MODELING OF HYDROGEN-BASED PLASMAS IN MICROWAVE
PLASMA-ASSISTED CHEMICAL VAPOR DEPOSITION REACTORS AT MODERATE
PRESSURES
By
Collin Stephen Meierbachtol

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
Electrical Engineering – Doctor of Philosophy
2013

ABSTRACT
MODELING OF HYDROGEN-BASED PLASMAS IN MICROWAVE
PLASMA-ASSISTED CHEMICAL VAPOR DEPOSITION REACTORS AT MODERATE
PRESSURES
By
Collin Stephen Meierbachtol
Microwave Plasma-Assisted Chemical Vapor Deposition (PACVD) systems are used in
the deposition of high quality diamond films. These systems have traditionally been operated
at less than 20% atmospheric pressure (atm), resulting in growth rates up to 5 µm/hr. Under
such conditions, the system operation and plasma behavior are well-understood and have
been successfully modeled. Recent experiments at pressures approaching 40% atm have
demonstrated faster growth rates and better quality samples. At these increased pressures,
the system operation and plasma behavior are not completely understood, with unusual
plasm abehavior sometimes observed. Experimental measurements within these systems can
be difficult, making numerical models attractive for aiding in understanding this behavior.
This thesis presents a multiphysics numerical model of Microwave PACVD systems, which
is accurate under these operating conditions. Electromagnetic field propagation, chemical
reactions, species diffusion, thermal processes, energy transfer, and convective flows are all
included in the multiphysics model. The model is verified against canonical problems and
validated against experimental data. Extensive numerical results are provided for different
operating conditions and system configurations.

Copyright by
COLLIN STEPHEN MEIERBACHTOL
2013

For Krista.

v

ACKNOWLEDGMENTS

Many have aided in my completing this dissertation, whether directly or indirectly, and
whether known or unknown to them. I extend my sincere gratitude to them for their contributions.
Thank you to the numerous teachers, coaches, mentors, and guides who have shared
support, encouragement, and knowledge with me over the years. You have challenged me to
discover and learn new things. You have pushed me to test myself and my limits. And you
have provided a foundation for my future growth.
Thank you to the many members of the Electromagnetics Research Group (EMRG) at
Michigan State University (both past and present) for your time, counsel, and collaboration
over the many years. Your friendships fostered an environment of openness and acceptance
that made it possible to try, fail, and succeed on many occasions, all while gaining knowledge
and experience.
Thank you to my committee members for their helpful advice and guidance, both with
respect to this dissertation and my research as a whole.
Thank you to Dr. Timothy Grotjohn for his collaborations and help with the direction of
my research and its review. This work would not have been possible without your continued
support and willingness to meet to discuss results, ideas, and future plans.
Thank you to Dr. Shanker Balasubramaniam for his guidance, willingness to discuss my
research, and general counsel over the last several years. Thank you for pushing me outside
my comfort zone and challenging me on many levels. Thank you for introducing me to the
world of computational science, and for instilling a sense of rigor and confidence along with
this.
Thank you to my family for their continued support throughout my life and in everything
I do. You provided the strong foundation upon which everything I know is built.
And finally, thank you to my wife, Krista, for her understanding, patience, and support.

vi

You pushed me when I needed to be pushed, and consoled me when I was weary. I could
not have made it through the countless long nights, frustrations, and without you.
And so, thank you again to all those listed here, and to the many more I may have
overlooked, for you all aided in the completion of this work. Thank you!

vii

TABLE OF CONTENTS

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

xii

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xiii

KEY TO SYMBOLS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxvii
CHAPTER 1
INTRODUCTION . . . . . . . . . . . . . . . . . . . . . .
1.1 Diamond . . . . . . . . . . . . . . . . . . . . . . .
1.1.1 History . . . . . . . . . . . . . . . . . . . .
1.1.2 Applications . . . . . . . . . . . . . . . . .
1.2 Synthetic Diamond . . . . . . . . . . . . . . . . .
1.2.1 High Pressure Production . . . . . . . . .
1.2.2 Low Pressure Production . . . . . . . . . .
1.3 MPACVD Systems . . . . . . . . . . . . . . . . .
1.3.1 History . . . . . . . . . . . . . . . . . . . .
1.3.2 Advantages of MPACVD Diamond Growth
1.3.3 Higher Pressure Experiments . . . . . . .
1.4 Multiphysics Plasma Models . . . . . . . . . . . .
1.4.1 Motivation . . . . . . . . . . . . . . . . . .
1.4.2 History . . . . . . . . . . . . . . . . . . . .
1.4.3 Current State of MPACVD Modeling . . .
1.5 Contributions . . . . . . . . . . . . . . . . . . . .

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1
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1
2
3
4
5
5
6
7
8
8
9
11
11

CHAPTER 2
PROBLEM STATEMENT . . . . . . . . . . . . . . . . .
2.1 Introduction to MPACVD Systems . . . . . . .
2.1.1 MPACVD System Configurations . . . .
2.1.2 MSU MPACVD System Overview . . . .
2.2 Basics of MPACVD operation . . . . . . . . . .
2.2.1 Detailed description of physical processes
2.2.2 Role of plasma in MPACVD . . . . . . .
2.3 Motivation for Numerical Model . . . . . . . . .

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12
12
12
15
17
17
20
21

. . . . . . . . . .
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and Assumptions
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24
24
25
25
25
27
27

CHAPTER 3
MATHEMATICAL FORMALISMS . . .
3.1 Choice of Temporal Frame . . .
3.2 Electromagnetics . . . . . . . .
3.2.1 Maxwell’s Equations . .
3.2.2 Physical Approximations
3.2.3 Final Coupled Form . .
3.2.4 Plasma Influence . . . .

viii

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28
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50
51

CHAPTER 4
SOLUTION PROCEDURE . . . . . . . . . . . . . .
4.1 Axisymmetric Approximation . . . . . . . .
4.2 Finite Difference . . . . . . . . . . . . . . .
4.3 Electromagnetics Module . . . . . . . . . . .
4.3.1 Finite-Difference Frequency-Domain .
4.3.2 Equation Discretization . . . . . . .
4.3.3 Boundary Conditions . . . . . . . . .
4.3.4 Matrix Representation . . . . . . . .
4.3.5 Solution Procedure . . . . . . . . . .
4.3.6 Reflected Power Minimization . . . .
4.4 Absorbed Power . . . . . . . . . . . . . . . .
4.5 Plasma Fluid Module . . . . . . . . . . . . .
4.5.1 Equation Discretization . . . . . . .
4.5.2 Boundary Conditions . . . . . . . . .
4.5.3 Matrix Representation . . . . . . . .
4.5.4 Solution Procedure . . . . . . . . . .
4.5.5 Solution Algorithm . . . . . . . . . .
4.6 Substrate Thermal Module . . . . . . . . . .
4.6.1 Equation Discretization . . . . . . .
4.6.2 Boundary Conditions . . . . . . . . .
4.6.3 Matrix Representation . . . . . . . .
4.7 Overall Solution Algorithm . . . . . . . . . .

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105
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111
113
114

CHAPTER 5
VERIFICATION AND VALIDATION . .
5.1 Verification . . . . . . . . . . . .
5.1.1 Electromagnetics Module .
5.1.2 Transport Equations . . .
5.1.3 Substrate Thermal Model
5.1.4 Complete Simulation . . .

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118
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126
126

3.3
3.4

3.5
3.6

3.2.5 Boundary Conditions . . . .
Absorbed Power . . . . . . . . . . .
Plasma Physics . . . . . . . . . . .
3.4.1 Approximations . . . . . . .
3.4.2 Mass Transport . . . . . . .
3.4.3 Momentum Transport . . .
3.4.4 Electron Energy Transport .
3.4.5 Total Energy Transport . .
3.4.6 Boundary Conditions . . . .
Electrical Conductivity . . . . . . .
Inclusion of Substrate Temperature
3.6.1 Heat Equation . . . . . . . .
3.6.2 Boundary Conditions . . . .

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5.2

Validation . . . . . . . . . . . . .
5.2.1 Electromagnetics Module .
5.2.2 Plasma Fluid Module . . .
5.2.3 Substrate Thermal Model
5.2.4 Complete Simulation . . .

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128
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137
146
147

CHAPTER 6
RESULTS . . . . . . . . . . . . . . . . . .
6.1 High-pressure example . . . . . .
6.1.1 Electromagnetics module .
6.1.2 Plasma fluid module . . .
6.2 Extreme high-pressure example .
6.2.1 Electromagnetics module .
6.2.2 Plasma fluid module . . .
6.3 Power and pressure combinations
6.4 Effects of reactor geometry . . . .
6.4.1 Substrate position . . . .
6.4.2 Reactor cavity height . . .
6.4.3 Reactor Configuration . .
6.5 Power Minimization . . . . . . . .
6.6 Convergence . . . . . . . . . . . .

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264

CHAPTER 7
ANALYSIS AND DISCUSSION . . . . . . . . .
7.1 Electromagnetics . . . . . . . . . . . .
7.2 Absorbed Power . . . . . . . . . . . . .
7.3 Plasma Fluid Module . . . . . . . . . .
7.3.1 Validation . . . . . . . . . . . .
7.3.2 Effect of average gas flow . . . .
7.4 Atomic Hydrogen Diffusion Coefficient
7.5 Conductivity . . . . . . . . . . . . . .
7.6 Stability . . . . . . . . . . . . . . . . .
7.7 Convergence . . . . . . . . . . . . . . .

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267
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CHAPTER 8
SUMMARY . . . . . . . . . . . . . . . . . .
8.1 Future work . . . . . . . . . . . . .
8.1.1 Further physics modules . .
8.2 Recommendations for experiments .

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302
304
305
306

APPENDICES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Appendix A: Collision Integrals . . . . . . . . . . . . . . . . . . . . . . . .
Appendix B: List of Chemical Reactions . . . . . . . . . . . . . . . . . . .
Appendix C: Physical Constants Calculations . . . . . . . . . . . . . . . .
Appendix D: Finite Difference Stencil for Momentum Transport Equation .

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Appendix E: FORTRAN files . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345
BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372

xi

LIST OF TABLES

Table 5.1 Lowest 20 analytic and calculated eigenvalues for a regular PEC cylinder waveguide with varying mesh refinement for the electromagnetics
module. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
Table 5.2 Lowest 20 analytic and calculated eigenvalues for filled cylindrical waveguide with varying mesh resolutions and homogeneous permittivity of
r = 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
Table 5.3 Lowest 20 analytic and calculated eigenvalues for a homogeneously filled
cylindrical waveguide with r = 1 and σe = 1 + j0 with varying mesh
ˆ
refinements. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
Table 5.4 Lowest 20 analytic and calculated eigenvalues for an empty cylindrical
cavity with varying mesh refinement. . . . . . . . . . . . . . . . . . . 125
Table 5.5 Lowest 20 analytic and calculated eigenvalues for an empty cylindrical waveguide with varying mesh refinements for the substrate thermal
module. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
Table 5.6 Comparison of analytic and measured skin depth values for wave propagation into a lossy material with r = 1 − j0.1 for mesh discretization
of ∆ = λ0 /20. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
Table 7.1 Maximum iteration limits imposed in plasma module solution to demonstrate convergence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284
Table A.1 Collision integral polynomial fit coefficients. . . . . . . . . . . . . . . . 309
Table B.1 List of Arrhenius chemical reactions. . . . . . . . . . . . . . . . . . . . 320
Table B.2 List of non-Arrhenius chemical reactions. All reactions depend on the
electron temperature, Te . . . . . . . . . . . . . . . . . . . . . . . . . . 321
Table B.3 List of surface chemical reactions. . . . . . . . . . . . . . . . . . . . . . 321

xii

LIST OF FIGURES

Figure 2.1

MSU MPACVD system schematic . . . . . . . . . . . . . . . . . . .

14

Figure 4.1

Generic mesh element with labeled geometry feature index pairs. . .

55

Figure 4.2

Staggered Yee mesh in two-dimensions. . . . . . . . . . . . . . . . .

56

Figure 4.3

Typical electromagnetics numerical solution domain mesh. . . . . . .

58

Figure 4.4

Magnetic field and permittivity sample locations in the discretized
magnetic field equation . . . . . . . . . . . . . . . . . . . . . . . . .

63

Location of the TF/SF regions in the electromagnetics solution domain for the MSU Reactor B configuration. Red is SF, all blue shades
represent TF. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

69

Figure 4.6

Locations of plasma module samples on numerical mesh. . . . . . . .

78

Figure 4.7

Numerical mesh within the plasma region for the plasma fluid module. 79

Figure 4.8

One-dimensional mass transport boundary condition schematic for
right-edge domain boundary. . . . . . . . . . . . . . . . . . . . . . .

