Table Of ContentComputational
Microelectronics
s.
Edited by Selberherr
Modelling of Interface
Carrier Transport for
Device Simulation
Dietmar Schroeder
Springer-Verlag Wien GmbH
Dr.-Ing. Dietmar Schroeder
Technische Elektronik
Technische Universitiit Hamburg-Harburg
Hamburg, Federal Republic of Germany
This work is subject to copyright.
All rights are reserved, whether the whole or part of the material is concerned, specifically
those of translation, reprinting, re-use of illustrations, broadcasting, reproduction
by photocopying machines or similar means, and storage in data banks.
© 1994 Springer-Verlag Wien
Originally published by Springer-Verlag Wien New York in 1994
Typesetting: Asco Trade Typesetting Ltd., Hong Kong
Printed on acid-free and chlorine free bleached paper
With 69 Figures
ISSN 0179-0307
ISBN 978-3-7091-7368-8 ISBN 978-3-7091-6644-4 (eBook)
DOl 10.1007/978-3-7091-6644-4
Preface
This book represents a comprehensive text devoted to charge transport
at semiconductor interfaces and its consideration in device simulation
by interface and boundary conditions. It contains a broad review of the
physics, modelling and simulation of electron transport at interfaces in
semiconductor devices. Particular emphasis is put on the consistent deriva
tion of interface or boundary conditions for semiconductor device simula
tion. The book is of interest with respect to a wide range of electronic
engineering activities, as process design, device design, process character
ization, research in microelectronics, or device simulator development. It is
also useful for students and lecturers in courses of electronic engineering,
and it supplements the library of technically oriented solid-state physicists.
The deepest roots of this book date back to the mid-seventies. Being a
student of electrical engineering, who was exposed for the first time to
the material of semiconductor device electronics, I was puzzled by noticing
that much emphasis was put on a thorough introduction and understand
ing of the basic semiconductor equations, while the boundary conditions
for these equations received very much less attention. Until today on
many occasions one could get the impression that boundary conditions are
unimportant accessories; they do not stand on their own besides the bulk
transport equations, although it is clear that they are of course a necessary
complement of these.
The problem of boundary conditions returned to me some years later when
our device simulation group at the Technical University of Hamburg
Harburg made some investigations on partitioning the simulation domain
with the goal to use transport models of different accuracy (and complexity)
in different regions of the device. Here, the problem arose that we had
a number of models, materials, and regions, a number of interfaces between
the regions, and a number of external boundaries, where we needed consis
tent interface and boundary conditions for all the occurring combinations.
This was the initial event for me to investigate from scratch the origin
and nature of interface and boundary conditions. The goal was to obtain
VI Preface
a consistent method to systematically derive such conditions for various
different situations. In fact, this book is the compiled result of this effort
of quite some years.
It was my intention to provide as much detail as necessary for the reader to
follow and verify the main reasoning without having to resort to referenced
material or having to guess about the subtleties and approximations during
a number of un-displayed intermediate mathematical steps. Some basic
understanding of semiconductor device physics is required for reading the
book, since these topics are only briefly recalled in Chapter 2. Thus, the
book should be useful to advanced students, who wish to deepen their
understanding of device physics and simulation. Part of the presented mat
ter represents indeed the content of lectures I held in recent years for stu
dents of electrical engineering at the Technical University of Hamburg
Harburg. At the same time, the book is intended to support the work of
engineers and scientists in the semiconductor industry and research la
boratories, who have to deal with problems of interface carrier transport
and appreciate a comprehensive collection of the material, with numerous
references to continuing literature for probing further.
It is my sincere wish to express my appreciation to the many people who
contributed to the book in various ways.
I am extremely grateful to Prof. R. Paul for his encouragement to begin
at all the activity of writing this book, for the opportunities he granted to
me to carry out the project, and for his careful reading of the manuscript.
My very special thanks are due to my wife Helga for her constant support
and encouragement during all these activities. Moreover, she draw mostly
all of the figures, and she never got tired to take care of my most pedantic
wishes for correction of many details.
I am particularly indebted to S. Selberherr for his interest in this work
and for his engagement as the editor of this series.
I like to express my thanks to Prof. G. Lautz for making me enthusiastic
for the field of semiconductor electronics by his excellent lectures on elec
tromagnetic fields and solid-state electronics. I would like to thank also
Prof. W. Schultz for conveying his style of getting to the roots of a subject,
as well as for his inspiring humorous brainwaves.
Many thanks are due to a great number of colleagues all over the world.
For many pleasant and interesting discussions I am very indebted to G.
Baccarani, F. Odeh, D. Ventura, A. Gnudi, W. Fichtner, G. Wachutka, S.
