LARGE-SIGNAL RF SIMULATION AND CHARACTERIZATION OF ELECTRONIC
DEVICES USING FERMI KINETICS TRANSPORT
By
Nicholas Charles Miller

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
Electrical Engineering – Doctor of Philosophy
2017

ABSTRACT
LARGE-SIGNAL RF SIMULATION AND CHARACTERIZATION OF ELECTRONIC
DEVICES USING FERMI KINETICS TRANSPORT
By
Nicholas Charles Miller
Design of radio frequency (RF) power amplifiers (PAs) for wireless communications requires
small- and large-signal data collected from the underlying transistors, including scattering parameters (S-Parameters) and load-pull (LP), to determine optimal impedance targets. High speed
devices operating with fundamental frequencies above 35 GHz present extreme challenges for measuring the harmonic signals resulting from nonlinear effects. Predictive physics based simulations
in conjunction with compact modeling capabilities are promising alternatives to expensive and
time-consuming measurements. To date, tools either exist in the electron transport domain or in
the behavioral modeling domain and a key goal is to treat these problems simultaneously because
they are strongly coupled at millimeter wave frequencies. Accurate physics based simulations of
high speed and high power transistors require proper treatment of hot-electron, self-heating, and
full-wave effects. The Boltzmann solver called Fermi kinetics transport (FKT) has been shown to
capture all of these important physical effects. FKT can approach the accuracy of Monte Carlo
methods while maintaining the computational efficiency of deterministic solvers. The latter trait
allows simulation of large electronic devices such as the output stages of PAs. Previous work
on FKT provided proof of concept results which demonstrated its versatility and accuracy as an
electronic device simulation framework.
The purpose and contribution of this thesis is the use of FKT as a predictive TCAD tool to generate RF data required for PA design. This work begins with a thorough investigation of the underlying physical equations and their numerical solution for electronic device simulations. Included in
this investigation is an analysis of the full-wave discretization technique called Delaunay-Voronoi
surface integration (DVSI), a derivation of the FKT device equations and their discretization in
energy- and real-space, and a detailed account on the numerical solution of the fully coupled non-

linear system of equations. The detail provided in this work is meant to provide future device engineers and researchers a thorough understanding of the numerical framework for their application
and simulation needs. The FKT device simulator is then applied to real device geometries to generate useful data for RF circuit designers. Extensions of the FKT method required for large-signal
LP simulations are presented with representative applications. Additionally, compact behavioral
models are extracted directly from FKT device simulations, enabling a computationally efficient
means for simulated LP data generation. The resulting TCAD tool is a promising simulation capability for high power RF transistor design and characterization. It is anticipated that PA design for
5G applications will use techniques like these in the near future.

To my wife, Samantha.

iv

ACKNOWLEDGEMENTS

First and foremost, I would like to sincerely thank my advisor Professor John Albrecht for all the
support and advice, both technical and personal, over my graduate school career. I would especially
like to thank him for all his hard work these past few months on making my thesis readable. I wish
to continue collaboration with him for many years.
Next, I extend a wealth of gratitude to my committee members Professor Timothy Grotjohn,
Professor Donald Morelli, and Professor John Verboncoeur for their advice and input to my graduate career.
I would like to sincerely thank Dr. Matt Grupen at the Air Force Research Laboratory Sensors
Directorate (AFRL) for his support and encouragement over the years and his time and effort
editing my thesis. This thesis is based on his pioneering work. Matt amazes me with his deep
technical discussions on solid state physics. However, they always seem to be laced with Seinfeld
references. I was never sure if that was his or my fault.
I would also like to thank Dr. Bob Fitch and Dr. Gregg Jessen at AFRL for their collaboration
and for furnishing measured data.
Many thanks to my colleagues at Michigan State University. In particular, I thank Marco
Santia, Dr. Nandan Tandon, and Matt Hodek for their years of support and encouragement.
I also thank our colleagues at Tech-X corporation, Dr. David Smithe and Dr. Kris Beckwith,
for their technical advice over the years.
Thank you to the SMART scholarship program for supporting my graduate education and
research and enabling my collaborations with and future at AFRL.
I would be remiss to not thank my undergraduate research advisor Professor Shanker Balasubramaniam for his time and support in helping me learn computational electromagnetics/physics.
Furthermore, I would like to thank Professor Ed Rothwell for connecting me with the CEM group
and starting my research career. Finally, I would like to thank Dr. Andrew Baczewski, Dr. Naveen
Nair, and Dr. Daniel Dault, each of whom devoted countless hours towards my fundamental un-

v

derstanding of computational electromagnetics/physics.
I would not be here without the support and encouragement of my parents, Michael and Debra
Miller. Thank you for all you have done!
Finally and most importantly, I thank my wife Samantha for all her love, support, and encouragement. It is to her that I dedicate this thesis.

vi

TABLE OF CONTENTS

LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xi

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xii
CHAPTER 1

INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1

CHAPTER 2 BACKGROUND . . . . . . . . . . . . . . . . . . . .
2.1 The Box Integration Method . . . . . . . . . . . . . . . . .
2.2 The Delaunay-Voronoi Mesh . . . . . . . . . . . . . . . . .
2.3 Systems of Nonlinear Partial Differential Equations . . . . .
2.3.1 Numerical Solution of Discrete Nonlinear Equations
2.3.1.1 Fixed Point Iteration Method . . . . . . . .
2.3.1.2 Newton’s Method . . . . . . . . . . . . .
2.3.1.3 Comparison of FPM and Newton’s Method
2.3.1.4 Other Iterative Methods . . . . . . . . . .
2.3.2 Mesh Convergence of Discretization Techniques . .
2.3.3 Stability of Discrete Equations . . . . . . . . . . . .
2.4 RF Amplifiers . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.1 Amplifier Figures of Merit . . . . . . . . . . . . . .
2.4.2 Linear Two-Port Models . . . . . . . . . . . . . . .
2.4.3 Large Signal Analysis . . . . . . . . . . . . . . . . .
2.5 X-Parameters . . . . . . . . . . . . . . . . . . . . . . . . .
2.5.1 Two ports, one harmonic . . . . . . . . . . . . . . .
2.5.2 Two ports, two harmonics . . . . . . . . . . . . . .
2.5.3 Two ports, K harmonics . . . . . . . . . . . . . . . .
2.6 Load-Dependent X-Parameters . . . . . . . . . . . . . . . .
2.6.1 Two ports, one harmonic . . . . . . . . . . . . . . .
2.6.2 Two ports, K harmonics . . . . . . . . . . . . . . . .

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

7
7
8
14
15
15
16
17
19
19
22
24
25
26
29
34
35
36
39
41
41
42

CHAPTER 3 DELAUNAY-VORONOI SURFACE INTEGRATION
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2 Physical Equations and Their Geometric Representation . . .
3.3 Maxwell’s Rotational Equations . . . . . . . . . . . . . . .
3.3.1 DVSI Discretization . . . . . . . . . . . . . . . . .
3.3.2 Dielectric Discontinuities . . . . . . . . . . . . . . .
3.3.3 Boundary Conditions . . . . . . . . . . . . . . . . .
3.3.3.1 Perfect Electrical Conductor . . . . . . . .
3.3.3.2 Impedance Boundary Condition . . . . . .
3.3.4 PEC Cavities with Ideal Dielectrics . . . . . . . . .
3.4 Poisson’s Equation . . . . . . . . . . . . . . . . . . . . . .
3.4.1 DVSI Discretization . . . . . . . . . . . . . . . . .

.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.

44
44
47
48
48
51
52
53
53
54
56
56

vii

.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.

58
58
59
59
60
60
61
62
63
64
66

CHAPTER 4 FERMI KINETICS TRANSPORT . . . . . . . . . . . . . . . . . . . .
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.1 Moments of the BTE . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.2 An Alternative Treatment of Heat Flow . . . . . . . . . . . . . . . .
4.1.3 Self Heating Effects . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 FKT Energy-Space Discretization . . . . . . . . . . . . . . . . . . . . . . .
4.2.1 Converting Reciprocal-Space Integration to Energy-Space Integration
4.2.2 Isosurface Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2.3 Piece-Wise Energy Power Laws . . . . . . . . . . . . . . . . . . . .
4.2.4 Incomplete Fermi Integrals . . . . . . . . . . . . . . . . . . . . . . .
4.3 FKT Real-Space Discretization . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.1 Continuity Equations . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.2 Particle and Energy Fluxes . . . . . . . . . . . . . . . . . . . . . . .
4.3.2.1 The Original SG Discretization . . . . . . . . . . . . . . .
4.3.2.2 SG Discretization of the FKT Particle Flux . . . . . . . . .
4.3.3 Heat Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

67
67
69
71
72
72
72
74
76
79
81
82
82
83
83
85
87

CHAPTER 5 NUMERICAL SOLUTION OF THE FKT EQUATIONS .
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2 The System of Nonlinear Equations . . . . . . . . . . . . . . .
5.2.1 Solution Variables . . . . . . . . . . . . . . . . . . . . .
5.2.1.1 Electron Gases in Real-Space . . . . . . . . .
5.2.1.2 Electron Gases in Energy-Space . . . . . . . .
5.2.1.3 Electromagnetics . . . . . . . . . . . . . . . .
5.3 The Nonlinear Solver: Newton’s Method . . . . . . . . . . . . .
5.4 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . .
5.4.1 Ohmic Metal Contact . . . . . . . . . . . . . . . . . . .
5.4.2 Lattice Heat Absorbing Boundary Condition . . . . . . .
5.4.3 Quasi-Static AC Impedance Boundary Condition . . . .
5.4.4 Full-Wave Voltage Port . . . . . . . . . . . . . . . . . .
5.4.5 Full-Wave AC Impedance Port . . . . . . . . . . . . . .
5.4.6 Heterojunctions and Thermionic Emission Boundaries .

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

88
88
88
90
91
92
93
93
95
95
96
96
97
97
97

3.5

3.6

3.4.2 Boundary Conditions for Poisson’s Equation . . . .
3.4.3 Spherical Region with Dirichlet Surfaces . . . . .
Complete Electromagnetics for Device Simulation . . . . .
3.5.1 DVSI Discretization . . . . . . . . . . . . . . . .
3.5.2 Device Boundary Conditions . . . . . . . . . . . .
3.5.2.1 Metal Contacts . . . . . . . . . . . . . .
3.5.2.2 Extrinsic Impedance Boundary Condition
3.5.2.3 Voltage Boundary Condition . . . . . . .
3.5.2.4 Two-Port Devices . . . . . . . . . . . .
3.5.3 Lossless Transmission Line . . . . . . . . . . . . .
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . .

viii

.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

5.5

5.6
5.7
5.8

5.9

5.4.7 Schottky Metal Contact . . . . . . . . . . . . . . . . . . . . . . . .
Semiconductor Device Simulation Work-Flow . . . . . . . . . . . . . . . .
5.5.1 Work-Flow Overview . . . . . . . . . . . . . . . . . . . . . . . . .
5.5.2 Equilibrium Solve . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.5.3 Static Solve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.5.4 Quasi-Static Solve . . . . . . . . . . . . . . . . . . . . . . . . . .
5.5.5 Full-Wave Solve . . . . . . . . . . . . . . . . . . . . . . . . . . .
Mesh Convergence of the Discrete FKT Equations . . . . . . . . . . . . . .
Stability of the Discrete FKT Equations . . . . . . . . . . . . . . . . . . .
Advanced Simulation Techniques . . . . . . . . . . . . . . . . . . . . . . .
5.8.1 The Linear Solve . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.8.2 Second-Order SG Discretization . . . . . . . . . . . . . . . . . . .
5.8.2.1 Derivation of the SG2 Particle Flux . . . . . . . . . . . .
5.8.2.2 Validation of the Equations: Particle Flux Reconstruction
5.8.2.3 Four Particle System . . . . . . . . . . . . . . . . . . . .
5.8.2.4 Flux Limiters . . . . . . . . . . . . . . . . . . . . . . . .
5.8.2.5 Higher-Order Simulation of a 1D Diode . . . . . . . . . .
Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

101
101
101
103
104
107
107
108
112
114
115
117
117
119
121
123
124
126

.
.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.
.

128
128
130
131
135
139
140
140
143
144
147
151
153

CHARACTERIZATION AND COMPACT MODEL EXTRACTION OF
GAN HEMTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
AFRL AlGaN/GaN HEMT . . . . . . . . . . . . . . . . . . . . . . . . . . . .
TLM Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
DC and Small-Signal Characterization . . . . . . . . . . . . . . . . . . . . . .
Large-Signal Characterization . . . . . . . . . . . . . . . . . . . . . . . . . .
XP Extraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
PA Design with the XP Model in ADS . . . . . . . . . . . . . . . . . . . . . .
7.7.1 Practical Considerations . . . . . . . . . . . . . . . . . . . . . . . . .
7.7.2 Class AB Amplifier . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Discussion and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.

156
156
156
159
161
166
170
176
177
178
183

CHAPTER 6 RF SIMULATIONS OF GAN HEMT TECHNOLOGY
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2 AlGaN/GaN HEMT Topology . . . . . . . . . . . . . . . . .
6.3 Thermal Equilibrium and Static Simulations . . . . . . . . . .
6.4 Small-Signal RF Simulations . . . . . . . . . . . . . . . . . .
6.5 Large-Signal RF Simulations . . . . . . . . . . . . . . . . . .
6.5.1 Quasi-Static AC Load Impedance for LP Simulations .
6.5.2 Computing Large-Signal FOMs . . . . . . . . . . . .
6.5.3 LP Simulations . . . . . . . . . . . . . . . . . . . . .
6.5.4 Class A Amplifier . . . . . . . . . . . . . . . . . . . .
6.5.5 Class AB Amplifier . . . . . . . . . . . . . . . . . . .
6.5.6 Quasi-Static vs. Full-Wave . . . . . . . . . . . . . . .
6.6 Discussion and Conclusions . . . . . . . . . . . . . . . . . . .

.
.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.
.

CHAPTER 7
7.1
7.2
7.3
7.4
7.5
7.6
7.7

7.8

ix

CHAPTER 8 CONCLUSION AND FUTURE RESEARCH . . . . . . . . . . . . . . . . 186
8.1 Future Research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
APPENDIX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

x

LIST OF TABLES

Table 5.1: A list of semiconductor device simulation BCs. . . . . . . . . . . . . . . . . . . 95
Table 5.2: Convergence of the static FKT solution variables in the GaN MESFET example with a gate-source bias of −0.5 V and drain-source bias of 4 V . . . . . . . . . 110
Table 5.3: Convergence of the I-V family of the GaN HEMT example calculated with
the static FKT solver. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
Table 5.4: Solution variable errors and approximate order of convergence of the SG1 and
SG2 FKT device simulations of the 1D diode. . . . . . . . . . . . . . . . . . . 126
Table 7.1: The incident and scattered harmonics of the AFRL GaN HEMT biased as a
class AB amplifier with a power available from the source of 11.6 dBm at
10 GHz. The large-signal tone at the fundamental frequency of the input port
is the only non-zero incident wave. . . . . . . . . . . . . . . . . . . . . . . . . . 172
Table 7.2: The incident and scattered harmonic phasors of the AFRL GaN HEMT biased
as a class AB amplifier with a power available from the source of 11.6 dBm at
10 GHz. The large-signal simulation is perturbed with a small-signal tone at
the second port and at the fundamental frequency. . . . . . . . . . . . . . . . . . 174
Table 7.3: The incident and scattered harmonic phasors of the AFRL GaN HEMT biased
as a class AB amplifier with a power available from the source of 11.6 dBm at
10 GHz. The large-signal simulation is perturbed with a small-signal tone at
the second port and at the fundamental frequency. This perturbation signal is
out of phase with the first perturbation signal. . . . . . . . . . . . . . . . . . . . 174
Table 7.4: The incident and scattered harmonic phasors of the AFRL GaN HEMT biased
as a class AB amplifier with a power available from the source of 11.6 dBm at
10 GHz. The large-signal simulation is perturbed with a small-signal tone at
the first port and at the fifth harmonic. . . . . . . . . . . . . . . . . . . . . . . . 175
Table 7.5: The incident and scattered harmonic phasors of the AFRL GaN HEMT biased
as a class AB amplifier with a power available from the source of 11.6 dBm at
10 GHz. The large-signal simulation is perturbed with a small-signal tone at
the first port and at the fifth harmonic. This perturbation is out of phase with
the first perturbation at the input port and the fifth harmonic. . . . . . . . . . . . 175
Table 7.6: The 20 largest XP coefficients extracted from the AFRL GaN HEMT biased
as a class AB amplifier with a power available from the source of 11.6 dBm at
10 GHz. The coefficients are sorted by their magnitude. . . . . . . . . . . . . . . 176
xi

LIST OF FIGURES

Figure 1.1: A system level diagram of an RFIC transmitter and receiver. The front-end
components are further decomposed into the specific types of amplifiers required for the different modules. . . . . . . . . . . . . . . . . . . . . . . . . .

1

Figure 2.1: (left) A cross-section of a cylindrical volume. (right) A polygonal approximation of the cylindrical volume with eight finite surfaces. . . . . . . . . . . .

7

Figure 2.2: (left) The Delaunay triangulation of a random set of ten points. The BowyerWatson algorithm was used to create this triangulation. The Delaunay triangles are illustrated in black and the Voronoi cells in red. Only the Voronoi
cells corresponding to interior Delaunay points are shown. Two circumcircles of the Delaunay triangles are also included. (right) One of the divergence
stencils corresponding to the shaded Voronoi cell is outlined with arrows. . . . .

8

Figure 2.3: The Delaunay triangulation (black lines) and the corresponding Voronoi diagram (red lines) of the basic outline of a MESFET. This mesh was generated
with the open source program Gmsh which uses the TetGen Delaunay triangulator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

Figure 2.4: A zoom of the Gmsh generated Delaunay mesh of the simple MESFET geometry. The gray shaded area is the semiconductor and the non-shaded region is the insulating substrate. (a) A coarse mesh which does not preserve
material interfaces and geometry boundaries. (b) A refined mesh which overmeshes the charge transport direction. . . . . . . . . . . . . . . . . . . . . . . 10
Figure 2.5: The (a) initial DV mesh, which is an input to the CVT algorithm and (b) the
resulting DV mesh from the CVT algorithm. . . . . . . . . . . . . . . . . . . . 11
Figure 2.6: The Delaunay triangulation (black lines) and the corresponding polygons of
the Voronoi diagram (red lines) of the basic outline of a MESFET. This mesh
was generated with an in-house code based upon the work of Conti. . . . . . . . 12
Figure 2.7: Two examples of mesh splitting at material interfaces. In both cases, the
primary edges separating the red and blue shaded Voronoi polygons represent
the material interface. (a) Material interface Voronoi splitting on a mesh
generated by the algorithm of Conti. (b) The same material interface Voronoi
splitting on a mesh generated by Gmsh. . . . . . . . . . . . . . . . . . . . . . 13

xii

Figure 2.8: An example of Voronoi diagram splitting in 3D. The primary edges are drawn
in black and the Voronoi edges are drawn in red. The interface of the Delaunay mesh is shaded in gray. The dashed red lines correspond to the Voronoi
cells in the plane of the interface. Two Voronoi cells in the interface are
shaded in red. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
Figure 2.9: A set of nonlinear equations and its solution. The first and second residuals are plotted with red and black surfaces, respectively and the solution is
illustrated with a blue circle. . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
Figure 2.10: Residual vector norm calculated at each iteration of the FPM and Newton’s
method. Newton’s method exhibits a quadratic rate of convergence. The
FPM demonstrates a linear rate of convergence. . . . . . . . . . . . . . . . . . 18
Figure 2.11: The Jacobian matrices of Eqns. (2.13) and (2.14) on a 10 and 11 node mesh,
respectively. The last solution variable in (b) is governed by a backward
difference approximation of Eqn. (2.11). . . . . . . . . . . . . . . . . . . . . . 20
Figure 2.12: Discrete solutions of Eqn. (2.14) on an N = 11 node mesh. The solutions are
plotted at each Newton iteration. The analytic solution is drawn as a dashed
black line. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
Figure 2.13: The error of the numerical solution of Eqn. (2.11) using a backward difference approximation (red line) and central difference approximation (blue
line) of the spatial derivative. The backward difference approximation exhibits first-order mesh convergence. The central difference exhibits secondorder mesh convergence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
Figure 2.14: The direction field of Eqns. (2.17) and (2.18) and four trajectories with different initial conditions. The initial condition of each trajectory is illustrated
with a solid circle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
Figure 2.15: Calculation of Eqn. (2.19) for each perturbation presented in Figure 2.14.
The five colors correspond to the five trajectories with different initial conditions. The black, green, and yellow lines indicate that the system of equations
is not stable. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
Figure 2.16: A linear two-port model of a DUT with input and output matching networks.
This model is for a single fundamental frequency. . . . . . . . . . . . . . . . . 26
Figure 2.17: The ADS layout of S-Parameter simulation of a GaAs FET circuit model. . . . 29

xiii

Figure 2.18: S-Parameters, reflection coefficients, and power gains of the GaAs FET circuit model simulated in ADS. The reflection coefficients on the Smith chart
are the dotted lines. The ADS variables GammaIn and GammaOut and the
power gain variables G_dB, GA_dB, and GT_dB correspond to Eqns. (2.34),
(2.35), (2.40), (2.41), and (2.42), respectively. . . . . . . . . . . . . . . . . . . 29
Figure 2.19: A nonlinear two-port model of a DUT with matching networks. . . . . . . . . . 30
Figure 2.20: The layout of the HB simulation of a GaAs FET circuit model in ADS. . . . . . 31
Figure 2.21: The output power “Pdel_dBm”, power gain “Gain”, transducer gain, and
PAE of the GaAs FET circuit model calculator with HB in ADS. The 1 dB
compression point is approximately 10 dBm input power. . . . . . . . . . . . . 32
Figure 2.22: The layout of the LP simulation of a GaAs FET circuit model in ADS. HB is
used to simulate the circuit and the load tuner (dashed red box) changes the
reflection coefficient presented to the output of the transistor. . . . . . . . . . . 32
Figure 2.23: LP simulation of the GaAs FET circuit model in ADS at a center frequency
of (left) 0.5 GHz and (right) 1 GHz. The red lines are the PAE contours and
the blue lines are the output power contours. The PAE and output power
FOMs are maximized with different load impedances. . . . . . . . . . . . . . . 33
Figure 3.1: A two-dimensional example of the staggered DVSI mesh with the (left) Delaunay triangles drawn in blue and (right) the Voronoi polygons drawn in
red. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
Figure 3.2: (left) Dual edges (shown as red arrows) associated with a primary edge (blue
arrow). The dual polygon and dual edges are comprised of the circumcenters
of each tetrahedron attached to the primary edge. (right) Primary polygon
and edges associated with one of the dual edges. . . . . . . . . . . . . . . . . . 49
Figure 3.3: An illustration of the decomposition of a dual polygon which spans multiple
dielectric regions. The dual polygon is split into multiple sub-polygons. Each
is completely contained within a single primary tetrahedron and assigned the
relative dielectric constant of that tetrahedron. The 3-D mesh was rotated
such that the primary edge points into the page. . . . . . . . . . . . . . . . . . 52
Figure 3.4: Illustrations of a primary edge on two different boundaries. . . . . . . . . . . . 53
Figure 3.5: Results for two PEC cavities with ideal dielectrics.

. . . . . . . . . . . . . . . 55

Figure 3.6: A visualization of the DVSI representation of Poisson’s equation in three
dimensions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

xiv

Figure 3.7: (left) A mesh of the spherical region with Dirichlet surfaces at r = 0.25 m
and r = 2 m. (right) The numerical solution Φ(r) of Poisson’s equation with
a simple Gaussian charge density and a Dirichlet potential of 0.25 V applied
at both surfaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
Figure 3.8: The two-port model of an electronic device. Input voltages are labeled with
a subscript 1 and output voltages are labeled with a subscript 2. The input
and output ports are ideal 50 Ω transmission lines. . . . . . . . . . . . . . . . . 63
Figure 3.9: (left) The transient input current (top) and output voltage (bottom) calculated
in the lossless transmission line. (right) The spectral reflection and transmission coefficients. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
Figure 4.1: (a) A mesh of the GaN IBZ and corresponding electron energy calculated
with EPM and (b) the resulting surfaces of constant energy. . . . . . . . . . . . 75
Figure 4.2: The (left) panel diagram of wurtzite GaN and (right) DOS calculated in the
first Γ-valley. The EPM is used to calculate the electron energy. DOS data
was obtained from Matt Grupen AFRL. . . . . . . . . . . . . . . . . . . . . . 75
Figure 4.3: The DOS isosurface integrations calculated in the first four wurtzite GaN
valleys. The DOS data was obtained from Matt Grupen AFRL. . . . . . . . . . 76
Figure 4.4: The Âľ(E) isosurface integrations calculated with different lattice temperature. The Âľ(E) data was obtained from Matt Grupen AFRL. . . . . . . . . . . 77
Figure 4.5: Approximation of the first Γ-valley DOS data with piece-wise energy power
laws. The solid black line is the isosurface integration data and the dashed
red line is the energy power law fit. Both data sets were obtained from Matt
Grupen AFRL. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
Figure 4.6: Approximation of the first Γ-valley µ(E) data with piece-wise energy power
laws. The solid black line is the isosurface integration data and the dashed
red lines are the energy power law fits. Both data sets were obtained from
Matt Grupen AFRL. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
Figure 4.7: The particle and kinetic energy flux from node 0 with Fermi distribution 0 to
node 1 with Fermi distribution 1 connected by a DV primary edge. . . . . . . . 87
Figure 5.1: (a) An example of a device mesh generated by the Sentaurus meshing tool.
(b) An illustration of the electron gases assigned to unique Voronoi polygons
in the DV mesh. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

xv

Figure 5.2: The (left) panel diagram of wurtzite GaN and (right) corresponding DOS
isosurface integral of the first Γ-valley. The energy-space of the first Γ-valley
is split into two distinct Fermi distributions doubling the number of electron
gases in the semiconductor mesh. . . . . . . . . . . . . . . . . . . . . . . . . . 92
Figure 5.3: The band diagram across a two material heterojunction. The electron affinity
of material A is greater than material B’s electron affinity. The relative Fermi
level solution variables are illustrated for each charge gas. . . . . . . . . . . . . 99
Figure 5.4: An example of the particle divergences associated to two distributions at a
heterojunction. Particle fluxes are calculated between two electron gases in
the same semiconductor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
Figure 5.5: A general work-flow diagram of a semiconductor device simulation. . . . . . . 102
Figure 5.6: The 3D GaN MESFET example used to demonstrate the device simulation
work-flow. The yellow regions at the top of the mesh are the Ohmic source
and drain and the Schottky gate contacts. The bottom surface of the mesh is
the Ohmic ground contact (not shown). The GaN channel is the dark green
region and the insulating substrate is the gray region. The volume labeled
“launch” represents the insulating region connected to the voltage port (the
red surface). This is how EM energy is injected into the domain (see Section 5.4 for more information on the port BCs). . . . . . . . . . . . . . . . . . 103
Figure 5.7: The thermal equilibrium solution variable profile of the 3D GaN MESFET
example. (a) The semiconductor band diagram illustrating the relative Fermi
level solution variable in the z-direction of the MESFET. (b) The spatial
profile of the electric potential solution variable. . . . . . . . . . . . . . . . . . 105
Figure 5.8: The I-V family calculated with the static solve. The gate-source biases start
at 0.5 V (top curve) and decrease by steps of 0.25 V . . . . . . . . . . . . . . . . 106
Figure 5.9: The spatial profile of the electric potential solution variable at a specific quiescent bias. The gate-source bias is −0.5 V and the drain-source bias is 4 V . . . 106
Figure 5.10: Simulated S-Parameters of the GaN MESFET. The operating point of the
device is a gate-source bias of −0.5 V and a drain-source bias of 4 V . The
top portion of the figure is a polar plot and the bottom half is a Smith chart
plot. The dashed lines represent the S-Parameters calculated with the quasistatic solver and the solid lines are the S-Parameters calculated with the fullwave solver. The full-wave simulation produces significantly different SParameters than the quasi-static solver. . . . . . . . . . . . . . . . . . . . . . . 109
Figure 5.11: The I-V families of the GaN HEMT calculated with the static solver on a
series of four meshes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
xvi

Figure 5.12: Deviation of the numerical enthalpy, Eqn. (5.60), for the GaN MESFET (red
line) and HEMT (black line) examples. Both equilibria are perturbed by displacing two randomly chosen electron gases from their thermal equilibrium
states. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
Figure 5.13: Five independent enthalpy analyses of the GaN MESFET example. Each
perturbation moves three randomly chosen electron gases from their thermal
equilibrium states. No perturbation was found to be unstable. The deviation from the thermal equilibrium enthalpy decayed exponentially near the
convergence tolerance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
Figure 5.14: An example of the Jacobian matrix calculated in the FKT device simulator.
The Jacobian matrix corresponds to a full-wave FKT solve. Dotted lines
mark the boundaries of the residual rows and solution variable columns in
the Jacobian matrix. The rows of the Jacobian correspond to Poisson’s equation, Ampere’s equation, Faraday’s equation, electron continuity, and energy
conservation residuals, respectively. . . . . . . . . . . . . . . . . . . . . . . . . 115
Figure 5.15: An example of the Jacobian matrix after RCM re-ordering. . . . . . . . . . . . 116
Figure 5.16: Flux reconstruction on a 10×10×10 structured cube mesh. An analytic profile is prescribed for the electric potential, relative electron Fermi level, and
electron temperature solution variables. The FKT particle fluxes are computed on all primary edges of the DV mesh and compared to analytic fluxes
to quantify a reconstruction error. . . . . . . . . . . . . . . . . . . . . . . . . . 120
Figure 5.17: Order analysis of the FKT particle flux reconstruction. The errors corresponding to the SG1 particle fluxes are illustrated in red and the SG2 errors
are illustrated in blue. As expected, the SG1 particle flux reconstructions exhibit first-order convergence and the SG2 particle flux reconstructions exhibit
second-order convergence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
Figure 5.18: The simulated static I-V of the four particle system with two interior DOF.
SG1 device simulation currents are plotted in black and SG2 currents are
plotted in red. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
Figure 5.19: The phase portrait of the four particle, two DOF system. The direction field
is plotted with black vectors and four trajectories are plotted with blue lines.
Initial conditions of the trajectories are plotted with blue circles. There are
three equilibria of the system of nonlinear equations, two of which are unstable. 122
Figure 5.20: The simulated thermal equilibrium band diagram of the 1D diode. The SG1
results are plotted with solid lines and the results computed with the SG2
solver are plotted with squares. . . . . . . . . . . . . . . . . . . . . . . . . . . 125

xvii

Figure 5.21: The simulated static I-V of the 1D diode. SG1 device simulation currents are
plotted in black and SG2 currents are plotted in red. . . . . . . . . . . . . . . . 125
Figure 6.1: Electron drift velocity vs. electric field of bulk GaN calculated with MC. . . . . 128
Figure 6.2: The 3D mesh used for FKT simulations of an AlGaN/GaN HEMT. . . . . . . . 131
Figure 6.3: The thermal equilibrium electric potential profile of the AlGaN/GaN HEMT
computed with the FKT device simulator. . . . . . . . . . . . . . . . . . . . . 132
Figure 6.4: The thermal equilibrium band diagram (calculated across the stack under the
Schottky gate) of the AlGaN/GaN HEMT computed with the FKT device
simulator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
Figure 6.5: The electric potential profile of the AlGaN/GaN HEMT in quiescent bias.
The gate-source bias is -1 V and the drain-source bias is 10 V. . . . . . . . . . . 133
Figure 6.6: The static I-V family of the AlGaN/GaN HEMT. The solid black lines represent FKT simulations with different gate biases and the red dots are the
measured equivalents. The gate biases range from 1 V to -3 V with steps of
1 V. This result was first presented by Matt Grupen AFRL. . . . . . . . . . . . 133
Figure 6.7: A comparison of the static I-V families of the AlGaN/GaN HEMT computed with the different FKT device solvers. The DD, FKT, FKT with lattice
heating, and the FKT solver with lattice heating and trapping effects are illustrated with dashed blue, dashed red, dashed green, and solid black lines,
respectively. The measurements are illustrated with red dots. . . . . . . . . . . 134
Figure 6.8: The transconductance of the AlGaN/GaN HEMT. The solid black line represents the drain current and the red line is the transconductance of the device.
The drain-source bias is 10 V. . . . . . . . . . . . . . . . . . . . . . . . . . . 135
Figure 6.9: The (a) AC input and output voltages and (b) AC input and output currents of
the AlGaN/GaN HEMT calculated with the quasi-static solver. The voltage
BC is assigned to the input port and the AC impedance port BC is assigned
to the output. For the quasi-static solver, the input port is the Schottky gate
metal contact and the output port is the Ohmic drain metal contact. . . . . . . . 136
Figure 6.10: The (a) AC input and output voltages and (b) AC input and output currents
of the AlGaN/GaN HEMT calculated with the full-wave solver. The voltage port BC is assigned to the input port and the AC impedance port BC is
assigned to the output. For the full-wave solver, the input port connects the
Schottky gate and the Ohmic ground and the output port connects the Ohmic
drain and the Ohmic ground. . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

xviii

Figure 6.11: Two S-Parameters of the AlGaN/GaN HEMT computed with the FKT device
simulator. The solid lines represent the full-wave results and the dashed lines
represent the quasi-static results. The quiescent bias of the device is a drainsource bias of 6 V and no gate-source bias. . . . . . . . . . . . . . . . . . . . 137
Figure 6.12: The simulated (black lines) and the measured (red squares) small-signal current gain of the AlGaN/GaN HEMT. . . . . . . . . . . . . . . . . . . . . . . . 138
Figure 6.13: The simulated available power gain (black) and transducer power gain (red)
of the AlGaN/GaN HEMT. Dashed lines represent the quasi-static power
gains and the solid lines represent the full-wave power gains. . . . . . . . . . . 139
Figure 6.14: The (a) input and (b) output TD voltages of the AlGaN/GaN HEMT calculated with the quasi-static solver. The operating point of the HEMT is -2 V
gate-source bias and 10 V drain-source bias. The large-signal tone (gatesource voltage) is 1 V in amplitude at 35 GHz. . . . . . . . . . . . . . . . . . . 141
Figure 6.15: The (a) input and (b) output TD currents of the AlGaN/GaN HEMT calculated with the quasi-static solver. The operating point of the HEMT is -2 V
gate-source bias and 10 V drain-source bias. The large-signal tone (gatesource voltage) is 1 V in amplitude at 35 GHz. . . . . . . . . . . . . . . . . . . 141
Figure 6.16: The magnitudes of the (a) input and (b) output voltage phasors of the AlGaN/GaN HEMT driven by a large-signal tone. These voltages are plotted in
dBm. The five harmonics are computed by Fourier transforms of the transient
voltages. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
Figure 6.17: The magnitudes of the (a) input and (b) output current phasors plotted in
dBm. These five harmonics are computed by Fourier transforms of the transient currents. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
Figure 6.18: An example of reflection coefficient samples of the Smith chart that can be
used for LP simulations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
Figure 6.19: LP contours calculated from FKT device simulations of the GaN HEMT biased as a class A amplifier (gate-source bias of 1 V and drain-source bias
of 10 V) with -2 dBm power available from the source at a fundamental frequency of (a) 15 GHz and (b) 25 GHz. The optimal loads for maximum transducer gain are 104 + j130 Ω at 15 GHz and 54 + j105 Ω at 25 GHz. The peak
gain is 9.3 dB at 15 GHz and 9.1 dB at 25 GHz. The step size of the contours
is 0.2 dB. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

xix

Figure 6.20: LP contours calculated from FKT device simulations of the GaN HEMT biased as a class A amplifier (gate-source bias of 1 V and drain-source bias
of 10 V) with -2 dBm power available from the source at a fundamental frequency of (a) 35 GHz and (b) 45 GHz. The optimal loads for maximum transducer gain are 25 + j82 Ω at 35 GHz and 16 + j67 Ω at 45 GHz. The peak
gain is 9.0 dB at 35 GHz and 8.8 dB at 45 GHz. The step size of the contours
is 0.2 dB. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
Figure 6.21: Simulated input power sweeps of the GaN HEMT biased as a class A amplifier (gate-source bias of 1 V and drain-source bias of 10 V) at a fundamental
frequency of (a) 15 GHz and (b) 25 GHz. The optimal loads determined from
the LP data are 104 + j130 Ω at 15 GHz and 54 + j105 Ω at 25 GHz. The
peak PAE resides at an input power of approximately 10 dBm for both frequencies. The device operating at 15 GHz achieves the highest overall peak
PAE of approximately 12%. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
Figure 6.22: Simulated input power sweeps of the GaN HEMT biased as a class A amplifier (gate-source bias of 1 V and drain-source bias of 10 V) at a fundamental
frequency of (a) 35 GHz and (b) 45 GHz. The optimal loads determined from
the LP data are 25 + j82 Ω at 35 GHz and 16 + j67 Ω at 45 GHz. The PAE
diminishes with increasing frequency. At 45 GHz, the peak PAE is between
2–3%. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
Figure 6.23: Simulated (a) output voltage transients and (b) output voltage spectral content
of the GaN HEMT biased as a class A amplifier at 15 GHz. The input powers
of the blue, green, and red lines are −9.8 dBm, −1.8 dBm, and 3.8 dBm, respectively. The transistor exhibits strong inter-modulation at each simulated
input power. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
Figure 6.24: LP data calculated from FKT device simulations of the GaN HEMT biased as
a class AB amplifier (gate-source bias of -2 V and drain-source bias of 10 V)
with -2 dBm power available from the source at a fundamental frequency of
(a) 15 GHz and (b) 25 GHz. The optimal loads for maximum transducer gain
are 77 + j155 Ω at 15 GHz and 30 + j124 Ω at 25 GHz. The peak gain is
11.1 dB at 15 GHz and 10.9 dB at 25 GHz. The step size of the transducer
gain contours is 0.2 dB. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
Figure 6.25: LP data calculated from FKT device simulations of the GaN HEMT biased as
a class AB amplifier (gate-source bias of -2 V and drain-source bias of 10 V)
with -2 dBm power available from the source at a fundamental frequency
of (a) 35 GHz and (b) 45 GHz. The optimal loads for maximum transducer
gain are 24 + j88 Ω at 35 GHz and 16 + j69 Ω at 45 GHz. The peak gain is
10.7 dB at 35 GHz and 10.5 dB at 45 GHz. The step size of the transducer
gain contours is 0.2 dB. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

xx

Figure 6.26: Simulated input power sweeps of the GaN HEMT biased as a class AB amplifier (gate-source bias of -2 V and drain-source bias of 10 V) at a fundamental
frequency of (a) 15 GHz and (b) 25 GHz. The optimal loads determined from
the LP data are 77 + j155 Ω at 15 GHz and 30 + j124 Ω at 25 GHz. The peak
PAE resides at an input power of approximately 0 dBm at 15 GHz and 5 dBm
at 25 GHz. The device operating at 15 GHz achieves the highest overall peak
PAE of approximately 15%. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
Figure 6.27: Simulated input power sweeps of the GaN HEMT biased as a class AB amplifier (gate-source bias of -2 V and drain-source bias of 10 V) at a fundamental
frequency of (a) 35 GHz and (b) 45 GHz. The optimal loads determined from
the LP data are 24 + j88 Ω at 35 GHz and 16 + j69 Ω at 45 GHz. The peak
PAE resides at an input power of approximately 8 dBm at both 35 GHz and
45 GHz. The PAE of the class AB amplifier does not diminish like the class
A amplifier at high frequency. At 45 GHz, the peak PAE of the device is
approximately 8 – 9%. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
Figure 6.28: Simulated (a) output voltage transients and (b) output voltage spectral content
of the GaN HEMT biased as a class AB amplifier at 15 GHz. The input powers of the blue, green, and red lines are −16.7 dBm, −4.9 dBm, and 2.8 dBm,
respectively. Strong inter-modulation is observed at each input power. . . . . . 150
Figure 6.29: Input power sweeps of the GaN HEMT biased as a class AB amplifier calculated with the quasi-static (dashed lines) and the full-wave (solid lines) FKT
device simulators. The gate-source bias is -2 V, the drain-source bias is 10 V,
the fundamental frequency is 10 GHz, and the load impedance is 50 Ω. . . . . . 151
Figure 6.30: Quasi-static (dashed lines) and full-wave (solid lines) frequency sweeps of
the GaN HEMT biased as a class AB amplifier. The gate-source bias is -2 V,
the drain-source bias is 10 V, and the load impedance is 50 Ω. Black lines
correspond to a power available from the source of -13 dBm and red lines
correspond to a power available from the source of -2 dBm. . . . . . . . . . . . 152
Figure 6.31: Summary of the optimal loads determined through FKT device simulations
of the AlGaN/GaN HEMT biased as class A and class AB amplifiers. The
fundamental frequency of both amplifiers was swept from 15–45 GHz with
5 GHz increments. The optimal load shifts in a counter-clockwise motion
across the Smith chart with increasing frequency. . . . . . . . . . . . . . . . . 154
Figure 7.1: A cross-section of AFRL’s standard GaN HEMT device. . . . . . . . . . . . . . 156
Figure 7.2: The DV mesh used to simulate the AFRL GaN HEMT in the FKT device
simulator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

xxi

Figure 7.3: A zoom of the GaN HEMT stack under the Schottky gate. The stack consists
of (from substrate to metal) a 1.9 Âľm GaN buffer, a 10 nm GaN channel, 1 nm
of AlN, 15.4 nm of AlGaN, and a 3 nm GaN cap. The mole fraction of the
AlGaN layer is x = 0.278. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
Figure 7.4: The physical parameters of the AFRL GaN HEMT. Most notable are the
polarization charge densities P at the heterojunctions. . . . . . . . . . . . . . . 158
Figure 7.5: An example of determining the approximate access resistance with TLM.
A series of resistors are measured/simulated and the total resistance of each
resistor is calculated. The slope of the data determines the sheet resistance
(access resistance) divided by the width of the resistors and the y-intercept
determines twice the contact resistance. . . . . . . . . . . . . . . . . . . . . . . 159
Figure 7.6: Simulated I-V curves of the TLM resistors computed with the FKT device
solver. The lengths of the resistors vary from 5 Âľm to 30 Âľm with 5 Âľm
increments. The I-V curve with the largest drain current at the drain-source
bias of 1 V corresponds to the L = 30 Âľm resistor. . . . . . . . . . . . . . . . . 160
Figure 7.7: The simulated (black dots) and measured (solid line) total resistances of the
GaN TLM resistors. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
Figure 7.8: Electron density across the GaN HEMT channel. The channel begins at
0.5 Âľm and ends at 3.5 Âľm. Five different polarization charge densities are
prescribed to the GaN/AlN heterojunction. The five regions are outlined with
vertical dashed lines. The contacts of the device are illustrated at the top of
the figure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
Figure 7.9: The thermal equilibrium band diagram of the AFRL GaN HEMT computed
with the FKT device simulator. . . . . . . . . . . . . . . . . . . . . . . . . . . 163
Figure 7.10: Simulated DC I-V curves (black lines) compared with AFRL measurements
(red dots). The gate-source bias ranges from +1 V to -3 V with 1 V steps. . . . . 163
Figure 7.11: Simulated DC drain current (solid black line) and transconductance (solid
red line) of the AFRL GaN HEMT. The measurements obtained from AFRL
are plotted with dashed lines. . . . . . . . . . . . . . . . . . . . . . . . . . . . 164
Figure 7.12: S-Parameters of the AFRL GaN HEMT. The solid lines represent the results computed with the quasi-static FKT device solver. The measured SParameters obtained from AFRL are drawn with dashed lines. . . . . . . . . . . 165
Figure 7.13: The small-signal current gain of the AFRL GaN HEMT. The FKT device
simulation data (black line) compares reasonably well with the measurements (red boxes). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
xxii

Figure 7.14: Input power sweep data of the AFRL GaN HEMT biased as a class AB amplifier at 10 GHz. The output power is plotted in black, the transducer gain
is plotted in red, and the PAE is plotted in blue. FKT device simulation data
are plotted with solid lines and measurements are plotted with squares. . . . . . 166
Figure 7.15: Simulated input power sweeps of the AFRL GaN HEMT at 10 GHz. The
gate-source bias is changed from -2.8 V to -2 V (dashed lines) and -1 V (dotdashed lines). The drain-source bias is held constant at 21 V. The gain and
PAE of the device is reduced at -1 V gate-source voltage, as expected. . . . . . . 167
Figure 7.16: Simulated input power sweeps of the AFRL GaN HEMT at 10 GHz. The
drain-source bias is changed from 21 V to 10 V (dashed lines) and 30 V (dotdashed lines). The gate-source bias is held constant at -2.8 V. The smallsignal gain of the device increases at 10 V drain-source bias. However, the
device operating at this bias compresses at a lower input power. . . . . . . . . 168
Figure 7.17: Simulated input power sweeps of the AFRL GaN HEMT at 10 GHz. The
source-drain spacing is changed from 3 Âľm to 4 Âľm (dashed lines) and 2 Âľm
(dot-dashed lines). These variations of the source-drain spacing have small
impacts on the device power metrics. . . . . . . . . . . . . . . . . . . . . . . 169
Figure 7.18: LP data generated from FKT device simulations of the AFRL GaN HEMT
biased as a class AB amplifier at 10 GHz. The gate-source bias is -2.8 V, the
drain-source bias is 10 V, and the power available from the source is 1.5 dBm.
Output power contours are illustrated with blue lines and PAE contours are
illustrated with red lines. The optimal load impedance for peak output power
is 57 + j81 Ω at 10 GHz and the load impedance 53 + j108 Ω at 10 GHz
corresponds to peak PAE. The peak output power is 19 dBm and the peak
PAE is 18.2%. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
Figure 7.19: The (a) input and (b) output voltages of the AFRL GaN HEMT calculated
with the quasi-static solver. The operating condition of the HEMT is a quiescent bias of a -2.8 V gate-source bias and 10 V drain-source bias. The
large-signal incident wave at the fundamental frequency of the input port is
1 V in magnitude at 10 GHz. . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
Figure 7.20: The (a) input and (b) output currents of the AFRL GaN HEMT calculated
with the quasi-static solver. The operating condition of the HEMT is a quiescent bias of a -2.8 V gate-source bias and 10 V drain-source bias. The
large-signal incident wave at the fundamental frequency of the input port is
1 V in magnitude at 10 GHz. . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

xxiii

Figure 7.21: The (a) input and (b) output voltages of the AFRL GaN HEMT calculated
with the quasi-static solver. The operating condition of the HEMT is a quiescent bias of a -2.8 V gate-source bias and 10 V drain-source bias. The solid
black lines represent the large-signal simulation and the dashed red lines represent the first perturbation simulation of the first incident harmonic at the
output port. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
Figure 7.22: The (a) input and (b) output currents of the AFRL GaN HEMT calculated
with the quasi-static solver. The operating condition of the HEMT is a quiescent bias of a -2.8 V gate-source bias and 10 V drain-source bias. The solid
black lines represent the large-signal simulation and the dashed red lines represent the first perturbation simulation of the first incident harmonic at the
output port. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
Figure 7.23: Layout of the XP model simulation in ADS. The components “P_1Tone”
and “Term” represent the input and output ports of the circuit. Component
“X2P” is the XP model which specifies and input file containing the model
information and coefficients. Current and power probes are placed in the
circuit to calculate the large-signal FOMs. . . . . . . . . . . . . . . . . . . . . 177
Figure 7.24: Validation of the input power dependent ADS XP model. This model is
extracted from FKT simulations of the AFRL GaN HEMT and imported into
ADS. Device parameters of the XP model include a gate-source bias of 2.8 V, a drain-source bias of 10 V, a fundamental frequency of 10 GHz, and a
load impedance of 50 Ω. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
Figure 7.25: Simulated LP data generated from ADS simulations of the AFRL HEMT XP
model with a power available from the source of (a) 3 dBm and (b) 6 dBm.
The power dependent XP model was extracted with a gate-source bias of 2.8 V, a drain-source bias of 10 V, a fundamental frequency of 10 GHz, and
a load impedance of 50 Ω. The optimal loads are 39 + j76 Ω at 3 dBm and
42 + j71 Ω at 6 dBm, both at the fundamental frequency. . . . . . . . . . . . . 180
Figure 7.26: Simulated LP data generated from ADS simulations of the AFRL HEMT XP
model with a power available from the source of (a) 9 dBm and (b) 12 dBm.
The power dependent XP model was extracted with a gate-source bias of 2.8 V, a drain-source bias of 10 V, a fundamental frequency of 10 GHz, and
a load impedance of 50 Ω. The optimal loads are 45 + j56 Ω at 9 dBm and
42 + j46 Ω at 12 dBm, both at the fundamental frequency. . . . . . . . . . . . . 181
Figure 7.27: A summary of the optimal load impedances for maximum output power determined from the LP simulations of the AFRL GaN HEMT XP model in
ADS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

xxiv

Figure 7.28: The (a) input and (b) output CW voltage signals calculated with the AFRL
GaN HEMT XP model in ADS. Only the AC components of the waveforms
are illustrated. The load impedance presented to the XP model is 42 + j46 Ω
at 10 GHz. This is the optimal load for maximum output power determined
from the LP simulations with a power available from the source of 12 dBm. . . 182
Figure 7.29: The (a) input and (b) output CW current signals calculated with the AFRL
GaN HEMT XP model in ADS. Only the AC components of the waveforms
are illustrated. The load impedance presented to the XP model is 42 + j46 Ω
at 10 GHz — the optimal load for maximum output power determined from
the LP simulations with a power available from the source of 12 dBm. . . . . . 182
Figure 7.30: A comparison of the simulated LP data generated directly from the FKT device simulations (red lines) and from the ADS simulations of the XP model
(blue lines). The reported FOM is the output power of the device. The magnitude of the large-signal incident phasor is held constant at 0.75 V for the
LP simulations. Note that this definition of the incident wave is different
than Eqns. (7.6) and (7.7). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183

xxv

CHAPTER 1
INTRODUCTION

Wireless communications is an ever-growing research area in the information age. Examples of
wireless communications include commercial applications such as cellular telephones, wireless local area network (WLAN) and Wi-Fi, global positioning system (GPS), and military applications
like radar sensors, electronic warfare (EW) systems, and satellite communications. A critical component common to all wireless systems is the radio frequency (RF) amplifier. As bandwidth and
efficiency criteria evolve with more sophisticated technology, so too must the underlying transistor technologies which comprise the integrated RF amplifiers as well as the circuit topologies that
exploit these capabilities.
At a system level, RF integrated circuits (RFICs) used for wireless communications are grouped
into two modules: the transmitter (Tx) and the receiver (Rx). A basic illustration of both RFIC
modules is presented in Figure 1.1. The transmission module is responsible for up-converting