83

Tangential velocity component sample locations surrounding the domain boundary. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

88

Figure 4.5

Figure 4.9

Figure 4.10

Plasma fluid module numerical solution flowchart . . . . . . . . . . . 108

Figure 4.11

Substrate temperature module solution domain (a) sampling mesh
and (b) material mapping for the MSU MPACVD Reactor B. . . . . 110

Figure 4.12

Complete simulation flowchart. . . . . . . . . . . . . . . . . . . . . . 116

Figure 5.1

Simple cylinder schematic for verifying Helmholtz matrix equation
solution in electromagnetics module. . . . . . . . . . . . . . . . . . . 119

Figure 5.2

Cylindrical waveguide with TM(z) polarized incident field for validating electromagnetics module solution. . . . . . . . . . . . . . . . 128

Figure 5.3

Cylindrical waveguide for quantifying reflections from PMLs and validating the electromagnetics module numerical solution. . . . . . . . 129

xiii

Figure 5.4

Real components of the (a) TF/SF source vector, (b) azimuthal magnetic (H/m), and (c) radial and (d) vertical electric field (V /m) intensities for the canonical waveguide solution. . . . . . . . . . . . . . 130

Figure 5.5

Cylindrical waveguide with lossless dielectric region for validating the
electromagnetics module numerical solution. . . . . . . . . . . . . . 132

Figure 5.6

Real components of the (a) TF/SF source vector, (b) azimuthal magnetic (H/m), and (c) radial and (d) vertical electric field (V /m) intensities for the lossless material ( r = 2) validation. . . . . . . . . . 133

Figure 5.7

Real components of the (a) TF/SF source vector, (b) azimuthal magnetic (H/m), and (c) radial and (d) vertical electric field (V /m) intensities for the lossy material ( r = 1 − j0.1) validation. . . . . . . 135

Figure 5.8

(a) Discretized MSU MPACVD schematic with magnitudes of the
(b) azimuthal magnetic (H/m), and (c) radial and (d) vertical electric
field (V /m) intensities for the validation fo the unfilled reactor. In (a),
red denotes the TF/SF interface, green the silica bell jar and subsrate
walls, and blue the plasma and reactor regions. White denotes metal
(PEC). The color scales for the electric fields are the same. . . . . . 136

Figure 5.9

Comparison of present to approximated atomic hydrogen mass diffusion coefficients as used by (a) Koldanov, et al. and (b) Mankelevich,
et al. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138

Figure 5.10

The canonical driven cavity system used to validate the momentum
transport equation. . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

Figure 5.11

Numerical vector velocity solution in units of m/s to the momentum
transport equation for the driven cavity canonical system in Cartesian
coordinates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

Figure 5.12

Validation of driven cavity numerical solution for two Reynolds numbers against reported data. . . . . . . . . . . . . . . . . . . . . . . . 141

Figure 5.13

The laminar flow through a pipe in the (a) Cartesian and (b) cylindrical coordinate frames. . . . . . . . . . . . . . . . . . . . . . . . . 142

Figure 5.14

Numerical vector velocity solutions in units of m/s to the momentum
transport equation for the laminar pipe flow canonical systems in (a)
Cartesian and (b) cylindrical coordinates. . . . . . . . . . . . . . . . 143

Figure 5.15

(a) Mass density (kg/m3 ) and (b) viscosity (P a s) distributions for
the validation of the variable density momentum transport solution.

xiv

144

Figure 5.16

Vector velocity solutions in the presence of (a) constant and (b) variable density and viscosity distributions to the momentum transport
equation. Units are given in m/s. . . . . . . . . . . . . . . . . . . . 145

Figure 5.17

Right toroid used for validating the substrate thermal module and
related boundary conditions. . . . . . . . . . . . . . . . . . . . . . . 146

Figure 5.18

Numerical solution for cylindrical toroid of thickness 1 m used to
validate the substrate thermal model. . . . . . . . . . . . . . . . . . 147

Figure 5.19

Schematic of MSU MPACVD reactor used for Er |wall measurements. 149

Figure 5.20

Comparison of measured and simulated |Er | component along the
outer wall of the MSU MPACVD reactor. . . . . . . . . . . . . . . . 149

Figure 5.21

Schematic of the modified MPACVD reactor used during the measurement of Tg and ne at low pressures. . . . . . . . . . . . . . . . . 150

Figure 5.22

Comparison between experimental and averaged numerical simulation
data of (a) gas temperature (K) and (b) electron density (cm−3 ) for
the modified MSU MPACVD reactor at 400 W. . . . . . . . . . . . 152

Figure 5.23

Validation of substrate temperature module simulation versus experimental data at low and high pressures in the MPACVD reactor. . . 153

Figure 6.1

Magnitude of electrical conductivity numerical solution for the MSU
MPACVD ”Reactor B” configuration operating at 2.2 kW and 180
Torr. Units are given in S/m. . . . . . . . . . . . . . . . . . . . . . 156

Figure 6.2

Plots of (a) radial and (b) vertical electric (V /m), (c), azimuthal
magnetic (H/m), and (d) total electric field (V /m) components from
the electromagnetics module. Shown for MSU MPACVD Reactor B
at a pressure of 180 Torr and absorbed power of 2.2 kW. . . . . . . 157

Figure 6.3

Plots of absorbed power density (a) across the full MPACVD reactor
and (b) a detail of the plasma region. Shown for MSU MPACVD
Reactor B at a pressure of 180 Torr and absorbed power of 2.2 kW.
The color scales are the same for both plots. Units are given in W/m3 .159

Figure 6.4

Molecular hydrogen mole fraction solution for the MSU MPACVD
Reactor B operating at 180 Torr and 2.2 kW. . . . . . . . . . . . . . 160

Figure 6.5

Atomic hydrogen mole fraction solution for the MSU MPACVD Reactor B operating at 180 Torr and 2.2 kW. . . . . . . . . . . . . . . 161

xv

Figure 6.6

First excited atomic hydrogen mole fraction solution for the MSU
MPACVD Reactor B operating at 180 Torr and 2.2 kW. . . . . . . . 162

Figure 6.7

Second excited atomic hydrogen mole fraction solution for the MSU
MPACVD Reactor B operating at 180 Torr and 2.2 kW. . . . . . . . 162

Figure 6.8

H + ion mole fraction solution for the MSU MPACVD Reactor B
operating at 180 Torr and 2.2 kW. . . . . . . . . . . . . . . . . . . . 163

Figure 6.9

+
H2 ion mole fraction solution for the MSU MPACVD Reactor B
operating at 180 Torr and 2.2 kW. . . . . . . . . . . . . . . . . . . . 164

Figure 6.10

+
H3 ion mole fraction solution for the MSU MPACVD Reactor B
operating at 180 Torr and 2.2 kW. . . . . . . . . . . . . . . . . . . . 164

Figure 6.11

H − ion mole fraction solution for the MSU MPACVD Reactor B
operating at 180 Torr and 2.2 kW. . . . . . . . . . . . . . . . . . . . 165

Figure 6.12

Electron mole fraction solution for the MSU MPACVD Reactor B
operating at 180 Torr and 2.2 kW. . . . . . . . . . . . . . . . . . . . 166

Figure 6.13

Vector average gas flow solution for the MSU MPACVD Reactor B
operating at 180 Torr and 2.2 kW. Color denotes speed while the
arrow denotes flow direction. Units are in m/s. . . . . . . . . . . . . 166

Figure 6.14

Electron temperature solution for the MSU MPACVD Reactor B operating at 180 Torr and 2.2 kW. Units are given in K. . . . . . . . . 168

Figure 6.15

Neutral species and ion temperature solution for the MSU MPACVD
Reactor B operating at 180 Torr and 2.2 kW. Units are given in K. . 169

Figure 6.16

Absolute pressure difference solution from 180 Torr (∼2.4 × 104 P a)
for the MSU MPACVD Reactor B operating at 180 Torr and 2.2 kW.
Units are given in P a. . . . . . . . . . . . . . . . . . . . . . . . . . . 170

Figure 6.17

Volumetric total number density solution for the MSU MPACVD
Reactor B operating at 180 Torr and 2.2 kW. Units are given in m−3 .170

Figure 6.18

Volumetric atomic hydrogen number density solution for the MSU
MPACVD Reactor B operating at 180 Torr and 2.2 kW. Units are
given in m−3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

Figure 6.19

Volumetric electron number density solution for the MSU MPACVD
Reactor B operating at 180 Torr and 2.2 kW. Units are given in m−3 .172

Figure 6.20

Substrate temperature solution for the MSU MPACVD Reactor B
operating at 180 Torr and 2.2 kW. Units are given in K. . . . . . . 173
xvi

Figure 6.21

Magnitude of electrical conductivity numerical solution for the MSU
MPACVD ”Reactor B” configuration operating at 2.5 kW and 300
Torr. Units are given in S/m. . . . . . . . . . . . . . . . . . . . . . 174

Figure 6.22

Plots of (a) radial and (b) vertical electric (V /m), (c), azimuthal
magnetic (H/m), and (d) total electric (V /m) field components from
the electromagnetics module. Shown for MSU MPACVD Reactor B
at a pressure of 300 Torr and absorbed power of 2.5 kW. . . . . . . 175

Figure 6.23

(a) Volumetric absorbed power density numerical solution (b) with
detail for the MSU MPACVD ”Reactor B” configuration operating
at 2.5 kW and 300 Torr. The same color scale applies to both. Units
are given in W/m3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

Figure 6.24

Molecular hydrogen mole fraction numerical solution for the MSU
MPACVD ”Reactor B” configuration operating at 2.5 kW and 300
Torr. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

Figure 6.25

Atomic hydrogen mole fraction numerical solution for the MSU MPACVD
”Reactor B” configuration operating at 2.5 kW and 300 Torr. . . . . 179

Figure 6.26

First excited atomic hydrogen mole fraction numerical solution for
the MSU MPACVD ”Reactor B” configuration operating at 2.5 kW
and 300 Torr. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

Figure 6.27

Second excited atomic hydrogen mole fraction numerical solution for
the MSU MPACVD ”Reactor B” configuration operating at 2.5 kW
and 300 Torr. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180

Figure 6.28

Proton ion mole fraction numerical solution for the MSU MPACVD
”Reactor B” configuration operating at 2.5 kW and 300 Torr. . . . . 181

Figure 6.29

+
H2 ion mole fraction numerical solution for the MSU MPACVD
”Reactor B” configuration operating at 2.5 kW and 300 Torr. . . . . 182

Figure 6.30

+
H3 ion mole fraction numerical solution for the MSU MPACVD
”Reactor B” configuration operating at 2.5 kW and 300 Torr. . . . . 182

Figure 6.31

H − ion mole fraction numerical solution for the MSU MPACVD
”Reactor B” configuration operating at 2.5 kW and 300 Torr. . . . . 183

Figure 6.32

Electron mole fraction numerical solution for the MSU MPACVD
”Reactor B” configuration operating at 2.5 kW and 300 Torr. . . . . 184

Figure 6.33

Vector average gas flow numerical solution for the MSU MPACVD
”Reactor B” configuration operating at 2.5 kW and 300 Torr. Units
given in m/s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185
xvii

Figure 6.34

Electron temperature numerical solution for the MSU MPACVD ”Reactor B” configuration operating at 2.5 kW and 300 Torr. Units given
in K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

Figure 6.35

Gas temperature numerical solution for the MSU MPACVD ”Reactor
B” configuration operating at 2.5 kW and 300 Torr. Units given in K.187

Figure 6.36

Difference in local pressure numerical solution for the MSU MPACVD
”Reactor B” configuration operating at 2.5 kW and 300 Torr. Units
given in P a. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188

Figure 6.37

Total gas volumetric number density numerical solution for the MSU
MPACVD ”Reactor B” configuration operating at 2.5 kW and 300
Torr. Units given in m−3 . . . . . . . . . . . . . . . . . . . . . . . . 189

Figure 6.38

Atomic hydrogen volumetric number density numerical solution for
the MSU MPACVD ”Reactor B” configuration operating at 2.5 kW
and 300 Torr. Units given in m−3 . . . . . . . . . . . . . . . . . . . 190

Figure 6.39

Electron volumetric number density numerical solution for the MSU
MPACVD ”Reactor B” configuration operating at 2.5 kW and 300
Torr. Units given in m−3 . . . . . . . . . . . . . . . . . . . . . . . . 190

Figure 6.40

Substrate temperature solution for the MSU MPACVD ”Reactor B”
configuration operating at 2.5 kW and 300 Torr. Units given in K. . 191

Figure 6.41

Radial electric field intensity solutions for (a) 180 Torr, 2.2 kW, (b)
200 Torr, 2.1 kW, (c) 220 Torr, 2.2 kW, and (d) 240 Torr, 2.3 kW.
Shown for MSU MPACVD Reactor B. Units are given in V /m. . . . 193

Figure 6.42

Vertical electric field intensity solutions for (a) 180 Torr, 2.2 kW, (b)
200 Torr, 2.1 kW, (c) 220 Torr, 2.2 kW, and (d) 240 Torr, 2.3 kW.
Shown for MSU MPACVD Reactor B. Units are given in V /m. . . . 194

