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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.

Inhaltsverzeichnis

Frontmatter

Introduction

1. Introduction

Abstract
The general themes of this book attempt to capture timely issues of modeling, theoretical computation, and mathematical theory. Engineers, scientists, and mathematicians now accept nonlinear systems of the scope developed here as basic for the analysis and prediction of the processes of charge transport, in a self-consistently determined electric field. In this brief introductory chapter, we shall expand upon the remarks of the preface, and discuss the goals of the book, and their development. Some historical detail will be furnished, insofar as it relates to the thematic aims.
Joseph W. Jerome

Modeling of Semiconductor Devices

Frontmatter

2. Development of Drift-Diffusion Models

Abstract
The basic principle underlying the function of a transistor element is that current flows between oppositely directed p/n junctions (or does not flow), depending upon controlling currents or voltages elsewhere in the element. A junction occurs when regions with contrasting doping characteristics abut in a semiconductor. Thus, a p-region is one with an excess of free hole carriers and an n-region, on the other hand, contains an excess of free electrons. Such charge imbalances are induced by the process of doping, whereby impurity concentrations are injected into pure crystalline semiconductor materials. From the standpoint of valence chemistry, the impurity atoms have numbers of electrons in their outer shells different from the semiconductor atoms. Preponderance of donor impurities creates n-regions as well as net concentrations of positive ions, and acceptor impurities create p-regions and negative ions. The net charge at any point in the device is obtained by combining the ions with the free electron and hole carriers. Thus, an electric field is created via the first Maxwell equation, often called the Poisson equation in the theory of electrostatics. There are also quantum mechanical interpretations related to the impurity injection process; these include energy band bending and wave packet identifications as classical particles (see [87, 122]).
Joseph W. Jerome

3. Moment Models: Microscopic to Macroscopic

Abstract
In this chapter, we shall present alternative models to that of drift-diffusion. More precisely, the models developed here are refinements, for which drift-diffusion is a limiting case. The two which we select are the classical hydrodynamic model, and a class of energy transport models. A quantum transport variant of the hydrodynamic model is briefly discussed at the conclusion of the chapter. For the reader who wishes to pursue these topics in greater detail, a special issue of VLSI DESIGN, edited by the author, has appeared (vol. 3, no. 2, 1995).
Joseph W. Jerome

Computational Foundations

Frontmatter

4. A Family of Solution Fixed Point Maps: Partial Decoupling

Abstract
For the remainder of the book, we restrict our attention to the drift-diffusion model. Moreover, we employ the so-called quasi-Fermi levels, originally introduced in Chap. 2, and designated below by v and w. For our purposes, we shall thus study variants of the following model, discussed in §2.5.
Joseph W. Jerome

5. Nonlinear Convergence Theory for Finite Elements

Abstract
In the following chapter (Chap. 6), a very powerful convergence theory will be established for numerical fixed point approximations. It is the goal of the present chapter to show that this framework encompasses the semiconductor device model in the case of constant mobility coefficients and zero recombination terms.
Joseph W. Jerome

Mathematical Theory

Frontmatter

6. Numerical Fixed Point Approximation in Banach Space

Abstract
As we have seen in the preceding chapters, the drift-diffusion model of a steady-state semiconductor device is formed by a system of three coupled partial differential equations (PDEs) for which discrete and continuous maximum principles exist. This system of PDEs is solved by a solution vector of three function components. Moreover, a fixed point mapping T can be defined. Although the definition of T is not unique, and various decouplings are possible, as was rigorously analyzed in Chap. 4, it is possible to achieve complete decoupling, via gradient equations, when the recombination term satisfies monotonicity properties, or is taken to be zero. This is carried out by solving each of these PDEs for its corresponding component, and substituting these components in successive PDEs in a Gauss-Seidel iterative fashion. Fixed points of such a mapping then coincide with solutions to the drift-diffusion model. Iteration with this mapping T defines an algorithm for the solution of the drift-diffusion model, typically termed Gummel iteration in the literature. It is really Picard iteration for the map T. The Lipschitz constant has been examined in detail in Chap. 4.
Joseph W. Jerome

7. Construction of the Discrete Approximation Sequence

Abstract
Since the fundamental paper of Moser (cf. [105]), it has been understood analytically that regularization is necessary as a postconditioning step in the application of approximate Newton methods, based upon the system differential map. A development of these ideas in terms of current numerical methods and complexity estimates was given by the author in [65]. The approach of Moser is often termed Nash-Moser iteration, because of the fundamental link to generalized implicit function theorems (cf. [108]), specifically, the Nash implicit function theorem. It was proposed by the author in [70], and analyzed further in [71], to use the fixed point map as a basis for the linearization, and thereby avoid the loss of derivatives phenomenon identified by Moser, and termed a numerical loss of derivatives in [65]. In the context of numerical analysis, this loss occurs because the approximation of the identity condition, involved in approximate Newton methods, is not robust with respect to differentiation up to the order of the nonlinear differential system (see (7.8) below). In this chapter, we shall discuss the implications of this fact. It is at the core of preferring the fixed point formulation to the differential formulation. We begin with the former. Our discussion of this case is brief, because of the developments of the preceding chapters.
Joseph W. Jerome

Backmatter

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