Transmitter
Data
source

Receiver

Modulator

Tx
front-end

Power
amplier

RF lter

Rx
front-end

Demodulator

RF lter

Low noise
amplier

Data
sink

Figure 1.1: A system level diagram of an RFIC transmitter and receiver. The front-end components
are further decomposed into the specific types of amplifiers required for the different modules.
(modulating) the signal to a specific frequency band, amplifying the signal with a desired amount
of gain, and filtering the signal to the allocated frequency band of the system. Once the signal is
received from the channel, the Rx module must filter out unwanted frequency content, amplify the
1

signal with the least amount of noise possible, then down-convert (demodulate) the signal to the
base-band frequency range.
Power amplifiers (PAs) are used to amplify the signal before transmission with the antenna and
are an important sub-component of the overall wireless communication process. Ever since the
first operational example in the 1940s, transistors have been an important part of practical PAs and
remain the only prospect for integrated amplification. The choice of semiconductors for the transistor technology has changed over time due to the changing demand of the PAs and the emergence
of higher performance semiconductors that can be readily integrated. The biggest breakthrough
which has allowed a wide expansion of semiconductor technology is the monolithic microwave
integrated circuit (MMIC) process. The gallium arsenide (GaAs) MMIC amplifier was reported
in 1976 and since then there has been tremendous progress in both LNAs and PAs [1]. The GaAs
MMIC PA is still an integral part of every cell phone, which lends a calibration to the impact of
the technology. Although GaAs was a major part of early RF semiconductor devices, there exist a wide range in transistor types and semiconductors. Included in the transistor types are the
GaAs and silicon carbide (SiC) metal-semiconductor field effect transistors (MESFETs), silicon
(Si) based metal-oxide-semiconductor FETs (MOSFETs), silicon germanium (SiGe) heterojunction bipolar transistors (HBTs) [2], and GaAs and gallium nitride (GaN) high electron mobility
transistors (HEMTs) [3].
GaN HEMTs are a popular PA transistor technology choice due to their high electron saturation velocity, large semiconductor band-gap, high two-dimensional electron gas (2DEG) density in
the channel, and high electron mobility [3]. A semiconductor with a large band-gap is superior to
that with a lower band-gap as the former will suffer from band-to-band impact ionization at much
higher electric fields, enabling higher operating voltages and greater output power in the same
form factor. Furthermore, with a high electron saturation velocity, GaN HEMTs can operate at
higher frequency compared to their GaAs counterparts which is critical for penetrating high power
applications at millimeter-wave frequencies and higher. Finally, in terms of device fabrication,
the ability of GaN technology to form heterojunctions allows for polarization induced high carrier

2

concentration and high electron mobility without modulation doping [4]. These two features allow for a high current density and low channel resistance, which also allows for high frequency
operation [5]. These characteristics of GaN HEMTs have allowed implementation of high power
and high frequency PAs [6, 7] and are important components for next generation radar and satellite
communication RF front-ends.
Independent of the transistor technology and process, a wide variety of PA circuit topologies
exist that have been developed over decades to exploit various physical benefits of each given
technology. The most simple analog PAs are class A, class B, class AB, and class C amplifiers.
The difference between the class A-C amplifiers is the operating point, with the class C amplifier
boasting the highest theoretical efficiency at the expense of linearity and vice versa for class A. The
next class types of PAs are the switching classes D and E. Class D amplifiers treat the transistor
as an ideal switch, whereas the class E amplifier treats the transistor as a non-ideal switch and
optimizes the load impedance for maximum efficiency [8]. More advanced types of amplifiers
are the class F and inverse class F amplifiers. Class F amplifiers operate with class B biasing
and are presented with a harmonic tuning matching network for high efficiency [9]. A transistor
operating with a class B bias and with optimized harmonic termination impedance is designated as
a class J amplifier. Advanced PA design includes broadband amplifier techniques and linearization
techniques. Techniques for increasing bandwidth include reactive/resistive matching amplifiers
and feedback amplifiers [1], as well as distributed amplifiers [10]. The techniques for amplifier
linearization include feedback techniques [11, 12], feed-forward techniques [13], and efficiency
enhancement techniques such as Chireix outphasing [14] and the Doherty amplifier [15].
Design of PAs is significantly more involved than the design of other types of amplifiers such
as the low noise amplifier (LNA) and specific applications require drastically different designs.
Basic requirements of PAs include high gain, high power added efficiency (PAE), and higher linearity [1]. Once the device technology, the type of PA, and the operating point are selected, the
design of the PA often requires measured and simulated load-pull (LP) data to create models that
will allow designed circuit performance to match the fabricated reality. LP refers to the evaluation

3

of figures of merit (FOMs) of a device under test (DUT) when the input and/or output loading
conditions are changed [16]. The FOMs, whether measured or simulated, are provided for single transistors or DUTs rather than an entire PA module. Once the response data are available
for multiple harmonics of the loading conditions, designers can then create the optimum loading
conditions to maximize the desired FOMs. Although the LP concept is relatively straightforward,
realizing the measurement system and producing the data can be incredibly time-consuming and
sophisticated. The use of mechanical tuners to systematically change the loading conditions presented to the DUT has been in use since the mid-1970s [17]. When LP data for multiple harmonics are required for proper PA design, the use of mechanical tuners is a laborious measurement
technique. Today, state-of-the-art measurement systems utilize active LP techniques for fast and
efficient multi-harmonic data generation. The first instances of active LP used the open-loop active
load, where a signal generator is used to inject incident waves to the DUT to mimic the scattered
waves generated by impedance mismatches [18]. Closed-loop technology offers an improved version of the original active LP technique, where the reflected waves are generated directly from the
incident waves through directional couplers [16]. The four-port differential time-domain (TD) LP
technique is the state-of-the-art and offers powerful measurement capabilities [19].
Measured LP is the standard high frequency characterization used throughout the industry, but
it does not come without drawbacks. The main difficulties of active LP are the high power and
linearity required by the load amplifiers [16]. Furthermore, the data generated are only valid for
a single DUT operating point and therefore bias and frequency sweeps are required for a broad
range of device performance data. An alternative approach for PA design is utilization of computer aided design (CAD) tools with nonlinear device models. PA design with CAD tools offers a
flexible and efficient approach to performance optimization and can be used to generate synthetic
LP data or entire PA module characterization. In order to generate accurate and useful results, the
nonlinear device model must be accurate over a wide range of operating points, frequency, loading
conditions, and large-signal excitations. These nonlinear models can be grouped into two main
categories: circuit based modeling and behavioral modeling.

4

Circuit based modeling of transistors has existed nearly as long as the devices have themselves.
Many small-signal two-port models exist for a wide variety of transistors [1]. These models can be
useful for circuit design, e.g., conjugate input matching. However, because PAs are driven by largesignal RF signal levels, these models do not accurately describe the nonlinear nature of the devices.
Nonlinear circuit models are therefore critical for large-signal RF design of PAs. An example of
a well-known nonlinear device model for GaN HEMT technology is the Angelov model [20, 21].
Keysight Technologies (a spin-off from Agilent Technologies), provides a circuit-based nonlinear
GaN HEMT model called the DynaFET model [22, 23], which has garnered considerable attention for nonlinear compact modeling. Both examples of nonlinear GaN HEMT models require
extensive measurements of devices to calculate parameters and train the models.
Opposite to circuit based modeling are behavioral nonlinear device models. Rather than develop equivalent circuits to represent the nonlinear transistor, behavioral modeling relies on approximations of the nonlinear analytic map between the input waves (voltages or currents) and
the scattered waves. An example of a nonlinear behavioral model applicable to GaN HEMTs is
the Polyharmonic Distortion Model (PHD) [24], which later was trademarked as the X-Parameter
(XP) model [25, 26] and was used for high-efficiency PA design [27].
Regardless of the choice of the nonlinear device model (circuit based or behavioral), the vast
majority of implementations utilize measurements to compute model parameters and/or train the
model. An alternative to measurement based model extraction is physics based or technology
computer aided design (TCAD) based model extraction. With TCAD based model extraction capabilities, the cost of sophisticated measurement systems and device fabrication can be overcome
to allow efficient PA characterization and design. It is critical that the numerical TCAD models
accurately describe the underlying physics of the semiconductor devices to produce quality and
useful results. To the author’s knowledge, there only exists one TCAD framework which can accurately produce nonlinear device models. This framework couples an accurate Monte Carlo (MC)
method with harmonic balance (HB) [28] and has been used to extract XPs [29]. Although this
TCAD framework boasts the accuracy of MC, it suffers from an immense computational burden

5

and cannot include full-wave electromagnetic (EM) effects in its present form, although it could
be added. A greater limitation is that MC physical domains are limited to sizes of a few microns
and PA output stages are often 10–100 times larger in reality.
The main contribution of this thesis is the utilization of a TCAD framework called Fermi Kinetics Transport (FKT), which captures hot-electron and full-wave effects, to simulate, characterize,
and extract XP models of state-of-the-art GaN HEMT technology. FKT, which is a deterministic Boltzmann solver, offers a promising numerical framework for accurate, robust, and efficient
electronic device simulation. The rest of this thesis is organized as follows. Background material required for this work is presented in Chapter 2. In Chapter 3, a complete discussion of the
full-wave EM discretization method called Delaunay-Voronoi Surface Integration (DVSI), which
is part of the FKT TCAD numerical framework, is presented. A thorough derivation of the governing equations of the FKT device simulator and their discretization in energy- and real-space
is presented in Chapter 4. Numerical details required for solving the nonlinear device equations
are presented in Chapter 5, including device simulation work-flow and boundary conditions. This
chapter also includes an investigation into some numerical characteristics of the FKT equations
and reports some advanced simulation techniques. Chapter 6 presents small- and large-signal RF
simulations of GaN HEMT technology from DC to mm-wavelengths. In Chapter 7, a state-ofthe-art GaN HEMT is characterized with the FKT device simulator. This chapter presents several
small- and large-signal simulations of the HEMT technology. XP extraction of the GaN HEMT is
also reported in this chapter. The compact model is imported into Advanced Design System (ADS)
to generate data for RF circuit designers. A conclusion and discussion of future research enabled
by this thesis are provided in Chapter 8.

6

CHAPTER 2
BACKGROUND

2.1

The Box Integration Method

Discretization of continuous systems of equations can be done in a multitude of ways. The
standard approach for discretizing continuous equations which govern charge transport in semiconductor devices is the box integration method [30]. The idea of the box integration method is
based upon Gauss’ divergence theorem and can be dated as far back as the 1960s [31, 32, 33].
Given an arbitrary vector field ~F, its divergence integrated over a volume is equal to the flux of the
vector field through the surface of the volume
ZZZ

∇ ·~FdV =

V

ZZ

n̂ ·~FdS.

(2.1)

S

Figure 2.1 illustrates the cross-section of a cylindrical volume. The 2D version of the surface
integral for the cylindrical region is
ZZ
S

Z2π

n̂ ·~FdS =

dφ ρ ρ̂ ·~F.

(2.2)

φ =0

Figure 2.1: (left) A cross-section of a cylindrical volume. (right) A polygonal approximation of
the cylindrical volume with eight finite surfaces.

7

For the discrete version of the cylindrical region shown in the right of Figure 2.1, the surface
integral becomes
ZZ

n̂ ·~FdS ≈ ∑

S

Z

i S
i

dSi n̂i ·~F ≈ ∑ Fi Ai ,

(2.3)

i

if the vector projections of the field ~F are approximated as constant over each surface of the polygon. This is the crux of the box integration method. Given a polyhedron which encloses a single
point in space, the divergence integrated over the polyhedron is represented by a summation of
fluxes across the polyhedron’s faces. A special type of mesh is required to produce these types of
polyhedra and is the topic of the following section.

2.2

The Delaunay-Voronoi Mesh

The basic principles of the box integration method were outlined in the previous section. To
properly discretize the divergence of a vector field with the box integration method, a special type
of mesh called a Delaunay mesh is required. A 2D Delaunay mesh is a set of polygons where
the circumcircle of any polygon contains no points of the mesh. The points of each polygon
lie on the perimeter of its respective circumcircle. The 3D analogue is a set of polyhedra with

Figure 2.2: (left) The Delaunay triangulation of a random set of ten points. The Bowyer-Watson
algorithm was used to create this triangulation. The Delaunay triangles are illustrated in black
and the Voronoi cells in red. Only the Voronoi cells corresponding to interior Delaunay points
are shown. Two circumcircles of the Delaunay triangles are also included. (right) One of the
divergence stencils corresponding to the shaded Voronoi cell is outlined with arrows.
8

circumspheres. An example of a Delaunay triangulation of a random set of points is shown in
Figure 2.2. The Bowyer-Watson algorithm [34, 35] was used to generate this triangulation. In the
left of Figure 2.2, the Delaunay triangles are drawn in black and the Voronoi cells (most commonly
referred to as the Voronoi diagram [36]) corresponding to the interior Delaunay nodes are drawn in
red. Two circumcircles which inscribe triangles are also drawn in gray. This simple triangulation
demonstrates the powerful relationship between the Delaunay triangles and the Voronoi polygons.
Each Voronoi polygon corresponds to a single Delaunay node in the mesh. The faces of this
Voronoi polygon (edges) correspond to edges which are connected to the Voronoi polygon’s node.
The normals of the Voronoi polygon’s faces point in the exact same direction as their corresponding
Delaunay edges, by construction. The importance of this relationship now becomes clear in terms
of the box integration method. A divergence of a vector field integrated over each Voronoi polygon
can be represented as a discrete summation of fluxes, defined on the Delaunay edges, across each
face of the Voronoi polygon. One divergence stencil corresponding to the shaded Voronoi polygon
is illustrated in Figure 2.2 (right).

Figure 2.3: The Delaunay triangulation (black lines) and the corresponding Voronoi diagram (red
lines) of the basic outline of a MESFET. This mesh was generated with the open source program
Gmsh which uses the TetGen Delaunay triangulator.
There exists a wide range of algorithms to generate Delaunay triangulations. Included is
the Bowyer-Watson algorithm, as well as other standard methods used in computational geometry [37, 38]. These methods are reliably used for the finite element method (FEM), among many
other computational methods. An out-of-the-box Delaunay meshing algorithm, however, does not
produce adequate meshes for simulating semiconductor devices with the box integration method.
9

To elaborate on the difficulties of generating a quality Delaunay-Voronoi (DV) mesh for discretization with the box integration method, a simple MESFET geometry is considered. The device
consists of a thin semiconductor resting upon a larger substrate. These types of features are common in electronic devices. Figure 2.3 illustrates an attempt at meshing a basic MESFET with an
open-source Delaunay meshing tool. The code is Gmsh [39], which uses the software TetGen [40]
as its Delaunay mesher. It is evident that the Voronoi polygons span across the material interfaces
as well as overall geometry boundaries. This is due to obtuse angles in triangles near the boundaries whose circumcenters fall outside of the domain boundary. A characteristic like this will not
allow proper simulation of electronic devices as the charge densities must be uniquely defined in
each material. Furthermore, numerical boundary conditions (BCs) rely on proper truncation of the
Voronoi cells at mesh boundaries.

(b)

(a)

Figure 2.4: A zoom of the Gmsh generated Delaunay mesh of the simple MESFET geometry. The
gray shaded area is the semiconductor and the non-shaded region is the insulating substrate. (a) A
coarse mesh which does not preserve material interfaces and geometry boundaries. (b) A refined
mesh which over-meshes the charge transport direction.

As an attempt to fix the Voronoi polygons, the mesh is refined in the semiconductor region.
Figure 2.4 illustrates (a) the original mesh and (b) the refined mesh in the top left corner of the
device. The refined mesh produces conforming Delaunay triangles in both regions as well as
Voronoi polygons which preserve geometry boundaries and can preserve material interfaces. This
mesh, however, is not optimal for simulating electronic devices. The transport direction in field

10

effect transistors is parallel to semiconductor/substrate interfaces or heterojunctions. Therefore,
the refined mesh in Figure 2.4b contains too many Delaunay edges which are perpendicular to the
transport direction. Generating a Delaunay mesh with standard FEM software which preserves
material interfaces and geometry boundaries will generate superfluous edges which leads to a very
large number of degrees of freedom (DOF).
An alternative to simple mesh refinement could be sophisticated algorithms which ensure that
circumcenters do not fall outside of their respective Delaunay triangles. One example is the centroidal Voronoi tessellation (CVT) algorithm [41]. The CVT algorithm attempts to move the points
of the Delaunay triangulation to the centroids of their respective Voronoi polygons. Figure 2.5

(a)

(b)

Figure 2.5: The (a) initial DV mesh, which is an input to the CVT algorithm and (b) the resulting
DV mesh from the CVT algorithm.
presents an example of the CVT algorithm. Figure 2.5a is the initial Delaunay (and corresponding Voronoi diagram) triangulation of a random set of points. These mesh points are then moved
according to the CVT algorithm to produce Figure 2.5b. Clearly, the CVT algorithm produces
quality DV elements in the interior of the mesh. It does not prevent circumcenters from falling
outside of the mesh boundary, however. Furthermore, because the algorithm is based upon moving
mesh nodes, the CVT algorithm will not preserve the original device geometry. Several papers
appear in the literature which address these problems. They report a stitching algorithm, which

11

aims to use the CVT algorithm in the interior of the geometry and stitch the resulting mesh onto a
boundary conforming and preserving mesh [42, 43, 44].
A considerable amount of work exists in DV meshing algorithms specifically designed for
semiconductor device geometries. A prominent algorithm which generates 3D Delaunay grids
suitable for complex semiconductor device structures was pioneered by Conti, Hitschfeld, and
Fichtner [45, 46, 47]. These algorithms decompose the global device geometry into sub-regions
which can be refined in terms of the mesh. The sub-regions automatically preserve material interfaces and mesh boundaries and the interiors of the sub-regions are allowed to have circumcenters
which fall outside of their respective element but still reside inside the sub-region. The commer-

Figure 2.6: The Delaunay triangulation (black lines) and the corresponding polygons of the
Voronoi diagram (red lines) of the basic outline of a MESFET. This mesh was generated with
an in-house code based upon the work of Conti.
cial device simulator Sentaurus Device, a product of Synopsys, utilizes a separate meshing code
which incorporates these algorithms in its meshing tools [48]. An example of a mesh generated
by an in-house Air Force Research Laboratory, Sensors Directorate (AFRL) code is presented in
Figure 2.6. Further progress on this meshing topic extends the work of Conti to produce boundary
conforming meshes [49, 50].
Finally, a discussion on splitting the Voronoi mesh at material interfaces and truncating at mesh
boundaries is required. Given that the triangles (2D) or tetrahedra (3D) on material interfaces or
mesh boundaries do not have obtuse angles “pointing” towards the interface/boundary, the Voronoi
mesh can be properly split or truncated. Figure 2.7 illustrates material interface truncation on (a) a
mesh generated by the algorithm of [45] and (b) a mesh generated by Gmsh [39]. In both instances,
the Voronoi cells corresponding to Delaunay nodes on the material interface have edges which are

12

(a)

(b)

Figure 2.7: Two examples of mesh splitting at material interfaces. In both cases, the primary edges
separating the red and blue shaded Voronoi polygons represent the material interface. (a) Material
interface Voronoi splitting on a mesh generated by the algorithm of Conti. (b) The same material
interface Voronoi splitting on a mesh generated by Gmsh.
exactly perpendicular to the material interface. The Voronoi edges intersect the Delaunay edges at
exactly the circumcenter of the Delaunay edges — the midpoint of the edge. In 3D, the edges of a
Voronoi polyhedron intersect at the circumcenters of the Delaunay triangles. The Voronoi diagram
at the mesh boundaries uses the exact same recipe for truncation. The portion of the Voronoi
cell which falls outside of the Delaunay mesh, however, is not included in the simulation domain.
Figure 2.8 demonstrates the truncation of the Voronoi diagram at a boundary shaded in gray. The

p2
d2

p1
A1

Figure 2.8: An example of Voronoi diagram splitting in 3D. The primary edges are drawn in black
and the Voronoi edges are drawn in red. The interface of the Delaunay mesh is shaded in gray. The
dashed red lines correspond to the Voronoi cells in the plane of the interface. Two Voronoi cells in
the interface are shaded in red.

13

Voronoi edges in the plane of the boundary are drawn as dashed red lines. This boundary could be
either a material interface or the end of the Delaunay mesh. If the boundary is a material interface,
the Voronoi cells on the interface are used to calculate thermionic emission across the boundary.
In the case of mesh boundaries, the Voronoi edges and cells on the plane are used for BCs. Both
of these cases will be discussed in subsequent sections.
The following list summarizes the requirements of meshes suitable for semiconductor device
simulation with the box integration method.
• The mesh must be Delaunay compliant, i.e., no nodes of any primary element fall within any
of the circumcircles/circumspheres.
• Both the Delaunay triangulation and the Voronoi diagram must preserve mesh boundaries
and material interfaces. In other words, the circumcenters of interior elements cannot cross
mesh boundaries or material interfaces.
If these two criteria are met, the Voronoi cells corresponding to mesh nodes on boundaries/interfaces will have perpendicular edges to these boundaries/interfaces. These features are utilized to
truncate the Voronoi diagram into material specific regions and the edges are used for device simulation BCs. An example of separating the Voronoi diagram into unique materials is illustrated in
Figure 2.7.

2.3

Systems of Nonlinear Partial Differential Equations

Simulating charge transport by means of the FKT equations amounts to solving nonlinear partial differential equations (PDEs). Before complicated systems of nonlinear equations are derived
and discretized, some important properties are discussed. Simple systems of nonlinear equations
are presented in this section to discuss central concepts. First, the important topic of solving
discrete nonlinear PDEs with iterative methods is presented. Then, characteristics of discrete nonlinear PDEs including convergence and stability are presented.

14

2.3.1

Numerical Solution of Discrete Nonlinear Equations

Subsequent chapters are devoted to the derivation and discretization of the nonlinear transport
equations. Once a discrete set of equations is assembled, they must be solved for the independent
variables. This section focuses on two prominent iterative methods, namely, the fixed point iteration method (FPM) and the Newton-Raphson or Newton’s method. Given a set of N independent
variables ~x = {x1 , x2 , ¡ ¡ ¡ , xN }T and N equations ~F = {F1 (~x), F2 (~x), ¡ ¡ ¡ , FN (~x)}T , the iterative
methods attempt to calculate the independent variables which satisfy
~F(~x) = 0.

(2.4)

A very naive attempt at solving Eqn. (2.4) is a simple guess-and-check. The independent variables
could be randomly sampled until they satisfy the residuals. However, there is no guarantee that this
will ever produce the correct solution variables. Rather than randomly choosing the independent
variables, the FPM and Newton’s method provide “updates” to the solution variables in a systematic way. The following sections provide details on how the solutions are updated with the specific
iterative methods.

2.3.1.1

Fixed Point Iteration Method

~ x) if it satisfies~xP =
A set of independent variables~xP is a fixed point of the nonlinear equations G(~
~ xP ). The FPM attempts to use the fixed point to update the independent variables. To do so, the
G(~
~ x). If this manipulation
original nonlinear equations ~F(~x) = 0 are algebraically converted to~x = G(~
is not possible, then the FPM cannot be used to iteratively solve the nonlinear equations. Given an
initial guess of the independent variables ~x0 , the FPM algorithm is

For m = 0 to M_max:
Compute FP = G(x_m)
Compute residual = x_m - FP
If |residual| < tolerance: break
Update solutions x_{m+1} = FP
15

Here, M_max represents a user-defined maximum iteration count and tolerance defines the convergence criterion of the nonlinear solver.
The benefit of using the FPM is that it only requires the algebraic manipulation of the nonlinear
equations (if that manipulation is possible). It does not require the expensive calculation of a
Jacobian matrix (see the following section on Newton’s method). One drawback of the FPM is its
slow convergence. The convergence of the FPM will be demonstrated in Section 2.3.1.3.

2.3.1.2

Newton’s Method

Newton’s method is a robust iterative solver for systems of nonlinear equations [51]. Given an
initial guess of the independent variables~x0 and the “change” in solution variables ∆~x =~xm+1 −~xm ,
the Newton algorithm is

For m = 0 to M_max:
Compute residual vector "F"
Compute Jacobian matrix "J"
If |F| < tolerance: break
Solve the linear system J*dx = -F
Update solutions x^{m+1} += dx
Again, M_max represents a user specified maximum iteration count of the Newton algorithm and

tolerance is the value that defines convergence of the nonlinear solver. The variable dx in the
algorithm symbolizes the change in the independent variables ∆~x.
This algorithm is a standard of solving nonlinear systems of equations because of its versatility
and its quadratic convergence. However, Newton’s method is known to fail if the initial guess is
far from the solution. Source stepping is a common approach to alleviating convergence issues
of Newton’s method. For example, an applied bias of a device simulation is added incrementally
and solutions to Newton’s method are fed into the next solve. This method can also be applied to
time stepping algorithms which require reduction of the time step. Newton’s method requires the

16

calculation of the Jacobian matrix in the linear system


∂ F1
∂ F1
∂ F1
 ∂ x1 ∂ x2 · · · ∂ xN   ∆x1


 ∂ F2 ∂ F2
∂ F2  
··· ∂x 
 ∂x
∆x2
∂ x2

N 
1

 .
.. 
 ...
...
 ..
. 


 ∂F

N ∂ FN · · · ∂ FN
∆xN
x
x
x
1

2

N








 F1 (~x) 




 F (~x) 

 2

 = − . .

 .. 






FN (~x)

(2.5)

Calculation of the Jacobian matrix and the solution of the linear system becomes expensive for
large DOF.

2.3.1.3

Comparison of FPM and Newton’s Method

A simple set of nonlinear equations is presented and solved with the FPM and Newton’s method.
This system of equations, with independent variables x and y, is
F1 (x, y) = 2x2 + ey − 7x = 0,

(2.6)

F2 (x, y) = 9y − ex = 0.

(2.7)

The 2×2 Jacobian matrix required for Newton’s method is


y
 4x − 7 e 
Ji j (x, y) = 
,
−ex 9

(2.8)

and the FPM functions are
G1 (x, y) = (2x2 + ey )/7,
G2 (x, y) = ex /9.

(2.9)
(2.10)

The residuals F1 (x, y) and F2 (x, y) and their solution F1 = F2 = 0 are illustrated in Figure 2.9.
q
Figure 2.10 presents the norm of the residual F12 + F22 at each iteration of the FPM and Newton’s method. The red and blue lines illustrate the norm of the residuals calculated at each iteration
of Newton’s method and the FPM, respectively. Newton’s method exhibits a quadratic rate of convergence and the FPM a linear rate of convergence. However, to achieve quadratic convergence,
17

Figure 2.9: A set of nonlinear equations and its solution. The first and second residuals are plotted
with red and black surfaces, respectively and the solution is illustrated with a blue circle.

|residual|

1.0e+02
1.0e+00

Newton's Method

1.0e-02

FPM

1.0e-04
1.0e-06
1.0e-08
1.0e-10
1.0e-12
1.0e-14
1.0e-16
0

5

10

15

20

25

Iteration
Figure 2.10: Residual vector norm calculated at each iteration of the FPM and Newton’s method.
Newton’s method exhibits a quadratic rate of convergence. The FPM demonstrates a linear rate of
convergence.
Newton’s method must calculate a Jacobian matrix and solve the linear system of equations at
each iteration. The FPM only requires the evaluation of the simple algebraic manipulation of the
nonlinear equations. Newton’s method is the preferred nonlinear solver for the discrete transport
equations in subsequent chapters due to its robustness and strong convergence.

18

2.3.1.4

Other Iterative Methods

Solving the linear system of equations at each Newton iteration becomes burdensome with many
DOF. Moreover, calculation of the Jacobian matrix can be difficult or even impossible in some
cases. To overcome these obstacles, a class of nonlinear solvers called “quasi-Newton methods”
was developed as an alternative to Newton’s method. Among these is Broyden’s method [52],
which only requires the calculation of the Jacobian at the first iteration. Broyden’s method still
requires a linear solve at each iteration, however. Another important class of nonlinear solvers is
the Newton-Krylov method [51]. Rather than directly solving the linear system of equations at each
iteration with an LU factorization, these methods utilize Krylov subspace-based linear solvers such
as GMRES. The calculation of the Jacobian can also be removed altogether with the Jacobian-free
Newton-Krylov (JFNK) methods [53].

2.3.2

Mesh Convergence of Discretization Techniques

Discretizing systems of PDEs requires approximations of continuous derivatives in space and time.
The box integration method is used for the FKT equations (see Section 2.1). An important characteristic of discretization techniques is the order of convergence of the methods. This is also known
as mesh convergence. Quantifying mesh convergence requires evaluating the error in the numerical solutions on a series of meshes. The numerical solutions are compared to analytic solutions.
If no analytic solutions are available, mesh convergence is quantified by using a highly discretized
solution as the “exact” or reference solution.
Consider the 1D nonlinear first-order differential equation
dy
+ 3y2 = 0,
dx

(2.11)

with the BC y(x = 0) = 1. The exact solution of the differential equation is
yexact (x) =

19

1
.
1 + 3x

(2.12)

Because the solution function is singular at x = −1/3, the domain is bounded to x > 0. This
differential equation is approximated with a backward difference
yi − yi−1
+ 3y2i = 0
∆x

(2.13)

yi+1 − yi−1
+ 3y2i = 0.
2∆x

(2.14)

and a central difference

Here, yi is the ith solution on a uniform 1D mesh with N nodes.
Eqns. (2.13) and (2.14) are solved with Newton’s method. The Jacobian matrices of both discrete equations are illustrated in Figure 2.11. Both Jacobian matrices are calculated at the first
Newton iteration with all initial solution variables set to zero. The (0, 0) element corresponds to
the Dirichlet BC residual y0 − 1.0 = 0.0. This BC is scaled by 5 in these plots. Figure 2.11a

(a)

(b)

Figure 2.11: The Jacobian matrices of Eqns. (2.13) and (2.14) on a 10 and 11 node mesh, respectively. The last solution variable in (b) is governed by a backward difference approximation of
Eqn. (2.11).
presents the backward difference Jacobian matrix on a 10 node mesh and Figure 2.11b presents
the same for the central difference approximation on an 11 node mesh. The additional node and
solution variable added to the central difference equations is solved with a backward difference approximation. The maximum and minimum values of the backward difference Jacobian correspond
to the solution variables with the factor 1/∆x. Figure 2.12 illustrates the discrete y(x) solutions
20

calculated with the central difference approximation at each Newton iteration. The mesh size is
N = 11. The analytic solution is drawn with a black dashed line.
1

Iteration 0
Iteration 1
Iteration 2
Iteration 3
Iteration 4
Iteration 5
Iteration 6
Analytic

y(x)

0.8
0.6
0.4
0.2
0
0

0.2

0.4

0.6

0.8

1

x
Figure 2.12: Discrete solutions of Eqn. (2.14) on an N = 11 node mesh. The solutions are plotted
at each Newton iteration. The analytic solution is drawn as a dashed black line.
To quantify the mesh convergence of solving Eqns. (2.13) and (2.14), the L2 error,
r

2
∑ yi − yi,exact
i
Îľk = r

2 ,
∑ yi,exact

(2.15)

i

is calculated on a series of meshes. The order of convergence p is determined by the relation
−p

Îľk < CNk ,

(2.16)

where Nk is the number of mesh points (DOF) on the kth mesh. Figure 2.13 presents the evaluation
of Eqn. (2.15) on a series of meshes. The number of nodes in the series of meshes ranges from
102 to 105 . Solutions are calculated from x = 0 to x = 1. The Dirichlet BC is y0 = 0. Errors
corresponding to the backward and central difference approximations are drawn in red and blue,
respectively. It is clear that the backward difference and central difference approximations are firstand second-order discretization techniques. Second-order convergence of the central difference
approximation is due to first-order error cancellation on the uniform mesh.

21

1.0e+00
Backward difference

1.0e-01

Central difference

1.0e-02
1.0e-03

1 st ord

er

Îľk

1.0e-04
1.0e-05
1.0e-06

2 nd o

rder

1.0e-07
1.0e-08
100

1000

10000

Degrees of freedom
Figure 2.13: The error of the numerical solution of Eqn. (2.11) using a backward difference approximation (red line) and central difference approximation (blue line) of the spatial derivative. The
backward difference approximation exhibits first-order mesh convergence. The central difference
exhibits second-order mesh convergence.
2.3.3

Stability of Discrete Equations

Stability theory is a mathematical framework for analyzing the stability of nonlinear PDE solutions.
The theory is prominent in systems theory and control engineering and is applied to the numerical
solution of the nonlinear FKT equations. An autonomous system dx
dt = f (x) has an equilibrium
point xe where f (xe ) = 0. The crux of stability theory [54] is as follows. The equilibrium point is
• stable if, for each perturbation of the equilibrium point, the time-integrated solutions |x(t)|
are bounded by a constant δ > 0.
• unstable if it is not stable.
• asymptotically stable if it is stable and the perturbation returns to the equilibrium point as
t → ∞.
To elaborate on stability analysis of nonlinear equations, consider the autonomous system
dx
− 10x + 5xy = 0,
dt
dy
− 3y − xy + 3y2 = 0.
dt
22

(2.17)
(2.18)

This nonlinear system has equilibrium points at (x, y) = (0, 0), (0, 1), and (3, 2). Figure 2.14
4

3

y

2

1

0

Equilibria
-1
-1

0

1

2

3

4

5

x
Figure 2.14: The direction field of Eqns. (2.17) and (2.18) and four trajectories with different
initial conditions. The initial condition of each trajectory is illustrated with a solid circle.
presents the direction field of Eqns. (2.17) and (2.18). The equilibria are plotted at (x, y) = (0, 0),
(0, 1), and (3, 2). The direction field is a vector field with components (dx/dt, dy/dt). The solid
lines in Figure 2.14 represent four trajectories of Eqns. (2.17) and (2.18) — the time evolution of
the integrated solutions (x(t), y(t)). Each solid line starts at a different initial condition. It is clear
from Figure 2.14 that the (0, 0) and (0, 1) equilibria are unstable. All trajectories diverge away
from these points. The (3, 2) equilibrium point is asymptotically stable.
Direction fields and trajectory plots provide intuitive results for 2×2 systems of nonlinear equations. These tools do not apply to large systems of equations, however. Rather, the time evolution
of a perturbation from an equilibrium point is analyzed by calculating


∆H(t) = H(t) − Hequilib.  .

(2.19)

Here, Hequilib. represents the equilibrium point of the N-dimensional system of nonlinear equations. H(t) represents the time evolution of the integrated equations. Figure 2.15 presents the
calculation of Eqn. (2.19) for each trajectory shown in Figure 2.14. All trajectories are compared
23

1e4
1e2

ΔH(t)

1
1e-2
1e-4
1e-6
1e-8
1e-10

0

2

4

6

8

10

Time-step
Figure 2.15: Calculation of Eqn. (2.19) for each perturbation presented in Figure 2.14. The five
colors correspond to the five trajectories with different initial conditions. The black, green, and
yellow lines indicate that the system of equations is not stable.
to the Hequilib. = (3, 2) equilibrium point. The perturbations drawn in red and blue converge to the
(3, 2) equilibrium. The black, green, and yellow perturbations, however, diverge from the equilibrium points. This analysis provides quantitative stability results for large systems of nonlinear
equations. The system of equations is stable if ∆H(t) → 0 as t → ∞ for all perturbations.

2.4

RF Amplifiers

RF amplifier design relies heavily on accurate characterization and modeling of transistors. The
transistor model must be valid over many operating conditions. For example, a circuit designer may
require a valid model for a wide range of biases, a broad frequency range, several input powers, and
various loading conditions. In what follows, the pertinent figures of merit (FOMs) for PA design
are discussed. Linear two-port models and their uses in PA design are described in Section 2.4.2
and a discussion of large-signal PAs is provided in Section 2.4.3.

24

2.4.1

Amplifier Figures of Merit

The important FOMs for RF PA design include power gain, output power, the 1 dB gain compression point, PAE, and many others [1, 55]. The amplifiers are typically biased for peak transconductance, given by the maximum of
gm =

∂ ID
.
∂VGS

(2.20)

Here, VGS is the gate-source bias and ID is the steady-state drain current. The transconductance is
calculated with a constant drain-source bias VDS . Power gain of amplifiers is defined as the ratio
of the output power to the input power. Gain is typically categorized as power gain G, available
power gain GA , and transducer power gain GT . The definitions of the gain metrics are
P
G= L.
Pin
Pavn
,
Pavs
P
GT = L ,
Pavs

GA =

(2.21)
(2.22)
(2.23)

Here, Pin is the power delivered to the network, Pavs is the power available from the source, Pavn
is the power available from the network, and PL is the power delivered to the load. Each gain is
equal to the maximized gain, GT = GA = GP = Gmax , if the input and output are both conjugate
matched to the device.
The output power is defined as the power delivered to the load PL . For an ideal amplifier,
the output power will increase linearly with respect to the input power. The output power of real
transistors, however, will begin to saturate at high input power. This effect is quantified with the
1 dB gain compression point which is the point at which the gain decreases by 1 dB.
One of the most important FOMs in PA design is PAE. The PA stage consumes the most DC
power in many wireless and Tx/Rx module applications and therefore it is critical to maximize
the PAE for the optimal PA design. The PAE is defined as the ratio of the output to input power
reference to the DC power level, or
P − Pin
PAE = L
.
PDC
25

(2.24)

This work will focus on the output power, power gain, and PAE for PA design purposes. There
are many other important RF amplifier FOMs, however. Included is bandwidth, noise figure,
input and output voltage standing wave ratio (VSWR), adjacent channel power ratio (ACPR), and
stability [56, 1, 55].

2.4.2

Linear Two-Port Models

Linear two-port models are the standard representation of small-signal RF transistor responses.
Figure 2.16 presents a general two-port model. Subscript 1 corresponds to the first port with
voltage V1 and current I1 . Here, the incident and scattered waves at both ports are

Vs

Input matching
network

50 Ί

+

A1

DUT

A2

B1

[S]

B2

+

V1

V2

ΓS

Output matching
network

I2

I1

Γout

Γin

50 Ί

ΓL

Figure 2.16: A linear two-port model of a DUT with input and output matching networks. This
model is for a single fundamental frequency.

A1,2 =

V1,2 + Z0 I1,2
,
2

(2.25)

B1,2 =

V1,2 − Z0 I1,2
.
2

(2.26)

The reflection coefficient looking into the generator is ΓS and Γin signifies the reflection coefficient
looking into port 1. The same notation is used at port 2. The reflection coefficients looking into
the generator and load are
Z − Z0
ΓS = S
,
ZS + Z0

(2.27)

Z − Z0
ΓL = L
.
ZL + Z0

(2.28)

The reference impedance is typically Z0 = 50 Ω.

26

Scattering parameters (S-Parameters) describe the linear relation between the incoming waves
A and outgoing waves B in the two-port model. The definition of S-Parameters is


 

 S11 S12   A1   B1 


=
,
S21 S22
A2
B2

(2.29)

where each coefficient is given by
S11 =
S21 =
S12 =
S22 =


B1 
,
A1 A =0
2

B2 
,
A1 A =0
2

B1 
,
A2 A =0
1

B2 
.
A2 A =0

(2.30)
(2.31)
(2.32)
(2.33)

1

The reflection coefficients looking into the input and output of the device are calculated from the
S-Parameters
S S Γ
Γin = S11 + 12 21 L ,
1 − S22 ΓL

(2.34)

S S Γ
Γout = S22 + 12 21 S ,
1−S Γ

(2.35)

11 S

The input and output powers can now be defined in terms of the reflection coefficients and the
S-Parameters. The power delivered to the network is
|VS |2 |1 − ΓS |2
(1 − |Γin |)2 ,
Pin =
2
8Z0 |1 − ΓS Γin |

(2.36)

the power available from the source is
|VS |2 |1 − ΓS |2


Pavs =
8Z0 1 − |Γ |2

(2.37)

S

the power available from the network is
Pavn =

|VS |2
|S21 |2 |1 − ΓS |2


8Z0 |1 − S Γ |2 1 − |Γ |2
out
11 S

27

(2.38)

and the power delivered to the load is


2
2
|VS |2 |S21 | 1 − |ΓL | |1 − ΓS |
.
PL =
8Z0 |1 − S22 ΓL |2 |1 − ΓS Γin |2
2

(2.39)

The small-signal gain can be calculated directly as


2
2
|S
|
|Γ
|
1
−
L
21
P

G= L = 
,
Pin
1 − |Γin |2 |1 − S22 ΓL |2

(2.40)

as well as the available power gain


2
|S21 | 1 − |ΓS |
Pavn

,
GA =
=
2
2
Pavs
|1 − S11 ΓS | 1 − |Γout |
2

(2.41)

and the transducer gain

GT =

PL
=
Pavs




|S21 |2 1 − |ΓS |2 1 − |ΓL |2
|1 − ΓS Γin |2 |1 − S22 ΓL |2

(2.42)

S-Parameters can also be converted to other representations of the linear two-port model, including the Z-, Y -, h-, and T - Parameters. For example, the small-signal current gain is one of the
h-parameters
h21 =

−2S21
.
(1 − S11 ) (1 + S22 ) + S12 S21

(2.43)

The impedance and admittance parameters are also useful for de-embedding and circuit model
extraction purposes.
S-Parameter simulation of a GaAs FET circuit model in ADS is presented to illustrate the use
of small-signal circuit models. The layout of the S-Parameter simulation in ADS is shown in
Figure 2.17. The gate to source bias is set by the parameter “Vlow” and the drain to source bias
is set by “Vhigh”. The simulation is over the 1–3 GHz frequency range and with input and output
port impedances of 25 Ω. Figure 2.18 presents the S11 and S22 S-Parameters, the input and output
reflection coefficients Γin and Γout , and the three power gains G, GA , and GT . The input and output
reflection coefficients differ from S11 and S22 because ΓS and ΓL are nonzero. Furthermore, the
three power gains produce different metrics over the frequency range.
28

I_Probe
Is_low

S-PARAMETERS
S_Param
SP1
Start=1 GHz
Stop=3 GHz
Step=0.01 GHz

I_Probe
Is_high

Vs_low

Var
Eqn

VAR
STIMULUS
Vhigh=4
Vlow=-1

Vs_high
L
L2
L=1 uH
R=

V_DC
SRC2
Vdc=Vlow

L
L1
L=1 uH
R=

V_DC
SRC1
Vdc=Vhigh

C
C2
C=1.0 uF
V_In
I_Probe
I_In

C
C1
C=1.0 uF

Term
Term2
Num=1
Z=25 Ohm

GaAsFET
FET1
Model=FLC301XP
Area=
Temp=
Trise=

Term
Term1
Num=2
Z=25 Ohm

Statz_Model
FLC301XP

1.4

1.2

1.0

0.9

0.5

2.0

1.8

0.6

1.6

0.7

0.8

Figure 2.17: The ADS layout of S-Parameter simulation of a GaAs FET circuit model.

35

0.4

3.0

0.3

4.0

0.2

20

10

5.0

4.0

3.0

1.8
2.0

1.6

1.4

1.2

0.9
1.0

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

20

GT_dB
GA_dB
G_dB

10

0.1

-20

-0.1

25

20

-10

GammaOut
GammaIn
S(2,2)
S(1,1)

30

5.0

-0.2

15

-5.0
-4.0

-0.3

10

-3.
0

4
-0.

-2.0

-1.8

-1.6

-1.4

-1.2

-1.0

-0.9

-0.8

-0.7

-0.6

-0.5

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

2.6

2.8

3.0

freq, GHz

freq (1.000GHz to 3.000GHz)

Figure 2.18: S-Parameters, reflection coefficients, and power gains of the GaAs FET circuit model
simulated in ADS. The reflection coefficients on the Smith chart are the dotted lines. The ADS
variables GammaIn and GammaOut and the power gain variables G_dB, GA_dB, and GT_dB
correspond to Eqns. (2.34), (2.35), (2.40), (2.41), and (2.42), respectively.
2.4.3

Large Signal Analysis

The linear two-port models of the previous section do not accurately describe the nonlinear behavior of transistors. Because the models are linear, they do not depend on the input power level or the

29

loading conditions of the device. For power levels comparable to or above the 1 dB compression
point, however, the measured S-Parameters will depend on input power level and loading condi-

f1

2f1

...

0

Kf1

Frequency

f1

2f1

...

Vs

Input matching
network

I1k
50 Ί

f1

2f1

...

Kf1

Frequency

I2k

+

A1k

DUT

A2k

F(Apk)

B2k

+

V1k

V2k
B1k

ΓS

0

Kf1

Frequency

Output matching
network

0

Kf1

~
~

...

Frequency

~
~

2f1

~
~

f1

~
~

0

I2k

V2k

I1k

V1k

tions of the device. Small-signal S-Parameters can therefore not be used for large-signal simulation

Γout

Γin

50 Ί

ΓL

Figure 2.19: A nonlinear two-port model of a DUT with matching networks.
of devices.
Figure 2.19 illustrates a generalization of the two-port model to include the K harmonic response of a nonlinear device. In this figure, the input voltages and currents are designated as V1k
and I1k , where the subscript 1 refers to port 1 and the subscript k is the index of the K harmonics.
The fundamental frequency is k = 1 and the “DC” term is k = 0. Port 2 uses the same notation.
Because transistors are inherently nonlinear, a large-signal applied to port 1 will generate multiple
harmonics at port 2. This is the inter-modulation effect.
The power definitions for the nonlinear two-port model in Figure 2.19 are different than those
for the linear analogue. Reflection coefficients ΓS , Γin , Γout , and ΓL are evaluated at the largesignal fundamental frequency (k = 1). The input power, power available from the source, and
power delivered to the load are
1 
∗ 	,
Pin = Re V11 I11
2
|VS |2
,
8RS
1 
∗ 	,
PL = Re V21 I21
2
Pavs =

30

(2.44)
(2.45)
(2.46)

respectively. The factor RS in the large-signal Pavs definition is the real component RS = Re {ZS }
of the source impedance [56]. VS is a large-signal tone at the fundamental frequency. The power
gain and PAE are
P
G= L,
Pin
P − Pin
PAE = L
,
PDC

(2.47)
(2.48)

respectively and the DC power dissipation is
PDC = V10 I10 .

