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Electromagnetic physical modeling of microwave devices and circuits

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ELECTROMAGNETIC PHYSICAL MODELING OF MICROWAVE DEVICES AND
CIRCUITS
by
Yasser A. Hussein
A Dissertation Presented in Partial Fulfillment
of the Requirements for the Degree
Doctor of Philosophy
ARIZONA STATE UNIVERSITY
August 2003
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UMI Number: 3094964
UMI
UMI Microform 3094964
Copyright 2003 by ProQuest Information and Learning Company.
All rights reserved. This microform edition is protected against
unauthorized copying under Title 17, United States Code.
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ELECTROMAGNETIC PHYSICAL MODELING OF MICROWAVE DEVICES AND
CIRCUITS
by
Yasser A. Hussein
has been approved
July 2003
APPROVED:
, Co-Chair
, Co-Chair
Supervisory Committee
ACCEPTED:
a M
Dean, Graduate College
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ABSTRACT
Electromagnetic physical modeling (global modeling) is the accurate approach to
model today’s high frequency active devices. The technique couples the device physics
and electromagnetics into a single package. There are many challenges in the
implementations o f the global modeling techniques. The most prominent ones are CPU­
time requirements, stability, and accuracy. Accordingly, global modeling techniques
should be based on efficient computer aided design (CAD) tools.
In this dissertation, a complete description of the global modeling technique is
provided, with an emphasis on a hydrodynamic model (HD) coupled with Maxwell’s
equations.
Moreover, electromagnetic physical modeling is carried out for complex
microwave structures. This includes closely packed microwave transistors and
multifinger transistors.
The second part of this dissertation deals with introducing new numerical
techniques to efficiently perform the global modeling approach. The new numerical
techniques are based on wavelets and genetic algorithms (GAs).
A genetic algorithm is employed to solve the equations that describe the
semiconductor transport physics in conjunction with Poisson’s equation. An objective
function is formulated, and most of the GA parameters are recommended to change
during the simulation. Furthermore, the effect of different GA parameters is analyzed.
The technique is validated by simulating a submicrometer field effect transistor (FET),
and then compared to successive over relaxation (SOR); showing the same degree of
accuracy along with a moderate speed of convergence.
iii
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Also in this dissertation, a new wavelet-based time-domain simulation approach
for large-signal physical modeling of high frequency semiconductor devices is presented.
The proposed approach solves the complete hydrodynamic model and Maxwell’s
equations on nonuninform multiresolution self-adaptive grids. The nonuniform grids are
obtained by applying wavelet transforms followed by hard thresholding. A general
criterion is mathematically defined for grid updating within the simulation. In addition,
an efficient thresholding formula is proposed and verified. Different numerical examples
are presented along with illustrative comparison graphs showing more than 75%
reduction in CPU time, while maintaining the same degree of accuracy achieved using a
uniform grid case.
iv
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I dedicate this dissertation to my parents for their deep love and support.
v
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ACKNOWLEDGMENTS
This dissertation would not have been completed without the will and blessing of
God, the most gracious, and the most merciful.
Next, I would like to express my gratitude to my advisor, Prof. Samir El-Ghazaly,
for his continuous support and guidance. This dissertation would not have been finished
without Prof. El-Ghazaly’s invaluable ideas, devoted time, and support. Also, I will
always be indebted to Prof. El-Ghazaly for accepting to be my advisor and helping me
get admitted to ASU. I was privileged working under his supervision and being a member
of his research group.
I would like to express my appreciation and gratitude to Prof. Steven Goodnick
for accepting to be my co-advisor. I would also like to thank Prof. Goodnick for his
support and interest in my research along with valuable discussions and suggestions
during the research presentations. My appreciation also goes to my committee members,
Prof. George Pan, Prof. Hans Mittelmann, and Dr. Rodolfo Diaz for their interest and
time devoted to the considerations of this dissertation. All of them taught me something
and I will always be grateful.
I would also like to express my appreciation to Prof. Saad Eid of Cairo
University, who introduced genetic algorithms to me.
I am also grateful to my teachers at ASU, especially Prof. Carl Gardner, Prof.
Christian Ringhofer, and Dr. Badawy El-Sharawy.
Sincere appreciation and respect are due to my past and present friends and
colleagues of the Telecommunication Research Center for their help and useful
discussions. In particular, I would like to thank Dr. Emmanuel Larique, Dr. Sebastien
vi
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Goasguen, Dr. Kai Liu, Mr. Munes Tomeh, and Mr. Ekram Bhuiyan. Many thanks to
my colleagues Mohammed Waliullah, Sung-Woo Lee, Yong-Hee Park, and Aly Aly.
This research has been supported by a grant provided by the U.S. Army Research
Office. I would like to thank them generously for their contributions.
vii
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TABLE OF CONTENTS
Page
LIST OF TABLES................................................................................................................... xi
LIST OF FIGURES................................................................................................................ xii
CHAPTER
1 INTRODUCTION..................................................................................................................1
1.1
Motivations and Obj ective of This Study............................................................... 1
1.2
Dissertation Overview..............................................................................................5
1.3
Original Contributions............................................................................................. 7
1.4
Publications............................................................................................................... 9
2 GLOBAL MODELING OF COMPLEX MICROWAVE STRUCTURES....................12
2.1
Introduction..............................................................................................................12
2.2
Equivalent Circuit Models..................................................................................... 13
2.3
Physics-Based Models ...........................................................................................14
2.4
The Hydrodynamic Model (H D )...........................................................................15
2.4.1
Hydrodynamic Model Limitations................................................................. 17
2.4.2
Single Gas Approximation.............................................................................. 19
2.4.3
The Relaxation Time Approximation........................................................... 20
2.4.4
Transport Parameters......................................................................................20
2.4.5
Boundary Conditions Implementation.......................................................... 22
2.4.6
Discretization of the HD Model.....................................................................25
viii
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CHAPTER
2.4.7
2.5
Page
2.4.6.1
Discretization of the Continuity Equation.......................................... 26
2.4.6.2
Discretization of the Energy Conservation Equation......................... 27
2.4.6.3
Discretization of the Momentum Conservation Equation.................27
Hydrodynamic Model DC Results.................................................................28
Full-Wave Physical Simulations............................................................................ 33
2.5.1
Error and Stability Analysis........................................................................... 36
2.5.2
Electromagnetic-Physical Coupling...............................................................37
2.5.3
Absorbing Boundary Conditions...................................................................38
2.5.3.1
MUR Absorbing Boundary Conditions.............................................. 38
2.5.3.2
PML Absorbing Boundary Conditions................................................39
2.6
Microwave Characteristics......................................................................................39
2.7
EM-Wave Propagation Effects.............................................................................. 42
2.8
Summary.................................................................................................................. 51
3 MODELING AND OPTIMIZATION OF MICROWAVE DEVICES AND CIRCUITS
USING GENETIC ALGORITHMS (GAS)......................................................................... 52
3.1
Introduction..............................................................................................................52
3.2
Genetic Algorithms: Literature Overview............................................................ 53
3.3
Problem Description............................................................................................... 58
3.4
Optimization Using Genetic Algorithms.............................................................. 61
3.5
Results And Discussions.........................................................................................69
3.5.1
DC Simulation Results....................................................................................77
ix
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CHAPTER
3.5.2
3.6
Page
AC Simulation Results...................................................................................79
Summary................................................................................................................... 81
4 A NEW WAVELET-BASED TIME-DOMAIN TECHNIQUE FOR MODELING AND
OPTIMIZATION OF HIGH-FREQUENCY ACTIVE DEVICES.................................... 82
4.1
Introduction...............................................................................................................82
4.2
Fundamentals of the M RTD ................................................................................... 84
4.2.1
Two Dimensional MRTD Scheme................................................................. 85
4.2.2
Battle-Lemarie Expansion Basis.................................................................... 88
4.3
Problem Description................................................................................................ 93
4.4
The Proposed Algorithm..........................................................................................95
4.5
Results And Discussions........................................................................................107
4.5.1
Hydrodynamic Model DC Simulation Results........................................... 107
4.5.2
Hydrodynamic Model AC Simulation Results........................................... 113
4.5.3
FDTD Simulation Results............................................................................117
4.6
Scheme Errors and Stability Analysis................................................................... 119
4.7
Summary................................................................................................................. 121
5 CONCLUSIONS AND FUTURE WORK...................................................................... 122
5.1
Conclusions............................................................................................................. 122
5.2
Future W ork............................................................................................................124
REFERENCES......................................................................................................................125
x
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LIST OF TABLES
Table
Page
2.1
Transistor parameters used in the simulations....................................................... 29
2.2
Multifmger transistor optimized-parameters used in the simulation................... 50
3.1
GA parameters used in the simulation.................................................................... 69
4.1
Coefficients a(i),b0(i),c0( i ) ...................................................................................91
4.2
Transistor parameters used in the simulation......................................................... 95
4.3
Grid adaptability o f the different variables for T0 = 1% ......................................103
4.4
Effect o f the threshold value on error and CPU-time for FDTD simulations..! 17
xi
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LIST OF FIGURES
1.1
Output power versus frequency of millimeter-wave devices: solid lines,
tubes; dashed line, solid-state devices. After Sieger et al. [1]................................4
2.1
A typical FET cross-section view............................................................................ 23
2.2
Carrier density distribution....................................................................................... 29
2.3
Electron energy distribution (eV).............................................................................30
2.4
The distribution of the x-direction electric field (V/cm).......................................30
2.5
The distribution of the y-direction electric field (V/cm).......................................31
2.6
The distribution o f the x-direction velocity (cm/sec.).........................................31
2.7
The distribution of the y-direction velocity (cm/sec.)......................................... 32
2.8
Potential distribution (V).......................................................................................... 32
2.9
I-V Characteristics.....................................................................................................33
2.10
Genetic view o f the electromagnetic computational domain.................................34
2.11
Attenuation constant as a function of frequency at different points along the
device width for the gate electrode......................................................................... 40
2.12
Effective dielectric constant as a function of frequency at different points
along the device width for the gate electrode........................................................ 41
2.13
Phase velocity as a function of frequency at different points along the device
width for the gate mode........................................................................................... 42
2.14
3D view of the simulated transistors (not to scale).................................................43
xii
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Figure
2.15
Page
Drain voltage (normalized) of the simulated transistor when EM-wave
propagation and electron-wave interaction are considered at different points
in the z-direction.......................................................................................................44
2.16
The potential of a passive electrode at different points in the z-direction
induced due to the proximity of an operating transistor excited by a
Gaussian signal.........................................................................................................45
2.17
Drain voltage (normalized) at z = 62.5mm when EM-wave propagation and
electron-wave interaction are considered. Solid line: transistor is simulated
alone. Dashed line: Source electrode of a second operating transistor is
0.5jum apart from the drain of the simulated transistor........................................ 45
2.18
Drain voltage (normalized) at z = 62.5mm when EM-wave propagation and
electron-wave interaction are considered. Solid line: transistor is simulated
alone.
Dotted line: Drain electrode of a second operating transistor is
0.5fim apart from the drain of the simulated transistor........................................ 46
2.19
Generic 3D view of the simulated multifmger transistors (not to scale), (a)
Single-finger transistor (lx 450pm). (b) Two-finger transistor (2x 225pm).
(c) Four-finger transistor (4x112.5pm).................................................................. 49
2.20
Output voltage for the simulated multifinger transistors when EM-wave
propagation and electron-wave interactions are considered.................................50
3.1
Cross-section of the simulated transistor................................................................ 60
xiii
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Figure
Page
3.2
Generic flowchart of the genetic algorithm.............................................................64
3.3
Flowchart of the randomly generated solution G .................................................. 65
3.4
A 5 by 5 grid example illustrating how the algorithm in Fig. 3.3 works...........66
3.5
Distance from the optimal solution versus number
of generations for
different mutation values......................................................................................... 73
3.6
Distance from the optimal solution versus number
of generations for
different number of crossover points..................................................................... 73
3.7
Distance from the optimal solution versus number
of generations for
different values of crossover widths....................................................................... 74
3.8
Distance from the optimal solution versus number
of generations for
different numbers of mutated elements.................................................................. 74
3.9
Distance from the optimal solution versus number
of generations for
different selection criteria........................................................................................ 75
3.10
Distance from the optimal solution versus number
of generations for
different probabilities of mutation..........................................................................75
3.11
Distance from the optimal solution versus number
of generations for
different population sizes........................................................................................ 76
3.12
Sample DC results obtained using the proposed algorithms, (a) Potential
distribution, (b) Carrier density distribution..........................................................77
3.13
AC gate and drain voltages obtained using the proposed algorithm ................... 78
4.1
Cross-section of the simulated transistor................................................................94
xiv
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4.2
Generic flowchart of the proposed algorithm......................................................... 97
4.3
(a) Normalized details coefficients for the electron energy at a certain
transverse cross-section, (b) Grid points marked on the actual curve for the
electron energy at the same transverse cross-section, (c) Normalized details
coefficients for the x-momentum at a certain transverse cross-section, (d)
Grid points marked on the actual curve for x-momentum at the same
transverse cross-section........................................................................................ 100
4.4
(a) Normalized details coefficients for the electron energy at a certain
longitudinal cross-section, (b) Grid points marked on the actual curve for the
electron energy at the same longitudinal cross-section, (c) Normalized
details coefficients for the x-momentum at a certain longitudinal crosssection. (d) Grid points marked on the actual curve for x-momentum at the
same longitudinal cross-section............................................................................ 101
4.5
Demonstration o f the procedure employed to obtain the nonuniform grid for
the y-direction electric and magnetic fields for FDTD simulations..................106
4.6
Remaining number of unknowns as a percentage versus the iteration number
for different initial threshold values......................................................................108
4.7
DC drains current convergence curves for the uniform grid and the proposed
wavelet-based non-uniform grids for different initial threshold values
4.8
109
DC potential distribution obtained by the proposed algorithm using a value
of the initial threshold equals to 1%..................................................................... I l l
xv
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4.9
Comparison between the uniform-grid and the proposed algorithm with a value
of the initial threshold equals to 1% (a) Gate-to-source capacitance versus
gate-to-source voltage, (b) Transconductance versus gate-to-source voltage. .114
4.10
Large-signal result obtained by the proposed algorithm for a value of the
initial threshold equals to 1%................................................................................ 116
4.11
AC output voltage for the uniform grid and the proposed wavelet-based nonuniform grids with different values of the initial threshold................................ 116
4.12
Potential of the gate at a specific cross-section versus time for the uniform
grid case and the proposed MRTD algorithm with different values of T0
xvi
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118
Chapter 1
Introduction
1.1
Motivations and Obj ective of This Study
The quest for high-power, lightweight, compact-size millimeter-wave components
is becoming more demanding. Fig. (1.1) shows a comparison of power levels of various
tube devices and solid-state devices. It is noteworthy to say that tubes produce highpower at millimeter-wave frequencies. However, they are becoming less desirable
because o f their reliability, bulky-size, and the need for high-voltage DC power supplies.
As the millimeter-wave components evolve, new modeling techniques should be
developed to accurately design and optimized these devices. As the frequency increases,
the wave-length of the operating frequency becomes comparable to device dimensions.
Moreover, the time-period of the operating frequency becomes comparable to electron
relaxation times. This forces us to address topics such as the electromagnetics and device
physics on more than an individual basis. This is achieved by developing coupled
electromagnetic-physics-based simulators or global modeling simulators [2],
Electromagnetic physical simulation is carried out by solving Maxwell’s
equations in conjunction with a physical device model. There are different physical
models with different range of validity and CPU-time requirements. For instance, the
hydrodynamic model (HD) and the drift-diffusion model (DD) are both derived from the
Boltzman transport equation (BTE) with different truncation order of moments and are
considered fluid-based physical models. The DD model is an approximation for the HD
model, which can be employed only if hot-electron and velocity overshoot phenomena do
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2
not take place. On the other hand, Monte Carlo simulations of the BTE are considered
particle-based models, which are much more accurate than fluid-based models. BTE
simulations are implemented for ultra small devices, where the assumptions to model
electrons as fluids break down. It is thus imperative to carefully choose each of these
physical models depending on the problem under consideration. It is not optimum to
employ, for instance, BTE simulations for devices where HD models are still valid and
accurate. The reason is BTE simulations need more CPU-time compared to HD models.
From the above, one can conclude that modeling is a very powerful tool that
should be used to design and optimize microwave devices and circuits. Accordingly, new
numerical modeling techniques should be developed to efficiently model today’s
microwave components and chips. The issues of stability, accuracy, and CPU-time of the
new numerical techniques should be carefully addressed and maintained.
In this dissertation, efficient numerical techniques have been developed and
successfully applied to Maxwell’s equations and the highly nonlinear HD model. The
numerical techniques are based on wavelets and genetic algorithms.
A genetic algorithm has been developed and successfully applied to solve
Poisson’s equations in conjunction with the HD model. The proposed technique solves
the equations that describe the semiconductor transport physics in conjunction with
Poisson’s equation, employing an adaptive real-coded GA.
An objective function is
formulated, and most o f the GA parameters are recommended to change during the
simulation. In addition, different methods for describing the way the GA parameters
change are developed and studied. The effect of GA parameters including mutation value,
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3
number o f crossover points, selection criteria, size of population, and probability of
mutation is analyzed. The technique is validated by simulating a submicrometer field
effect transistor (FET), and then compared to successive over relaxation (SOR); showing
the same degree of accuracy along with a moderate speed of convergence. The purpose
o f this study was to introduce a new vision for a genetic algorithm capable of optimizing
real value functions with a considerably large number of variables. This study also
represents a fundamental step toward applying GAs to Maxwell’s equations in
conjunction with the hydrodynamic model (HDM), aiming to develop an optimized and
unconditionally stable global-modeling simulator
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Fig. 1.1. Output power versus frequency of millimeter-wave devices: solid lines, tubes;
dashed line, solid-state devices. After Sieger et al. [1].