Figure 6.43

Absorbed power densities for (a) 180 Torr, 2.2 kW, (b) 200 Torr, 2.1
kW, (c) 220 Torr, 2.2 kW, and (d) 240 Torr, 2.3 kW. Shown for MSU
MPACVD Reactor B. Units are given in W/m3 . . . . . . . . . . . . 195

Figure 6.44

Atomic hydrogen mole fraction solutions at (a) 180 Torr, 2 kW and
(b) 200 Torr, 2.1 kW from the plasma module. Shown for MSU
MPACVD Reactor B. . . . . . . . . . . . . . . . . . . . . . . . . . . 197

Figure 6.45

Atomic hydrogen mole fraction solutions at (a) 220 Torr, 2.2 kW
and (b) 240 Torr, 2.3 kW from the plasma module. Shown for MSU
MPACVD Reactor B. . . . . . . . . . . . . . . . . . . . . . . . . . . 198

xviii

Figure 6.46

Electron densities at (a) 180 Torr, 2 kW and (b) 200 Torr, 2.1 kW
from the plasma module. Shown for MSU MPACVD Reactor B. Units
are in m−3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

Figure 6.47

Electron densities at (a) 220 Torr, 2.2 kW and (b) 240 Torr, 2.3
kW from the plasma module. Shown for MSU MPACVD Reactor B.
Units are in m−3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200

Figure 6.48

Vector average gas flows at (a) 180 Torr, 2 kW and (b) 200 Torr, 2.1
kW from the plasma module. Shown for MSU MPACVD Reactor B.
Units are in m/s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

Figure 6.49

Vector average gas flows at (a) 220 Torr, 2.2 kW and (b) 240 Torr,
2.3 kW from the plasma module. Shown for MSU MPACVD Reactor
B. Units are in m/s. . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

Figure 6.50

Electron temperature solutions at (a) 180 Torr, 2 kW and (b) 200
Torr, 2.1 kW from the plasma module. Shown for MSU MPACVD
Reactor B. Units are in K. . . . . . . . . . . . . . . . . . . . . . . . 204

Figure 6.51

Electron temperature solutions at (a) 220 Torr, 2.2 kW and (b) 240
Torr, 2.3 kW from the plasma module. Shown for MSU MPACVD
Reactor B. Units are in K. . . . . . . . . . . . . . . . . . . . . . . . 205

Figure 6.52

Gas temperature solutions at (a) 180 Torr, 2 kW and (b) 200 Torr,
2.1 kW from the plasma module. Shown for MSU MPACVD Reactor
B. Units are in K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

Figure 6.53

Gas temperature solutions at (a) 220 Torr, 2.2 kW and (b) 240 Torr,
2.3 kW from the plasma module. Shown for MSU MPACVD Reactor
B. Units are in K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

Figure 6.54

Plots of radial electric field intensities for substrate positions of (a) 0
mm, (b) -2 mm, (c) -4 mm, and (d) -6 mm from the electromagnetics
module. Shown for MSU MPACVD Reactor B at a pressure of 180
Torr and absorbed power of 2.2 kW. Units are given in V /m. . . . . 211

Figure 6.55

Plots of vertical electric field intensities for substrate positions of (a) 0
mm, (b) -2 mm, (c) -4 mm, and (d) -6 mm from the electromagnetics
module. Shown for MSU MPACVD Reactor B at a pressure of 180
Torr and absorbed power of 2.2 kW. Units are given in V /m. . . . . 212

Figure 6.56

Plots of absorbed power densities for substrate positions of (a) 0 mm,
(b) -2 mm, (c) -4 mm, and (d) -6 mm from the electromagnetics
module. Shown for MSU MPACVD Reactor B at a pressure of 180
Torr and absorbed power of 2.2 kW. Units are given in W/m−3 . . . 213

xix

Figure 6.57

Atomic hydrogen mole fraction numerical solutions for substrate positions of (a) 0 mm and (b) -2 mm for the MSU MPACVD ”Reactor
B” configuration operating at 180 Torr and 2.2 kW. . . . . . . . . . 215

Figure 6.58

Atomic hydrogen mole fraction numerical solutions for substrate positions of (a) -4 mm and (b) -6 mm for the MSU MPACVD ”Reactor
B” configuration operating at 180 Torr and 2.2 kW. . . . . . . . . . 216

Figure 6.59

Total power deposited on top substrate surface due to hydrogen bombardment for various vertical substrate positions in the MSU MPACVD
Reactor ”B” configuration operating at 180 Torr and 2.2 kW. . . . . 217

Figure 6.60

Electron mole fraction numerical solutions for substrate positions of
(a) 0 mm and (b) -2 mm for the MSU MPACVD ”Reactor B” configuration operating at 180 Torr and 2.2 kW. . . . . . . . . . . . . . 218

Figure 6.61

Electron mole fraction numerical solutions for substrate positions of
(a) -4 mm and (b) -6 mm for the MSU MPACVD ”Reactor B” configuration operating at 180 Torr and 2.2 kW. . . . . . . . . . . . . . 219

Figure 6.62

Average gas flow velocity numerical solutions for substrate positions
of (a) 0 mm and (b) -2 mm for the MSU MPACVD ”Reactor B”
configuration operating at 180 Torr and 2.2 kW. Units are given in
m/s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220

Figure 6.63

Average gas flow velocity numerical solutions for substrate positions
of (a) -4 mm and (b) -6 mm for the MSU MPACVD ”Reactor B”
configuration operating at 180 Torr and 2.2 kW. Units are given in
m/s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

Figure 6.64

Electron temperature numerical solutions for substrate positions of
(a) 0 mm and (b) -2 mm for the MSU MPACVD ”Reactor B” configuration operating at 180 Torr and 2.2 kW. Units are given in K. . 222

Figure 6.65

Electron temperature numerical solutions for substrate positions of
(a) -4 mm and (b) -6 mm for the MSU MPACVD ”Reactor B” configuration operating at 180 Torr and 2.2 kW. Units are given in K. . 223

Figure 6.66

Gas temperature numerical solutions for substrate positions of (a) 0
mm and (b) -2 mm for the MSU MPACVD ”Reactor B” configuration
operating at 180 Torr and 2.2 kW. Units are given in K. . . . . . . 225

Figure 6.67

Gas temperature numerical solutions for substrate positions of (a) -4
mm and (b) -6 mm for the MSU MPACVD ”Reactor B” configuration
operating at 180 Torr and 2.2 kW. Units are given in K. . . . . . . 226

xx

Figure 6.68

Power reflection coefficient for several substrate positions for the MSU
MPACVD ”Reactor B” configuration operating at 180 Torr and 2.2
kW. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227

Figure 6.69

Plots of radial electric field intensities for reactor heights of of (a) 20.0
cm, (b) 20.2 cm, (c) 20.4 cm, and (d) 20.6 cm from the electromagnetics module. Shown for MSU MPACVD Reactor B at a pressure
of 180 Torr and absorbed power of 2.2 kW. Units are given in V /m.

228

Figure 6.70

Plots of vertical electric field intensities for reactor heights of of (a)
20.0 cm, (b) 20.2 cm, (c) 20.4 cm, and (d) 20.6 cm from the electromagnetics module. Shown for MSU MPACVD Reactor B at a
pressure of 180 Torr and absorbed power of 2.2 kW. Units are given
in V /m. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229

Figure 6.71

Plots of absorbed power density distributions for reactor heights of
of (a) 20.0 cm, (b) 20.2 cm, (c) 20.4 cm, and (d) 20.6 cm from the
electromagnetics module. Shown for MSU MPACVD Reactor B at a
pressure of 180 Torr and absorbed power of 2.2 kW. Units are given
in W/m3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230

Figure 6.72

Plots of atomic hydrogen mole fraction solutions for reactor heights
of of (a) 20.0 cm and (b) 20.2 cm within the MSU MPACVD Reactor
B at a pressure of 180 Torr and absorbed power of 2.2 kW. . . . . . 232

Figure 6.73

Plots of atomic hydrogen mole fraction solutions for reactor heights
of of (a) 20.4 cm and (b) 20.6 cm within the MSU MPACVD Reactor
B at a pressure of 180 Torr and absorbed power of 2.2 kW. . . . . . 233

Figure 6.74

Plots of absorbed power density distributions for reactor heights of
of (a) 20.0 cm and (b) 20.2 cm within the plasma region. Shown for
MSU MPACVD Reactor B at a pressure of 180 Torr and absorbed
power of 2.2 kW. Units are given in W/m3 . . . . . . . . . . . . . . . 235

Figure 6.75

Plots of absorbed power density distributions for reactor heights of
of (c) 20.4 cm and (d) 20.6 cm within the plasma region. Shown for
MSU MPACVD Reactor B at a pressure of 180 Torr and absorbed
power of 2.2 kW. Units are given in W/m3 . . . . . . . . . . . . . . . 236

Figure 6.76

Plots of electron mole fraction solutions for reactor heights of of (a)
20.0 cm and (b) 20.2 cm within the MSU MPACVD Reactor B at a
pressure of 180 Torr and absorbed power of 2.2 kW. . . . . . . . . . 237

Figure 6.77

Plots of atomic hydrogen mole fraction solutions for reactor heights
of of (a) 20.4 cm and (b) 20.6 cm within the MSU MPACVD Reactor
B at a pressure of 180 Torr and absorbed power of 2.2 kW. . . . . . 238
xxi

Figure 6.78

Plots of average gas
(a) 20.0 cm and (b)
at a pressure of 180
given in m/s. . . . .

flow velocity solutions for reactor heights of of
20.2 cm within the MSU MPACVD Reactor B
Torr and absorbed power of 2.2 kW. Units are
. . . . . . . . . . . . . . . . . . . . . . . . . . . 239

Figure 6.79

Plots of average gas
(a) 20.4 cm and (b)
at a pressure of 180
given in m/s. . . . .

flow velocity solutions for reactor heights of of
20.6 cm within the MSU MPACVD Reactor B
Torr and absorbed power of 2.2 kW. Units are
. . . . . . . . . . . . . . . . . . . . . . . . . . . 240

Figure 6.80

Plots of gas temperature solutions for reactor heights of of (a) 20.0 cm
and (b) 20.2 cm within the MSU MPACVD Reactor B at a pressure
of 180 Torr and absorbed power of 2.2 kW. Units are given in K. . . 241

Figure 6.81

Plots of gas temperature solutions for reactor heights of of (a) 20.4 cm
and (b) 20.6 cm within the MSU MPACVD Reactor B at a pressure
of 180 Torr and absorbed power of 2.2 kW. Units are given in K. . . 242

Figure 6.82

Plot of the reflected power ratio for various reactor heights within the
MSU MPACVD Reactor B at a pressure of 180 Torr and absorbed
power of 2.2 kW. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243

Figure 6.83

Plots of radial electric field intensities for MSU MPACVD reactor
configurations (a) A, (b) B, and (c) C from the electromagnetics
module. Shown for a pressure of 150 Torr and absorbed power of 2
kW. Units are given in V /m. . . . . . . . . . . . . . . . . . . . . . . 245

Figure 6.84

Plots of vertical electric field intensities for MSU MPACVD reactor
configurations (a) A, (b) B, and (c) C from the electromagnetics
module. Shown for a pressure of 150 Torr and absorbed power of 2
kW. Units are given in V /m. . . . . . . . . . . . . . . . . . . . . . . 246

Figure 6.85

Plots of absorbed power density for MSU MPACVD reactor configurations (a) A, (b) B, and (c) C from the electromagnetics module.
Shown for a pressure of 150 Torr and absorbed power of 2 kW. Units
are given in W/m3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248

Figure 6.86

Plots of atomic hydrogen mole fraction for MSU MPACVD reactor
configurations (a) B and (b) C from the plasma fluid module. Shown
for a pressure of 150 Torr and absorbed power of 2 kW. . . . . . . . 249

Figure 6.87

Plots of electron mole fraction for MSU MPACVD reactor configurations (a) B and (b) C from the plasma fluid module. Shown for a
pressure of 150 Torr and absorbed power of 2 kW. . . . . . . . . . . 251

xxii

Figure 6.88

Plots of vector average gas flow for the MSU MPACVD reactor configurations (a) A and (b) B from the plasma fluid module. Shown for
a pressure of 150 Torr and absorbed power of 2 kW. Units are given
in m/s. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252

Figure 6.89

Plot of vector average gas flow for the MSU MPACVD reactor configurations C from the plasma fluid module. Shown for a pressure of
150 Torr and absorbed power of 2 kW. Units are given in m/s. . . . 253

Figure 6.90

Plots of electron temperature for MSU MPACVD reactor configurations (a) B and (b) C from the plasma fluid module. Shown for a
pressure of 150 Torr and absorbed power of 2 kW. Units are given in
K. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254

Figure 6.91

Plots of gas temperature for MSU MPACVD reactor configurations
(a) B and (b) C from the plasma fluid module. Shown for a pressure
of 150 Torr and absorbed power of 2 kW. Units are given in K. . . . 255

