Table Of ContentPeter A. Markowich
Christ ian A. Ringhofer
Christ ian Schmeiser
Semi.
conducto r
Equations
Springer-Verlag Wien N e w \ f a r k
Semiconductor
Equations
P. A. Markowich
C. A. Ringhofer
C. Schmeiser
Springer-Verlag Wien New York
Peter A. Markowich
Fachbereich Mathematik
Technische Universitat, Berlin
Christian A. Ringhofer
Department of Mathematics
Arizona State University, Tempe, Arizona, USA
Christian Schmeiser
Institut fur Angewandte und Numerische Mathematik
Technische Universitat, Wien, Austria
This work is subject to copyright.
Al l rights are reserved, whether the whole or part of the material is concerned,
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© 1990 by Springer-Verlag Wien
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Printed in Austria by Novographic, Ing. W. Schmid, A-1238 Wien
Printed on acid-free paper
With 33 Figures
Library of Congress Cataloging-in-Publication Data. Markowich, Peier A., 1956- Semiconductor equations Peter A. Markowich,
Christian Ringhofer, Christian Schmeiser. p. cm. Includes bibliographical references. I S B N 0-387-82157-0 (U.S.) 1. Semiconductors
Mathematical models. I. Ringhofer, Christian, 1957- . II . Schmeiser, Christian, 1958- . I I I . Title. TK7871.85.M339. 1989.
621.381'52—dc20
ISBN 3-211-82157-0 Springer-Verlag Wien-New York
ISBN 0-387-82157-0 Springer-Verlag New York-Wien
Preface
In recent years the mathematical modeling of charge transport in semi-
conductors has become a thr iv ing area in applied mathematics. The drift
diffusion equations, which constitute the most popular model for the simula-
t ion of the electrical behavior of semiconductor devices, are by now mathe-
matically quite well understood. As a consequence numerical methods have
been developed, which allow for reasonably efficient computer simulations
in many cases of practical relevance. Nowadays, research on the drift diffu-
sion model is of a highly specialized nature. I t concentrates on the explora-
t ion of possibly more efficient discretization methods (e.g. mixed finite
elements, streamline diffusion), on the improvement of the performance of
nonlinear iteration and linear equation solvers, and on three dimensional
applications.
The ongoing miniatur izat ion of semiconductor devices has prompted a shift
of the focus of the modeling research lately, since the drift diffusion model
does not account well for charge transport in ultra integrated devices.
Extensions of the drift diffusion model (so called hydrodynamic models) are
under investigation for the modeling of hot electron effects in submicron
MOS-transistors, and supercomputer technology has made it possible
to employ kinetic models (semiclassical Boltzmann-Poisson and Wigner-
Poisson equations) for the simulation of certain highly integrated devices.
The focus of this book is the presentation of the hierarchy of semiconductor
models ranging from kinetic transport equations to the drift diffusion equa-
tions. Particular emphasis is given to the derivation of the models and the
physical and mathematical assumptions used therefore. We do not go into
the mathematical technicalities necessary for a detailed analysis of the
models but rather sacrifice rigour for the sake of conveying the basic prop-
erties and features of the model equations. The mathematically interested
reader is encouraged to consult the references for in-depth investigations
of specific subjects.
We address applied mathematicians, electrical engineers and solid state phy-
sicists. The exposition is accessible to graduate students in each of the three
fields. In particular, we hope that this book w i l l be useful as a text for
advanced graduate courses in this area and we urge students to work the
vi Preface
problems, which can be found at the end of each chapter, for a deeper
penetration of the material.
We are grateful to our colleagues U . Ascher, J. Batt, F. Brezzi, P. Degond,
P. Deuflhard, D . Ferry, W. Fichtner, L . Gastaldi, P. Kasperkovitz, N . Kluks -
dahl, D . M a r i n i , H . Neunzert, F. Nier, R. O 'Mal ley, F. Poupaud, T. Seidman,
S. Selberherr, H . Steinriick, P. Szmolyan, and T. Taylor for many hours of
stimulating discussions. I n particular we are indebted to A. Arno ld , N .
Mauser, P. Pietra, and R. WeiB for reading large parts of the manuscript.