(2.49)

An HB simulation layout of the GaAs FET model in ADS is illustrated in Figure 2.20. The
fundamental frequency of the simulation is 1 GHz and the power available from the source Pavs
is swept from 0 dBm to 25 dBm. Each FOM is plotted over the Pavs sweep in Figure 2.21. The
x-axis of the plot is the input power Pin , in dBm, resulting from the applied Pavs . Here, the output
power Pout is the solid black line, the power gain G is the solid red line, the transducer gain GT is
a dashed pink line, and the PAE is the solid blue line. The output power linearly responds to the
input power at small-signal levels. However, the transistor begins to exhibit strong nonlinearity at
the 1 dB compression point. This is approximately Pin = 10 dBm input power.
I_Probe
Is_low

I_Probe
Is_high

HARMONIC BALANCE
HarmonicBalance
HB1
Freq[1]=RFfreq GHz
Order[1]=5
SweepVar="Pavs"
Start=0
Stop=40

Vs_low

V_DC
SRC2
Vdc=Vlow

L
L2
L=1 uH
R=

L
L1
L=1 uH
R=

V_DC
SRC1
Vdc=Vhigh
C
C2
C=1.0 uF

V_In
I_Probe
I_In
P_1Tone
PORT1
Num=1
Z=25 Ohm
P=dbmtow(Pavs)
Freq=RFfreq GHz

Var
Eqn

Vs_high

VAR
STIMULUS
Pavs=16 _dBm
RFfreq=1 _GHz
Vhigh=4
Vlow=-1

P
I

C
C1
C=1.0 uF

GaAsFET
FET1
Model=FLC301XP
Area=
Temp=
Trise=

P_Probe
P_Load

Term
Term1
Num=2
Z=25 Ohm

Statz_Model
FLC301XP

Figure 2.20: The layout of the HB simulation of a GaAs FET circuit model in ADS.
Characterizing large-signal responses of transistors is most commonly accomplished by plotting constant FOM contours on a Smith chart as a function of the load reflection coefficient, i.e.,
31

50

PAE
TransducerGain
Gain
Pdel_dBm

40

30

20

10

0

-10
-10

-5

0

5

10

15

20

25

30

Pin_dBm

Figure 2.21: The output power “Pdel_dBm”, power gain “Gain”, transducer gain, and PAE of the
GaAs FET circuit model calculator with HB in ADS. The 1 dB compression point is approximately
10 dBm input power.
I_Probe
Is_low

I_Probe
Is_high

Vs_low

V_DC
SRC2
Vdc=Vlow

L
L2
L=1 uH
R=

L
L1
L=1 uH
R=

V_DC
SRC1
Vdc=Vhigh
C
C2
C=1.0 uF

V_In
I_Probe
I_In
P_1Tone
PORT1
Num=1
Z=Z_s
P=dbmtow(Pavs)
Freq=RFfreq

Vs_high

P
I

C
C1
C=1.0 uF

GaAsFET
FET1
Model=FLC301XP
Area=
Temp=
Trise=

P_Probe
P_Load

S1P_Eqn
S1
S[1,1]=LoadTuner
Z[1]=Z0

PARAMETER SWEEP
Var
Eqn

VAR
STIMULUS
Pavs=22 _dBm
RFfreq=1 GHz
Vhigh=4
Vlow=-1

ParamSweep
Sweep1

HARMONIC BALANCE
HarmonicBalance
HB1
Freq[1]=RFfreq
Order[1]=5

Statz_Model
FLC301XP
Trise=
Imelt=

Figure 2.22: The layout of the LP simulation of a GaAs FET circuit model in ADS. HB is used to
simulate the circuit and the load tuner (dashed red box) changes the reflection coefficient presented
to the output of the transistor.
load-pull (LP). The FOM is plotted at the fundamental frequency. LP data is readily measured
using systems ranging from computer-controlled electro-mechanical stub tuners to modern active

32

2.0

1.4

1.6

1.8

1.0

1.2

0.9

Maximum
Power
Delivered,
dBm

31.11

-1.8
-2.0

-1.6

-1.4

-1.2

-1.0

-0.9

-0.6

indep(PAE_contours_p) (0.000 to 29.000)
indep(Pdel_contours_p) (0.000 to 37.000)

20

10

5.0

4.0

3.0

1.4
1.6
1.8
2.0

1.2

0.7
0.8
0.9
1.0

0.6

0.5

0.4

-0.5

4
-0.

-1.8
-2.0

-1.6

-1.4

-1.2

-1.0

-0.9

-0.8

-0.7

-0.5

0.3

-0.3

-3.0
-0.6

-0.2

-0.8

-4.0
4
-0.

0.8

0.7

0.6

0.5

31.70

-0.1

-0.7

-5.0
-0.3

20
0.2

20

10

5.0

4.0

3.0

1.4
1.6
1.8
2.0

1.2

-10

-0.2

10

0.1

Pdel_contours_p
PAE_contours_p

1.8

2.0

1.4

1.6

1.0

1.2

0.9

0.8

0.7
0.8
0.9
1.0

0.6

0.5

0.4

0.3

0.2

0.1

-0.1

Maximum
Power
Delivered,
dBm

-3.0

-20

Pdel_contours_p
PAE_contours_p

0.1

20

49.12

5.0

-4.0

0.2

10

4.0

-5.0

5.0

3.0

-10

69.26

-20

0.3

4.0

0.1

0.4

3.0

Maximum
Power-Added
Efficiency, %

0.2

0.6

Maximum
Power-Added
Efficiency, %
0.3

0.5

PAE (thick) and Delivered
Power (thin) Contours

0.4

0.7

PAE (thick) and Delivered
Power (thin) Contours

indep(PAE_contours_p) (0.000 to 31.000)
indep(Pdel_contours_p) (0.000 to 37.000)

Figure 2.23: LP simulation of the GaAs FET circuit model in ADS at a center frequency of (left)
0.5 GHz and (right) 1 GHz. The red lines are the PAE contours and the blue lines are the output
power contours. The PAE and output power FOMs are maximized with different load impedances.
LP test benches [16]. PA designers create suitable matching networks for the impedance design
target synthesized from the LP data [57].
ADS offers a suite of simulation examples in its DesignGuide to help their users. Included
in the kit is an example of LP simulation of the GaAs FET nonlinear circuit model. Figure 2.22
illustrates the GaAs FET LP simulation in ADS. This layout is similar to Figure 2.20. In the LP
simulation, the “S1P_Eqn” component presents a series of reflection coefficients to the output port
of the transistor. The HB simulation is repeated for each load impedance to calculate the output
power and PAE. Finally, contours of each FOM are plotted on the Smith chart in Figure 2.23.
The LP contours in Figure 2.23 are calculated at a fundamental frequency of (left) 0.5 GHz
and (right) 1 GHz. Both LP simulations were driven by 22 dBm power available from the source.
Clearly, the load impedance which maximizes the power delivered to the load does not maximize
the PAE at 0.5 GHz. The simulations at 1 GHz also exhibit the same effect. Furthermore, the optimal impedances are different at each fundamental frequency. This simple example demonstrates
some of the difficulties of the PA design process.

33

2.5

X-Parameters

The PHD behavioral model is reviewed and discussed. This model was later trademarked
under Agilent Technologies which is now Keysight Technologies [26]. XP modeling is a black-box
frequency-domain technique and is the mathematically correct super set of S-Parameters [25]. The
PHD model was first presented in [58] and later derived in [24]. The XP (PHD) model was later
used extensively for black-box modeling of nonlinear devices [59, 60, 27, 61]. Using the notation
in Figure 2.19, the XP model is now presented and discussed. A general nonlinear model describes
the scattered waves at all ports in relation to the incident waves at all ports. Because the model is
nonlinear, however, each scattered wave harmonic depends on all incident wave harmonics at each
port. To simplify the model while retaining important features, the nonlinear function is linearized
around the large-signal tone A11 . The general form of the XP model is
B pm = ∑ S pq,mn Pm−n Aqn + ∑ Tpq,mn Pm+n A∗qn
qn

(2.50)

qn

with Tp1,m1 = 0 ∀(q, n). The incident and scattered waves are calculated from the port voltages
and currents by
Vqn + Z0 Iqn
,
2
Vpm − Z0 I pm
B pm =
.
2
Aqn =

(2.51)
(2.52)

Here, the subscript q refers to the port number and n the harmonic index. For two-port nonlinear
models, q = 1, 2. The terms S pq,mn and Tpq,mn are the coefficients of the XP model and P =
phase {A11 } is the phase of the large-signal tone.
Calculating the coefficients of the XP model is similar to calculating S-Parameters. For the SParameters, two simulations are required to calculate Eqns. (2.30) – (2.33). Because the scattered
waves B pm depend on both the incident waves Aqn and the complex conjugate of the incident waves
Aqn , the XP model requires (4K − 1) simulations/measurements to calculate the 2K × (4K − 1)
coefficients, S pq,mn and Tpq,mn , of a two-port model. This is demonstrated through the following
examples.

34

2.5.1

Two ports, one harmonic

The XP model with q = 1,2 ports and k = 1 harmonics is
B11 = S11,11 A11 + S12,11 A21 + T12,11 P2 A∗21 ,

(2.53)

B21 = S21,11 A11 + S22,11 A21 + T22,11 P2 A∗21 .

(2.54)

There are two incident waves and scattered waves at each port in this model. The incident wave
A11 is the large-signal tone and A21 is a small-signal tone at the output port. This notation is used
throughout the rest of this work.
If the large-signal tone A11 is the only non-zero incident wave, i.e., A21 = 0, then the S11,11
and S21,11 coefficients are trivially calculated. However, the remaining four XP coefficients are
not calculated with a non-zero A21 and A11 = 0. The XP coefficients are only valid for one largesignal input A11 . Therefore, this term must remain constant when calculating the coefficients of
the model.
The four remaining coefficients are calculated with two more simulations/measurements. Each
simulation is driven by the large-signal tone A11 and a small-signal tone A21 . The A21 incident
wave of the second simulation must be 90◦ out of phase with the first simulation in order to calculate the remaining XP coefficients. The two simulations yield four equations. These are Eqns.
(2.53) and (2.54) with two orthogonal incident wave sets. The remaining XP coefficients are calculated by solving the 4×4 linear system outlined in the following.
Calculating the six XP coefficients of a two-port, one-harmonic model is outlined with the
following. The notation ai represents a set of incident waves corresponding to a single simulation/measurement of the nonlinear device. For the rest of this work, the large-signal tone is a
S
cosine. This simplifies the model with P = 1. The notation AC
qn and Aqn is used to represent two

out of phase incident waves. The superscript C represents a cosine wave and the superscript S a
sine wave. Finally, the scattered waves calculated with the simulation/measurement ai are Bqn (ai ).
With these definitions, the XPs are calculated by:
1. Apply a0 = {A11 , A021 = 0}
35

• Simulate/measure the {B11 (a0 ), B21 (a0 )} scattered waves
• Compute S11,11 = B11 (a0 )/A11 and S21,11 = B21 (a0 )/A11 .
S
2. Apply a1 = {A11 , AC
21 } and a2 = {A11 , A21 }

• Simulate/measure the {B11 (a1 ), B21 (a1 )} and {B11 (a2 ), B21 (a2 )} scattered waves
• Compute S12,11 , S22,11 , T12,11 , T22,11 by solving the following linear system











2.5.2

AC
21

 ∗
AC



0

21  
∗ 

C
0
A
0 AC

21
21

 ∗


AS21 0
AS
0

21  

∗ 
S
S
0 A21
0
A
21
0





S12,11  
 

S22,11 
 
=
 

T12,11 
 
 
T22,11



B11 (a1 ) − S11,11 A11 

B21 (a1 ) − S21,11 A11 


B11 (a2 ) − S11,11 A11 


B21 (a2 ) − S21,11 A11

(2.55)

Two ports, two harmonics

The general algorithm for calculating the coefficients of the XP model becomes more apparent
with a two-port, two-harmonic example. This system of equations is
B11 = S11,11 A11 + S12,11 A21 + S11,12 A12 + S12,12 A22 + T12,11 A∗21 + T11,12 A∗12 + T12,12 A∗22
B12 = S11,21 A11 + S12,21 A21 + S11,22 A12 + S12,22 A22 + T12,21 A∗21 + T11,22 A∗12 + T12,22 A∗22
B21 = S21,11 A11 + S22,11 A21 + S21,12 A12 + S22,12 A22 + T22,11 A∗21 + T21,12 A∗12 + T22,12 A∗22
B22 = S21,21 A11 + S22,21 A21 + S21,22 A12 + S22,22 A22 + T22,21 A∗21 + T21,22 A∗12 + T22,22 A∗22
Computing the 28 XP coefficients of the two-port, two-harmonic model is described in the following. Again, the notation ai = {A11 , A12 , A21 , A22 } represents a single simulation/measurement of
the nonlinear device. The coefficients are calculated by:
1. Apply a0 = {A11 , 0, 0, 0}
• Simulate/measure the {B11 (a0 ), B12 (a0 ), B21 (a0 ), B22 (a0 )} scattered waves

36

• Compute S11,11 = B11 (a0 )/A11 , S11,21 = B12 (a0 )/A11 , S21,11 = B21 (a0 )/A11 , and
S21,21 = B22 (a0 )/A11
S
2. Apply a1 = {A11 , AC
12 , 0, 0} and a2 = {A11 , A12 , 0, 0}

• Simulate/measure the {B11 (a1 ), B12 (a1 ), B21 (a1 ), B22 (a1 )} and {B11 (a2 ), B12 (a2 ),
B21 (a2 ), B22 (a2 )} scattered waves
• Compute S11,12 , S11,22 , S21,12 , S21,22 , T11,12 , T11,22 , T21,12 , T21,22 by solving the
following linear system


AC
12

0

0

0

∗
(AC
12 )

0

0























0





C
C
∗

0 A12 0
0
0
(A12 )
0
0



C )∗
0
0 AC
0
0
0
(A
0

12
12


C
C
∗
0
0
0 A12
0
0
0
(A12 )  


S
S
∗

A12 0
0
0 (A12 )
0
0
0



0 AS12 0
0
0
(AS12 )∗
0
0



S
S
∗
0
0 A12 0
0
0
(A12 )
0


S
S
∗
0
0
0 A12
0
0
0
(A12 )


B (a ) − B11 (a0 )
 11 1



 B12 (a1 ) − B12 (a0 ) 




 B (a ) − B (a ) 
 21 1
21 0 




 B22 (a1 ) − B22 (a0 ) 


=

 B11 (a2 ) − B11 (a0 ) 




 B (a ) − B (a ) 
 12 2
12 0 




 B21 (a2 ) − B21 (a0 ) 


B22 (a2 ) − B22 (a0 )

S
3. Apply a3 = {A11 , 0, AC
21 , 0} and a4 = {A11 , 0, A21 , 0}

37

S11,12
S11,22
S21,12
S21,22
T11,12
T11,22
T21,12
























T21,22

(2.56)

• Simulate/measure the {B11 (a3 ), B12 (a3 ), B21 (a3 ), B22 (a3 )} and {B11 (a4 ), B12 (a4 ),
B21 (a4 ), B22 (a4 )} scattered waves
• Compute S12,11 , S12,21 , S22,11 , S22,21 , T12,11 , T12,21 , T22,11 , T22,21 by solving the
following linear system


AC
21

0

0

0

∗
(AC
21 )

0

0

0






0
0
0
0
0
0



C )∗
0
0
0
0
(A
0
0 AC

21
21


∗
C
C
0
0
0
(A21 )  
0
0
0 A21


S
S
∗

A21 0
0
0 (A21 )
0
0
0



0 AS21 0
0
0
(AS21 )∗
0
0



S
S
∗
0
0 A21 0
0
0
(A21 )
0


S
S
∗
0
0
0 A21
0
0
0
(A21 )


B (a ) − B11 (a0 )
 11 3



 B12 (a3 ) − B12 (a0 ) 




 B (a ) − B (a ) 
 21 3
21 0 




 B22 (a3 ) − B22 (a0 ) 

=


 B11 (a4 ) − B11 (a0 ) 




 B (a ) − B (a ) 
 12 4
12 0 




 B21 (a4 ) − B21 (a0 ) 


B22 (a4 ) − B22 (a0 )























AC
21

∗
(AC
21 )

S12,11
S12,21
S22,11
S22,21
T12,11
T12,21
T22,11
























T22,21

(2.57)

S
4. Apply a5 = {A11 , 0, 0, AC
22 } and a6 = {A11 , 0, 0, A22 }

• Simulate/measure the {B11 (a5 ), B12 (a5 ), B21 (a5 ), B22 (a5 )} and {B11 (a6 ), B12 (a6 ),
B21 (a6 ), B22 (a6 )} scattered waves
• Compute S12,12 , S12,22 , S22,12 , S22,22 , T12,12 , T12,22 , T22,12 , T22,22 by solving the
following linear system

38



AC
22

0

0

0

∗
(AC
22 )

0

0

0





C )∗

0
0
0
0
0
(A
0 AC
22
22



C )∗
0
0
0
0
(A
0
0 AC

22
22


∗
C
C
0
0
0
(A22 )  
0
0
0 A22



0
0
0
0
0 (AS22 )∗
AS22 0



∗
S
S
0
0
0
0
(A22 )
0 A22 0



∗
S
S
0
0
0
(A22 )
0
0 A22 0


∗
S
S
0
0
0
(A22 )
0
0
0 A22


B (a ) − B11 (a0 )
 11 5



 B12 (a5 ) − B12 (a0 ) 




 B (a ) − B (a ) 
 21 5
21 0 




 B22 (a5 ) − B22 (a0 ) 


=

 B11 (a6 ) − B11 (a0 ) 




 B (a ) − B (a ) 
 12 6
12 0 




 B21 (a6 ) − B21 (a0 ) 


B22 (a6 ) − B22 (a0 )























2.5.3

S12,12
S12,22
S22,12
S22,22
T12,12
T12,22
T22,12























T22,22

Two ports, K harmonics

The system of equations of a two-port, K-harmonic XP model is
B11 = S11,11 A11 + · · · + S12,1K A2K + T12,11 A∗21 + · · · + T12,1K A∗2K
..
.
B1K = S11,K1 A11 + · · · + S12,KK A2K + T12,K1 A∗21 + · · · + T12,KK A∗2K
B21 = S21,11 A11 + · · · + S22,1K A2K + T22,11 A∗21 + · · · + T22,1K A∗2K
..
.
B2K = S21,K1 A11 + · · · + S22,KK A2K + T22,K1 A∗21 + · · · + T22,KK A∗2K

39



(2.58)

The 2K × (4K − 1) coefficients of the XP model are calculated by
1. Apply a0 = {A11 , 0, ¡ ¡ ¡ , 0}
• Simulate/measure the B pm (a0 ) = {B11 (a0 ), · · · , B1K (a0 ), B21 (a0 ), · · · , B2K (a0 )} scattered waves
• Compute S11,11 , · · · , S11,K1 , S21,11 , · · · , S21,K1 :

S11,11 = B11 (a0 )/A11
..
.
S11,K1 = B1K (a0 )/A11
S21,11 = B21 (a0 )/A11
..
.
S21,K1 = B2K (a0 )/A11
C
S
S
2. Apply aC
i j = {A11 , 0, ¡ ¡ ¡ , Ai j , ¡ ¡ ¡ , 0} and ai j = {A11 , 0, ¡ ¡ ¡ , Ai j , ¡ ¡ ¡ , 0}
C
C
C
C
• Simulate/measure the B pm (aC
i j ) = {B11 (ai j ), ¡ ¡ ¡ , B1K (ai j ), B21 (ai j ), ¡ ¡ ¡ , B2K (ai j )}

and B pm (aSij ) = {B11 (aSij ), ¡ ¡ ¡ , B1K (aSij ), B21 (aSij ), ¡ ¡ ¡ , B2K (aSij )} scattered waves
• Compute the S pi,m j = {S1i1 j , · · · , S1i,K j , S2i,1 j , · · · , S2i,K j } and Tpi,m j = {1i,1 j , · · · ,
T1i,K j , T2i,1 j , ¡ ¡ ¡ , T2i,K j } coefficients by solving the following linear system


DC
ij

∗
(DC
i j)







B pm (aC
i j ) − B pm (a0 )



  S pi,m j  


=

DSij (DSij )∗
Tpi,m j
B pm (aSij ) − B pm (a0 )
 
 ∗
C I, (DC )∗ = AC
where DC
=
A
ij
i j I, and I is the 2 × K identity matrix.
ij
ij



3. Repeat # 2 for 2K − 1 simulations to calculate all 2K × (4K − 1) XP coefficients.

40

(2.59)

2.6

Load-Dependent X-Parameters

The XP model of the previous section can be used to calculate the response of a nonlinear
device to the inputs A pk . The XP coefficients are valid for a single large-signal tone A11 and
any small-signal perturbations A pk , pk 6= 11. The XP model can be made load-dependent by the
substitution
A2k = ΓL,k B2k

(2.60)

Here, port 2 is the output port and ΓL,k is the reflection coefficient presented to the output port. The
load reflection coefficient is specified for each harmonic. The reflection coefficient substitution and
its consequences on calculating the output of the nonlinear model is discussed with the following
examples. Again, the large-signal tone A11 is a cosine and the phase term is P = 1.

2.6.1

Two ports, one harmonic

The system of equations of a two-port, one-harmonic XP model with a load reflection coefficient
is
B11 = S11,11 A11 + S12,11 A21 + T12,11 A∗21
B21 = S21,11 A11 + S22,11 A21 + T22,11 A∗21
A21 = ΓL,1 B21
The XP coefficients are computed with the algorithm described in the previous section. The reflection coefficient substitution yields
B11 − S12,11 Γ1 B21 = S11,11 A11 + T12,11 A∗21 ,
B21 − S22,11 Γ1 B21 = S21,11 A11 + T22,11 A∗21 .
The scattered waves can no longer be calculated with a simple summation in Eqn. (2.50). The XP
coefficients are calculated by solving the linear system


 

∗
 1 −S12,11 Γ1   B11   S11,11 A11 + T12,11 A21 
=


 
.
∗
0 1 − S12,11 Γ1
B21
S21,11 A11 + T22,11 A21
41

(2.61)

2.6.2

Two ports, K harmonics

The load-dependent XP model is generalized to two-ports and K harmonics. The system of equations for this XP model is
B11 = S11,11 A11 + · · · + S12,1K A2K + T12,11 A∗21 + · · · + T12,1K A∗2K
..
.
B1K = S11,K1 A11 + · · · + S12,KK A2K + T12,K1 A∗21 + · · · + T12,KK A∗2K
B21 = S21,11 A11 + · · · + S22,1K A2K + T22,11 A∗21 + · · · + T22,1K A∗2K
..
.
B2K = S21,K1 A11 + · · · + S22,KK A2K + T22,K1 A∗21 + · · · + T22,KK A∗2K
A21 = Γ1 B21
..
.
A2K = ΓK B2K
The reflection coefficient substitution yields
B11 − S12,11 Γ1 B21 · · · − S12,1K ΓK B2K = S11,11 A11 + · · · + S11,1K A1K + · · · + T12,1K A∗2K
..
.
B1K − S12,K1 Γ1 B21 · · · − S12,KK ΓK B2K = S11,K1 A11 + · · · + S11,KK A1K + · · · + T12,KK A∗2K
B21 − S22,11 Γ1 B21 · · · − S22,1K ΓK B2K = S21,11 A11 + · · · + S21,1K A1K + · · · + T22,1K A∗2K
..
.
B2K − S22,K1 Γ1 B21 · · · − S22,KK ΓK B2K = S21,K1 A11 + · · · + S21,KK A1K + · · · + T22,KK A∗2K

The scattered wave outputs of the nonlinear device are calculated by solving the general linear
system










∗
∗
 I −S1   B1i   S11,i j A1 j + T11,i j A1 j + T12,i j A2 j 
=


 

0 I − S2
B2i
S21,i j A1 j + T21,i j A∗2 j + T22,i j A∗2 j

42

(2.62)




Here, the ith rows of the K × K matrices S1 and S2 are S12,i1 Γ1 , · · · , S12,iK ΓK and S22,i1 Γ1 ,

· · · , S22,iK ΓK , respectively. The matrix I is the K × K identity matrix.

43

CHAPTER 3
DELAUNAY-VORONOI SURFACE INTEGRATION

3.1

Introduction

DVSI is a computational electromagnetics (CEM) method that operates on unstructured meshes
and is amenable to moving charges. The latter is paramount for any electronic device solver. This
method has been successfully utilized to simulate semiconductor devices with full-wave electromagnetics coupled simultaneously to a novel Boltzmann equation solver for the charge transport
including a full band-structure description for the electronic states [62]. Because the focus of that
paper was the simulation of electronic devices and comparisons with other charge transport methods, complete detail of the EM discretization method was not included. Therefore, the purpose of
this chapter is to present the numerical method of DVSI in sufficient detail to allow for independent
implementations and to ensure the reproducibility of key results in the emerging literature.
In order to introduce DVSI in a CEM context, consider the two dominant numerical approaches: integral equation (IE) solvers and differential equation (DE) solvers. Both have a common goal of computing EM scalar and/or vector fields over a domain of interest. CEM methods
can further be decomposed into two sub-categories: frequency-domain (FD) and TD methods.
Because the ultimate goal is to develop a CEM method which couples to nonlinear electronic
transport which is inherently in the time-domain, frequency domain methods are not considered in
this discussion.
IE methods seek to solve an EM problem in which the unknown quantity of interest lies within
an integral. These unknown quantities are typically surface currents on the boundaries of the computational domain. Electric and magnetic fields are computed through scalar and vector potentials.
An IE is most often derived from Maxwell’s equations using the Green’s function approach [63].
The crux of the approach is deriving a Green’s function which satisfies the linear Helmholtz equation subject to point sources. IEs are then built upon the principle of linear superposition. Although

44

this method is a staple of antenna design and other radiation (free-space) applications, the Green’s
function method cannot be applied when the point source is inherently nonlinear and is very limited
when the dielectric medium is inhomogeneous. Other methods for deriving IEs from Maxwell’s
equations may produce viable options for coupling to electronic transport, but they are unknown
to the author. For a more detailed discussion of IEs, please see [63] and references therein.
TD-DE methods are much better suited for representing and solving full-wave EM phenomena
in electronic devices. Unlike IE methods, DE methods seek EM solutions directly from Maxwell’s
equations in differential form. The two most common TD-DE categories can be classified as unstructured and structured grid methods. Even though these two methods are both DE solvers,
they differ greatly in their computational approach and sophistication. Unstructured grid (mesh)
methods first decompose the computational domain into a collection of elements which are typically tetrahedra. The governing equations are then written in a form suitable for discretization.
This amounts to writing Maxwell’s equations in curl-curl form for the well known Finite Element
Method Time Domain (FEMTD) [63] or adding flux terms to Maxwell’s equations for the Discontinuous Galerkin Time Domain (DGTD) method [64]. Both of these methods seek a variational
solution by approximating the quantities of interest with collections of basis functions which are
carefully tailored to satisfy the governing equations and BCs, e.g., vector finite elements. When
coupling these discretization methods to charge transport, it is not clear how to construct a basis
set which approximates an equation with exponentially nonlinear dependence on the quantity of
interest, such as the exponential relationship between carrier density in a semiconductor and the
potential.
A separate discretization scheme, called the Finite Volume Method (FVM) [65], is a DGTD
method with zeroth order basis functions. This method does not require a basis set to describe
the solution variables and therefore does not have equivalent limitations with regard to nonlinear
dependencies. However, special care must be taken when deriving the flux terms. An exhaustive
search has yielded no viable options for representing full-wave EM in an electronic device simulator using DGTD, although attempts to construct the necessary flux terms are undoubtedly an active

45

area of research in the DG numerical methods community.
The other class of the DE methods is a structured grid method and is most commonly referred
to as Finite Difference Time Domain (FDTD). It does not require a basis set since all partial
derivatives are approximated with Taylor Expansions [63]. This makes FDTD a viable method
to couple full-wave EM to nonlinear charge transport and implementations have been reported
in the literature [66]. One drawback to using FDTD, however, is that the method represents the
spatial computational domain with structured grids. Even if these structured grids are spaced nonuniformly, it still becomes burdensome when representing the complicated features in modern
electronic devices, e.g., wrap-around gates or recessed Ohmic contacts.
The DVSI method in this work approximates Maxwell’s equations in integral form with piecewise constant vector projections and tessellates the computational domain with an unstructured,
staggered mesh. The fact that DVSI tessellates the computational domain with an unstructured
mesh is what makes DVSI well suited for these purposes. Parts of DVSI can be seen in the literature, including the well known Box Integration Method [30] in the computational electronics
community and the Finite Integration Method (FIT) [67] and the Co-Volume Method [42] which
are used for linear EM applications. It is worth mentioning that in certain limits of geometry, i.e.,
rectangular mesh elements, the DVSI method collapses exactly to a traditional Yee cell, which is
the classical FDTD co-volume solution technique for this class of problems [68, 69]. DVSI is a
fully general version for arbitrary geometries and completely unstructured discretization similar to
an unstructured form of FDTD [70]. In regards to an unstructured FDTD formulation, DVSI is
preferred for electronic device simulation due to its amenability to realistic device BCs. This will
be discussed in subsequent sections.
In what follows, a brief discussion of the governing equations and the staggered meshes is
presented in Section 3.2. Then, complete discussions of the discretization of Maxwell’s rotational
equations and Poisson’s equation are presented in Sections 3.3 and 3.4, respectively. Each section
contains benchmark tests that demonstrate and validate the numerical methods through physically
relevant and quantitative test cases. Section 3.5 combines these methods and introduces the nec-

46

essary additional information that results in a numerical framework for representing full-wave EM
for the target engineering application of electronic device simulation. This section also includes a
benchmark test to validate the complete discretization of the relevant Maxwell’s equations.

3.2

Physical Equations and Their Geometric Representation

The relevant physical quantities needed to simulate full-wave EM for the purposes of electronic
device simulation are the electric field ~E (~r,t), the magnetic field ~H (~r,t), and the electric potential
ÎŚ (~r,t). Total electric fields are separated into rotational and irrotational components, i.e., their
Helmholtz decomposition ~E (~r,t) = ~Erot (~r,t) + ~Eirr (~r,t). The Coulomb Gauge, ∇ · ~A = 0, is used
throughout and is mathematically convenient. The technique is started by expressing Maxwell’s
equations in their standard integral forms
ZZ
S



I

∂ ~
Erot (~r,t) − ∇Φ (~r,t) +~J (~r,t) dS = ~H (~r,t) · d~`,
n̂ · ε
∂t

(3.1)

Γ

ZZ
S

−

I
∂ ~H (~r,t)
dS = − ~Erot (~r,t) · d~`,
n̂ · µ
∂t

(3.2)

Γ

ZZ

n̂ · ε∇Φ (~r,t) dS =

ZZZ

ρ (~r,t) dV.

(3.3)

V

S

As alluded to in the introduction, the computational domain is represented with staggered unstructured meshes known as the Delaunay mesh and its dual Voronoi diagram. Figure 3.1 illustrates
a two-dimensional example of the staggered meshes. The left highlights the Delaunay triangles in
blue and the right highlights the Voronoi polygons in red. In three dimensions, the Delaunay mesh
is comprised of finite edges, triangles, and tetrahedra which are called primary elements. The
Voronoi diagram is made up of finite edges, polygons, and polyhedra which are then called dual
elements. The relationship between the primary mesh and the dual mesh is exploited to discretize
Maxwell’s equations and is the focus of subsequent sections. The open source software TetGen
was used to generate all Delaunay-compliant meshes for this work. For an in-depth discussion on
mesh generation, there is extensive computer science literature that is actively seeking DelaunayVoronoi mesh optimization. These are compliant not only with DVSI, but also with engineering
47

numerical methods across disciplines, e.g., solutions to the Navier-Stokes equations within the
marker-and-cell algorithm.

3.3

Maxwell’s Rotational Equations

To begin the discussion of solving Maxwell’s equations in a bounded region with DVSI, the
rotational equations (i.e., Ampere’s and Faraday’s laws) are discretized and the pertinent BCs for
electronic device simulation are applied to obtain results. Two benchmark examples are presented
which validate the numerical method quantitatively through comparisons with analytic results.

3.3.1

DVSI Discretization

Solving Maxwell’s rotational equations, with proper BCs, is convenient with the relationship between the Delaunay and Voronoi meshes. When solving the rotational equations, it is assumed
that the electric field ~E is the total field comprised of both a rotational and irrotational component.
Ultimately, in Section 3.5 the fields are decomposed into rotational and irrotational components,
but initially only with total field versions of Eqns. (3.1) – (3.2).

Figure 3.1: A two-dimensional example of the staggered DVSI mesh with the (left) Delaunay
triangles drawn in blue and (right) the Voronoi polygons drawn in red.

48

Figure 3.2: (left) Dual edges (shown as red arrows) associated with a primary edge (blue arrow).
The dual polygon and dual edges are comprised of the circumcenters of each tetrahedron attached
to the primary edge. (right) Primary polygon and edges associated with one of the dual edges.

ZZ
S

#
I
~
~
∂ E (r,t) ~
~
+ J (r,t) dS = ~H (~r,t) ¡ d~`,
n̂ · ε
∂t
"

(3.4)

Γ

ZZ
S

n̂ · µ

∂ ~H (~r,t)
dS = −
∂t

I

~E (~r,t) ¡ d~`.

(3.5)

Γ

Electric fields are represented as spatially constant vector projections on primary edges. Likewise, magnetic fields are represented as spatially constant vector projection quantities on dual
edges. These relationships are depicted in Figure 3.2. The main points of DVSI discretization of
the rotational equations are:
1. A mesh edge is always perpendicular to its associated polygon.
2. The solution variables representing electric and magnetic fields are spatially constant vector
projections on mesh edges.
3. Materials reside within primary tetrahedra ensuring that primary edges do not cross dielectric
boundaries, which is critical for applying BCs.

49

4. The entire volume retains the permeability of free space (we do not consider magnetic inhomogeneity as it is rare in semiconductor devices).
With this geometric description and these assumptions about electric and magnetic fields, the integral form of Maxwell’s equations can now be written as the semi-discrete equations


∂ Ei
+ Ji Ai = ∑ H j L j ,
Îľi
∂t
j
Âľ0

∂Hj
A = − ∑ Ei Li ,
∂t j
i

(3.6)

(3.7)

where Ei is short-hand notation representing the temporally dependent scalar field projection
E (~ri ,t). The ith primary edge has length Li and the jth dual edge has the length L j . Areas Ai
and A j represent the polygons associated with mesh edges. The semi-discrete form of Maxwell’s
equations can be integrated in time with any method such as Forward Euler, Backward Euler, or
the Crank-Nicolson method. The fully discrete Maxwell’s equations are now written as

 

 


N(Îą)
n+1
A
n
A
−Dil
(1 − α) Aim   El   Ji
Ai 
ιAim   El
 
 Dil

+
 = 0.
+


F
n
n+1
(1
−
Îą)
F
−D
H
0
H
ÎąF jl D F
jl
m
m
jm
jm
(3.8)
The superscript n represents the “known” field quantity at a previous time step and n + 1 represents
the next “unknown” field quantity in the time-marching method. The diagonal matrices are defined
as DilA = εi Ai δil /∆t and D F
jm = µ0 A j δ jm /∆t, the Ampere matrix is defined as Aim = −cim Lm , and
the Faraday matrix is defined as F jl = c jl Ll . The tensors in the Ampere and Faraday matrices are
collections of Âą1 to ensure a properly closed line integral around associated edges. For example,
the Ampere constants are assigned to make primary edge vector projections follow the right hand
rule and the Faraday constants are assigned to make dual edge vector projections follow the left
hand rule. The constant Îą reflects the choice of the time integration method. Values Îą = 0, 1, 1/2
will yield Forward Euler, Backward Euler, and Crank-Nicolson schemes, respectively. Current
density time index N(Îą) is set to n for Forward Euler and n + 1 for Backward Euler and CrankNicolson.
50

One important note about DVSI is that for simulations in which the geometry of interest is
on the order of one or many wavelengths, this method would require a dense mesh to keep the
spatially constant approximations valid. Because this paper is meant to highlight the numerical
aspects of DVSI, dense discretizations are used to evaluate canonical CEM benchmarks which
have analytic solutions. Simulating electronic devices does not retain this meshing requirement,
since the majority of devices are inherently sub-wavelength in size. This is a tremendous meshing
advantage of DVSI for semiconductor devices that are sub-wavelength in terms of obtaining fullwave solutions without over-meshing.
Also, since fluxes are defined as constant over polygons, an edge which does not physically
touch its associated face is still valid as long as the area is computed correctly and the electric flux
accounts for regions of different dielectric properties as described in the subsequent section. This is
because the edge is closer to its associated dual face than any other edge due to the Delaunay nature
of the mesh. This implies the need for strict adherence to Delaunay-compliant meshes. Validity
of these mesh relationships also extends to node / volume relationships as well as boundary edges.
Experience has shown that commercial tools claim to have < 1% non-Delaunay compliance in
three dimensions. Given the previous statements on mesh requirements, even a small fraction of
non-Delaunay compliant elements can lead to erroneous results with Box Integration or DVSI.

3.3.2

Dielectric Discontinuities

Dielectric discontinuities are easily addressed with DVSI. The area and dielectric constant of the
ith electric flux term is computed as a summation of the Ni local relative dielectric constants and
areas of the electric flux, namely
Ni

Îľi Ai = Îľ0

∑ εr,k Ak .

(3.9)

k=0

Sub-polygons are created within the main dual polygon where each sub-polygon resides completely within a single primary tetrahedron. An example of sub-polygon decomposition of the
electric flux area is shown in Figure 3.3. The first three sub-polygons reside within one material,
i.e., Îľr,1 = Îľr,2 = Îľr,3 and the other two are in another material with Îľr,4 = Îľr,5 . The total elec51

Figure 3.3: An illustration of the decomposition of a dual polygon which spans multiple dielectric
regions. The dual polygon is split into multiple sub-polygons. Each is completely contained within
a single primary tetrahedron and assigned the relative dielectric constant of that tetrahedron. The
3-D mesh was rotated such that the primary edge points into the page.
tric flux area is then calculated using Eqn. (3.9). Even though this example only illustrates five
sub-polygons, the process is completely general to more complicated mesh elements.

3.3.3

Boundary Conditions

BCs are essential for any numerical simulation. In terms of DVSI, BCs are applied to primary
and dual edges (as well as primary nodes in subsequent sections) which reside on the boundaries
of the mesh. Each boundary dual edge connects the mid-point of its associated primary edge to
the circumcenters of the boundary primary triangles associated with the same primary edge — see
Figure 3.4 for an illustration. These relationships hold true for triangles which do not physically
contain their circumcenters.
To properly analyze devices, BCs must be introduced to emulate sources and to truncate the
computational domain. The former is accomplished by applying voltages across metal contacts
in an electronic device. Before advancing to voltage BCs, examples are presented where EM
fields will be excited by electric current densities to facilitate quantitative analysis of DVSI against
52

canonical examples.

3.3.3.1

Perfect Electrical Conductor

One important BC for CEM and electronic device simulations is the perfect electrical conductor
(PEC). A DVSI implementation of PEC edges is quite straightforward. Rotational electric field
edges which lie on PEC boundaries are always tangential to the surface. Because the physics states
that n̂×~E = 0 and n̂· ~H = 0, the electric field edges tangential to and magnetic field edges normal to
PEC boundaries are simply removed from the DVSI solution set. Figure 3.4 illustrates the removal
of DVSI edges at PEC interfaces.

Figure 3.4: Illustrations of a primary edge on two different boundaries.

3.3.3.2

Impedance Boundary Condition

An impedance BC, otherwise known as a plane wave or first-order absorbing BC (ABC), exploits
the relationship between TEM electric and magnetic fields or plane waves, given by
~H = 1 k̂ × ~E.
Z
Here, Z =

p

(3.10)

Âľ/Îľ is the impedance of the matched exterior region. Utilizing this relationship

between electric and magnetic fields requires knowledge of the wave vector. A simple resolution
of this issue is assuming that the wave vector is in the direction of the boundary’s normal k̂ ≈ n̂.
This proves to be a convenient simplification as n̂ × p̂i , where p̂i is the unit vector of the primary
edge, is exactly the unit direction of in-plane dual magnetic field edge. The fully discrete DVSI

53

equations with the impedance BC are




D Ail


+ ÎąI1

ÎąF jl

Eln+1
n+1
Hm



ιAim  



F jm
D
 



N(Îą)
A
n
Ai 
 −D il + β I1 β Aim   El   Ji
+
+

 = 0. (3.11)
F
n
β F jl
−D jm
Hm
0



The impedance BC tensor takes the form I1 = −d̂ f · n̂ f × p̂i L f δil δlb /Z, where the in-plane dual
edge with length L f and unit vector d̂ f is associated with the boundary primary edge with the unit
vector p̂i . There are always two boundary primary triangles (with different normals n̂ f ) associated
with a boundary primary edge. This implies that there are two in-plane dual edges associated with
the ith primary edge and is why a separate index f is applied to these edges. The first Kronecker
delta ensures that the impedance boundary affects diagonal matrix elements and the second ensures
that only primary electric field edges associated with the impedance boundary (with index b) are
updated. The term β = (1 − α) is defined for brevity.
This BC serves two important functions in a device simulator. The first is a truncation to any
open regions, i.e., the end of the physical device. The other is a model of an external load. This
is a critical feature when dealing with devices that interface with other circuit components. In
the device situation, impedance BCs are far more important than sophisticated methods of making
perfectly matched layers or radiation BCs.

3.3.4

PEC Cavities with Ideal Dielectrics

To demonstrate the validity of this numerical framework, two PEC cavity results are presented and
discussed. All numerical examples in this section solve the purely rotational fields of Maxwell’s
equations. To excite fields in the cavities, a current density is introduced on primary edges.
These examples test all the aforementioned numerical aspects of the rotational solver except the
impedance BC. This BC will be tested in Section 3.5.

54

The mathematical form of the current density used for these numerical examples is a modulated
Gaussian, or
"

#
(t − t0 )2
Ji (t) = cos [ω0 (t − t0 )] exp −
.
2σ 2

(3.12)

Here, ω0 = 2π f0 is the center angular frequency of the Gaussian, σ = 2.335/(2πBW ) is Gaussian
width associated with the bandwidth BW , and t0 = 6σ is the Gaussian delay. For each subsequent
result, the time step was calculated with ∆t = 1/ (S fmax ), where fmax is the highest frequency in
the simulation and S is a unit-less scale factor used to control the size of the time step. For each of
the following benchmarks, the time step scale was chosen as S = 400.
The first result is a cubic PEC cavity filled 40% with air and 60% with bulk semiconductor
(Îľr = 5.9, which represents bulk GaN). The dielectric discontinuity lies in the z = 0.4 m plane.
Figure 3.5 (left) displays the computed resonant frequencies of the half filled cavity compared

Figure 3.5: Results for two PEC cavities with ideal dielectrics.
with the analytic resonant frequencies [71] (dotted lines). Two different meshes were used to
compute the DVSI results A and B. Mesh A is comprised of 6000 tetrahedra resulting in 7830
DOF and mesh B contains 20,250 tetrahedra resulting in 27,720 DOF. Local maxima of both mesh
results were calculated to quantitatively assess the numerical method. Then, the error between the

55

computed resonant frequencies fn,i and the analytic frequencies fa,i was calculated with
r

2
∑ fn,i − fa,i
i
r
ε = 100 ×
.
2
∑ fa,i

(3.13)

i

The global errors for the first eight resonant frequencies computed with meshes A and B were
Îľ = 1.36% and Îľ = 0.59%, respectively. This demonstrates the increasing accuracy of the higher
resonant frequencies with mesh density, as expected.
To test the framework’s ability to conform to curved features, the second result is a section of
a cylindrical cavity illustrated in the inset of Figure 3.5 (right). The inner diameter is d1 = 0.2 m
and the outer diameter is d2 = 0.6 m. The entire cavity is air filled. Figure 3.5 (right) shows the
numerically computed resonant frequencies in comparison with analytic solutions [71]. The error
between the analytic and the numerical resonant frequencies was again computed and global error
for the first five frequencies of the cylindrical sector cavity was Îľ = 0.68%. The fourth computed
resonant frequency displayed the worst individual error of Îľ = 1.25%. This is attributed to the
mesh density in the cylindrical directions of the geometry, which was not uniform in the ρ, φ , and
z directions.

3.4

Poisson’s Equation

The next step in developing a DVSI full-wave EM framework is to discretize Poisson’s equation. This section is solely devoted to solving irrotational problems where only the electric potential
is the solution variable.

3.4.1

DVSI Discretization

The Delaunay-Voronoi mesh provides a convenient physical representation of Poisson’s equation
−

ZZ

n̂ · ε∇Φ (~r,t) dS =

ZZZ
V

S

56

ρ (~r,t) dV.

(3.14)

As discussed in Section 3.2, a bounded domain is tessellated with two interweaving meshes. Each
primary node, which is a point of the Delaunay mesh, is bounded by a dual polyhedron. An
example is shown in Figure 3.6.

Figure 3.6: A visualization of the DVSI representation of Poisson’s equation in three dimensions.
The key to discretizing Poisson’s equation within DVSI is the fact that all primary edges connected to a primary node are always normal (or perpendicular) to the dual polygons of the bounding
dual polyhedron. This is analogous to the divergence of the electric field out of the enclosing dual
polyhedron. The gradient is discretized as a finite difference on the qth primary edge associated
with node k. Using these facts, Poisson’s equation is written as
Pkp ÎŚnp


Akq  n
n
≡ ∑ εkq
Φk − Φq = ρknVk .
Lkq
q

(3.15)

Here, the summation index q runs over all primary edges attached to primary node k. The same
time notation is used as the previous section, i.e., ÎŚnk = ÎŚ (~rk ,tn ) is the previous time step solution.
Eqn. (3.15) states that electric scalar potentials are assigned to the nodes of a primary edge and the
spatially constant electric field projection is defined as a finite difference of these potentials over
57

the primary edge with length Lkq . These potentials are driven by the charge density ρk defined
inside the kth dual polyhedron with volume Vk .

3.4.2

Boundary Conditions for Poisson’s Equation

As with Maxwell’s rotational equations, Poisson’s equation cannot be solved without imposing
BCs. Truncation of the computational domain is handled through Ampere’s law and therefore only
Dirichlet BCs will be discussed. This boundary will be used extensively to create a meaningful
voltage source to drive electronic devices.
A Dirichlet BC specifies the values that the solution must take. In the case of Poisson’s equation, these solutions are the electric scalar potentials. For all Dirichlet nodes listed with index b
associated with the kth primary node, Poisson’s equation will take the form
A
Pkp Φnp = ρknVk + ∑ εkb kb Φnb .
Lkb
b

3.4.3

(3.16)

Spherical Region with Dirichlet Surfaces

To demonstrate the validity of the DVSI Poisson solver while also displaying the usefulness of
DVSI’s unstructured grid, a spherical region with two Dirichlet surfaces at r = 0.25 m and r = 2 m
is solved. A simple Gaussian charge model ρ (~r) = exp (−~r ·~r) is applied in the spherical region.
This allows the DVSI solver results to be compared to an analytic solution, derived by integrating
the Laplacian in spherical coordinates with the Gaussian source term and a Dirichlet voltage of
0.25 V applied on both the inner and outer surfaces.
Figure 3.7 displays a slice of the DVSI mesh (left) and a cross section of the DVSI scalar
potential solution (right). Mesh refinement near the center of the sphere was performed. The error
was computed between the DVSI numerical solution and the analytic solution using Eqn. (3.13).
Now, fn,i is replaced with ÎŚn,i , the numerical potential at node ~ri , and fa,i is replaced with ÎŚa,i
which is the analytic solution at the same location. For the mesh shown in Figure 3.7 (left), the error

58

Figure 3.7: (left) A mesh of the spherical region with Dirichlet surfaces at r = 0.25 m and r = 2 m.
(right) The numerical solution Φ(r) of Poisson’s equation with a simple Gaussian charge density
and a Dirichlet potential of 0.25 V applied at both surfaces.
was Îľ = 2.5%, showing that the numerical solution, which resides on a completely unstructured
mesh, agrees well with the analytic solution.

3.5

Complete Electromagnetics for Device Simulation

With the derivations and examples presented in Sections 3.3 and 3.4, a full-wave EM DVSI
framework can now be presented. Here, the electric field, magnetic field, and the electric potential
are solution variables that are solved simultaneously. Eqns. (3.1) – (3.3) completely describe EM
phenomena in an electronic device.

3.5.1

DVSI Discretization

Proper simulation of electronic devices with full-wave effects requires solving the system of equations discussed in Sections 3.3 and 3.4 and shown in Eqns. (3.1) – (3.3). For the rest of this section,
the rotational electric field will drop the “rot” subscripts and all electric field variables are assumed
to be purely rotational. The fully discrete system of equations is

59



DilA



A
Pip

ÎąAim


F jm
 ιF
0

jl D

0
0
ÎąPkp








Eln+1
n+1
Hm
ÎŚn+1
p








−D Ail

β Aim



+
 β F jl

0

−D F
jm
0

A
−P ip



Eln



  Hn
0
 m

β Pkp
ÎŚnp





 
 
+
 
 

N(Îą)
Ji
Ai

0
N(Îą)

−ρk

Vk




 = 0. (3.17)



A couples
Eqn. (3.17) is a direct descendant of Eqns. (3.8) and (3.15). The additional tensor Pip

the rotational electric and magnetic fields to the static electric potential. This tensor takes the form


A Φn ≡ εi Ai Φn − Φn , where the ith primary edge points from node 1 with potential Φ
Pip
i,1
p
i,1
i,2
∆tL
i

to node 2 with potential ÎŚi,2 .

3.5.2

Device Boundary Conditions

BCs for the complete EM equations will now be presented utilizing the information described in
the previous two sections. Each BC previously described is now presented in the context of an
electronic device simulation. Additional requirements needed for relevance to device simulation
are also highlighted. A numerical example of a transmission line that requires these BCs for
excitation and propagation is compared with analytic solutions.