Figure 6.92

Plots of radial electric field intensities for MSU MPACVD Reactor B
with (a) minimized and (b) non-minimized reflected powers from the
electromagnetics module. Units are given in V /m. . . . . . . . . . . 257

Figure 6.93

Plots of vertical electric field intensities for MSU MPACVD Reactor
B with (a) minimized and (b) non-minimized reflected powers from
the electromagnetics module. Units are given in V /m. . . . . . . . . 257

Figure 6.94

Plots of absorbed power densities for MSU MPACVD Reactor B with
(a) minimized and (b) non-minimized reflected powers from the electromagnetics module. Units are given in W/m3 . . . . . . . . . . . . 259

Figure 6.95

Plots of atomic hydrogen mole fraction for MSU MPACVD Reactor
B with (a) minimized and (b) non-minimized reflected powers from
the plasma fluid module. . . . . . . . . . . . . . . . . . . . . . . . . 260

Figure 6.96

Plots of electron number density for MSU MPACVD Reactor B with
(a) minimized and (b) non-minimized reflected powers from the plasma
fluid module. Units are given in m−3 . . . . . . . . . . . . . . . . . . 261

Figure 6.97

Plots of energy flux from (a) thermal conduction and (b) atomic hydrogen bombardment at the substrate surface. Units are given in
W/m2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263

Figure 6.98

Numerical transport equation residuals plotted against total iteration
number for the MSU MPACVD Reactor B operating at 180 Torr and
2 kW. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265

xxiii

Figure 6.99

Numerical transport equation residuals plotted against total iteration
number for the MSU MPACVD Reactor B operating at 300 Torr and
3 kW. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265

Figure 7.1

Vector energy (a) convection and (b) conduction fluxes within the
plasma region for the MSU MPACVD Reactor B operating at 180
Torr and 2.2 kW. Units are given in kg/s3 . . . . . . . . . . . . . . . 271

Figure 7.2

Vector mass (a) convection and (b) diffusion fluxes within the plasma
region for the MSU MPACVD Reactor B operating at 180 Torr and
2.2 kW. Units are given in kg/m2 /s. . . . . . . . . . . . . . . . . . 272

Figure 7.3

(a) Atomic hydrogen mole fraction and (b) gas temperature (K)
within the plasma region for the MSU MPACVD Reactor B operating at 180 Torr and 2.2 kW. . . . . . . . . . . . . . . . . . . . . . 276

Figure 7.4

(a) Atomic hydrogen mole fraction and (b) gas temperature (K) calculated using the Mankelevich approximated atomic hydrogen diffusion coefficient within the plasma region for the MSU MPACVD
Reactor B operating at 180 Torr and 2.2 kW. . . . . . . . . . . . . . 277

Figure 7.5

(a) Atomic hydrogen mole fraction and (b) gas temperature (K) calculated using the Koldanov approximated atomic hydrogen diffusion
coefficient within the plasma region for the MSU MPACVD Reactor
B operating at 180 Torr and 2.2 kW. . . . . . . . . . . . . . . . . . 278

Figure 7.6

Total residual in plasma matrix equations for differing allowed maximum iteration limits during Gauss-Seidel update. . . . . . . . . . . 284

Figure 7.7

Initial (a) electron and (b) gas temperature solutions during cold start
conditions for plasma solution module. Units are given in K. . . . . 286

Figure 7.8

Total residual in plasma matrix equations for differing initial solutions
provided to the plasma fluid module. . . . . . . . . . . . . . . . . . 287

Figure 7.9

Plots of electric field intensities for (a) radial component at coarse
grid spacing, (b) radial component at fine grid spacing, (c) vertical
component at coarse grid spacing, and (d) vertical component at find
grid spacing throughout the MSU MPACVD Reactor B at 180 Torr
and 2.2 kW. All units are given in V /m. . . . . . . . . . . . . . . . . 289

Figure 7.10

Plots of absorbed power density in the plasma region of MSU MPACVD
Reactor B at 180 Torr and 2.2 kW for mesh spacings of approximately
(a) 1.2 and (b) 0.8 mm. Units are given in W/m3 . . . . . . . . . . . 290

xxiv

Figure 7.11

Plots of absorbed power density in the plasma region of MSU MPACVD
Reactor B at 180 Torr and 2.2 kW for mesh spacings of approximately
(a) 0.6 and (b) 0.5 mm. Units are given in W/m3 . . . . . . . . . . . 291

Figure 7.12

Plots of atomic hydrogen mole fraction in the plasma region of MSU
MPACVD Reactor B at 180 Torr and 2.2 kW for mesh spacings of
approximately (a) 1.2 and (b) 0.8 mm. . . . . . . . . . . . . . . . . 292

Figure 7.13

Plots of atomic hydrogen mole fraction in the plasma region of MSU
MPACVD Reactor B at 180 Torr and 2.2 kW for mesh spacings of
approximately (a) 0.6 and (b) 0.5 mm. . . . . . . . . . . . . . . . . 293

Figure 7.14

Plots of electron mole fraction in the plasma region of MSU MPACVD
Reactor B at 180 Torr and 2.2 kW for mesh spacings of approximately
(a) 1.2 and (b) 0.8 mm. . . . . . . . . . . . . . . . . . . . . . . . . . 294

Figure 7.15

Plots of electron mole fraction in the plasma region of MSU MPACVD
Reactor B at 180 Torr and 2.2 kW for mesh spacings of approximately
(a) 0.6 and (b) 0.5 mm. . . . . . . . . . . . . . . . . . . . . . . . . . 295

Figure 7.16

Plots of electron temperature in the plasma region of MSU MPACVD
Reactor B at 180 Torr and 2.2 kW for mesh spacings of approximately
(a) 1.2 and (b) 0.8 mm. Units are given in K. . . . . . . . . . . . . 297

Figure 7.17

Plots of electron temperature in the plasma region of MSU MPACVD
Reactor B at 180 Torr and 2.2 kW for mesh spacings of approximately
(a) 0.6 and (b) 0.5 mm. Units are given in K. . . . . . . . . . . . . 298

Figure 7.18

Total plasma fluid module transport equation matrix residuals for
four mesh resolutions corresponding to the MSU MPACVD Reactor
B configuration at 180 Torr and 2.2 kW. . . . . . . . . . . . . . . . 299

Figure 7.19

Total simulation run times (in seconds) for four mesh resolutions
corresponding to the MSU MPACVD Reactor B configuration at 180
Torr and 2.2 kW. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300

Figure A.1

Experimental data and corresponding numerical approximation for
the collision integral of type 1 in the H − H+ interaction. . . . . . . 313

Figure A.2

Experimental data and corresponding numerical approximation for
the collision integral of type 2 in the H − H+ interaction. . . . . . . 313

Figure A.3

Experimental data and corresponding numerical approximation for
the collision integral of type 1 in the H − e interaction. . . . . . . . 314

xxv

Figure A.4

Experimental data and corresponding numerical approximation for
the collision integral of type 2 in the H − e interaction. . . . . . . . 314

Figure A.5

Experimental data and corresponding numerical approximation for
the collision integral of type 1 in the H2 − e interaction. . . . . . . . 315

Figure A.6

Experimental data and corresponding numerical approximation for
the collision integral of type 1 in the H2 self interaction. . . . . . . . 315

Figure A.7

Experimental data and corresponding numerical approximation for
the collision integral of type 2 in the H2 self interaction. . . . . . . . 316

Figure A.8

Experimental data and corresponding numerical approximation for
the collision integral of type 1 in the H self interaction. . . . . . . . 316

Figure A.9

Experimental data and corresponding numerical approximation for
the collision integral of type 2 in the H self interaction. . . . . . . . 317

Figure A.10 Experimental data and corresponding numerical approximation for
the collision integral of type 1 in the H − H2 interaction. . . . . . . 317
Figure A.11 Experimental data and corresponding numerical approximation for
the collision integral of type 2 in the H − H2 interaction. . . . . . . 318

xxvi

KEY TO SYMBOLS

SYMBOL

DESCRIPTION

UNITS

a

Acceleration vector

m/s2

α

Plasma chemical species designation

B

Magnetic flux density

c

Speed of light in vacuum

cs

Molar concentration of species s

mol/m3

ctot

Total molar concentration

mol/m3

Cp

Specific heat capacity

J/kg K

D

Electric flux density

C/m2

Da

Ambipolar diffusion coefficient

m2 /s

Ds

Diffusion coefficient for species s

m2 /s

Dsr
(1)
∆s,r
(2)
∆s,r

Binary diffusion coefficient between species s and r

m2 /s

W b/m2
m/s

Modified collision integral of the first kind
Modified ollision integral of the second kind

ms
m−s

Gradient operator

1/m

·

Divergence operator

1/m

×

Curl operator

1/m

E

Electric field intensity

V /m

Ea

Chemical reaction activation energy

Er

Radial electric field component

V /m

Et

Tangential electric field component

V /m

Ez

Vertical electric field component

V /m

EEDF

Electron Energy Distribution Function

abbr.

ˆ

Complex electromagnetic permittivity

F/m

Free space electromagnetic permittivity

F/m

0

xxvii

J

r

Relative electromagnetic permittivity

v,s
(1)
v,s
(2)
v,s

Vibrational mode energy of species s

J

Vibrational mode electron equilibrium energy of species s

J

Vibrational mode neutral equilibrium energy of species s

J

f

Electromagnetic wave frequency

f

Plasma distribution function

g

Gravitational acceleration

hs

Enthalpy of chemical species s

H

Magnetic field intensity

Hz

m/s2
J
H/m

Imaginary component operator
j

Imaginary unit

J0

Zeroeth order bessel function of first kind

Je

Electric current density

A/m2

Jm

Magnetic current density

V /m2

k

Electromagnetic wavenumber magnitude

1/m

k

Electromagnetic wavenumber

1/m

kb

Boltzmann’s Constant

J/K

kr

Radial electromagnetic wavenumber mangitude

1/m

kr,s

Chemical reaction rate constant for r and s

kz

Vertical electromagnetic wavenumber mangitude

λ

Electromagnetic wavelength

m

λ0

Free space electromagnetic wavelength

m

λD

deBye length of plasma

m

λe

Electron thermal conductivity coefficient

W/m − K

λg

Neutral species thermal conductivity coefficient

W/m − K

λr

Rotational modes thermal conductivity coefficient

W/m − K

λt

Translational modes thermal conductivity coefficient

W/m − K

xxviii

1/m

λtr

Summation of λt + λr

W/m − K

λv

Vibrational modes thermal conductivity coefficient

W/m − K

log

Natural logarithm function

log10

Base-10 logarithm function

m

Mass

kg

me

Electron rest mass

kg

ms

Mass of species s

kg

M

Plasma molar mass

kg/mol

¯
M

Average molar mass of gas mixture

kg/mol

Ms

Molar mass of species s

kg/mol

n

Volumetric number density

1/m3

ne

Electron volumetric number density

1/m3

nH

Atomic hydrogen volumetric number density

1/m3

Nav

Avagadro constant

1/mol

Ns

Number of individual chemical species in gas mixture

η1

Stoichiometric coefficient prior to chemical reaction

η2

Stoichiometric coefficient following chemical reaction

¯ (1,1)
π Ωs,r
¯ (2,2)
π Ωs,r

Collision integral between s and r of first kind

m2

Collision integral between s and r of second kind

m2

p

Pressure

Pa

pe

Electron momentum

kg − m/s

pα

Pressure of species α

Pa

pe

Electron pressure

Pa

p0

Standard atmospheric pressure

Pa

PACVD

Plasma-Assisted Chemical Vapor Deposition

abbr.

PEC

Perfect Electric Conductor

abbr.

¯
P

Time-averaged absorbed microwave power density
xxix

W/m3

W/m3

Pabs

Steady absorbed microwave power density

Ptot

Total absorbed microwave power density

π

pi

ˆ
φ

Azimuthal direction unit vector

q

Electron charge

C

qα

Vector heat flux

W/m2

qs

Charge on species s

Q

Quality factor

Qet

Qij from electrons to Hydrogen translational modes

W/m3

Qev

Qij from electrons to Hydrogen vibrational modes

W/m3

Qf

Volumetric flow rate

QH

Heat flux

W/m2

Qij

Volumetric power transfer density from i to j

W/m3

Qrad

Volumetric power loss due to radiation

W/m3

Qtv

Qij from Hydrogen translational to vibrational modes

W/m3

r
ˆ

Radial direction unit vector

r

Generic spatial vector

R

Gas constant

Re

Reynolds number

ρ

Volumetric mass density

kg/m3

ρs

Volumetric mass density of species s

kg/m3

ρe

Electron volumetric mass density

kg/m3

W

C

sccm

m
J/K − mol

Real component operator
W/m2

S

Electromagnetic Poynting vector

sccm

Standard cubic centimeter-per-minute (see Qf )

abbr.

SOR

Successive Overrelaxation

abbr.