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5
Wavelets have been applied, for the first time in literature, to the HD model along
with extending it to Maxwell’s equations. This has achieved the ultimate goal by
developing a fast and unified modeling approach for FDTD and physical simulations
suitable for the global modeling technique. The developed approach solves the partial
differential equations PDEs of either the HD model or Maxwell’s equations on
nonuniform multiresolution self-adaptive grids obtained using wavelets. A CPU-time
reduction of 80% is achieved while maintaining the same degree of accuracy obtained
using the original techniques.
Finally, this dissertation also represents electromagnetic physical modeling results
of complex microwave structures. This includes closely packed millimeter-wave
transistors simulated simultaneously. In additions, electromagnetic-physical modeling of
high-power and frequency multifinger transistors has been carried out and the
preliminary results are given based on ad-hoc optimizations.
1.2
Dissertation Overview
This dissertation is organized as follows:
Chapter two presents global modeling results of complex microwave structures
along with an overview of the details of how global modeling is performed. The chapter
begins with an introduction followed by a comparison between circuit and physics-based
models. Then, the HD model equations are presented in details including the
implementation of single gas and relaxation-time approximations. Transport parameter
estimation is also provided. Different discretization schemes employed for the HD model
are conferred along with boundary condition implementation. Elements of the full-wave
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6
physical simulation are also described. Implementation of the different boundary
conditions for FDTD simulation is provided. Stability and accuracy are discussed. The
coupling between the FDTD model and physical model is then explained. The second
part of chapter two presents full-wave physical simulations of complex microwave
structures. This includes two closely packed millimeter-wave transistors simulated
simultaneously and multifinger transistors. This chapter also presents the results of the
characteristics of high-frequency transistors including attenuation, phase-velocity, and
effective-dielectric constant. The chapter is then wrapped up with a summary.
Chapter three presents an approach for global modeling of microwave devices and
circuits using genetic algorithms. The chapter begins with an introduction and an
overview of genetic algorithms. The details of the proposed genetic-based algorithm are
provided. Complete set of results along with their discussions are given. Different
numerical examples are presented to study the effect of various algorithm parameters.
Finally, chapter summary is provided.
Chapter four presents a new time-domain simulation approach for large-signal
physical modeling of high frequency semiconductor devices, using wavelets. The
proposed approach solves the complete hydrodynamic model and Maxwell’s equations
on nonuniform multi-resolution self-adaptive grids. The nonuniform grids are conceived
by applying wavelet transforms followed by hard thresholding. Chapter four begins with
an introduction followed by an overview of the multiresolution time-domain (MRTD)
techniques. Then, the problem under consideration is described. Details of algorithm
implementation for both the HD model and FDTD simulation are also presented. A
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7
comprehensive set of results is included along with illustrative comparison graphs.
Chapter summary is then provided.
Chapter five gives the overall conclusions of this dissertation. Future work and
possible research extensions are also provided.
1.3
Original Contributions
The research work presented in this dissertation has led to the following
contributions:
A complete genetic-based software package to optimize functions with a large
number of unknowns has been developed. An entirely new concept, namely fitnessdependent GA parameters, has been introduced and implemented. A novel vision for
formulating the objective function is also provided. Two versions of the software
package were developed using C/C++ and Fortran programming languages. This
research represents a fundamental contribution in which GAs can be employed to
optimize functions with a large number of unknowns or to solve problems that have
stability constraints in order to have unconditionally stable algorithm. Genetic algorithms
are also superior over traditional gradient-based algorithms in that they do not get stuck
in local minima. Thus, genetic algorithms are suitable of finding a global solution for
problems with multiple extreme.
A wavelet-based algorithm has been developed from the scratch. The algorithm
solves the hydrodynamic partial differential equations on nonuniform self-adaptive grids
obtained using wavelets. This is the first time in literature to introduce a wavelet-based
HD model simulator. Moreover, an efficient grid-updating criterion has been introduced.
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8
The algorithm is general and independent of the problem and equations under
consideration, which is found suitable to solve a wide range of problems. The developed
software package is implemented in Fortran. An 80% reduction in CPU-time has been
achieved using the proposed algorithm. The reason is the proposed wavelet-based
algorithm removes the redundancies of the original formulations.
The same wavelet-based algorithm developed for the HD model is extended to
three-dimensional (3-D) FDTD simulation. The developed algorithm is general that it
can be applied to any type o f problem including inhomogeneous and anisotropy media.
This is in contrast o f the developed MRTD approaches available today. These
approaches require careful formulation for each type of problem along with being very
difficult to be extended to 3-D problems. This underlines the generality and versatility of
the proposed technique. Fortran is used to develop the software package of the
algorithm.
Electromagnetic physical modeling of complex microwave structures is presented
in this dissertation. For the first time in literature, two closely packed millimeter-wave
transistors are simulated simultaneously employing a coupled electromagnetic-physical
model. The results show that EM-wave propagation effects should be considered, not
only inside the device, but around it as well. This study is a fundamental step toward
electromagnetic physical modeling of several microwave components simultaneously.
Electromagnetic physical modeling of high-power and frequency multifinger
transistors is presented in this dissertation. The preliminary results of this dissertation
show that at very high frequency, several phenomena with strong impact on the device
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9
behavior start to emerge, such as phase velocity mismatches, electron-wave interaction,
and attenuation. The results suggest that contemporary microwave devices should be
optimized to minimize these effects or possibly take advantage of in favor of improved
device characteristics. The results also recommend multifinger transistors as potential
alternatives to conventional transistors. This is achieved by using multiple-finger gates of
less width instead of a single-gate device. Furthermore, this dissertation underlines the
enhanced microwave characteristics of multifinger transistors attributable to reducing
attenuation and EM-wave propagation effects along the device width.
1.4
Publications
The work associated with this dissertation resulted in the following publications [2]-[17]:
•
Y. A. Hussein, M. Wali, and S. M. El-Ghazaly,” Efficient Simulators and Design
Techniques for Global Modeling of High-Frequency Active Devices,” in Advances in
RF Design, Editor: J. Kiang, Kluwer Academic Publishers, in press (to appear 2003).
•
Yasser A. Hussein and Samir M. El-Ghazaly, “ Extending Multiresolution Time
Domain Technique (MRTD) To The Simulation of High-Frequency Active Devices,”
IEEE Transactions on Microwave Theory and Techniques, in press and to appear July
2003.
•
Yasser A. Hussein and Samir M. El-Ghazaly, “ Global Modeling of Microwave
Devices and Circuits Using a Genetic-Based Optimization Technique,” IEEE
Transactions on Microwave Theory and Techniques, to appear January 2004.
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•
Yasser A. Hussein and Samir M. El-Ghazaly, “ Global Modeling of Microwave
Devices Using a New Multiresolution-Time Domain (MRTD) Technique,” IEEE
Microwave and Wireless Components Letters, submitted.
•
Yasser A. Hussein, Samir M. El-Ghazaly, and Stephen M. Goodnick,“ An Efficient
Electromagnetic-Physics-Based
Technique
For Accurate
Modeling
of High-
Frequency Multifinger Transistors,” IEEE Transactions on Microwave Theory and
Techniques, to appear December 2003.
•
Yasser A. Hussein, Samir M. El-Ghazaly, and Stephen Goodnick, “A New WaveletBased Technique for Full-wave Physical Simulation of Millimeter-wave Transistors,”
presented at the MTT-s International Microwave Symposium (long paper),
Philadelphia 2003.
•
Yasser A. Hussein, Samir M. El-Ghazaly, Yong-Hee Park, and Stephen Goodnick,
“EM-Wave Effects On Closely Packed Microwave Transistors Using a Fast TimeDomain Simulation Approach,” presented at the MTT-s International Microwave
Symposium (longpaper), Philadelphia 2003.
•
Yong-Hee Park, Yasser A. Hussein, Samir El-Ghazaly, Vijay Nair, and Herb
Goronkin, “Effect o f Bonding-Wire On Electrically Tunable Microstrip Antennas,”
invited for Presentation at APS, Columbus-Ohio 2003.
•
Samir M. El-Ghazaly, Stephen Goodnick, Yasser A. Hussein, et-al., “Hierarchy of
Global Modeling Simulations: From Circuit-Based to Physics-Based Models,”
presented at the Asia-Pacific Microwave Conference Workshop, Japan, Nov. 2002.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
•
Yasser A. Hussein and
Samir M. El-Ghazaly,
“Global Modeling of Active
Microwave Devices Incorporating a Novel Time-Domain Large-Signal FullHydrodynamic Physical Simulator Using Wavelet-Based Adaptive Grids,” presented
at the MTT-s International Microwave Symposium (longpaper), Seattle, WA, U.S.A.,
June 2002.
•
Samir M. El-Ghazaly, Stephen Goodnick, Yasser A. Hussein, et-al., “Discretization
and Circuit-Based Simulation of High Frequency Devices and Circuits Including
Distributed Effects,” presented at the MTT-s International Microwave Symposium
Workshop, Seattle, WA, U.S.A., June 2002.
•
Yasser A. Hussein and
Samir M. El-Ghazaly,
“Global Modeling of Active
Microwave Devices Using Genetic Algorithms,” presented at APS/URSI, San
Antonio, TX, U.S.A., June 2002.
•
Yasser A. Hussein and
Samir M. El-Ghazaly,
“ Global Modeling of Active
Microwave Devices Using Wavelets,” presented at APS/URSI, San Antonio, TX,
U.S.A., June 2002.
•
Yasser A. Hussein and Samir M. El-Ghazaly, “ Large-Signal Physical Modeling of
Active Microwave Devices Using an Adaptive Real-Coded Genetic Algorithm,”
presented at APS, San Antonio, TX, U.S.A., June 2002.
•
Yasser A. Hussein and Samir M. El-Ghazaly, “Global Modeling of Compact HighSpeed Circuits,” US Army Research Office Annual Reports (DAAD19-99-1-0194),
May 2002/2003.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Chapter 2
Global Modeling of Complex Microwave Structures
2.1
Introduction
Modeling is a very powerful tool since it is very helpful to know the device
characteristics before fabrication. This would save us time and money because fabrication
will be carried out at the end of the design cycle. Accordingly, optimization and design of
microwave devices should be based on modeling techniques that incorporate every aspect
including the physics of the device as well as electromagnetic-wave propagation effects.
Modem high performance electronics are based on technologies such as
monolithic microwave integrated circuits (MMICs), with a large number of closely
packed passive and active structures, and several levels of transmission lines and
discontinuities. These devices operate at high speeds, frequencies, and often over very
broad bandwidths. It is thus perceptible that the design of MMICs should be based on
robust design tools that would simulate all the circuit elements simultaneously. The
possibility of achieving this type of modeling is addressed by global circuit modeling that
has been demonstrated in [18]-[45].
Device Models can be classified into two main categories: physical device models
and equivalent circuit models. Each model has its own advantages and disadvantages.
The choice of each approach depends on the problem under consideration. Generally
speaking, circuit models are more suitable for large-scale circuit design and optimization.
On the other hand, physics-based models are used for device development, optimization,
and characterization. However, the increase of the operating frequency along with the
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13
small size of today’s chips have led to the importance to employ physical models for the
design and optimizations of MMICs.
In the following sections, an overview of both models, namely the physical and
equivalent circuit models, will be provided. Accuracy and range of validity of each model
will also be given.
2.2
Equivalent Circuit Models
Equivalent circuit models are based on the electrical performance of the device at
its terminals. The elements of the equivalent circuit models can be extracted either by
measurements or by simulation using physics-based models. One of the advantages of
equivalent circuit models is that they are easy to implement. Thus, they are suitable for
large-scale design and optimization. Furthermore, equivalent circuit models are very
efficient in terms of CPU-time, i.e., computationally efficient. However, at very high
frequency, circuit models cannot obtain the correct device characteristics. The reason is
circuit models elements are extracted at a specific operating condition. This includes,
biasing, operating frequency, power, and temperature.
Another limitation of circuit
models is they do not relate circuit element values to the physical and process parameters,
such as doping profile, device geometry, mobility, and effective mass. The equivalent
circuit models also break down for high-frequency devices for the following reasons [46]:
•
Device dimensions become very small and comparable to the operating frequency
wavelength. The distributed effects become very important and must be modeled.
This is achieved by electromagnetic simulation, i.e., full-wave simulations.
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14
•
The electron transit and relaxation times become comparable to the operating
frequency time period. Accordingly, the coupling between the electromagnetic
wave and the physics of the device becomes very significant.
•
For high power application, in which time-varying fields are comparable to the
dc bias fields, the EM-wave and electron interactions become highly nonlinear
with several harmonics. This can be accounted for by employing a coupled
electromagnetic-physics-based simulator.
2.3
Physics-Based Models
Physical models provide insight to the device operation and able to accurately
obtain the device characteristics at different operating conditions. The physical models
provide the important link between the physical and process parameters (doping profile,
gate length, mobility, etc.) and electrical performance parameters (dc characteristics, RF
tranconductance, resistances, capacitances, etc.). From the above-mentioned reasons,
physics-based models are very suitable to model and optimize today’s devices and
circuits over a wide-band of frequencies.
Physics-based models can be classified into two categories: particle-based models
and fluid-based models. The first category is represented by the Monte-Carlo technique.
On the other hand, the second category is based on a set of conservation equations
obtained by estimating the different moments of the Boltzmann’s Transport Equation
(BTE). These models usually involve several approximations, from the more complex to
the simplest, they are: Full-Hydrodynamic models, Energy models, and Drift-Diffusion
models.
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15
The drift-diffusion model includes a drift velocity controlled by the electric field
and diffusion down carrier density gradients.
In a homogeneous system, the drift-
diffusion model is reduced to Ohm’s law. This assumes that the microscopic distribution
of momentum and energy over the charge carriers at any time and location equals to that
found in a large sample with a DC field equals to the local instantaneous field. This
assumption breaks down for submicron devices, where carrier transport is predominantly
non-stationary. For submicron devices, non-stationary effects such as hot electron effect
and velocity overshoot start to emerge and should be accounted for in device modeling.
Semi-classical device models have been developed to include energy and
momentum relaxation effects at the same time provide CPU-time efficient models. These
models deal with charge carriers as classical particles, which are derived from quantum
models.
In the next section, a complete description of the Full-hydrodynamic model (HD)
will be provided. The HD is accurate for devices with gate length less than 0.5
micrometer and larger than 0.1 micrometer. The HD is a semi-classical model because
the descriptions used for the band structure and scattering processes are generated by
quantum mechanics.
2.4
The Hydrodynamic Model (HD)
The transistor model used in this chapter is a two-dimensional (2-D) full-
hydrodynamic large-signal physical model [47]. The active device model is based on the
moments o f the Boltzmann’s Transport Equation obtained by integrating over the
momentum space. The integration results in a strongly coupled highly nonlinear set of
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16
partial differential equations, called the conservation equations or the HD. These
equations provide a time-dependent self-consistent solution for carrier density, carrier
energy, and carrier momentum, which are given as follows [48]-[50].
•
current continuity
— + V.(»w) = 0.
dt
C2-1)
energy conservation
d(ne)
»
•
+ qnu.E + V.(nu(£ + K BT )) = - n(e ^
tb00
(2.3)
x-momentum conservation
^ dt +
^
4ox ^
xm(s)
<2-4>
In the above equations, n is the electron concentration, o is the electron velocity,
E is the electric field, e is the electron energy, s0 is the equilibrium thermal energy,
and p is the electron momentum. The energy and momentum relaxation times are given
by rs and zm_respectively. Similar expression is obtained for the .y-direction momentum.
The current density J is estimated from the FHD using (2.5).
J (t) = -qnv(t).
(2.5)
The low field mobility is given by the empirical relation [51]:
,, _
0 “ ~1 +
8000
^
(A V 10-")
cm2
V i’
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(2.6)
17
The above model accurately describes all the non-stationary transport effects by
incorporating energy dependence into all the transport parameters such as effective mass
and relaxation times.
2.4.1
Hydrodynamic Model Limitations
It is very crucial to choose the appropriate semiconductor device model to
accurately and efficiently model ultrasmall semiconductor devices. The reason is each
model has its own range of validity and CPU-time requirements. In general,
semiconductor device modeling can be categorized into four approaches: classical
transport, semi-classical transport, quantum transport, and atomic-level transport [51].
The classical transport is valid for devices of 1 pm-scale where all transport phenomena
are assumed to change more slowly than the carrier energy relaxation and the electron
temperature is assumed to be equal to the lattice temperature. The drift-diffusion model is
an example o f a classical transport modeling approach. On the other hand, semi-classical
transport models are employed for devices of scale between 0.1 pm and 1 pm. These
models include some phenomena that take place in ultrasmall devices such as hot
electron and velocity overshoot. The hydrodynamic model and Monte Carlo method are
considered semi-classical transport approaches and are derived from Boltzmann’s
transport equation
(BTE). They are semi-classical because scattering processes and
descriptions of energy bands are generated using quantum mechanics. However, for
devices of scale less than 0.1 pm where the device size approaches the coherent length of
electrons, semi-classical transport models will no longer be valid. Accordingly, quantum
transport models (QT) should be employed to accurately model such devices. These
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18
models perfectly capture quantum interference effects such as tunneling and energy
quantization. Furthermore, since the electron’s flight time across the device region
becomes extremely short, the uncertainty relation in terms of energy and time are also
included in QT models. Transmission coefficient, Wigner function, and Green’s function
are examples of quantum transport modeling approaches. Further downscaling of devices
requires employing Atomic-level transport modeling approaches. These are valid for
devices o f scale less than 10 A0. It is noteworthy to say that quantum correction terms
can be added to the HD model and BTE to have a CPU-time efficient model that includes
quantum transport. These models are quantum hydrodynamic (QHD) or BTE with
quantum corrections [51]-[53].
