Table Of ContentComputational
Microelectronics
Edited by
S. Selberherr
Arokia Nathan
Henry Baltes
Microtransducer CAD
Physical and Computational
Aspects
SpringerWienN ewYork
Prof. Dr. Arokia Nathan
Dept. of Electrical and Computer Engineering
University of Waterloo
Waterloo, Ontario N2L 301, Canada
Prof. Dr. Henry Baltes
Physical Electronics Laboratory
ETH Hoenggerberg
CH-8093 Zurich, Switzerland
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.
© 1999 Springer-Verlag/Wien
Softcover reprint of the hardcover 1 st edition 1999
Typesetting: Thomson Press (India) Ltd., New Delhi 110001
Printing: Adolf Holzhausens Nfg. OesmbH, A-1070 Wien
Binding: Papyrus, A-1100 Wien
Printed on acid-free and chlorine-free bleached paper
SPIN: 10637710
With 137 Figures
ISSN 0179-0307
lSBN-13: 978-3-7091-7321-3 e-lSBN-13: 978-3-7091-6428-0
DOl: 10.1007/978-3-7091-6428-0
To Nandanee & Gabriella
Preface
Semiconductor microtransducers have been investigated and developed for
more than three decades while their numerical simulation has been
underway for less than half that time. Integrated Si microtransducers are
realized using microfabrication techniques similar to those for standard
ICs. Unlike IC devices, however, microtransducers must interact with their
environment, so their numerical simulation is considerably more complex.
While the design of ICs aims at suppressing "parasitic effects", micro
transducers thrive on the exploitation and optimization of the one or
the other such effect. The challenging quest for physical models and
simulation tools enabling microtransducer CAD is the topic of this book.
The book is intended as a text for graduate students in Electrical Engi
neering and Physics and as a reference for CAD engineers in the micro
system industry.
The authors have been involved in microtransducer modeling since
the days of the magnetic sensor modeling tool ALBERTINA, whose
development was started in 1983 at the University of Alberta, Edmonton,
Canada. We gratefully acknowledge our partners in those pioneering
efforts: Prof. H. G. Schmidt-Weinmar and Prof. W. Allegretto.
Since then, many colleagues experienced in the art of numerical
modeling of semiconductor IC devices have generously shared their
valuable insights and tools. It is our pleasure to thank Prof. G. Baccarani,
University of Bologna, Prof. S. G. Chamberlain, DALSA Inc., Prof. R.
Dutton, Stanford University, Prof. W. Fichtner, ETH-Zurich, Prof. S. C.
Jain, IMEC, Prof. D. 1. Roulston, University of Waterloo, Prof. M. Rudan,
University of Bologna, Prof. S. Selberherr, University of Vienna (whose
book, Analysis and Simulation of Semiconductor Devices, initiated our
numerical sensor simulation efforts), and last, but not least, Prof. S. D.
Senturia of MIT for his pioneering efforts on CAD of MEMS.
We also would like to acknowledge the contributions to this book made
by our colleagues, former and present research associates, and PhD
students: Dr. K. Aftatooni, dpiX, Inc., Palo Alto; Dr. K. Benaissa, Texas
Instruments, Richardson; Dr. T. Dravia, University of Waterloo; Dr. M.
Ershov, University of Aizu; Dr. 1. Funk, Boston Group, Zurich; Prof. 1. G.
VIII Preface
Korvink, University of Freiburg; M. Kulas, University of Waterloo; Dr. Y.
Lu, IBM, San Jose; Prof. T. Manku, University of Waterloo; M. Nagata,
Yamatake Corp.; Dr. N. 0, DALSA Inc.; Prof. O. Paul, University of
Freiburg, Germany; Prof. T. P. Pearsall, University of Washington; Dr. H.
Pham, University of Waterloo; Q. Ma, University of Waterloo; Dr. C.
Riccobene, AMD, Sunnyvale; M. Stevens, Nortel, Ottawa; and Dr. N. R.
Swart, Analog Devices, Wilmington. Our sincere appreciation goes to Prof.
