Ebook: Analysis of Charge Transport: A Mathematical Study of Semiconductor Devices

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.

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The book is organized into seven chapters, with the introduction serving as a comprehensive guide. The remaining chapters are divided in pairs, under the format of modeling, computational foundations, and mathematical theory. The modeling chapters provide an elementary discussion of the drift-diffusion system for semiconductors, and introduce an hierarchy of moment models, based upon the Boltzmann transport equation. The final four chapters deal exclusively with the drift-diffusion model.

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The book is organized into seven chapters, with the introduction serving as a comprehensive guide. The remaining chapters are divided in pairs, under the format of modeling, computational foundations, and mathematical theory. The modeling chapters provide an elementary discussion of the drift-diffusion system for semiconductors, and introduce an hierarchy of moment models, based upon the Boltzmann transport equation. The final four chapters deal exclusively with the drift-diffusion model.
Content:
Front Matter....Pages I-XI
Introduction....Pages 1-6
Front Matter....Pages 7-7
Development of Drift-Diffusion Models....Pages 9-26
Moment Models: Microscopic to Macroscopic....Pages 27-50
Front Matter....Pages 51-51
A Family of Solution Fixed Point Maps: Partial Decoupling....Pages 53-88
Nonlinear Convergence Theory for Finite Elements....Pages 89-122
Front Matter....Pages 123-123
Numerical Fixed Point Approximation in Banach Space....Pages 125-142
Construction of the Discrete Approximation Sequence....Pages 143-154
Back Matter....Pages 155-167

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The book is organized into seven chapters, with the introduction serving as a comprehensive guide. The remaining chapters are divided in pairs, under the format of modeling, computational foundations, and mathematical theory. The modeling chapters provide an elementary discussion of the drift-diffusion system for semiconductors, and introduce an hierarchy of moment models, based upon the Boltzmann transport equation. The final four chapters deal exclusively with the drift-diffusion model.
Content:
Front Matter....Pages I-XI
Introduction....Pages 1-6
Front Matter....Pages 7-7
Development of Drift-Diffusion Models....Pages 9-26
Moment Models: Microscopic to Macroscopic....Pages 27-50
Front Matter....Pages 51-51
A Family of Solution Fixed Point Maps: Partial Decoupling....Pages 53-88
Nonlinear Convergence Theory for Finite Elements....Pages 89-122
Front Matter....Pages 123-123
Numerical Fixed Point Approximation in Banach Space....Pages 125-142
Construction of the Discrete Approximation Sequence....Pages 143-154
Back Matter....Pages 155-167
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