σe
ˆ

Complex electric conductivity

S/m

xxx

m2

σχ

Collision cross section between electrons and neutrals

Te

Electron temperature

K

Tg

Neutral and ion species temperature

K

Tv

Hydrogen vibrational modes temperature

K

t

Instantaneous time

s

τ

Time constant

s

τ
ˆ

Kinematic viscosity tensor

θv,s

Characteristic vibration temperature of species s

ˆ
U

Internal thermal energy per unit mass

µ

Kinematic viscosity of gas mixture

µ0

Permeability of free space

µr

Relative permeability

v

Average vector gas velocity

m/s

vr

Radial component of average vector gas flow

m/s

vth

Electron thermal velocity

m/s

vz

Vertical component of average vector gas flow

m/s

νef f

Effective collision rate between electrons and H

Hz

νs

Rate of chemical production of species s

Hz

Ws

Net molar production rate of chemical species s

mol/m3 − s

Wsurf,s

Net surface molar production rate of chemical species s

mol/m2 − s

ω

Angular electromagnetic wave frequency

xs

Molar fraction of species s

z
ˆ

Vertical direction unit vector

Z+

Set of all positive integers

Pa
K
J/kg
kg/m − s
H/m

xxxi

rad/s

CHAPTER 1
INTRODUCTION
1.1
1.1.1

Diamond
History

Diamond is a unique material. It is just one of several stable forms of carbon. It is chemically
inert with pure samples appearing transparent and colorless in the optical wavelengths.
Diamond has long been of interest to mankind. It was used as an abrasive for many thousands
of years. It is revered for its brilliance and relative scarcity, and is sometimes used for
monetary trade. To this day, diamond remains a precious gem in common circulation.

1.1.2

Applications

Diamond is also known for its uncommon mechanical hardness and inherent physical properties. This led to various industrial applications over the last half century including industrial
coatings such as saw blades and cutting tools, and applications to hard surfaces and materials.
The optical and thermal properties of diamond have more recently been utilized for
various scientific and technological purposes. Diamond lenses and lens coatings have been

1

developed for use in high power lasers and systems. Their use is required under such conditions where material degradation under normal operating conditions is a concern.
Diamond is also known to exhibit semiconducting properties under the right conditions
with a relatively wide bandgap. Therefore, the fact that it is considered a semiconductor allows for applications as transistors and electrical devices. In fact, diamond has been utilized
in the production and research of diodes. Taking advantage of its unique thermal properties,
such devices are able to operate at much higher power densities without degradation compared to similar silicon or III-V semiconductor materials. Thus, they could have significant
impacts in the high power transistor industry.

1.2

Synthetic Diamond

Historically, diamond was only obtainable naturally. However, with the increased cost of
mining and other economic and ethical factors, natural diamond is becoming more expensive
and difficult to find and produce in large quantities. An alternative to natural diamond
discovery is to produce diamond via synthetic methods. These diamond samples are known
as synthetic diamond. Synthetic diamond is becoming a viable option, due in part to its
increasing quality and predictable rate of production. However, several considerable hurdles
stand in its way of replacing the need for natural diamond mining. The current bottleneck
in this process lies in producing large quantities of high quality synthetic diamond at an
acceptable rate. This requires improvements in both the quality and production rate for
the diamond synthesis process, meaning the underlying physical processes in such systems
much be better understood. By developing accurate simulations, this understanding can be
greatly advanced at lower costs. Thus, the drive for more efficient, accurate, and powerful
simulations is underway.
In order to provide suitable context for the present work, a brief history of synthetic
diamond and related modeling research is required. Michigan State University (MSU) has

2

played a major role in the development of Microwave Plasma-Assisted Chemical Vapor Deposition (MPACVD) reactors for diamond film growth [1–5]. A good review of the history
of MPACVD, and synthetic diamond growth itself, is given in [3] and is paraphrased in the
following paragraphs. A history of related numerical modeling will also be given.

1.2.1

High Pressure Production

Synthetic diamond growth is divided into two categories: high pressure (HPHT) and low
pressure deposition. Here, high pressure growth (via high pressure, high-temperature or
HPHT) represents pressures exceeding several thousand times greater than standard atmospheric pressure. The basics of both are discussed in the following paragraphs.
The objective of the HPHT process is to mimic conditions necessary for natural diamond
growth. During natural growth, carbon in the earth’s mantle is heated and pressurized
to extreme conditions. This causes the atoms in the carbon to align at the atomic level,
resulting in a highly ordered crystal structure. Only under these extreme conditions is
diamond produced naturally.
In order to recreate these conditions in HPHT growth, a quantity of graphite is used as
a seed. This graphite is then pressurized under the weight of a diamond-tipped anvil and
heated in a furnace. Pressures produced during HPHT growth often exceed several GPa, with
temperatures reaching several thousand Kelvin. These pressures and temperatures roughly
mimic those produced in the earth’s mantle during natural growth. After a length of time,
the graphite has been aligned on the atomic level, and transforms to diamond. The HPHT
method results in small, volumetric, and pure diamond samples. Large synthetic diamonds
are difficult to produce via HPHT due to the difficulties associated with applying an extreme
pressure uniformly over a brittle substance. Moreover, the diamonds produced are similar
to those mined naturally resulting in polyhedral shapes. HPHT often results in very pure
synthetic diamond. Of course, this is dependent on the purity of the initial graphite sample
and the cleanliness of the system during growth.
3

1.2.2

Low Pressure Production

Low pressure synthetic diamond production is often achieved via some application of the
Chemical Vapor Deposition (CVD) process. The CVD method has been used in the production of numerous materials, and is not limited to diamond. For example, CVD is popular in
the semiconductor industry, where it is often used to produce aluminum or arsenide-based
compounds for use in transistors. Its advantages include growth within a relatively controlled
environment, and the capability of large-area deposition.
The fundamental physical process for material deposition and production in CVD is
the chemical reactions of gas species at the growth surface. During the CVD process, one
or a mixture of gases flow into a chamber containing the growth surface and a chemical
precursor. The chamber is typically under vacuum, with operating pressures of µTorr to
atmospheric pressure. This gas flows into the chamber, and chemical reactions take place
between gases and the precursor at the substrate. The newly-produced chemical species
then migrate toward the growth surface and attach, forming a new material layer. At the
same time, more chemical reactions are taking place with more species migrating toward the
surface. This process is indefinitely repeatable, only ceasing once the gas flow is stopped or
new species no longer move to the surface.
In the case of diamond production, the feed gas is typically a mixture comprising of
predominantly hydrogen with a small percentage (<10%) methane [1, 2, 4, 5]. The methane
contributes the carbon necessary for diamond production, while the hydrogen is used as a
chemical reaction precursor for the production of various other species. These are complex
reactions that are not completely understood. However, a significant amount of research
has been conducted in this area, and generally well-known approximations to empirical data
are often made. For example, it is known the standard CVD production method does not
supply the energy necessary for some of the important precursor chemical reactions required
for diamond growth at the substrate surface. A plasma is thus utilized to transfer energy to

4

those species requiring activation during chemical reactions. The result is diamond deposition
via plasma-enhanced or plasma-assisted CVD (PACVD).
Even in the PACVD process, there are many well-known and studied deposition methods.
These include hot filament, direct current, and Microwave PACVD (MPACVD). In all of
these systems, the basic principle of the plasma capturing and transferring energy to the
necessary chemical reactions remains. The basic principle of MPACVD systems is that
energy is transfered to the much less massive electrons via their high frequency oscillations
induced by the microwave electric fields. The energy is then transferred to the other heavy
species via chemical reactions, allowing for carbon, and therefore diamond, deposition at the
growth surface. A much more detailed description of this process is provided in Chapter 2.
MPACVD will be exclusively considered as the primary diamond film growth method from
hereon.

1.3
1.3.1

MPACVD Systems
History

In order to provide context for MPACVD modeling, a brief history related to the development
of synthetic diamond and MPACVD systems is provided in the following paragraphs. After
many unsuccessful attempts in the first half of the twentieth century, Bundy, et. al. were
the first to report the successful synthetic production of diamond in 1955 [6], using a HPHT
process involving pressurizing a molten carbon sample toward the thermodynamic diamond
transition. This ushered in the era of HPHT diamond growth, which is still used today.
In low pressure diamond deposition, carbon is atomically deposited at the growth surface.
Thus, the goal is to produce free carbon atoms in a stable manner. The first report of low
pressure diamond growth can be traced to Eversole, who patented the growth mechanism in
1962 [7, 8]. Hydrocarbon gas was passed over a diamond seed at very low pressures (<1 Torr)
and moderate temperatures (600-1000 C). This resulted in diamond depositions of several
5

milligrams, and signaled the first successful synthesis of diamond at low pressures [8].
Several years later, Angus et. al. reported the first successful synthetic diamond growth
specifically via Chemical Vapor Deposition (CVD) [9]. Pure methane gas, along with mixtures of methane and hydrogen gases, were heated to approximately 1300 K. Pressures
ranging from roughly <1% to almost 60% atmosphere were used in experimental runs of
several hours. This resulted in diamond deposition in upwards of several milligrams.
The rate of diamond deposition was greatly increased when Spitsyn et. al. and Matsumoto et. al. reported their findings around 1980 of several µm/hr [10–12]. Their contributions include the selection of a methane-to-hydrogen ratio around 1:100, which is still used
today.
The following year, the Matsumoto group was the first to report high frequency plasma
enhanced deposition of diamond [13]. Growth on non-diamond surfaces including silicon were
reported for a methane fraction of 1-3%. Absorbed powers ranged from 300-700 Watts at
relatively low temperatures (<1000 C). Pressures were restricted to <60 Torr. This resulted
in a maximum growth rate of 3 µm/hr; comparable to their previous works.
MPACVD has become a prominent growth technique for diamond deposition since then.
Several distinct reactor designs have been developed [2, 3, 14–16]. Various additional gases
and mixing ratios have been reported, leading to improved deposition rates [17–19].
In the mid 2000s, work began on investigating growth at higher pressures [4, 5, 20]. It
was realized that such conditions increase the deposition rate and diamond quality [4, 5].
Thus, the MSU MPACVD reactor operating at higher pressures (>20% atmosphere) will be
used as the example reactor for the numerical simulations from hereon.

1.3.2

Advantages of MPACVD Diamond Growth

MPACVD systems have many advantages over similar PACVD diamond deposition methods.
High deposition rates extending into the tens of microns per hour have been reported [5,
21]. This is comparable, if not higher, than similar PACVD systems mentioned previously.
6

Moreover, the system geometry is often tunable, allowing for efficient and maximized growth
over the characteristic lengthy experimental run time [5]. Finally, their behavior at lower
pressure conditions is relatively well understood with several decades of data to draw from
[4, 5, 21, 22]. This allows for improved overall efficiency and deposition rates.

1.3.3

Higher Pressure Experiments

MPACVD systems have traditionally operated at pressures not exceeding roughly 150 Torr.
This resulted in stable plasma behavior for long time periods. Growth rates of up to 5 µm/hr
have been reported when operating with up to 3 kW of microwave power. The rather lengthy
period of constant operation led to many improvements in the reactor design and tuning,
ultimately improving the diamond quality and growth rate. At the same time, numerical
models were developed to aid in the understanding of their operation. These models have
been successful in predicting MPACVD system behavior at these pressures. These have
also been employed to aid in the design of new reactor designs meant to run at the same
pressures.
Recent experiments conducted at higher pressures (approaching 300 Torr) have been performed in the last several years with the majority of this work being carried out at Michigan
State University [5, 21]. While maintaining several kilowatts of power, the operating pressure
often reaches between 200 and 300 Torr [5, 21]. Experimental results have confirmed higher
growth rates at such conditions [5]. The sample quality at these pressures is also higher
than similar diamond samples grown at lower pressure conditions. This suggests that simply
increasing the pressure during diamond deposition may lead to improved deposition rates
and sample quality.
However, the plasma behavior and properties at these operating conditions are not completely understood. For example, during higher pressure MPACVD operation, the plasma
has been observed to lift off the substrate apparently influenced by some buoyant force [5].
It has also been known to move around the plasma region, even forming below and out7

side of the substrate pedestal toward the reactor floor. These observations have led to the
hypothesis that the gas flow may be the cause of such adverse effects. The hydrogen and
hydrocarbon gas mixture flows throughout the plasma region regardless of pressure. At a
constant calibrated flow rate, its inlet flow speed is expected to decrease with increasing operating pressure. At the same time, the overall flow speed within the plasma is expected to
increase with increasing pressure due to increased gas temperature gradients. This is important to the overall plasma behavior and properties, and may influence the plasma position.
However, it is nearly impossible to empirically measure such effects during system operation
due to the high temperatures present, and the fact that such a measurement would likely
influence the plasma properties itself. Thus, numerical models would provide the necessary insight into these important processes and quantify their effects. Unfortunately, unlike
the standard lower pressure conditions, no numerical models currently exist that accurately
describe the plasma behavior under these operating conditions.