3.5.2.1

Metal Contacts

Metal contacts are critically important BCs for electron devices and are both the regions where
voltages (or currents) are applied to bias and drive the device as well as the regions that interact
with external circuitry (e.g., driving loads). These contacts, in terms of a physical device, are
certainly not perfect electrical conductors, but are sufficiently conducting to be treated as such in
the majority of technologically relevant cases. Extensions to non-ideal contacts are an interesting
research topic when the distributed effects of finite conductivity are important. For the purposes of

60

developing the numerical methods of DVSI and to allow comparison with analytic solutions, only
the ideal cases when metals are PECs are considered.
The PEC BC was previously discussed in Section 3.3 and the only additional information
needed is the definition of electric potentials. Because a PEC defines an equipotential surface, a
Dirichlet BC is assigned to any potentials residing on this boundary. Two PEC surfaces can then
be used to define a voltage, where one conductor is the reference potential which could be ground
(or other reference potential offset).

3.5.2.2

Extrinsic Impedance Boundary Condition

The impedance BC is important for device simulation for several reasons. First, it can properly
emulate resistive sheets which can be used to represent external loads. Second, the impedance BC
can be used as a first-order domain truncation. An important point of the impedance BC is that it
affects the total electric field. The discretized linear system including the impedance BC is







DilA



Eln+1
n+1
Hm



+ ιI1 ιAim Pip + ιI2  






F
ÎąF jl
D jm
0




0
0
ÎąPkp
ÎŚn+1
p



A
n
 −Dil + β I1 β Aim −Pip + β I2   El 



  Hn 
F
+
β
F
−D
0

 m 
jl
jm



0
0
β Pkp
ÎŚnp



N(Îą)
n+1 − β Φn
J
A
+
I
ιΌ
i
2
i
b
b


+
0




N(Îą)
n
−ρk
Vk + Pkp δ pb ιΌn+1
−
β
ÎŚ
b
b




 = 0. (3.18)





Again, the impedance BC tensor takes the form I1 = −d̂ f · n̂ f × p̂i L f δil δlb /Z and the second


impedance tensor is defined as I2 = −d̂ f · n̂ f × p̂i L f δil δlb / (Li Z), where Li is the length of
the ith primary edge. The potentials in the source vector, with index b, account for any primary

61

nodes which lie on corners of the computational domain which connect Dirichlet boundaries and
impedance boundaries directly.

3.5.2.3

Voltage Boundary Condition

One critical BC that may be difficult to implement without incorporating Poisson’s equation is the
voltage source, a staple of device engineering. This excitation boundary allows the application
of voltages across different device contacts, such as the gate and drain of a field effect transistor
(FET). It also allows a simulator to operate in conditions that replicate those in a physical device.
The voltage boundary can be any shape, but for simplicity it is assumed that the voltage boundary is a rectangular region connecting two PECs. The two PEC contacts are assigned constant
potentials and the overlap between the contacts and the voltage excitation serves as Dirichlet line
segments for the solution of Poisson’s equation.
Coupling the voltage boundary into the rotational electric and magnetic fields is now considered. To do so, the voltage boundary is first declared as free of rotational electric fields. Dual
magnetic fields in the voltage plane are then declared as unknowns and they are solved for using
Ampere’s law, Eqn. (3.1). Because dual magnetic field edges in the voltage plane correspond to
exactly one rotational electric field edge, Ampere’s law for the null rotational electric field edge is
applied to solve for the in-plane dual magnetic field edge. This is mathematically written, with the
Backward Euler time integration scheme, as


n+1 + P A Φn+1 − Φn + J n+1 A = 0.
Aim Hm
i
p
p
ip
i

(3.19)

In this version of Ampere’s law for the voltage boundary, the line integrals of magnetic fields
include the dual magnetic field edges in the voltage boundary plane. This is a subtle, but highly
practical issue for device simulation.
With the use of the Poisson Dirichlet BC and the voltage boundary Ampere’s law for the in-

62

plane magnetic fields, the discrete linear system with the voltage excitation is written as


DilA

A
Pip



Eln+1
n+1
Hm



ÎąAim






 ιF


F
D
0



jl
jm



0
0
ÎąPkp
ÎŚn+1
p


A
A
n
 −Dil β Aim −Pip   El


  Hn
F
+
β
F
−D
0

 m
jl
jm


0
0
β Pkp
ÎŚnp





 
 
+
 
 

N(Îą)
A
Ji
Ai + Pib



n
ÎŚn+1
−
ÎŚ
b
b

0

N(Îą)
n
−ρk
Vk + Pkb ιΌn+1
b − β Φb




 = 0.

 
(3.20)

The electric potentials added to the source vector reflect any Dirichlet voltages.

3.5.2.4

Two-Port Devices

In experiments and circuits, transistors are driven by time dependent voltages applied across contacts. For example, given a FET with source, gate, and drain metal contacts, the device operates
with a voltage bias applied from the drain to the source and an oscillating voltage applied across
the gate and the source. In terms of the complete DVSI equations, these form the voltage BCs.
The applied voltage v1 (t) and the simulated current i1 (t) are related to the port voltages in Fig +
−
− 
ure 3.8 by v1 (t) = v+
1 (t) + v1 (t) and i1 (t) = v1 (t) − v1 (t) /Z1 [62]. It is assumed that the input
and output ports are 50 Ω transmission lines. Given the two-port device network in Figure 3.8,

Figure 3.8: The two-port model of an electronic device. Input voltages are labeled with a subscript
1 and output voltages are labeled with a subscript 2. The input and output ports are ideal 50 Ω
transmission lines.
inward and outward propagating waves are defined. These waves can then be used to characterize

63

the device with S-parameters, calculated with
vÂą
n (t) =

1
[vn (t) Âą Zn in (t)] ,
2

(3.21)

V − (ω)
,
S11 (ω) = 1+
V1 (ω)

(3.22)

V − (ω)
S21 (ω) = 2+
.
V1 (ω)

(3.23)

Here, the spectral voltages Vn± (ω) are the Fourier transforms of the transient voltages v±
n (t). The
output current i2 (t) is the transient current across the load ZL . Customarily, the load Z2 represents
the characteristic impedance of a high frequency probe impedance matched to a network analyzer
so that the voltage v+
2 (t) = 0. For demonstrative purposes, the load and output impedances are
chosen to be purely resistive.
Before the transient voltages and currents are Fourier transformed, they must be computed in
the simulation. Voltage drops are computed with a contiguous set of primary edges between metal
contacts. An integral of the total electric field starting from the reference contact yields the proper
voltage drop. Similarly, current is computed by integrating a contiguous, straight line of dual edges
on which magnetic fields are defined. The line integral of the magnetic fields is parallel to the metal
contacts and in the direction which corresponds to outward propagating power.

3.5.3

Lossless Transmission Line

To demonstrate the validity of the complete set of EM equations, a simple lossless transmission line
is analyzed. This is a good candidate as it will require a transient voltage excitation, will test the
impedance BC, and will be quantitatively validated via analytic S-parameters. The computational
domain is bounded by two PEC plates located at the z = 0 and z = 0.1 mm planes, a voltage
boundary at the x = 0 plane, a ZL = 50 Ω impedance boundary at the x = 1 mm plane, and two
“open” boundaries at the y = 0 and y = 0.1 mm planes. These open boundaries declare in-plane
magnetic fields to be null. The DVSI transmission line is excited with a Gaussian voltage drop


(t−t0 )2
are
between the PEC plates. The parameters of the Gaussian voltage v(t) = V0 exp −
2
2σ

64

V0 = 0.5 V, σ = 2.5 × 10−14 s, and t0 = 10σ . The time step is chosen as ∆t = 0.1σ and the
number of time steps is 100,000.

Figure 3.9: (left) The transient input current (top) and output voltage (bottom) calculated in the
lossless transmission line. (right) The spectral reflection and transmission coefficients.
Figure 3.9 shows (left) the transient input current and output voltage and (right) the magnitude
of the reflection and transmission coefficients |S11 (ω)| and |S21 (ω)|, respectively. The solid lines
are the analytic values [63]. Because this transmission line is lossless, the relation |S11 (ω)|2 +
|S21 (ω)|2 = 1 is true for all frequencies. A good quantitative analysis of the full-wave transmission
line simulation is the computed deviation of this relation from unity. Using the data presented
in Figure 3.9 (right) and a modified version of Eqn. (3.13), the total deviation from unity was
Îľ = 3.7%. This result is a worst case scenario considering the fact that it was computed up to
600 GHz — twice the frequency corresponding to the electrical size of the transmission line.
Electronic devices, which are the target application of this numerical framework, rarely operate at
wavelengths below the electrical size of its gate width.
We now simulate the lossless transmission line on a series of meshes to estimate the DVSI
method’s order of convergence. Each mesh refinement decreases the edge length in the propagation
direction. The number of tetrahedra range from 480 to 9600 and the total number of DOF ranges
from 854 to 15,864. The DOF reflect the number of primary edges which define electric fields, the
number of dual edges which define magnetic fields, and the number of primary nodes which define

65

electric potentials. The lossless transmission line operates with the same inputs as the previous
example except with a modified Gaussian bandwidth of σ = 10−12 s. The solution computed
on each mesh was compared to the finest mesh solution to produce an error given by a modified
version of Eqn. (3.13). These errors were then used to compute an error bound given by
−p

|Sk − L| < CNk ,

(3.24)

where Sk are the k solutions computed on each mesh, L is the solution computed on the finest
mesh, Nk is the number of edges in the propagation direction for each mesh, and p is the order
of convergence. The computed order of convergence was p ≈ 2.0. As expected, this result is
consistent with the order of convergence of the traditional Yee cell algorithm [70].

3.6

Conclusion

DVSI is an appropriate numerical framework for discretizing full-wave EM and coupling to
charge transport. With the approximations outlined in this chapter, this numerical method is
amenable to nonlinear charge transport on unstructured grids. The complete discretization of
Maxwell’s equations within DVSI was presented as well as the proper BCs for device simulation.
Several benchmarks were presented to demonstrate the rotational and irrotational field descriptions independently and simultaneously. DVSI agreed well with analytic solutions for a number of
numerical tests, including propagation in a transmission line using a voltage excitation. The importance of having a Delaunay-compliant mesh was emphasized as a requirement of the method in
general. Although the fields are approximated to be constant vector projections on mesh edges, this
is an appropriate level of discretization for electronic devices where the physical dimensions are
substantially sub-wavelength. Finally, the framework allows for the solution of charge transport
on the same mesh as the EM problem which is not typical of device solvers.

66

CHAPTER 4
FERMI KINETICS TRANSPORT

4.1

Introduction

Charge transport in semiconductors can be determined by the solution of the semi-classical
Boltzmann transport equation (BTE) [72]
~F
∂ f (~k,~r,t)
+~v · ∇r f (~k,~r,t) + · ∇k f (~k,~r,t) =
∂t
h̄

∂ f (~k,~r,t)
∂t

!
.

(4.1)

coll

The BTE is a seven-dimensional integro-differential equation and describes how a collection of
particles responds to electromagnetic fields and scattering potentials. The distribution function
f (~k,~r,t) gives the occupation probability at the real-space location~r and reciprocal-space location
~k at time t. It can be found by solving Eqn. (4.1). Clearly, approximations are required in order to
yield tractable solutions.
A powerful statistical method for solving the BTE is the MC technique [73]. MC was first
presented as a statistical approach to solving general integro-differential equations [74]. Since
then, it has seen a wide variety of applications and is one of the preferred methods for accurate
numerical simulation of semiconductor devices. Many varieties exist in the literature including the
ensemble MC (EMC) method [75]. However, the EMC’s stochastic algorithm is very computationally expensive. The cellular MC (CMC) implementation of the EMC method was developed
to reduce the burdensome computational demands [76]. Despite scattering rate pre-computation
and tabulation, CMC is still computationally intensive and can lead to long simulation times for
large, complex devices. Combining MC charge transport with full-wave EM using, for example,
the FDTD method [66], further increases the computational demands.
An alternative approach to solving the BTE is through deterministic methods. These rely on
taking moments of the BTE. One of the first deterministic Boltzmann solvers for semiconductor
device simulations was presented by Scharfetter and Gummel [77]. This work presented physi-

67

cal approximations to the phenomenological drift-diffusion (DD) model of charge transport. The
physical size of the device technology was still relatively large and the numerical solution of the
DD model produced satisfactory results. However, reduced device features produce larger electric
fields and hot-electron effects, such as electron velocity saturation, rendering the DD approximation insufficiently accurate.
Two seminal works which extended the DD model to include carrier heating were presented by
Stratton [78] and Blotekjaer [79]. These methods utilized different approximations of the BTE.
Stratton proposed that the distribution function be split into even and odd components while
Blotekjaer derived moments of the BTE without any assumptions on the form of the distribution function. Both methods required closure relations to generate a linearly independent system
of equations [80]. A common closure relation is the prescription of electronic heat flow with
Fourier’s law [80]. Another common approximation used to specify momentum and energy in hydrodynamic models is parabolic band-structure [80]. However, this approximation will reduce the
accuracy of the device solver when the electric field is large. To the best of the author’s knowledge,
no hydrodynamic or energy-transport model can include complete electronic band-structure. Some
device solvers employ fitting parameters such as field- and temperature-dependent mobilities as an
attempt to overcome the limitations of the parabolic band-structure [81].
FKT is a deterministic Boltzmann solver which has shown promising results in the literature.
Historically, FKT was conceived by seeking an alternative method for calculating the electronic
heat flow. Rather than use the combination of Fourier’s law and an approximate thermal conductivity [80], it was proposed that a thermodynamic identity could be used as a closure relation
for energy-transport models [82]. This closure relation provided a robust Boltzmann solver and
it was later shown that FKT was completely amenable to electronic band-structure [83] and fullwave EM [62]. In particular, a version of FKT with GaAs band-structure, quantum scattering
formalisms, and hot-electron effects was shown to reproduce the electron velocity saturation calculated by sophisticated MC methods [83] in a fraction of the computational time. FKT was also
shown to accurately simulate a GaAs MESFET and GaN HEMT from DC up through mm-wave

68

frequencies without adjustable calibration parameters [83, 84].
In the rest of this chapter, a detailed discussion of the FKT device simulation framework is
presented. In Section 4.1.1, a detailed derivation of the governing equations of charge transport is
presented. Energy- and real-space discretization techniques are presented in Sections 4.2 and 4.3,
respectively. Finally, Section 4.4 concludes this chapter.

4.1.1

Moments of the BTE

Following the method of Stratton [78], the collision term of the BTE is replaced with a momentum
relaxation time approximation. The distribution function is split into even (equilibrium) f0 (~k,~r,t)
and odd f1 (~k,~r,t) functions in ~k-space. This yields


~E
~
∂
f
(
k,~
r,t)
∂ f (~k,~r,t)

0
~
~
+~v · ∇r f (k,~r,t) − q · ∇k f (k,~r,t) =


∂t
h̄
∂t

coll

f (~k,~r,t)
− 1
.
τk

(4.2)

Here, the real-space external force ~F is only due to an electric field, i.e., ~F = −q~E. Eqn. (4.2)
is solved with the method of moments. Electron density and electron kinetic energy density are
defined as
1
n = 3 d~k f0 (~k),
4π
Z
1
En = 3 d~kE(~k) f0 (~k),
4π
Z

(4.3)
(4.4)

and the electron flux density and electron kinetic energy flux density are defined as
~Jn = 1
4π 3
~Kn = 1
4π 3

Z

Z

d~k~v f1 (~k),

(4.5)

d~kE(~k)~v f1 (~k).

(4.6)

The symmetric (equilibrium) distribution function is the well-known Fermi-Dirac distribution
1

f0 (~k) =


exp

E(~k)−F
kB T

,



(4.7)

+1

which represents an electron gas with Fermi level F and temperature T . The term kB is Boltzmann’s constant. The non-equilibrium distribution function will be approximated with piece-wise
69

equilibrium distribution functions. Moments of the BTE will provide the non-equilibrium nature
of the electron gases.
Solving the BTE with the method of moments amounts to integrating Eqn. (4.2) over all
reciprocal-space. Using a general ~k-space dependent operator O(~k), the BTE can be separated
into symmetric (even) and antisymmetric (odd) equations, or
"
 #
Z∞
~
∂
f
E
1
∂
f
0
0 
~kO(~k)
d
+~
v
¡
∇
f
−
q
= 0,
¡
∇
f
−
r
1
∂t
∂t coll
h̄ k 1
4π 3

(4.8)

−∞

1
4π 3

Z∞
−∞

"

#
~
f
∂
f
E
d~kO(~k) 1 + 1 +~v · ∇r f0 − q · ∇k f0 = 0.
τk
∂t
h̄

(4.9)

The multiplication of an even function with an odd function results in an odd function. Furthermore, the derivatives of even and odd functions (with respect to the dimension in which it is even
or odd) result in odd and even functions, respectively. When an odd operator is used for a moment of the BTE, the LHS of Eqn. (4.8) is exactly zero. The LHS of Eqn. (4.9) is exactly zero
when an even operator is used for a moment of the BTE. Taking moments of the BTE with the
operators {1,~v, E(~k), E(~k)~v} yields the electron continuity equation, particle flux density, energy
conservation equation, and the kinetic energy flux density
∂n
+ ∇ ·~Jn +Cn = 0
∂t
Z


~Jn = 1
~k~vτk q~E · ∇k f0 −~v · ∇r f0 ,
d
h̄
4π 3
∂ En
+ ∇ ·~Sn + q~E ·~Jn +CE = 0,
∂t
Z
q

~Kn = 1
~
~
~
d kE(k)~vτk E · ∇k f0 −~v · ∇r f0 .
h̄
4π 3

(4.10)
(4.11)
(4.12)
(4.13)
~

~

Jn
Kn
Appendix A.1 gives an in-depth derivation of Eqns. (4.10) – (4.13). The terms τk ∂∂t
and τk ∂∂t

can be ignored for devices operating up to the mm-wavelength regime as τk  ∂t. For sub-mmwavelength devices, i.e., terahertz devices, these terms must be considered.

70

4.1.2

An Alternative Treatment of Heat Flow

Eqns. (4.10) – (4.13), along with the definitions (4.3) – (4.7), are not a closed system of equations.
The third moment of the BTE yields the kinetic energy flux density ~Kn . But, the total energy flux
density ~Sn = ~Kn + ~Hn is a combination of the kinetic energy flux density and the heat flux density
or heat flow. Therefore, a prescription of the heat flow is required.
A common approximation of the electronic heat flow used in many other deterministic Boltzmann solvers is Fourier’s law [80]
~Hn = −κ(T )∇T,

(4.14)

where T is the temperature of the distribution of electrons and Îş(T ) is a fictitious electronic thermal conductivity. The thermal conductivity is typically approximated by the Wiedemann-Franz
law [80]

Îş(T ) =

5
−p
2




kB 2
qÂľnT.
q

(4.15)

Here, p is a tune-able correction factor.
In [80], it is reported that numerical issues in the device solver could be attributed to Fourier’s
law. It also reports that incorporating higher moments of the BTE may help alleviate some numerical issues. This thesis will focus on a different closure relation. In the seminal paper by Grupen
[82], an alternative treatment of heat flow is presented. An analogy of the thermodynamic identity
kB T ∆σ = ∆En − F∆n,

(4.16)

is used to calculate heat flow between Fermi distributions. This treatment of heat flow produces
an extremely stable numerical framework. In subsequent sections which describe the spatial discretization of FKT, the second law of thermodynamics will be used to calculate the discrete heat
flux density. It will be shown that the thermodynamic heat flow is easily obtained from the discrete
particle and kinetic energy flux densities.

71

4.1.3

Self Heating Effects

Transistors used in PA circuits are heavily biased. Self heating effects are therefore critical for
accurately simulating the devices. The final governing equation in the global FKT system is the
lattice energy conservation equation [84]
ρC p

∂ TL
+ ∇ · κ∇TL −CE = 0,
∂t

(4.17)

where TL is the temperature of the lattice and ρ, C p , κ are the mass density, specific heat, and lattice
thermal conductivity of the semiconductor crystal, respectively. The energy collision operator CE
is the source term of the lattice heating equation.

4.2

FKT Energy-Space Discretization

The first step in solving Eqns. (4.10) – (4.13) is the discretization of reciprocal-space. To
avoid discretizing three-dimensions, reciprocal-space is converted to energy-space. Section 4.2.1
provides these details. The calculation of isosurface integrals is then presented in Section 4.2.2.
Finally, the approximation and calculation of the piece-wise energy windows are presented in
Sections 4.2.3 and 4.2.4. These steps are pre-computations in terms of the FKT device simulator.
The resulting data corresponding to specific bulk materials are imported into the device simulator.

4.2.1

Converting Reciprocal-Space Integration to Energy-Space Integration

The densities, fluxes, and collision operators were presented in Section 4.1.1. Re-writing these
equations with band-structure dependencies yields
n=

1
4π 3

Z

d~k f0 (E(~k)),

1
En = 3 d~kE(~k) f0 (E(~k)),
4π
Z


~Jn = 1
~k~vτk q~E · ∇k f0 (E(~k)) −~v · ∇r f0 (E(~k)) ,
d
h̄
4π 3
Z


~Kn = 1
~kE(~k)~vτk q~E · ∇k f0 (E(~k)) −~v · ∇r f0 (E(~k)) .
d
h̄
4π 3

(4.18)

Z

72

(4.19)
(4.20)
(4.21)

Cn

=

1

Z Z

4π 3

0
d~k d~kW~ ~ 0


2

n

k,k



0
~
~
nq + 1 f0,i (E(k)) 1 − f f ,0 (E(k ))



~k ~k
i f


o
0
~
~
−nq f f ,0 (E(k )) 1 − fi,0 (E(k)) ,

CE

=

1
4π 3

Z Z


2

0
d~k d~kW~ ~ 0 E(~k)
k,k

n

(4.22)





0
nq + 1 f0,i (E(~k)) 1 − f f ,0 (E(~k ))

~k ~k
i f


o
0
−nq f f ,0 (E(~k )) 1 − fi,0 (E(~k)) .

(4.23)

Here, W~ ~ 0 is a scattering probability between two ~k-space states computed from Fermi’s golden
k,k

rule and nq = 1/ (exp (h̄ω/kB TL ) − 1) is the phonon occupation number with energy h̄ω [84]. The
variable TL is the lattice temperature. The initial electron state has an occupation probability fi,0
and the final state has an occupation probability f f ,0 . Using the sifting property of the delta funcdf

df

tion, applying the relation ∇k f0 (E) = dE0 ∇k E = dE0 h̄~v, and using the delta function composition
property, the densities, fluxes, and collision operators are
Z

n=

D(E) f0 (E)dE,

(4.24)

D(E) f0 (E)EdE,

(4.25)



d
f
(E)
0
¾ (E) ¡ q~E
− ∇r f0 (E) dE,
dE

(4.26)



d
f
(E)
0
− ∇r f0 (E) EdE,
¾ (E) ¡ q~E
dE

(4.27)

Z

En =
E

~Jn =

Z

~Kn =

Z

E

E

Cn

Z

=

G(E)








	
nq + 1 f0,i (E) 1 − f f ,0 (E − h̄ω) − nq f f ,0 (E − h̄ω) 1 − fi,0 (E) dE, (4.28)

E

CE

Z

=

G(E)








	
nq + 1 f0,i (E) 1 − f f ,0 (E − h̄ω) − nq f f ,0 (E − h̄ω) 1 − fi,0 (E) EdE,

E

(4.29)

73

where




d~Si
,

~
|∇k E(k)| ~k=~k
i

Z
~v(~k)~vT (~k)τk (~k) 
1
.
µ (E) = 3 ∑ d~Si

4π i
|∇k E(~k)| ~k=~k
i


Z Z


1
1
0


G(E) =
,
d~S d~SW~ ~ 0


k,k |∇ E(~k)| 
~ 
k
~k=~k |∇k E(k)| ~k=~k
1
D(E) = 3 ∑
4π i

1

Z

~k ~k
i f

i

(4.30)

(4.31)

(4.32)

f

Here, Âľ (E) is a tensor. Appendix A.2 provides an in-depth derivation of the reciprocal-space to
energy-space conversion.

4.2.2

Isosurface Integrals

Eqns. (4.30) – (4.32) require integration over surfaces of constant electron eigen-energy, i.e., calculation of isosurface integrals. The calculation of bulk GaAs isosurface integrals for the FKT
device simulator was first presented in [83]. It was later applied to bulk GaN in [84]. This work
will present the process of determining surfaces of constant energy and the resulting isosurface
integrations following the seminal work of Grupen.
Reciprocal space must be discretized to determine the surfaces of constant electron energy and
calculate the isosurface integrals. To this end, a mesh of the irreducible edge of the Brillouin zone
(IBZ) is generated and the band-structure is calculated on the nodes of the mesh. A k-space mesh
of the wurtzite IBZ is presented in Figure 4.1a. Also illustrated is the electron energy calculated
with the empirical pseudopotential method (EPM) [85]. Surfaces of constant energy are illustrated
in Figure 4.1b [84]. An outline of the wurtzite IBZ is included with the energy isosurfaces. Details
for calculating the isosurfaces of electronic band-structure are provided in [83].
The isosurfaces are divided into zones according to valleys of the GaN band-structure. An
example of the first Γ-valley zone is presented in the panel diagram in Figure 4.2 (left). This
zone is shaded in red. The DOS isosurface integral calculation, Eqn. (4.30), is also presented in
Figure 4.2 (right) for the same zone. DOS data was obtained from Matt Grupen AFRL [84]. This

74

(b)
(a)

Figure 4.1: (a) A mesh of the GaN IBZ and corresponding electron energy calculated with EPM
and (b) the resulting surfaces of constant energy.
10

Energy (eV)

Energy (eV)

8
6
4
2
0
-2

6.5
6
5.5
5
4.5
4
3.5
3
0 1 2 3 4 5 6

K

Γ

A

L

H

D(E) (x1021 cm-3 eV-1)

Figure 4.2: The (left) panel diagram of wurtzite GaN and (right) DOS calculated in the first Γvalley. The EPM is used to calculate the electron energy. DOS data was obtained from Matt
Grupen AFRL.
process is repeated for the other band-structure valleys. Figure 4.3 presents the DOS integrations
for the first four valleys of wurtzite GaN. The data was obtained from Matt Grupen AFRL [84].
As described in [62] and [84], the isosurface integrals depend on FKT device simulator solution
variables. Isosurface integration data must therefore be sampled over a range of FKT solution

75

D(E) (x 1021 cm-3 eV-1)

10
9
8
7
6
5
4
3
2
1
0

U
M

Γ1
0

0.5

1

Γ3
1.5

2

2.5

3

Energy (eV)
Figure 4.3: The DOS isosurface integrations calculated in the first four wurtzite GaN valleys. The
DOS data was obtained from Matt Grupen AFRL.
variables. This data is interpolated in the device simulator to calculate the quantities required for
residual and Jacobian construction. An example of the FKT solution variable dependence of the
Âľ(E) isosurface integral is presented in Figure 4.4. Three integrations are presented with lattice
temperatures of 300 K, 400 K, and 500 K. This data was obtained from Matt Grupen AFRL [84].
All isosurface integrals which depend on FKT solution variables are sampled in a similar manner.

4.2.3

Piece-Wise Energy Power Laws

The isosurface integration data of the previous section can be incorporated into the FKT device
simulator in various ways. This work will focus on the method of Grupen [83]. Isosurface integration data is approximated with piece-wise energy power laws. Energy power laws allow the use of
fast numerical Fermi-Dirac integral routines in the device simulator.
With the piece-wise energy power law approximation of the isosurface integrals, the densities,

76

Îź(E) (x 1025 cm-1 s-1 eV-1)

1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0

300 K
400 K
500 K

0

0.5

1

1.5

2

2.5

3

Energy (eV)
Figure 4.4: The Âľ(E) isosurface integrations calculated with different lattice temperature. The
Âľ(E) data was obtained from Matt Grupen AFRL.
fluxes, and collision operators are

n≈∑

Eb,k
Z

kE


Îą
k
Ak E − Eρ,k
f0 (E)dE ≡ ∑ nk ,

(4.33)

k

a,k



Eb,k


Z



Îą +1
k
En ≈ ∑ Eρ,k nk +
Ak E − Eρ,k
f0 (E)dE ,



k 

(4.34)

Ea,k

~Jn ≈ ∑

Ed,k
Z

Dk E − EJ,k

kE
c,k


β

k



d f0 (E)
~
qE
− ∇r f0 (E) dE ≡ ∑~Jn,k ,
dE
k



Ed,k




Z



β +1
d
f
(E)
0
~Kn ≈ ∑ EJ,k~Jn,k +
~
Dk E − EJ,k k
qE
− ∇r f0 (E) dE ,


dE

k 
Ec,k

77

(4.35)

(4.36)

Cn ≈ ∑

Eh,k
Z

Gk (E − Es,k )γk







nq + 1 f0,i (E) 1 − f f ,0 (E − h̄ω)

kE
g,k


	
− nq f f ,0 (E − h̄ω) 1 − fi,0 (E) dE ≡ ∑ Ckn ,

(4.37)

k


Eh,k

Z






E
n
C ≈ ∑ Es,kCk +
Gk (E − Es,k )γk +1 nq + 1 f0,i (E) 1 − f f ,0 (E − h̄ω)

k 
Eg,k



	 
− nq f f ,0 (E − h̄ω) 1 − fi,0 (E) dE .



(4.38)

Here, the flux isosurface integral matrix is µ (E) ≈ µ(E)I with~v(~k)~vT (~k) ≈ v(~k)2 I. The matrix I

D(E) (x 1021 cm-3 eV-1)

is the identity matrix. Figure 4.5 illustrates an approximation of the first Γ-valley DOS data. The

9
8
7
6
5
4
3
2
1
0
0

0.5

1

1.5

2

2.5

3

Energy (eV)
Figure 4.5: Approximation of the first Γ-valley DOS data with piece-wise energy power laws. The
solid black line is the isosurface integration data and the dashed red line is the energy power law
fit. Both data sets were obtained from Matt Grupen AFRL.
DOS integration data is plotted with a solid black line and the energy power law fits are plotted
with dashed red lines. Figure 4.6 illustrates the approximation of the first Γ-valley µ(E) data.
78

Îź(E) (x 1025 cm-1 s-1 eV-1)

1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0
0

0.5

1

1.5

2

2.5

3

Energy (eV)
Figure 4.6: Approximation of the first Γ-valley µ(E) data with piece-wise energy power laws. The
solid black line is the isosurface integration data and the dashed red lines are the energy power law
fits. Both data sets were obtained from Matt Grupen AFRL.
Here, the integration data is plotted with a solid black line and the energy power law fits are plotted
with dashed red lines. Because the Âľ(E) isosurface integral depends on solution variables of the
FKT device simulator, energy power laws are required for each data set corresponding to samples
of the solution variables.

4.2.4

Incomplete Fermi Integrals

The final component of the energy-space discretization is the evaluation of the energy integral over
each piece-wise segment. The electron density and particle flux density in a single energy segment
are

Eb,k
Z


Îą
k
Ak E − Eρ,k
f0 (E)dE,

nk =

(4.39)

Ea,k

~Jn,k =

Ed,k
Z



d
f
(E)
0
Dk E − EJ,k k q~E
− ∇r f0 (E) dE,
dE

β

Ec,k

79

(4.40)

Eqns. (4.39) and (4.40) are incomplete Fermi integrals of order ιk and βk . With the substitutions
d f0 (E)
d f (E)
=− 0
,
dE
dF
Îľ=
Ρρ,k =

(4.41)

E − Ec
,
kB T

(4.42)

F − Ec − (Eρ,k − Ec )
,
kB T

(4.43)

F − Ec − (EJ,k − Ec )
,
kB T
Ea,k − Eρ,k
ak =
,
kB T
Eb,k − Eρ,k
,
bk =
kB T
Ec,k − EJ,k
ck =
,
kB T
Ed,k − EJ,k
dk =
,
kB T

ΡJ,k =

(4.44)
(4.45)
(4.46)
(4.47)
(4.48)

the piece-wise densities and fluxes become


nk = Ak (kB T )ιk +1 Fιk Ρρ,k , ak , bk ,

(4.49)



En,k = Eρ,k nk + Ak (kB T )ιk +2 Fι +1 Ρρ,k , ak , bk ,
k
"
#!
βk +1




(k
T
)
B
~Jn,k = −qDk ~E (kB T )βk F 0 ηJ,k , ck , dk + ∇r
Fβ ΡJ,k , ck , dk
,
βk
k
q

~Kn,k = EJ,k~Jn,k − qDk ~E (kB T )βk +1 F 0
βk +1 ΡJ,k , ck , dk
#!
"


(kB T )βk +2
Fβ +1 ΡJ,k , ck , dk
.
+ ∇r
k
q

(4.50)
(4.51)




(4.52)

Here, the incomplete Fermi integral of order Îą and its derivative are
Zb

Fι (Ρ, a, b) =
a

ξι
dÎľ,
exp (ε − η) + 1

80

(4.53)

and
0

Fι (Ρ, a, b) =

dFÎą
aÎą
bÎą
= αFα−1 (η, a, b) +
−
.
dΡ
exp (ε − η) + 1 exp (ε − η) + 1

(4.54)

The series expansion method [86] is preferred for the fast numerical evaluation of the incomplete
Fermi integral of arbitrary order and parameter.
Because of the multiplication of the initial and final occupation probabilities, fi,0 and f f ,0 , the
collision operators, Eqns. (4.37) and (4.38), cannot be represented as Fermi integrals. Therefore,
the numerical routines for fast Fermi integral evaluation are not applicable for calculating the
collision operators. Numerical quadrature of the piece-wise energy windows is a suitable method
for evaluating the collision operator. The calculations of the collision operators are
Cn ≈ ∑ ∑ Gk (Ei − Es,k )γk
k







nq + 1 f0,i (Ei ) 1 − f f ,0 (Ei − h̄ω)

i


	
− nq f f ,0 (Ei − h̄ω) 1 − fi,0 (Ei ) wi ∆Ei ,

(4.55)

(
CE ≈ ∑ Es,kCkn + ∑ Gk (Ei − Es,k )γk +1







nq + 1 f0,i (Ei ) 1 − f f ,0 (Ei − h̄ω)
i
k




	
− nq f f ,0 (Ei − h̄ω) 1 − fi,0 (Ei ) wi ∆Ei ,



(4.56)

where wi is the weight of the quadrature rule. Specific quadrature implementations include fast
trapezoidal rules and Gaussian quadrature rules.

4.3

FKT Real-Space Discretization

After the discretization of energy-space, the next step in solving Eqns. (4.10) – (4.13) is discretizing real-space. The following provides the discretization details following the common practices of other deterministic Boltzmann solvers [87]. In particular, the box integration method [30]
is used for the continuity equations and the Scharfetter-Gummel (SG) discretization technique [77]
is used for the particle fluxes.

81

4.3.1

Continuity Equations

Discretizing the continuity equations requires integration over the semiconductor device domain.
The integral form of the continuity equations on the DV mesh are
ZZZ 
V

ZZZ 
V


ZZ
∂n
n
+C dV +
n̂ ·~Jn dS = 0,
∂t

(4.57)

S



∂ En
+ q~E ¡~Jn +CE dV +
∂t

ZZ

n̂ ·~Sn dS = 0,

(4.58)

S

and

ZZ
∂ TL
E
−C dV +
n̂ · κ∇TL dS = 0.
ρC p
∂t

ZZZ 
V

(4.59)

S

Here, ~E = ~Erot − ∇Φ is the total electric field defined on a primary edge of the DV mesh. Using
the box integration method [30] and backward Euler time discretization, the continuity equations
become
"

#
t
nt+1
−
n
n,t+1
t+1 A = 0,
i
i
+Ci
Vi + ∑ Jn,i
j ij
∆t
j
#
" t+1
t


En,i − En,i
E,t+1
t+1
t+1
+Ci
Vi + q ∑ Et+1
L
j Jn,i j Ai j + ∑ Sn,i j Ai j = 0.
j
∆t
j
j
"
#
t+1 − T t

Îş A
TL,i
ij ij
L,i
E,t+1
t+1
t+1
ρiC p,i
− ∑ Ci
Vi + ∑ TL,i − TL, j
= 0.
∆t
Li j
i
j

(4.60)

(4.61)

(4.62)

These transport equations are coupled in space and time with the discrete full-wave EM equations
derived in Chapter 3.

4.3.2

Particle and Energy Fluxes

In Section 4.2.4, the particle flux density and kinetic energy flux density equations were discretized
in terms of energy-space. Eqns. (4.51) and (4.52) are written without the arguments of the incomplete Fermi integrals for brevity. The fluxes in the kth piece-wise energy window are
"
#!
βk +1
(k
T
)
B
~Jn,k = −qDk ~E (kB T )βk F 0 + ∇r
Fβ
,
βk
k
q

82

(4.63)

and
"

(kB T )βk +2
~Kn,k = EJ,k~Jn,k − qDk ~E (kB T )βk +1 F 0
+
∇
Fβ +1
r
βk +1
k
q

#!
.

(4.64)

The kinetic energy flux density differs from the particle flux density by an order of energy. Therefore, the derivation of the real-space particle flux density discretization is directly amenable to the
kinetic energy flux density.

4.3.2.1

The Original SG Discretization

The method of Scharfetter and Gummel [77] is utilized to discretize the fluxes. The original SG
discretization technique was applied to the phenomenological DD equation. Given the projection
of the DD equation onto a primary edge of the DV mesh


kT dn(r)
Jn (r) = −µn n(r)E(r) +
,
q dr

(4.65)

where r is the spatial dimension along the edge, the particle flux Jn (r) and the electric field E(r) are
approximated as spatially constant on the edge [77]. The mobility Âľn and electron temperature T
are assumed to be spatially constant within a specific material region. With these approximations,
Eqn. (4.65) is a linear first-order differential equation in terms of n(r). Its solution on the ith
primary edge, with the BCs n(r = r0 ) = n0 and n(r = r1 ) = n1 , is
Jn,i =

Âľn kB T
[B(ξ )n0 − B(−ξ )n1 ] ,
qL

(4.66)

where
Ξ=

qE
L.
kB T

(4.67)

Appendix A.3 provides the complete derivation of the original SG discretization of the phenomenological DD model.

4.3.2.2

SG Discretization of the FKT Particle Flux

The SG discretization of the phenomenological DD model approximated the flux and electric field
as spatially constant on a primary edge and solved a linear first-order differential equation with
83

density BCs at the nodes of the primary edge. The FKT particle flux requires approximations to
its spatially dependent coefficients in order to use the same process. The projection of the kth
piece-wise FKT particle flux on the jth primary edge is
β jk

Jn, jk (r) = −qD jk E j (r) (kB T (r))

"
#!
β jk +1
d
(k
T
(r))
B
Fβ0 (r) +
Fβ (r) ,
jk
jk
dr
q

(4.68)

where r represents the spatial coordinates along the edge from r0 to r1 . The index k is associated
to the primary edge with index j for all subsequent particle fluxes. With the assumption that the
particle flux and electric field projections are spatially constant on the primary edge, i.e., Jn, jk (r) ≈
Jn, jk and E j (r) ≈ E j , there are two possible discrete particle fluxes (out of many ways to discretize
the equation) resulting from SG discretization. They are the generalized Einstein relation form
E =
Jn,
jk


 

qD jk
Einn, jk ave B(ξn,E, jk )NE,0, jk − B(−ξn,E, jk )NE,1, jk ,
Lj

(4.69)

with the density
β
NE,m, jk = (kB Tm ) jk Fβ0 ,

(4.70)

jk

and Bernoulli function argument
1

Ξn,E, jk =




Einn, jk ave



E j + ∆Einn, jk L j ,

(4.71)

and the inverse generalized Einstein relation form




qD jk kB T
I
Jn, jk =
B(ξn,I, jk )NI,0, jk − B(−ξn,I, jk )NI,1, jk ,
Lj
q ave

(4.72)

with the density
β
NI,m, jk = (kB Tm ) jk Fβ ,

(4.73)

jk

and the Bernoulli function argument
1

Ξn,I, jk =  k T 
B
q



−1
Einn, jk



k T
Ej +∆ B
q
ave


L j.

(4.74)

ave

Appendix A.3.2 gives a complete derivation of both SG discretizations of the particle flux density
and the definitions of the generalized Einstein relation form, Eqns. (34) – (36), and the inverse
generalized Einstein relation form, Eqns. (43) – (46).
84

The SG discretization of the particle flux density is directly applicable to the kinetic energy
flux density. The generalized Einstein relation form of the discrete kinetic energy flux density is
E =
Kn,
jk


 

qD jk
E ,
EinE, jk ave B(ξE,E, jk )EE,0, jk − B(−ξE,E, jk )EE,1, jk + EJ, jk Jn,
jk
Lj

(4.75)

with the density
β +1
EE,m, jk = (kB Tm ) jk Fβ0 +1 ,
jk

(4.76)

and Bernoulli function argument
1

ΞE,E, jk =




EinE, jk ave



E j + ∆EinE, jk L j ,

(4.77)

and the inverse generalized Einstein relation form is
I
Kn,
jk

qD jk
=
Lj






kB T
I ,
B(ξE,I, jk )EI,0, jk − B(−ξE,I, jk )EI,1, jk + EJ, jk Jn,
jk
q ave

(4.78)

with the density
β +1
EI,m, jk = (kB Tm ) jk Fβ +1 (r),
jk

(4.79)

and the Bernoulli function argument
1

ΞE,I, jk =  k T 
B
q



Ein−1
E, jk



k T
Ej +∆ B
q
ave


L j.

(4.80)

ave

The generalized Einstein relation EinE, jk and the inverse generalized Einstein relation Ein−1
E, jk of
the kinetic energy flux density differ from their particle flux counterparts by an order β jk . For
example, the kinetic energy inverse is

4.3.3

F0
(r)
β jk +1
−1
EinE, jk (r) = F
.
β jk +1 (r)

Heat Flow

The heat of a thermodynamic process is determined by the second law of thermodynamics. The
second law states that the entropy of a thermodynamic system always increases, or
kB T ∆σ = ∆En − F∆n = En (T2 ) − En (T1 ) − F [n(T2 ) − n(T1 )] .

85

(4.81)

Here, ∆En is the internal energy of the system and F∆n is the free energy [88]. The quantity
kB T ∆σ is the heat of the thermodynamic process. With T2 > T1 , the heat of this thermodynamic
process is positive.
Heat flow is required for closing the moments of the BTE. Therefore, an analogue of Eqn.
(4.81) is needed. The internal and free energy rates are exactly the kinetic energy flux and the
particle flux multiplied by the Fermi level, respectively. Physically, the discretized particle and
kinetic energy flux densities represent the rates of particle and energy exchange between two nonequilibrium Fermi distributions connected by a DV primary edge. The fluxes are
Jn, jk = J0, jk (T j,0 , T j,1 ) − J1, jk (T j,0 , T j,1 ),

(4.82)

Kn, jk = K0, jk (T j,0 , T j,1 ) − K1, jk (T j,0 , T j,1 ),

(4.83)

and

where J0, jk (T j,0 , T j,1 ) represents particles flowing from distribution 0 with relative Fermi level
(F − Ec )0 and temperature kB T0 to distribution 1 with relative Fermi level (F − Ec )1 and temperature kB T1 . J1, jk (T j,0 , T j,1 ) represents the opposite flow of particles from distribution 1 to
distribution 0. Figure 4.7 presents an illustration of the particle and kinetic energy exchange. Both
terms of the fluxes depend on the Fermi distribution temperature at each node. This is due to the
approximations made to the coefficients of the particle fluxes.
With the definitions (4.82) and (4.83), the heat flow is a direct analogue of Eqn. (4.81). The
heat flowing from distribution 0 to 1 across the DV primary edge is
Hn, jk = K0, jk (T j,0 , T j,1 ) − K0, jk (T j,1 , T j,1 )


− (F − Ec )0 J0, jk (T j,0 , T j,1 ) − J0, jk (T j,1 , T j,1 )
− K1, jk (T j,0 , T j,1 ) + K1, jk (T j,0 , T j,0 )


+ (F − Ec )1 J1, jk (T j,0 , T j,1 ) − J1, jk (T j,0 , T j,0 ) .

(4.84)

The first four terms of the heat flow represent the internal and free energy rates of distribution 0
and the last four represent the internal and free energy of distribution 1. The signs of each term are
86

J = J0 - J1
K = K 0 - K1
n0

n1

Real-Space Transport

J0
Fermi
Distribution 0

K0

Fermi
Distribution 1

kBT0, (F-Ec)0

J1

kBT1, (F-Ec)1

K1
Figure 4.7: The particle and kinetic energy flux from node 0 with Fermi distribution 0 to node 1
with Fermi distribution 1 connected by a DV primary edge.
determined by the definition that heat flows from distribution 0 to distribution 1 (or from primary
node 0 to primary node 1). If T0 > T1 , then Eqn. (4.84) is a positive scalar quantity on the primary
edge. If T1 > T0 , then Eqn. (4.84) is a negative scalar quantity on the primary edge. Each case
results in heat flow from higher temperature to lower temperature which is a consequence of the
second law of thermodynamics.

4.4

Conclusion

FKT is a deterministic Boltzmann solver which enforces a thermodynamic heat flow as its
closure relation. A thorough derivation of the FKT equations and their discretization in energyand real-space was presented in this chapter. Section 4.2 provided the energy-space discretization
techniques including the calculation of isosurface integrals, approximating the resulting integrations with piece-wise energy power laws, and converting the energy integrals into incomplete
Fermi-Dirac integrals. Real space discretization techniques were presented in Section 4.3. The
box integration method was used to approximate divergence operators on the DV mesh and the SG
method was applied to discretize the FKT particle and energy flux densities. Finally, a detailed
description of the heat flow algorithm was reported in Section 4.3.3.

87

CHAPTER 5
NUMERICAL SOLUTION OF THE FKT EQUATIONS

5.1

Introduction

A detailed derivation of the FKT device simulation equations and their discretization in energyand real-space was presented in Chapter 4. This chapter focuses on solving the system of nonlinear
FKT equations. In what follows, a detailed description of the nonlinear solver, the BCs used for device simulations, and important characteristics of the system of nonlinear equations are presented.
The mesh convergence and stability of the FKT equations are investigated. Finally, advanced simulation techniques including higher-order discretization methods are presented.

5.2

The System of Nonlinear Equations

The governing equations of the FKT device simulator are Poisson’s equation, Ampere’s and
Faraday’s equations, electron continuity, energy conservation, and lattice heating. After real- and
energy-space discretization, the residuals for n-type unipolar devices are




+
t+1 V
t+1 εi j Ai j − q
N
−
n
(5.1)
Poisi ≡ ∑ Φt+1
−
ÎŚ
∑ D,i,m i,m i,m = 0,
i
j
Li j
m
j
#
"
t+1
t − Φt
ÎŚt+1
−
ÎŚ
ÎŚ
i,0
i,1
i,0
i,1 Îľi Ai
t+1 A −
t+1
Ampi ≡ Et+1
− Etrot,i −
− q ∑ Jn,i,m
i,m ∑ H j L j = 0,
rot,i +
Li
Li
∆t
m
j
(5.2)
Hit+1 − Hit
Fari ≡
µ0 Ai + ∑ Erot, j L j = 0,
(5.3)
∆t
j
"
#
t
nt+1
−
n
t+1
i +Cn,t+1 V +
ElConti ≡ i
(5.4)
i ∑ Jn,i j Ai j = 0,
i
∆t
j
" t+1
#
t


En,i − En,i
E,t+1
t+1
t+1 A +
t+1
EnConsi ≡
+Ci
Vi + q ∑ E j L j Jn,i
(5.5)
i j ∑ Sn,i j Ai j = 0,
j
∆t
j
j
LattConsi ≡ ρiC p,i

t+1 − T t
TL,i
L,i

∆t



t+1 − T t+1 κi j Ai j = 0.
Vi,m + ∑ TL,i
L, j
Li j
j

E,t+1

Vi − ∑ Ci,m
m

88

(5.6)

Densities, fluxes, and collision operators with the subscript m are associated to unique Fermi distributions at the ith semiconductor node of the DV mesh. The electron number and energy densities
(with the time step superscript t removed for brevity) are


ni = ∑ Aik (kB Ti )αik +1 Fαik ηρ,ik , aik , bik ≡ ∑ nik ,

(5.7)

o
n

Îąik +2
Fι +1 Ρρ,ik , aik , bik ,
En,i = ∑ Eρ,ik nik + Aik (kB Ti )
ik

(5.8)

k

k

k

and the particle and kinetic energy flux densities are


 
qD jk
Einn, jk ave B(ξn, jk )N0, jk − B(−ξn, jk )N1, jk ≡ ∑ Jn,i jk ,
Lj
k
k

Jn,i j = ∑

β
Nm, jk = (kB Tm ) jk Fβ0

1

Ξn, jk =




jk



ΡJ,m, jk , c jk , d jk ,

(5.10)



E j + ∆Einn, jk L j ,

Einn, jk ave



 

qD jk
EinE, jk ave B(ξE, jk )E0, jk − B(−ξE, jk )E1, jk ,
Kn,i j = ∑ EJ, jk Jn,i jk +
Lj
k


β +1
Em, jk = (kB Tm ) jk Fβ0 +1 ΡJ,m, jk , c jk , d jk ,
jk
1

ΞE, jk =

(5.9)

(5.11)
(5.12)
(5.13)



E j + ∆EinE, jk L j .