We thank U . Schweigler for skillfully typing a por t ion of the manuscript.
We are indebted to the Centre des Mathematiques Appliquees of the Ecole
Polytechnique, Palaiseau, France, and to the Inst i tut fur Technische Elek-
t ron ik of the Technische Universitat Hamburg-Harburg where a part of the
manuscript was wri t ten. The first and the th i rd author acknowledge support
from the "Osterreichischer Fonds zur Forderung der wissenschaftlichen
Forschung" under the grant numbers P6771 and P4919, respectively. The
second author acknowledges support by the Nat iona l Science Foundation,
U S A under grant no. P M S 880-1153.
Paris, 1989
Peter A. M a r k o w i c h Christ ian A. Ringhofer
Fachbereich Mathemat ik Department of Mathematics
Technische Universitat Berlin Arizona State University
StraBe des 17. Juni 136 Tempe, A Z 85287, U S A
D-1000 Berlin 12
Christian Schmeiser
Inst i tut fiir Angewandte und Numerische Mathemat ik
Technische Universitat Wien
Wiedner HauptstraBe 8-10
A-1040 Wien, Austria
Contents
In t r o d u c t i o n
1 K i n e t i c T r a n s p o r t M o d e l s for Semiconduc to rs 3
1.1 Int roduc t ion 3
1.2 The (Semi-)Classical Liouvi l le Equation 4
Particle Trajectories 5
A Potential Barrier 5
The Transport Equation 8
Particle Ensembles 8
The Ini t ia l Value Problem 9
The Classical Hami l ton ian 11
The Semi-Classical Liouvi l le Equation 12
Magnetic Fields 16
1.3 The Boltzmann Equation 16
The Vlasov Equation 17
The Poisson Equation 21
The Whole Space Vlasov Problem 23
Bounded Position Domains 24
The Semi-Classical Vlasov Equation 25
Magetic Fields—The Maxwel l Equations 26
Collisions—The Boltzmann Equation 28
The Semi-Classical Boltzmann Equation 30
Conservation and Relaxation 32
L o w Density Approx imat ion 33
The Relaxation Time Approx imat ion 34
Polar Optical Scattering 35
Particle-Particle Interaction 36
1.4 The Quantum Liouvi l le Equation 36
The Schrodinger Equation 37
Tunneling 38
Particle Ensembles and Density Matrices 40
Wigner Functions 41
viii Contents
The Quantum Transport Equat ion 42
Pure and Mixed States 44
The Classical L i m i t 48
Nonnegat ivi ty of Wigner Functions 50
A n Energy-Band Version of the Quan tum Liouvi l le Equation 52
1.5 The Quantum Boltzmann Equat ion 57
Subensemble Density Matrices 58
The Quantum Vlasov Equat ion 60
The Poisson Equat ion 63
The Quantum Vlasov Equat ion on a Bounded Position
D o m a i n 65
The Energy-Band Version of the Quan tum Vlasov Equation 65
Collisions 67
1.6 Applications and Extensions 68
Mult i -Va l l ey Models 69
Bipolar M o d e l 71
Tunnel ing Devices 75
Problems 77
References 80
2 F r o m K i n e t i c t o F l u i d D y n a m i c a l M o d e l s 83
2.1 Int roduc t ion 83
2.2 Small Mean Free Path—The Hi lber t Expansion 85
2.3 M o m e n t Methods—The Hydrodynamic M o d e l 90
Derivat ion of the Dri f t Diffusion M o d e l 91
The Hydrodynamic Mode l 92
2.4 Heavy D o p i n g Effects—Fermi-Dirac Distr ibut ions 94
2.5 H i g h Field Effects—Mobil i ty Models 95
2.6 Recombination-Generation Models 98
Problems 101
References 102
3 T h e D r i f t D i f f u s i o n E q u a t i o n s 104
3.1 Int roduc t ion 104
3.2 The Stationary Dr i f t Diffusion Equations 108
3.3 Existence and Uniqueness for the Stationary Dri f t Diffusion
Equations 110
3.4 Forward Biased P-N Junctions 116
The Equ i l ib r ium Case 118
The No n - E q u i l i b r i u m Case 125
Asymptot ic Val id i ty in the One-Dimensional Case 129
Velocity Saturation Effects—Field Dependent Mobil i t ies 130
3.5 Reverse Biased P-N Junctions 133
Moderately Reverse Biased P-N Junctions 134
Contents ix
P-N Junctions Under Extreme Reverse Bias Condit ions 134
The One-Dimensional Problem 135
The Two-Dimensional Case 141
3.6 Stability and Condi t ioning for the Stationary Problem 143
3.7 The Transient Problem 148
3.8 The Linearization of the Transient Problem 149