Miiller, A. Schenk, 1. Nylander, G. Nanz, B. Meinerzhagen, M. Lundstrom,
M. Schubert, P. Gough, S. Battersby, R. Kuvecke, W. Schilders, P. Hemker,
P. van der Zeeuw, H. Molenaar, W. Joppich, R. Constapel, C. Ringhofer,
R. Brunetti, V. Axelrad, J. Bloedel, A. Neureuther, A. Wong, G. Chin, W.
Dietrich, A. Hintz, W. Schoenmaker, M. Mosko, K. Jensen, and many
others.
I benefited from many fruitful discussions with colleagues and students
Preface VII
at the Department of Technical Electronics of the Technical University of
Hamburg-Harburg. I particularly appreciate discussions with A. Stelter,
P. Conradi, M. Weber, S. Ebmeyer, S. Becker, M. Brandstetter, and o.
Kalz. Special thanks are due to my students A. Bergemann, R. Philipps,
S. Zschiegner, and T. Ostermann, who carried out much of the numerical
work, and helped to clarify many topics by their critical questions.
I like to express very special thanks to Davide, Gipo, Alberto, Antonella,
Eleonora, and ZsoIt for the warm and friendly atmosphere during a visit of
mine for some weeks at the University of Bologna in October 1991.
I close this preface with the hope that the reader will acknowledge and
benefit from all the efforts that went into the completion of the book. I
have done my best to create this work, and I wish it might be useful to
many colleagues in our profession.
Technical University of Hamburg-Harburg Dietmar Schroeder
Hamburg, August 1993
Contents
1 Introduction 1
2 Charge Transport in the Volume 3
2.1 Quantum Transport 4
2.2 Semi-Classical Transport 6
2.3 Balance Equations 9
2.4 Poisson Equation 14
2.5 Necessity of Boundary Conditions 14
2.5.1 Enumeration of Boundary Conditions for the Boltz
mann Equation 17
2.5.2 Enumeration of Boundary Conditions for the Poisson
and Balance Equations 19
3 General Electronic Model of the Interface 20
3.1 Energy Band Structures at the Interface 20
3.2 Interface States 24
3.3 Surface Charge and Dipole Densities 26
3.4 Electrostatic Interface Conditions 31
4 Charge Transport Across the Interface 37
4.1 Full Quantum Transport 37
4.1.1 Boundary Conditions 37
4.1.2 Interface Conditions 41
4.2 Semi-Quantum Interface Transport 42
4.2.1 Concepts 42
4.2.2 The Relation of the Wave Vectors 47
4.2.3 The Interface Condition 50
4.3 Semi-Classical Interface Transport 53
4.4 Interface Conditions for the Balance Equations 55
x
Contents
4.5 Semi-Classical Interface Conditions for the Balance
Equations 58
4.6 General Band Structures 61
4.6.1 Transition 61
4.6.2 Reflection 63
4.6.3 An Example 65
4.6.4 The Interface Condition for General Bands 68
4.7 Non-Conservative Interface Conditions 70
4.8 Multiple Bands and Interface States 74
5 Semiconductor-Insulator Interface 79
5.1 Interface Conditions 79
5.2 Boundary Conditions 81
5.2.1 Continuity Equations 81
5.2.2 Poisson Equation 89
5.3 Virtual Boundaries 90
6 Metal-Semiconductor Contact 93
6.1 Band Model and Current Transport Mechanisms 93
6.1.1 Band Model 93
6.1.2 Transport Mechanisms 97
6.2 Non-Ideal Contact 100
6.2.1 Transmission Probability 102
6.2.2 Setting Up the Boundary Condition 106
6.2.3 Approximation of the Fermi-Dirac Distribution 111
6.2.4 Evaluation of the Contact Current 113
6.2.5 Boundary Condition for the Poisson Equation 119
6.2.6 Computation of the Maximum Field 124
6.2.7 Minority Carriers and Bipolar Model 128
6.2.8 Summary 134
6.2.9 Discussion 138
6.3 Schottky Contact 141
6.4 Ohmic Contact 148
7 Semiconductor Heterojunction 154
7.1 Heterojunction Devices 156
7.1.1 Real-Space Transfer Transistor 157
7.1.2 Hetero-Bipolar Transistor 159
7.2 Interface Conditions for the Heterojunction 160
7.3 Literature Review 168
Contents XI
8 M OSFET Gate 172
8.1 Surface Scattering 172
8.2 Surface Mobility Degradation 176
9 Discretization 185
9.1 Volume Discretization 185
9.2 Boundary Conditions 189
9.3 Interface Conditions 195
9.3.1 One Element at the Interface 198
9.3.2 Two Nodes on the Interface 198
9.3.3 Two Elements at the Interface 199
Appendices 200
A Transformation of k-Vectors 200
B Conservation of Transverse Momentum 202
C Calculation of of YfT 204
D Approximation of Surface Mobility 207
Bibliography 209
Index 218