(5.14)
EinE, jk ave
These are the “generalized Einstein” particle fluxes derived in Appendix A.3.2. The script E is



implied. The index k is associated to the primary edge with index j. This primary edge corresponds


to the stencil associated to the ith primary node. The functions Ein∗, jk ave and ∆Ein∗, jk , which
are discussed in Appendix A.3.2, are




F
β
1  kB T jk 
∆Einn, jk =

Lj
q Fβ0 

jk j,1

and



F
β
kB T jk 
−
q Fβ0 





F
β


1  kB T jk 
Einn, jk ave = 
2
q Fβ0 

jk j,1

89



,

(5.15)

jk j,0



F
β
kB T jk 
+
q Fβ0 

jk j,0



.

(5.16)

Here, the notation “ j, 0” represents node zero of the jth primary edge. The Einstein functions for
the kinetic energy flux density are




F
β
+1
1  kB T jk 
∆EinE, jk =

Lj
q Fβ0 +1 
jk
and

j,1



F
β
+1
kB T jk 
−
q Fβ0 +1 
jk





F
β
+1


1  kB T jk 
EinE, jk ave = 
2
q Fβ0 +1 
jk

j,1



,

(5.17)

j,0



F
β
+1
kB T jk 
+
q Fβ0 +1 
jk



.

(5.18)

j,0

See Section 4.2 for a detailed discussion on the energy-space discretization and Section 4.3 for
details on the real-space discretization. With the definitions









 qD jk
Einn, jk ave Tp, j , Tq, j B(Ξn, jk Tp, j )N0, jk Tp, j ,
J0, jk Tp, j , Tq, j =
Lj

(5.19)










 qD jk
Einn, jk ave Tp, j , Tq, j B(−ξn,k Tp, j )N1, jk Tp, j ,
J1, jk Tp, j , Tq, j =
Lj

(5.20)










 qD jk
K0, jk Tp, j , Tq, j =
EinE, jk ave Tp, j , Tq, j B(ΞE, jk Tp, j )E0, jk Tp, j ,
Lj

 qD jk








K1, jk Tp, j , Tq, j =
EinE, jk ave Tp, j , Tq, j B(−ξE, jk Tp, j )E1, jk Tp, j ,
Lj

(5.21)
(5.22)

the heat flow and total energy flux densities are

Hn,i j = ∑ K0, jk (T j,0 , T j,1 ) − K0, jk (T j,1 , T j,1 )
k



− (F − Ec )0 J0, jk (T j,0 , T j,1 ) − J0, jk (T j,1 , T j,1 )
− K1, jk (T j,0 , T j,1 ) + K1, jk (T j,0 , T j,0 )

	
+(F − Ec )1 J1, jk (T j,0 , T j,1 ) − J1, jk (T j,0 , T j,0 ) ,
Sn,i j = Kn,i j + Hn,i j .

5.2.1

(5.23)
(5.24)

Solution Variables

The solution variables of Eqns. (5.1) – (5.24) are the electric potential Φt+1
i , rotational electric
t+1
field Et+1
rot,i , magnetic field Hi , the Fermi level relative to the conduction band minimum in each

90

t+1 of a discrete electron gas, and the lattice
material (F − Ec )t+1
i , the electron temperature (kB T )i

temperature (kB TL )t+1
i . Assignment of the electron gas solution variables in real- and energy-space
and electromagnetics solution variables are next presented.

5.2.1.1

Electron Gases in Real-Space

One or more electron gases are assigned to each Voronoi polygon (2D) or polyhedron (3D) in the
DV mesh located in a semiconducting material. If a semiconductor mesh node lies on a material
interface, then the Voronoi cell is split into sub-regions. Each sub-region corresponding to a unique
material is assigned a separate electron gas. Primary nodes on the boundary of the mesh are
truncated in a similar manner. Section 2.2 provides a discussion on material interface and mesh
boundary splitting of the Voronoi diagram.

Material 2

Material 1

A
C

Material 3
Material 2

B

Material 3

(b)
(a)

Figure 5.1: (a) An example of a device mesh generated by the Sentaurus meshing tool. (b) An
illustration of the electron gases assigned to unique Voronoi polygons in the DV mesh.
Figure 5.1a presents an example of a DV mesh on which electron gases are assigned. Three
Voronoi polygons are drawn in Figure 5.1b. The Voronoi polygon associated to gas A is completely
enclosed in material 2 and therefore the entire polygon shaded in red represents the volume of the
gas. Because gas B is associated to a node on the boundary of the mesh, the Voronoi polygon is
91

10

Energy (eV)

Energy (eV)

8
6
4
2
0
-2

6.5
6
5.5
5
4.5
4
3.5
3

Γ1
0 1 2 3 4 5 6

K

Γ

A

L

H

DOS (x1021 cm-3 eV-1)

Figure 5.2: The (left) panel diagram of wurtzite GaN and (right) corresponding DOS isosurface
integral of the first Γ-valley. The energy-space of the first Γ-valley is split into two distinct Fermi
distributions doubling the number of electron gases in the semiconductor mesh.
truncated at the boundary. Node C is an important case in semiconductor device meshes. Because
this mesh node resides on a material interface, it receives separate gases associated to the two
unique materials. The Voronoi volume of gas C3 (the gas associated to material 3) is shaded in
pink and the Voronoi volume of gas C2 (the gas associated to material 2) is shaded in red. This
procedure generalizes to interfaces of an arbitrary number of materials.

5.2.1.2

Electron Gases in Energy-Space

Energy-space discretization was presented in Section 4.2. The band-structure of a bulk semiconductor is incorporated into the FKT device simulator through piece-wise energy power law fits of
the isosurface integrals. For example, the power law fits are used to calculate the electron density
and energy density, Eqns. (4.49) and (4.50). These densities are associated to a sub-region of the
Voronoi cell corresponding to a unique material as described in the previous section.
The energy-space corresponding to an electron gas in a unique material can further be discretized into multiple Fermi distributions. Typically, valleys in the semiconductor band-structure
are assigned separate distribution functions. Scattering between the distributions is calculated with

92

the collision operators Cn and CE [83]. Figure 5.2 presents an illustration of a panel diagram and
the DOS isosurface integral, Eqn. (4.30), of wurtzite GaN [84]. The band-structure was calculated
with EPM [89]. The DOS data was obtained from Matt Grupen AFRL [84]. Only the first Γ-valley
is included in the DOS calculation and the energy-space of the valley is split into two separate
Fermi distributions labeled Γ1 and Γ2 . The separation of the first Γ-valley into two distinct Fermi
distributions yields two sets of solution variables (F − Ec )t+1 and (kB T )t+1 at each semiconductor mesh node. Real-space transport only occurs between the same distributions in energy-space.
Therefore, there is no particle flux between the Γ1 and Γ2 distributions. Rather, particles can drift
and diffuse between Γ1 at mesh node 1 and Γ1 at mesh node 2 and then scatter from Γ1 to Γ2 at
mesh node 2. The same is true for energy transport.

5.2.1.3

Electromagnetics

The solution variables corresponding to full-wave EM are the rotational electric field Erot,i and the
magnetic field Hi . The rotational electric field vector projections are defined on the primary edges
of the Delaunay mesh and magnetic field vector projections are defined on the edges of the Voronoi
diagram. A derived solution variable is the total electric field. On the jth primary edge, the total
electric field is
E j = Erot, j +

Φ j,0 − Φ j,1
,
Lj

(5.25)

where the subscript j, 0 corresponds to node zero of the primary edge. The primary edge points
from node 0 to 1. A complete discussion of the full-wave EM discretization, DVSI, is presented in
Chapter 3.

5.3

The Nonlinear Solver: Newton’s Method

The coupled nonlinear system, Eqns. (5.1) – (5.24), is solved with Newton’s method. See
Section 2.3.1.2 for a detailed description on the general Newton’s method solution of systems of
nonlinear equations. The pseudo-code outlining the Newton algorithm in Section 2.3.1.2 illustrates
the need to compute the residual vector and Jacobian matrix at each iteration. The residual vector

93

is simply the concatenation of the discrete nonlinear Eqns. (5.1) – (5.5). These equations are
represented as Poisi , Ampi , Fari , ElConti , EnConsi , and LattConsi , respectively. The Jacobian
matrix Ji j is the partial derivative of the ith residual with respect to the jth solution variable.
The Newton linear system of an FKT device simulation is now discussed. The global nonn
o
linear system consists of Nphi electric potential solution variables ÎŚ1 , ÎŚ2 , ..., ÎŚNphi , NF eleco
n
tron relative Fermi level solution variables (F − Ec )1 , (F − Ec )2 , ..., (F − Ec )NF , NT electron
o
n
temperature solution variables kB T1 , kB T2 , ..., kB TNT , and NT L lattice temperature solution vario
n
ables kB TL,1 , kB TL,2 , ..., kB TL,NT L . If full-wave EM is included in the device simulation, there
n
o
are an additional NE rotational electric field solution variables Erot,1 , Erot,2 , ..., Erot,NE and NH
n
o
magnetic field solution variables H1 , H2 , ..., HNH .
A linear system is solved at each Newton iteration. The linear system is comprised of the Poisson residuals Poisi , then the Ampere residuals Ampi , then Fari , ElConti , EnConsi , and LattConsi .
With this arrangement, the full-wave FKT Newton linear system is



∂ Pois1
∂ Pois1
∂ Pois1
¡
¡
¡


∂ Φ1
∂ Φ2
∂ kB TL,N
∆Φ1
TL 




∂ Pois2
∂ Pois2
∂ Pois


··· ∂k T 2
∆Φ2
∂ Φ1
∂ Φ2


B
L,NT L 


..


..
..
..
.
.


.
.


 ∂ LattConsN

∂ LattConsN
∂ LattConsN
TL
TL ¡¡¡
TL
∆kB TL,NT L
kB TL,1

kB TL,2

kB TL,N
TL













 = −









Pois1
Pois2
..
.
LattConsNT L





.




(5.26)

The solution of Eqn. (5.26) determines the change in solution variables {∆Φ1 , · · · , ∆Erot,1 , · · · ,
o
∆H1 , · · · , ∆(F − Ec )1 , · · · , ∆kB T1 , · · · , ∆kB TL,NT L . The global solution variable set is updated
according to
ÎŚm+1
= ÎŚm
1 + ∆Φ1
1
..
.
m
Em+1
rot,1 = Erot,1 + ∆Erot,1

..
.
m+1 = k T m
kB TL,N
B L,NT L + ∆kB TL,NT L .
TL

94

(5.27)

5.4

Boundary Conditions

A list of the BCs required for electronic device simulations is presented in Table 5.1. Typical
BCs include metal contacts and port BCs used to provide or absorb EM energy.
B.C.

Erot

ÎŚ

Ohmic
ÎŚCN +Va
Schottky
ÎŚSB +Va
Lattice Heat Absorbing
Soln.
AC Impedance (Ohmic)
ÎŚt+1
ACI
AC Impedance (Schottky)
ÎŚt+1
ACI
Full-wave Voltage
Soln.
Full-wave Impedance
Soln.

F − Ec

kB T

kB TL

(F − Ec )CN
Soln.
Soln.
(F − Ec )CN
Soln.
Soln.
Soln.

kB TL
Soln.
Soln.
kB TL
Soln.
Soln.
Soln.

Soln.
Soln.
Soln.
Soln.
Soln.
Soln.
Soln.

H

0
0
0
0
Soln. Soln.
0
0
0
0
0
Soln.
Soln. Himp

Table 5.1: A list of semiconductor device simulation BCs.

5.4.1

Ohmic Metal Contact

An essential metal BC found in almost every semiconductor device simulation is the Ohmic contact. Charge neutrality is enforced at a distribution on an interface between a semiconductor and
an Ohmic contact. The solution variable F − Ec of a charge distribution on an Ohmic contact interface is fixed to the charge neutral value (F − Ec )CN and maintains the electric potential ΦCN +Va .
Here, Va is a voltage applied to the Ohmic contact. The charge neutral quantities are determined
by the numerical solution of


+
Poisi |ohm. ≡ ∑ ND,i,m
− nt+1
i,m Vi,m = 0,

(5.28)

ΦCN = −χe + ΦM + (F − Ec )CN .

(5.29)

m

and the assignment

Here, the electron affinity χe and the work-function ΦM are properties of the semiconductor and
metal, respectively.

95

5.4.2

Lattice Heat Absorbing Boundary Condition

The heat absorbing (HA) BC is meant to emulate lattice heat flow out of the device domain. Without this BC, the simulated devices will heat up to nonphysical levels. Outgoing lattice heat flow is
represented by the outward lattice temperature gradient (∇TL )HA on a boundary mesh node. The
gradient flux is through the in-plane Voronoi polygons associated to primary nodes on the mesh
boundary (see Figure 2.8). The total HA BC is
"
#
t+1 − T t
TL,i
L,i
E,t+1
− ∑ Ci
LattConsi |HA ≡ ρiC p,i
Vi
∆t
i

Îş A
ij ij
t+1 − T t+1
+ [n̂ · (∇TL ) κA]HA = 0.
+ ∑ TL,i
L, j
L
i
j
j

(5.30)

Here, n̂ represents the normal of the mesh boundary. The lattice temperature gradient at the mesh
nodes of the HA boundary are reconstructed from the vector projections of the lattice temperature
gradients onto the primary edges associated to the mesh nodes.

5.4.3

Quasi-Static AC Impedance Boundary Condition

The quasi-static solver is excellent for lower frequency simulations where full-wave effects are
negligible. In order to calculate useful quantities including S-Parameters with the quasi-static
solver, the AC impedance must be used as a termination opposite the excitation contact. The
AC impedance BC is defined on metal contacts in the quasi-static solver. The integrated current
flowing into the AC impedance contact is
it+1 = ∑ Jkt+1 Ak .

(5.31)

k

Here, Jk represents the particle flux density on the kth primary edge pointing into the AC impedance
metal contact. The area Ak is that of the in-plane Voronoi polygon associated to the primary edge
(see Figure 2.8). This integrated current specifies a Dirichlet BC on the metal contact’s potential
via Ohm’s law, i.e.,


ÎŚt+1
i 

ACI

= ZACI it+1
ACI ,

t+1 − it=0 represents only the AC component of the integrated current.
where it+1
ACI ≡ i

96

(5.32)

5.4.4

Full-Wave Voltage Port

The voltage BC is a specific case of the port BC in which EM energy enters the full-wave solver.
The voltage port connects two metal contacts in the DUT. The electric field in the voltage port is
strictly irrotational, i.e., Erot,i = 0. Ampere’s law is solved on voltage port primary edges. The BC
is
iÎľA
h
t+1 A −
t+1
t+1
t
|
Ampi volt. ≡ Ei,irr − Ei,irr i i − q ∑ Jn,im
im ∑ H j L j = 0.
∆t
m

(5.33)

j

Φi,0 −Φi,1
Here, the irrotational electric field is Et+1
where the subscript i, 0 corresponds to node
L
i,irr =
i

zero of the

ith

primary edge. The primary edge points from node 0 to 1. This BC serves as the

governing equation for the magnetic field defined on the ith in-plane dual edge associated to a
primary edge on the boundary (see Figure 2.8). This magnetic field is included in the line integral
of Eqn. (5.33).

5.4.5

Full-Wave AC Impedance Port

The full-wave AC impedance (FWACI) BC is an energy sink port BC connecting two metal contacts. The magnetic field HFWACI is defined on an in-plane dual edge associated to a primary edge
on the boundary. The magnetic field is equal to the total electric field on the primary edge divided
by the boundary’s impedance ZFWACI . The BC is


t=0 − Z
t+1
t=0
FWACIi ≡ Et+1
−
E
H
−
H
FWACI
i
i,FWACI = 0,
i
i,FWACI

(5.34)

In a manner similar to the quasi-static AC impedance BC, the subtraction of the solutions calculated
at the first time step ensures that the BC only affects the AC components of the solution variables.
This emulates a bias tee circuit.

5.4.6

Heterojunctions and Thermionic Emission Boundaries

As outlined in Section 5.2.1.1, one or more electron gases are assigned to semiconductor mesh
nodes residing on heterojunctions. Figure 5.3 illustrates an example of a heterojunction between
97

two semiconductors. In this example, each electron gas is approximated with a single Fermi distribution in energy-space. Because the two materials have different electron affinities, there is a
discontinuity in the conduction band across the heterojunction and thermionic emission is used to
calculate real-space transport across the discontinuity. With both distributions’ Fermi levels referenced to the higher conduction band, the particle flux and kinetic energy flux density thermionic
emissions from semiconductor 1 to 2 are
1→2 (T ) =
Jn,T
E 1

1 m∗
(kB T1 )2
2π 2 h̄3

Z∞

dεz ln [1 + exp (η̃1 − εz )] ,

(5.35)

dεz εz ln [1 + exp (η̃1 − εz )] ,

(5.36)

0

and
∗
1→2 (T ) = 1 m (k T )3
Kn,T
B 1
1
E
π 2 h̄3

Z∞
0

with the parameter
Ρ̃1 =

(Fn − Ec )1 + Ec,1 − Ec,high
.
kB T1

(5.37)

A numerical quadrature routine is required to evaluate Eqns. (5.35) and (5.36). The high side
conduction band edge Ec,high is associated to the charge distribution in the material with the smaller
electron affinity at the heterojunction. The total thermionic emission particle, kinetic energy, heat,
and total energy flux densities are
1→2 (T ) − J 2→1 (T ),
Jn,T E = Jn,T
E 1
n,T E 2

(5.38)

1→2 (T ) − K 2→1 (T ),
Kn,T E = Kn,T
E 1
n,T E 2

(5.39)

1→2 (T ) − K 1→2 (T )
Hn,T E = Kn,T
E 1
n,T E 2
h
i
1→2 (T ) − J 1→2 (T )
− (F − Ec,high )1 Jn,T
E 1
n,T E 2
2→1 (T ) + K 2→1 (T )
− Kn,T
E 2
n,T E 1
h
i
2→1 (T ) − J 2→1 (T ) ,
+ (F − Ec,high )2 Jn,T
E 2
n,T E 1

(5.40)

and
Sn,T E = Kn,T E + Hn,T E .
98

(5.41)

Distribution A1

Distribution B1

Mesh Nodes

Electron Energy

(Fn-Ec)A1

(Fn-Ec)B1

Material A

Ec
Fn

Material B

Ev
Real Space

Figure 5.3: The band diagram across a two material heterojunction. The electron affinity of material A is greater than material B’s electron affinity. The relative Fermi level solution variables are
illustrated for each charge gas.
The thermionic particle flux and total energy flux densities are added to the electron continuity
and energy conservation divergences, Eqns. (5.4) and (5.5). Figure 5.4 illustrates the particle divergence for the two unique charge distributions defined on a heterojunction mesh node. There is only
a BTE particle flux (Jn,i j and Kn,i j ) between charges associated to the same material. Thermionic
emission connects the two charge distributions at the heterojunction. With charge distributions A1
and B1 at the heterojunction mesh node, the electron continuity and energy conservation equations
for both distributions are
ElContA1 |HJ ≡

!
t
nt+1
−
n
t+1
t+1
A1 +Cn,t+1 V +
A1
A1 ∑ Jn,A1 j AA1 j + Jn,T E AT E = 0,
A1
∆t
j

(5.42)

ElContB1 |HJ ≡

!
t
nt+1
−
n
t+1
t+1
B1 +Cn,t+1 V +
B1
B1 ∑ Jn,B1 j AB1 j − Jn,T E AT E = 0,
B1
∆t
j

(5.43)

99

Material A

Material B

∇⋅Jn,A1

∇⋅Jn,B1

Thermionic Emission
Across Heterojunction

Figure 5.4: An example of the particle divergences associated to two distributions at a heterojunction. Particle fluxes are calculated between two electron gases in the same semiconductor.

EnConsA1 |HJ ≡

t+1 − E t
En,A1
n,A1

∆t

!
E,t+1

+CA1

VA1



t+1
t+1
+ q ∑ Et+1
L
j Jn,i j AA1 j + ∑ Sn,A1 j Ai j
j
j

j

t+1
+ St+1
n,T E AT E + Epot Jn,T E AT E = 0.

EnConsB1 |HJ ≡

t+1 − E t
En,B1
n,B1

∆t

(5.44)

!
E,t+1

+CB1

VB1



t+1
t+1
t+1
+ q ∑ Et+1
L
j Jn,B1 j AB1 j + ∑ Sn,B1 j Ai j − Sn,T E AT E = 0.
j
j

(5.45)

j

The last term in Eqn. (5.45) represents the potential energy required to overcome the barrier Epot =
χA − χB with χA > χB . The area AT E corresponds to the Voronoi polygon in the heterojunction
plane associated to the primary node on the interface (see Figure 2.8).

100

5.4.7

Schottky Metal Contact

Another important BC is the rectifying Schottky metal contact. This metal-semiconductor interface
is a special type of heterojunction. The important characteristic of the BC is the Schottky barrier
ΦSB . With no applied bias, the solution variable F − Ec of a charge distribution at this interface
equals −qΦSB . In the typical way, an applied bias which raises the Fermi level in the Schottky
metal will increase the potential barrier of the metal-semiconductor interface. An applied bias
which lowers the Fermi level will decrease the potential barrier. Transport across the discontinuity
caused by the Schottky barrier is calculated with thermionic emission. The energy −qΦSB and
the lattice temperature kB TL are used to compute the thermionic emission from the metal into the
semiconductor.

5.5

Semiconductor Device Simulation Work-Flow

In this section, the device simulation work-flow is discussed. Device solves include equilibrium, static, quasi-static, and full-wave simulations. Each device solve will be reviewed and
discussed.

5.5.1

Work-Flow Overview

An overview of the semiconductor device simulation work-flow is provided in Figure 5.5. The
input to any device simulation is the mesh of the DUT. Included in the DUT mesh are material
specifications, dopant profiles and quantities, and BCs including Ohmic/Schottky contacts. The
solve types are grouped into three distinct classes: Equilibrium, static, and transient solves.
The Newton linear system corresponding to a specific device solve will be presented in block
matrix form. For example, the matrix corresponding to the partial derivative of the ith Poisson
∂ Pois
residual with respect to the jth electric potential solution variable is ∂ Φ i . The ith diagonal of this
j

matrix is
Îľi j Ai j
∂ Poisi
=∑
.
∂ Φi
L
i
j
j

101

(5.46)

The derivative of the ith Poisson residual with respect to the jth Fermi level solution variable is
∂ nt+1
∂ Poisi
i,m
= q∑
Vi,m .
∂ (F − Ec ) j
m ∂ (F − Ec ) j

(5.47)

This Jacobian element requires partial derivatives of the densities with respect to the relative Fermi
levels which amounts to derivatives of incomplete Fermi integrals in terms of the derived solution
variables Ρ. The numerical routine which calculations the incomplete Fermi integrals also returns
the derivative of the Fermi integrals.

Static
solve

Thermal equilibrium prole
Energy (eV)

Equilibrium
solve

Metal contacts
Port B.C.s

Quiescent bias, static I-V family, etc.
IDS

DUT mesh

Materials
Dopant proles
Boundary conditions

VDS

Transient
solve

S-Parameters, power sweep, load-pull, etc.

Figure 5.5: A general work-flow diagram of a semiconductor device simulation.
The following sections describe the solve types in more detail. To do so, a practical example
is utilized — a GaN MESFET. Figure 5.6 presents an illustration of the 3D MESFET including
the material definitions and the metal contacts. The Ohmic source and drain and the Schottky
gate are the yellow regions at the top of the mesh. The Ohmic ground at the bottom of the mesh
is not shown. The dark green region is the GaN channel and the gray region is the insulating

102

substrate. The transport direction (from source to drain) is along the x-axis in this example. The
substrate/semiconductor interface is perpendicular to the z-axis.
A complete device simulation is conducted with the 3D GaN MESFET example. First, the
thermal equilibrium profile is calculated with the equilibrium solver. Then, the quiescent bias is
calculated and finally the S-Parameters are calculated with the quasi-static and full-wave solvers.

Schottky Gate

Ohmic Drain

Ohmic Source

n-GaN

Launch
Input Port

Substrate

Ohmic Ground

Figure 5.6: The 3D GaN MESFET example used to demonstrate the device simulation workflow. The yellow regions at the top of the mesh are the Ohmic source and drain and the Schottky
gate contacts. The bottom surface of the mesh is the Ohmic ground contact (not shown). The GaN
channel is the dark green region and the insulating substrate is the gray region. The volume labeled
“launch” represents the insulating region connected to the voltage port (the red surface). This is
how EM energy is injected into the domain (see Section 5.4 for more information on the port BCs).

5.5.2

Equilibrium Solve

The equilibrium solve is the first component of the device simulation work-flow. This produces the
electric potential and electron relative Fermi level profiles in thermal equilibrium. Only Poisson’s
equation, Eqn. (5.1), is solved to produce the thermal equilibrium solution profiles. The Newton
linear system of this solve is


∂ Poisi
∂Φj




∆Φ j


=−


Poisi

.

(5.48)

Because the conduction band Ec will vary across the device in thermal equilibrium, the relative
Fermi level solution variable F − Ec must also be updated after each Newton iteration. This can be
103

accomplished by amending the diagonals of the Jacobian to be
t+1

∂ ni,m
Îľ i j Ai j
∂ Poisi
=∑
+q∑
Vi,m
∂ Φi
L
∂
ÎŚ
i
j
i
m
j
∂ nt+1
Îľ i j Ai j
i,m
=∑
−q∑
Vi,m
m ∂ (Ec )i
j Li j
∂ nt+1
Îľ i j Ai j
i,m
=∑
+q∑
Vi,m .
m ∂ (F − Ec )i
j Li j

(5.49)

The relation Ec = −χ − Φ dictates that a positive change in the electric potential corresponds to
a negative change in the conduction band. Furthermore, the electron Fermi level F is spatially
constant at thermal equilibrium which allows for the final substitution in Eqn. (5.49). The thermal
equilibrium solution variables are updated as
ÎŚim+1 = ÎŚm
i + ∆Φi

(5.50)

(F − Ec )im+1 = (F − Ec )m
i + ∆Φi .

(5.51)

Here, the superscript m represents the Newton iterations.
After the Poisson stage of the thermal equation solve, the rest of the steady-state equations are
added one at a time to allow all solution variables to relax to their numerical thermal equilibrium.
The Newton linear system of the final equilibrium stage, comprised of the residuals Poisi , ElConti ,
EnConsi , and LattConsi , is shown in Eqn. (5.52). The additional stages of the thermal equilibrium
solves do not drastically change the electric potential or relative Fermi level solution variables.
The semiconductor band diagram and the electric potential profile of the 3D GaN MESFET
example in thermal equilibrium are shown in Figures 5.7a and 5.7b, respectively. As shown in
Figure 5.6, the metal contact located at the top middle of the device is a Schottky contact. In this
example, the Schottky barrier is ÎŚSB = 1.0 V .

5.5.3

Static Solve

The next solve in the semiconductor device simulation work-flow is the static or steady-state solve.
Typically, an external voltage or current bias is applied to metal contacts to produce steady-state
104

Ec

-4

Energy (eV)

-4.5

F

-5
-5.5
-6
-6.5
-7

Ev

-7.5
-8

Electric Potential

-8.5
0

0.02

0.04

0.06

0.08

0.1

MESFET z-direction (Îźm)

(b)

(a)

Figure 5.7: The thermal equilibrium solution variable profile of the 3D GaN MESFET example.
(a) The semiconductor band diagram illustrating the relative Fermi level solution variable in the
z-direction of the MESFET. (b) The spatial profile of the electric potential solution variable.
current flow. The static solver computes steady-state data including the current-voltage (I-V) family and transconductance. The quiescent bias of a transistor is also calculated with the static solver.
As described in previous sections, a linear system is solved at each Newton iteration in order
to update the FKT device simulation solution variables. The linear system of the mth static solve
Newton iteration is

∂ Poisi
∂Φj



 ∂ ElConti
 ∂Φ

j

 ∂ EnConsi
 ∂Φ
j


0

∂ Poisi
∂ (F−Ec ) j
∂ ElConti
∂ (F−Ec ) j

∂ Poisi
∂ kB T j
∂ ElConti
∂ kB T j

∂ EnConsi
∂ (F−Ec ) j

∂ EnConsi
∂ kB T j

∂ LattConsi
∂ (F−Ec ) j

∂ LattConsi
∂ kB T j





∆Φ j

∂ ElConti  

∂ kB TL, j   ∆(F − Ec ) j

∂ EnConsi  
 ∆kB T j
∂ kB TL, j 

∂ LattConsi 
∆kB TL, j
0

∂ kB TL, j





Poisi









 ElConti 




 = −
.
 EnCons 



i 



LattConsi
(5.52)

The static FKT device simulation equations include Eqns. (5.1), (5.4), (5.5), and (5.6) with all time
∂ → 0. The Jacobian matrix, Eqn. (5.52), is more complex than the
derivatives set to zero, i.e., ∂t

equilibrium Jacobian matrix. For example, the middle diagonal matrix element requires the partial
derivative of the collision operator and the particle flux with respect to the relative Fermi level
solution variable. This Jacobian matrix element is
t+1
n,t+1
∂ Jn,i
∂Ci
∂ ElConti
j
=
Vi + ∑
Ai j .
∂ (F − Ec ) j ∂ (F − Ec ) j
∂
(F
−
E
)
c
j
j

(5.53)

Derivatives of incomplete Fermi integrals are required for evaluating this Jacobian matrix element.
105

The particle flux is exponentially dependent on ratios of Fermi integrals making this matrix element
highly nonlinear in terms of the solution variables.
Two static simulations of the GaN MESFET are now presented. Figure 5.8 illustrates the static

Drain Current ID (mA)

3
2.5
2
1.5
1
0.5
0
0

0.5

1

1.5

2

2.5

3

3.5

4

Drain-Source Bias VDS (V)
Figure 5.8: The I-V family calculated with the static solve. The gate-source biases start at 0.5 V
(top curve) and decrease by steps of 0.25 V .

Electric Potential

Figure 5.9: The spatial profile of the electric potential solution variable at a specific quiescent bias.
The gate-source bias is −0.5 V and the drain-source bias is 4 V .
I-V family of the MESFET. The gate-source bias ranges from VGS = 0.5 V to -1.5 V with a 0.25 V
step. The drain-source bias is swept from VDS = 0 to 4 V for each gate bias. The electric potential
solution variable profile of the MESFET with a quiescent bias of VGS = −0.5 V and VDS = 4 V is
presented in Figure 5.9.
106

5.5.4

Quasi-Static Solve

After the quiescent bias of the device is calculated by the static solve, the quasi-static solve can
be used to simulate the RF response without the inclusion of full-wave EM. The RF response
metrics of transistors include small-signal S-Parameters. The simulated S-Parameters across a
broad frequency range can be very useful to circuit designers for impedance matching at input and
output ports in standard commercial software including ADS.
The linear system of the mth Newton iteration of the quasi-static simulation retains the same
form as Eqn. (5.52). However, because the time derivatives are now included in Eqns. (5.1), (5.4),
(5.5), and (5.6), the Jacobian matrix is different. As an example, the middle diagonal element of
the quasi-static Jacobian matrix is
n,t+1

∂Ci
∂ nt+1
1
i
+
∂ (F − Ec ) j ∆t ∂ (F − Ec ) j

∂ ElConti
=
∂ (F − Ec ) j
+∑
j

t+1
∂ Jn,i
j

A .
∂ (F − Ec ) j i j

!
Vi
(5.54)

The time step ∆t is typically chosen according to the driving frequency of the electronic device
simulation. A standard choice is ∆t = 1/(100 × f0 ), where f0 is the fundamental frequency of the
device simulation.

5.5.5

Full-Wave Solve

Full-wave and hot-electron effects are captured in an electronic device simulation by solving Eqns.
(5.1) – (5.24). This solver can produce simulated S-Parameters which reflect the high frequency
parasitics in an electronic device. These parasitics become important at high frequency. The

107

Newton linear system corresponding to a full-wave FKT device simulation is


∂ Poisi
∂Φj

∂ Poisi
∂ (F−Ec ) j
∂ Ampi
∂ (F−Ec ) j

0
0


∂ Ampi
∂ Ampi
 ∂ Ampi
 ∂Φ
∂ Erot, j
∂Hj
j


∂
Far
∂
Fari

i
0
0

∂ Erot, j
∂Hj

 ∂ ElConti ∂ ElConti
∂ ElConti

0
 ∂Φj
∂ Erot, j
∂ (F−Ec ) j

 ∂ EnConsi ∂ EnConsi
∂ EnConsi
 ∂Φ
0
∂
E
∂
(F−Ec ) j

j
rot, j

∂ LattConsi
0
0
0
∂ (F−Ec ) j




∆Φ j


 Poisi 




 ∆E

 Amp

rot, j




i












∆H j
Fari
 = −

×




 ∆(F − Ec ) j 
 ElConti 








 ∆k T

 EnCons 
B j
i 







∆kB TL, j
LattConsi

∂ Poisi
∂ kB T j
∂ Ampi
∂ kB T j

0
∂ ElConti
∂ kB T j
∂ EnConsi
∂ kB T j
∂ LattConsi
∂ kB T j

0






0




0


∂ ElConti 

∂ kB TL, j 

∂ EnConsi 

∂ kB TL, j 


∂ LattConsi
∂ kB TL, j

(5.55)

Full-wave simulations use the same choice of time step as quasi-static simulations. An example
of the Jacobian matrix calculated in a full-wave simulation of the GaN MESFET is illustrated in
Figure 5.14. The lattice heating equation is not included in this simulation.
Figure 5.10 presents the RF S-Parameters calculated with the quasi-static (dashed lines) and
full-wave (solid lines) solvers. The frequency sweep is 1 to 200 GHz. The full-wave solver captures
different small-signal parasitics in the MESFET. Proper treatment of the high frequency parasitics
is paramount for accurate circuit model extraction.

5.6

Mesh Convergence of the Discrete FKT Equations

The mesh convergence of the discrete FKT device simulation equations is investigated. This
numerical characteristic provides insight into how the errors of the device simulation solution variables converge with mesh refinement. Simulation of large electronic devices requires meshing
strategies in order to yield accurate results without insurmountable computational demands. Fol-

108

20

10

5.0

4.0

3.0

1.6
1.8
2.0

S12

1.4

1.2

0.9
1.0

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

S21

-20
-0.1

S22

-5.0

S11

-4.0

3
-0.

-10

-0.2

-3.
0
8
-1.

-1.6

-1.4

-1.2

-1.0

-0.9

-0.8

-0.7

-0.6

-0.
5

Quasi-static
Full-wave

0
-2.

4
-0.

Figure 5.10: Simulated S-Parameters of the GaN MESFET. The operating point of the device is
a gate-source bias of −0.5 V and a drain-source bias of 4 V . The top portion of the figure is a
polar plot and the bottom half is a Smith chart plot. The dashed lines represent the S-Parameters
calculated with the quasi-static solver and the solid lines are the S-Parameters calculated with the
full-wave solver. The full-wave simulation produces significantly different S-Parameters than the
quasi-static solver.
lowing Section 2.3.2, convergence of the discrete FKT equations is quantified by evaluating the ith
relative L2 error

v


u
u ∑ w j ũi j,k − ui j,k 2
uj
,
Îľik = u
t
∑ w j u2i j,k

(5.56)

j

on a series of meshes. Here, ũi j,k and ui j,k are the ith numerical and analytic solutions in the jth
element of the kth mesh, respectively. The weights w j are specified by the specific error calculation.
The global error of all solution variables in the device simulation is calculated as
Îľk =

r

∑ εik2 .

(5.57)

i

The order of convergence of the FKT equations is determined by the relation
−p
|Îľk | < CNk k ,

109

(5.58)

Nk

Îľk

pk

1041
1646
3034
6283

0.12896775
0.03903999
0.01340268
0.00572172

3.00
2.22
1.38
1.28

Table 5.2: Convergence of the static FKT solution variables in the GaN MESFET example with a
gate-source bias of −0.5 V and drain-source bias of 4 V .
Nk

Îľk

∆εk

2277
2893
4513
5535

0.05570511
0.02115591
0.00024652
0.00002885

-0.03454920
-0.02090938
-0.00021767

Table 5.3: Convergence of the I-V family of the GaN HEMT example calculated with the static
FKT solver.
where Nk is the number of DOF in the mesh, pk is the approximate order of convergence, and C is
a constant.
Two numerical examples are presented to provide insight into the convergence of the discrete
FKT equations. The first is a static simulation of the GaN MESFET. After a quiescent bias of
VGS = −0.5 V and VDS = 4 V is calculated with the static solver, the global solution variables
are saved on a series of five meshes. The first and last meshes are the most coarse and dense,
respectively. Solutions on the dense mesh are used as the “analytic solutions” ui j,k in Eqn. (5.56).
Calculating the global solution variable error Îľk requires interpolation of the nodal solution
variables on the dense mesh to the nodes of the series of meshes. To this end, linear interpolation
is used for its simplicity and efficiency in the post-processing. After the dense mesh solution
variables are interpolated to the series of meshes, the global solution variable error is calculated
with numerical quadrature. Therefore, the weights w j in Eqn. (5.56) represent the volumes of the
quadrature simplices.
Table 5.2 reports the integrated errors for each mesh in the series. The first column lists the
DOF corresponding to each mesh. The second and third columns list the solution variable errors

110

Îľk and the approximate order of convergence pk for each mesh in the series. According to these
results, the FKT device simulator exhibits approximately first-order convergence, i.e., pk = 1.
This is due to the SG discretization of the particle fluxes. The crux of the flux discretization is the
approximation of the particle fluxes as spatially constant across the primary edges of the DV mesh.
This approximation implies first-order mesh convergence.
A pragmatic example of FKT device simulation convergence is an investigation of the I-V
family calculated by static simulation of a GaN HEMT on a series of meshes. The GaN HEMT
is discussed in detail in Chapter 6. The device’s currents are an important output of the device
simulator. Therefore, it is a useful study to determine how the simulated currents are affected by
mesh refinement. The error calculation of this example requires evaluation of Eqn. (5.56) at each
drain-source bias. The I-V curves on the dense mesh are used as the “analytic” solutions. Table 5.3
lists the DOF, global errors in the I-V curves εk , and the corresponding change in the errors ∆εk
compared to the previous mesh in the series. Figure 5.11 presents the calculated I-V curves on the

Drain Current ID (mA)

series of four meshes. The change in the device’s response becomes negligible after the third mesh

90
80
70
60
50
40
30
20
10
0
-10

Mesh 1
Mesh 4

0

2

4

6

8

10

Drain-Source Bias VDS (V)
Figure 5.11: The I-V families of the GaN HEMT calculated with the static solver on a series of
four meshes.

111

refinement. Furthermore, only the linear region of the device’s I-V family is moderately affected
by the mesh refinement. This type of mesh refinement may not be necessary when simulating the
RF response at the peak transconductance of the saturated region of the I-V family.

5.7

Stability of the Discrete FKT Equations

Stability of nonlinear differential equations is a deep and rich subject in systems theory and
engineering [54]. One part of stability theory is the analysis of equilibrium points. The analysis
dx (t)

starts with a given equilibrium point of the autonomous system dti

= Fi (xi (t)), i.e., Fi = 0. The

system of nonlinear differential equations are said to be stable if for a given perturbation of the
equilibrium, the solutions return to the equilibrium point. A detailed discussion on stability of
nonlinear equations was presented in Section 2.3.3.
The following examples analyze the stability of the discrete FKT equations. The equilibrium

1.0e+03

HEMT

1.0e+02
1.0e+01

MESFET

ΔHt+1

1.0e+00
1.0e-01
1.0e-02
1.0e-03

Convergence Tolerance

1.0e-04
1.0e-05
1.0e-06
0

5

10

15

20

25

Time-step
Figure 5.12: Deviation of the numerical enthalpy, Eqn. (5.60), for the GaN MESFET (red line) and
HEMT (black line) examples. Both equilibria are perturbed by displacing two randomly chosen
electron gases from their thermal equilibrium states.

112

1.0e+03
1.0e+02
1.0e+01

ΔHt+1

1.0e+00
1.0e-01
1.0e-02
1.0e-03

Convergence Tolerance

1.0e-04
1.0e-05
1.0e-06
0

5

10

15

20

25

30

35

Time-step
Figure 5.13: Five independent enthalpy analyses of the GaN MESFET example. Each perturbation
moves three randomly chosen electron gases from their thermal equilibrium states. No perturbation
was found to be unstable. The deviation from the thermal equilibrium enthalpy decayed exponentially near the convergence tolerance.
points are the solutions of the static solver at thermal equilibrium, i.e., no external biases. After
the thermal equilibrium of the specific device example is computed, the total numerical enthalpy
of the system is calculated as


H ≡ ∑ En,i + kB Ti ni Vi .

(5.59)

i

The enthalpy is calculated from the electron kinetic energy density En,i , electron temperature kB Ti
and electron density ni defined in the DUT. The summation represents integration over the elements of the DV mesh. Perturbation amounts to displacing the FKT solutions from their thermal
equilibrium states. For these purposes, two electron gases are randomly chosen and their relative
Fermi levels and temperatures are perturbed. The perturbed solutions are used as initial conditions
to the quasi-static solver. A transient simulation with no external biases is solved until the discrete residuals reach the global numerical tolerance of 10−9 . The deviation of the total numerical

113

enthalpy is quantified as




∆H t+1 = H t+1 − Hequilib.  ,

(5.60)

where H t+1 is calculated at each time step of the quasi-static solver and Hequilib. is the numerical
enthalpy of the thermal equilibrium solutions.
Two GaN device examples, a MESFET and a HEMT, are chosen to numerically demonstrate
the stability of the discrete FKT equations. The GaN HEMT is discussed in detail in Chapter 6.
Both device examples are simulated as closed systems, i.e., lattice heating is not included. Two
electron gases are randomly chosen and perturbed from their thermal equilibrium states. Figure 5.12 presents the deviation of the enthalpy ∆H t+1 for both device examples. The red line
represents the enthalpy deviation versus time for the GaN MESFET example and the black line
presents the same for the GaN HEMT example. An approximate convergence line is illustrated in
the figure. Figure 5.13 presents five more enthalpy deviation simulations of the MESFET example. Each perturbation moved an additional random electron gas from its equilibrium state. No
perturbation simulation is found to be unstable. Furthermore, each enthalpy deviation approaches
an exponential rate of decay near the convergence tolerance. This provides insight into the stability
and robustness of the FKT device simulation equations.

5.8

Advanced Simulation Techniques

Simulating large electronic devices requires high computational efforts. For example, the DOF
corresponding to a four-finger transistor could quadruple compared to a single gate transistor. Furthermore, devices with exotic materials like gallium oxide (β -Ga2 O3 ) could require many Fermi
distributions associated to band-structure valleys which would further increase the DOF. To accommodate the simulation of large and/or complex electronic devices, the TCAD tool must include
advanced techniques to help alleviate some of the computational burden.

114

5.8.1

The Linear Solve

The most time consuming portion of a simulation of a large electronic device is the Newton linear
solve. The Jacobian of an FKT device simulation with an arbitrary unstructured DV mesh is a
sparse matrix. The sparsity comes from the fact that the discretization stencils only depend on
nearest neighbors in the DV mesh. Figure 5.14 illustrates an example of an FKT device simula-

ÎŚj

Erot,j

Hj

(F - Ec)j

kBTj

Poisi
Ampi

Fari
ElConti
EnConsi
Figure 5.14: An example of the Jacobian matrix calculated in the FKT device simulator. The
Jacobian matrix corresponds to a full-wave FKT solve. Dotted lines mark the boundaries of the
residual rows and solution variable columns in the Jacobian matrix. The rows of the Jacobian
correspond to Poisson’s equation, Ampere’s equation, Faraday’s equation, electron continuity, and
energy conservation residuals, respectively.
tion Jacobian. This Jacobian matrix corresponds to a full-wave simulation of the GaN MESFET
first presented in Section 5.5.1 and illustrated in Figure 5.6. Lattice heating was not included in
this FKT device simulation. The DV mesh of the MESFET is a standard example of a general
device simulation mesh. Therefore, this Jacobian can be considered a standard example of an FKT
device simulation Jacobian matrix. The sub-matrix blocks in the Jacobian matrix in Figure 5.14

115

500 1000 1500 2000 2500
500
1000
1500
2000
2500

Figure 5.15: An example of the Jacobian matrix after RCM re-ordering.
correspond to the five equations of the FKT device simulation. The sub-matrix blocks are outlined
with dashed lines. The governing equations of the full-wave simulation are Poisson’s equation,
Ampere’s equation, Faraday’s equation, electron continuity, and energy conservation. Located in
∂ Pois

the top left corner is the Poisson sub-matrix block corresponding to the elements ∂ Φ i .
j
∂ Far

One note about the Jacobian matrix in Figure 5.14 is about the Faraday sub-matrix block ∂ Φ i .
j
According to Eqn. (5.55), this sub-matrix should be zero. However, there are non-zero elements in
this matrix block in Figure 5.14. These non-zero elements correspond to the full-wave voltage port
BC described in Section 5.4.4. The voltage port BC requires magnetic field solution variables in
the plane of the boundary. These in-plane magnetic field variables are governed by Ampere’s law
and their residuals are located at the magnetic field solution variable rows. This dependence on
Ampere’s law gives rise to the non-zero Jacobian elements corresponding to the electric potential.
The linear solve computation time can be reduced by re-ordering the solution variables to re-

116

duce the bandwidth of the Jacobian matrix. The bandwidth of a matrix is defined as the largest
distance between the diagonal element and an off-diagonal element in a row. The overall bandwidth is the maximum distance of all rows. A banded matrix is one with a “reasonably small”
bandwidth. A standard method to reduce the matrix bandwidth is the reverse Cuthill-McKee algorithm (RCM) [90]. RCM aims to permute the sparse matrix into a banded matrix form with as small
of a bandwidth as possible. An implementation of the RCM algorithm is applied to the Jacobian
matrix of the GaN MESFET example. The resulting banded matrix is illustrated in Figure 5.15.
The bandwidth of the RCM re-ordered Jacobian matrix is 296.

5.8.2

Second-Order SG Discretization

Reduction of the Newton linear system size is another approach to alleviating some of the computational burden of simulating large devices. This could be accomplished by reducing the number
of mesh elements in the simulation. Coarsening the mesh, however, could have adverse affects
including inaccuracies in the simulated results and possible convergence issues.
A higher-order numerical framework would allow mesh coarsening without infringing on the
integrity of the simulated results. In Section 5.6, the mesh convergence of the FKT device simulator was presented and discussed. The FKT equations exhibit first-order mesh convergence due
to the SG discretization of the particle fluxes. Fluxes and electric field solution variables are approximated as spatially constant along the primary edges of the DV mesh which yields a first-order
method. This section expands upon the SG discretization to develop a second-order method. In
the following, the first-order particle flux device simulations are termed SG1 and the simulations
with the second-order fluxes are termed SG2.

5.8.2.1

Derivation of the SG2 Particle Flux

The new discretization of the FKT particle flux is derived and discussed. This discretization is an
extension of the original SG technique [77] which was presented in Section 4.3.2 and is derived in

117

Appendix A.3. The derivation starts with the phenomenological DD equation
Jn (r) = a1 n(r) + a2

dn(r)
,
dr

(5.61)

where a1 and a2 are constant on a primary edge of the DV mesh. The flux Jn (r) is approximated
with a second-order polynomial which yields
J2 (r − r0 )2 + J1 (r − r0 ) + J0 = a1 n(r) +

dn(r)
.
dr

(5.62)

Here, J0 , J1 , and J2 are coefficients of the polynomial. The original SG discretization technique
solved an inhomogeneous linear differential equation for the electron density. Then, BCs were
applied to yield the discrete form of the particle flux. This discretization follows the same approach.
Eqn. (5.62) is a linear first-order differential equation with a spatially dependent inhomogeneity.

dn

dn
The solution of Eqn. (5.62) with the BCs n(r = r0 ) = n0 , n(r = r1 ) = n1 , dr  = dx0 , and
r0

dn  = dn1 for J ≡ J (r − r )2 + J (r − r ) + J is
n
2
0
1
0
0
dr 
dx
r1



dn0
dn1
a2
B1 (Ξ )n0 + LB3 (Ξ )
− B2 (ξ )n1 − LB4 (ξ )
.
Jn =
L
dx
dx

(5.63)
a

Here, L is the length of the primary edge, the argument of the B functions is Ξ = a1 L, and the B
2
functions are
x (4 + ex (x − 4) + 3x)
,
4 (2 + ex (x − 2) + x)

(5.64)

x (4 + x + ex (3x − 4))
B2 (x) =
,
4 (2 + ex (x − 2) + x)

(5.65)

B1 (x) =

B3 (x) = −

4 + ex (x − 4) + x(x + 3)
,
4 (2 + ex (x − 2) + x)

(5.66)

B4 (x) = −

4 + x − ex (4 + x(x − 3))
.
4 (2 + ex (x − 2) + x)

(5.67)

Eqn. (5.63) is the second-order discretization of the DD flux. In this derivation, the flux is approximated with a second-order spatial polynomial. Because a second-order spatial polynomial exhibits
third-order mesh convergence, the overall discretization should be third-order. However, the FKT
particle will be second-order due to approximations made to the coefficients a1 and a2 . The density
m
gradient terms dn
dx are approximated with central differences across the primary edges.

