Monte Carlo Simulation of Semiconductor Devices

343 Pages·1993·6.868 MB·English

by  C. Moglestue (auth.)

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To my wiJe, Dela, and children Andrew and Helen
First edition 1993
© 1993 C. Moglestue
Originally published by Chapman & Hali in 1993
Softcover reprint of the hardcover 1s t edition 1993
ISBN 978-90-481-4008-4 ISBN 978-94-015-8133-2 (eBook)
DOI 10.1007/978-94-015-8133-2
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vi Contents
4.3 Phonon Selection Rules 84
4.4 Phonon-Electron Interaction 87
4.4.1 Acoustic Phonon Scattering 90
4.4.2 Interva/ley Scattering 95
4.4.3 Polar Optical Phonon (Fröhlich) Scattering 98
4.4.4 Acoustic Piezoelectric Scattering 100
4.4.5 Higher Order Scattering 101
4.4.6 Refinement of the Scattering Rate Formulae 102
4.5 Scattering from Electric Charges 103
4.5.1 Ionised Impurity Scattering 103
4.5.2 Remote Polar Optical and Ionised Impurity Scattering 105
4.5.3 Carrier-Carrier Scattering 106
4.6 Scattering from Neutral Imperfections 109
4.6.1 A/loy Scattering 109
4.6.2 Neutral Impurity Scattering 110
4.6.3 Dislocation Scattering III
4.7 Impact Ionisation III
4.8 Trapping and Release of Carriers 112
4.9 Time-Dependent Scattering 113
5. The Monte Carlo Method 115
5.1 Introduction 115
5.2 Generation of Random Numbers 115
5.3 Non-uniform Random Numbers 116
5.4 The Time of Free Flight 119
5.5 Selection of Scattering Events 121
5.6 Aigorithms to Shorten the Search for Time of Free Flight 123
5.7 Choice of Scattering Angles 126
5.8 Motion of Particles in the Local Electromagnetic Field 128
6. Simulation of Bulk Properties of Solids 130
6.1 Introduction 130
6.2 The Drift Velocity 132
6.3 The Carrier Distribution 144
6.4 Energy and Momentum Relaxation 149
6.5 Free Flight Path 153
6.6 Two-Dimensional Transport at Interfaces 154
7. The Field Equation 161
7.1 Introduction 161
7.2 Choice of Mesh 162
7.3 Assignment of Charge to the Mesh 164
7.4 The Fast Fourier Transform 166
7.5 The Boundary Conditions 170
8. Steady State Simulation of Devices 173
8.1 Introduction 173
8.2 Definition of the Device Geometry and the Field-Adjusting
Time Step 176
8.3 Contacts and Surface Charges 180
Contents vii
8.3.1 The Schottky Contact 181
8.3.2 The Ohmic Contacts 183
8.3.3 Free Surfaces 185
8.4 Initialisation of a Simulation 185
8.5 The Superparticle 186
8.6 The Steady State Characteristics 187
8.7 Negative and Positive Differential Resistivity 194
8.8 Wandering Gunn Domains 195
8.9 Luminescence 198
8.10 Heating 199
9. AIternating Current, Microwaves 202
9.1 Introduction 202
9.2 The Fourier Transform, Alternating Current Characteristics 202
9.3 Small Signal Analysis 205
9.4 The Equivalent Circuit 208
9.5 Gain 211
9.6 The Influence of Stray Fields 213
10. Composite Material Devices 216
10.1 Introduction 216
10.2 The Heterojunction 217
10.3 The Heterojunction Transistor 220
10.4 Tunnelling 224
10.5 Confined States. Transport in Quantum Wells 228
10.6 Problems Associated with Comparison with Experimental Data 230
10.7 Experimental Verification of the Particle Model 232
10.8 Field Emission 239
11. Ambipolar Devices 243
11.1 Introduction 243
11.2 The p-n Junction 243
11.3 Photodiodes and Detectors 247
11.4 Effects of a-Radiation on a Transistor 255
11.5 The Bipolar Transistor 258
11.6 Spontaneous Recombination of Electron-Hole Pairs 261
11.7 Stimulated Photon Emission, Lasers 265
12. Noise 267
12.1 Introduction 267
12.2 The Minimum Intrinsic Noise Figure 268
12.3 Theory of Noise 270
12.3.1 The Correlation Current 271
12.3.2 The Fluctuation Field 277
12.3.3 Summary of Noise Theory 282
12.4 Computer Experimental Verification 284
12.5 Turbulence and Chaos 289
13. Computers: Scope of Modelling 291
13.1 Introduction: Aspects of Modelling 291
viii Contents
13.2 Limitations to the Model 292
13.3 Computer Requirements 293
13.4 A Short History of Monte Carlo Modelling 295
13.5 Why Call it the Monte Carlo Particle Model? 296
Appendix. Useful Constants 299
References 300
List of Symbols 311
Index 322
Preface
Particle simulation of semiconductor devices is a rather new field which has
started to catch the interest of the world's scientific community. It represents a
time-continuous solution of Boltzmann's transport equation, or its quantum
mechanical equivalent, and the field equation, without encountering the usual
numerical problems associated with the direct solution. The technique is based
on first physical principles by following in detail the transport histories of indi
vidual particles and gives a profound insight into the physics of semiconductor
devices. The method can be applied to devices of any geometrical complexity and
material composition. It yields an accurate description of the device, which is
not limited by the assumptions made behind the alternative drift diffusion and
hydrodynamic models, which represent approximate solutions to the transport
equation. While the development of the particle modelling technique has been
hampered in the past by the cost of computer time, today this should not be held
against using a method which gives a profound physical insight into individual
devices and can be used to predict the properties of devices not yet manufactured.
Employed in this way it can save the developer much time and large sums of
money, both important considerations for the laboratory which wants to keep
abreast of the field of device research. Applying it to al ready existing electronic
components may lead to novel ideas for their improvement. The Monte Carlo
particle simulation technique is applicable to microelectronic components of any
