Table Of ContentComputational
Microelectronics
Edited by S. Selberherr
The Stationary
Semiconductor
Device Equations
Peter A. Markowich
Springer-Verlag Wien GmbH
Doz. Dr. Peter A. Markowich
Institut fUr Angewandte und Numerische Mathematik
Technische Universităt Wien, Austria
This work is subject to copyright.
AII rights are reserved, whether the whole or part of the material is concemed,
specifically those of translation, reprinting, re-use of illustrations, broadcasting,
reproduction by photocopying machine or similar means, and storage in data banks.
© 1986 by Springer-VerlagWien
Originally published by Springer-Verlag Wien-New York in 1986
Softcover reprint ofthe hardcover Ist edition 1986
With 40 Figures
ISSN 0179-0307
ISBN 978-3-211-99937-0 ISBN 978-3-7091-3678-2 (eBook)
DOI 10.1007/978-3-7091-3678-2
Preface
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.
With this book I want to provide device modelers with an overview of mathemati
cal results and discretisation techniques in order to make it possible for them to
design more efficient simulation programs and to assess and interpret simulation
results profoundly. Also I would like to introduce mathematicians to the semicon
ductor device problem and encourage them to contribute to this fairly new re
search area.
I want to express my gratitude to those, who contributed to this book. First of all
I thank Richard Weiss, who thaught me applied mathematics and numerical
analysis and encouraged me to write this book. I am also extremely grateful to
Christian Ringhofer for introducing me to the semiconductor device problem and
for many stimulating discussions. I thank Siegfried Selberherr for many hours of
discussing device physics and simulation with me. I am also indebted to Wolfgang
Agler, Uri Ascher, Frank de Hoog, Andrea Franz, Gerhard Franz, Wolfgang
Griebel, Andreas Griewank, Werner Jiingling, Erasmus Langer, Peter Pichler,
Hans Potzl, Christian Schmeiser, Herbert Steinriick, Fritz Straker, Peter
Szmolyan, Vidar Thomee, Christoph Uberhuber and Milos Zlamal for various
VI Preface
suggestions on the material. I am grateful to John Nohel of the Mathematics
Research Center of the University of Wisconsin-Madison for letting me enjoy the
stimulating atmosphere at the MRC as often as possible and to Neil Trudinger of
the Centre for Mathematical Analysis of the Australian National University for
inviting me to the CMA. A large part of the research for Chapter 5 was done there.
I thank Ms. Ursula Schweigler for typing skillfully the largest part of the manu
script and Ms. Monika Mazic for proofreading the galleys.
Finally, I want to express my gratitude to my wife Sylvia for her patience while I
was working on this book.
Vienna, November 1985
Peter A. Markowich
Contents
1. Introduction
A Review of Semiconductor Device Modeling
About This Book 2
Semiconductors 3
References 5
2. Mathematical Modeling of Semiconductor Devices 7
2.1 Derivation of the Basic Semiconductor Device Equations 7
Maxwell's Equations 7
Poisson's Equation 8
Continuity Equations 10
Current Relations 11
Summary 13
2.2 Parameter Models 14
The Doping Profile 14
The Recombination-Generation Rate 16
Carrier Mobility Models 18
2.3 Boundary Conditions and Device Geometry - The Static Problem 19
Boundary Conditions 19
The Static Problem 24
Geometric Properties of Devices 24
2.4 Dependent Variables and Scaling - Transformation of State Variables 25
The Singular Perturbation Scaling 26
References 29
3. Analysis of the Basic Stationary Semiconductor Device
Equations 31
3.1 Preliminaries 31
3.2 Existence of Solutions 33
3.3 Global Regularity 41
3.4 Uniqueness for Small Voltages 43
VIII Contents
3.5 Global Continuous Branches of Solutions 48
The Zero-Recombination-Generation Case 49
Voltage-Current Characteristics 52
Non-vanishing Recombination-Generation Terms - Impact Ionisation 54
3.6 Approximate Solutions 56
Approximation Analysis by Decoupling 57
Iteration Schemes 60
References 66
4. Singular Perturbation Analysis of the Stationary Semiconductor
Device Problem 68
4.1 The Device Equations As Singular Perturbation Problem 68
4.2 A Singularly Perturbed Second Order Model Problem 72
Structure of Solutions 73
Slow and Fast Scales 76
4.3 Asymptotic Expansions 80
Local Coordinate Transformations 81
The Outer Expansion 83
The Junction Layer Expansion 83
