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.

1. Introduction.- 1.1 Modeling.- 1.2 Computational Foundations.- 1.3 Mathematical Theory.- 1.4 Summary.- I. Modeling of Semiconductor Devices.- 2. Development of Drift-Diffusion Models.- 2.1 Descriptive Background.- 2.2 Modeling Overview.- 2.3 Scaling and Junction Width Estimation.- 2.3.1 System and Scalings.- 2.3.2 Example and Heuristic Analysis.- 2.4 The Scharfetter-Gummel Discretization.- 2.4.1 Variational Calculus.- 2.4.2 Piecewise Constant Flux.- 2.5 A Model for Drift-Diffusion.- 2.5.1 The Mobility Relations.- 2.5.2 Boundary Conditions and Current-Voltage Relations.- 3. Moment Models: Microscopic to Macroscopic.- 3.1 The Hydrodynamic Model.- 3.1.1 Charge, Momentum and Energy Transport Equations.- 3.1.2 Moment Closure and Relaxation Relations.- 3.2 Calibration with the Mechanics of Charged Fluids.- 3.2.1 Conservation of Mass and Energy.- 3.2.2 The Momentum Subsystem.- 3.3 Subsonic Linearization Analysis in One Dimension.- 3.4 Energy Transport Models and Stokes’ Flow.- 3.4.1 The Steady-State System.- 3.4.2 Exponential Variables in One Dimension.- 3.4.3 Maximum Principles.- 3.4.4 The Fixed Point Map.- 3.5 Modeling Issues.- 3.5.1 Regimes Defined by Damping.- 3.6 A Glimpse of the Quantum Hydrodynamic Model.- II. Computational Foundations.- 4. A Family of Solution Fixed Point Maps: Partial Decoupling.- 4.1 Contraction Property of the Gummel Map in Two Dimensions.- 4.1.1 A Framework for the Gummel Map.- 4.1.2 The Principal Result.- 4.1.3 The Supporting Lemmas and Hypotheses.- 4.2 General Case: A Framework of Weighted Spaces.- 4.3 Existence and Maximum Principles for Uf.- 4.4 Admissible Lag and the Mapping VW f.- 4.4.1 Admissible Lagging of the Continuity Subsystem.- 4.4.2 Uniqueness and Definition of the Map VWf.- 4.5 A Variational Inequality for the Current Continuity Subsystem.- 4.5.1 Abstract Inequality Formulation.- 4.5.2 The Concrete Variational Inequality.- 4.6 Equivalence with the Current Continuity Subsystem.- 4.7 Compactness and Continuity of VWf and Fixed Points of Tf.- 4.7.1 Compactness.- 4.7.2 Continuity.- 4.7.3 The Gummel Map and its Fixed Points.- 4.8 Technical Properties of Norms and Mappings.- 4.8.1 Norm Equivalence.- 4.8.2 Enhanced Continuity for the Subsystem Map.- 5. Nonlinear Convergence Theory for Finite Elements.- 5.1 Definitions of the Composite Mappings of T.- 5.2 The Numerical Map Tn.- 5.2.1 The Composite Finite Element Maps.- 5.2.2 The Discrete Maximum Principles.- 5.2.3 The Numerical Fixed Point Map.- 5.3 Approximation Theory in Energy Norms and Pointwise Norms.- 5.3.1 Approximation Theory for Gradient Equations.- 5.3.2 Convergence Properties of Tn in Energy Norms.- 5.3.3 Convergence Properties of Tn in the Pointwise Norm.- 5.4 A Calculus for the System Mappings.- 5.4.1 The Map U: Differentiability Properties.- 5.4.2 The Mappings V and W: Differentiability Properties.- 5.5 The Mappings Uh, Vh, and Wh.- 5.5.1 The Mapping Uh.- 5.5.2 The Mappings Vh and Wh.- 5.6 Summary of Results for T and Tn.- 5.7 Verification of the General Hypotheses.- 5.7.1 Verification of the ‘A Priori’ Estimates.- 5.7.2 Verification of the ‘A Posteriori’ Estimates.- 5.8 Final Convergence Results.- III. Mathematical Theory.- 6. Numerical Fixed Point Approximation in Banach Space.- 6.1 Linear Theory: Staircase to the Nonlinear Theory.- 6.2 Nonlinear Estimation and the Operator Calculus.- 6.2.1 ‘A Priori’ Estimates and Asymptotic Linearity.- 6.2.2 ‘A Posteriori’ Estimates.- 6.3 Approximate Fixed Points via Newton’s Method.- 6.4 The Inf-Sup Theory As a Special Case.- 7. Construction of the Discrete Approximation Sequence.- 7.1 The Fixed Point Map as Smoother.- 7.1.1 Solution of the Central Approximation Problem.- 7.2 Smoothing for Newton Iteration: Differential Maps.- 7.2.1 Framework for the Postconditioning Iteration.- 7.2.2 The Superlinear Convergence Theorem.- References.