arbitrary shape and complexity.
Since the purpose of this book is to explain the method, the reader does not
need to have any previous knowledge of particle simulation. The book is intended
for the research physicist, the electrical engineer and the graduate student who
wishes to learn more about this new modelling technique. The method will first
be explained in general terms, then in the form of illustrative examples. Readers
will also draw from the author's experience in modelling so that they can write
their own model software.
To make it easier for the reader to appreciate the method, the book opens with
a few chapters reviewing the essence of solid state physics. Here we discuss crystal
vibrations, the dynamics of electrons and the interaction between electrons and
the lattice, an important ingredient in transport theory. The material should be
sufficient to furnish the reader with the necessary physical background, but no
rigorous proof of the theory will be given as this can al ready be found in the
existing literature. The main topic, i.e. the Monte Carlo particle model, will then
be explained in elementary terms and then enlarged upon in the remainder of the
book. The book also contains elements of electrical engineering but it is neither
intended to be a textbook of this nor of transport theory. Most of the simulated
results which are included as examples to illustrate the method have been
published elsewhere, but there are also some results that have not yet appeared
in the literature. So me examples have also been selected to illustrate the power
and the potential of the method.
x Prejace
The author's simulated results have been obtained from the Science and Engi
neering Research Council's IBM 360/195 at Appleton-Rutherford Laboratory;
the VAX machines at GEC Hirst Research Centre at Wembley; the VAX installa
tions at the Fraunhofer Institute of Applied Solid State Physics at Freiburg im
Breisgau (Germany); and at the CRA Y XMP at the Naval Research Laboratory
in Washington DC. All these organisations are herewith acknowledged for their
generosity in donating computer time.
The symbols adorning the heads of each chapter represent Mayan numerals,
and originate from buildings at Quirigua and Palenque (Ifrah, 1981).
I would also like to express my gratitude towards Dr Feenstra et al. of IBM,
Yorktown Heights, and Dr J. Rosenzweig for permission to reproduce photo
graphs. I would also like to thank my colleagues Dr A. Axmann, Dr M. Berroth,
Dr J. Braunstein, Dr U. Nowotny and Dr J. Rosenzweig for refereeing the manu
script and Dr Prof. H. Rupprecht for his encouragement to produce this book.
Finally I greatly acknowledge Mrs C. Hindrichs for typing the manuscript.
Freiburg im Breisgau, 1992
1
The Foundation of Modelling
1.1 INTRODUCTION
Solid state electronics has had an enormous impact on our lives and now modern
civilisation is almost unimaginable without it. Almost all our machines make use
of electronic controls; computers enable caIculations of complexity which were
previously impracticable; administration is supported by information processing
machinery. Financial institutions like banks, insurance companies and the govern
ment rely increasingly on computers. Medium size and even small firms utilise
electronic book-keeping and stock control as this promises to be more reliable and
accurate than the traditional methods.
The amount of data and the speed with which it is communicated between
humans and machines increases steadily. Modern electronic entertainment is also
a way of transmitting information, although it is a one-way process. Future
development will probably see this changing into a dialogue.
The speed of communication and the complexity of information processing
will undoubtedly increase in future years. This makes it necessary to employ faster
circuits, and therefore faster components. There seems to be an unwritten law
stating that the capacity of whatever we build will be exploited to the full and
that we are then going to require more. This spurs us on to develop increasingly
sophisticated systems. Further applications of electronics are envisaged such as
video telephones, high definition television, anti-collision and guiding equipment
for road vehicles etc. The quest for new markets drives many institutions to invent
alternative uses for electronic equipment.
This development has been enabled by the steady miniaturisation of indivi
dual components, and by making their manufacture cheaper. The miniaturisa
tion started by making the first transistors smaller; today, quantum mechanical
effects such as electron tunnelling through barriers and conduction through
narrow channels with quantised transport are being taken advantage of when
developing better devices. This process will only stop when we reach the limits
set by the laws of physics and it is almost certain that we will come up against
them.
Such a development puts new requirements on the device physicist and engi
neer. Unfortunately, there is a great gap between these two groups of researchers
because of their different training. In our opinion this gap should be bridged and
we hope this book goes so me way towards achieving this.
Considering a single electronic component like a transistor or a diode, the usual
approach is to consider the current as a fluid. AIthough this is only a figurative
description, it does give resuIts that can be verified experimentally. In reality, the
current consists of individual particles that move through the device. Their motion
consists of a sequence of free f1ight terminated by scattering. The paths of these