The HD model presented in this dissertation is a semi-classical model approximation to
many particle quantum-mechanical problems. The HD model is an approximation for
BTE, which is valid only under the following assumptions [54]:
•
The free carriers in the device are considered as point particles, with well-defined
position and momentum. This is contrary to the quantum first exclusion principle.
Quantum effects are included implicitly in the effective mass and other properties
in the scattering rates.
•
The number of carriers in the device is large enough that a statistical description is
appropriate.
•
The carriers can be considered uncorrelated and thus the n-particle distribution
function can be written as a product of n single-particle distribution functions.
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19
•
Collisions instantaneously change the momentum of carriers but not their
positions.
It is noteworthy to underline the fact that the hydrodynamic model is an
approximation for BTE, which has further approximations. For instance, truncated
moment expansion in terms of local variables and problems in non-stationary transport
such as velocity overshoot.
2.4.2
Single Gas Approximation
The conservation equations are obtained by suitable integration of the
Boltzmann’s transport equation over the momentum space and averaging over the
multivalley conduction band [55]. Certainly, in semiconductors with multi-valley energy
structures, such as GaAs, there exists one electron gas with a different distribution
function for each of its conduction band valleys. As a result, the BTE and the
hydrodynamic equations are separately valid for each of the conduction band valleys. To
reduce the number of partial differential equations that need to be solved, the single
electron gas approximation is usually used. This consists of using average quantities over
all main valleys for the carrier density momentum and energy [56]. For instance, the new
quantities used in the single gas approximation are defined as follows:
n = Y s ni
(2.7)
T =Y r i
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20
The summations are taking over all conduction band valleys.
2.4.3
The Relaxation Time Approximation
The time relaxation approximation used for the collision terms is given by:
(O f}
V
/ collission
f~ fo
(2.8)
^
Where / 0 is the distribution function under equilibrium conditions and z is the
characteristic time that describes how the distribution function decays or relaxes back to
its equilibrium state as the driving forces are removed. This approximation is very
essential because it makes the solution of the BTE much easier. In this study, minority
charge transport as well as generation and recombination are neglected since we are
dealing with unipolar devices such as MESFETs.
2.4.4
Transport Parameters
The solution of the balance equations requires different transport parameters such
as the ensemble energy and momentum relaxation times, the electron effective mass,
mobility and temperature.
Two major approaches are employed to estimate these parameters. The first
approach is to guess the form of the distribution and then use the balance equations to
solve for the parameters in this function. The most generally used guess is the displaced
Maxwellian. The other approach to evaluate these transport parameters is based on
Ensemble Monte Carlo simulations. The parameters are evaluated under steady-state
conditions in bulk GaAs. Several simulations are carried out using different electric field
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21
and impurity concentration profiles. The resulting statistical information can then be
empirically fitted in analytical expressions suitable for numerical simulations. Following
this, the electron energy can be expressed in terms of the steady-state applied field E as:
e = 0 .3 1 6 -
0.316
fo r
l + (0.2£„)3!
g = 0.308 +3.353 fe *
£ ,,< 1 2 .5
kV
cm
(2.9)
! 2 '5^
10J
fo r E ss> 12.5 —
cm
While, the electron saturation velocity is given by:
1.53 + 0.9 (lO"4 Ess )
Dss
(2.10)
19
1+ 10'4 £ ss
and the electron drift velocity is given by:
MqE ss + ( 4 5 0 0 ) ~ 6 u ^ ( £ w )
vd =
(2 .11)
1 + (4 5 0 0 )-6 ( E s s )
where ju0 is the low-field mobility and is given by:
8000
Mo -
(2.12)
i + V ^ . i o - 17
The electron temperature Te is expressed as:
T =
0.085 1 -
for
s< 03eV
r e + 0.08x23
1+
0.115
T. =
0.05 + 0.03^1 +
^ - 0 . 3 x16
0.075 .
(2.13)
elsewhere
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22
The electron effective mass is approximated by:
0.0623m0
for Ess < 1.9944kV/cm
[0.0623 + 0.00526 (Ess - 1.944)] m0
for 1.944 < Ess < 4.167kV/cm
m* = J [0.074+ 0.0216 (Ea - 4.167)] m0
for 4.167 < E SS < 6.194kV/cm
[0.1172 + 0.01053 (Ess - 6.194)] m0
for 6.194 < Ess <8.33kV/cm
[0.1397 +0.005
for Ess > 8.33kV/cm
-8.33)]m 0
(2.14)
Finally, the electron mobility and the energy and momentum relaxation times are
respectively evaluated as:
(2.15)
(2.16)
T m { s h rnH s)M s)
(2.17)
q
2.4.5
Boundary Condition Implementations
In addition to the transport equations, a set of consistent boundary and initial
conditions is needed in order to have a unique solution to the problem. The surface and
contact properties provide the type boundaries to be used. These are generally mixed type
boundaries due to the presence of current free boundaries as well as Ohmic and Schottky
contacts.
At the surfaces were no current flow is assumed to exist, Neumann boundary
conditions are used for both the potential and carrier density. In this case, the gradient of
both the potential and the carrier density are set to zero in order to ensure zero drift and
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23
diffusion currents, respectively. Referring the 2D MESFET geometry of Fig. (2.1), those
boundaries are given as follows:
dx
^= o
dx
dV
dn
for boundaries parallel to the y axis,
(2.18)
=
0
for boundaries parallel to the x axis
Source
Gate
Drain
X
Active Layer
Buffer Layer
Fig. 2.1. A typical FET cross-section view.
The values for the potential at the metal contacts of the different device electrodes
are described by Dirichlet boundary conditions defined by the applied bias voltages. The
potential value at the semiconductor side of the Schottky contact is given by V = Vg - Vb.
Where Vg is the voltage applied externally at the gate electrode and Vh is the potential
barrier height. Voltages at the source and drain are defined by the applied bias voltages at
the two electrodes.
The boundary conditions for the carrier density at the source and drain contacts
can be defined using either Neumann or Dirichlet boundaries. In the Dirichlet
boundaries, a constant equilibrium carrier concentration, usually equal to the doping
concentration, is used. The boundary condition at the gate is often specified using
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24
thermionic emisson-diffusion theory [57], where the equilibrium electron density is given
by (2.19).
(2.19)
where N c is the effective density of states. This type of boundary condition, however,
results in a net gate current, which is found to slow the convergence rate to the steadystate solution [61]. Therefore, as an alternative, the boundary condition is set such that
there is zero gate-current.
In other words, the electron density at the gate is computed using (2.20).
qn/uE + D — = 0
dy
(2 .20)
where /u is the electron mobility and D = K bT/u. For the electron energy, Neumann type
boundaries are used everywhere.
The final solution is obtained in a self-consistent evaluation of the three
conservation equations in conjunction with Poisson’s equation. It was found that the
order in which these equations are solved is critical to the stability of the solution.
Poisson’s equation is used to update the electric field. Because the momentum
relaxation time is about one order of magnitude shorter than the dielectric and energy
relaxation times, the momentum equation is solved first using the new values of the
electric field. Then, the carrier energy is updated and used to compute the new transport
parameters. Finally, the continuity equation is solved for a new carrier density
distribution. This process is repeated until a self-consistent solution is obtained.
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25
2.4.6
Discretization of the HD Model
In this section, a finite-difference (FD) discretization scheme will be presented for
the hydrodynamic model. Stability and accuracy will also be discussed.
To have a numerical solution of the semiconductor model, a two-dimensional
rectangular grid that covers the computational domain is generated. The variables of the
transport equations are defined on a staggered grid. The scalar variables such as potential,
electron density, electron energy, and transport parameters are defined at the basic nodes
( i ,j ) . On the other hand, vector quantities such as electron velocity, electric field, and
current density are defined at the complementary nodes (z +1 / 2, j +1 / 2). These
arrangements separate the component of the vector variable and facilitate a convenient
way for decoupling the variables and expressing the approximations of the space
derivatives using the FD method.
Before carrying out a discretization scheme, a very important question needs to be
answered. What discretization scheme should we use: Implicit or Explicit. The fastest
answer would be to use implicit methods since there are no restrictions on the time-step
value used. Accordingly, one can use a large time step and the CPU-time will be reduced
dramatically. However, explicit methods have been found to outperform implicit methods
despite the time-step size restrictions for the following reasons:
•
Employing implicit discretization methods result in full matrices that need more
computational effort to be solved, i.e., more CPU-time.
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26
•
Error is a function of the time-step used. Accordingly, a large time-step should
not be employed because this increases error.
•
Implicit methods introduce diffusion. For a long time of simulation, this diffusion
accumulates and damps our solution. Eventually, the simulator will have a zero
output.
From the above, it is clear that there is no benefit achieved using implicit methods.
Thus, all equations in this study are represented or discretized using explicit FD schemes.
2.4.6.1
Discretization of the Continuity Equation
The continuity equation is expanded and written as:
dn _
d(nvx)
d(nvy)
(2.21)
dt
dx
dy
Using an FD scheme along with employing Upwinding, the space derivatives are
approximated as:
d(nox)
dx
d(nuy) ^
dy
1 f H U j) v x (i, j ) - n(i -1 , j)u x (i -1 , y)] [1 + S(i,j)] +1
2Ax\[n(i + l , j ) vx(i + l j ) - n ( i , j ) u x( i , j ) ] [ l - S( i , j ) \ J
i
fH U j ) v y(Uj ) ~ < i , j ~ l ) o yO', j ~ 1)][1 + T (i, y')] +1 (2.23)
2 A y [ H U j + l)vy(i, j + T) -n( i , j ) uy( i , j ) ] \ l - T( i , j ) ] J
S(i, j) = 1
ifox( i, j )>0.
S(i, j ) = - l
T(i,j) = 1
i f vx 0, y) < 0.
i f uy(i,j) > 0.
T(i,j) = - 1
i f vy(i,j) < 0.
On the hand, the time derivative is approximated using forward Euler as:
(2.24)
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27
2.4.62
Discretization of the Energy Conservation Equation
To enhance the stability of the FD scheme of the energy conservation equation,
the following procedure is followed. The energy conservation equation can be written as:
- =a -b s
(2.25)
dt
The above equation is a first order differential equation and has a closed form
solution given by:
* 0 = £ + m /„)
b
b
(2.26)
Rearranging the above equation and setting At = t - t 0 and t0 = k A t , along with
using a first order approximation of the potential leads to:
(2.27)
2.4.6.3
Discretization of the Momentum Conservation Equation
The momentum conservation equation is highly nonlinear compared to the other
two equations. This makes the momentum conservation equation very sensitive to the
smallest errors. In this study, LAX method is employed to achieve a higher degree of
accuracy. For instance, the derivatives of the electron velocity in x and y directions are
carried out as follow s.
f e l l
I dt ) i+l j
2
At
L “ (i +1 / 2, j ) - i [ y (i + 3 / 2, J ) + u ‘ (i - 1/2, j ) ] |
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a28)
28
f
\ \
diPy)
dt 'IJ+,. i
2
i - | u,‘" (i, j +1 / 2 ) - i [ o / ( i j + 3 / 2) + u ‘ ( I J - 1/ 2)] [
At
(2.29)
It is noteworthy to mention here that Upwinding is employed for the last two
terms on the right hand side of (2.28) and (2.29).
2.4.7
Hydrodynamic Model DC Results
DC results obtained by coupling Poisson’s with the HD model are presented in
this section. The DC results represent the initial condition for ac simulations, which are
carried out by coupling Maxwell’s equations with the HD model. To demonstrate the
potential of this approach, it is applied to an idealized MESFET structure, which is
discretized by a mesh of 64 Ar by 32 Ay with At = 0.001 p s. Forward Euler is adopted as
an explicit finite-difference method. In addition, Upwinding is employed to have a stable
finite-difference scheme. The space step sizes are adjusted to satisfy Debye length, while
the time step value At is chosen to satisfy the Courant-Friedrichs-Levy (CFL) condition.
First, DC simulations are performed, and the current density is calculated using (2.5).
DC excitation is performed by forcing the potential to be equal to the applied voltages to
the electrodes (i.e., Dirichlet boundary conditions). Table (2.1) summarizes the transistor
parameters used in the simulation. Figures (2.2) -(2.9) show the DC distributions and
results obtained using the HD model presented in the previous section.
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29
TABLE 2.1
TRANSISTOR PARAMETERS USED IN THE SIMULATIONS
Drain and source contacts
Gate-source separation
Gate-drain separation
Device thickness
Device length
Gate length
Device Width
Active layer thickness
Active layer doping
Schottky barrier height
DC gate-source voltage
DC drain-source voltage
0.5 pm
0.5 pm
1.0 pm
0.8 pm
2.8 pm
0.3 pm
250 pm
0.2 pm
2 x l0 17 cm'
0.8 V
-0.5 V
3.0 V
*10
2
1
0,5
0
Y
X
Fig. 2.2. Carrier density distribution.
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30
Fig. 2.3. Electron energy distribution (eV).
x 10
3
2
■o
1
u- 0
a
■6
Fig. 2.4. The distribution of the x-direction electric field (V/cm).
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31
*
x 10
Fig. 2.5. The distribution of the y-direction electric field (V/cm).
7
X 10
■
:
r"
Fig. 2.6. The distribution of the x-direction velocity (cm/sec.).
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32
x 10
£
Fig. 2.7. The distribution of the y-direction velocity (cm/sec.).
x
Fig. 2.8. Potential distribution (V).
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33
Vgs= -0.25
200
£ 150
£
Vgs= -0.75
c
<D
C 100
s
Q
Vgs= -2.75
0
0.5
1
1.5
2
2.5
Drain Voltage (V)
3
4
3.5
Fig. 2 .9 .1-V Characteristics.
2.5
Full-Wave Physical Simulations
The full-wave or electromagnetic simulation presented in this section is based on
Yee’s algorithm [62]. The algorithm begins with descritizing Maxwell’s equations along
with arranging terms. Brief description of the algorithm is provided below. More details
can be found in [63]. Maxwell’s equations are given by (2.30).
V x jE = - j u ^ dt
V x H =e $ £ - + J
dt
i
Expanding,
N
r ....
dz
1
I
dE y = _Lr dH X
dt
e dz
dE y
dx
dE = j_
dt
e
dHz
dx
- J ,
S’
1
dEx 1
dz
i
_L ' d E X
M [ dy
II
=
I
M
dx
1
I
= ±_ f dE z
dt
dEx
dt
I
d H x = JL dE y
dt
M dz
dHz
dt
(2.30)
dHx
- JZ
dy
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34
SOURC
GATE
DRAIN
ACTIVE LAYER
BUFFER LAYER
Fig. 2.10. Genetic view of the electromagnetic computational domain.
Yee defines the grid coordinates as:
(.i , j , k ) = ( i A x , j A y , k A z )
(2-32)
where Ax, Ay, and Az are the actual grid separations. Any function of space and time is
then written as:
F " ( i , j , k ) = (iAx, j A y , k A z , n A t ) .
(2.33)
Where At is the time increment and n is the time index. The spatial temporal
derivatives of F are written using central difference approximations as:
&
= ----------- 2 -------Ax---------- 2 -------
d F ' ( i , j , k ) _ dF" * ( i , j , k ) - d F n^ ( i , j , k )
dt
At
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<2-34)
(2.35)
35
The above equation are applied to the six scalar equations resulting in six coupled
explicit finite difference equations:
K k i j +i*
+
+ i ) = K ' H i j + 1 ,1 + 1 )
Az
At
■
(2.36)
E ; ( i , j + i , k + ± - ) - E : ( i , j , k + ±-)
H
; h
n
i
+
L
j
ik
+ i.) =
+ L j , k
+
1)
E : ( i + i , j , k + ± ) - E : ( i j , k + ±-)
At
+
(2.37)
E nx (i + 2 , J , k + 1) - E nx (i + ~ k , j , k )
n ;n H i + ± j + 1 *) = # ; " * ( / + 1 / + i1 *)
E:(i + ± , j + i , k ) - E : a + ± , j , k )
At
At
+
£;(/■ + i , j + \ , k ) - E ; ( i , j + L , k )
(2.38)
Ax
E:"(i + ^ J , k ) = E : ( i + L j , k ) +
At
H ] \ i + j1- , j. -+^1i *) - #;**(,■ + l j -
1-,*)
at
H n;H i +
+ + ) - H n;H i + i y , t - h
Az
-
2(i + ± , j , k )
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(2.39)
36
At
H "H iJ + \,k + \)~ H p Q .j+ ^ .k
-
\)
AZ
1
1
(2.40)
E : +' ( i , J , k + ± ) = E ; ( i , j , k + ±:) +
At
e (i,j,k + ^ )
H 7 J(i + \ , j , k + |1) - H p ( i - j - , j , k + £ )
Ax
H n; k h j + ± r , k + b - a : m , j - \ i, k + \ )
(2.41)
Ay
2.5.1
Error and Stability
The FDTD Yee-Based numerical scheme is second-order accurate, both in space
and time, because second difference discretization is employed. Accordingly, the error
introduced by this scheme is dispersive. This error is minimized by choosing the spatial
step-size to be much smaller than the wavelength X . As a rule of thumb, the spatial
increments are chosen to be less than A /1 0 . On the other hand, the stability of the Yeebased FDTD algorithm is based on the Courant-Friedrichs-Lewy (CFL) condition given
by (2.41).