118

The discrete flux, Eqn. (5.63), is similar to the original discrete particle flux, Eqn. (28), derived
in the appendix. There are two sets of terms which represent particles flowing from node r0 and
node r1 of the primary edge. This fact allows the heat flow algorithm to be applied to the secondorder discrete flux. Finally, the discretization of the general DD flux is applied to the generalized
Einstein FKT particle flux with the substitutions


a1 → −qD jk E j + ∆Einn, jk ,

(5.68)



a2 → −qD jk Einn, jk ave .

(5.69)

and

Because the second-order discretization is arranged in a similar manner to the first-order discretization, the argument of the B functions is the same as the argument of the Bernoulli functions in the
generalized Einstein flux. This argument is
Ξn, jk =

1



Einn, jk ave



E j + ∆Einn, jk L j .

(5.70)

In the following, validation and simulation results are presented with the SG2 particle fluxes.

5.8.2.2

Validation of the Equations: Particle Flux Reconstruction

Validation of the second-order particle flux discretization in the FKT device simulator is performed
by prescribing an analytic spatial profile to solution variables and calculating the discrete fluxes on
primary edges of the DV mesh. The discrete particle flux is compared to the analytic particle flux
to quantify a reconstruction error. The L2 error described in Section 5.6 is used.
An example of a flux reconstruction with an analytic solution variable profile is illustrated in
Figure 5.16. The left of the picture presents the structured cube mesh on which a flux reconstruction
error is calculated. Also shown in the figure is the analytic electric potential solution variable
profile. The reconstructed FKT particle fluxes are presented on the right of Figure 5.16. The
vector plot is an interpolation of the nodal vector data loaded into a visualization software. Scalar
vector projections onto primary edges of the DV mesh are compared to analytic vector projections
119

Figure 5.16: Flux reconstruction on a 10×10×10 structured cube mesh. An analytic profile is
prescribed for the electric potential, relative electron Fermi level, and electron temperature solution variables. The FKT particle fluxes are computed on all primary edges of the DV mesh and
compared to analytic fluxes to quantify a reconstruction error.
to calculate the global L2 error. This process is repeated on a series of meshes to quantify the mesh
convergence of the FKT particle flux reconstruction.
Order analysis results are presented in Figure 5.17 for the SG1 and SG2 FKT particle flux
reconstructions. Here, the L2 errors of the reconstructed particle fluxes are presented versus the
maximum primary edge length of the DV meshes. The errors corresponding to the SG1 particle
fluxes are plotted in red and the errors corresponding to the SG2 particle fluxes are plotted in blue.
As expected, the SG1 particle flux reconstructions exhibit first-order convergence. This result also
demonstrates that the SG2 particle fluxes derived in this section exhibit second-order mesh convergence. As described in Section 5.8.2.1, the particle flux is approximated with a second-order
polynomial and discretized by integrating the inhomogeneous first-order linear differential equation. Even though the particle flux is approximated with a third-order mesh convergent polynomial,
the overall particle flux is second-order due to approximations of the flux coefficients. The flux
coefficient approximations are described in Chapter 4 and Appendix A.3.

120

1.0e+00

L2 Error

1.0e-01

SG1
SG2

1.0e-02
1.0e-03

er
st
1 ord
nd

2

er
ord

1.0e-04
0.1

1

10

Maximum Edge Length h
Figure 5.17: Order analysis of the FKT particle flux reconstruction. The errors corresponding to
the SG1 particle fluxes are illustrated in red and the SG2 errors are illustrated in blue. As expected,
the SG1 particle flux reconstructions exhibit first-order convergence and the SG2 particle flux
reconstructions exhibit second-order convergence.
5.8.2.3

Four Particle System

Simulation of electronic devices with the SG2 FKT particle flux discretization begins with a simple
1D mesh with four nodes. The two boundary nodes are designated as Ohmic contacts. This
simulation has two interior DOF. A dopant discontinuity exists at the middle of the device. The first
two nodes are doped with 1012 cm−3 donors and the second two are doped with 1014 cm−3 donors.
Figure 5.18 presents the I-V curves computed with the SG1 and SG2 FKT device simulators. The
device currents calculated with the SG1 simulator are plotted in black and the device currents
calculated with the SG2 simulator are plotted in red. The SG2 device simulator failed to converge
at approximately Va = 0.2 V. Furthermore, the slope of the anode current changes sign before the
simulator crashes.
The previous results hint at a stability issue of the SG2 particle flux. To investigate this issue,
the phase portrait of the two DOF system is calculated for stability analysis. Figure 5.19 presents
the direction field drawn with black vectors and four trajectories drawn with blue lines. The x-axis

121

Ia (A)

1.0e-05
1.0e-06
1.0e-07
1.0e-08
1.0e-09
1.0e-10
1.0e-11
1.0e-12
1.0e-13
1.0e-14

SG1
SG2
0

0.2

0.4

0.6

0.8

1

Va (V)
Figure 5.18: The simulated static I-V of the four particle system with two interior DOF. SG1
device simulation currents are plotted in black and SG2 currents are plotted in red.

1.0

N1 (x 1016 cm-3)

0.8
0.6
0.4
0.2
0.0

0.5

1.0

1.5

2.0

N0 (x 1017 cm-3)
Figure 5.19: The phase portrait of the four particle, two DOF system. The direction field is plotted
with black vectors and four trajectories are plotted with blue lines. Initial conditions of the trajectories are plotted with blue circles. There are three equilibria of the system of nonlinear equations,
two of which are unstable.

122

represents the density N0 of the first particle and the y-axis represents the second particle’s density
N1 . The trajectories represent the integrated SG2 nonlinear equations with four different initial
conditions drawn with blue circles. There are several equilibria of the SG2 system of equations.
The direction field indicates that two of the equilibria are unstable. These multiple and unstable
equilibria are a known issue of higher-order methods in the computational fluid dynamics (CFD)
community.

5.8.2.4

Flux Limiters

A well known mathematical result from the CFD community is Godunov’s theorem. It states that
linear numerical schemes for solving PDEs, having the property of not generating new extrema,
can be at most first-order accurate [91]. The converse of this states that higher-order accurate
numerical methods will generate new extrema. This fact is numerically demonstrated with the
SG2 phase portrait in Figure 5.19. New extrema are generated by increasing the particle flux order
from first to second. These equations are no longer monotone and therefore not unique and are
unstable.
A substantial effort in the CFD community was devoted to the development of high-resolution
schemes for fluid-flow simulations. Specifically, Harten developed a measure of discrete variation in the solution fields which is called total variational diminishing (TVD) [92]. A numerical
technique called a “flux limiter” was developed in order to overcome the restrictions imposed by
Godunov’s theorem and enforce the monotonicity criterion set forth by Harten. Several versions
of flux limiters exist in the literature with a concise summary provided in [93].
A flux limiter attempts to diminish spurious oscillations in solutions near discontinuities and/or
sharp changes in the solution domain. The oscillations are a consequence of the new extrema
generated by the higher-order method. In terms of the FKT device simulator, this includes sharp
changes in electron density at dopant fronts and inflection points in the electric potential near metal
contacts. The particle fluxes near these device features are restricted to enforce monotonicity.
A smooth variation between low and high order is enforced based on numerical criteria. The

123

functional form of this variation is


J = Jlow − φ (r) Jlow − Jhigh .

(5.71)

Here, Jlow represents the lower-order, monotonic flux and Jhigh represents the higher-order flux.
The function φ (r) is the flux limiter. If φ (r) = 0, then the overall flux is first-order. A second-order
flux results from φ (r) = 1. Standard practice enforces a specific functional form for the flux limiter
φ (r) which smoothly varies from 0 to 1 [93]. An example of a flux limiter is the van Albada form
r2 + r
φ (r) = 2
.
r +1

(5.72)

Here, the term r represents successive gradients of electron density on a 1D mesh, or
N − Ni−1
.
ri = i
Ni+1 − Ni

5.8.2.5

(5.73)

Higher-Order Simulation of a 1D Diode

The flux limiter described in the previous section is applied to the SG2 particle fluxes in the FKT
device simulator. The Jlow term of Eqn. (5.71) represents the SG1 particle flux and Jhigh represents
the SG2 particle flux. The SG2 particle flux is given in Eqn. (5.63).
A 1D n+ -n diode is simulated with the SG2 FKT device simulator. The dopant density of the
n+ region is 3 × 1017 cm−3 donors. The second region is lightly doped with 1012 cm−3 donors.
Figure 5.20 presents an illustration of the thermal equilibrium band diagram calculated with the
SG1 and SG2 device simulators. The conduction and valence bands are drawn in black and the
electron Fermi level F is drawn in red. In this figure, the SG1 results are illustrated with solid lines
and the SG2 results are illustrated with squares. This result demonstrates that the SG2 equations
properly reproduce the thermal equilibrium band diagram.
The simulated I-V curves of the 1D diode are presented in Figure 5.21. Here, the SG1 results
are plotted in black and the SG2 results are plotted in red. The SG2 current is similar to the current
calculated with the SG1 device simulator. The difference resides in the fact that the SG2 device
simulation is more accurate and therefore provides different results.
124

Energy (eV)

-4.6
-4.8
-5
-5.2
-5.4
-5.6
-5.8
-6
-6.2
-6.4
-6.6

Ec
F

Ev

0

2

4

6

8

10

Diode x-direction (Îźm)

Ia (A)

Figure 5.20: The simulated thermal equilibrium band diagram of the 1D diode. The SG1 results
are plotted with solid lines and the results computed with the SG2 solver are plotted with squares.

1.4e-06
1.2e-06
1.0e-06
8.0e-07
6.0e-07
4.0e-07
2.0e-07
0.0e+00
-2.0e-07

SG1
SG2
0

0.2

0.4

0.6

0.8

1

Va (V)
Figure 5.21: The simulated static I-V of the 1D diode. SG1 device simulation currents are plotted
in black and SG2 currents are plotted in red.
Table 5.4 presents the error in the FKT solution variables versus the number of DOF. In a
manner similar to the mesh convergence analysis in Section 5.6, the solutions on each mesh are

125

compared to the solutions on the most dense mesh. The first column of Table 5.4 lists the DOF
for each mesh. The second through fifth columns present the relative error of the SG1 solutions,
the approximate order of convergence of the SG1 errors, the relative error of the SG2 solutions,
and the approximate order of convergence of the SG2 errors, respectively. As indicated by the
DOF

Îľk (SG1)

pk (SG1)

Îľk (SG2)

pk (SG2)

30
60
120
180

2.1e-1
1.4e-1
7.7e-2
3.5e-2

1.2
1.2
1.2
1.2

3.7e-1
2.7e-1
1.1e-1
2.4e-2

1.8
1.8
1.8
1.8

Table 5.4: Solution variable errors and approximate order of convergence of the SG1 and SG2
FKT device simulations of the 1D diode.
results in Table 5.4, the FKT device simulator with the SG2 particle fluxes exhibits higher-order
convergence than the FKT device simulator with the SG1 particle fluxes. The order of convergence
is not p = 2.0, however, because the flux limiter varies between 0 and 1 across the DV mesh. This
yields particle fluxes between first- and second-order and the overall order of convergence for the
1D diode example is pk = 1.8.

5.9

Conclusion

A detailed description of the numerical solution of the discrete FKT device simulation equations was presented in this chapter. The discrete system of nonlinear equations was summarized
in Section 5.2, a detailed description of the FKT solution variables was presented in Section 5.2.1,
and the Newton’s method approach to solving the nonlinear equations was presented in Section 5.3.
BCs used for simulating electronic devices were presented and discussed in Section 5.4. An
overview of the semiconductor device simulation work-flow was presented in the next section.
This included examples of the Jacobian matrices for static, quasi-static, and full-wave FKT device
simulations. The mesh convergence and the stability of the FKT device simulation equations were
then analyzed in Sections 5.6 and 5.7, respectively. The results presented in these sections provided
quantitative evidence that the FKT device solver is first-order convergent and numerically robust

126

and stable for all device examples that were investigated. Finally, advanced simulation techniques
were reported in the final section. Some aspects of the Newton linear solve were presented in Section 5.8.1. An extension of the SG discretization scheme was then presented which increased the
mesh convergence to nearly second-order. Section 5.8.2 provided derivation and implementation
details of the second-order SG scheme.

127

CHAPTER 6
RF SIMULATIONS OF GAN HEMT TECHNOLOGY

6.1

Introduction

Physics based modeling of GaN HEMT PAs offers a promising alternative to high frequency
and large-signal measurements. However, the physics based TCAD tool must accurately capture
the physical processes in the device to provide useful data to RF circuit designers. The salient
features required for accurate large-signal RF simulations include electronic band-structure, hotelectron effects, self-heating, scattering, trapping, and full-wave EM.
PA devices with GaN technology are scaled to deep sub-micron gate lengths. This allows high
frequency operation of the TR module. However, because the devices are scaled in physical size
but the applied voltages cannot be scaled in a similar way, the GaN transistors are subject to large
electric fields. This causes the electrons in the channel of the device to heat up substantially and
begin to emit phonons. This is the hot-electron effect. Figure 6.1 illustrates the electron drift ve-

Drift Velocity (x 107 cm/s)

locity as a function of the applied electric field calculated with MC [94, 95]. The electron velocity

3.5
3
2.5
2
1.5
1
0.5
0
0

100

200

300

400

500

600

Electric Field (kV/cm)
Figure 6.1: Electron drift velocity vs. electric field of bulk GaN calculated with MC.

128

peaks at approximately 200 kV/cm then begins to decrease in value. This is velocity overshoot and
saturation — a result of the hot-electron effect. It is critical that the TCAD simulation tool captures
hot-electron effects in order to produce accurate data.
The hot-electron effect can be mathematically explained with the second moment of the BTE.
The Joule’s heat source term in the energy conservation equation drives the electron temperature
and is eventually balanced by the collision operator sink term. Because the large electric fields
create hot electrons in the device which in turn interact with the lattice, it is necessary to include
lattice heating in the device simulator. The lattice energy conservation equation provides a modest
account for phonon temperature in the device.
Both physical effects, as well as full-wave EM, can be incorporated into deterministic and
stochastic device simulators. As outlined in the introduction, two important categories of charge
transport solvers are the hydrodynamic and MC methods. Both solvers have been applied to GaN
HEMT simulations.
Hydrodynamic solvers are widely used in the device simulation community for analysis and
characterization of power transistors. An AlGaN/GaN HEMT with surface traps was investigated
with the commercially available DESSIS hydrodynamic software in [96]. This contribution did
not include RF simulations, however. Self-heating and hot-electron effects were studied in AlGaN/GaN double-channel HEMTs in [97], but also did not include RF simulations. Finally, the
two-dimensional device simulator Minimos-NT [98] was used to accurately simulate both the static
and small-signal response of several GaN HEMTs. Hydrodynamic solvers attempt to incorporate
electronic band-structure and hot-electron effects through fitting parameters. For example, a fielddependent mobility is tuned in [98] to model velocity saturation in GaN. Furthermore, hydrodynamic solvers rarely incorporate full-wave EM, with the exception of a 2D solver used to simulate
THz plasma waves [99].
Many EMC and CMC methods have successfully simulated the responses of GaN HEMTs.
Chronologically, this list (which is not claimed to be complete) begins with quantum corrected
full-band CMC simulations [100]. Self-heating effects in sub-micron GaN HEMTs were calculated

129

with an electro-thermal MC method [101] and another MC method investigated the influence of
source-gate distance on AlGaN/GaN HEMT performance [102]. Two separate electro-thermal MC
analyses of GaN HEMTs were also presented in [103] and [104]. Finally, a full-band CMC solver
was applied to large-signal RF simulations of GaN HEMTs in [28]. Although the MC simulations
are accurate, the inclusion of full-wave EM is rare with the exceptions of [66] and [105]. A fullwave MC method, however, will suffer from long computational times.
FKT is a promising deterministic Boltzmann solver for large-signal RF GaN HEMT simulations. Features included and physical effects captured in the FKT device simulator are:
• Electronic band-structure calculated with EPM
• Hot-electron effects determined by higher moments of the BTE
• Self-heating effects by inclusion of the lattice heating equation
• Quantum mechanical scattering computed by Fermi’s golden rule
• Full-wave electromagnetics calculated with Ampere’s and Faraday’s laws
The derivation, discretization, and analysis of the FKT equations was presented in Sections 3 – 5.
In the following sections, a high speed GaN HEMT is simulated with the FKT device TCAD tool.

6.2

AlGaN/GaN HEMT Topology

Thermal equilibrium, static, and RF small- and large-signal FKT simulations of a single gate
AlGaN/GaN HEMT are presented. The AlGaN/GaN HEMT is a high frequency device which
demonstrated a record fmax of 230 GHz at a specific quiescent bias [106]. This device received
considerable attention from the device simulation community. It was first used to study the effects
of threading dislocations using the CMC method [107]. The GaN HEMT was also featured in an
Institute of Electrical and Electronic Engineers (IEEE) magazine article describing modeling and
simulation of terahertz devices with the CMC technique [108]. FKT device simulations of this
device were also presented in [84] and [109].

130

The layout of the AlGaN/GaN HEMT is illustrated in Figure 6.2. This is the 3D mesh used
for the FKT simulations. The stack of the device is 11 nm of GaN and 13 nm of Al0.32 Ga0.68 N.
The interfaces of the stack are oriented in the z-direction of the device. Detailed illustrations of
the HEMT stack are reported in [106, 107, 108]. The stack sits on a GaN buffer which is grown
on a SiC substrate. The Ohmic source and drain contacts of the transistor are the yellow regions

Ohmic
Source
n+-GaN

Schottky Gate
GaN

Ohmic
Drain

AlGaN

Input Port

Launch

SiC

Figure 6.2: The 3D mesh used for FKT simulations of an AlGaN/GaN HEMT.
at the top left and right of the device. The middle contact is the Schottky gate. Not included in
the illustration is the Ohmic ground contact at the bottom of the mesh. The SiC launch is located
at the front of the device. The input port BC of the launch is shaded in red. Not illustrated is the
output port BC connecting the Ohmic drain and the Ohmic ground metal contacts.

6.3

Thermal Equilibrium and Static Simulations

The thermal equilibrium of the HEMT is computed first with the FKT device simulator. As
outlined in previous sections, the thermal equilibrium solution variables are the initial guess of the
first static solve Newton iteration. Figure 6.3 presents the thermal equilibrium electric potential
profile in the transistor.

The Schottky T-gate of the device has an electric potential of approx-

imately 0.65 V and is shaded in green. The barrier of the Schottky gate is ÎŚSB = 1.2 eV. The
blue region surrounding the T-gate is air. The Ohmic contacts both have an electric potential of

131

Electric Potential

Figure 6.3: The thermal equilibrium electric potential profile of the AlGaN/GaN HEMT computed
with the FKT device simulator.

Ec

Energy (eV)

-4

F

-5
-6
-7

Ev

-8
-9
0

5

10

15

20

25

30

HEMT z-direction (nm)
Figure 6.4: The thermal equilibrium band diagram (calculated across the stack under the Schottky
gate) of the AlGaN/GaN HEMT computed with the FKT device simulator.
approximately 1.1 V — the red shaded regions.
A band diagram of the GaN HEMT computed with the FKT device simulator is presented in
Figure 6.4. This result is calculated across the HEMT stack under the Schottky gate. The beginning
of the GaN channel is located just before the 5 nm position in the band diagram. The AlGaN/GaN
heterojunction is located at 15 nm in the plot. The polarization charge density resulting from this
heterojunction yields the 2DEG of the HEMT. Highly degenerate electron gases reside near this
junction in the GaN channel.
Static simulations of the AlGaN/GaN HEMT are presented based on the work in [84]. An

132

example of the quiescent bias of the HEMT calculated with FKT is illustrated in Figure 6.5. The
static solver consists of Eqns. (5.1), (5.4), (5.5), and (5.6). Trapping effects are also included to
properly determine the mobile electron densities in the GaN channel [62, 84]. Figure 6.6 presents
the static I-V family of the AlGaN/GaN HEMT computed with the FKT device solver.

The

Electric Potential

Drain Current ID (mA/mm)

Figure 6.5: The electric potential profile of the AlGaN/GaN HEMT in quiescent bias. The gatesource bias is -1 V and the drain-source bias is 10 V.

1400
1200
1000
800
600
400
200
0
-200
0

2

4

6

8

10

12

14

16

Drain-Source Bias VDS (V)
Figure 6.6: The static I-V family of the AlGaN/GaN HEMT. The solid black lines represent FKT
simulations with different gate biases and the red dots are the measured equivalents. The gate
biases range from 1 V to -3 V with steps of 1 V. This result was first presented by Matt Grupen
AFRL.

133

Drain Current ID (mA/mm)

FKT

2500

DD

2000
FKT + Lattice
1500
FKT + Lattice + Traps
1000
500
0
0

2

4

6

8

10

12

14

16

Drain-Source Bias VDS (V)
Figure 6.7: A comparison of the static I-V families of the AlGaN/GaN HEMT computed with the
different FKT device solvers. The DD, FKT, FKT with lattice heating, and the FKT solver with
lattice heating and trapping effects are illustrated with dashed blue, dashed red, dashed green, and
solid black lines, respectively. The measurements are illustrated with red dots.
gate biases range from VGS = 1 V to -3 V with steps of 1 V. The drain-source bias, VDS , is swept
from 0 to 15 V at each gate bias. FKT simulations are illustrated with solid black lines and the
measurements [107] are illustrated with red dots.
Proper treatment of the electronic band-structure, the hot-electron and self-heating effects, the
scattering, and the deep traps in the simulation framework is critical for accurately reproducing
the measured data of the device. Figure 6.7 presents a comparison of several different device
simulations. The DD (no energy transport, scattering, lattice heating or traps), FKT (no lattice
heating or traps), FKT with lattice heating (no traps), and the complete device simulator are drawn
with dashed blue lines, dashed red lines, dashed green lines, and solid black lines, respectively.
Again, the measurements [107] are illustrated with red dots. The DD and FKT simulations severely
overestimate the drain currents. The FKT simulation with lattice heating provides better results.
Finally, the complete device simulation exhibits excellent agreement with the measurements. The
deep traps are required to properly model the channel resistance.
Finally, the transconductance of the AlGaN/GaN HEMT is calculated. The gate-source bias,

134

VGS , is swept from 1 V to device pinch-off at a specific drain-source bias, VDS . The transconductance is calculated with Eqn. (2.20). Figure 6.8 presents the transconductance of the HEMT at a
drain-source bias of VDS = 10 V. Here, the solid black line illustrates the drain current, ID , and

Drain Current ID (mA)

90

25

80
20

70
60

15

50
40

10

30
20

5

10
0

Transconductance (mS)

red line represents the transconductance, gm . This simulation is particularly useful to RF circuit

0
-4

-3

-2

-1

0

1

Gate-Source Bias VGS (V)
Figure 6.8: The transconductance of the AlGaN/GaN HEMT. The solid black line represents the
drain current and the red line is the transconductance of the device. The drain-source bias is 10 V.
designers. First, it can be imported as the transconductance component of a small- or large-signal
equivalent circuit. It is also used to determine the quiescent bias for RF simulations/measurements.
The device is typically operated at peak transconductance to provide the largest power gain.

6.4

Small-Signal RF Simulations

Simulations of the AlGaN/GaN HEMT with low input power are next presented. The linear
two-port simulation of the device follows the discussion in Section 2.4.2. To this end, the SParameters of the device are calculated with the quasi-static and full-wave solvers and used to
quantify the small-signal power gain and small-signal current gain.
Calculating the S-Parameters, Eqns. (2.30) – (2.33), requires two FKT device simulations. For
the first FKT simulation, a voltage port is assigned to the input of the device and an AC impedance
port is assigned to the output of the device. The impedance presented to the output is chosen

135

such that ΓL = 0. This ensures that A2 = 0 and allows the calculation of S11 and S21 . The input
port of the quasi-static simulation is the Schottky gate contact and the output port is the Ohmic
drain contact. For full-wave simulations, the input port is the boundary connecting the Schottky
gate and Ohmic ground contacts and the output port is the boundary connecting the Ohmic drain
and the Ohmic ground contacts. The second simulation reverses the port BCs and presents the
AC impedance to the input device such that ΓS = 0. The final two S-Parameters S12 and S22 are
calculated from this simulation. Because the reference impedance is Z0 = 50 Ω, both input and

0.1
0.08
0.06
0.04
0.02
0
-0.02
-0.04
-0.06

V1(t)
V2(t)

0

10

20

30

Current (mA)

Voltage (V)

output AC impedances are 50 Ω.

40

50

Time (ps)

3
2.5
2
1.5
1
0.5
0
-0.5
-1
-1.5
-2

I1(t)
I2(t)

0

10

20

30

40

50

Time (ps)

(a)

(b)

Figure 6.9: The (a) AC input and output voltages and (b) AC input and output currents of the
AlGaN/GaN HEMT calculated with the quasi-static solver. The voltage BC is assigned to the
input port and the AC impedance port BC is assigned to the output. For the quasi-static solver, the
input port is the Schottky gate metal contact and the output port is the Ohmic drain metal contact.
Figures 6.9a and 6.9b present the transient input and output signals of the AlGaN/GaN HEMT
computed with the quasi-static solver. Figures 6.10a and 6.10b present the same results computed
with the full-wave solver. The input port is a voltage BC and the output port is terminated with a
50 Ω AC impedance BC. The Fourier transforms of these signals are used to calculate the incident
waves, Eqn. (2.25), and scattered waves, Eqn. (2.26). The S11 and S21 S-Parameters are calculated
with Eqns. (2.30) and (2.31), respectively. The relation between the output voltage V2 (t) and output
current I2 (t) deserves special attention. The AC impedance BC requires a specific relationship
between its integrated voltage and current. As discussed in Sections 5.4.3 and 5.4.5, the current

136

V1(t)
V2(t)

0

20

40

60

Current (mA)

Voltage (V)

0.12
0.1
0.08
0.06
0.04
0.02
0
-0.02
-0.04
-0.06

80

100

1
0.8
0.6
0.4
0.2
0
-0.2
-0.4
-0.6

I1(t)
I2(t)

0

20

40

Time (ps)

60

80

100

Time (ps)

(a)

(b)

Figure 6.10: The (a) AC input and output voltages and (b) AC input and output currents of the
AlGaN/GaN HEMT calculated with the full-wave solver. The voltage port BC is assigned to the
input port and the AC impedance port BC is assigned to the output. For the full-wave solver,
the input port connects the Schottky gate and the Ohmic ground and the output port connects the
Ohmic drain and the Ohmic ground.
is prescribed to be exactly −V2 (t)/50. This relationship is qualitatively observed in Figures 6.9

0.5

2.0

1.0

and 6.10.

5.0

0.2

10

10
20

5.0

2.0

1.0

0.5

20

-10
-20

S11

-1.0

0
-2.

-0.
5

S22

-5.0

-0.2

Figure 6.11: Two S-Parameters of the AlGaN/GaN HEMT computed with the FKT device simulator. The solid lines represent the full-wave results and the dashed lines represent the quasi-static
results. The quiescent bias of the device is a drain-source bias of 6 V and no gate-source bias.

Two S-Parameters, S11 and S22 , are presented in Figure 6.11. Because the source and load
137

impedances are chosen to be ZS = ZL = 50 Ω, these S-Parameters represent the input and output reflection coefficients, i.e., Γin = S11 and Γout = S22 . The S-Parameters are plotted versus
frequency. Dashed lines represent the quasi-static results and solid lines represent the full-wave
results. Substantial differences exist between the quasi-static and full-wave S-Parameters at high
frequency. These differences are due to inductive coupling of the metal contacts.

25
Quasi-static
Full-wave
Measured

h21 (dB)

20
15
10
5
0
10

100

Frequency (GHz)
Figure 6.12: The simulated (black lines) and the measured (red squares) small-signal current gain
of the AlGaN/GaN HEMT.
A comparison of the simulated and measured small-signal current gain is reported in Figure 6.12. The small-signal current gain is calculated with Eqn. (2.43). The dashed black line
represents the quasi-static result and the solid black line is the full-wave result. The red squares
represent the measurements [107]. It is evident that treatment of full-wave EM is required in the
FKT simulator in order to properly capture the device parasitics at high frequency. This allows the
device solver to compute accurate S-Parameters and provide useful data for RF circuit designers
to synthesize high frequency matching networks for the GaN HEMT.
Finally, the small-signal power gain of the AlGaN/GaN HEMT is presented. Figure 6.13 reports the available power gain (black lines) calculated with Eqn. (2.41) and the transducer power
gain (red lines) calculated with Eqn. (2.42). The dashed lines represent the quasi-static power gains
and the solid lines are the power gains calculated with the full-wave solver. This result provides

138

25

GA

Quasi-static
Full-wave

Gain (dB)

20

GT

15
10
5
0
-5
-10
0

20

40

60

80

100

Frequency (GHz)
Figure 6.13: The simulated available power gain (black) and transducer power gain (red) of the AlGaN/GaN HEMT. Dashed lines represent the quasi-static power gains and the solid lines represent
the full-wave power gains.
further evidence that the full-wave solver is crucial for simulating the small-signal response of
GaN HEMT technology at high frequency.

6.5

Large-Signal RF Simulations

As outlined in Section 2.4.3, S-Parameters do not accurately model the nonlinear response
of high power transistors. Large-signal simulations of the transistors, including LP and power
sweeps, are therefore used for PA design. LP data determine the optimal loading conditions for
maximizing specific FOMs and power sweep results provide further information for optimal operating conditions. This section focuses on utilizing the FKT device solver framework for largesignal RF simulations of GaN HEMT technology. The quasi-static AC impedance is tailored to
simulate transistors with arbitrary reflection coefficients in the following subsection. Then, a discussion on calculating the large-signal FOMs with the FKT device solver is presented. Finally, this
section concludes with LP and power sweep simulations of the GaN HEMT and comparisons of
quasi-static and full-wave large-signal results.

139

6.5.1

Quasi-Static AC Load Impedance for LP Simulations

A critical component of large-signal RF simulations is the load impedance presented to the output
port of the transistor. Because the load reflection coefficient is sampled over the Smith chart, the
FKT device solver must incorporate impedances with reactive components. Section 5.4.3 discussed
the quasi-static AC impedance BC. This BC uses the total integrated current flowing into the
boundary, designated it+1 , to specify a Dirichlet condition on the electric potential. The Dirichlet
condition is


ÎŚt+1
i 

ACI

= ZACI it+1
ACI ,

(6.1)



where ÎŚt+1
i ACI represents the electric potential on the quasi-static AC impedance BC. To sample
the upper half of the Smith chart, the load is specified as a series RL impedance. In the TD, the BC
is
ÎŚt+1
= Rit+1 + L
i

it+1 − it
,
∆t

(6.2)

where R and L are the real and imaginary components of the load impedance ZL at the fundamental
frequency. The subscript ACI is dropped for brevity. The inverse of Eqn. (2.28) gives the load
impedance in terms of the reflection coefficient, or
ZL =

1 + ΓL
Z .
1 − ΓL 0

(6.3)

Sampling the lower half of the Smith chart requires RC load impedances which are implemented
in a similar manner.

6.5.2

Computing Large-Signal FOMs

The large-signal FOMs of the devices are computed at a single operating condition. Specifically,
the GaN HEMT at a given quiescent bias is driven by a single sinusoid with a large-signal amplitude. Furthermore, a load-impedance is presented to the output port of the device with the AC
quasi-static BC.
Figures 6.14 and 6.15 present the transient voltages and currents at both the input and output
ports calculated with the quasi-static FKT device solver. The quiescent bias of the GaN HEMT is
140

-0.5

CW

-1.5

V2(t) (V)

V1(t) (V)

-1

-2
-2.5
-3
-3.5
0

50

100

150

200

250

300

12
11.5
11
10.5
10
9.5
9
8.5
8

CW

0

50

Time (ps)

100

150

200

250

300

Time (ps)

(a)

(b)

Figure 6.14: The (a) input and (b) output TD voltages of the AlGaN/GaN HEMT calculated with
the quasi-static solver. The operating point of the HEMT is -2 V gate-source bias and 10 V drainsource bias. The large-signal tone (gate-source voltage) is 1 V in amplitude at 35 GHz.
0.015

0.045

CW

0.01

0.035

0.005

I2(t) (A)

I1(t) (A)

CW

0.04

0
-0.005

0.03
0.025
0.02

-0.01

0.015

-0.015

0.01
0

50

100

150

200

250

300

Time (ps)

0

50

100

150

200

250

300

Time (ps)

(a)

(b)

Figure 6.15: The (a) input and (b) output TD currents of the AlGaN/GaN HEMT calculated with
the quasi-static solver. The operating point of the HEMT is -2 V gate-source bias and 10 V drainsource bias. The large-signal tone (gate-source voltage) is 1 V in amplitude at 35 GHz.
VGS = −2 V and VDS = 10 V. The HEMT is driven by a large-signal sinusoid with a 1 V amplitude
at 35 GHz fundamental frequency. The load reflection coefficient presented to the output port is
ΓL = 0.44 − 0.17 j at 35 GHz. The device is simulated for 10 cycles of the driving tone to ensure
that all transient signals are continuous wave (CW).
The CW portion of the transient signals are Fourier transformed to calculate their spectral content. These signals are interpolated such that the first discrete frequency component corresponds
to the zeroth or DC frequency and the second discrete frequency component corresponds to the

141

40

V2(f) (dBm)

V1(f) (dBm)

35
30
25
20
15
10
5
0
-5
-10
-15
-20

30
20
10
0

0

50

100

0

150

50

100

150

Frequency (GHz)

Frequency (GHz)

(b)

(a)

10
5
0
-5
-10
-15
-20
-25
-30
-35

I2(f) (dBm)

I1(f) (dBm)

Figure 6.16: The magnitudes of the (a) input and (b) output voltage phasors of the AlGaN/GaN
HEMT driven by a large-signal tone. These voltages are plotted in dBm. The five harmonics are
computed by Fourier transforms of the transient voltages.

0

50

100

150

15
10
5
0
-5
-10
-15
-20
-25
0

Frequency (GHz)

50

100

150

Frequency (GHz)

(a)

(b)

Figure 6.17: The magnitudes of the (a) input and (b) output current phasors plotted in dBm. These
five harmonics are computed by Fourier transforms of the transient currents.
fundamental frequency of the large-signal tone. Figures 6.16 and 6.17 present the first five harmonics of the voltages and currents at the input and output ports of the GaN HEMT driven by the
large-signal RF tone. The magnitudes of the spectral voltages and currents are plotted in dBm.
The first harmonic of each plot is used to calculate the FOMs of the PA. The power available
from the source and the input power of this particular operating condition are Pavs = −2.04 dBm
and Pin = −2.83 dBm, respectively. The transistor generates an output power of Pout = 4.33 dBm
which corresponds to a power gain of G = 7.17 dB and a transducer power gain of GT = 6.39 dB.

142

6.5.3

LP Simulations

Optimal load impedances corresponding to specific operating conditions (bias, frequency, input
power) are determined through LP simulations. To this end, the load reflection coefficient presented to the output port of the device is swept over a wide range of the Smith chart. Figure 6.18

0.5

2.0

1.0

m3
impedance = Z0 * (1.007 + j0.662)

m3

10
20
10
20

5.0

2.0

1.0

0.2
0.5

5.0

-10
-20

-0.2

-1.0

0
-2.

-0.
5

-5.0

Figure 6.18: An example of reflection coefficient samples of the Smith chart that can be used for
LP simulations.
presents a collection of reflection coefficients that can be used for LP simulations. The marker
“m3” points to a reflection coefficient corresponding to an impedance of ZL = 50.4 + j33.1 Ω.
The GaN HEMT is simulated with the FKT device solver for each reflection coefficient in the
LP sample set. All desired FOMs are calculated from the CW voltages and currents. Finally,
these FOMs are loaded into ADS and contours of each FOM are plotted on the Smith chart. The
following sections will utilize this work flow for PA design.

143

5.0

-10
-20

-10
-20

-0.2

-1.0

Pavs = -2 dBm
f0 = 25 GHz

0
-2.

-0.
5

0
-2.

-1.0

-0.
5

-5.0

Pavs = -2 dBm
f0 = 15 GHz

-5.0

-0.2

10
20

5.0

2.0

20

1.0

10

20

0.5

10

10
20

5.0

2.0

0.2
1.0

5.0

0.2

0.5

2.0

0.5

2.0

GT = 9.1 dB
ZL = 54 + j105 Ί

1.0

1.0

0.5

GT = 9.3 dB
ZL = 104 + j130 Ί

(a)

(b)

Figure 6.19: LP contours calculated from FKT device simulations of the GaN HEMT biased as
a class A amplifier (gate-source bias of 1 V and drain-source bias of 10 V) with -2 dBm power
available from the source at a fundamental frequency of (a) 15 GHz and (b) 25 GHz. The optimal
loads for maximum transducer gain are 104 + j130 Ω at 15 GHz and 54 + j105 Ω at 25 GHz. The
peak gain is 9.3 dB at 15 GHz and 9.1 dB at 25 GHz. The step size of the contours is 0.2 dB.
6.5.4

Class A Amplifier

Large-signal simulation results of the GaN HEMT biased as a class A amplifier are presented in
this section. The first step in the PA design process is LP simulation with moderate input power.
Multiple LP data sets are collected from the GaN HEMT simulations at various fundamental frequencies. The quiescent bias of the amplifier is VGS = 1 V and VDS = 10 V. The amplifier is driven
by a power available from the source of Pavs = −2 dBm.
Figure 6.19a presents the LP data calculated from FKT device simulations of the class A amplifier operating at 15 GHz. The FOM is the transducer gain of the device. Peak gain resides at
a load impedance of ZL = 104 + j130 Ω at the fundamental frequency. The LP data set of the
device operating at 25 GHz is next reported in Figure 6.19b. The load impedance corresponding
to maximum transducer gain shifts to ZL = 54 + j105 Ω at 25 GHz. The last two sets of LP data
are presented in Figures 6.20a and 6.20b. The former is calculated with a fundamental frequency
of 35 GHz and the latter with 45 GHz.

144

5.0

-10
-20

-10
-20
-0.2

-1.0

Pavs = -2 dBm
f0 = 45 GHz

0
-2.

-0.
5

0
-2.

-1.0

-0.
5

-5.0

Pavs = -2 dBm
f0 = 35 GHz

-5.0

-0.2

10
20

5.0

2.0

20

1.0

10

20
0.5

10
10
20

5.0

2.0

0.2
1.0

5.0

0.2

0.5

2.0

0.5

2.0

GT = 8.8 dB
ZL = 16 + j67 Ί

1.0

1.0

0.5

GT = 9.0 dB
ZL = 25 + j82 Ί

(a)

(b)

Figure 6.20: LP contours calculated from FKT device simulations of the GaN HEMT biased as
a class A amplifier (gate-source bias of 1 V and drain-source bias of 10 V) with -2 dBm power
available from the source at a fundamental frequency of (a) 35 GHz and (b) 45 GHz. The optimal
loads for maximum transducer gain are 25 + j82 Ω at 35 GHz and 16 + j67 Ω at 45 GHz. The peak
gain is 9.0 dB at 35 GHz and 8.8 dB at 45 GHz. The step size of the contours is 0.2 dB.
The next step in the PA design process is sweeping the input power with the optimal load
impedance determined from the LP data. Because the input power depends on the loading condition as well as the intrinsic device parasitics, it is not trivial to sweep this quantity in the FKT
device simulator. Therefore, the power available form the source, Pavs , is swept across a broad
range and the resulting FOMs are plotted versus the calculated input power, Pin .
Figure 6.21a presents the input power sweep of the class A amplifier operating at a fundamental frequency of 15 GHz. In this figure, the output power, Pout , is plotted in black, the transducer
gain, GT , is plotted in red, and the PAE is plotted in blue. The power gain of the device is relatively constant at low input power. The 1 dB compression point is approximately -10 dBm. Figures 6.21b, 6.22a, and 6.22b report the input power sweeps of the HEMT at 25 GHz, 35 GHz, and
45 GHz, respectively. These simulations provide quantitative trends in the gain and efficiency of
the GaN HEMT versus fundamental frequency. The HEMT is more efficient at lower frequency.
Maximum PAE is achieved at an input power of approximately 10 dBm at 15 GHz. The gain of the

145

device is vastly reduced to approximately 2–3 dBm at 45 GHz. These high power results could be

f0 = 15 GHz
ZL = 104 + j130 Ί

14

Pout

12

15

10
10

GT

PAE

8
6
4

5

2
0

0
-30

-20

-10

0

10

20

f0 = 25 GHz
ZL = 54 + j105 Ί

14

Pout

15

12
10

PAE

GT

10

8
6

PAE (%)

20

PAE (%)
Power (dBm), Gain (dB)

Power (dBm), Gain (dB)

useful to circuit designers for determining trade-offs to meet specific criteria.

4

5

2
0

20

0
-30

-20

-10

0

10

Input Power (dBm)

Input Power (dBm)

(a)

(b)

20

f0 = 35 GHz
ZL = 25 + j82 Ί

14

Pout

12

15

10

GT

10

8
6

PAE

4

5

2
0

0
-30

-20

-10

0

10

20

20

f0 = 45 GHz
ZL = 16 + j67Ί

14

Pout

15

12
10

GT

10

8
6

PAE (%)

20

PAE (%)
Power (dBm), Gain (dB)

Power (dBm), Gain (dB)

Figure 6.21: Simulated input power sweeps of the GaN HEMT biased as a class A amplifier
(gate-source bias of 1 V and drain-source bias of 10 V) at a fundamental frequency of (a) 15 GHz
and (b) 25 GHz. The optimal loads determined from the LP data are 104 + j130 Ω at 15 GHz
and 54 + j105 Ω at 25 GHz. The peak PAE resides at an input power of approximately 10 dBm
for both frequencies. The device operating at 15 GHz achieves the highest overall peak PAE of
approximately 12%.

4

5

PAE

2

0

0
-30

-20

-10

0

10

Input Power (dBm)

Input Power (dBm)

(a)

(b)

20

Figure 6.22: Simulated input power sweeps of the GaN HEMT biased as a class A amplifier (gatesource bias of 1 V and drain-source bias of 10 V) at a fundamental frequency of (a) 35 GHz and (b)
45 GHz. The optimal loads determined from the LP data are 25 + j82 Ω at 35 GHz and 16 + j67 Ω
at 45 GHz. The PAE diminishes with increasing frequency. At 45 GHz, the peak PAE is between
2–3%.

146

50

Pin = -9.8 dBm
Pin = -1.8 dBm
Pin = 3.8 dBm

V2(f) (dBm)

V2(t) (V)

45
40
35
30
25
20
15
10
5
0

Pin= 3.8 dBm
Pin = -1.8 dBm
Pin = -9.8 dBm

40
30
20
10

0

50

100

150

200

250

300

350

0
-10

0

10

20

30

40

50

60

70

80

Frequency (GHz)

Time (ps)

(b)

(a)

Figure 6.23: Simulated (a) output voltage transients and (b) output voltage spectral content of
the GaN HEMT biased as a class A amplifier at 15 GHz. The input powers of the blue, green,
and red lines are −9.8 dBm, −1.8 dBm, and 3.8 dBm, respectively. The transistor exhibits strong
inter-modulation at each simulated input power.
The transient output voltages of the class A amplifier corresponding to various input powers
are presented in Figure 6.23a. The fundamental frequency of these HEMT simulations is 15 GHz.
Figure 6.23b illustrates the spectral content of the output voltages. The blue, green, and red lines
represent the input powers Pin = −9.8 dBm, −1.8 dBm, and 3.8 dBm, respectively. Significant
inter-modulation is observed in the response of the device at each input power.

6.5.5

Class AB Amplifier

The GaN HEMT is next biased as a class AB amplifier to improve efficiency. The gate-source bias
is VGS = −2 V. This operating point is between the linear class A bias and the nonlinear class B
bias. The DC drain current is smaller at this bias resulting in a higher PAE.
The optimal load impedance for peak transducer gain of the class AB amplifier is determined with LP data calculated from FKT device simulations. Four LP data sets are presented
in Figures 6.24a – 6.25b. The operating frequencies of the FKT device simulations are 15 GHz
– 45 GHz with increments of 5 GHz. The optimal loading conditions for maximum transducer
gain are highlighted in each figure. This impedance shifts across the Smith chart in a counterclockwise fashion as the fundamental frequency increases. At 15 GHz, the optimal loading condi-

147

5.0

-10
-20

-0.2

-1.0

Pavs = -2 dBm
f0 = 25 GHz

0
-2.

-0.
5

0
-2.

-1.0

-0.
5

-5.0

Pavs = -2 dBm
f0 = 15 GHz

-5.0

-0.2

10
20

5.0

2.0

1.0

20

0.5

10

20

10
20

5.0

2.0

10

-10
-20

0.2
1.0

5.0

0.2

0.5

2.0

0.5

2.0

GT = 10.9 dB
ZL = 30 + j124 Ί

1.0

1.0

0.5

GT = 11.1 dB
ZL = 77 + j155 Ί

(a)

(b)

Figure 6.24: LP data calculated from FKT device simulations of the GaN HEMT biased as a class
AB amplifier (gate-source bias of -2 V and drain-source bias of 10 V) with -2 dBm power available
from the source at a fundamental frequency of (a) 15 GHz and (b) 25 GHz. The optimal loads for
maximum transducer gain are 77 + j155 Ω at 15 GHz and 30 + j124 Ω at 25 GHz. The peak gain is
11.1 dB at 15 GHz and 10.9 dB at 25 GHz. The step size of the transducer gain contours is 0.2 dB.
tion is ZL = 77 + j155 Ω. As the frequency increases from 25 GHz and 35 GHz, the optimal load
impedance decreases from ZL = 30 + j124 Ω to ZL = 24 + j88 Ω. Finally, at 45 GHz, the optimal
load impedance for peak transducer gain is ZL = 16 + j69 Ω.
Power sweeps of the GaN HEMT biased as a class AB amplifier with the optimal loading
conditions are next presented. The power available from the source, Pavs , is swept across a broad
range and the resulting FOMs are plotted versus the calculated input power, Pin . Figure 6.26a
reports the input power sweep of the HEMT at 15 GHz. The load impedance determined from the
LP data is ZL = 77 + j155 Ω. The output power, Pout , is plotted in black, the transducer gain, GT ,
is plotted in red, and the PAE is plotted in blue. Figures 6.26b – 6.27b report the power sweep
results for the amplifier at 25, 35, and 45 GHz, respectively. The load impedances used in these
simulations are ZL = 30 + j124 Ω, 24 + j88 Ω, and 16 + j69 Ω, respectively.
The power sweep results demonstrate that the class AB amplifier exhibits better efficiency
than the class A amplifier, particularly at high frequency. At 45 GHz, the class AB amplifier

148

5.0

10
20

5.0

2.0

1.0

20

0.5

10

20

10
20

5.0

10

-10
-20

-0.2

0
-2.

-0.
5

-1.0

Pavs = -2 dBm
f0 = 45 GHz

-1.0

0
-2.

-0.
5

-5.0

Pavs = -2 dBm
f0 = 35 GHz

-5.0

-0.2

-10
-20

2.0

0.2
1.0

5.0

0.2

0.5

2.0

0.5

2.0

GT = 10.5 dB
ZL = 16 + j69 Ί

1.0

1.0

0.5

GT = 10.7 dB
ZL = 24 + j88 Ί

(a)

(b)

20

f0 = 15 GHz
ZL = 77 + j155 Ί

Pout

15

15

PAE

GT
10

10

5

5

0
-40

0
-30

-20

-10

0

10

20

20

20

f0 = 25 GHz
ZL = 30 + j124 Ί

Pout

15

GT

15

PAE

10

10

5

5

0
-40

PAE (%)

20

PAE (%)
Power (dBm), Gain (dB)

Power (dBm), Gain (dB)

Figure 6.25: LP data calculated from FKT device simulations of the GaN HEMT biased as a class
AB amplifier (gate-source bias of -2 V and drain-source bias of 10 V) with -2 dBm power available
from the source at a fundamental frequency of (a) 35 GHz and (b) 45 GHz. The optimal loads for
maximum transducer gain are 24 + j88 Ω at 35 GHz and 16 + j69 Ω at 45 GHz. The peak gain is
10.7 dB at 35 GHz and 10.5 dB at 45 GHz. The step size of the transducer gain contours is 0.2 dB.