R. Homsey, University of Waterloo, and S. Taschini, ETH Zurich, for critical
reading of the manuscript, and Prof. W. Huang, Prof. V. Karanassios, and
Prof. A. Khandani, University of Waterloo, for stimulating discussions.
This text evolved from a series of courses offered to graduate students
from Electrical Engineering and Physics; 1994, 1995, and 1996 at the
University of Waterloo, 1995 at the ETH Zurich, where it received signi
ficant additions, and from a short course given at the CAD for MEMS
Workshop, Zurich in 1997. Much of the material in the book can be pre
sented in about 40 hours of lecture time. The text begins with an illustrative
example to highlight the goals and benefits of microtransducer CAD.
Chapter 2 summarizes the model equations describing electrical transport in
semiconductor devices and microtransducers in the absence of external
fields. Models treating the effects of the external radiant, magnetic, thermal,
and mechanical fields on electrical transport are described systematically in
Chapters 3 to 6. Chapter 7 presents an abridged version of solid structural
and fluid mechanics. Here, we focus only on pertinent model equations and
boundary conditions. Model equations and boundary conditions relevant to
the various types of mechanical microactuation are discussed in Chapter 8.
Chapter 9 concludes with a glimpse into simulation techniques for the
mixed-signal microsystem, i.e., microtransducer plus circuitry. Where
possible, we have supplemented the model equations with tables and/or
graphs of process-dependent material data to enable the CAD engineer to
carry out simulations even when reliable material models are not available.
This book would not have been written without the support of our
institutions, ETH Zurich and the University of Waterloo, who granted
sabbatical leaves, and provided hospitality, to both of us in 1995 and 1996.
H. Baltes would like to thank Prof. G. Kovacs for the hospitality at Stanford
University in 1996, which greatly benefited the book.
We are grateful to A. Nathan's wife, Nandanee, who not only prepared
the manuscript by combining competence with devotion to the task, but
also tolerated a living room and basement that were littered with books,
papers, and manuscript pages over the grueling two-year period. As Mark
Twain remarked in The Adventures of Huckleberry Finn (1885). "There
ain't no more to write about, and I am rotten glad of it, because if I'd 'a'
knowed what a trouble it was to make a book I wouldn't 'a' tackled it".
Waterloo and Ziirich, October 1997 Arokia Nathan and Henry Baltes
Contents
Notation XIV
1 Introduction
1.1 Modeling and Simulation of Microtransducers
1.2 Illustrative Example 7
1.2.1 Thermal Flow Sensor 7
1.2.2 Thermal Sensors and Actuators 10
1.2.3 Goals and Benefits of Modeling and Simulation 12
1.3 Progress in Microtransducer Modeling 14
1.4 References 17
2 Basic Electronic Transport 30
2.1 Poisson's Equation 31
2.2 Continuity Equations 33
2.3 Carrier Transport in Crystalline Materials and
Isothermal Behavior 33
2.3.1 Transport Relations 33
2.3.2 Carrier Concentrations 35
2.3.3 Doping-Induced Band Gap Narrowing 38
2.3.4 Temperature-Dependence of Band Gap Energy 40
2.3.5 Carrier Mobility and Matthiessen's Rule 40
2.3.6 Generation-Recombination 46
2.4 Electrical Conductivity and Isothermal Behavior in
Polycrystalline Materials 49
2.4.1 Doping-Dependence 50
2.4.2 Temperature-Dependence 55
2.5 Electrical Conductivity and Isothermal Behavior in Metals 55
2.6 Boundary and Interface Conditions 60
2.6.1 Ohmic Contacts 61