At = -------
1
Ay2 Az2
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37
2.5.2
Electromagnetic-Physical Coupling
The full-wave solution is obtained by coupling the semiconductor and the
electromagnetic solutions with the appropriate initial conditions. The semiconductor
solution provides the electronic state inside the active device and evaluates the current
density. The electromagnetic solution, on the other hand, updates the electric and
magnetic fields everywhere inside the structure in response to the electron current
densities and applied excitation. The semiconductor part is considered a lossy dielectric
with a nonlinear conductivity, which is time, field, and space dependent.
This
conductivity is obtained from the solution of the active device model. To incorporate the
initial conditions of the solution, the above equations are modified to:
Ay
6 (/ + J , j , k )
+
Az
T o ta l
+
(2.42)
Az
+
T o ta l( n + \ )
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(2.43)
38
E r ( i , j , k + ±-) = E : ( i , j , k + ± ) +
At
e (if j + L }k)
H n;* ( i + l ; ' t + 1 ) - H ^ ( J - \ , j , k +
Ax
(2.44)
H">V,j +
j
2.5.3
+ f> - K h i , j - \ , k + 1)
+
Ky
: c ( i , j , k + ± ) ~ j ; " ' 1"* ' (,•, j , k + 1 )
Absorbing Boundary Conditions
Boundary conditions should be implemented since it is imperative to have a
confined computational domain. The main idea of the absorbing boundary conditions
(ABCs) is to have no reflection boundary surfaces. In this chapter, two methods that
have been developed to implement ABCs will be discussed, namely, MUR and perfectly
matched layer (PML) absorbing boundaries.
2.5.3.1
MUR Absorbing Boundary Conditions
Conceptually, the MUR ABC is based on the one-way wave equation. This allows
EM-wave propagation only in one direction [58]. The authors in [58] proposed two
different schemes based on accuracy, namely first order and second order MUR ABCs.
The first order and second order MUR ABCs are obtained using one term and two terms
of the Taylor expansion, respectively. Despite working very well for scattering problems
with uniform media, MUR ABCs are found to result in significant reflection for
multilayered media. On the other hand, the PML method is found to have better
performance for a broad type of problems.
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39
2.5.3.2
PML Absorbing Boundary Conditions
PML absorbing boundary conditions has been proposed by Berenger [62]-[64].
Berenger adds a new degree of freedom to control loss and impedance matching by
splitting the scalar field components into two sub-components. Following this, regardless
of the angle incidence, all waves observe the PML region to have the same characteristic
impedance as of the computational domain immediately before the PML domain.
2.6
Microwave Characteristics
To study the characteristics of transistors at high frequency, a time-domain
Guassian signal is applied between the source and gate electrodes. The input and output
time-domain signals are observed at different points along the width of the device. The
characteristics of the device are then estimated. For example, the propagation constant y
can be evaluated as:
F ( g?, z + 1 )
(2.45)
Where F(co,z) is the Fourier Transform of the time-domain signal. The
attenuation and propagation constants are evaluated as the real and imaginary parts of y,
respectively. Fig. (2.11) shows the attenuation constant as a function of frequency at
different points along the device width. Considering Fig. (2.11), it should be noticed that
the attenuation constant increases with frequency as well as from point to point along the
device width.
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40
60
co 40
0 30
—
100
120
z1 = 200 um
z2 = 300 um
z3 = 400 um
140
Frequency (GHz)
Fig. 2.11. Attenuation constant as a function of frequency at different points along the
device width for the gate electrode.
The phase velocity oph and effective dielectric constant s r can be estimated using
Equations (2.46)-(2.47), respectively.
»ph = J3
-
8 =
(2.46)
(2.47)
(2 n f ) 2
Where /? is the propagation constant, c is the free-space wave velocity, and co is
the frequency in rad./sec. Figures (2.12)-(2.13) show the effective dielectric constant
and phase velocity versus frequency at different points along the device width,
respectively. The results shown in these figures are mainly due to the change of the
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41
distribution of the electric field as a function of frequency and distance.
—
—
3.25
z1 = 200 um
z2 = 300 um
z3 = 400 um
3.2
| 3.15
</)
s
o
0
0)
a>
Q
a>
•§ 3.1
1 3.05
£W
2.95
20
40
60
80
Frequency ( GHz)
100
120
140
Fig. 2.12. Effective dielectric constant as a function of frequency at different points along
the device width for the gate electrode.
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42
x 10s
1.75
— z1 = 200 um
■— z2 = 300 um
—" z3 = 400 um
1.74
1.73
8 172
1.68
1.67
1.66
20
40
60
80
Freauencv (GHz)
100
120
140
Fig. 2.13. Phase velocity as a function of frequency at different points along the device
width for the gate mode.
2.7
EM-Wave Propagation Effects
In this section, a full-wave physical simulator is developed to model two closely
packed millimeter-wave transistors and multifinger transistors. Fig. (2.14) gives a 3D
view of the simulated transistors. The simulated devices are biased to V.as = 3.0V and
Vgs = -0.1V. The gate-length for the transistors is set to 0.2jum . The DC distributions
are obtained by solving the active device model only. A sinusoidal signal is employed in
the AC simulations with peak value of 100 mV and frequency of 80 GHz, respectively.
The two transistors shown in Fig. (2.14) are identical. First, full-wave simulations are
carried out for one transistor only, and the results are depicted in Fig. (2.15). Considering
this figure, one should observe the variations of the output voltage with distance along
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43
the device width. This demonstrates the importance of coupling the EM-waves with the
semiconductor transport physics for accurate modeling of millimeter-wave transistors.
Now, we turn our attention to full-wave simulation of the two transistors shown in Fig.
(2.14). First, we assume that one of the transistors is operating, while the other transistor
is not. Fig. (2.16) depicts the simulation results, which emphasize the significance to
include the EM-wave propagation effects, not only inside the device, but around it as
well. In fact, this is the basic theory of operation of multifinger transistors. Ideally, the
non-operating transistor should have a zero drain potential, however due to the proximity
of an operating transistor, an induced voltage that varies along the device width is
introduced.
TrmsistoE
Separation
Transistor#!
Transistor #2
2.1 pm
^
0 3 (Jm 0.4 (fin 0 2 (fin 0 3 (fin 0 3 (fint 0 3 (fin; Two possible oases:
«-^
Scnircel
Gatel
------------- +1 Brain2 or Souios2
D ran l .
,
Doped GaAs (2xl0l?end)
Undqped GaAs
«* **
•
OS (fin
I
Fig. 2.14. 3D view of the simulated transistors (not to scale).
Next, the two transistors in the configuration shown in Fig. (2.14) are simulated,
assuming that both transistors are now operating. There may be two cases to consider.
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44
The first case is to assume the drains of the two transistors are adjacent to each other (the
case of multifinger transistors). While, the other case is to consider the drain of one of the
transistors is adjacent to the source of the other transistor. Figures (2.17)-(2.18) show the
simulation results. The first conclusion that can be drawn out of the two figures is that the
proximity of an operating transistor affects the output voltage due to the EM-wave
propagation. Furthermore, the EM-wave effects for the case of two adjacent drain
electrodes is much larger than the other case. This is expected, since the drain electrode
has the amplified output signal. It is important to mention that the results in Figures
(2.15), (2.17), and (2.18) are normalized such that the effect of the increase of
Transconductance with width is not included. The reason is that we are interested only in
investigating the effect of EM-wave propagation.
0 .0 8
0 .0 6
0 .0 4
0.02
0
Q -0 .0 2
i\
-0 .0 4
-0 .0 6
—
—
—
-0 .0 8
0
z
z
z
z
=
=
=
=
6 2 .5 um
1 2 5 um
1 8 7 .5 um
2 5 0 um
0 .5
1
1.5
T im e (s)
2
2 .5
3
x 1Q-n
Fig. 2.15. Drain voltage (normalized) of the simulated transistor when EM-wave
propagation and electron-wave interaction are considered at different points in the zdirection.
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45
0.01
z1 —7 8 .1 2 um
z 2 - 5 8 .5 9 um
z 3 = 1 9 .5 3 um
0 .0 0 5
-0 .0 0 5
5-
CO
-
0.0 1
-0 .0 1 5
-
0.02
-0 .0 2 5 -
-0 .0 3
T im e (s)
Fig. 2.16. The potential of a passive electrode at different points in the z-direction
induced due to the proximity of an operating transistor excited by a Gaussian signal.
0.06
0.04
0.02
<D
O)
CO
Ia
Q
-
0.02
-0 .04
-0.06
T im e (s)
•11
x 10'
Fig. 2.17. Drain voltage (normalized) at z = 62.5mm when EM-wave propagation and
electron-wave interaction are considered. Solid line: transistor is simulated alone.
Dashed line: Source electrode of a second operating transistor is 0.5/um apart from the
drain of the simulated transistor.
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46
0.1
0.08
0.06
0.04
0.02
'«c
-
0.02
-
0.04
-
0.06
-
0.08
-
0.1
0.2
0.4
0.6
0.8
1 .2
1. 6
T im e (s)
1.8
x 10'11
Fig. 2.18. Drain voltage (normalized) at z = 6 2 . 5 when EM-wave propagation and
electron-wave interaction are considered. Solid line: transistor is simulated alone. Dotted
line: Drain electrode of a second operating transistor is 0.5ftm apart from the drain of the
simulated transistor.
Multifinger transistors have proven better performance over conventional
transistors, especially at very high frequency [65]-[75]. However, till now, modeling of
such devices did not account for EM-wave effects as well as electron-wave interactions
using a fully numerical simulator. Accordingly, it is indispensable to present analysis of
multifinger transistors based on a coupled electromagnetic-physics-based simulator.
It is clear from the previous section results that EM-wave propagation and
electron-wave interaction change the device characteristics at high frequency.
Accordingly, different structure shapes and configurations need to be employed to
minimize these effects, aiming to improve the device performance, especially at high
operating power and frequency. A possible solution is to use multiple gate-fingers of
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47
shorter lengths. In this manner, EM-wave propagation effects along the device width are
minimized. Moreover, attenuation is reduced as a result of reducing the gate metallic
resistance. Thus large number of fingers is better in terms of reducing attenuation and
wave-propagation effects along the device width. However, large number of fingers
means that attenuation and EM-wave propagation effects are increased along the feeding
line. Moreover, more fingers may cause more EM-waves interference. Thus, EM-wave
synchronization for the multiple fingers is crucial for maximum power and minimum
interference. It is noteworthy to say that EM-wave phase-velocity mismatches is due to
the different applied voltages to the electrodes and also due to the unsymmetrical shape
of the structure. Therefore, the number of fingers and distance between gate-fingers
should be optimized simultaneously.
Moreover, for the case of the four-finger transistors, the shape and size of the air­
bridge connecting the different fingers affect the high-frequency characteristics of the
transistor. Considering Fig. (2.19.c), it should be noticed that new capacitances between
the air-bridge and transistor electrodes Cair bridge are created. This would definitely change
the EM-wave phase velocities and as a result change the device behavior. Thus an
optimal air-bridge structure and size should be employed as well. Furthermore, the air­
bridge should not be fragile in order not to break easily, which represents an extra
constraint that needs to be included in our optimization problem. The feeding line shape
represents also a parameter that needs to be considered for circuit-matching issues.
In this study, ad-hoc optimization is performed to obtain near-optimal transistor
parameters based on the above criteria. Table (2.2) shows the new parameters for the
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48
optimized multifinger transistors, and Fig. (2.19) gives a generic 3D view of the
simulated multifinger transistors.
Output voltages for the simulated multifinger transistors are shown in Fig. (2.19).
Considering this figure, one should observe that the voltage-gain increases when using
four-finger transistors. In addition, the shape of the output signal for the four-finger
transistor case appears to be much better, which means fewer harmonics.
Design and optimization of high-frequency multifinger transistors need a
tremendous research work. It requires, as a backbone, a very efficient numerical
simulator that includes EM-wave propagation and electron-wave interactions. The
simulator should be accurate and most importantly fast in order to be suitable for
optimizing complex microwave structures. It is our belief that this paper presents this
type of simulator. This paper also presents, for the first time in literature, the preliminary
numerical results of electromagnetic physical simulation of multifinger transistors, based
on ad-hoc optimization. The future research work will employ rigorous optimization
techniques to obtain the optimal multifinger transistor structure based on the model
presented in this paper. Moreover, measurements will be carried out and compared to the
results achieved by our model.
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O u u n a l Lijmar (D o p id G*As)
B»n*> U jm lU n fcfw d -jaA fi
(a)
C h a n n e l L ay « f (D o p e d G aA s)
(b)
HuftVr 1 *y**i f
•"'l.d im 'J L .iy e r d ' 0 {>«d G a A s !
Fig. 2.19. Generic 3D view of the simulated multifinger transistors (not to scale).
(a) Single-finger transistor (lx 450|jm). (b) Two-finger transistor (2x 225p.m).
(c) Four-finger transistor (4x112.5|am).
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TABLE 2.2
MULTIFINGER TRANSISTOR OPTIMIZED-PARAMETERS
Drain and source contacts
Gate-source separation
i
0.5 pm
0.5 pm
0.4 pm
Gate-drain separation
0.4 pm
D evice thickness
D evice length
Gate length
2.1 pm
0.2 pm
1x450, 2x225, 4x112.5 pm
D evice W idth
A ctive layer thickness
i
0.1 pm
2x1017 cm'3
0.8 V
-0.2 V
3.0 V
(50 GHz
A ctive layer doping
S chottky b amer height
DC gate-source voltage
DC drain-source voltage
Operating frequency
□ .4
0 .3
0.2
0.1
0
-
0.1
-
0.2
S in g le Fin ger: 1 x 4 5 0 um
- 0 .3
T w o F in g e rs : 2 x 2 2 5 um
F o u r F in g e rs : 4 x 1 1 2 .5 um
- 0 .4
0
0.2
0 .4
0.6
0.0
1
1.2
1 .4
T im e (s e c o n d s )
1.6
^g-11
Fig. 2.20. Output voltage for the simulated multifinger transistors when EMpropagation and electron-wave interactions are considered.
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51
2.8
Summary
In this chapter, electromagnetic physical simulation is described in details. The
different parts o f the problem are provided. The emphasize is on an HD model coupled
with Maxwell’s equations. A complete set of DC and AC results is presented.
The preliminary results of this chapter show that at very high frequency, several
phenomena with strong impact on the device behavior start to emerge, such as phase
velocity mismatches, electron-wave interaction, and attenuation. The results suggest that
contemporary microwave devices should be optimized to minimize these effects or
possibly take advantage of in favor of improved device characteristics. The results also
recommend multifinger transistors as potential alternatives to conventional transistors.
This is achieved by using multiple-finger gates of less width instead of a single-gate
device. Furthermore, this chapter underlines the enhanced microwave characteristics of
multifinger transistors attributable to reducing attenuation and EM-wave propagation
effects along the device width. The future research work will involve employing rigorous
optimization techniques to obtain the optimal multifinger transistor structure based on the
electromagnetic-physical model presented in this chapter. Moreover, measurements will
be carried out and compared to the results achieved by our model.
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Chapter 3
Modeling and Optimization of Microwave Devices and Circuits Using Genetic
Algorithms
3.1
Introduction
This chapter presents a new approach for the simulation and optimization of
microwave devices, using a genetic algorithm (GA). The proposed technique solves the
equations that describe the semiconductor transport physics in conjunction with Poisson’s
equation, employing an adaptive real-coded GA. An objective function is formulated,
and most o f the GA parameters are recommended to change during the simulation. In
addition, different methods for describing the way the GA parameters change are
developed and studied. The effect of GA parameters including the mutation value,
number of crossover points, selection criteria, size of population, and probability of
mutation is analyzed. The technique is validated by simulating a submicrometer field
effect transistor (FET), and then compared to successive over relaxation (SOR); showing
the same degree of accuracy along with a moderate speed of convergence. The purpose
o f this chapter is to introduce a new vision for a genetic algorithm capable of optimizing
real value functions with a considerably large number of variables. This study also
represents a fundamental step toward applying GAs to Maxwell’s equations in
conjunction with the hydrodynamic model (HDM), aiming to develop an optimized and
unconditionally stable global-modeling simulator.
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53
3.2
Genetic Algorithms: Literature Overview
The early pioneers of science were as much interested in biology and psychology
as in electronics, and they looked to natural structure as guiding metaphors for how to
achieve their visions. It should be no surprise that from the earliest days computers were
applied not only to calculating missile trajectories and deciphering military codes but also
to modeling the brain, mimicking human learning, and simulating biological evolution.
These biological motivated computing activities have been developed over the
years, but since the early 1980’s they have undergone a rebirth in the computation
research community. The first has gone into the field of neural networks, the second into
machine learning, and third into what is called evolutionary computation of which
genetic algorithms are the most prominent example. GAs were invented by John Holland
in the 1960s, in which the author described the genetic algorithm as an abstraction of
biological evolution, and gave a theoretical framework for adaptation under the GA.
Holland’s GA is a method for moving from a population of chromosomes to a new
population by using a kind of natural selection along with the genetics-inspired operators
of crossover, mutation, and inversion. Each chromosome consists of genes, each gene
being an instance of a particular allele (e.g., 0 or 1). The selection operator chooses those
chromosomes in the population that will be allowed to reproduce, and on average the
fitter chromosomes produce more offspring than the less fit ones.
Crossover exchanges subparts of two chromosomes, roughly mimicking
biological recombination between two single-chromosome ‘haploid’ organisms. While,
mutation randomly changes the allele values of some locations in the chromosome, and
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54
an inversion reverses the order of a contiguous section of the chromosome, thus
rearranging the order in which genes are arrayed [76].