0
-30

-20

-10

0

Input Power (dBm)

Input Power (dBm)

(a)

(b)

10

20

Figure 6.26: Simulated input power sweeps of the GaN HEMT biased as a class AB amplifier
(gate-source bias of -2 V and drain-source bias of 10 V) at a fundamental frequency of (a) 15 GHz
and (b) 25 GHz. The optimal loads determined from the LP data are 77 + j155 Ω at 15 GHz and
30+ j124 Ω at 25 GHz. The peak PAE resides at an input power of approximately 0 dBm at 15 GHz
and 5 dBm at 25 GHz. The device operating at 15 GHz achieves the highest overall peak PAE of
approximately 15%.

149

Pout

15

GT

15

PAE

10

10

5

5

0
-40

0
-30

-20

-10

0

10

20

20

20

f0 = 45 GHz
ZL = 16 + j69 Ί

Pout

15

15

GT
10

10

5

PAE (%)

20

f0 = 35 GHz
ZL = 24 + j88 Ί

PAE (%)
Power (dBm), Gain (dB)

Power (dBm), Gain (dB)

20

5

PAE
0
-40

0
-30

-20

-10

0

10

Input Power (dBm)

Input Power (dBm)

(a)

(b)

20

Figure 6.27: Simulated input power sweeps of the GaN HEMT biased as a class AB amplifier
(gate-source bias of -2 V and drain-source bias of 10 V) at a fundamental frequency of (a) 35 GHz
and (b) 45 GHz. The optimal loads determined from the LP data are 24 + j88 Ω at 35 GHz and
16 + j69 Ω at 45 GHz. The peak PAE resides at an input power of approximately 8 dBm at both
35 GHz and 45 GHz. The PAE of the class AB amplifier does not diminish like the class A amplifier
at high frequency. At 45 GHz, the peak PAE of the device is approximately 8 – 9%.
50

35
Pin = -16.7 dBm
Pin = -4.9 dBm
Pin = 2.8 dBm

25

Pin = 2.8 dBm
Pin = -4.9 dBm
Pin = -16.7 dBm

40

V2(f) (dBm)

V2(t) (V)

30
20
15
10

30
20
10

5
0
0

50

100

150

200

250

300

350

Time (ps)

0
-10

0

10

20

30

40

50

60

70

80

Frequency (GHz)

(a)

(b)

Figure 6.28: Simulated (a) output voltage transients and (b) output voltage spectral content of the
GaN HEMT biased as a class AB amplifier at 15 GHz. The input powers of the blue, green, and red
lines are −16.7 dBm, −4.9 dBm, and 2.8 dBm, respectively. Strong inter-modulation is observed
at each input power.
demonstrates 10% PAE while the class A amplifier only demonstrates between 2 and 3%. It is
concluded that the class AB amplifier is preferred for RF PA design due to its higher efficiency.
Several transient output voltages of the input power sweep at 15 GHz are presented in Figure 6.28a. Figure 6.28b illustrates the spectral content of the output voltages. The blue, green, and

150

red lines correspond to the input powers Pin = −16.7 dBm, −4.9 dBm, and 2.8 dBm, respectively.
Significant inter-modulation is observed in the response of the class AB amplifier at each input
power.

6.5.6

Quasi-Static vs. Full-Wave

Small-signal GaN HEMT results computed with the quasi-static and full-wave FKT device simulators were presented in Section 6.4. The full-wave simulations of the HEMT at high frequency exhibited vastly different parasitic effects than the quasi-static simulations. Proper treatment of high
frequency parasitics is critical for accurate large-signal RF simulations. Comparisons of large-

20

f0 = 10 GHz
Quasi-static
Full-wave

15
10

ZL = 50 Ί

Pout

10
8

PAE
GT

6
4

5

PAE (%)

Power (dBm), Gain (dB)

signal RF GaN HEMT simulations with the quasi-static and full-wave solvers are next presented.

2

0
-35 -30 -25 -20 -15 -10 -5

0
0

5 10

Input Power (dBm)
Figure 6.29: Input power sweeps of the GaN HEMT biased as a class AB amplifier calculated
with the quasi-static (dashed lines) and the full-wave (solid lines) FKT device simulators. The
gate-source bias is -2 V, the drain-source bias is 10 V, the fundamental frequency is 10 GHz, and
the load impedance is 50 Ω.
An input power sweep of the AlGaN/GaN HEMT biased as a class AB amplifier at 10 GHz is
reported in Figure 6.29. The gate-source bias is VGS = −2 V and the drain-source bias is VDS =
10 V. The load impedance presented to the output of the device is ZL = 50 Ω. Quasi-static results
151

Transducer Gain (dB)

14
Quasi-static
Full-wave

12
10

Pavs = -13 dBm
Pavs = -2 dBm

8
6
4
2
0
0

20

40

60

80

100

Fundamental Frequency (GHz)
Figure 6.30: Quasi-static (dashed lines) and full-wave (solid lines) frequency sweeps of the GaN
HEMT biased as a class AB amplifier. The gate-source bias is -2 V, the drain-source bias is 10 V,
and the load impedance is 50 Ω. Black lines correspond to a power available from the source of
-13 dBm and red lines correspond to a power available from the source of -2 dBm.
are illustrated with dashed lines and full-wave results are illustrated with solid lines. The fullwave result has a significantly different 1 dB compression point at Pin = −4 dBm compared to the
quasi-static 1 dB compression point at Pin = −10 dBm. Furthermore, the PAE computed with the
full-wave solver is vastly different than the PAE computed with the quasi-static solver. Clearly,
proper treatment of full-wave EM is essential for accurately capturing the high frequency response
of devices.
Figure 6.30 presents a set of simulated frequency sweeps of the same class AB amplifier.
The fundamental frequency of the large-signal sinusoidal input tone is swept from 1–100 GHz.
Quasi-static simulation results are illustrated with dashed lines and full-wave simulation results
are illustrated with solid lines. Black lines correspond to a power available from the source of
Pavs = −13 dBm and red lines correspond to Pavs = −2 dBm. As expected, the full-wave simulations exhibit a vastly different frequency response compared to the quasi-static simulations. The
peak transducer gain of the HEMT is located at a fundamental frequency of approximately 55 GHz

152

according to the full-wave results. Quasi-static simulations do not provide the same qualitative
features in the gain.

6.6

Discussion and Conclusions

This chapter applied the FKT TCAD framework to small- and large-signal RF simulations of
an AlGaN/GaN HEMT. First, a detailed description of the HEMT was presented along with simulated thermal equilibrium results (i.e., no external bias). Next, static FKT simulations of the GaN
HEMT were presented and compared to measurements [107]. Specifically, Figure 6.6 presented
the static I-V family computed with the FKT device simulator. These results demonstrated excellent agreement with measurements. Figure 6.7 presented an interesting comparison of the I-V
families calculated with different versions of the FKT device simulator. This comparison demonstrated that electronic band-structure, hot-electron effects, self-heating effects, and trapping effects
all require proper treatment in the simulator in order to re-produce measured data.
The third section of this chapter focused on small-signal RF device simulations with the FKT
TCAD tool. A detailed description was provided for simulating S-Parameters of electronic devices.
Both quasi-static and full-wave simulation results were provided. The full-wave results differ from
quasi-static at high frequency. It is concluded that the full-wave solver is essential for capturing
the high frequency parasitics of devices. The main result of the small-signal RF section, which
was first calculated and presented in [84], is the small-signal current gain in Figure 6.12. The
simulations show excellent agreement with the measurements. The upward bending of the current
gain is due to the inductive coupling of the metal contacts. This high frequency parasitic effect is
captured with the full-wave device solver.
Large-signal RF simulations of the GaN HEMT were next presented in this chapter. Because
large-signal RF simulations require reactive load impedances, the beginning of the section focused
on the quasi-static AC impedance BC and its application as a series RL load. Next, a detailed
description of the post-processing required to calculate PA FOMs was presented. Finally, several
large-signal simulations of the GaN HEMT were reported for different amplifier classes and op-

153

1.2

1.0

0.9

0.5

2.0

1.8

0.6

1.6

Increasing f0

1.4

0.7

0.8

erating frequencies. The LP data calculated from the device simulations were used to determine

0.4
3.0
0.3

4.0

5.0

0.2

10
0.1
20

10

5.0

4.0

3.0

1.6
1.8
2.0

1.4

1.2

0.9
1.0

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

20

-20

-0.1

-10

-0.2

-5.0
-4.0

3
-0.

-3.
0
8
-1.

-1.6

-1.4

-1.2

-0.9

-1.0

-0.7

-0.8

-0.6

-0.
5

0
-2.

.4
-0

Class A
Class AB

Figure 6.31: Summary of the optimal loads determined through FKT device simulations of the
AlGaN/GaN HEMT biased as class A and class AB amplifiers. The fundamental frequency of
both amplifiers was swept from 15–45 GHz with 5 GHz increments. The optimal load shifts in a
counter-clockwise motion across the Smith chart with increasing frequency.
the optimal loading conditions. Figure 6.31 summarizes the optimal load reflection coefficients
determined by the FKT device simulations. The fundamental frequencies were 15, 25, 35, and
45 GHz for both the class A and class AB amplifiers. The optimal reflection coefficients shift in a
counter-clockwise motion across the Smith chart with increasing fundamental frequency. Reflection coefficients corresponding to the class A amplifier are illustrated with red stars and blue stars
represent the class AB amplifier’s optimal reflection coefficients. Input power sweep simulations
were then presented using the load impedances corresponding to peak transducer gain.
Finally, the large-signal RF response of the GaN HEMT was simulated with the quasi-static and
full-wave FKT device solvers in Section 6.5.6. The full-wave results exhibited different characteristics compared to the quasi-static results. This is due to the high frequency parasitics of the device
which are not captured with the quasi-static solver. Full-wave simulations will be paramount for
analysis of next generation devices operating at high frequency.

154

The FKT device simulator could be a valuable asset for high frequency design of GaN HEMT
PAs. This chapter was devoted to demonstrating its use through a wide range of simulations that
could be useful for RF circuit design. LP measurements at high frequency are expensive. The FKT
device simulator is therefore an excellent candidate for calculating supplemental high frequency
and high power LP data.

155

CHAPTER 7
CHARACTERIZATION AND COMPACT MODEL EXTRACTION OF GAN HEMTS

7.1

Introduction

The culmination of this thesis is the characterization and compact model extraction of a stateof-the-art AlGaN/GaN HEMT fabricated and measured at AFRL. This chapter is arranged as
follows. In the first section, a description of the GaN HEMT is presented. Characterizations of
the HEMT, including small- and large-signal simulations, are reported in the next several sections.
Finally, an XP model is extracted from FKT device simulations and used for PA design in ADS.

7.2

AFRL AlGaN/GaN HEMT

The AlGaN/GaN HEMT is fabricated and measured at AFRL. All measurements of this device
were provided by Bob Fitch AFRL and Gregg Jessen AFRL [6]. Figure 7.1 presents an illustration
of the HEMT stack. The gate length of the device is 0.14 Âľm and the width is 75 Âľm. Total

Figure 7.1: A cross-section of AFRL’s standard GaN HEMT device.

156

periphery of the device is 300 µm — four gate fingers × 75 µm unit gate length. The stack of the
HEMT consists of several layers. A 1.9 Âľm GaN buffer is first grown on the SiC substrate. The
remaining stack consists of a 10 nm GaN channel, 1 nm of aluminum nitride (AlN), 15.4 nm of
Al0.278 Ga0.722 N, and a 3 nm GaN cap.

Drain

Source
Gate

n+-GaN

Channel

n+-GaN

GaN Buffer
Figure 7.2: The DV mesh used to simulate the AFRL GaN HEMT in the FKT device simulator.

Gate
AlGaN
GaN Channel

GaN Cap
AlN

GaN Buffer
Figure 7.3: A zoom of the GaN HEMT stack under the Schottky gate. The stack consists of (from
substrate to metal) a 1.9 Âľm GaN buffer, a 10 nm GaN channel, 1 nm of AlN, 15.4 nm of AlGaN,
and a 3 nm GaN cap. The mole fraction of the AlGaN layer is x = 0.278.
The DV mesh on which the FKT device equations are solved is presented in Figures 7.2
and 7.3. This was generated with in-house meshing codes developed by Matt Grupen AFRL to
simulate high speed electronic devices [62, 84]. These codes were developed following the work
of Conti [46] described in Section 2.2. The silicon nitride (SiN) passivation on top of the GaN
157

cap and the air above the passivation layer are not illustrated in Figure 7.2. The n+ -GaN regions
in Figure 7.2 mimic the contact resistances of the source and drain Ohmic metal-semiconductor
interfaces.
As discussed previously, GaN HEMT technology is an excellent transistor choice for PAs due
to its high power and high frequency capabilities. The HEMT does not require modulation doping
because of spontaneous and piezoelectric polarization [3, 5]. The polarization charge densities,
however, are not necessarily known after fabrication and must be determined for the FKT device
simulations. The following sections on characterization of the HEMT discuss the determination of
the unknown physical parameters. The physical parameters of the AFRL GaN HEMT are presented
in Figure 7.4.
Gate
HG3
Hsource

Source

HGaN

Psurface

HG2

HG1

ÎŚSB

GaN
n+
GaN

HAlGaN

Al0.278GaN0.722

HAlN

AlN

Ndefects

2 DEG

GaN Channel

Lcap

LGF

PAlGaN/GaN
PAlN/AlGaN
PGaN/AlN

LGF
SiN

Source

Drain

Gate

HGaNBuffer

GaN Buffer

Lsource

Lgate

Ldrain

LSG

LGD

Figure 7.4: The physical parameters of the AFRL GaN HEMT. Most notable are the polarization
charge densities P at the heterojunctions.

158

7.3

TLM Characterization

The unknown physical parameters of the AFRL GaN HEMT can be determined through what
are known as transmission line measurements (TLM) [110]. For TLM analysis, the Schottky gate
of the HEMT is removed to yield a resistor. A series of GaN resistors with variable spacing

RT
Measured/simulated
Fit

2RC

L
-2LT

L1 L2 L3 L4 L5

Figure 7.5: An example of determining the approximate access resistance with TLM. A series of
resistors are measured/simulated and the total resistance of each resistor is calculated. The slope
of the data determines the sheet resistance (access resistance) divided by the width of the resistors
and the y-intercept determines twice the contact resistance.
between the Ohmic source and drain contacts are measured/simulated. The total resistance of each
device is calculated with Ohm’s law. An illustration of the set of total resistances is provided in
Figure 7.5. The total resistance is given by
R
RT = S (L + 2LT ) ,
W

(7.1)

where RS is the sheet resistance of the 2DEG, L is the length between the Ohmic contacts, LT is
the transfer length, and W is the width of the resistors [110]. A linear fit of the data determines the
slope and y-intercept. The sheet resistance (also called the access resistance) is calculated from the
slope of the data by
RS =

∆RT (L)
×W,
∆L
159

(7.2)

and the contact resistance (the resistance due to the Ohmic metal-semiconductor interface) is calculated from the y-intercept by
1
RC = RT (L = 0).
2

(7.3)

0.035
0.03

L = 30 Îźm

ID (A)

0.025
0.02
0.015

1 / RT

0.01
0.005
0
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

VDS (V)
Figure 7.6: Simulated I-V curves of the TLM resistors computed with the FKT device solver. The
lengths of the resistors vary from 5 Âľm to 30 Âľm with 5 Âľm increments. The I-V curve with the
largest drain current at the drain-source bias of 1 V corresponds to the L = 30 Âľm resistor.
The polarization sheet charge densities at the heterojunction interfaces are tuned to produce the
best agreement between the simulated and measured total resistances of the GaN TLM resistors.
To this end, several GaN resistors are simulated with the FKT device solver. The lengths of the
resistors vary between 5 Âľm and 30 Âľm with 5 Âľm increments. The width of each resistor is 81 Âľm.
Figure 7.6 presents the simulated I-V curves of the resistors. The drain-source bias, VDS , is swept
from 0 to 1 V, however, the resistance is calculated at low bias.
The total resistances of the devices are presented in Figure 7.7. The approximate access resistance and contact resistance calculated from the data are RS ≈ 304 Ω and RC ≈ 5.5 Ω. These
simulated results agree well with measurements and provide a good physical parameter set for RF
simulations with the FKT TCAD tool.

160

RT (Ί)

130
120
110
100
90
80
70
60
50
40
30
20

Simulated
Measured

5

10

15

20

25

30

L (Îźm)
Figure 7.7: The simulated (black dots) and measured (solid line) total resistances of the GaN TLM
resistors.

7.4

DC and Small-Signal Characterization

The initial physical parameter set determined by the TLM simulations is tuned for agreement
with DC and small-signal measurements. DC I-V measurements provided by AFRL [6] are first
used to tune the polarization charge densities at the heterojunctions. To improve the accuracy
of the simulations, the polarization charge densities are allowed to vary along the channel. This
provides additional DOF to fit the simulated I-V data onto the measurements. The variation of
the 2DEG density across the GaN HEMT channel is presented in Figure 7.8. The source contact
ends at 0.5 Âľm and the drain contact begins at 3.5 Âľm in the plot. The Schottky T-gate resides at
the middle of the plot and the gate arms extend outward. The channel is approximated with five
piece-wise polarization charge densities. The first polarization charge density covers the region
between the source contact and the beginning of the T-gate arm. The second, third, and fourth
regions follow the dimensions of the Schottky T-gate, i.e., the second starts at the beginning of the
Schottky arm and ends at the beginning of the gate stem (the portion of the gate that is attached
to the GaN cap). Finally, the fifth region extends from the end of the Schottky gate arm to the

161

T-gate

Source

Drain

Electron Density (cm-3)

1e+21

1e+20

1e+19

1e+18
0

0.5

1

1.5

2

2.5

3

3.5

4

HEMT Channel (Îźm)
Figure 7.8: Electron density across the GaN HEMT channel. The channel begins at 0.5 Âľm and
ends at 3.5 Âľm. Five different polarization charge densities are prescribed to the GaN/AlN heterojunction. The five regions are outlined with vertical dashed lines. The contacts of the device are
illustrated at the top of the figure.
beginning of the drain contact. These regions are illustrated in Figure 7.8 with dotted lines.
The thermal equilibrium band diagram of the HEMT computed with the FKT device simulator
is presented in Figure 7.9.

The conduction band minimum is plotted with a black line and the

electron Fermi level is plotted with a red line. This band diagram is calculated across the stack
(from the GaN buffer to the Schottky gate). The 2DEG resides at 5 µm — the GaN/AlN interface.
The simulated I-V family of the AFRL GaN HEMT is presented in Figure 7.10. Here, the
black lines represent the data computed with the FKT device simulator and the red dots represent
the AFRL measurements. The drain-source bias, VDS , is swept from 0 to 10 V and the gate-source
bias, VGS , is swept from +1 V to -3 V with 1 V steps. FKT simulations exhibit good agreement with
the measurements. One physical effect that is not captured by the FKT device simulator is called
the “kink effect”. It is suggested that the kink effect could be induced by hot-electron trapping and
field-assisted de-trapping via donor-like traps in the GaN buffer layer [111]. This trapping process

162

-2

GaN AlN

Energy (eV)

-2.5

GaN

AlGaN

-3
-3.5
-4
Ec

-4.5
F

-5
-5.5
0

5

10

15

20

25

HEMT Stack (nm)
Figure 7.9: The thermal equilibrium band diagram of the AFRL GaN HEMT computed with the
FKT device simulator.

1200

ID (mA/mm)

1000
800
600
400
200
0
-200
0

2

4

6

8

10

VDS (V)
Figure 7.10: Simulated DC I-V curves (black lines) compared with AFRL measurements (red
dots). The gate-source bias ranges from +1 V to -3 V with 1 V steps.
is much slower than the fundamental period of RF signals. Therefore, proper treatment of the kink
effect is not required large-signal RF simulations of the GaN HEMT.

163

0.3
Simulated
Measured

0.25

0.25

0.2

0.2

0.15

0.15

0.1

0.1

0.05

0.05

0

0

-0.05

-0.05
-6

-5

-4

-3

-2

-1

0

Transconductance (mS)

Drain Current ID (mA)

0.3

1

Gate-Source Bias VGS (V)
Figure 7.11: Simulated DC drain current (solid black line) and transconductance (solid red line) of
the AFRL GaN HEMT. The measurements obtained from AFRL are plotted with dashed lines.
Figure 7.11 illustrates the drain current (black lines) and the transconductance (red lines) versus
the gate-source bias, VGS , of the AFRL GaN HEMT. FKT device simulations are drawn with solid
lines and the AFRL measurements are drawn with dashed lines. FKT device simulations exhibit
good agreement with the measurements.
The S-Parameters S11 and S22 are presented in Figure 7.12. The solid lines represent the results
computed with the quasi-static FKT device simulator. Measured S-Parameters are drawn with
dashed lines. Simulations agree well with measurements at lower frequency. At higher frequency,
however, the simulations lose accuracy. Full-wave simulations are required for accuracy at high
frequency.
The AFRL GaN HEMT is further characterized by simulating the small-signal current gain.
Figure 7.13 presents the simulated h21 (black line) and the measured small-signal current gain
(red boxes). FKT simulations exhibit more gain than the measured device at lower frequency.
Furthermore, the slope of the measured current gain changes around 20–30 GHz. This is due to
inductive coupling of the metal contacts. High frequency parasitic effects, including the inductive

164

2.0

1.5

1.2

1.0

0.8
0.5

Simulated
Measured

3.0

5.0

0.2

10

10
20

5.0

2.0

1.0

S11

-20

0.5

20

S22

-10

-0.2

-5.0
-1.5

-1.2

-1.0

-0.8

0
-2.

-0.
5

-3.0

Figure 7.12: S-Parameters of the AFRL GaN HEMT. The solid lines represent the results computed
with the quasi-static FKT device solver. The measured S-Parameters obtained from AFRL are
drawn with dashed lines.

25
Simulated
Measured

h21 (dB)

20
15
10
5
0
10

100
Frequency (GHz)

Figure 7.13: The small-signal current gain of the AFRL GaN HEMT. The FKT device simulation
data (black line) compares reasonably well with the measurements (red boxes).
coupling of the metal contacts, can be captured with the full-wave FKT device simulator. This is
the topic of future research.

165

7.5

Large-Signal Characterization

Large-signal simulations of the AFRL GaN HEMT with the FKT device simulator are presented in this section. The device was measured at AFRL. The HEMT operates as a class AB
amplifier at 10 GHz.
An input power sweep of the AFRL GaN HEMT is illustrated in Figure 7.14. The simulated
output power, Pout , is plotted with a black line, the simulated transducer gain, GT , is plotted with
a red line, and the simulated PAE is plotted with a blue line. Measurements of Pout , GT , and
PAE are presented with black, red, and blue squares, respectively. The gate-source bias is VGS =
−2.8 V and the drain-source bias is VDS = 21 V. The load reflection coefficient is ΓL = 0.37 +
j0.60 Ω at 10 GHz. The reflection coefficient looking into the source is ΓS = −0.27 + j0.61 Ω

40
60
f0 = 10 GHz
35 ZL = 33 + j80 Ί
50
30
40
Pout
25
20
30
GT
15
20
10
10
5
PAE
0
0
-20 -15 -10 -5 0 5 10 15 20

PAE (%)

Power (dBm), Gain (dB)

at 10 GHz. The FKT device simulations presented in Figure 7.14 agree well with measurements

Input Power (dBm)
Figure 7.14: Input power sweep data of the AFRL GaN HEMT biased as a class AB amplifier at
10 GHz. The output power is plotted in black, the transducer gain is plotted in red, and the PAE
is plotted in blue. FKT device simulation data are plotted with solid lines and measurements are
plotted with squares.
at low input power. However, the simulations deviate from the measurements at higher input
power. Most noticeably, the simulations reach compression at a significantly lower input power
166

than the measurements. This could be due to the load impedance presented to the output of the
device. The simulations use a series RL load impedance, but this may not be the precise impedance
that is experimentally presented to the HEMT. Full-wave FKT device simulations may also be
required for better agreement with power sweep measurements. Simulated and measured PAE also
deviate at higher input power. The FKT device simulator overestimates the static drain current at

25
20

f0 = 10 GHz
VGS = -2.8 V
VGS = -2.0 V
VGS = -1.0 V

ZL = 33 + j80 Ί

30
25

Pout

20

15

GT

15

10

10

PAE

5
0
-20 -15 -10

PAE (%)

Power (dBm), Gain (dB)

VDS = 21 V leading to a lower simulated efficiency.

5
0

-5

0

5

10

15

Input Power (dBm)
Figure 7.15: Simulated input power sweeps of the AFRL GaN HEMT at 10 GHz. The gate-source
bias is changed from -2.8 V to -2 V (dashed lines) and -1 V (dot-dashed lines). The drain-source
bias is held constant at 21 V. The gain and PAE of the device is reduced at -1 V gate-source voltage,
as expected.
Large-signal characterization of the AFRL GaN HEMT continues by analyzing the power
sweep results of the device with different operating points. Figure 7.15 illustrates simulated input
power sweeps of the HEMT with different gate-source biases, VGS . The output power, Pout , transducer gain, GT , and the PAE are plotted with black, red, and blue lines, respectively. The fundamental frequency is 10 GHz and the load and source reflection coefficients are ΓL = 0.37 + j0.60 Ω
at 10 GHz and ΓS = −0.27 + j0.61 Ω at 10 GHz, respectively. In Figure 7.15, the gate-source bias
VGS is changed from -2.8 V to -2 V (dashed lines) and -1 V (dot-dashed lines). As expected, the

167

20

f0 = 10 GHz
VDS = 21 V
VDS = 10 V
VDS = 30 V

ZL = 33 + j80 Ί

30

Pout

25
20

15

GT

15

10

10

5

PAE

0
-20 -15 -10

PAE (%)

Power (dBm), Gain (dB)

25

5
0

-5

0

5

10

15

Input Power (dBm)
Figure 7.16: Simulated input power sweeps of the AFRL GaN HEMT at 10 GHz. The drain-source
bias is changed from 21 V to 10 V (dashed lines) and 30 V (dot-dashed lines). The gate-source bias
is held constant at -2.8 V. The small-signal gain of the device increases at 10 V drain-source bias.
However, the device operating at this bias compresses at a lower input power.
PAE decreases as the gate-source bias increases. A larger drain current, ID , results from a larger
gate-source bias. An increase in the DC power consumed at the drain results in a decrease in the
PAE. The quantitative decrease in the device efficiency determined by the FKT device simulations
may be useful for RF circuit designers.
Figure 7.16 presents several input power sweep simulations with varying drain-source bias,
VDS . The output power, Pout , transducer gain, GT , and the PAE are plotted with black, red, and blue
lines, respectively. The fundamental frequency and loading conditions are the same as the previous
results. Here, the drain-source bias, VDS , is changed from 21 V to 10 V (dashed lines) and 30 V
(dot-dashed lines). Decreasing VDS has substantial effect on each device FOM. The low power
transducer gain increases and the device compresses at a lower input power compared to VDS =
21 V. The PAE also increases because the DC drain current is reduced. FKT device simulations
of the HEMT with a drain-source bias VDS = 30 V exhibit similar effects. The transducer gain and
PAE are lower than the VDS = 21 V simulations.

168

20

ZL = 33 + j80 Ί

f0 = 10 GHz
LSD = 3 Îźm
(+ 1.0 Îźm)
(- 1.0 Îźm)

30

Pout

25
20

15

GT

15

10

10

PAE

5
0
-20 -15 -10

PAE (%)

Power (dBm), Gain (dB)

25

5
0

-5

0

5

10

15

Input Power (dBm)
Figure 7.17: Simulated input power sweeps of the AFRL GaN HEMT at 10 GHz. The sourcedrain spacing is changed from 3 Âľm to 4 Âľm (dashed lines) and 2 Âľm (dot-dashed lines). These
variations of the source-drain spacing have small impacts on the device power metrics.
The effect of the source-drain length, LSD = LSG + Lgate + LGD (see Figure 7.4), on the largesignal power metrics is investigated with the results in Figure 7.17. The output power, transducer
gain, and PAE are calculated from three FKT device simulations with different spacing between
the Ohmic contacts. The first simulation (solid lines) is the nominal source-drain length of LSD =
3.0 Âľm and the other two add/subtract 1.0 Âľm. The LSD = 4.0 Âľm input power sweep results are
plotted with dashed lines and the LSD = 2.0 Âľm input power sweep results are plotted with dotdashed lines. The transducer gain of the device is reduced by increasing the source-drain distance.
In a similar manner, reducing the source-drain distance increases the transducer gain. However, a
substantial decrease in the source-drain spacing (-33.3%) only moderately increases the low input
power gain from 14.48 dB to 14.93 dB — a 3.1% increase. The +33.3% increase in the sourcedrain spacing yields a 2.9% decrease in gain from 14.48 dB to 14.06 dB. These simulation results
could be valuable for RF circuit designers. Furthermore, measuring these results would require
fabricating several more devices.
Finally, LP data are generated from FKT device simulations of the AFRL HEMT. Figure 7.18

169

PAE = 18.2 %
ZL = 53 + j108 Ί
2.0

1.0

0.5

Pout = 19.0 dBm
ZL = 57 + j81 Ί

5.0

0.2

10
10
20

5.0

2.0

1.0

0.5

20

-10
-20
-0.2

0
-2.

-0.
5

-5.0

-1.0

Pavs = 1.5 dBm
Figure 7.18: LP data generated from FKT device simulations of the AFRL GaN HEMT biased as
a class AB amplifier at 10 GHz. The gate-source bias is -2.8 V, the drain-source bias is 10 V, and
the power available from the source is 1.5 dBm. Output power contours are illustrated with blue
lines and PAE contours are illustrated with red lines. The optimal load impedance for peak output
power is 57 + j81 Ω at 10 GHz and the load impedance 53 + j108 Ω at 10 GHz corresponds to peak
PAE. The peak output power is 19 dBm and the peak PAE is 18.2%.
reports the LP data of the HEMT biased as a class AB amplifier at 10 GHz. The gate-source bias
is VGS = −2.8 V and the drain-source bias is VDS = 10 V. A power available from the source of
Pavs = 1.5 dBm drives the FKT device simulations. The output power contours are illustrated with
blue lines and the PAE contours are illustrated with red lines. Based on the LP data, the optimal
load impedance for the peak output power of Pout = 19 dBm is ZL = 57 + j81 Ω at 10 GHz. A
maximum PAE of 18.2% is achieved with a load impedance of ZL = 53 + j108 Ω at 10 GHz.

7.6

XP Extraction

Compact transistor models allow efficient PA design. Models can be imported into commercial
CAD tools and used for simulated impedance matching, LP, power sweeps, etc. An example of
a powerful CAD tool is ADS. In this section, the procedure for extracting an XP model with the
FKT device simulator is presented and discussed.

170

The procedure for extracting a general XP model was presented in Section 2.5. A two-port
model with K harmonics requires 4K − 1 simulations to calculate the 2K × (4K − 1) XP coefficients. This work extracts an XP model of the AFRL GaN HEMT biased as a class AB amplifier
at 10 GHz. The gate-source bias is VGS = −2.8 V and the drain-source bias is VDS = 10 V. The
large-signal input tone at the fundamental frequency has a power available from the source of
Pavs = 11.6 dBm. Large-signal reflection coefficients looking into the source and into the load are
ΓS = 50 Ω at 10 GHz and ΓL = 50 Ω at 10 GHz, respectively. These operating conditions result in

0

14

-1

12

-2

10

V2(t) (V)

V1(t) (V)

an input power of Pin = 4.4 dBm.

-3
-4
-5

8
6
4

-6

2
0

100

200

300

400

500

Time (ps)

0

100

200

300

400

500

Time (ps)

(a)

(b)

Figure 7.19: The (a) input and (b) output voltages of the AFRL GaN HEMT calculated with the
quasi-static solver. The operating condition of the HEMT is a quiescent bias of a -2.8 V gate-source
bias and 10 V drain-source bias. The large-signal incident wave at the fundamental frequency of
the input port is 1 V in magnitude at 10 GHz.
The first XP extraction simulation applies only the large-signal tone A11 at the input of the
device. The large-signal tone remains the same for all subsequent XP extraction simulations. The

	

	
XP coefficients S11,11 , S11,21 , ¡ ¡ ¡ , S11,K1 and S21,11 , S21,21 , ¡ ¡ ¡ , S21,K1 are calculated
with the transient voltages and currents resulting from the first FKT simulation. Figures 7.19
and 7.20 present the input and output voltages and currents of the AFRL GaN HEMT biased as
a class AB amplifier with a power available from the source of Pavs = 11.6 dBm at 10 GHz. The
incident harmonics, A pk , and scattered harmonics, B pk , calculated with Eqns. (2.51) and (2.52) are
presented in Table 7.1. The transient gate voltage is chosen such that A11 is a pure cosine with a

171

0.2
0.15

I2(t) (A)

I1(t) (A)

0.08
0.06
0.04
0.02
0
-0.02
-0.04
-0.06
-0.08

0.1
0.05
0
-0.05

0

100

200

300

400

500

0

100

Time (ps)

200

300

400

500

Time (ps)

(a)

(b)

Figure 7.20: The (a) input and (b) output currents of the AFRL GaN HEMT calculated with the
quasi-static solver. The operating condition of the HEMT is a quiescent bias of a -2.8 V gate-source
bias and 10 V drain-source bias. The large-signal incident wave at the fundamental frequency of
the input port is 1 V in magnitude at 10 GHz.
2 V amplitude. Inspection of Table 7.1 confirms that the transient gate voltage and current yield a
k

fk (GHz)

A1k (V)

A2k (V)

B1k (V)

1
2
3
4
5

10.0
20.0
30.0
40.0
50.0

1.000 0◦
-

-

0.850
0.164
0.069
0.009
0.020

−93◦
116◦
−124◦
153◦
−37◦

B2k (V)
2.834
0.285
0.416
0.078
0.069

117◦
77◦
−173◦
129◦
−77◦

Table 7.1: The incident and scattered harmonics of the AFRL GaN HEMT biased as a class AB
amplifier with a power available from the source of 11.6 dBm at 10 GHz. The large-signal tone at
the fundamental frequency of the input port is the only non-zero incident wave.
pure cosine large-signal incident wave A11 . Simulations with this form of the large-signal incident
wave allow easy extraction of the XP coefficients because the phase of A11 is P = 1.
The remaining XP coefficients are calculated following the procedure described in Section 2.5.
The incident harmonics are perturbed to calculate the higher-order XP coefficients. The largesignal tone A11 is not perturbed. Any change to A11 requires re-calculation of the XP model
coefficients. The small-signal incident wave A21 is perturbed first. The GaN HEMT is simulated
	
twice with two sets of non-zero incident waves {A11 , A21 } and {A11 , A∗21 . The tone A∗21 of
the second perturbation simulation is 90◦ out of phase with the A21 tone of the first perturbation
172

14

-1

12

-2

10

V2(t) (V)

V1(t) (V)

0

-3
-4
-5

8
6
4

-6

2
0

100

200

300

400

500

0

100

Time (ps)

200

300

400

500

Time (ps)

(a)

(b)

0.08
0.06
0.04
0.02
0
-0.02
-0.04
-0.06
-0.08

0.25
0.2

I2(t) (A)

I1(t) (A)

Figure 7.21: The (a) input and (b) output voltages of the AFRL GaN HEMT calculated with the
quasi-static solver. The operating condition of the HEMT is a quiescent bias of a -2.8 V gatesource bias and 10 V drain-source bias. The solid black lines represent the large-signal simulation
and the dashed red lines represent the first perturbation simulation of the first incident harmonic at
the output port.

0.15
0.1
0.05
0
-0.05

0

100

200

300

400

500

Time (ps)

0

100

200

300

400

500

Time (ps)

(a)

(b)

Figure 7.22: The (a) input and (b) output currents of the AFRL GaN HEMT calculated with the
quasi-static solver. The operating condition of the HEMT is a quiescent bias of a -2.8 V gatesource bias and 10 V drain-source bias. The solid black lines represent the large-signal simulation
and the dashed red lines represent the first perturbation simulation of the first incident harmonic at
the output port.
simulation. Figure 7.21 presents the (a) input and (b) output voltages and Figure 7.22 illustrates
the (a) input and (b) output currents of the two A21 perturbation simulations. The first perturbation
applies a pure cosine as the small-signal A21 incident wave and the second applies a pure sine as
the small-signal A∗21 incident wave. The spectral incident and scattered waves, A pk and B pk , of the

173

k

fk (GHz)

A1k (V)

A2k (V)

B1k (V)

1
2
3
4
5

10.0
20.0
30.0
40.0
50.0

1.000 0◦
-

0.050 0◦
-

0.846
0.165
0.069
0.008
0.020

−93◦
116◦
−123◦
155◦
−36◦

B2k (V)
2.815
0.299
0.412
0.077
0.070

117◦
78◦
−172◦
129◦
−74◦

Table 7.2: The incident and scattered harmonic phasors of the AFRL GaN HEMT biased as a
class AB amplifier with a power available from the source of 11.6 dBm at 10 GHz. The largesignal simulation is perturbed with a small-signal tone at the second port and at the fundamental
frequency.
k

fk (GHz)

A1k (V)

A2k (V)

B1k (V)

1
2
3
4
5

10.0
20.0
30.0
40.0
50.0

1.000 0◦
-

0.050 −90◦
-

0.856
0.165
0.070
0.009
0.020

−93◦
115◦
−124◦
153◦
−38◦

B2k (V)
2.833
0.290
0.417
0.078
0.070

118◦
74◦
−172◦
130◦
−78◦

Table 7.3: The incident and scattered harmonic phasors of the AFRL GaN HEMT biased as a
class AB amplifier with a power available from the source of 11.6 dBm at 10 GHz. The largesignal simulation is perturbed with a small-signal tone at the second port and at the fundamental
frequency. This perturbation signal is out of phase with the first perturbation signal.
first and second A21 perturbation simulations are presented in Tables 7.2 and 7.3, respectively. For
both simulations, the magnitude of the A21 phasor is 0.05 V. As confirmed by the data presented
in Tables 7.2 and 7.3, the second perturbation is 90◦ out of phase with the first perturbation. The
linear system solved to calculate the XP coefficients corresponding to the A21 terms is singular if
the two perturbations are not orthogonal. Section 2.5.3 provides details for calculating the S p2,m1 =
{S12,11 , ¡ ¡ ¡ , S12,51 , S22,11 , ¡ ¡ ¡ , S22,51 } and Tp2,m1 = {T12,11 , ¡ ¡ ¡ , T12,51 , T22,11 , ¡ ¡ ¡ , T22,51 } XP
coefficients given the two A21 perturbation simulations.
The perturbation procedure is repeated for all remaining higher-order incident harmonics, A pk ,
to calculate their respective XP coefficients. The fifth incident harmonic at the input port, A15 , is
included in the remaining perturbation simulations. Tables 7.4 and 7.5 present the incident and
scattered wave phasors for the two orthogonal perturbation simulations. A pure cosine is applied
for the first A15 perturbation and a pure sine is applied for the second simulation. Tables 7.4
174

k

fk (GHz)

A1k (V)

A2k (V)

B1k (V)

1
2
3
4
5

10.0
20.0
30.0
40.0
50.0

1.000 0◦
0.050 0◦

-

0.873
0.170
0.071
0.014
0.041

−93◦
115◦
−125◦
147◦
−123◦

B2k (V)
2.854
0.283
0.424
0.084
0.065

117◦
73◦
−175◦
120◦
−69◦

Table 7.4: The incident and scattered harmonic phasors of the AFRL GaN HEMT biased as a class
AB amplifier with a power available from the source of 11.6 dBm at 10 GHz. The large-signal
simulation is perturbed with a small-signal tone at the first port and at the fifth harmonic.
k

fk (GHz)

A1k (V)

A2k (V)

B1k (V)

1
2
3
4
5

10.0
20.0
30.0
40.0
50.0

1.000 0◦
0.050 −90◦

-

0.873
0.170
0.075
0.012
0.023

−93◦
114◦
−125◦
118◦
101◦

B2k (V)
2.860
0.279
0.441
0.079
0.095

117◦
73◦
−174◦
117◦
−78◦

Table 7.5: The incident and scattered harmonic phasors of the AFRL GaN HEMT biased as a class
AB amplifier with a power available from the source of 11.6 dBm at 10 GHz. The large-signal
simulation is perturbed with a small-signal tone at the first port and at the fifth harmonic. This
perturbation is out of phase with the first perturbation at the input port and the fifth harmonic.
and 7.5 confirm that the two signals are orthogonal. The XP coefficients S p1,m5 = {S11,15 , ¡ ¡ ¡ ,
S11,55 , S21,15 , ¡ ¡ ¡ , S21,55 } and Tp1,m5 = {T11,15 , ¡ ¡ ¡ , T11,55 , T21,15 , ¡ ¡ ¡ , T21,55 } are calculated
with the two A15 perturbation simulations.
The two-port, five-harmonic XP model of the GaN HEMT biased as a class AB amplifier
with a power available from the source of Pavs = 11.6 dBm at 10 GHz has 190 coefficients. The
q
2 
2
first 20 XP coefficients (sorted by the magnitude of the coefficients, S pq,mn  + Tpq,mn  ) are
presented in Table 7.6. As expected, the largest coefficient, S21,11 , corresponds to the gain of the
device. Because Tp1,m1 = 0 ∀(q, n), the fundamental scattered wave phasor at the output port is
related to the incident wave phasors by
B21 = S21,11 A11 + S21,12 A12 + ¡ ¡ ¡ + S22,1K A25
+ T21,12 A∗12 + · · · + T22,1K A∗25 .

(7.4)

The coefficients in Table 7.6 with indices (p, m) = (2, 1) contribute to the fundamental scattered
175

p

q

m

n

S pq,mn

2
2
2
2
2
2
2
1
1
1
1
2
2
2
1
2
2
2
2
2
2

1
1
1
1
2
1
2
1
1
1
1
2
2
1
1
2
2
1
2
1
1

1
2
1
1
1
1
1
3
1
5
4
1
3
3
2
5
3
1
1
4
3

1
2
3
4
5
2
4
3
1
5
4
3
3
3
2
5
4
5
2
4
4

2.834
1.143
0.773
0.697
0.605
0.757
0.638
0.872
0.850
0.845
0.834
0.557
0.711
0.695
0.806
0.793
0.677
0.496
0.530
0.650
0.489

Tpq,mn
117◦
82◦
57◦
94◦
102◦
119◦
100◦
−137◦
−93◦
−150◦
−148◦
97◦
−136◦
94◦
−127◦
−118◦
177◦
109◦
148◦
56◦
162◦

0.328
0.777
0.627
0.657
0.472
0.616
0.077
0.037
0.020
0.621
0.401
0.425
0.106
0.180
0.401
0.593
0.391
0.105
0.379

1◦
25◦
13◦
16◦
−30◦
14◦
160◦
−150◦
25◦
12◦
88◦
96◦
60◦
171◦
92◦
17◦
8◦
−10◦
90◦

Table 7.6: The 20 largest XP coefficients extracted from the AFRL GaN HEMT biased as a class
AB amplifier with a power available from the source of 11.6 dBm at 10 GHz. The coefficients are
sorted by their magnitude.
wave phasor at the output port, B21 . There is clearly inter-modulation between the input and output
harmonics of this device. Eight of the top 20 largest XP coefficients correspond to the output term
B21 . For a linear device, the only non-zero XP coefficient corresponding to B21 would be S21,11 .

7.7

PA Design with the XP Model in ADS

An XP model was extracted from FKT device simulations of the AFRL GaN HEMT in the
previous section. In this section, the HEMT XP model is loaded into ADS for simple PA design.
Figure 7.23 presents the XP simulation layout in ADS. An input file containing the XP information and coefficients is loaded into the component labeled “X2P”. Because there are some small
notational differences between the XP model in ADS and the XP model presented in this work, the

176

HARMONIC BALANCE
HarmonicBalance
HB1
Freq[1]=RFfreq GHz
Order[1]=1

Var
Eqn

VAR
VAR1
RFfreq=10
Pavs=12.9225

V_In
I_Probe
I_In
P_1Tone
in
Num=1
Z=50 Ohm
P=dbmtow(Pavs)
Freq=RFfreq GHz

1

P

V_Out

2

Ref

DC_Block
DC_Block2

DC_Block
DC_Block1

I

I_Probe
I_Out

P_Probe
P_Load

Term
out
Num=2
Z=50 Ohm

X2P
XNP1
File="fkt_afrlHemt_K5.xnp"
ModelFundamentalFreq[1]=

Figure 7.23: Layout of the XP model simulation in ADS. The components “P_1Tone” and “Term”
represent the input and output ports of the circuit. Component “X2P” is the XP model which
specifies and input file containing the model information and coefficients. Current and power
probes are placed in the circuit to calculate the large-signal FOMs.
next section will focus on several practical considerations for the following ADS simulations.

7.7.1

Practical Considerations

Several practical considerations are presented in order to seamlessly apply the extracted XP model
of this chapter to PA design in ADS. The XP model in Keysight’s ADS is defined as
bik = XikB (|a11 |) Pk +

∑



S
k−l
T
k+l
∗
|)
|)
Xik, jl (|a11 P a jl + Xik, jl (|a11 P a jl ,

(7.5)

( j,l)6=(1,1)

where i, j are port indices, k, l are harmonic indices, a jl are incident wave phasors, bik are scattered
S
wave phasors, P is the phase of the large-signal phasor a11 , XikB is the B-type XP coefficient, Xik,
jl
T is the T-type XP coefficient [26].
is the S-type XP coefficient, and Xik,
jl

The first difference between the ADS XP model and the XP model in this work is the definition
of the incident and scattered wave phasors. In this work, the phasors are defined in Eqns. (2.51)
and (2.52). The ADS definitions of the incident and scattered waves are
V jl + Z j I jl
a jl = q
 	,
8Re Z j

(7.6)

Vik − Zi∗ Iik
bik = p
.
8Re {Zi }

(7.7)

177

Here, Zi is the complex impedance of port i. The normalization of the ADS phasors follows the
“power definition” [26].
Notationally, the ADS XP model, Eqn. (7.5), differs from the XP model presented in this
work, Eqn. (2.50). First, the port and harmonic indices of the ADS XP coefficients are arranged
differently than the ones presented in this work. However, this poses no issue when loading the
coefficients into ADS. A more important distinction is the B-type XP coefficient, XikB (|a11 |). Using
the ADS port and harmonic indices i, j, k, l, these coefficients are related to the S p1,q1 terms by
XikB (|a11 |) Pk = Si1,k1 A11 Pk .

(7.8)

The X B coefficients define the product of the large-signal phasor, A11 , and the S pm,qn coefficients.
This simple yet important distinction could yield significant differences in the gain computed from
the XP model simulations.

7.7.2

Class AB Amplifier

An XP model is extracted from AFRL GaN HEMT simulations over a range of input power. ADS
offers a wide range of independent variables of the XP model including fundamental frequency and
quiescent bias. However, the operating point, VGS = −2.8 V and VDS = 10 V, and the fundamental
frequency of 10 GHz are constant in this XP model. The load impedance of the XP extraction
simulations is ZL = 50 Ω.
Validation of the XP model imported into ADS is first presented. The simulated FOMs of the
ADS XP model are compared with the FOMs calculated directly from the FKT device simulations.
This is a simple validation test as the XP model is extracted from the same FKT device simulations.
Figure 7.24 presents the input power sweep results calculated in ADS and calculated directly from
the FKT device simulations. The reported FOMs are output power, Pout , (black), power gain, GP ,
(red), and PAE (blue). The FKT device simulation FOMs are plotted with solid lines and the ADS
FOMs are plotted with squares. This result validates that the XP model is properly imported into
ADS and that there is no mismatch in power definitions described in Section 7.7.1.

178

20

ZL = 50 Ί

f0 = 10 GHz

20

Pout

GP

15

15
10
10

PAE
FKT Sim.
ADS XP Sim.

5
0
-10

PAE (%)

Power (dBm), Gain (dB)

25

5
0

-5

0

5

10

Input Power (dBm)
Figure 7.24: Validation of the input power dependent ADS XP model. This model is extracted from
FKT simulations of the AFRL GaN HEMT and imported into ADS. Device parameters of the XP
model include a gate-source bias of -2.8 V, a drain-source bias of 10 V, a fundamental frequency of
10 GHz, and a load impedance of 50 Ω.
To demonstrate the versatility of the XP model, several LP data sets are calculated in ADS
with different input powers. These are proof-of-concept results generated from the XP model
extracted at ΓL = 0. Figures 7.25a – 7.26b present LP data generated from ADS simulations
of the AFRL GaN HEMT XP model with Pavs = 3, 6, 9, and 12 dBm, respectively. These LP
simulations determine the optimal load for maximum output power. At Pavs = 3 dBm, the optimal
load is ZL = 39 + j76 Ω at 10 GHz. The optimal load for Pavs = 6 dBm is ZL = 42 + j71 Ω at
10 GHz. Finally, the optimal loads for Pavs = 9 dBm and Pavs = 12 dBm are ZL = 45 + j56 Ω and
ZL = 42 + j46 Ω, respectively, both at 10 GHz. The optimal load impedances corresponding to the
powers available from the source of 3, 6, 9, and 12 dBm yield output powers of Pout = 17.8, 19.4,
20.0, and 20.4 dBm, respectively. A summary of the LP data versus input power is illustrated in
Figure 7.27. Each reflection coefficient plotted on the Smith chart corresponds to the optimal load
impedance determined from the ADS LP simulations. The optimal reflection coefficient shifts in a
clockwise motion across the Smith chart as the input power increases.

