The Stationary Semiconductor Device EquationsSpringer Science & Business Media, 12/12/1985 - 195 من الصفحات In the last two decades semiconductor device simulation has become a research area, which thrives on a cooperation of physicists, electrical engineers and mathe maticians. In this book the static semiconductor device problem is presented and analysed from an applied mathematician's point of view. I shall derive the device equations - as obtained for the first time by Van Roosbroeck in 1950 - from physical principles, present a mathematical analysis, discuss their numerical solu tion by discretisation techniques and report on selected device simulation runs. To me personally the most fascinating aspect of mathematical device analysis is that an interplay of abstract mathematics, perturbation theory, numerical analysis and device physics is prompting the design and development of new technology. I very much hope to convey to the reader the importance of applied mathematics for technological progress. Each chapter of this book is designed to be as selfcontained as possible, however, the mathematical analysis of the device problem requires tools which cannot be presented completely here. Those readers who are not interested in the mathemati cal methodology and rigor can extract the desired information by simply ignoring details and proofs of theorems. Also, at the beginning of each chapter I refer to textbooks which introduce the interested reader to the required mathematical concepts. |
المحتوى
Introduction | 1 |
About This Book | 2 |
Semiconductors | 3 |
References | 5 |
Mathematical Modeling of Semiconductor Devices | 7 |
Poissons Equation | 8 |
Continuity Equations | 10 |
Current Relations | 11 |
44 The ZeroSpaceCharge Approximation | 93 |
45 The Layer Problems | 95 |
46 Representation and Structure of Solutions | 100 |
47 Extensions | 109 |
Graded Junctions | 110 |
Strongly Onesided Junctions | 112 |
Fielddependent Mobilities Velocity Saturation | 115 |
48 Scaling Revisited and Conditioning | 120 |
Summary | 13 |
22 Parameter Models | 14 |
The RecombinationGeneration Rate | 16 |
Carrier Mobility Models | 18 |
23 Boundary Conditions and Device Geometry The Static Problem | 19 |
The Static Problem | 24 |
24 Dependent Variables and Scaling Transformation of State Variables | 25 |
The Singular Perturbation Scaling | 26 |
References | 29 |
Analysis of the Basic Stationary Semiconductor Device Equations | 31 |
32 Existence of Solutions | 33 |
33 Global Regularity | 41 |
34 Uniqueness for Small Voltages | 43 |
35 Global Continuous Branches of Solutions | 48 |
The ZeroRecombinationGeneration Case | 49 |
VoltageCurrent Characteristics | 52 |
Nonvanishing RecombinationGeneration Terms Impact Ionisation | 54 |
36 Approximate Solutions | 56 |
Approximation Analysis by Decoupling | 57 |
Iteration Schemes | 60 |
References | 66 |
Singular Perturbation Analysis M of the Stationary Semiconductor Device Problem | 68 |
42 A Singularly Perturbed Second Order Model Problem | 72 |
Structure of Solutions | 73 |
Slow and Fast Scales | 76 |
43 Asymptotic Expansions | 80 |
Local Coordinate Transformations | 81 |
The Outer Expansion | 83 |
Boundary Layer Expansions | 87 |
A Ohmic Contacts | 88 |
B Neumann Segments | 89 |
D SemiconductorOxide Interfaces | 90 |
Conditioning of the Continuity Equations | 124 |
References | 129 |
Discretisation of the Stationary Device Problem | 131 |
51 Construction of Discretisations | 132 |
Discretisation of Poissons Equation | 133 |
C Finite Difference Schemes | 137 |
Discretisation of the Current Relations and Continuity Equations | 139 |
A Weak Formulations | 140 |
B The Failure of the Standard Discretisation | 141 |
C Finite Element ScharfetterGummel Discretisations | 143 |
D The Finite Difference ScharfetterGummel Schemes | 148 |
52 Solutions of Static Discretisations | 149 |
The Coefficient Matrices | 150 |
53 Convergence of Discretisations | 151 |
Decoupled Convergence Analysis | 152 |
Classical Convergence Theory | 154 |
Convergence Analysis of the SGSchemes | 158 |
Convergence Analysis for Poissons Equation | 162 |
Current Density and Outflow Current Approximations | 167 |
54 Extensions and Conclusions | 169 |
Schottky Contacts | 170 |
Adaptive MeshConstruction | 171 |
References | 176 |
Numerical Simulation A Case Study | 178 |
References | 183 |
Appendix | 184 |
Mathematical Notation | 185 |
B Scalars Vectors and Matrices | 186 |
D Landau Symbols | 187 |
F Functionalanalytic Notations | 189 |
191 | |
طبعات أخرى - عرض جميع المقتطفات
The Stationary Semiconductor Device Equations <span dir=ltr>P.A. Markowich</span> لا تتوفر معاينة - 2014 |
The Stationary Semiconductor Device Equations <span dir=ltr>P.A. Markowich</span> لا تتوفر معاينة - 2010 |
عبارات ومصطلحات مألوفة
applied bias approximation assume asymptotic expansions basic semiconductor device boundary conditions Boundary Value Problems carrier concentrations computation continuity equations convergence analysis current density current relations Debye length defined denote depend derive device models device simulation discrete solutions doping profile electric field error estimate finite difference grad grid hi+1 holds hole concentration implies iteration junction layer layer terms Lemma Lipschitz continuous Markowich Math mathematical mesh methods mobilities MOS-transistor Neumann Nonlinear numerical obtain Ohmic contacts one-dimensional order of magnitude p-region parameter physical piecewise linear pn-diode pn-junction Poisson's equation proof recombination-generation rate recombination-generation term reduced solution resp right hand side Ringhofer satisfies scaled Schottky contacts segments Selberherr semiconductor device equations Semiconductor Device Models semiconductor device problem semiconductor-oxide interface singular perturbation Theorem thermal equilibrium u₁ unique v₁ variable vector voltage zeroth order ΘΩΝ ΧΕΩ