In this section, an overview of GAs will be provided. The emphasize will be on
the studies carried out in our field, i.e., microwave and antennas. GAs have proven a
great deal o f flexibility for solving different types of problems. The genetic-based
algorithms are unconditionally stable and not likely to get stuck in local minima when
employed for solving any type of problems.
In [77], the authors presented a new technique to obtain the material parameters
for each single layer in a multiplayer structure, using GAs. The GA for parameter
extraction from the reflection or transmission measurements is based on a simplified
evolutionary strategy. The GA is briefly described and verified by measurements in the
frequency range from 115 to 145 GHz. The parameters obtained using the GA show good
agreements with reference values estimated by other researchers.
In [78], the authors introduced a GA for parameter extraction for RF on-chip
inductors. The details of applying the developed GA in extracting the model parameters
are discussed. For a set of inductors, the meaningfulness of extracted parameters is
considered in the procedure of extraction. The S-parameters, inductance, and Q value can
then be set as the fitting targets simultaneously. The values of the model parameters are
extracted by a fully automatic programmed GA procedure, which shows excellent
agreement with measured data.
In [79], the authors presented a novel procedure for synthesizing multilayered
radar absorbing coating, using GAs. The algorithm was successfully applied to the
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55
synthesis o f wide-band absorbing coating in the frequency ranges 0.2-2 GHz and 2-8
GHz.
hi [80], a computational approach to the imaging of an imperfectly conducting
cylinder is presented, using GAs. Based on the boundary condition and the measured
scattered field, a set of nonlinear integral equations is derived and the imaging problem is
formulated into an optimization problem. The GA is then employed to find the global
solution of the cost function. Numerical results demonstrated that, even when the initial
guess is far away from the exact one, good reconstruction is obtained. In such a case, the
gradient-based methods often get trapped in a local minimum. Numerical results show
that multiple incident directions permit good reconstruction of shape and, to a lesser
extent, conductivity in the presence of noise in measured data.
In [81], topology and dimensions of line-segment circuits are expressed by a set
o f parameters, which describe the way the structural growth of line-segment circuits. The
sets of parameters are then optimized by GAs to specify specifications. In the GA
process, to reduce computational time, a circuit is decomposed into lines and
discontinuous elements. The S-parameters are then synthesized to obtain the response of
the circuit. Three filters and a power divider are designed and tested.
In [82], a graphical analysis of the impedance matching problem for the
multilayer dielectric and magnetic coating of metallic surfaces, for normal plan wave
incidence, is presented using GA. Methods for visual design using Smith-chart-type
graphical tools, which can complement computationally intensive optimization, are
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56
derived. The problem of estimating the required permittivity and permeability for a given
frequency and thickness is also discussed.
In [83], a possible application of global optimization techniques based on a GA to
imaging biological bodies exposed to interrogating microwaves has been evaluated by
means o f numerical forward simulations. For the text case considered (a human
abdomen), it is shown that a reduced investigation domain can be used with limited errors
on the measured scattered data. Once the domain is selected, the global optimization
technique can then be applied. The authors also proposed a hybrid technique that employs
both gradient methods and GAs.
In [84], the authors presented a procedure for synthesizing broadband microwave
absorbers incorporating frequency selective surface (FSS) screens embedded in dielectric
media using a binary coded GA. The GA simultaneously and optimally chooses the
material in each layer, thickness of each layer, FSS screen periodicity in the x- and ydirections, its placement within the dielectric composite, and the FSS screen material.
Furthermore, the GA generates the cell structure of the FSS screen. The result is a
multilayer composite that provides maximum absorbing of both TE and TM waves of a
prescribed range of frequencies and incident angles. This technique automatically places
an upper bound on the total thickness of the composite.
In [85], the authors report on the structure of a large-signal neural network (NN)
high electron mobility (HEMT) model as determined by a pruning technique and a
genetic algorithm. By representing the configuration of a standard multilayered neural
network as a chromosome, the optimum configuration of a model is obtained through a
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57
simulated evolution process.
For this approach, the configuration of an NN that
simultaneously represents seven intrinsic elements of an equivalent circuit was also
shown for comparison to previous work.
In [86], the study is aimed at exploring the possibility of using a microwave
approach based on a genetic algorithm to detect a defect inside a known host object,
starting from the knowledge of the scattered field, the problem solution is recast as a twostep procedure. After defining a cost function depending on the geometric parameters of
the crack, a minimization procedure based on a hybrid-coded genetic algorithm is
applied. The influence of the noise as well as the geometry of the defect on the crack
detection and reconstruction is investigated. Moreover, the numerical effectiveness of the
iteration approach is examined.
More research studies employing GAs can be found in [87]-[120]. The above
overview represents a very good introduction to GA application in our field. This also
represents an excellent starting point for those interested to use GAs.
Some optimization problems have multiple local minima. Where methods based
on steepest descent would fall in one of these local minima, resulting in a different
solution. GA’s are random algorithms and researchers have found their generality and
that they are unconditionally stable. GAs are thus suitable to find the global solution for
problems having multiple minima [76]. Furthermore, in many problems requiring solving
systems of linear equations Ax = b, the matrix A has a large Condition Number. For these
problems, standard methods will not be able to get the correct solution. For instance,
solving Poisson’s equation on a nonuniform grid. In this case, genetic-based algorithms
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58
would outperform standard methods. This is because standard methods work fine only for
well-posed problems (problems with A having a small condition number). On the other
hand, a genetic-based algorithm converges independent of the condition number o f A .
From the above, it is motivating to make an effort to apply GAs to Maxwell’s
equations or the hydrodynamic model (HDM), aiming to develop an optimized and
unconditionally stable algorithm. It is noteworthy to say that the main purpose of this
chapter is to lay the foundation of a genetic algorithm capable of optimizing real value
problems, with a considerably large number of unknowns.
In this stage of the work, we will demonstrate that genetic algorithms can be
applied to the hydrodynamic model (HDM) in conjunction with Poisson’s equation to
accurately model submicrometer gate devices, with less stability constraints. Ultimately,
a hydrodynamic model should be implemented with equations that would have numerical
stability restrictions such as Maxwell’s equations rather than Poisson’s equation in order
to obtain a self-consistent simulation of electromagnetic-wave propagation effects,
employing an optimized and unconditionally stable algorithm.
3.3
Problem Description
The transistor model used in this work is a two dimensional (2-D) full-
hydrodynamic large-signal physical model. The active device model is based on the
moments of the Boltzmann’s Transport equation obtained by integrating over the
momentum space. The integration results in a strongly coupled highly nonlinear set of
partial differential equations, called the conservation equations. These equations provide
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59
a time-dependent self-consistent solution for carrier density, carrier energy, and carrier
momentum, which are given as follows.
•
current continuity
— + V.(«u) - 0.
dt
•
(3.1)
energy conservation
(3.2)
•
x-momentum conservation
(3.3)
In the above equations, n is the electron concentration, v is the electron velocity, E
is the electric field, s is the electron energy, s0 is the equilibrium thermal energy and p
is the electron momentum. The energy and momentum relaxation times are given by z£
and rm, respectively. Similar expression can be obtained for the y-direction momentum.
The three conservation equations are solved in conjunction with Poisson’s equation:
(3.4)
where <j) is the electrostatic potential, q is the electron charge, e is the dielectric constant,
N d is the doping concentration, and n is the carrier concentration at any given time. The
total current density distribution J inside the active device at any time t is given as:
J ( t) = -qnu(t).
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(3.5)
60
The low field mobility is given by the empirical relation:
(3.6)
On the other hand, the mobility for large-signal simulations is calculated as
ju = vd / Ess, where vd is estimated using (3.7).
(3.7)
In the above equation, vd is the electron drift-velocity, ju0 is low field mobility given
by Eq. (3.6), uss and Ess are the steady-state electron velocity and electric field,
respectively. It is significant to note that both u„ and Ess are functions of energy, and
they get updated each time a new energy distribution is estimated using the
hydrodynamic model.
The above model accurately describes all the non-stationary transport effects by
incorporating energy dependence into all the transport parameters such as the effective
mass and relaxation times. Fig. (3.1) shows the cross-section of the simulated structure.
05 pm
0 3 Mm
Drain
Doped GaAs C2iclOl , cm'^
t
28 Mm
Fig. 3.1. Cross-section of the simulated transistor.
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61
3.4
Optimization Using Genetic Algorithms
In this section, we apply a genetic algorithm for the solution of the boundary value
problem for the distribution of potential across an FET.
A generic flowchart of the algorithm is shown in Fig. (3.2). The first step is to
read the matrix A and vector b of the system of linear equations, Ax = b, which are
derived from Poisson’s equation [121]. A population of random solutions (chromosomes)
representing the vector solution x is then initialized. Next, the objective function
(fitness) is estimated for all the chromosomes that have been randomly generated. Based
on the fitness, two parents are generated either by roulette wheel selection or by
tournament selection methods.
Mutation and crossover are then performed on the
selected parents to generate two children. Replacement is conducted by comparing the
fitness o f children with their parents, and the worst two chromosomes are removed from
the population. The best chromosome is identified based on fitness, and finally a check is
carried out against a certain stopping criteria to either stop the simulator, or to perform
another iteration. The details o f implementing the proposed algorithm are as follows.
•
Step One: Initialization
Initialization is done by randomly generating M chromosomes representing the
GA population. Real encoding is adopted for our problem. Each chromosome contains n
genes, which corresponds to the variables in the vector solution x . The generated random
numbers have a range associated with the applied DC voltages to the device electrodes.
•
Step Two: E valuate Fitness
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62
Each chromosome is evaluated based on an objective function. The objective
function is developed in a way that it accurately determines how close the randomly
generated solutions are to the optimal solution.
Dealing with 2D Poisson’s equation means that the matrix A has one to five
elements in each row (sparse matrix) [121]. Based on this, it was found that the vector
solution x with size n - N xN y should be fully constructed by knowing only the
2N x elements close to the electrodes following the flowchart given in Fig. (3.3). Fig.
(3.4) shows a five-by-five-grid to illustrate the implementation of the algorithm shown in
Fig. (3.3). Each square in Fig. (3.4) represents a grid-point. In Fig. (3.4), the elements
inside the gray squares are used to estimate the next element, which is inside the darker
square. For instance, .% is estimated as a linear combination of % ,
and Xj5,
following the algorithm in Fig. (3.3). This process is repeated until all elements in our
domain are estimated, as shown in Fig. (3.4). It is worth mentioning that the proposed
numbering sequence is crucial for the correct estimation.
On the other hand, the
2N x elements needed to implement the algorithm in Fig. (3.3) can be randomly generated
with a minimal error based on the following. Since boundary conditions must be
satisfied, then the values of the potential are known precisely at the boundaries.
Moreover, an estimate for the value of the potential near boundaries can be randomly
generated bearing in mind that they would have very close values to the potential at the
electrodes. The randomly generated solution g along with the second norm of A x - b are
included in the objective function given by Eq. (3.8), which needs to be maximized for a
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63
minimum value o f the dominator. In Eq. (3.8), g is the randomly generated solution,
while a is a scalar representing the weight of the norm criterion. The value of a is
chosen to be 10 %, by trial and error.
F itn es s
= --------------- = = = = = ----------
1+
—
+D
n
Z
D =
I*/ - 8:
------------
ci = Y aiJXJ~bi
7=1
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( 3 .8)
64
Start
In itia liz a tio n
E valu ate F it n e s s
R e g e n e r a tio n
C r o ss o v e r
M u ta tio n
R e p la c e m e n t B a s e d o n F itn ess
Id e n tify B e s t C h r o m o so m e
F itn e s s <
STCR
No
Ye
Fig. 3.2. Generic flowchart of the genetic algorithm.
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2NX Values for
Randomly Guess
Potential
I
Obtain the Remaining N - 2 N X Potential Values
as Follows:
J = 1:^-2
i = N - j : N y - j : s t ep = - N y
8 l ~l ~
elseif
O
Al
~
i
if (i + N y < N and i
X At.kSk
^ i,i
:k = i,i + l , i - N y ,i + N y
(i + N y > N )
b> ~
S t-1 -
Z A>,kSk
*
: k = i, i + 1, i - N
i,i
e ls e
bt
Si-i
en d
~ Z Ai.k8t
*
: k = i, i + 1, / + N
Ai,i
loops
r
Supply the Initial Guess g to the Main Program
(
Stop
)
Fig. 3.3. Flowchart of the randomly generated solution G.
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*-x
*5
*10
*15
*4
*9
*14
*5
*4
*10
i
*9
66
*13
*20
*25
*14
*19
*241
|* 1 8 1
*5
*10
*4
*9
, *15 ^
*14
*23
*20 ,.....*25
,*19
*24
f ir * " .....
*131
*18
*23
*25
*24
*23
*5
*10
§X A
Xq
|* 3 |
*8
L.
y
'
*15
*20
*25
X]A
X\ n
iy
*24
*13
*18
*23
*5
*10
*15
*20
*25
*4
*9
*14
*19
*24
*3
*8
*13
*18
*23
*2
*7
*12
*17
*22
*1
*6
*11
*16
*21
Fig. 3.4. A 5 by 5 grid example illustrating how the algorithm in Fig. 3.3 works.
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67
Step Three: Regeneration
•
Two methods are used for parent regeneration, namely, Roulette Wheel and
Tournament selections. The details of implementing each method are described below.
A. Roulette Wheel Selection
Parents in the Roulette Wheel method are chosen randomly according to their
fitness. As the name implies, the method imitates the Roulette Wheel game, where the
thrown dice would most probably end being in the slot with the largest area. Following
this, one can conclude that the chromosome with the largest fitness value is most likely to
be chosen because it has the largest slot size or equivalently the largest fitness.
B. Tournament Selection
In this method, two groups from the population are randomly selected, sub­
populations. It is worth mentioning that the population size is chosen randomly as well.
The best chromosome from each of the randomly generated sub-populations is chosen to
represent a parent.
•
Step Four: Crossover
Now, two parents have been selected for their genes to be crossed over and
mutated. Crossover is conducted by first randomly selecting a crossover point within the
chromosome. Two children are then conceived by mixing the genes of the two parents at
the crossover point. At this moment, two different parameters can be analyzed. The first
parameter is the number of crossover points, i.e., more than one crossover point can be
achieved. The other parameter is the number of genes involved in each crossover point,
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68
and will be denoted by the crossover width. The effects of both parameters are studied
and included in the results section.
•
Step Five: Mutation
Mutation is carried out by randomly changing one or more genes (variables)
of the created offspring. We then have two mutation parameters to study their effect. The
first one is the number of mutated genes or variables within the chromosome. The other
parameter deals with the value of mutation.
•
Step Six: Replacement
Replacement is performed by comparing the fitness of the parents with their
offspring. The best two chromosomes out of the four are included in the population for
the next iteration.
•
Step Seven: Ending the Algorithm
The best chromosome is identified at each iteration and error is calculated as the
second norm of A x - b . This error is checked against a predefined value, if satisfied, the
simulator stops and prints the final results.
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69
3.5
Results and Discussions
In this section, the effect of different GA parameters on the algorithm behavior is
investigated. The GA parameters used in the simulation are summarized in Table (3.1).
TABLE 3.1
_______ GA PARAMETERS USED IN THE SIMULATION_________
MU
NC
CW
NM
SC
NPOP
PM
Mutation factor or value
Number o f crossover points
Crossover width
Number o f mutated variables
Selection criterion
Size o f population
Probability o f mutation_____
The default values are 0.1, 1, 1, 1, 1, 100, and 1 for MU, NC, CW, NM, SC,
NPOP, and PM respectively.
It is significant to note here that the developed algorithm is implemented as a
subroutine to solve the system of linear equations, Ax = b. Poisson’s equation is then
coupled to the HDM equations as a subroutine. The coupling is carried out as follows.
First, the hydrodynamic equations are solved to get the updated value for carrier density
n . The updated carrier density is then plugged into Poisson’s equation, resulting in a new
system of linear equations. The new system of linear equations is passed to the geneticbased Poisson solver to solve for x , i.e., the updated value of the potential. The potential
is differentiated to get the electric filed. Finally, the updated value of the electric field is
plugged into the HDM to estimate the updated value of carrier density. This process is
repeated until the stopping criterion is satisfied.
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70
Fig. (3.5) shows the distance from the optimal solution versus number of
generations for different values of MU. The mutation value of any gene (variable) is
proposed as follows.
Childnew(i) = Childold(i) + M U ■RND
(3 .9 )
Where MU is the mutation factor and RND is a random number between -1 and
1. Fig. (3.5) shows that the best result is obtained when the mutation value is dependent
on the fitness. The reason is that as the value of the fitness increases, which indicates
being very close to the optimal solution, the mutation factor decreases. In this manner,
the mutation value is changed in the correct way for a faster convergence. Moreover,
introducing a random feature along with the dependence of MU on the fitness does not
enhance the convergence. A general conclusion is that smaller values of MU are observed
to have better convergence curves.
Figures (3.6) and (3.7) illustrate the effect of the number of crossover points and
crossover width, respectively. From Fig. (3.6), it can be concluded that larger number of
crossover points is the right choice for better accuracy along with higher speed of
convergence. Moreover, choosing the number of crossover points to change randomly
within simulation does not improve the algorithm. On the other hand, considering Fig.
(3.7), it is apparent that rate of convergence of the genetic algorithm is independent of the
crossover width value.
Fig. (3.8) shows the effect of the number of mutated elements on the convergence
and accuracy of the proposed algorithm. It can be observed that as the number of the
mutated variables decreases, the convergence and accuracy of the algorithm are
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71
improved. The best curve is obtained for a number of mutated variables equals to one
percent.