179

5.0

2.0

1.0

0.5

10
20

5.0

10
20

-10
-20

-0.2

0
-2.

0
-2.

-0.
5

-5.0
-0.
5

-5.0

-0.2

10
20

-10
-20

0.2
2.0

0.2

1.0

5.0

10
20

5.0

0.5

2.0

0.5

2.0

Pout = 19.4 dBm
ZL = 42 + j71 Ί

1.0

1.0

0.5

Pout = 17.8 dBm
ZL = 39 + j76 Ί

(a)

(b)

-1.0

Pavs = 6 dBm

-1.0

Pavs = 3 dBm

Figure 7.25: Simulated LP data generated from ADS simulations of the AFRL HEMT XP model
with a power available from the source of (a) 3 dBm and (b) 6 dBm. The power dependent XP
model was extracted with a gate-source bias of -2.8 V, a drain-source bias of 10 V, a fundamental
frequency of 10 GHz, and a load impedance of 50 Ω. The optimal loads are 39 + j76 Ω at 3 dBm
and 42 + j71 Ω at 6 dBm, both at the fundamental frequency.
The transient input and output CW signals resulting from an ADS simulation of the XP model
with the optimal load impedance at an input power of Pin = 5.4 dBm are presented in Figures 7.28
and 7.29. The input and output CW voltages are presented in Figures 7.28a and 7.28b and the input
and output CW currents are illustrated in Figures 7.29a and 7.29b, respectively. The DC offset is
not included in the CW signals. Clearly, the XP model reaches compression at the input power of
Pin = 5.4 dBm as the output waveforms exhibit characteristics of a class B amplifier.
Finally, a comparison of the simulated LP data calculated directly from the FKT device simulations and the LP data computed with the XP model in ADS is reported in Figure 7.30. The
magnitude of the large-signal incident phasor is held constant at |A11 | = 0.75 V for the LP simulations. Note that this definition of the incident wave follows Eqn. (2.51) and is different than
Eqns. (7.6) and (7.7). Contours of the output power FOM computed from the FKT device simulations are plotted with red lines and output power contours computed from the ADS XP simulations
are plotted with blue lines. Simulated LP data generated from the ADS XP simulations do not

180

5.0

2.0

0.5

10
20

5.0

10
20

1.0

0.2

-10
-20

-0.2

0
-2.

0
-2.

-0.
5

-5.0
-0.
5

-5.0

-0.2

10
20

-10
-20

2.0

0.2

1.0

5.0

10
20

5.0

0.5

2.0

0.5

2.0

Pout = 20.4 dBm
ZL = 42 + j46 Ί

1.0

1.0

0.5

Pout = 20.0 dBm
ZL = 45 + j56 Ί

(a)

(b)

-1.0

Pavs = 12 dBm

-1.0

Pavs = 9 dBm

1.4

1.2

1.0

0.9

0.5

2.0

1.8

0.6

1.6

0.7

0.8

Figure 7.26: Simulated LP data generated from ADS simulations of the AFRL HEMT XP model
with a power available from the source of (a) 9 dBm and (b) 12 dBm. The power dependent XP
model was extracted with a gate-source bias of -2.8 V, a drain-source bias of 10 V, a fundamental
frequency of 10 GHz, and a load impedance of 50 Ω. The optimal loads are 45 + j56 Ω at 9 dBm
and 42 + j46 Ω at 12 dBm, both at the fundamental frequency.

0.4

3.0

0.3

4.0

5.0

0.2

Increasing Pin

10

0.1

20

10

5.0

4.0

3.0

1.6
1.8
2.0

1.4

1.2

0.9
1.0

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

20

-20
-0.1

-10

-0.2

-5.0
-4.0

3
-0.
-3.
0

0
-2.

8
-1.

-1.6

-1.4

-1.2

-1.0

-0.9

-0.7

-0.8

-0.6

-0.
5

.4
-0

Figure 7.27: A summary of the optimal load impedances for maximum output power determined
from the LP simulations of the AFRL GaN HEMT XP model in ADS.

181

V2(t) (V)

V1(t) (V)

2
1.5
1
0.5
0
-0.5
-1
-1.5
-2
-2.5
0

50

100

150

200

6
5
4
3
2
1
0
-1
-2
-3
-4
0

50

100

Time (ps)

150

200

Time (ps)

(a)

(b)

0.04
0.03
0.02
0.01
0
-0.01
-0.02
-0.03
-0.04
-0.05

I2(t) (A)

I1(t) (A)

Figure 7.28: The (a) input and (b) output CW voltage signals calculated with the AFRL GaN
HEMT XP model in ADS. Only the AC components of the waveforms are illustrated. The load
impedance presented to the XP model is 42 + j46 Ω at 10 GHz. This is the optimal load for
maximum output power determined from the LP simulations with a power available from the
source of 12 dBm.

0

50

100

150

200

Time (ps)

0.06
0.04
0.02
0
-0.02
-0.04
-0.06
-0.08
-0.1
0

50

100

150

200

Time (ps)

(a)

(b)

Figure 7.29: The (a) input and (b) output CW current signals calculated with the AFRL GaN
HEMT XP model in ADS. Only the AC components of the waveforms are illustrated. The load
impedance presented to the XP model is 42 + j46 Ω at 10 GHz — the optimal load for maximum output power determined from the LP simulations with a power available from the source of
12 dBm.
exactly replicate the LP data generated from the FKT device simulations. Clearly, the presence
of a non-zero reflection coefficient alters the large-signal operating point of the HEMT. The XP
model needs to be sampled over a range of load impedances to generate accurate LP data. A loaddependent XP model is available in ADS and the software interpolates the XP coefficients across

182

1.8

2.0

1.4

1.6

1.2

1.0

0.9

0.8

0.5

0.6

0.7

Pout - FKT
Pout - ADS XP
0.4

3.0

0.3

4.0

5.0

0.2

10
0.1

20

10

5.0

4.0

3.0

1.4
1.6
1.8
2.0

1.2

0.7
0.8
0.9
1.0

0.6

0.5

0.4

0.3

0.2

0.1

20

-20

-0.1

-10

-0.2

-5.0
-4.0

3
-0.

-3.
0
8
-1.
0
-2.

-1.6

-1.4

-1.2

-0.9

-1.0

-0.7

|A11| = 0.75 V
-0.8

-0.6

-0.
5

4
-0.

Figure 7.30: A comparison of the simulated LP data generated directly from the FKT device
simulations (red lines) and from the ADS simulations of the XP model (blue lines). The reported
FOM is the output power of the device. The magnitude of the large-signal incident phasor is held
constant at 0.75 V for the LP simulations. Note that this definition of the incident wave is different
than Eqns. (7.6) and (7.7).
the load impedance independent variable [112].

7.8

Discussion and Conclusions

This chapter presented a range of results that demonstrated the ability of the FKT device simulator to generate LP data for RF circuit design. The GaN HEMT used in the FKT simulations was
presented in Section 7.2. A detailed description of the device layout was illustrated in Figures 7.1
– 7.4. Section 7.3 described the TLM characterization process and applied it to estimate unknown
physical parameters in the GaN HEMT. The HEMT was further characterized in Section 7.4 by
analyzing its DC and small-signal response. Comparison of the simulated and the measured I-V
family was reported in Figure 7.10. The drain current, ID , and the transconductance, gm , of the
HEMT versus the gate-source bias, VGS , was presented in Figure 7.11. Simulations agreed well
with the measurements. S-Parameters and the small-signal current gain were presented in Figures 7.12 and 7.13, respectively. The simulated DC and small-signal results provide confidence
183

that the FKT device simulator is adequately capturing the overall response of the AFRL GaN
HEMT.
Large-signal simulations of the AFRL GaN HEMT with the FKT device simulator were presented in Section 7.5. A simulated input power sweep of the HEMT was compared to measurements in Figure 7.14. The impact of the operating point and other device features on the power
response of the device were also investigated in Section 7.5. The simulated low power response
of the device agreed well with measurements. However, the simulated high power response of the
device did not agree with measurements. First, the simulated PAE was significantly lower than
what was measured. This was expected as the simulated quiescent drain current was larger than
measurements. Second, the simulated device compresses at a lower input power level compared
to the measured device. These issues are the topic of future research which will aim to accurately
reproduce the high power response of the AFRL GaN HEMT.
The remaining sections of this chapter focused on extraction and application of XP models.
Section 7.6 provided a detailed description of the XP extraction process of the AFRL GaN HEMT
simulations. Included in this section were data tables containing the incident and scattered wave
phasors of the FKT device simulations. The XP extraction process injects incident wave signals at
different harmonics of the device ports to calculate the coefficients of the model. The data tables
provided quantitative examples of the signal injection process. Table 7.6 reported the 20 largest
coefficients of the high power AFRL GaN HEMT XP model extracted at 10 GHz.
Section 7.7 presented proof-of-concept results demonstrating the use of the extracted XP model
in ADS. The XP model was extracted over a range of input power levels at 10 GHz. The ADS
XP model was used to generate the LP data reported in Figures 7.25a – 7.26b. ADS simulations
of the XP model provided computationally efficient means to generate useful data for RF circuit
design. Frequency and operating point dependencies could be incorporated into the XP model.
This would be accomplished by extracting XP coefficients from FKT device simulations over a
range of fundamental frequencies and DC biases. ADS interpolates the XP coefficients as functions
of the input power, fundamental frequency, and DC bias independent variables. Loading conditions

184

must also be included in the XP model. Extraction of an XP model with operating condition
dependencies from FKT device simulations is now possible with the work presented in this chapter.
The FKT device simulator is an excellent candidate for high frequency and high power simulations of GaN HEMT technology. Simulations of the HEMT technology can provide supplemental
data for RF circuit design. Furthermore, the device simulator can generate measurement intensive
data including LP at high frequency and frequency and/or power dependent LP. The simulation
framework is amenable to XP model extraction which enables efficient power sweep and LP simulations in commercial CAD software like ADS.

185

CHAPTER 8
CONCLUSION AND FUTURE RESEARCH

This thesis presented high frequency and high power GaN HEMT simulations using the FKT
device simulation TCAD tool. A thorough derivation and discretization of governing equations
were presented along with a detailed discussion on solving the nonlinear device equations with
the corresponding BCs. The FKT device simulator was used to generate useful data for RF circuit
designers including S-Parameters, high frequency LP, and high frequency input power sweeps.
Many of these simulated results would be difficult to measure at a fundamental frequency above
35 GHz. Finally, an XP model was extracted from FKT device simulations of a state-of-the-art
GaN HEMT fabricated at AFRL. The XP model was imported into ADS for computationally
efficient LP simulations.

8.1

Future Research

The main contribution of this work is utilizing the FKT device simulation TCAD tool developed
by Grupen [84] to generate useful data for RF circuit designers. A list of research topics that are
now possible based on the contributions of this thesis are
• Accurate simulation of the AFRL GaN HEMT at high input power
• Full-wave LP simulations
• Frequency, DC bias, and load dependent XP extraction
• Full-wave XP extraction
• Harmonic mixing simulations of state-of-the-art devices

186

APPENDIX

187

FKT DERIVATIONS

A.1

Moments of the BTE

In Section 4.1.1, moments of the BTE were separated into even and odd components. These
components are
1
4π

Z∞
−∞

1
4π

"

 #
~E

∂
f
∂
f
0
0

d~kO(~k)
+~v · ∇r f1 − q · ∇k f1 −
= 0,
h̄
∂t
∂t coll
Z∞

−∞

(1)

"

#
~
f
E
∂
f
d~kO(~k) 1 + 1 +~v · ∇r f0 − q · ∇k f0 = 0.
h̄
τk
∂t

(2)

Eqns. (1) and (2), in conjunction with the moment operators {1,~v, E(~k), E(~k)~v}, are now used to
derive the governing equations of FKT. It is worth stating that the left hand side of Eqns. (1) and
(2) are exactly zero for integration over reciprocal-space with odd and even moment operators,
respectively. For all of the following derivations, the ~k-space integrals are over the entire domain
and therefore the limits of integration will be dropped.
Two vector calculus identities will be used in the following derivations. They are the scalarscalar product gradient chain rule
∇ (ψφ ) = φ ∇ψ + ψ∇φ ,

(3)

and the vector-scalar product chain rule
 
∇ · ψ~A = ~A · ∇ψ + ψ∇ · ~A.

A.1.1

(4)

Zeroth moment — Electron Continuity

The electron continuity equation is derived from Eqn. (1) with the unity moment operator. Using
Eqn. (4.3), integration of the first term yields
1
4π 3

Z

∂f
∂
d~k 0 =
∂t
∂t



1
4π 3

188

Z


∂n
~
d k f0 ≡
.
∂t

(5)

Integration of the second term, using Eqns. (4) and (4.5), yields the divergence term of the electron
continuity equation
1
4π 3

Z



1
d~k~v · ∇r f1 = ∇r ·
4π 3

Z



d~k~v f1 ≡ ∇ ·~Jn .

(6)

The term with the reciprocal-space gradient is more involved. Using, Eqn. (4), as well as the
divergence theorem, this term integrates to
−

q
4π 3 h̄

Z

q
d~k~E · ∇k f1 = − 3
4π h̄

Z

d~Ak n̂ · ~E f1 = 0.

(7)

This term vanishes as the non-equilibrium distribution function goes to zero much faster than the
surface area goes to infinity. Physically, this term must vanish as there are never an infinite number
of particles in any system. Finally, the collision term is written as a collision operator

Z

1
∂
f
0

~k
≡ Cn ,
d
∂t coll
4π 3

(8)

and is treated using Fermi’s golden rule. The zeroth moment of the BTE is
∂n
+ ∇ ·~Jn +Cn = 0.
∂t
A.1.2

(9)

Second Moment — Energy Conservation Equation

The electron energy conservation equation is derived from Eqn. (1) with the energy moment operator E(~k). First, Eqn. (4.4) is used for integration of the first term. This yields


Z
Z
1
∂ f0
∂
1
∂ En
~
~
~
~
d
kE(
k)
=
d
kE(
k)
f
≡
.
0
∂t
∂t 4π 3
∂t
4π 3

(10)

Next, using Eqns. (4) and (4.6), the divergence term of the energy conservation equation results
from the integration
1
4π 3

Z

d~kE(~k)~v · ∇r f1 = ∇r ·



1
4π 3

Z


~
~
d kE(k)~v f1 ≡ ∇ ·~Sn .

(11)

Unlike the zeroth moment of the BTE, the second will yield a non-zero reciprocal-space gradient
term. Using Eqns. (3) and (4) and the divergence theorem, the third term integrates to
Z
Z
Z

 
q
q
~
~
~
~
~
~
~
~
~
− 3
d kE · E(k)∇k f1 = − 3
d Ak n̂ · EE(k) f1 − d kE · ∇k E(k) f1 .
4π h̄
4π h̄
189

(12)

The first term of Eqn. (12) goes to zero for the same reasons as Eqn. (7). Then, with the relation
h̄~v = ∇k E(~k) and Eqn. (4.5), the fourth term integrates to
−

q
4π 3 h̄

Z

d~k~E · E(~k)∇k f1 = q~E ·~Jn .

(13)

This is the Joule’s heat component of the energy conservation equation. Finally, the collision term
is written as a collision operator
1
4π 3

Z



∂
f
0

d~kE(~k)
≡ CE .
∂t coll

(14)

The energy conservation equation is
∂ En
+ ∇ ·~Sn + q~E ·~Jn +CE = 0.
∂t
A.1.3

(15)

First and Third Moments — Particle and Kinetic Energy Flux Densities

The first and third moments do not require much attention before discretization. Using Eqns. (4.5)
and (4.6) and the approximation that density of states, d~k, is time-independent, the particle and
kinetic energy flux densities are

~

Z


~
~Jn + τk ∂ Jn = 1
~k~vτk q~E · ∇k f0 −~v · ∇r f0 ,
d
∂t
h̄
4π 3

(16)

Z
q

~
~Kn + τk ∂ Kn = 1
~
~
~
d kE(k)~vτk E · ∇k f0 −~v · ∇r f0 .
h̄
∂t
4π 3

(17)

~

Jn
Kn
and τk ∂∂t
are approximately zero when τk  ∂t. This proves to be an excellent
The terms τk ∂∂t

approximation for up to sub-mm-wavelength devices.

A.2

Reciprocal-Space to Energy-Space Conversion

After taking moments of the BTE, the densities and fluxes require integration over the first
Brillouin zone of reciprocal-space. Rather than engage in such a tedious calculation, FKT aims to
make use of a couple well known delta function properties. Chapter 8 of [113] provides a detailed
discussion of a similar derivation.

190

A.2.1

Two Delta Function Properties

There are two properties of the delta function which are needed to convert the reciprocal-space
integration to energy-space integration. The first is the well-known sifting property of the delta
function. This is

Z∞

n(t)δ (t − T )dt = n(T )

(18)

−∞

The second identity is the composition of the delta function with another function
Z∞

δ ( f (~r)) g(~r)d~r = ∑

Z∞

i −∞

−∞


g(~r) 
d~Si ,
|∇ f (~r)| ~r=~r

(19)

i

where ~ri and d~Si represent the surface and surface differential element corresponding to a root
1 δ (x) and a
of the scalar function f (~r). This identity can be derived with the identity δ (ax) = |a|

Taylor expansion at the function’s roots.

A.2.2

Electron Density Conversion

To illustrate the conversion from reciprocal-space integration to energy-space integration, the electron density is derived. The flux equations follow the same derivation. Using the sifting property,
Eqn. (18), the electron density becomes
Z∞ 

n=
−∞

1
4π 3

Z


~
d k f0 (E) δ (E − E(~k))dE,

and Eqn. (19) is used to convert the delta function to



Z∞
Z

1

 1 ∑ d~Si
 f0 (E)dE.
n=

3
~

4π i
|∇k E(k)| ~k=~k
−∞
i

(20)

(21)

The surfaces with differential elements d~Si are defined by the roots Ei of E − E(~k) = 0, i.e.,
surfaces of constant electron eigen-energy or “isosurfaces.”

191

A.3

SG Discretization

In Section 4.3.2, the discrete FKT particle flux was presented. This appendix includes the
complete derivation of the SG discretization of both the phenomenological DD model and the
FKT particle flux.

A.3.1

Discretization of the Classic DD Model

The vector projection of the phenomenological DD model for semiconductor device simulation
defined on a primary edge of a DV mesh is
Jn (r − r0 ) = a1 n(r − r0 ) + a2

dn(r − r0 )
,
dr

(22)

where r0 and r1 are the nodes of the primary edge and a1 and a2 are constant over the primary
edge. Given the approximation that the particle flux and electric field are spatially constant on the
primary edge, i.e., Jn (r − r0 ) ≈ Jn and E(r − r0 ) ≈ E, the DD model becomes a first-order linear
differential equation
dn(r − r0 )
.
dr
i
h
a
Its solution, with the integrating factor p(r − r0 ) = exp a1 (r − r0 ) , is
2


J0
a1
n(r − r0 ) =
+Cexp − (r − r0 ) .
a1
a2
J0 = a1 n(r − r0 ) +

(23)

(24)

Here, C is an integration constant. Determining the constants J0 and C require the two BCs
n(r − r0 )|r = n0 and n(r − r0 )|r = n1 . Applying these BCs yields the two equations
0

1

J
n0 = 0 +C,
a1

(25)



J0
a1
n1 =
+Cexp − (r1 − r0 ) .
a1
a2

(26)

and

Solving Eqn. (25) for C, substituting into Eqn. (26), and re-arranging for J0 yields
h
i
a1
n1 − n0 exp − a (r1 − r0 )
2
h
i .
J0 = a1
a1
1 − exp − a (r1 − r0 )
2

192

(27)

x
With the definition of the Bernoulli function B(x) = exp(x)−1
, the discrete particle flux is
 


 
a2
a1
a1
Jn = J0 = −
L n0 − B − L n1 .
B
L
a2
a2

(28)

Here, the length of the primary edge is L = r1 − r0 . The Bernoulli function has the asymptotic
forms
lim B(x) → 0,

(29)

lim B(x) → 1.

(30)

x→∞
x→0

Also, as the Bernoulli function argument approaches infinity, the Bernoulli function goes to negative infinity as −x.

A.3.2

Discretization of the FKT Particle Flux

With a spatially constant particle flux and electric field, the projection of the kth piece-wise FKT
particle flux on the jth primary edge is
β jk

Jn, jk = −qD jk E j (kB T (r))

#!
"
β jk +1
(k
T
(r))
d
B
Fβ (r) .
Fβ0 (r) +
jk
jk
dr
q

(31)

The spatial coordinate r is defined on a primary edge from r0 to r1 . Solving Eqn. (31) requires
choosing the dependent variable of the linear first-order differential equation. The following subsections explore two possibilities.

A.3.2.1

Generalized Einstein Relation Form

If the dependent variable of Eqn. (31) is
β
NE, jk (r) = (kB T (r)) jk Fβ0 (r),
jk

193

(32)

then the FKT particle flux becomes



F
(r)
β
d k T (r) jk
N
(r)
Jn, jk = −qD jk E j NE, jk (r) +  B
dr
q Fβ0 (r) E, jk
jk






F
(r)
F
(r)
β
β
dN
(r)
d k T (r) jk 
k T (r) jk  E, jk 
= −qD jk E j +  B
NE, jk (r) +  B
.
0
dr
q Fβ (r)
q Fβ0 (r)
dr
jk

jk

(33)
To make Eqn. (33) a linear first-order differential equation, the following approximations are made.
With the definition of the generalized Einstein Relation
kB T (r) Fβ jk (r)
,
Einn, jk (r) =
q Fβ0 (r)

(34)

jk

these approximations are


 




F
(r)
F
(r)
F
(r)
kB T (r) β jk  
d  kB T (r) β jk  1  kB T (r) β jk 
−
≈ 
 = ∆Einn, jk ,
dr
q Fβ0 (r)
L
q Fβ0 (r) 
q Fβ0 (r) 
jk

jk

jk

r1

(35)

r0

and


 


F
(r)
F
(r)
F
(r)


kB T (r) β jk
kB T (r) β jk  
1  kB T (r) β jk 
=
Ein
+
≈


n, jk ave .
q Fβ0 (r) 2
q Fβ0 (r) 
q Fβ0 (r) 
jk

jk

jk

r1

(36)

r0

Eqn. (33) is solved in the same manner as Eqn. (23) with the substitutions


a1 → −qD jk E j + ∆Einn, jk ,

(37)

and


a2 → −qD jk Einn, jk ave .
(38)


The discrete particle flux, with the BCs NE, jk (r)r = NE,0, jk and NE, jk (r)r = NE,1, jk , is
0

E =
Jn,
jk

1


 

qD jk
Einn, jk ave B(ξn,E, jk )NE,0, jk − B(−ξn,E, jk )NE,1, jk ,
Lj

(39)

with
Ξn,E, jk =

1



Einn, jk ave
194



E j + ∆Einn, jk L j .

(40)

A.3.2.2

Inverse Generalized Einstein Relation Form

In the previous subsection, the discrete particle flux with the generalized Einstein relation was
derived. If, however, the dependent variable of Eqn. (31) is
β
NI, jk (r) = (kB T (r)) jk Fβ (r),

(41)

jk

then the FKT particle flux becomes




Fβ0 (r)
d kB T (r)
jk
Jn, jk = −qD jk E j
NI, jk (r) +
NI, jk (r) 
Fβ (r)
dr
q
jk

 0





Fβ (r)
dN
(r)
k T (r)
d kB T (r) 
I, jk
jk
.
NI, jk (r) + B
Ej +
= −qD jk 
Fβ (r)
dr
q
q
dr

(42)

jk

Again, to make Eqn. (42) a linear first-order differential equation, the following approximations
are made. With the definition of the inverse generalized Einstein Relation
Fβ0 (r)
jk

Ein−1
n, jk (r) =

,

(43)


 !


kB T (r) 
kB T
kB T (r) 
−
=∆
,
q r
q r
q

(44)

Fβ (r)
jk

these approximations are
d
dr



kB T (r)
q



1
≈
L

kB T (r) 1
≈
q
2
and

1

0


 ! 

kB T (r) 
kB T (r) 
kB T
+
=
,
q r
q r
q ave
1

0


 


F
(r)
F
(r)
β jk
   −1 
1  β jk 
jk
  = Ein
≈ 
+ 0
n, jk ave .
Fβ0 (r) 2 Fβ0 (r) 
Fβ (r) 
jk
jk
jk
r1
r0
Fβ (r)

(45)



(46)

Again, Eqn. (33) is solved in the same manner as Eqn. (23), but with the substitutions


a1 → −qD jk Ein−1
n, jk



k T
Ej +∆ B
q
ave


,

(47)

and

a2 → −qD jk
195


kB T
.
q ave

(48)



The discrete particle flux, with the BCs NI, jk (r)r = NI,0, jk and NI, jk (r)r = NI,1, jk , is
0
1
I
Jn,
jk

qD jk
=
Lj






kB T
B(ξn,I, jk )NI,0, jk − B(−ξn,I, jk )NI,1, jk ,
q ave

(49)

with
1

Ξn,I, jk =  k T 
B
q



Ein−1
n, jk



k T
Ej +∆ B
q
ave

ave

196


L j.

(50)

BIBLIOGRAPHY

197

BIBLIOGRAPHY

[1]

I. Bahl, Fundamentals of RF and microwave transistor amplifiers.

Wiley, 2009.

[2]

J. D. Cressler, “SiGe HBT technology: A new contender for Si-based RF and microwave
circuit applications,” IEEE Transactions on Microwave Theory and Techniques, vol. 46,
no. 5, pp. 572–589, 1998.

[3]

U. K. Mishra, P. Parikh, and Y.-f. Wu, “AlGaN / GaN HEMTs - An overview of device
operation and applications,” Proceedings of the IEEE, vol. 90, no. 6, pp. 1022–1031, 2002.

[4]

O. Ambacher, “Role of spontaneous and piezoelectric polarization induced effects in groupIII nitride based heterostructures and devices,” Phys. Stat. Sol. (B), vol. 381, pp. 381–390,
1999.

[5]

U. K. Mishra, L. Shen, T. E. Kazior, and Y.-F. Wu, “GaN-based RF power devices and
amplifiers,” Proceedings of the IEEE, vol. 96, no. 2, pp. 287–305, 2008.

[6]

R. C. Fitch, D. E. Walker, A. J. Green, S. E. Tetlak, J. K. Gillespie, R. D. Gilbert, K. A.
Sutherlin, W. D. Gouty, J. P. Theimer, G. D. Via, K. D. Chabak, and G. H. Jessen, “Implementation of high-power-density X-band AlGaN/GaN high electron mobility transistors in
a millimeter-wave monolithic microwave integrated circuit process,” IEEE Electron Device
Letters, vol. 36, no. 10, pp. 1004–1007, 2015.

[7]

G. C. Barisich, A. C. Ulusoy, E. Gebara, and J. Papapolymerou, “A reactively matched
1.0-11.5 GHz hybrid packaged GaN high power amplifier,” IEEE Microwave and Wireless
Components Letters, vol. 25, no. 12, pp. 811–813, 2015.

[8]

D. Sokal and A. Sokal, “Class E - A new class of high-efficiency tuned single-ended switching power amplifiers,” IEEE Journal of Solid-State Circuits, vol. SC-10, no. 3, pp. 168–176,
1975.

[9]

I. Song, A. C. Ulusoy, M. A. Oakley, I. Ju, M.-K. Cho, and J. D. Cressler, “Inverse classF X-band SiGe HBT power amplifier with 44% PAE and 24.5 dBm peak output power,”
Microwave and Optical Technology Letters, vol. 58, no. 12, pp. 2868–2871, 2016.

[10] J. B. Beyer, S. N. Prasad, R. C. Becker, J. E. Nordman, and G. K. Hohenwarter, “MESFET distributed amplifier design guidelines,” IEEE Transactions on Microwave Theory and
Techniques, vol. MTT-32, no. 3, pp. 268–275, 1984.
[11] J. L. Dawson and T. H. Lee, “Automatic phase alignment for a fully integrated cartesian
feedback power amplifier system,” IEEE Journal of Solid-State Circuits, vol. 38, no. 12, pp.
2269–2279, 2003.
[12] D. K. Su and W. J. McFarland, “An IC for linearizing RF power amplifiers using envelope
elimination and restoration,” IEEE Journal of Solid-State Circuits, vol. 33, no. 12, pp. 2252–
2258, 1998.
198

[13] A. M. Smith and J. K. Cavers, “A wideband architecture for adaptive feedforward linearization,” in Vehicular Technology Conference, 1998, pp. 2488–2492.
[14] M. P. van der Heijden, M. Acar, J. S. Vromans, and D. A. Calvillo-cortes, “A 19W highefficiency wide-band CMOS-GaN class-E chireix RF outphasing power amplifier,” in Microwave Symposium Digest, 2011.
[15] A. Agah, H.-t. Dabag, B. Hana, P. M. Asbeck, J. F. Buckwalter, and L. E. Larson, “Active
millimeter-wave phase-shift Doherty power amplifier in 45-nm SOI CMOS,” IEEE Journal
of Solid-State Circuits, vol. 48, no. 10, pp. 2338–2350, 2013.
[16] A. Ferrero and M. Pirola, “Harmonic load-pull techniques,” IEEE Microwave Magazine, no.
June, pp. 116–123, 2013.
[17] J. M. Cusack and S. M. Perlman, “Automaic load contour mapping for microwave power
transistors,” IEEE Transactions on Microwave Theory and Techniques, vol. MTT-22, no. 12,
pp. 1146–1152, 1974.
[18] Y. Takayama, “A new load-pull characterization method for microwave power transistor,”
in IEEE MTT-S International Microwave Symposium Digest, 1976, pp. 218–220.
[19] M. Marchetti, M. J. Pelk, K. Buisman, W. C. E. Neo, M. Spirito, and L. C. N. D. Vreede,
“Active harmonic load-pull with realistic wideband communications signals,” IEEE Transactions on Microwave Theory and Techniques, vol. 56, no. 12, pp. 2979–2988, 2008.
[20] I. Angelov, H. Zirath, and N. Rorsman, “A new empirical nonlinear model for HEMT
and MESFET devices,” IEEE Transactions on Microwave Theory and Techniques, vol. 40,
no. 12, pp. 2258–2266, 1992.
[21] I. Angelov, V. Desmaris, K. Dynefors, P. Å. Nilsson, N. Rorsman, and H. Zirath, “On the
large-signal modelling of AlGaN / GaN HEMTs and SiC MESFETs,” in Gallium Arsenide
and Other Semiconductor Application Symposium, 2005.
[22] J. Xu, J. Hom, M. Iwamoto, and D. E. Root, “Large-signal FET model with multiple time
scale dynamics from nonlinear vector network analyzer data,” in Microwave Symposium
Digest, 2010, pp. 417–420.
[23] J. Xu, R. Jones, S. A. Harris, T. Nielsen, and D. E. Root, “Dynamic FET model - DynaFET
- for GaN transistors from NVNA active source injection measurements,” in Microwave
Symposium, 2014, pp. 2–4.
[24] J. Verspecht and D. E. Root, “Polyharmonic distortion modeling,” IEEE Microwave Magazine, no. June, pp. 44–57, jun 2006.
[25] D. Root, J. Verspect, J. Horn, and M. Marcu, “X-Parameters: Characterization, modeling, and design of nonlinear RF and microwave components,” Cambridge University Press,
2013.
[26] Keysight, “www.keysight.com,” 2017.

199

[27] Y. Wang, T. S. Nielsen, O. K. Jensen, and T. Larsen, “X-parameter based GaN device modeling and its application to a high-efficiency PA design,” pp. 1–4, 2014.
[28] D. Guerra, “GaN HEMT modeling and design for millimeter and sub-millimeter wave
power amplifiers through Monte Carlo particle-based device simulations,” Ph.D. dissertation, 2011.
[29] D. Guerra, M. Saraniti, D. K. Ferry, and S. M. Goodnick, “X-parameters and efficient physical device simulation of mm-wave power transistors with harmonic loading,” International
Microwave Symposium, no. 2, pp. 1–4, 2014.
[30] A. Franz, G. Franz, S. Selberherr, C. Ringhofer, and P. Markowich, “Finite boxes - A generalization of the finite-difference method suitable for semiconductor device simulation,”
IEEE Transactions on Electron Devices, vol. 30, no. 9, pp. 1070–1082, sep 1983.
[31] G. E. Forsythe and W. R. Wasow, Finite-difference methods for partial differential equations. New York: Wiley, 1960.
[32] L. Fox, Numerical solution of ordinary and partial differential equations, 1st ed., Oxford,
1962.
[33] Y. Kurihara and J. J.L. Holloway, “Numerical integration of a nine-level global primitive
equations model formulated by the box method,” Mon. Wea. Rev., vol. 95, no. 8, pp. 509–
530, 1967.
[34] A. Bowyer, “Computing Dirichlet tessellations,” The Computer Journal, vol. 24, no. 2, pp.
162–166, feb 1981.
[35] D. F. Watson, “Computing the n-dimensional Delaunay tessellation with application to
Voronoi polytopes,” The Computer Journal, vol. 24, no. 2, pp. 167–172, feb 1981.
[36] F. Aurenhammer, “Voronoi diagrams — A survey of a fundamental geometric data structure,” ACM Computing Surveys (CSUR), vol. 23, no. 3, pp. 345–405, sep 1991.
[37] D. T. Lee and B. J. Schachter, “Two algorithms for constructing a Delaunay triangulation,”
International Journal of Computer & Information Sciences, vol. 9, no. 3, pp. 219–242, jun
1980.
[38] L. P. Chew, “Constrained Delaunay triangulations,” in Proceedings of the third annual symposium on Computational geometry - SCG ’87. New York, New York, USA: ACM Press,
1987, pp. 215–222.
[39] C. Geuzaine and J.-F. Remacle, “Gmsh: A 3-D finite element mesh generator with built-in
pre- and post-processing facilities,” International Journal for Numerical Methods in Engineering, vol. 79, no. 11, pp. 1309–1331, sep 2009.
[40] H. Si, “TetGen, a Delaunay-based quality tetrahedral mesh generator,” ACM Transactions
on Mathematical Software, vol. 41, no. 2, pp. 1–36, feb 2015.

200

[41] Q. Du, V. Faber, and M. Gunzburger, “Centroidal Voronoi tessellations: Applications and
algorithms,” SIAM Review, vol. 41, no. 4, pp. 637–676, jan 1999.
[42] Z. Q. Xie, O. Hassan, and K. Morgan, “Tailoring unstructured meshes for use with a 3D
time domain co-volume algorithm for computational electromagnetics,” Int. J. Numer. Meth.
Engng, vol. 87, no. July 2010, pp. 48–65, 2011.
[43] I. Sazonov, D. Wang, O. Hassan, K. Morgan, and N. Weatherill, “A stitching method for the
generation of unstructured meshes for use with co-volume solution techniques,” Computer
Methods in Applied Mechanics and Engineering, vol. 195, no. 13-16, pp. 1826–1845, feb
2006.
[44] S. Walton, O. Hassan, and K. Morgan, “Advances in co-volume mesh generation and mesh
optimisation techniques,” Computers & Structures, vol. 181, pp. 70–88, mar 2017.
[45] N. Hitschfeld, P. Conti, and W. Fichtner, “Mixed element trees: A generalization of modified octrees for the generation of meshes for the simulation of complex 3-D semiconductor
device structures,” IEEE Transactions on Computer-Aided Design of Integrated Circuits and
Systems, vol. 12, no. 11, pp. 1714–1725, 1993.
[46] P. Conti, N. Hitschfeld, and W. Fichtner, “An octree-based mixed element grid allocator for
the simulation of complex 3-D device structures,” IEEE Transactions on Computer-Aided
Design of Integrated Circuits and Systems, vol. 10, no. 10, pp. 1231–1241, 1991.
[47] P. Conti, M. Tomizawa, and A. Yoshii, “Generation of oriented three-dimensional Delaunay
grids suitable for the control volume integration method,” International Journal for Numerical Methods in Engineering, vol. 37, no. 19, pp. 3211–3227, 1994.
[48] Synopsys, “Mesh Generation Tools User Guide,” 2013.
[49] J. Krause, “On boundary conforming anisotropic Delaunay meshes,” Ph.D. dissertation,
2001.
[50] N. Hitschfeld, L. Villablanca, J. Krause, and M. C. Rivara, “Improving the quality of meshes
for the simulation of semiconductor devices using Lepp-based algorithms,” International
Journal for Numerical Methods in Engineering, vol. 58, no. 2, pp. 333–347, 2003.
[51] C. T. Kelley, Solving nonlinear equations with Newton’s method.
2003.

Philadelphia: SIAM,

[52] C. G. Broyden, “A class of methods for solving nonlinear simultaneous equations,” Mathematics of Computation, vol. 19, no. 92, p. 577, oct 1965.
[53] D. Knoll and D. Keyes, “Jacobian-free Newton-Krylov methods: A survey of approaches
and applications,” Journal of Computational Physics, vol. 193, no. 2, pp. 357–397, 2004.
[54] H. K. Khalil, Nonlinear systems, 3rd ed.

Upper Saddle River: Prentice Hall, 2002.

[55] D. Pozar, Microwave engineering, 4th ed.

201

Wiley, 2005.

[56] G. Gonzalez, Microwave transistor amplifiers: Analysis and design, 2nd ed. Upper Saddle
River: Prentice Hall, 1997.
[57] S. C. Cripps, RF power amplifiers for wireless communications, 2nd ed. Norwood: Artech
House, 2006.
[58] D. E. Root, J. Verspecht, D. Sharrit, J. Wood, and A. Cognata, “Broad-band poly-harmonic
distortion (PHD) behavioral models from fast automated simulations and large-signal vectorial network measurements,” IEEE Transactions on Microwave Theory and Techniques,
vol. 53, no. 11, pp. 3656–3664, 2005.
[59] J. Horn, D. E. Root, and G. Simpson, “GaN device modeling with X-parameters,” 2010.
[60] A. M. PelĂĄez-pĂŠrez, S. Woodington, M. FernĂĄndez-barciela, P. J. Tasker, and J. I. Alonso,
“Application of an NVNA-based system and load-independent X-parameters in analytical
circuit design assisted by an experimental search algorithm,” IEEE Trans. Microwave Theory and Tech., vol. 61, no. 1, pp. 581–586, 2013.
[61] Y. Wang, T. S. Nielsen, O. K. Jensen, and T. Larsen, “GaN HEMT device modeling using
X-parameters with emphasis on complexity reduction of harmonic load-pull measurement,”
2015.
[62] M. Grupen, “Full wave electromagnetics and hot electron transport with electronic band
structure for high speed semiconductor device simulation,” IEEE Trans. Microwave Theory
and Tech., vol. 62, no. 12, pp. 2868 – 2877, 2014.
[63] J.-M. Jin, Theory and computation of electromagnetic fields.

Wiley, 2010.

[64] J. Hesthaven and T. Warburton, “Nodal high-order methods on unstructured grids,” J. Computat. Phys., vol. 181, no. 1, pp. 186–221, sep 2002.
[65] R. J. LeVeque, Finite volume methods for hyperbolic problems.
Press, 2002.

Cambridge University

[66] K. J. Willis, S. C. Hagness, and I. Knezevic, “Multiphysics simulation of high-frequency
carrier dynamics in conductive materials,” Journal of Applied Physics, vol. 110, pp. 1–15,
2011.
[67] T. Weiland, “Time domain electromagnetic field computation with finite difference methods,” International Journal of Numerical Modeling: Electronic Networks, Devices, and
Fields, vol. 9, pp. 295–319, 1996.
[68] K. S. Yee, “Numerical solution of initial value problems of Maxwells equations,” IEEE
Transactions on Antennas and Propagation, vol. 14, pp. 302–307, 1966.
[69] N. K. Madsen, “Divergence preserving discrete surface integral methods for Maxwell’s curl
equations using non-orthogonal unstructured grids,” Journal of Computational Physics, vol.
119, pp. 34–45, 1995.

202

[70] A. Taflove and S. C. Hagness, Computational electrodynamics - The finite-difference timedomain method, 3rd ed. Artech House, 2005.
[71] R. F. Harrington, Time-harmonic electromagnetic fields.

Wiley-IEEE Press, 2001.

[72] K. Hess, Advanced theory of semiconductor devices, 2000, vol. 1.
[73] C. Jacoboni, Theory of electron transport in semiconductors: A pathway from elementary
physics to nonequilibrium green functions, 2010.
[74] N. Metropolis and S. Ulam, “The Monte Carlo method,” Journal of the American Statistical
Association, vol. 44, no. 247, pp. 335–341, 1949.
[75] M. Fischetti and S. Laux, “Monte Carlo analysis of electron transport in small semiconductor devices including band-structure and space-charge effects,” Physical Review B, vol. 38,
no. 14, pp. 9721–9745, 1988.
[76] M. Saraniti and S. M. Goodnick, “Hybrid fullband cellular automaton/Monte Carlo approach for fast simulation of charge transport in semiconductors,” IEEE Transactions on
Electron Devices, vol. 47, no. 10, pp. 1909–1916, 2000.
[77] D. L. Scharfetter and H. K. Gummel, “Large-signal analysis of a silicon read diode oscillator,” IEEE Transactions on Electron Devices, vol. ED-16, no. 1, pp. 64–77, 1969.
[78] R. Stratton, “Diffusion of hot and cold electrons in semiconductor barriers,” Physical Review, vol. 126, no. 6, pp. 2002–2014, 1962.
[79] K. Blotekjaer, “Transport equations for electrons in two-valley semiconductors,” IEEE
Transactions on Electron Devices, vol. ED-17, no. 1, pp. 38–47, 1970.
[80] T. Grasser, T. W. Tang, H. Kosina, and S. Selberherr, “A review of hydrodynamic and
energy-transport models for semiconductor device simulation,” Proceedings of the IEEE,
vol. 91, no. 2, pp. 251–274, feb 2003.
[81] Synopsys, “Sentaurus device user guide,” 2013.
[82] M. Grupen, “An alternative treatment of heat flow for charge transport in semiconductor
devices,” Journal of Applied Physics, vol. 106, no. 12, p. 123702, 2009.
[83] ——, “Energy transport model with full band structure for GaAs electronic devices,” Journal of Computational Electronics, vol. 10, no. 3, pp. 271–290, may 2011.
[84] ——, “GaN high electron mobility transistor simulations with full wave and hot electron
effects,” IEEE Transactions on Electron Devices, vol. 63, no. 8, pp. 3096–3102, 2016.
[85] T. K. Bergstresser and M. L. Cohen, “Electronic structure and optical properties of hexagonal CdSe, CdS, and ZnS,” Physical Review, vol. 164, pp. 1069–1080, 1967.
[86] M. Goano, “Series expansion of the Fermi-Dirac integral over the entire domain of real j
and x,” Solid-State Electronics, vol. 36, no. 2, pp. 217–221, 1993.

203

[87] S. Selberherr, Analysis and simulation of semiconductor devices.
1984.
[88] K. Seeger, Semiconductor physics: An introduction, 9th ed.

Springer-Verlag Wien,

Springer-Verlag, 2004.

[89] S. Sridharan, A. Christensen, A. Venkatachalam, S. Graham, and P. D. Yoder, “Temperatureand doping-dependent anisotropic stationary electron velocity in wurtzite GaN,” IEEE Electron Device Letters, vol. 32, no. 11, pp. 1522–1524, nov 2011.
[90] J. George and J.-H. Liu, Computer solution of large sparse positive definite systems. Prentice Hall, 1981.
[91] S. Godunov, “Different methods for shock waves,” Ph.D. dissertation, Moscow State University, 1954.
[92] A. Harten, “High resolution schemes for hyperbolic conservation laws,” Journal of Computational Physics, vol. 49, no. 3, pp. 357–393, mar 1983.
[93] M. S. Darwish and F. Moukalled, “TVD schemes for unstructured grids,” International
Journal of Heat and Mass Transfer, vol. 46, no. 4, pp. 599–611, 2003.
[94] B. Benbakhti, A. Soltani, K. Kalna, M. Rousseau, and J. C. De Jaeger, “Effects of selfheating on performance degradation in AlGaN/GaN-based devices,” IEEE Transactions on
Electron Devices, vol. 56, no. 10, pp. 2178–2185, 2009.
[95] S. Forster, B. Ubochi, and K. Kalna, “Monte Carlo simulations of electron transport in bulk
GaN,” in 2017 International Workshop on Computational Nanotechnology, 2017.
[96] A. Brannick, N. A. Zakhleniuk, B. K. Ridley, L. F. Eastman, J. R. Shealy, and W. J. Schaff,
“Hydrodynamic simulation of surface traps in the AlGaN/GaN HEMT,” Microelectronics
Journal, vol. 40, no. 3, pp. 410–412, 2009.
[97] X. D. Wang, W. D. Hu, X. S. Chen, and W. Lu, “The study of self-heating and hot-electron
effects for AlGaN/GaN double-channel HEMTs,” IEEE Transactions on Electron Devices,
vol. 59, no. 5, pp. 1393–1401, 2012.
[98] S. Vitanov, V. Palankovski, S. Maroldt, R. Quay, S. Murad, T. RĂśdle, and S. Selberherr,
“Physics-based modeling of GaN HEMTs,” IEEE Transactions on Electron Devices, vol. 59,
no. 3, pp. 685–693, 2012.
[99] S. Bhardwaj, B. Sensale-Rodriguez, H. G. Xing, and J. L. Volakis, “Full-wave hydrodynamic model for predicting THz emission from grating-gate RTD-gated plasma wave
HEMTs,” in 2015 73rd Annual Device Research Conference (DRC), 2015.
[100] S. Yamakawa, S. Goodnick, S. Aboud, and M. Saraniti, “Quantum corrected full-band cellular Monte Carlo simulation of AlGaN/GaN HEMTs,” Journal of Computational Electronics,
vol. 3, no. 3-4, pp. 299–303, 2004.

204

[101] T. Sadi, R. W. Kelsall, and N. J. Pilgrim, “Investigation of self-heating effects in submicrometer GaN/AlGaN HEMTs using an electrothermal Monte Carlo method,” IEEE Transactions
on Electron Devices, vol. 53, no. 12, pp. 2892–2900, 2006.
[102] S. Russo and A. D. Carlo, “Influence of the source-gate distance on the AlGaN/GaN HEMT
performance,” IEEE Transactions on Electron Devices, vol. 54, no. 5, pp. 1071–1075, 2007.
[103] S. Sridharan, A. Venkatachalam, and P. D. Yoder, “Electrothermal analysis of AlGaN/GaN
high electron mobility transistors,” Journal of Computational Electronics, vol. 7, no. 3, pp.
236–239, 2008.
[104] A. Ashok, D. Vasileska, O. L. Hartin, and S. M. Goodnick, “Electrothermal Monte Carlo
simulation of GaN HEMTs including electron-electron interactions,” IEEE Transactions on
Electron Devices, vol. 57, no. 3, pp. 562–570, 2010.
[105] S. N. Tenneti, N. K. Nahar, and J. L. Volakis, “Full-wave modeling of THz RTD-gated GaN
HEMTs,” Infrared Physics and Technology, vol. 72, pp. 221–228, 2015.
[106] T. Palacios, A. Chakraborty, S. Heikman, S. Keller, S. P. DenBaars, and U. K. Mishra,
“AlGaN / GaN high electron mobility transistors with InGaN back-barriers,” IEEE Elec,
vol. 27, no. 1, pp. 13–15, 2006.
[107] F. A. Marino, N. Faralli, T. Palacios, D. K. Ferry, S. M. Goodnick, and M. Saraniti, “Effects
of threading dislocations on AlGaN / GaN high-electron mobility transistors,” IEEE Trans.
Electron Devices, vol. 57, no. 1, pp. 353–360, 2010.
[108] S. M. Goodnick and M. Saraniti, “Modeling and simulation of terahertz devices,” IEEE
Microwave Magazine, vol. 13, no. 7, pp. 36–44, 2012.
[109] N. C. Miller, J. D. Albrecht, and M. Grupen, “Large-signal RF GaN HEMT simulation using
Fermi kinetics transport,” in 2016 74th Annual Device Research Conference (DRC). IEEE,
jun 2016, pp. 1–2.
[110] D. K. Schroder, Semiconductor material and device characterization, 3rd ed.
NJ, USA: Wiley-Interscience, 2005.

Hoboken,

[111] M. Wang and K. J. Chen, “Kink effect in AlGaN/GaN HEMTs induced by drain and gate
pumping,” IEEE Electron Device Letters, vol. 32, no. 4, pp. 482–484, 2011.
[112] G. Simpson, J. Horn, D. Gunyan, and D. E. Root, “Load-Pull + NVNA = Enhanced Xparameters for PA designs with high mismatch and technology-independent large-signal
device models,” in 72nd Microwave Measurement Symposium, 2008, pp. 88–91.
[113] N. Ashcroft and N. Mermin, “Solid state physics,” 1976.

205