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Semiconductor Models : Their Mathematical Study and Approximation

Emphasis in this paper is upon the development of the mathematical properties of the drift-diffusion semiconductor device model, especially steady-state properties, where the system is incorporated into a fixed point mapping framework. An essential feature of the paper is the description of the Krasnosel’skii calculus, the appropriate extension to nonlinear equations and systems of the famous inf-sup saddle point approximation theory for linear equations. The introductory section describes certain scaling effects and the association with numerical methods constructed via exponential fitting. The fixed point and numerical fixed maps are introduced in the next section, while the finite element approximation theory is described in the third section. The fourth section deals with the evolution system and modular pre-algorithms defined by Newton’s method, combined with a fully implicit time semidiscretization. More general moment models are introduced in the fifth section, and some historical comments appear in the final section. Specific algorithms and implementation issues are not discussed. The principal section headings are as follows. 1. The Drift-Diffusion Model, Scaling, and Exponential Fitting. 2. Steady-State: The Fixed Point and Numerical Fixed Point Maps. 3. A Nonlinear Finite Element Convergence Theory. 4. The Evolution System and Newton’s Method. 5. More General Moment Models: A Review. 6. Epilogue: Historical Perspective.