This complies with nature, since biological mutation hits only a very small
number of genes. Moreover, changing the number of the mutated variables randomly
throughout the simulation introduces a reasonable improvement over the one percent
mutation case.
Fig. (3.9) demonstrates how the choice of parent selection method affects the
algorithm performance.
It can be pointed out that roulette wheel selection has the best
performance. Furthermore, employing a hybrid technique does not improve the
algorithm. For instance, error reaches 10”3 in almost 500 generations when roulette
wheel selection is employed, whereas 1200 generations are needed for tournament
selection to reach the same value of error. It is worth mentioning here that roulette wheel
selection inherently uses some sort of elitism. Employing elitism may or may not be
useful depending on the problem under consideration. The main reason for roulette wheel
selection producing better results is the choice of the objective function given by (3.8). It
is important to mention that tournament selection is known to produce better results over
the roulette wheel method. However, this is not general, and the numerical example
provided in this chapter emphasizes that.
Fig. (3.10) shows the effect of the probability of mutation. It should be noticed
that increasing the probability o f mutation has a positive effect on the performance o f the
algorithm, and the mutation probability is not crucial to the algorithm as long as it is
relatively high. Fig. (3.11) shows the effect of the population size on the algorithm
convergence. This figure emphasizes that the population size is not a critical parameter.
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72
The reason is that the proposed objective function given by (3.8) inherently allows
elitism. This makes the proposed genetic algorithm independent of population size.
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73
10
— ■ MU
MU
— MU
MU
10'1
=
=
=
=
0.9
0.1
RND*(1-FIT)
0.1*(1-FIT)
10'2
10"
10'
-5
10'
-7
0
500
1000
1500
2000
2500
3000
3500
number of generations
Fig. 3.5. Distance from the optimal solution versus number of generations for different
mutation values.
—
NC =
NC =
NC =
NC =
1
3
10
5*RND
o 10
500
1000
1500
2000
number of oenerations
2500
3000
3500
Fig. 3.6. Distance from the optimal solution versus number of generations for different
number of crossover points.
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74
—
—
CW = 1
CW = 10
CW = 30
o 10'
10
500
1000
1500
2000
number of generations
2500
3000
3500
Fig. 3.7. Distance from the optimal solution versus number of generations for different
values o f crossover widths.
NM = 1
— NM = 3
— NM = 10
— NM = RND*5
o 10'
500
1000
1500
2000
number of generations
2500
3000
3500
Fig. 3.8. Distance from the optimal solution versus number of generations for different
numbers of mutated elements.
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75
SC = 1, random group selection
SC = 2, roulette wheel selection
SC = INT (1.5*RND+1), hybrid
o 10'
500
1000
1500
2000
number of generations
2500
3000
3500
Fig. 3.9. Distance from the optimal solution versus number of generations for different
selection criteria.
N POP
N POP
N POP
— - N POP
=
=
=
=
10
30
70
100
10
o 10'
500
1000
1500
2000
number of generations
2500
3000
3500
Fig. 3.10. Distance from the optimal solution versus number of generations for different
population sizes.
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76
—
PM = 0.1
PM = 0.5
PM = 0.8
o 10'
500
1000
1500
2000
number of generations
2500
3000
3500
Fig. 3.11. Distance from the optimal solution versus number of generations for different
population sizes.
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77
3.5.1
DC Simulation Results
To demonstrate the potential of the proposed approach, it is applied to an
idealized MESFET structure, which is discretized by a mesh of 32 Ar by 32 Ay
with At = 0.001 p s. Forward Euler is adopted as an explicit finite difference method. In
addition, upwinding is employed to have a stable scheme. The space step sizes are
adjusted to satisfy Debye length, while the time step value At is chosen to satisfy the
Courant-Friedrichs-Levy (CFL) condition. While, Poisson’s equation is solved using the
proposed algorithm.
Fig. (3.12.a) shows the potential distribution obtained using the proposed
algorithm. This graph demonstrates that boundary conditions are satisfied at the
electrodes. For instance, the value of the potential at the gate equals to -1.3 volts, which
is the applied DC voltage minus the Schottky barrier height. While, Fig. (3.12.b) shows
the carrier density distribution. It is significant to indicate that the proposed algorithm
gives precisely the same results obtained when SOR algorithm is employed. The
comparison results between the algorithms are not provided because their results coincide
exactly on each other. It is noteworthy to say that the purpose of this section is to show
that genetic algorithms can be applied to solve real value problems having a large number
of unknowns, with a very high degree of accuracy. The speed of convergence is not an
objective at this stage o f the work. Ultimately, the developed genetic algorithm needs to
be applied to equations that have stability constraints in order to have unconditionally
stable algorithm, or to solve problems that traditional optimization techniques cannot
solve (problems with multiple local minima).
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78
x 10
2.5
2(/>-
1.5
c
b
J<D
a;
'E
S 0.5
•0.5
0 0
X
2.5
|<o
c
4)
o
Q.
1.5
0.5
•0.5
Fig. 3.12. Sample DC results obtained using the proposed algorithms, (a) Potential
distribution, (b) Carrier density distribution.
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79
3.5.2
AC Simulation Results
The AC excitation applied to the gate electrode is given as:
v ss( 0 = Vgs0 + A v gssin (a )t)
(3.10)
where Vgs is the DC bias applied to the gate electrode, Augs is the peak value of the AC
signal (0.2 volts), and co is the frequency of the applied signal in rad. I sec. The
frequency used in the simulation is 60 GHz.
First, the DC distribution is obtained by solving Poisson’s equation, using the
proposed algorithm in conjunction with the three hydrodynamic conservation equations.
Then, a new value of Vgs is calculated using (3.10). The new value of Vgs is used to
update Poisson’s equation to get the new potential distribution. The electric field is then
estimated and used to update the variables in the conservation equations. This process is
repeated every A/ until t = tmax. The current density is obtained using (3.5). The current
density calculated on the plan located midway between the drain and gate is integrated to
obtain the total current. The output voltage is estimated by multiplying the total current
by the resistance that defines the DC operating point (Q point) of the transistor. Fig.
(3.13) shows the AC gate and drain voltages. A maximum gain of 9 dB is achieved.
Moreover, it is observed that there is an output delay of about 1p s that represents the
time required for the transistor to respond to the input signal.
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80
D.e
- -
vds
vgs
0.6
D.4
0.2
s
E
05B
“
0n
"5
>
-
0.2
-0.4
-
0.6
-0.0
0
5
10
15
20
25
30
35
Tim e (ps)
Fig. 3.13. AC gate and drain voltages obtained using the proposed algorithm.
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40
81
3.6
Summary
In this chapter, a new technique is developed for solving the hydrodynamic model
(HDM) in conjunction with Poisson’s equation, using an adaptive real-coded genetic
algorithm. Several GA design parameters have been studied to illustrate their effects on
the algorithm convergence. The novelty of the proposed technique comes from the
genetic algorithm itself. This has been achieved by developing a very efficient objective
function, along with introducing completely new concepts such as fitness-dependent GA
parameters. Moreover, the problem this chapter presents is a new application of genetic
algorithms. In addition, the proposed genetic algorithm should outperform standard
methods for several types of problems. For instance, finding the global solution of
optimization problems having multiple local minima, and problems where matrices have
large Condition Numbers. This study also represents a fundamental step toward applying
GAs to Maxwell’s equations in conjunction with the HDM, aiming to develop an
optimized and unconditionally stable global-modeling simulator.
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Chapter 4
A New Wavelet-Based Time-Domain Technique For Modeling and Optimization Of
High-Frequency Active Devices
4.1
Introduction
In this chapter, a new time-domain simulation approach for large-signal physical
modeling o f high frequency semiconductor devices, using wavelets, is presented. The
proposed approach solves the complete hydrodynamic model and Maxwell’s equations
on nonuninform multiresolution adaptive grids. The nonuniform grids are obtained by
applying wavelet transforms followed by hard thresholding. This allows forming fine and
coarse grids in locations where variable solutions change rapidly and slowly,
respectively. A general criterion is mathematically defined for grid updating within the
simulation. In addition, an efficient thresholding formula is proposed and verified. The
developed technique is validated by simulating a submicrometer FET. Different
numerical examples are presented along with illustrative comparison graphs, showing
more than 75% reduction in CPU time, while maintaining the same degree of accuracy
achieved using a uniform grid case. Tradeoffs between threshold values, CPU-time, and
accuracy are discussed.
Global modeling is a tremendous task that involves advanced numerical
techniques and different algorithms. As a result, it is computationally expensive [21-49].
Therefore, there is an urgent need to present a new approach to reduce the simulation
time, while maintaining the same degree of accuracy presented by global modeling
techniques. One approach is to adaptively refine grids in locations where the unknown
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83
variables vary rapidly. Such technique is called multiresolution time domain (MRTD),
and a very attractive way to implement it is to use wavelets [122]-[123].
The MRTD approach has been successfully applied to fmite-difference timedomain (FDTD) simulations of passive structures [124]-[134]. However, for the active
devices, that are characterized by a set of coupled and highly nonlinear partial differential
equations, applying the same approach would become quite time consuming [135].
Several different approaches for solving partial differential equations (PDE’s) using
wavelets have been considered. It has been observed by several authors that nonlinear
operators such as multiplication are too computationally expensive when conducted
directly on a wavelet basis. One of the approaches for solving PDE’s is the Interpolating
Wavelets technique presented in [136], in which the nonlinearities are dealt with using
the so-called sparse point representation (SPR). Interpolating Wavelets have been
successfully applied to the simple drift diffusion active device model [137]-[139]. Being
primarily developed for long-gate devices, the drift diffusion model leads to inaccurate
estimations
of device internal distributions and microwave
characteristics
for
submicrometer devices [54], It is worth mentioning that in [137], the author proposed a
new technique to solve simple forms of Hyperbolic PDE’s using an Interpolating Wavelet
scheme. These PDEs can represent Maxwell’s equations or the simple drift-diffusion
model but not the complete hydrodynamic model. Thus, a new approach to apply
wavelets to the hydrodynamic model PDEs is needed, along with extending it to
Maxwell’s Equations, for accurate modeling of submicrometer devices, while achieving a
CPU-time reduction.
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84
In this chapter, a unified approach to apply wavelets to the full hydrodynamic
model and Maxwell’s equations is developed. The main idea is to take snapshots of the
solution during the simulation, and apply wavelet transform to the current solution to
obtain the coefficients of the details. The coefficients of the details are then normalized,
and a threshold is applied to obtain a nonuniform grid. Two independent grid-updating
criteria are developed for the active and passive parts of the problem. Moreover, a
threshold formula that is dependent on the variable solution at any given time has been
developed and verified. A comprehensive set of results is included along with illustrative
comparison graphs.
4.2
Fundamentals of the MRTD
The construction o f biorthogonal wavelet bases relies on the notation of
multiresolution analysis [140]. This notation gives a formal description of the intuitive
idea that every signal can be constructed by a successive refinement, by iteratively adding
details to an approximation. The coefficients of the approximations are given by:
+00
a x[n ,m ]= $ x(t)<pnm(t)d t
(4 ^
-oo
where <pnm (t) is the family of dilates and translates of the scaling function formed as
(pnm{t) = 2 m,1(p{2mt - n )
(4.2)
On the other hand, (4.3) gives the coefficients of the details.
+00
d x[n ,m ]= \ x ( t) y / nm(t)d t
(4.3)
—oo
where y/nm (t) is the family of dilates and translates of the wavelet function defined as:
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85
( / „ ( 0 = 2’ ' V ( 2 ’ < - « )
(4'4)
While some wavelets such as Daubechies are asymmetrical [140], it is possible to
create symmetric wavelets with compact support by using two sets of wavelets, one to
compose the signal and the other to construct it. Such wavelets are called Biorthogonal
[141].
Complete coverage of the theory and applications of wavelets in electromagnetic
and device modeling can be found in the first book on the subject by George Pan [142].
4.2.1
Two Dimensional MRTD Scheme
For simplicity, the authors in [124] proposed a two-dimensional scheme for a
homogeneous lossless medium with the permittivity e and permeability p. Assuming no
variation along the y-direction, the Maxwell’s equations for the 2D TMZ mode were
written as:
dEx
1 dH
s r =- s ^ r
SHy
l
dt
p
dEz dEx )
- r - T 1
ox
dz I
dEz
1 dH
- r - =—
dt
e dx
(4-5)
(4-6)
(4. 7)
To derive the 2D MRTD scheme, the electric and magnetic field components
incorporated in these equations were expanded in a series of scaling and wavelet
functions in both x- and z- directions and in pulse functions in time [125]:
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
86
Ex(x ,z,t) =
+00
£
(*¥.(*)
2
k,l,m =-oo
+QO
+00
2rz - 1
£,/,m =-oo r2=0 p z =0
+oo
+oo 2r* —1
+ Z
E
L
£ /-w A (
o^
2( ^ , ( z)
k ,l,m = -oo ^ = 0 p x=0
+oo
+oo
(4.8)
2 ^ /z - l
+£ £ £
k,l,m=-x> rx ,rz =0 p x ,p z =0
Ez(x ,z,t) =
+oo
k ,l,m = - oo
+00
+
+00 2rz- l
I
k j , m = - oo r2=0 p z =0
+oo
+oo 2 ^ -1
Z Z k K ^ T 2 K ( 0 ^ X(X)<f>m-U2(z)
+ Z
k,l,m=-<x> rx =0 p x =0
+oo
+ Z
+oo
^
2'*,/z —1
Z
Z
k ^ : ^ K { t ) w ^ PS x ) y 'tm .P,^ ')
k j , m =-oo ^ , ^ = 0 p x ,p z =0
H y(x ,z,t) =
+00
k+inHi'-lf2>
m-l/2K+l/2(t)$l-l/2(X)<
/>m-\n(Z)
Z
k ,l,m = - oo
+Q0
+oo 2 ^ -1
+£ I £
k ,l,m = - oo f2=0 p z=0
+oo
+ Z
+oo 2j* - 1
Z Z * + l / 2^ / - ^ A +l/2(0 ^ /2W ^ ,-l/ 2(^)
k ,l,m =-oo rx=0 p x=0
+co
+oo
2r* ,fz -1
+£ £ £
k ,l,m =-oo r*,r2=0 p x ,p z =0
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(4.10)
87
Where cp„ (x) = cp((x / Ax) - n) and \yrn (x) = 2 r/2 v|/0 (2r [(x / Ax) - « ] - / ? ) represent
the Battle-Lamerie scaling function and r -resolution wavelet function, respectively.
Since higher resolutions of wavelets are shifted and dilated version of the zero resolution,
their domain will be a fraction of that of the zero-resolution wavelet. Thus there is going
to be more than one higher resolution coefficient for each MRTD cell. Specially, for the
arbitrary r resolution for the n cell to the x -direction, there exists 2r wavelet coefficients
located at x / Ax = n + (p / 2r+l), p = 0,.... ,2r -1 . This is the reason for the summation of
the p terms for each resolution r in the above expansion.
and k+l
with
k = x ,y ,z and p,v = <j),v)/ are the coefficients for the field expansions in terms of scaling
and wavelet functions.
Schemes based on a displacement of H and E by (I!2 )(rx'rz)+1 instead of 54 could
provide slightly improved numerical dispersion and stability characteristics. However, for
simplicity reasons, the authors in [125] used Yee’s convention for the derivation of the
their MRTD equations. The indices /, m, k are the discrete space and time indexes related
to the space and time coordinates. For an accuracy of 0.1%, the above summations are
truncated to a finite number of terms determined by the dispersion and stability
requirements (typically their maximum value is between 22-26). The time domain
expansion function hk (t) is defined as:
=
— *)
At
with the rectangular pulse function:
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(4.11)
The magnetic-field components are shifted by half a discretization interval in the
space and time domains with respect to the electric field components (leap-frog).
Upon inserting the field expansions, Maxwell’s equations are sampled using pulse
functions as time-domain test functions and scaling/wavelet functions as space-domain
test functions. For the sampling in the time domain, the following integrals were
employed [124]:
+00
(4.13)
-00
and
(4.14)
-0 0
where 5k is the Kroenecker symbol
4.2.2
1
fo r k = k
0
fo r k ^ k
(4.15)
Battle-Lemarie Expansion Basis
Sampling in the space domain is obtained by use of orthogonal relationships for
the Battle-Lemarie scaling and wavelet functions [123]:
+ 00
(4.16)
—00
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89
+00
J <Vm(.x W m \P'(x ) dx = W r , p
and
(4.17)
+00
J<P m‘,p'(x >prm ,p(x ) dx = ^r,r^m,m^p,p'A x
(4.18)
-00
The integrals containing derivatives can be approximated by the following
expressions:
+•00
jV * (x) d(Pm' ^ 1 ~X) dx K Y s t - n
(4-19)
-CO
with
oo A
a ( 0 = -^ Jl <KI) l2^ sin[ W + 1 / 2 ) ] ^
(4.20)
and
+ l\k
/
\ d§m'+H2,p(x )
,
V ’ "''-''2
< L 0 ) ------------------ d x * y
j
/ ■
xs
d r ^ P ^ m+i,m'
(4.2 1)
with
00
d r ( i , p ) = - f 2 - ’-/ 2 $ „ ( 5 ) J 0 ( 5 / 2 ' - ) s i n [ « i + 0 . 5 + P / 2 ' + l / 2 ' ' * 1) ] ^
n J
(4.22)
and
+00
Jf
d $m'+ l / l ( *
ax
)
e r O ,P )&M ^
J~Ul=~nc,rl1
<4-23>
— 00
with
00
cr ( i , p ) = -
f 2 - W2 i „ ( O v 0 ( V 2 ' ' ) s i n [ « ( + 0 . 5 + p / 2 r + l / 2 " 1)]a'4 (4.24)
71 J
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90
and
too
1
rl ( \
^ + 1 /2 ^ 2 ( x ) ,
V m , p l ( X ) --------------- ~ -----------d x
dx
b,r\,r2,2
m+i , m
rib , r l , r 2 , l
(4.25)
with
br\,r2(i’P l ’P T >=
00
-
f|vj)0 ( ^ / 2 ' ' 1) 11 v|;0 (^ /2 '" 2) | ^sin [^ (i + 0.5 + p 2 ! 2 rl - p H 2 rX + l / 2 r2+1 - l / 2 /’1+1)]c/^
71 J
0
(4.26)
For the remainder of this section, the authors in [124] considered only an
expansion as a series of scaling and zero-resolution wavelet functions. Hints for the
enhancement of additional wavelet resolutions will be presented where needed. Since for
the zero resolution ( r = 0 ), there is only one wavelet coefficient per cell ( p = 0 ), the
p symbols will be omitted from the definition of the b,c,d coefficients. The coefficients
a{i), b0(/), c 0 (/) are given in Table (4.1) [125].