Boundary Layer Expansions 87
A. Ohmic Contacts 88
B. Neumann Segments 89
C. Schottky Contacts 89
D. Semiconductor-Oxide Interfaces 90
4.4 The Zero-Space-Charge Approximation 93
4.5 The Layer Problems 95
4.6 Representation and Structure of Solutions 100
4.7 Extensions 109
Graded Junctions 110
Strongly One-sided Junctions 112
Field-dependent Mobilities - Velocity Saturation 115
4.8 Scaling Revisited and Conditioning 120
Rescaling Poisson's Equation 120
Conditioning of the Continuity Equations 124
References 129
5. Discretisation of the Stationary Device Problem 131
5.1 Construction of Discretisations 132
Discretisation of Poisson's Equation 133
A. The Ritz-Galerkin Approximation 133
B. Piecewise Linear Finite Element Approximation 133
C. Finite Difference Schemes 137
Contents IX
Discretisation of the Current Relations and Continuity Equations 139
A. Weak Formulations 140
B. The Failure of the Standard Discretisation 141
C. Finite Element Scharfetter-Gummel Discretisations 143
D. The Finite Difference Scharfetter-Gummel Schemes 148
5.2 Solutions of Static Discretisations 149
The Coefficient Matrices 150
5.3 Convergence of Discretisations 151
Decoupled Convergence Analysis 152
Classical Convergence Theory 154
Convergence Analysis of the SG-Schemes 158
Convergence Analysis for Poisson's Equation 162
Current Density and Outflow Current Approximations 167
5.4 Extensions and Conclusions 169
Semiconductor-Oxide Interfaces 169
Schottky Contacts 170
Adaptive Mesh-Construction 171
References 176
6. Numerical Simulation - A Case Study 178
References 183
Appendix 184
Notation of Physical Quantities 184
Mathematical Notation 185
A. Device Geometry 185
B. Scalars, Vectors and Matrices 186
C. Functions 186
D. Landau Symbols 187
E. Linear Spaces of Functions and Norms 187
F. Functionalanalytic Notations 189
Subject Index 191
1
Introduction
A Review of Semiconductor Device Modeling
Semiconductor device modeling started in the early fifties just after Van
Roosbroeck had formulated the so-called fundamental semiconductor device
equations, a nonlinear system of partial differential equations, which descri
bes potential distribution, carrier concentrations and current flow in arbi
trary semiconductor devices (see [1.32]). In the early stages highly simplified
one-dimensional models accessible to direct analytic treatment were used in
order to understand device characteristics and to improve device design (see
[1.28], [1.29]). The trend towards miniaturisation in VLSI and device design,
mainly caused by the increasing demand for fast computers with large
storage, rendered the simplified models and consequently the fully analytic
approach obsolete. Instead, the emphasis shifted towards numerical simula
tion techniques, i.e. the computational solution of the semiconductor device
equations based on numerical discretisation methods. This approach was
suggested by Gummel [1.9] for the bipolar transistor. De Mari [1.3], [1.4]
applied the fully computational approach to pn-junction diodes. It became
clear very soon that standard methods and theories of discretisation techni
ques are inappropriate because they require an enormous amount of com
puter resources in order to give reasonably accurate results when modeling
practically relevant devices. The main reason for this is that the equations
are stiff and allow for solutions of locally different behaviour. The stiffness
problem was - to a certain extent - overcome by the ingenuity of Scharfetter
and Gummel, who developed a nonstandard, special purpose discretisation
method, which - sometimes in a modified and extended way - is being used
up to now (see [1.25]).
In the seventies mathematicians started to pay attention to the fundamental
semiconductor device equations. At first the emphasis was on proving exis
tence, uniqueness and dependence-on-data results by employing abstract
methods from the theory of partial differential equations (see [1.19], [1.20])
and on the 'classical' convergence analysis of discretisation methods (see
[1.21], [1.22]). At the same time the understanding of device performance
improved and the interaction of mathematical and physical insight enhanced
the development of feasable numerical techniques. This made it possible to
simulate complicated, intrinsically multi-dimensional devices (see [1.27] for