1.4
1.4.1

Multiphysics Plasma Models
Motivation

Although the end result in the MPACVD process is the atomic deposition of carbon over
a surface, understanding the complex processes occurring in the plasma above the surface
is of significant importance. For example, characterizing its chemical behavior can lead to
estimations of the rate of atomic carbon production and its constituents. The localized
density of various species may also be determined. Thus, the behavior of the plasma and its
related processes will remain the primary focus from here on.

8

1.4.2

History

The development of numerical simulations of PACVD systems allowed for the first time the
prediction of these systems before they were physically built. This resulted in the ability to
predict growth mechanisms in terms of geometries and system conditions before any hardware was developed. Recognizing these possibilities, numerical models and their accurate
development have received much attention. A brief background on their development will
now be presented.
High frequency induction plasmas were some of the first systems to be numerically modeled, with the first reported in the late 1960s. One of the first multiphysics models coupling
both electromagnetic fields with high frequency plasmas was introduced by Armstrong and
Ranz in 1968 [23]. Analytic derivations were used to model the fields within a radio frequency induction coupled plasma. Constant temperature profiles were assumed. Power loss
through both radiative and convective processes were included.
Improving upon the constant temperature profile assumption, Miller and Ayen proposed
a similar two-dimensional model for a similar system the next year [24]. This was a finitedifference numerical model, but did not couple with the imposed electromagnetic fields.
Temperature and current distributions were plot against various geometry parameters, and
compared to experimental agreement. Boulos et. al. later developed a coupled, multiphysics
numerical model that accurately coupled both electromagnetic fields, temperature profiles,
and flows [25, 26].
In the mid-1980s, Mostaghimi et. al. introduced multiphysics, fully two-dimensional
models [27, 28]. Charge, momentum, and energy were all explicitly conserved in the plasma
flow module. Two years later, Mostaghimi et. al. introduced a second temperature specifically governing the electrons [29]. This was developed for argon plasmas in a radio frequency
inductively coupled plasma torch. These became the first instances of a two-dimensional
multiphysics model for high frequency coupled plasmas.

9

One of the first plasma models describing the CVD process was introduced by Pfender
et. al. in the mid 1990s [30]. This work described the reactive flows for a supersonic plasma
jet nozzle. About the same time, this group also introduced the Thermal Plasma CVD
simulation [31]. This model described the chemical and thermal properties of methanehydrogen gas mixtures for supersonic and high speed CVD reactors for diamond growth.
In the mid-1990s, Hassouni et al. reported on their development of a one-dimensional
MPACVD reactor model [32]. In 1999, Hassouni et. al. reported a two-dimensional, coupled multiphysics numerical model specifically designed for hydrogen-based low pressure
MPACVD reactors [16]. A finite-difference time domain (FDTD) model was used to update
the electromagnetic fields, from which the absorbed power density was calculated. This was
then fed to the plasma module, where the two temperature model used by Mostaghimi was
again employed. Eight different hydrogen species, and electron distributions were solved via
a Gauss-Seidel line relaxation method. This model was only valid at lower pressures (<20%
atmosphere), where convective and viscous forces were assumed to be negligible and thus
neglected.
The work by Hassouni et. al. has been recently improved upon by including a onedimensional hydrocarbon module [33] and temporal response [34]. However, these models
remain valid only within the lower pressure regime. This is one of the significant limitations
of the current simulation capabilities. Thus, a primary objective of this work is to develop
a model capable of simulating systems at higher pressures.
Previous MPACVD models have been developed which include convective flows and forces
[35–39]. Some of these models have reported results extending above the lower pressure limit
of 150 Torr [36, 38, 39]. Time-dependent convective flow models were used to treat the nonlinear convection terms. Vector flow fields were provided in several of these works [35–37].
The only fully coupled, multiphysics MPACVD simulation was that of Koldanov et al. [35].
However, the reported pressures remained below 100 Torr.

10

1.4.3

Current State of MPACVD Modeling

As stated previously, current MPACVD models omitting or ignoring convective flows and
momentum conservation remain limited to lower pressures (< 150 Torr) [16, 33, 34]. Those
that do include convective flows either do not include electromagnetic fields in a coupled
manner [36–38], or do not extend to the higher pressures where convective flows are expected
to influence the plasma properties [35]. No known reports of a fully coupled, multiphysics
MPACVD model capable of simulating plasma behavior at pressures exceeding 150 Torr while
also including convective flows is known to exist. The primary thrust of the current work
is to develop a fully coupled, multiphysics model which includes momentum conservation in
the plasma modeling scheme.

1.5

Contributions

The objectives of this thesis are related to the development of a plasma simulation that can
operate at higher pressures (up to 300 Torr). Specifically, these objectives will include the
following, listed in order of importance:
1. Including convective flow forces via the momentum transport equation (average gas
flow, convective and buoyant forces, local pressure gradients, etc.)
2. Developing an internal substrate temperature model
3. Developing a more stable and numerically efficient simulation
These objectives are discussed in more detail in the following chapters.

11

CHAPTER 2
PROBLEM STATEMENT
The basic operation, various configurations of, and important characteristics pertaining to
MPACVD reactors are now discussed. Since the beginning of their development several
decades ago, MPACVD reactors have been studied extensively with the goal of improving
both the diamond deposition rate and sample quality. This resulted in a number of different approaches. The following section details the principal MPACVD designs that have
been reported. Extending from these designs is a seemingly endless stream of systems with
numerous configurations, operating modes, and intricacies, each with their own advantages
and disadvantages. An overview of these systems is now presented.

2.1
2.1.1

Introduction to MPACVD Systems
MPACVD System Configurations

Several different MPACVD system configurations have been developed and reported over
the last several decades. Many of these MPACVD systems are presented elsewhere in the
literature including detailed descriptions of their operation and performance [4, 5, 16, 21,
22, 33, 34, 40–44]. For reasons of operation familiarity and easy access to geometric and
experimental data, the class of Michigan State University MPACVD reactors are chosen
12

as the primary systems of choice from hereon. These will be described in the following
paragraphs.
The Michigan State University MPACVD reactor is based upon a main cylindrical cavity
with a substrate pedestal at its base. A schematic of the MSU MPACVD system is provided
in Fig. 2.1.
In the MSU reactor, the microwave energy is fed into the system via a coaxial waveguide
located at the top of the main chamber. This configuration is designed to support a TM103
like dominant mode in the microwave frequency range. The reactor height is thus chosen to
be roughly 3λ0 /2, with λ0 being the free space electromagnetic wavelength at the excitation
frequency [4]. With an input microwave power frequency of 2.45 GHz, this corresponds to
a reactor height on the order of 20 cm. This ensures reasonable power matching across the
feed-cavity interface, leading to efficient power absorption by the plasma.
The diamond sample and growth surface is located at the top edge of the pedestal. As
will be noticed from the proceeding configurations, this growth surface is often raised from
the bottom of the main reactor chamber. This serves to improve the coupling between the
plasma and the incident microwave power by effectively positioning the plasma within the
electromagnetic mode structure [4, 5].
Other research groups have been known to use this reactor configuration outside of Michigan State University. For example, experimental data measured from, and simulation results
based upon, a similar MPACVD reactor have been reported by Koldanov, et al. [35]. Similarities can also be found between the MSU MPACVD reactor and those developed by other
research groups. For example, a similar reactor configuration was developed by the CVD
Diamond group at the University of Paris LIMHP. Many common characteristics are observed, including a cylindrical main cavity, a microwave coaxial feed at the top of the main
reactor chamber, a raised deposition surface pedestal, and a main chamber height dependent
on half-integer free space wavelengths [16, 34].

13

Figure 2.1: Saggital view of the MSU MPACVD system. [4]

14

2.1.2

MSU MPACVD System Overview

What follows is a description of the main components and their purpose during nominal
system operation. Those geometry components labeled in Fig. 2.1, along with the important
omitted components during labeling, are described.
Microwave power is generated external to the system via a magnetron source. This power
is delivered to the PACVD chamber via a waveguide structure. The microwave power is then
introduced into the main cavity via the coaxial waveguide located at the top the of reactor
chamber as denoted by the thatched arrow pointing along −ˆ. The coaxial waveguide diz
mensions are chosen to provide the maximum power transfer between it and the main reactor
chamber at the designed microwave operating frequency of 2.45 GHz. The excitation probe
located at the center of the coaxial waveguide forms the center conductor, is adjustable, and
is used to tune the reflected (and ultimately absorbed) power during operation. Minimizing
the reflected power necessarily improves the power absorption by the plasma and decreases
the power lost to the reactor walls via resistive loading.
The main reactor cavity height is also adjustable. Like the antenna probe, the cavity
height tunes the amount of power absorbed by the plasma by altering the electromagnetic
field intensity and mode formation within the main cavity. Electromagnetic fields in the cavity form a standing wave pattern. This pattern depends heavily on the reactor configuration,
its various geometry parameters, and the microwave discharge properties.
The growth surface and substrate rest on a raised cooling stage. This cooling stage acts
to regulate the substrate temperature during deposition via an internal closed water loop.
Its surface temperature can exceed 1500 K due to power deposited via species collisions
and thermal conduction from the much hotter plasma. Its position is also tunable through
the adjustment of shims located at its base. However, these cannot be changed during the
deposition process. Instead, it must be adjusted between experiment runs. Similar to the
main cavity height and probe positions, the vertical position of the substrate greatly affects

15

the plasma properties and position.
Another main component in the MSU MPACVD reactors is the silica bell jar. This bell
jar (sometimes referred to as the quartz dome) acts to contain the hydrogen-hydrocarbon
gas mixture separated from the rest of the reactor cavity, which is at atmospheric pressure.
It also serves to confine the plasma and its available formation region to the volume in the
immediate vicinity of the substrate. In an electromagnetic sense, the bell jar is translucent to
the electromagnetic fields, with a (essentially) purely real component of permittivity. Losses
across this material during electromagnetic power transfer and scattering within the reactor
are negligible. However, it does affect the electromagnetic field patterns, and so its position
remains fixed.
To the immediate outside of the cooling stage outer radius is a small vacuum gap where
the excess hydrocarbon feed gas exits the plasma region. This is depicted in Fig. 2.1 by the
downward arrows labeled Processing Gas Exit on either side of the cooling stage structure. It
allows the hydrocarbon gas to escape the chamber to avoid accumulation during deposition.
The physical system pressure is measured downwind from this outlet, which is an important
measurement used to characterize the system during deposition. Plasma is not expected to
form in this region, and would be undesirable.
Forming the outer boundary of the processing gas exit region is the substrate silica wall.
This is labeled as the Fused Silica Tube in Fig. 2.1. Its main purpose is to form the outer
wall of the gas exit region with minimal effect on the electromagnetic field pattern. This
is achieved by using the same silica or quartz material as in the bell jar and keeping its
thickness to a minimum. Its thickness and inner radius with respect to the outer pedestal
wall is fixed. However, it always joins with outer substrate lip at its top edge.
With the feed gas exit point detailed, the feed gas entry point must be described. Experimentally, the hydrocarbon gas flows through several small holes distributed azimuthally
toward the base of the reactor. This is labeled as the Feedgas Input in Fig. 2.1 at its bottom
left corner.

16

The last of the primary components to describe in the MSU MPACVD reactor are the
outer baseplates. These baseplates are locate at either side of Fig. 2.1 below the substrate
vertical position. They partition the reactor into two separate electromagnetic regions: the
main reactor where the TM103 mode is supported, and the lower region below the substrate
where TEM modes are known to be supported [5, 21]. With the adjustable cavity height
setting its upper boundary, the top surface of the baseplate sets its lower electromagnetic
boundary. These also affect the field formation below the substrate, which in turn affects
the plasma properties via power efficiency and coupling.

2.2

Basics of MPACVD operation

A detailed description of the important physical processes and their resulting effects on
the typical MPACVD reactor and corresponding plasma is provided. This will include the
important chemical reactions taking place both in the bulk plasma region, as well as a
description of those important to the deposition process itself. The latter occur only at the
growth surface and are important to both the successful and efficient deposition of diamond.