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91
TABLE 4.1
COEFFICIENTS a(i),b0(i),c0(i)
1:
«(*)
0
L m n m u 7 m
m
m m
2.47253977327429
0.
I
-0.153878843323872
0.9502282774123074
-4.659725793402785E-O2
2
S.96O630332468729OB-02
0.1060591000788887
M s a m o s is s s m i & o a
8
-2 9291577898008906*02
9.392437777679437B-02
*3.69t9&774O974982B-02
3.14M 4447S216036& 03
2 .05744M I98775452E-02
4
5
-8.1844023252837126-03
1.3493860087091086*02
•ia isa o 3 ia o 8 0 4 9 6 7 & 4 n
S
4Jf57585fi52354830& 03
.2.8S89418100947S2B4J3
SJ7O 877725270031E-03
I
-2.3423053566494616-03
2.7780805141155296-03
-3.202621363952005 B-03
8
1,182877717O42020E-03
4 .m M 6 t§ 7 W S S 6 E - «
1.71408684956O89OE-03
0
-6.716635O68590737&O4
7J718O 730937770JE-04
-0.1765O84384& 4196604
10
3.583500907489797& 04
4.4M 267305845043B-O 4
4.9117M 74807M 8B 4M
11
'1 .9 3 1 3 2 M 8 4 7 U 7 N E -M
1.9527114191049006-04
-2.629253O13538502E-04
12
1.0198277670578696-04
-1.021304423384722E-04
1.407386855875626E-04
13
*5.0139431835184346*05
5.531259273804269E-05
-7.5338406895730006*05
14
2M 4 m m m m m E rm
-2.94733O468OME31E-05
4.0331462310996741-05
In [124], the stability limit for the 2D MRTD scheme based only on the scaling
functions’ expansion (S-MRTD) is given by:
At < -----------------cH " , C l a i n l
(4.27)
(Ax)2
(Ay)2
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92
where Ax and Ay are the cell dimensions and c = —} = is the velocity of the light in the
yjjuz
modeled medium.
For Ax = Ay = A , the above stability criterion gives:
A‘ur,m £
(4.28)
cV 2 £ M K i ' ) l
It is known [60] that
A' fdtd ~
j~ t------------ —
(4.29)
Cf ( A x F + (Av)r
For Ax = Ay = A , the above FDTD stability criterion reduces to
^
fd td
—
;V2
(4.30)
The above equations show that, for the same discretization size, the upper bounds
of the time steps of FDTD and S-MRTD are comparable and related through some factor.
The stability analysis can be generalized easily to three-dimensional case [124].
It is very clear from the above analysis that the proposed MRTD technique is not
simple to derive and implement. Moreover, the author applied the technique to a very
simple problem. This means that more complex derivations are needed for practical
problems. Accordingly, general approaches should be developed in order to remove the
redundancies of the original formulations. This can be implemented by solving the
original PDEs without modifications on nonuniform grids obtained using wavelets.
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93
4.3
Problem Description
The transistor model used in this work is a 3D full-hydrodynamic large-signal
electromagnetic-physical model. The active device model is based on the moments of the
Boltzmann’s Transport Equation obtained by integrating over the momentum space. The
integration results in a strongly coupled highly nonlinear set of partial differential
equations, called the conservation equations. These equations provide a time-dependent
self-consistent solution for carrier density, carrier energy, and carrier momentum, which
are given as follows.
current continuity
f
+ V'<ro) = 0'
(4.31)
■ energy conservation
- + qnv.E + V.(ffl>(s + K bT) ) = dt
x-momentum conservation
Ot
dx
~ Sq)
x8(e)
=
(4.32)
(4.33)
In the above equations, n is the electron concentration, u is the electron
velocity, E is the electric field, e is the electron energy, e0 is the equilibrium thermal
energy and p is the electron momentum. The energy and momentum relaxation times are
given by ze and r mj respectively. Similar expression is obtained for the y-direction
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94
momentum. The three conservation equations are solved in conjunction with Maxwell’s
equations:
(4.34)
W xH = $ £-+ J
dt
(4.35)
where E is the electric field, H is the magnetic field, D is the electric flux density, and
B is the magnetic flux density. The fields in Maxwell’s Equations are updated using the
current density /estim ated by Eq. (10).
J(t) = -qnu(t).
(4.36)
The above model accurately describes all the non-stationary transport effects by
incorporating energy dependence into all the transport parameters such as effective mass
and relaxation times, along with including EM-wave effects. Fig. (4.1) shows the crosssection of the simulated structure with parameters summarized in Table (4.2).
05 pm
1
Source
□3 pm
Gate
Drain
DopedGaAs (2k10' ' cm’')
t
y
□S pm
UndopedGaAs
21 pm
Fig. 4.1. Cross-section of the simulated transistor.
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95
TABLE 4.2
TRANSISTOR PARAMETERS USED IN THE SIMULATION
Drain and source contacts
0.5 pm
Gate-source separation
0.5 pm
Gate-drain separation
1.0 pm
Device thickness
0.8 pm
Device length
2.8 pm
Gate length
0.3 pm
Device Width
250 pm
Active layer thickness
0.2 pm
Active layer doping
2 x 10 1 cm'
Schottky barrier height
0.8 V
DC gate-source voltage
-0.5 V
DC drain-source voltage
3.0 V
4.4
The Proposed Algorithm
Fig. 2 shows the flow chart of the proposed algorithm. A uniform grid is defined
at the beginning of the simulation. Equations 5 through 9 are then solved in the sequence
shown in the flow chart to update the grid of the different variables at the new iteration
with the following criterion.
1 I
C
r
— x
I max,min
max,min
xm
l a x , m in
■
£
Hydro
The updating criterion checks if the solution of the variable x has changed by
8 since last iteration using wavelet transform. The subscripts c and / designate quantities
defined in the current time and last time where wavelet transform is performed,
respectively. The subscript “max, min” indicates that the maximum and minimum of the
variable x are checked with Relation (4.37) at the same time. It is worth mentioning here
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96
that boundary grid-points are not included for the maximum or minimum checking. The
value o f S used in the simulation is 0.1. If (4.37) is satisfied, wavelet transform is
performed on the current variable solution followed by thresholding to obtain a new
nonuniform grid for the variable x. Biorthogonal wavelets are used with notation BI03.1
to point out three vanishing moments for the mother wavelet and only one vanishing
moment for the scaling function. The nonuniform grids of the different variables are then
combined into only one nonuniform grid for the next iteration. The above steps are
repeated until the stopping criterion is satisfied.
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97
Initialization
Define a Uniform Grid (Nx by Ny)
Evaluate Fields
Solve Momentum Eq.
Solve Energy Eq.
Update Transport Parameters
Solve Continuity Eq.
No
Perform Wavelet Transform (BIO.3.1) on
the Current Solution of the Variable x and
Obtain the Coefficients of the Details.
Normalize the Coefficients of the Details.
Calculate the Threshold Value.
Remove Grid Points of Values Less Than
the Threshold.
Combine Variable Grids into One Nonuniform Grid
No
Stopping
Criterion
Satisfied?
Yes
( ^ ^ S to p ~ ~ ^ )
Fig. 4.2. Generic flowchart of the proposed algorithm.
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98
It should be noted that magnitude ranges of the variables used in the simulations
vary dramatically. For instance, carrier density per cm'3, is on the order of 1017, while,
energy expressed in eV, is on the order of 0.5. Accordingly, the threshold value should
be dependent on the variable solution at any given iteration. The proposed threshold
formula is given by (4.38).
T = I jl
N
I t *
(4.38)
/=i
In this equation, T0 is the initial threshold value, d? s are the coefficients of the
details, and A is the number of grid points in the x- or y-direction. Hence, the value of the
threshold T depends mainly on the variable solution at any given time rather than being
fixed. The values of T0 used in the simulation are 0.001, 0.01, and 0.05, respectively.
In this paper, a new technique to conceive the nonuniform grids using wavelets has been
developed. The main idea is to apply wavelet transform to the variable solution at any
given time to obtain the coefficients of the details, which are then normalized to its
maximum. Only grid points where the value of the normalized coefficients of the details
are larger than the threshold value given by Eq. (4.38) are included. Figures 4.3 and 4.4
illustrate different examples of the technique employed to obtain the nonuniform grids for
electron energy and x-momentum solutions at a specific cross-section. Fig. (4.3)
exemplifies how the proposed algorithm obtains the nonuniform grid using transverse
compression only. For instance, Fig. (4.3.a), shows the normalized amplitude of the
coefficients of the details for the electron energy. While, Fig. (4.3.b.) marks the grid
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99
points remaining after thresholding the normalized coefficients of the details using (4.38).
It is observed that the proposed technique accurately removes grid points in the locations
where variable solutions change very slowly.
Fig. (4.4) shows the method adopted to obtain the nonuniform grid using
longitudinal compression only. Considering Figures (4.3) and (4.4), one can conclude
that the compression in the longitudinal cross-sections is much more than that in the
transverse cross-sections. This is consistent with the fact that the physical changes in the
longitudinal cross-sections are much slower compared to those in the transverse crosssections.
Fig. (4.5) shows the procedure employed to obtain the nonuniform grid of the
electron energy. The process is achieved by obtaining two separate grids for the
transverse and longitudinal compressions, respectively. Then the two grids are combined
together using logical ‘AND’ to conceive the overall grid for the electron energy at this
given time. The same process is conducted for the other variables including xmomentum, y-momentum, carrier density, and potential whenever (4.37) is satisfied. The
separate grids of our variables are then combined using logical ‘OR’ to obtain the overall
grid for the next iterations.
The overall grid obtained needs further processing in order to define a finitedifference scheme on it. The simplest way to achieve that is to have the same number of
grid points for the parallel cross-sections, while the number of grid points in the
longitudinal cross-sections and the transverse cross-sections need not to be the same.
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100
Following the above procedure, it was found that boundary conditions
implementation, including Ohmic and Schottky contacts, does not need special treatment.
They can be treated similar to the standard finite-difference (FD) scheme. The only issue
the algorithm needs to keep track of is identifying the new boundaries of the metallic
contacts for each new grid, which is straightforward.
0.8
0.8
0.6
0.6
0.2
0.2
Normalized
A m p litu d e
0.4
0.4
=5 - 0.2
z -0.4
- 0.2
- 0.6
-0.4
- 0.6
- 0.8
10
20
30
40
50
80
10
20
30
Grid P aints
G rid P oints
(a)
(c)
40
50
40
50
80
0.16
0.14
0.12
Electron
Energy
(ev)
V 4
0.08
0.06
0.04
0.02
10
20
30
40
Grid P ointi
(b)
50
80
10
20
30
Grid Points
(d)
Fig. 4.3. (a) Normalized details coefficients for the electron energy at a certain transverse
cross-section, (b) Grid points marked on the actual curve for the electron energy at the
same transverse cross-section, (c) Normalized details coefficients for the x-momentum at
a certain transverse cross-section, (d) Grid points marked on the actual curve for xmomentum at the same transverse cross-section.
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Normalized Amplitude
101
0)
"O
3
Q.
E
<
73
Q
N)
"E!5
Z
a
-0.5
10
20
Grid Points
30
40
50
60
40
50
60
Grid Points
(c)
x 10
0.3
Electron Energy (ev)
0.25
0.2
H 10
0.15
0.05
10
20
30
40
Grid Points
(b)
50
60
10
20
30
Grid Points
id )
Fig. 4.4. (a) Normalized details coefficients for the electron energy at a certain
longitudinal cross-section, (b) Grid points marked on the actual curve for the electron
energy at the same longitudinal cross-section, (c) Normalized details coefficients for
the x-momentum at a certain longitudinal cross-section, (d) Grid points marked on the
actual curve for x-momentum at the same longitudinal cross-section.
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102
Table (4.3) shows the evolution of the nonuniform grids. It can be observed that
the number of grid points for the overall grid increases as time advances. The reason is
that at the beginning of the simulation the solution is not completely formed yet. As the
time marches, more grid points are needed to incorporate the changes in the solution.
Furthermore, the different variable grids should not be updated at the same rate. For
instance, it is apparent that the potential needs not to be updated at the same rate as the
other variables. Notice that Table (4.3) is used for illustration purposes to demonstrate the
way the different variable grids change. In the actual simulation, the potential grid is
updated a few times at the beginning of the simulation, and then it remains unchanged.
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103
TABLE 4.3
GRID ADAPTABILITY OF THE DIFFERENT VARIABLES FOR T0 = 1%
Variable
Potential
Carrier Density
Energy
x-Momentum
y -Momentum
All Variables
Potential
Carrier Density
Energy
x-Momentum
y -Momentum
All Variables
Potential
Carrier Density
Energy
x-Momentum
y -Momentum
All Variables
Potential
Carrier Density
Energy
x-Momentum
y -Momentum
All Variables
Potential
Carrier Density
Energy
x-Momentum
y -Momentum
All Variables
Unknowns
Unknowns
Remaining After
Remaining After
Longitudinal
Transverse
Compression
Compression
(%)
(%)
Time Iteration # 1 2 0
7.74
5.69
14.92
6.54
17.63
39.65
16.06
43.39
17.53
16.11
22.36
65.14
Time Iteration # 250
8.59
5.88
16.70
13.69
39.21
23.00
19.09
43.65
20.02
19.46
61.94
28.93
Time Iteration # 480
6.27
9.23
21.51
17.16
43.99
28.88
23.44
38.57
26.20
26.76
58.84
36.25
Time Iteration # 5 9 0
6.04
9.64
29.88
18.24
48.85
31.88
41.91
29.08
37.04
32.91
62.36
44.73
Time Iteration # 730
7.01
11.13
34.08
16.55
39.77
35.64
41.91
27.98
51.46
34.84
62.84
58.96
Total
Unknowns
Remaining
(%)
0.63
2.64
8.54
9.18
3.78
14.43
0.76
5.18
12.26
10.64
7.95
20.58
0.90
7.71
15.72
12.72
13.53
25.05
0.93
10.61
18.43
16.21
20.36
31.74
1.15
8.42
18.43
13.32
25.00
36.43
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104
Now, we turn our attention to Maxwell’s Equations. The passive part of the FET
represents a co-planar structure in which a 3D FDTD is developed to solve for the
electric and magnetic fields. The current density estimated from the active device
conversation equations is used to update the variables in Maxwell’s Equations.
It is importance to state that the same approach developed to obtain the
nonuniform grid for the variables of the conversation equations is applied to Maxwell’s
Equations as well. However, a different updating mechanism should be developed to
keep track of the wave propagation within the passive structure. The following is the
algorithm developed for the grid updating of FDTD simulations.
Step 1: Construct a 3D matrix M that has only 0’s and l ’s, based whether or not we
have a non-zero solution of the field at this location. For example, “ 1” is assigned if a
non-zero field solution exists, and “0 ” elsewhere.
Step 2: Estimate the value of dFDTD (FDTD grid-updating factor) as:
y (v m new ® m old''i,j,k
..)..,
jLmd
I i *
_________
(4.39)
FDTD
Where M newand M old are the matrices constructed using step one for the current,
and old solutions of the fields, respectively. N xd, N yd, and N zd are the number of grid
points in x , y , and z directions, respectively.
Step 3: Check c ^ ^ ’s value against a predefined value, for example 5%.
Step 4: If satisfied, move the grid to z = z + d z . Where dz is proportional to 4FDTD-
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105
Step 5: t = t + dt
Fig. (4.5) illustrates examples of how the nonuniform grids are obtained for the
magnetic and electric fields at a specific cross-section for FDTD simulations. For
instance, Fig. (4.5.a) shows the normalized amplitude of the coefficients of the details for
the electric field. Fig. (4.5.b) marks the grid points remaining after thresholding the
normalized coefficients of the details using (4.38).
It should be observed that the
proposed technique accurately removes the grid points in locations where variable
solutions change very slowly. This would have an effect of reducing the CPU-time by
removing the redundant grid-points introduced by the original formulation. The total grid
of the electric field can be achieved by obtaining two separate grids for the transverse and
longitudinal compressions, respectively. Then the two grids are combined together using
logical ‘AND’ to conceive the overall grid for the electric field at this given time. It is
worth mentioning here that the excitation wave exists at the source plan at all times, and
the technique proposed here is generic that can be applied also to a short pulse
propagating in the computational domain.
