution of the drift diffusion semiconductor equations are based on finite difference techniques, or on certain combinations of finite elements and finite differences. The extreme gradients which occur in semiconductor devices have motivated several attempts to exploit perturbation analysis in order to improve the numerical scheme. Selberherr, Markowich et. al. showed the singular perturbation nature of the drift diffusion equations. Their analytic approximations of diodes influenced some aspects of the Minimos and Bambi simulators. Asymptotic analyses of MOSFET were presented by Brews and by Ward. These solutions are mainly valid for long channels, and their accuracy is limited.

Implicit one‐step methods for the system of differential equations arising from a space discretisation of the semiconductor equations are considered. It is shown that mere spectral conditions like A‐stability or L‐stability do not give a reliable answer to the behaviour of the numerical solution. Rather, positivity arguments for the corresponding rational matrix functions play an important role.

We present an abstract mathematical and numerical analysis for Drift‐Diffusion equation of heterojunction semiconductor devices with Fermi‐Dirac statistic. For the approximation, a mixed finite element method is considered. This can be profitably used in the investigation of the current through the device structure. A peculiar feature of this mixed formulation is that the electric displacement D and the current densities jn and jp for electrons and holes, are taken as unknowns, together with the potential φ and quas‐Fermi levels φn and φp. This enably D, jn and jp to be determined directly and accurately. For decoupled system, existence, uniqueness, regularity and stability results of the approximate solution are given. A priori and a posteriori error estimates are also presented. A nonlinear implicit scheme with local time steps is used. This algorithm appears to be efficient and gives satisfactory results. Numerical results for an heterojunction bipolar transistor, In two dimension, are presented.

The paper deals with a spatial discretization of transient semiconductor device equations. The method can be regarded as a combination of FDM‐ and FEM‐ideas. In the first part of the paper the method is described and—for a weakly acute triangulation—existence, uniqueness, non‐negativity, stability and conservativity of the semidiscrete solution are proved. The second part contains an error estimation under stronger assumptions on the regularity of the analytical solution and on the uniformity of the triangulation respectively. A linear convergence rate is obtained.

Two new mixed finite element schemes for discretizing current continuity equations are presented. Together with the good features of the already‐known mixed scheme (current preservation and good approximation of sharp shapes), they provide M‐matrices, even when a zero order term is present in the equations.

When considering the one dimensional semiconductor device equations, the condition number turns out to be so large that a numeric solution of the equation is meaningless. Applying singular perturbation methods to the linearized model the solution can be separated into two parts: One, which is very sensitive to perturbations, and one, which is robust against perturbations of the equations.

An extension of the Scharfetter‐Gummel discretization scheme is presented which is designed for electrothermal semiconductor device equations including avalanche generation terms. The scheme makes explicit use of the exponential character of solutions, and reduces to the standard Scharfetter‐Gummel scheme in the isothermal non‐avalanche case.