3.9 Existence for the Nonlinear Problem 153
Asymptotic Expansions for the Transient Dri f t Diffusion
Equations 156
3.10 Asymptotic Expansions on the Diffusion Time Scale 157
3.11 Fast Time Scale Expansions 162
The Case of a Bounded In i t i a l Potential 163
Fast Time Scale Solutions for General In i t i a l Data 165
Problems 171
References 172
4 Devices 175
4.1 Int roduc t ion 175
Static Voltage-Current Characteristics 176
4.2 P - N Diode 180
The Depletion Region in Thermal Equi l ib r ium 181
Strongly Asymmetric Junctions 185
The Voltage-Current Characteristic Close to Thermal
Equi l ib r ium 188
H i g h In jec t ion—A Mode l Problem 191
Large Reverse Bias 192
Avalanche Breakdown 195
Punch Through 197
4.3 Bipolar Transistor 198
Current Gain Close to Thermal Equ i l ib r ium 199
4.4 P I N - D i o d e 202
Thermal Equi l ib r ium 203
Behaviour Close to Thermal Equ i l ib r ium 206
4.5 Thyris tor 208
Characteristic Close to Thermal Equ i l ib r ium 210
Forward Conduct ion 212
Break Over Voltage 215
4.6 M I S Diode 218
Accumulat ion 221
Deplet ion—Weak Inversion 222
Strong Inversion 223
4.7 M O S F E T 225
Derivat ion of a Simplified Mode l 228
A Quasi One-Dimensional Mode l 230
Computa t ion of the One-Dimensional Electron Density 231
X Contents
Computa t ion of the Current 233
4.8 G u n n Diode 235
Bulk Negative Differential Conduct ivi ty 237
Traveling Waves 238
The Gunn Effect 240
Problems 242
References 243
A p p e n d i x 245
Physical Constants 245
Properties of Si at Room Temperature 245
Subject I n d e x 246
Introduction
Semiconductor device modeling spans a wide range of areas in solid state
physics, applied and computat ional mathematics. The involved topics range
from the most basic principles of kinetic transport in solids over statistical
mechanics to complicated bifurcation problems in the mathematical de-
scription of certain devices and to numerical methods for part ial differential
equations.
This book tries to give an overview of the involved models and their
mathematical treatment. I t addresses, on one hand, the engineer and the
physicist interested in the mathematical background of semiconductor device
modeling. O n the other hand it can be used by the applied mathematician
to familiarize himself (herself) w i th a field which has immediate and techno-
logically relevant applications and gives rise to a whole variety of interesting
mathematical problems. The scope of semiconductor device modeling is
clearly interdisciplinary. Quanti tat ive answers are needed to describe de-
vices and these answers can be obtained from a variety of different physical
models.
We start from the most basic physical principles for kinetic transport of
charged particles. Then we discuss a hierarchy of simplified model equations
culminat ing in the drift diffusion equations which are the most widely used
model today.
I n order to make this book accessible to as wide a range of readers as
possible the emphasis has been placed on concepts, and mathematical details
have been replaced by references to the corresponding literature.
In the first Chapter the classical and quantum mechanical transport models
in ensemble phase space and single particle phase space are discussed.
Furthermore it is shown how the quantum mechanical models can be
incorporated into the classical transport picture via the so called semi-
classical models. The solution of transport equations in phase space is a very
complex task. Therefore, simplified equations for integral quantities, such
as particle and energy densities, are frequently used. These simplified equa-
tions are part ial differential equations in position space only. The derivation
of these equations, i.e. the hydrodynamic models and finally the drift diffu-
sion equations, is the subject of Chapter 2. Chapter 3 is devoted to a
