## About this book

### Introduction

This book addresses the mathematical aspects of semiconductor modeling, with particular attention focused on the drift-diffusion model. The aim is to provide a rigorous basis for those models which are actually employed in practice, and to analyze the approximation properties of discretization procedures. The book is intended for applied and computational mathematicians, and for mathematically literate engineers, who wish to gain an understanding of the mathematical framework that is pertinent to device modeling. The latter audience will welcome the introduction of hydrodynamic and energy transport models in Chap. 3. Solutions of the nonlinear steady-state systems are analyzed as the fixed points of a mapping T, or better, a family of such mappings, distinguished by system decoupling. Significant attention is paid to questions related to the mathematical properties of this mapping, termed the Gummel map. Compu tational aspects of this fixed point mapping for analysis of discretizations are discussed as well. We present a novel nonlinear approximation theory, termed the Kras nosel'skii operator calculus, which we develop in Chap. 6 as an appropriate extension of the Babuska-Aziz inf-sup linear saddle point theory. It is shown in Chap. 5 how this applies to the semiconductor model. We also present in Chap. 4 a thorough study of various realizations of the Gummel map, which includes non-uniformly elliptic systems and variational inequalities. In Chap.

### Keywords

Abweichungs-Diffusionssystem Boltzmann transport equation Boltzmann-Transportmodell Drift-diffusion system Energie Gummel Iteration Scharfetter-Gummel discretization System decoupling calculus differential equation energy transport model finite element method inf-sup Theorie linearization mixed boundary value problem nichtlineare finite Elemente-Konvergenz-Theorie nonlinear finite element convergence theory numerical fixed point map p/n Junktion transistor

#### Authors and affiliations

- 1.Department of MathematicsNorthwestern UniversityEvanstonUSA

### Bibliographic information