2.2.1

Detailed description of physical processes

The collisions between electrons and heavy species result in the production of new species
via chemical reactions. Many of these chemical reactions either occur at very low rates or in
relatively few numbers, and so are not discussed in detail. However, they will be necessary
to track as they may represent precursor production paths for various important hydrogen
and hydrocarbon species in the MPACVD plasma. These chemical reaction data are listed
in several external sources for further reading [45].
The electron collisions with molecular hydrogen dominate MPACVD plasmas, and typically lead to the excitation of an excited vibrational mode [16, 45]. If it contains enough
energy, the molecular hydrogen will dissociate into atomic hydrogen via the following chem-

17

ical reaction
H2 + e− → 2H + e−

(2.1)

where H2 , H, and e− represent the molecular and atomic hydrogen, and electron chemical species, respectively. Eq. (2.1) represents one of the primary methods through which
atomic hydrogen is produced in MPACVD systems. This statement holds true regardless of
operating conditions such as pressure or absorbed power levels.
At higher operating pressures, three body recombination becomes important. It is also
at these conditions, where the electrons and heavy species approach the same temperature,
that energy is also transferred from the electrons to the hydrogen and hydrocarbon species
directly via elastic collisions [45]. Furthermore, the reversible production and consumption
of atomic hydrogen with other heavy species is also possible [45].
MPACVD plasmas are typically considered to be under local equilibrium conditions in
the bulk region. Thus, the production of atomic hydrogen within the plasma must be
balanced by its consumption either in the bulk plasma or at the reactor boundaries. The
recombination of atomic hydrogen requires upwards of several seconds at typical MPACVD
operating conditions [46]. This suggests its primary consumption or extraction takes place
via recombination and attachment at the substrate and reactor walls.
In the presence of methane, the atomic hydrogen may also recombine to form molecular
hydrogen and hydrocarbon species via the two reactions

H + CH3 + M → CH4 + M
H + CH4

CH3 + H2

(2.2)

(2.3)

where M represents a heavy molecule. However, due to the small percentage of methane
typically present in the feed gas mixture within MPACVD systems, these reactions represent
only a fraction of the total recombination of the atomic hydrogen.
As a result, the atomic hydrogen predominantly diffuses and is lost via recombination
18

at the diamond surface. This recombination rate is vital not only to the plasma properties
in order to balance its concentration in the bulk region, but also in the diamond deposition
process. As such, it has received much attention and has been measured by several groups
[47–51]. The resulting recombination reaction

2H → H2

(2.4)

is exothermic, depositing between 20 and 100 kcal/mol on the diamond surface, which
may represent the primary substrate heating mechanism in many plasma-enhanced diamond
deposition systems [45, 52].
The chemical reactions involving hydrocarbons is likewise complex and lengthy to fully
describe. These species are similarly produced via collisions between heavy species, primarily
atomic hydrogen which was described earlier to dominate the radicals within the gas mixture.
Depending on operating conditions and physical location within the plasma, acetylene and
the methyl radical CH3 are known to be abundant in comparison to other hydrocarbon
radicals [33, 34, 45]. Methane dominates even atomic hydrogen in mole fraction in the bulk
plasma region at lower operating powers and pressures [45, 53, 54]. However, for the higher
pressure operating conditions which are of particular interest here, the atomic hydrogen
is known to dominate hydrocarbon radicals [45, 53, 54]. Thus, the hydrocarbon chemistry
becomes rather complex with rates of production and consumption depending heavily on the
operating conditions. Further insight and detail regarding these hydrocarbon processes in
the bulk plasma region are provided elsewhere [45].
Of course, to this point the process of carbon transfer to the diamond surface has remained untouched. That is, the production of atomic hydrogen and its recombination at the
surface, along with the production of carbon radicals in the bulk plasma have been discussed.
However, the method by which carbon recombines at the growth surface ultimately resulting
in diamond deposition has not been discussed. This would of course require a hydrocarbon

19

molecule to diffuse to the diamond surface as the presence of at least one carbon atom is
required for sustainable deposition. A series of experiments at Rice University in the early
1990s concluded that it is in fact the methyl radical CH3 which is primarily responsible for
the deposition of diamond [45, 55–57]. This result was independently confirmed by others,
with further experiments suggesting other radicals may also contribute but ultimately result
in slower growth with worse quality [45, 58, 59]. Likewise, atomic carbon may play a small
role in the deposition process, but results have not been conclusive of this conjecture [45].
Thus, it appears both acetylene and the methyl radical CH3 recombination at the diamond
surface contribute to the deposition process.
At this point, the chemistry at the diamond surface must be addressed. Due to the high
atomic hydrogen diffusion at the sample surface, the majority of this area is hydrogenated
and nonreactive [45]. However, for the small fraction that remains reactive, the surface sites
are governed by a balance between the following two reactions

Cd H + H → Cd ∗ +H2

(2.5)

Cd ∗ +H → Cd H

(2.6)

where Cd H represents a hydrogen-terminated surface site and Cd ∗ is likewise a surface site
without hydrogen terminated [45]. Goodwin et al. showed that Eq. (2.6) is the dominant
hydrogen termination reaction at typical atomic hydrogen mole fraction levels [60]. Surface
chemistry at these open sites is equally important and discussed elsewhere [45].

2.2.2

Role of plasma in MPACVD

From the above description, it is clear chemical reactions between species play an important
role in the plasma dynamics within MPACVD reactors. Interactions between these physical
processes are important not only for the correct operation of the system, but also for the
successful deposition of diamond. Aside from chemical reactions, the transport of various
20

physical quantities via several kinetic phenomena are equally important in MPACVD systems. These processes include the transport of mass, momentum, and energy throughout
the gas flow region via convection, the diffusion of mass and conduction of heat, and energy
lost via radiative processes. For example, species are transported throughout the plasma via
diffusion and convection processes. This overall process results in a redistribution of mass
according to the reactor geometry and energy distribution. Moreover, the average gas flow
also affects the flow of species within the plasma, with the quantification of its exact distribution and significance being one of the primary objectives of this work. Similarly, energy is
redistributed from hotter to cooler regions according to the laws of thermodynamics. Finally,
both the de-excitation of excited electrons and their acceleration result in a net energy loss
via radiation. Thus, several complex and important physical phenomena are present within
MPACVD systems. These are all interacting on different time and length scales making
their interdependence and properties difficult to predict. As a result, these processes must
be treated carefully in order to accurately capture the total system behavior.
From the above descriptions, it is clear the role of the plasma in MPACVD systems is to
produce and direct the important reactive species toward the diamond surface. Moreover,
this plasma must create and maintain the correct physical conditions required for diamond
deposition. At the same time, these plasma properties and environmental conditions must
remain consistent throughout the duration of the deposition process in order to maximize
the growth rate and maintain high quality. It is this combination of complex interrelated
dependencies and conditions which make the deposition of diamond difficult.

2.3

Motivation for Numerical Model

The system described above represents an extremely complex and nonlinear set of interacting
processes. While each individual process is typically well understood and characterized, the
interactions between the dynamics within a complex material environment is not necessarily

21

well understood. Moreover, the basic understanding or prediction of the system is not easy
and often is unavailable using pen and paper calculations. Instead, a numerical approach
is typically used to predict, or simulate, these MPACVD systems. This same approach is
developed here, as it provides the most approachable and applicable method for predicting
the system behavior under a variety of conditions. Unlike physical MPACVD systems, a
simulation is easily adjusted in terms of several variables including geometry, operating
conditions, and system makeup in order to evaluate future designs.
Aside from the numerous advantages associated with a numerical model of the MPACVD
system, several concerns often arise. Many of these concerns are directly related to the
particular system at hand, and are listed below. For example, one difficulty in the current
simulation is capturing the inherently nonlinear behavior of the entire system in a numerically
efficient manner. That is, the formation and stability of the plasma even during experiments
is sometimes difficult to ensure. Coupling this need with reasonable numerical efficiency
is also a difficult task. This nonlinearity of the system behavior is directly related to the
interdependence of the many physical processes. For instance, the electromagnetic field
propagation is heavily influenced by the plasma position and properties. Moreover, the
power absorption, which is also dependent on the electromagnetic field structure, strongly
influences the plasma properties and formation. These are just a few examples of this strong
interdependence between physical processes resulting in nonlinear system behavior.
On top of predicting future reactor configurations and operating conditions, the numerical simulation of MPACVD reactors provides important insight into the behavior of the
system not otherwise available from experiments. That is, the observation and analysis of
diamond deposition in MPACVD reactors is often limited to qualitative data. Aside from
substrate temperatures and optical images captured via optical cameras, the direct measurement of important plasma properties such as gas temperature and electron number density
are difficult to achieve [40]. Moreover, it may also be difficult to single out important precursor species from others present in the plasma to take measurements. Numerical simulation

22

of these systems provides a tool for accessing this information easily and efficiently. It also
allows for the reevaluation of MPACVD system performance under a change of reactor configurations and operating conditions. What may take several days to even weeks to perform
experimentally, numerical simulations can often predict in a matter of hours or even minutes. Improved efficiency in the evaluation of reactor performance is expected to result in
many more reactor configurations and operating conditions, which in turn may lead to more
efficient diamond deposition of higher quality.
The motivation behind the coupled, multiphysics numerical simulation of the MPACVD
reactors has been discussed, with numerous advantages over experimentation listed. These
include the further and ease of analysis of important physical properties, along with the
ability to more quickly evaluate changes to reactor designs and configurations. As a first
step in the simulation process, the mathematical framework must be derived. This includes
the governing mathematical equations and applicable boundary conditions.

23

CHAPTER 3
MATHEMATICAL FORMALISMS
Many complex and interrelated physical processes are present in MPACVD reactors during
standard operation. Physically, these processes are all ongoing simultaneously and dependent
on one another. In order to mimic this interdependence between physical processes, the
most accurate solution is to simulate these in a multiphysics framework. This necessitates
the understanding of each individual process, as well as their interdependence, in order to
accurately describe the entire system. The result is an accurate model of the entire system.
The numerous equations governing the physical operation of the MPACVD system will be
derived, with its theoretical foundation presented, in the following paragraphs.

3.1

Choice of Temporal Frame

The characteristic time scales of inherent plasma and electromagnetic processes within
MPACVD systems can range from picoseconds to milliseconds or more. On the other hand,
MPACVD systems typically operate for several hours, days, or even weeks in order to deposit
appreciable amounts of diamond. Thus, it is the long-term, effectively steady behavior of
the system that is important to capture in the current simulation. As a result, this steady
behavior and thus time independence is applied across the entire simulation.

24

3.2

Electromagnetics

A derivation of the equations governing the electromagnetic fields and their propagation
within the MPACVD system is now presented. As stated previously, these electromagnetic
fields are assumed to be purely time-harmonic.

3.2.1

Maxwell’s Equations

The frequency domain, point form of the four oft-referenced Maxwell’s Equations govern the
electromagnetic fields within the MPACVD reactor. These are

· D = ρe

(3.1a)

·B =0

(3.1b)

× E = −jω B

(3.1c)

× H = Je + jω D

(3.1d)

where D and B are the electric and magnetic flux densities, respectively, E and H are the
electric and magnetic field intensities, respectively, ρe the volumetric electric charge density,
Je is the electric current density, j the imaginary unit, and ω the radial electromagnetic
excitation frequency. Before manipulating and simplifying the above expressions given in
Eq. (3.1), several assumptions and approximations are applied.

3.2.2

Physical Approximations and Assumptions

Charge accumulation is neglected within dielectrics, and all metal surfaces are assumed
to behave as perfect electric conductors (PECs). The electromagnetic field solutions are
assumed to be purely oscillatory in form, resulting in trivial solutions to both Eq. (3.1a) and
Eq. (3.1b).
25

The magnetic and electric flux densities are related to their corresponding field intensities
by introducing the following material properties

D = 0 rE

(3.2a)

B = µ0 µr H

(3.2b)

where 0 and µ0 are the permittivity and permeability of free space, and r and µr are the
relative permittivity and permeability, respectively.
All materials present within the MPACVD system are assumed to be non-magnetic,
resulting in µr = 1 applied everywhere. This reduces Eq. (3.1) to Eq. (3.3) below.

× E = −jωµ0 H

(3.3a)

× H = Je + jω 0 r E

(3.3b)

Ohm’s Law relates the electric current density to the electric field intensity and the material
dependence via Eq. (3.4)
Je = σe E
ˆ

(3.4)

where σe is the complex electrical conductivity. This results in the final form of Ampere’s
ˆ
Law as given below in Eq. (3.5).

× H = (ˆe + jω 0 r ) E
σ

(3.5)

Eq. (3.3a) and Eq. (3.5) represent a set of two coupled, linear equations that completely
govern all electromagnetic field scattering, propagation, and absorption within the MPACVD
system. They must be solved simultaneously for accurate electromagnetic field calculation.

26

3.2.3

Final Coupled Form

Eq. (3.3a) and Eq. (3.5) could thusly be solved in their present form via iterative procedures.
Instead, Eq. (3.3a) and Eq. (3.5) may be solved simultaneously via one equation by back
substituting one solution variable into its corresponding equation.
To begin this process, the electric field intensity is first solved from Eq. (3.5)

E=

1
(ˆe + jω 0 r )
σ

×H

(3.6)

which is then operated upon by the curl operator

×E =

×

1
(ˆe + jω 0 r )
σ

× H = −jωµ0 H

(3.7)

Eq. (3.7) may then be simplified and expressed in terms of the excitation frequency after
multiplying through by jω 0 . The result is the final electromagnetic field intensity governing
equation given in Eq. (3.8) below.





×

1
−j σe
ˆ
ω 0 + r

×−

ω 2
H=0
c

(3.8)

The electric field intensity is readily calculated from the magnetic field intensity solution to
Eq. (3.8) via Eq. (3.6).