2.6.2 Schottky Contacts 63
2.6.3 Insulators and Interfaces 63
2.6.4 Outer Boundaries 64
2.7 The External Fields - What Do They Influence? 64
2.8 References 65
x Contents
3 Radiation Effects on Carrier Transport 69
3.1 Reflection and Transmission of Optical Signals 71
3.1.1 Single- and Multi -Layer Thin Film Systems 72
3.2 Modeling Optical Absorption in Intrinsic Semiconductors 74
3.2.1 Band-to-Band Transitions 75
3.2.2 Absorption Coefficient 77
3.3 Absorption in Heavily-Doped Semiconductors 78
3.3.1 Band-to-Band Absorption Coefficient 79
3.3.2 Free Carrier Absorption Coefficient 82
3.4 Optical Generation Rate and Quantum Efficiency 82
3.5 Low Energy Interactions with Insulators and Metals 84
3.5.1 Refractive Index and Extinction Coefficient 84
3.6 High Energy Interactions and Monte Carlo Simulations 89
3.6.1 Photoelectric Effect, Compton Scattering,
and Pair Production 89
3.6.2 Ionization Yield 92
3.6.3 Photon Attenuation Coefficients 93
3.6.4 Monte Carlo Simulations 94
3.7 Model Equations for Radiant Sensor Simulation 96
3.8 Illustrative Simulation Example - Color Sensor 97
3.9 References 100
4 Magnetic Field Effects on Carrier Transport 104
4.1 Galvanomagnetic Transport Equation 106
4.1.1 Galvanomagnetic Effects 111
4.2 Galvanomagnetic Transport Coefficients 113
4.2.1 Magnetic Field Dependence 113
4.2.2 Electric Field Dependence 121
4.3 Equations and Boundary Conditions for Magnetic
Sensor Simulation 124
4.3.1 Unipolar Analysis 125
4.3.2 Bipolar Analysis 129
4.4 Illustrative Simulation Example - Micromachined
Magnetic Vector Probe 133
4.5 References 136
5 Thermal Non-Unifonnity Effects on Carrier Transport 140
5.1 Non-Isothermal Effects 142
5.1.1 The Seebeck, Peltier, and Thomson Effects 143
5.1.2 Wiedemann-Franz Law 145
5.2 Electrothermal Transport Model 147
5.2.1 Governing Equations 147
5.2.2 Boundary Conditions 149
5.3 Electrical and Thermal Transport Coefficients 152
Contents Xl
5.3.1 The Seebeck Coefficient in Semiconductors and
Metals 152
5.3.2 Thermal Conductivity in Semiconductors, Metals,
and Dielectrics 161
5.3.3 Specific Heat in Semiconductors, Metals, and
Dielectrics 172
5.4 Electro-Thermo-Magnetic Interactions 174
5.5 Heat Transfer in Thermal Microstructures 177
5.5.1 Governing Equations for Convective Heat Transfer 178
5.5.2 Zero Flow Two-Dimensional Heat Transfer
Coefficient 180
5.5.3 Thermal Conductivity of Gases 182
5.5.4 Radiative Heat Transfer 185
5.5.5 Model Simplification for Quasi Three-Dimensional
Analysis 186
5.6 Summary of Equations and Computational Procedure 191
5.7 Illustrative Simulation Example - Micro Pirani Gauge 193
5.8 References 196
6 Mechanical Effects on Carrier Transport 202
6.1 Piezoresistive Effect 204
6.1.1 Piezoresistance Coefficients in Monocrystalline
Semiconductors 205
6.1.2 Doping- and Temperature-Dependence of
Piezoresistance Coefficients 209
6.1.3 Non-Linear Piezoresistance Coefficients 211
6.1.4 Piezoresistance Coefficients in Poly crystalline
Semiconductors 212
6.2 Strain and Electron Transport 215
6.2.1 Conduction Band 215
6.2.2 Electron Mobility and Piezoresistance 219
6.3 Strain and Hole Transport 221
6.3.1 Valence Band 221
6.3.2 Hole Mobility and Piezoresistance 224
6.4 Piezojunction Effect 231
6.5 Effects of Stress Gradients 232
6.5.1 Electron Transport 234
6.5.2 Hole Transport 236
6.5.3 Phonon Transport and Heat Flux 238
6.5.4 Thermodynamic Consideration of
Electro-Thermo-Mechanical Interactions 238
6.6 Galvano-Piezo-Magnetic Effects 240
6.6.1 Piezo-Hall Coefficients 240
6.7 The Piezo Drift-Diffusion Transport Model 243