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106
0.5
0.5
04
03
02
% 0:3
0I
1 02
0
-0.3
too
120
d istan ce
mtwms «f
distance, in term s of grid points
(a)
(c)
xtCF
1,005
f
2, 4]015
-002
E
-0.03
-0.04
a
40
si
ao
too
distancemtermsofgridpoints
(b)
120
20
40
60
80
distance in ttrm s of grid points
100
120
(d)
Fig. 4.5. Demonstration of the procedure employed to obtain the nonuniform grid for the
y-direction electric and magnetic fields for FDTD simulations.
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107
4.5
Results And Discussions
4.5.1
Hydrodynamic Model DC Simulation Results
The approach presented in this chapter is general and it can be applied to any
unipolar transistor. To demonstrate the potential of this approach, it is applied to an
idealized MESFET structure, which is discretized by a mesh of 64 Ar by 64 Ay
with At = 0.001 p s . Forward Euler is adopted as an explicit fmite-difference method. In
addition, Upwinding is employed to have a stable fmite-difference scheme. The space
step sizes are adjusted to satisfy Debye length, while the time step value At is chosen to
satisfy the Courant-Friedrichs-Levy (CFL) condition. First, DC simulations are
performed following the flow chart given by Fig. (4.2), and the current density is
calculated using (4.36). DC excitation is performed by forcing the potential to be equal to
the applied voltages to the electrodes (i.e., Dirichlet boundary conditions).
Fig. (4.6) shows the percentage remaining number of unknowns or grid points
versus the time-iteration number for different initial threshold values. Notice that as the
threshold value increases, the remaining number of grid points decreases. At the end of
the simulation and for an initial threshold value of 0 . 1%, the remaining number of
unknowns is almost 70%, whereas for an initial threshold value of 1%, the remaining
number o f unknowns is about 30%. The remaining number of unknowns is very sensitive
to the initial threshold value in a way that small changes in T0 results in considerable
changes in the remaining number of unknowns.
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108
100
-V- To=0.001
To=0.01
T d= 0 .0 5
to
c
so
c
c
->
01
-Q
E
3
z
c
c
CO
E
0!
□e
Iteration N u m b er
Fig. 4.6. Remaining number of unknowns as a percentage versus the iteration number for
different initial threshold values.
Furthermore, it is observed that the remaining number of unknowns change
during the simulation and this is associated with the grid adaptability used in the
simulation. In the next section, we will study the effect of the initial threshold value on
the final result accuracy as well the trade off between accuracy and CPU-time.
It is important to note that a suitable approach to investigate the capabilities of the
proposed technique is to compare it to the uniform-grid algorithm. In this case, the new
simulator will be accurately evaluated. Since both algorithms, the wavelet-based and the
uniform one will run on the same computer. In addition, both algorithms will have the
same discretization schemes and the exact semiconductor parameters.
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109
Fig. (4.7) shows the drain current convergence curves versus the CPU-time in
seconds for the cases of the uniform grid and the proposed wavelet-based adaptive grids
with different initial threshold values T0. Fig. (4.7) demonstrates that using the proposed
wavelet-based grids approach reduces the CPU-time dramatically. For instance, there is a
reduction of about 75% in CPU-time over the uniform grid case for the initial threshold
value of 1%, while the DC drain current error is within 1%. In addition, increasing the
initial threshold beyond certain value has a negative effect on the accuracy of the final
results.
-
0.1
-
0.2
-0.3
2-04
c -O.i
-0.7
-
0.8
-0.9
—
0
20
40
60
Uniform Grid
W avelet-Based Adaptive Grids With To=0.1%
W avelet-Based Adaptive Grids With To=1%
. W avelet-Based Adaptive Grids With To=5%
80
100
CPU-Time (Seconds)
120
140
160
180
Fig. 4.7. DC drain current convergence curves for the uniform grid and the proposed
wavelet-based non-uniform grids for different initial threshold values.
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110
This is apparent for T0 equals to 5%, where there is no agreement between the
results achieved using the uniform grid case and the wavelet-based nonuniform grids.
The reason is that using large values of T0 implies that more grid points are removed,
including important grid points that will have a negative effect on the final result. On the
other hand, using a very small threshold values implies redundant grid points. In
summary, there should be an optimal value of T0 such that both the CPU-time and error
are minimized. In this work, T0of 1% is suggested to have a considerable reduction in
CPU-time, while keeping error within an acceptable range.
Fig. (4.8) shows the potential distribution obtained using the proposed algorithm
with T0 equals to 1%. This graph demonstrates that boundary conditions are satisfied at
the electrodes. For instance, the value of potential at the gate is equal to -1.3 volts, which
is the applied DC voltage minus the Schottky barrier height.
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Ill
Fig. 4.8. DC potential distribution obtained by the proposed algorithm using a value of
the initial threshold equals to 1%.
The values of the elements of the small-signal equivalent-circuit model are
computed from the variations in voltages, currents, and charges due to small changes in
the DC bias voltages and/or currents. For instance, the small signal gate-to-source
capacitance and transconductance are computed as:
AQ
C
g
8s
AV
gs
(4.40)
AI
(4.41)
gs
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112
where Q
o
is the charge on the gate electrode and V
o*5
is the gate-to-source voltage. The
charge Qg is calculated using the integral form of Gauss’s law. An important figure of
merit that can be evaluated from these parameters is the device unit-gain cut-off
frequency given by (4.42).
f
_
J t ~ 2nC
The values of C
gs
and g
m
(4.42)
gs
are plotted against the applied gate-to-source voltage
for both the uniform-grid case and the proposed algorithm. Fig. (4.9) shows the
comparison, where the good agreement between the proposed algorithm and the uniformgrid case should be observed. Using the proposed algorithm and for V
found to be 263 mS/mm, and C
= -0 .5 V , g m is
to be 0.47 pF/mm. In this case, the unit-gain cut-off
frequency is calculated using (4.42) to be 90 GHz.
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113
4.5.2
Hydrodynamic Model AC Simulation Results
The AC excitation applied to the gate electrode is given as:
V (t) = V
+Au
sin(*yf)
gs
gso
gs
(4.43)
where VgS0 is the DC bias applied to the gate electrode, A ogS is the peak value of the
AC signal (0.1 volts), and a is the frequency of the applied signal in rad./sec. The
frequency used in the simulation is 60 GHz. AC excitation is implemented in the same
manner as in DC excitation. However, values of the gate potential are obtained at the new
time t using (4.43).
First, the DC solution is obtained by solving Poisson’s equation in conjunction
with the three hydrodynamic conservation equations. Then, a new value gate-source
voltage is calculated using (4.43). This new value is used to update Poisson’s equation for
the new voltage distribution, and consequently the new electric field. The electric field is
then used to update the variables in the conservation equations. This process is repeated
every A t following the proposed algorithm given in Fig. (4.2), until t - t m a x . The
current density is obtained using (4.36). The current density calculated on the plan
located midway between the drain and gate is integrated to obtain the total current. The
output voltage is estimated by multiplying the total current by the resistance that defines
the DC operating point (Q point) of the transistor.
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114
0.5
Uniform Grid
Wavelet-Based Grid
0.48
0.46
0.44
0.42
E
E
u_
& 0.4
(/)
O)
o
0.38
0.36
0.34
0.32
0.3
-
- 0.8
1.6
fal
280
260
240
220
200
160
140
120
- Uniform Grid
" Wavelet-Based Grid
100
80
•1.8
- 1.6
-1.4
•1.2
•1
- 0.8
.6
-O .i
Vgs (V)
fbl
Fig. 4.9. Comparison between the uniform-grid and the proposed algorithm with a value
of the initial threshold equals to 1% (a) Gate-to-source capacitance versus gate-to-source
voltage, (b) Transconductance versus gate-to-source voltage.
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115
Fig. (4.10) shows the output voltage obtained using the proposed algorithm with
T0=\% . A gain of 11 dB is achieved. Moreover, it is observed that there is an output
delay of about 1 ps that represents the time required for the transistor to respond to the
input signal.
Fig. (4.11) shows the output voltage for the uniform-grid case and the proposed
algorithm with different initial threshold values jf.
The purpose of the figure is to
emphasize that an optimal value of the threshold should be employed to maintain the
required accuracy, while keeping CPU-time as minimum as possible. It is observed that
using different values of T0 affects the accuracy of the solution. For instance, using a
large value of T0 results in a completely different solution, and this means the scheme for
this special case is inaccurate. This is apparent for the case of T0 equals to 5%. Because
employing a large value o f T0 results in removing significant grid points, which degrades
the final results. Similar to DC simulations, the existence of an optimal value for T0 is
suggested. Furthermore, it is noticed that there is no significant difference in terms of
accuracy between the two cases of T0= 0 . 1% and T0=\% . The mean relative error
obtained for the two cases is in the order of 3 ~ to 4%. This suggests that using T0equals
to 1% be the right choice in terms of both accuracy and CPU-time.
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116
0.5
— Vgs
..... Vds
0.4
/
0.3
0.2
s
±2
o
>
-
0.1
-
0.2
-0.3
-0.4
-0.5
0.5
1.5
Time
2.5
■it
x10'
Fig. 4.10. Large-signal result obtained by the proposed algorithm for a value of the initial
threshold equals to 1%.
—
—
Uniform Grid
W avelet-Based Adaptive Grids To=0.1%
W avelet-Based Adaptive Grids To=1.0%
. . . . . W avelet-Based Adaptive Grids To=5.0%
0.5
0.4
0.3
0.2
0.1
-
0.2
-0.3
-0.4
-0.5
0
0.5
1
1.5
Time (Seconds)
2
2.5
3
x 1 0 '11
Fig. 4.11. AC output voltage for the uniform grid and the proposed wavelet-based nonuniform grids with different values of the initial threshold.
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117
4.5.3
FDTD Simulation Results
A 3D Yee-based FDTD code is developed, with the proposed algorithm
employed. A Guassian excitation pulse is applied to evaluate the algorithm over a wide
range o f frequencies. Table (4.4) shows that as threshold value increases, CPU-time and
error introduced decreases as well. It is noteworthy to point out that using an initial
threshold value equals to 10% seems to reduce error along with the CPU-time. However,
considering Fig. (4.12), one should conclude that using T0 equals to 10% introduces
dispersion, which is a serious type of error. Accordingly, an initial threshold value of 5%
is recommended in terms of both CPU-time and error for FDTD simulations. It is
important to emphasize that the passive and active parts of the problem have different
optimal threshold values. This is expected since the variables in the conservations
equations are highly nonlinear compared to the fields obtained when solving Maxwell’s
Equations. Research leading to the work presented in this chapter can be found in [2][15].
TABLE 4.4
EFFECT OF THE THRESHOLD VALUE ON ERROR AND CPU-TIME FOR FDTD
SIMULATIONS
T
-Lo
CPU-Time
(Seconds)
0.0 (Uniform Grid)
0 . 1%
1 .0 %
5.0%
10 .0 %
744.90
300.17
205.92
155.10
111.05
Error
2-norm
co-norm
0.0873%
0.0871%
0.0778%
0.0473%
8.80%
8.75%
7.69%
3.66%
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118
0.8
—
—
0.7
Uniform Grid
MRTD, To=1%
MRTD, To=5%
MRTD, To=10%
0.6
0.5
S
0.4
0.2
0.1
-
0.1
0.5
2.5
3.5
time in seconds
Fig. 4.12. Potential of the gate at a specific cross-section versus time for the uniform grid
case and the proposed MRTD algorithm with different values of T0.
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119
4.6
Scheme Errors and Stability Analysis
It is important to mention here that the simulation and physical times are
completely separate entities. The simulation time required to model a specific physical
process should vary depending on the technique implemented in the simulation.
The purpose of this section is to demonstrate that the mechanism by which error is
introduced when employing the proposed wavelet-based technique is different than of the
uniform-grid case. The local truncation error for the uniform grid case is dependent, in
general, on the mesh spacing ( Ar and Ay), and the time step used A t . On the other hand,
the local truncation error for the wavelet-based nonuniform grids approach depends on
how accurately the important grid points are reserved as well as the time step used. This
suggests that the local truncation errors, due to spatial discretization, for the uniform grid
case and the wavelet-based nonuniform grids are different. The local truncation error
accumulates from iteration to iteration. The total truncation or discretization error is thus
dependent on the number of iterations used (space and time iterations combined).
Accordingly, one can conclude that the total error introduced by the wavelet-based
technique due to the local discretization errors accumulating during the simulation, may
or may not be larger than that of the uniform grid case, at least for the two cases of
T0= 0 . 1% and 7^=1%. The reason is the number of iterations required reaching the steady
state solution for the uniform grid case is much larger than that of the proposed
algorithm. In summary, the total error introduced depends on the local truncation error
along with the number of iterations required to reach the final solution. This explains the
results in the paper comparison figures, where it would be difficult to draw a precise
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120
conclusion of which technique is more accurate. This is because for each case or curve,
the number o f iterations required to obtain the steady state solution and the local
discretization errors are different. The problem of identifying the most accurate solution
becomes even more difficult since we are dealing with a highly nonlinear problem.
It is worth mentioning here that the proposed algorithm does not have any
stability constraints if At is chosen to satisfy the CFL condition at the beginning of the
simulation. The reason is, as the simulation progresses, the spatial distances employed
become even larger than the ones introduced at the beginning. This represents an extra
benefit o f using the proposed algorithm that it does not need any time-step At change
while the simulation is in progress.
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121
4.7
Summary
In this chapter, a new wavelet approach has been developed and successfully
applied to a 2D full hydrodynamic large-signal simulator and 3-D FDTD simulator. The
proposed algorithm solves the highly nonlinear PDEs that characterize the semiconductor
device behavior and Maxwell’s equations on nonuniform grids. The nonuniform grids are
conceived by applying wavelet transforms to the variable solution followed by
thresholding. It is found that each variable has its own grid at any given time, and the
grids of the different variables need not to be updated at the same rate. A reduction of
75% in CPU-time is achieved compared to a uniform grid case with an error of 2% on the
DC drain current for a 1% initial threshold value. Furthermore, the same CPU-time
reduction has been achieved for AC simulations with a mean relative error of order 3 to
4%. Moreover, an 80% CPU-time reduction is obtained for FDTD simulations with a
0.1% average error on the potential. It has been observed that tradeoffs exist among the
threshold value, CPU-time, and accuracy, suggesting an optimal value of the threshold.
The proposed algorithm efficiently solves both DC and large-signal AC problems.
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Chapter 5
Conclusions and Future Work
5.1
Conclusions
This dissertation presents a new approach for the simulation and optimization of
microwave devices, using a genetic algorithm (GA). The proposed technique solves the
equations that describe the semiconductor transport physics in conjunction with Poisson’s
equation, employing an adaptive real-coded GA. An objective function is formulated,
and most of the GA parameters are recommended to change during the simulation. In
addition, different methods for describing the way the GA parameters change are
developed and studied. The effect of GA parameters including the mutation value,
number of crossover points, selection criteria, size of population, and probability of
mutation is analyzed. The technique is validated by simulating a submicrometer field
effect transistor (FET), and then compared to successive over relaxation (SOR), showing
the same degree of accuracy along with a moderate speed of convergence. The purpose
o f this study is to introduce a new vision for a genetic algorithm capable of optimizing
real value functions with a considerably large number of variables. This study also
represents a fundamental step toward applying GAs to Maxwell’s equations in
conjunction with the hydrodynamic model (HDM), aiming to develop an optimized and
unconditionally stable global-modeling simulator.
Moreover, this dissertation presents a new wavelet approach that has been
developed and successfully applied to a 2D full hydrodynamic large-signal physical
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123
simulator and 3D FDTD simulator. The proposed algorithm solves the highly nonlinear
PDEs that characterize the semiconductor device behavior on nonuniform multiresolution
self-adaptive grids. The nonuniform grids are conceived by applying wavelet transforms
to the variable solution followed by thresholding. It is found that each variable has its
own grid at any given time, and the grids of the different variables need not to be updated
at the same rate. A reduction of 75% in CPU-time is achieved compared to a uniform grid
case with an error of 2% on the DC drain current for a 1% initial threshold value.
Furthermore, the same CPU-time reduction has been achieved for AC simulations with a
mean relative error of order 3 to 4%. Moreover, an 80% CPU-time reduction is obtained
for FDTD simulations with a 0.1% average error on the potential. It has been observed
that tradeoffs exist among the threshold value, CPU-time, and accuracy, suggesting an
optimal value of the threshold. The proposed algorithm efficiently solves both DC and
large-signal AC problems.
Furthermore, in this dissertation the potential of high power and frequency
multifinger transistors is demonstrated, using a completely numerical coupled
electromagnetic-physical simulator. The preliminary results of this study show that at
very high frequency, several phenomena with strong impact on the device behavior start
to emerge, such as phase velocity mismatches, electron-wave interaction, and attenuation.
The results suggest that contemporary microwave devices should be optimized to
minimize these effects or possibly take advantage of in favor of improved device
characteristics. The results also recommend multifinger transistors as potential
alternatives to conventional transistors. This is achieved by using multiple-finger gates of
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124
less width instead o f a single-gate device. Furthermore, this dissertation underlines the
enhanced microwave characteristics of multifinger transistors attributable to reducing
attenuation and EM-wave propagation effects along the device width.
5.1
Future Work
There are many possible future research ideas that can be carried out. For instance:
•
A possible future research work is to employ rigorous optimization techniques to
obtain the optimal multifinger transistor structure based on the electromagneticphysical model presented in this dissertation. Moreover, measurements can be
carried out and compared to the results achieved by our model.
•
To accurately model nano-devices, a possible future research work may involve
developing a global modeling simulator by carrying out Monte Carlo simulations
in conjunction with Maxwell’s equations. Moreover, measurements can be
performed to validate the algorithm.
•
Another possible work extension is to model complete microwave chips having
several components simultaneously, using the coupled electromagnetic-physicsbased simulator introduced in this dissertation.
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125
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