3.2.4

Plasma Influence

Note in Eq. (3.8) the electromagnetic fields are coupled to the plasma influence through the
complex conductivity, σe , inside the curl operator. This term is calculated from the plasma
ˆ
properties themselves. Also note the spatial dependence of this conductivity is correctly
treated and remains inside the curl operator. This will remain important to the overall
accuracy of the coupled solution.
27

3.2.5

Boundary Conditions

With the equations governing electromagnetic energy propagation inside the MPACVD reactor volume now derived, the conditions necessary to enforce at the reactor boundaries are
discussed. The assumption of PECs at all outer reactor interfaces requires zero tangential
electric fields at these surfaces. Moreover, the same tangential fields must be continuous
across dielectric material interfaces. These accurately capture all necessary material boundary conditions applied to electromagnetic fields.
With the governing equations for the electromagnetic field propagation and scattering
now stated, the equation used to calculate the absorbed power distribution may now be
derived. This will depend on the electric field intensity as derived in the following section.

3.3

Absorbed Power

The electromagnetic absorbed power density is important to the behavior of the plasma
discharge in MPACVD systems. Here, it is calculated via a transformation of the timeaveraged absorbed power over one wave cycle to the frequency domain as given in Eq. (3.9).
1
{σe }|E0 |2
Pabs =
2

(3.9)

where Pabs is the absorbed microwave power density, {σe } denotes the real component
of the complex electrical conductivity, and |E0 |2 is the square of the complex electric field
magnitude. This power density distribution is then passed to the plasma solution module.
The total absorbed microwave power may be calculated via

Ptot =

Pabs d3 r

28

(3.10)

where Ptot represents the total absorbed power and

d3 r represents a volumetric integration

over the entire plasma region.

3.4

Plasma Physics

The governing equations describing both the macroscopic behavior of the plasma in the
presence of the microwave power density and average gas flow are derived in the following
section.

3.4.1

Approximations

The approximations applied to the plasma solution module are now listed. These include
the reasoning behind their application and an argument for their validity. Their anticipated
impact on the overall solution process will also be discussed.
Arguably the most important approximation applied when typically modeling MPACVD
plasmas is the assumption of a Maxwellian Electron Energy Distribution Function (EEDF).
As electrons represent the primary species via which the majority of the microwave power
is transferred to the plasma, their energy distribution function is important when characterizing these systems. By approximating this distribution with a well-known function, the
distribution function may be stated analytically. This greatly reduces the number of required
calculations at each solution step, in turn allowing for much faster overall computation run
times. Thus, approximating the EEDF is advantageous to the overall success and numerical
efficiency of the simulation.
For the plasmas supported in MPACVD reactors, it has been shown that the EEDF may
be approximated via a Maxwellian distribution [32, 61]. That is, Hassouni, et al. developed
a one-dimensional transport model located along the axis of symmetry for a MPACVD
system [32, 61]. Simulation results were compared against measured experimental data.
It should be noted that, as reported by Hassouni, et al., assuming a Maxwellian EEDF

29

overestimates the ionization rate in the plasma, and slightly underestimates the dissociation
rate constants used in the mass transport equations [61]. This results in slightly lower atomic
hydrogen densities when compared with experimental data. Moreover, a two-temperature
EEDF model did appear to provide slightly better agreement with experimental results and
avoided this increased atomic hydrogen mole fraction solution [62]. These simulated results
remained in agreement with the experimental data due to large measurement error [61].
The resulting vibrational and gas temperatures also closely matched the experimental data.
Thus, a Maxwellian EEDF was deemed valid in describing the EEDF for MPACVD systems
and is again applied here. The velocity and energy distributions of all other ions and neutral
species are also assumed to be Maxwellian. This is a common approximation, as the energy
of these much more massive species is often much lower than that of the electrons, and is
less likely influenced by external electromagnetic fields, collisions between species, and other
factors.
The energies, and thus average temperatures, of the electrons and ions/neutrals must
be tracked and solved simultaneously to ensure accuracy during plasma simulations. Due
to their large differences in mass, electrons and ions/neutrals represent two very different
systems in terms of kinetic energy. The resultant energies, and thus temperatures, associated
with these two species are often different, suggesting their separate solution and tracking
during the simulation process. As a result, the well-known two-temperature method as
proposed by Mostaghimi, et al. is applied [29]. This method more accurately captures these
important plasma characteristics than similar single temperature models. It also avoids the
unnecessary tracking of individual energies for each separate species within the plasma.
MPACVD systems operating at lower pressures have been well-characterized under the
assumption of negligible or zero gas flow influence on the plasma. This has been confirmed via
countless numerical simulations agreeing with experimental data where this flow is simply
neglected [16, 34]. Although momentum transfer is ignored under these conditions, mass
transfer must still take place. This is accomplished through diffusive mass flow, where the

30

plasma particles are transfered throughout the plasma region due to macroscopic gradients
in density. In the absence of an average plasma flow or for a negligible hydrocarbon gas flow,
this net flow of mass is due purely to these diffusion processes. Thus, the plasma behavior
and rate of mass transfer during the deposition process is assumed to be dominated by
diffusion forces.
For the currently considered higher pressure operating conditions, these diffusion forces
are still expected to dominate the transfer of mass and particles within the plasma. However,
the average plasma gas flow is not necessarily expected to remain negligible, and is included
in the solution process.
Both electric and magnetic fields influence the behavior of charged species within the
plasma according to the Lorentz force equation

F = −qα E + vα × B

(3.11)

where qα is the charge associated with species α. Since the species velocities within MPACVD
reactors are expected to remain relatively low, the resulting force due to the cross product
of the electron velocity and magnetic flux density may be argued as negligible compared
to that of the electric field. Thus, the force on charged particles due to the magnetic flux
density is neglected from hereon, simplifying the calculations to follow.
The distributions of all physical quantities, and thus variables, contained in the simulation
are approximated as spatially continuous [63]. This approximation suggests the application
of a fluid-based simulation method, which assumes a statistical distribution of species and
temperatures sampled spatially across pre-determined locations. However, discontinuities
may sometimes exist in the numerical solution distribution of certain simulation variables.
These are avoided through the increased numerical sampling rate or artificial dampening of
the nonlinear solution. For the densities and temperatures present in MPACVD systems,
this approximation of a continuous and statistical distribution is valid, and will be applied

31

from hereon.
The particular list of chemical species to track during the simulation must be made.
From the typical MPACVD feed gas mixture of molecular hydrogen and methane, numerous
chemical species are produced and exist within the plasma [33, 34]. These are produced during the microwave power transfer from the electromagnetic field to the electrons, which then
transfer energy to various other ions and neutral species. Some of these species are important
precursors to carbon attachment at the growth surface and thus vital to the diamond deposition process [45]. Hundreds of chemical reactions are known to exist in MPACVD systems,
and have been characterized elsewhere [16, 34]. The choice of which chemical reactions and
species to include in the simulation have been scrutinized and demonstrated to accurately
capture the important physics [16, 35, 64]. Here, for the purpose of a much simpler numerical
simulation, only hydrogen-based ions and neutral species are considered and tracked [16].
This greatly reduces the number of species and chemical reactions to track and update,
simplifies the code structure and solution procedure, and improves the overall numerical
efficiency of the simulation. Moreover, since the methane percentage used in the MPACVD
deposition process often remains below 10%, the majority of the important electron-heavy
species chemical reactions and higher density species are expected to be accurately captured.
In bounded plasma systems, a sheath is known to form at material boundaries. This
sheath can have a large impact on the overall behavior and properties of the plasma. It is
often on the order of a deBye length, and thus can exhibit length scales much smaller than
the overall dimensions of the corresponding plasma. This in turn, often requires very fine
numerical resolution in order to accurately simulate. Several approaches for avoiding this
fine numerical sampling surrounding this plasma boundary have been proposed, including
the logical sheath boundary condition [65, 66]. This has been proven to accurately capture
the important physics of the sheath region while maintaining sampling rates larger than the
deBye length.
In the present model, the boundary conditions will include forcing zero change in the

32

electron energy, and applying modified species surface reaction and recombination rates.
This is a crude approximation to the behavior of the electron energy within the sheath, but
avoids the necessity of fine numerical sampling. It also ignores any sort of surface charging
which may be present in MPACVD reactors, and may in fact play an important role in the
plasma behavior. That is, due to their much greater mobility, the electrons reach the plasma
boundaries first and negatively charge these surfaces.
As a result, the charging of the quartz dome in particular could result in electrostatic
forces which attract the positive ions and thereby affecting the overall plasma position and
behavior. Moreover, the behavior of the species within the sheath layer at the substrate
surface is expected to play a vital role in accurately modeling the diamond deposition process
and estimating its growth rate. Thus, a more detailed and accurate sheath model would serve
as an important improvement on the present numerical solution methods and is left for future
implementations of the simulation.

3.4.2

Mass Transport

The mass transport equation represents first and simplest of the three macroscopic transport
equations governing the plasma behavior. It relates the net change in each species mass due
to chemical reactions with its diffusion throughout the plasma. The mass transport equation
may be derived from Generalized Transport Equation[67] by substituting the species mass,
mα , for the average general macroscopic variable. Doing so, the mass transport equation
becomes
F
∂
∂
(nα < ζ >α )+ ·(nα < ζv >α )−nα <
· (v) ζ >α =
(nα < ζ >α )
(3.12)
∂t
mα
∂t
coll
with several simplifications readily applied.
First, it is recognized the species mass is a single, scalar value with its average value
being equal to itself (< m >α = mα ). The product of nα < m >α was shown to be written

33

as the volumetric mass density of the species, ρα . Moreover, the species velocity, vα , was
already used to represent its average velocity in velocity distribution space, and so is taken
as equal to < v >α . Finally, the gradient of species mass in velocity space is assumed to be
zero, resulting in the simplified transport equation in Eq. (3.13)
∂ρα
+
∂t

· (ρα vα ) =

∂ρα
= Wα
∂t coll

(3.13)

where Wα represents the volumetric rate production of mass for species α. Note that the
average species velocity, vs , is its total average velocity and remains individual to each
chemical species.
Within a multi-component plasma, the total velocity of any species may be treated as a
summation of its individual velocity and the average velocity of the plasma as a whole. As
a result, the individual species velocity is represented as a change from the average gas, or
reference, velocity. This may be expressed mathematically as

vs = v + Vs

(3.14)

where v is the average velocity of the plasma as a whole, and Vs is the species velocity
associated with a particular species, s. In the present case, this species velocity represents
the diffusion velocity.
Substituting the linear summation of the species diffusion and average plasma velocities
for the average species velocity in Eq. (3.13) results in Eq. (3.15)
∂ρs
+
∂t

· ρs Vs +

· (ρs v) = Ws

(3.15)

where the average plasma flow velocity v is typically a known quantity, or may even be
negligible for low-flow systems as was often assumed in previous lower pressure MPACVD
systems [16, 34]. The diffusion velocity for each species is often difficult to calculate or

34

measure directly. Instead, it may be calculated as related to the species diffusion coefficient
via
Vs = −Ds xs

(3.16)

where Ds is the diffusion coefficient associated with species s, and xs is its mole fraction.
The mole fraction is calculated as the fraction of specie number density against the total
number density of the plasma
xs =

ns
n

(3.17)

where ns is the volumetric number density of specie s at any particular position, and n
represents the total volumetric number density of the plasma.
With the diffusion velocity formed in terms of the diffusion coefficient and mole fraction,
this quantity may be substituted back into Eq. (3.15). Finally, since a steady-state solution
is assumed, the time rate of change in mass is also dropped, resulting in Eq. (3.18)

−

· (ρs Ds xs ) +

· (ρs v) = Ws

(3.18)

where Eq. (3.18) represents the final form of the mass transport equation.
Here,

· represents the total flux through a closed surface surrounding an infinitesimal

volume. The left-hand-side terms represent the flux of mass out of a closed surface via
diffusion and convection. The right-hand-side of Eq. (3.18) represents the net volumetric
production or consumption rate of mass. Eq. (3.18) governs the change in mass for all
chemical species contained in the plasma. Thus, it must be solved Ns times, where Ns is
the total number of unique chemical species tracked in the plasma.

3.4.3

Momentum Transport

With the diffusion velocity of each individual species known, the average velocity of the gas
flow within the plasma region must likewise be updated. Its behavior is governed separately

35

by the momentum transport equation. This equation has been traditionally ignored in previous lower pressure MPACVD simulations [16, 34]. This was due to its expected negligible
contribution to the overall influence on the plasma properties and location. However, this
equation must be included in the overall solution scheme when pressure is increased, as the
influence of these forces on the plasma behavior are not expected to remain negligible.
The momentum transport equation represents a balance between the rate of change in
momentum with the flux of momentum flowing through a closed surface surrounding an
infinitesimal volume and the summation of all external forces acting upon said volume. The
generalized transport equation is again used as the staring point in this derivation, but
this time the species momentum, mα v, is substituted into the General Transport Equation
[67]. As with the mass transport equation derivation, the resultant terms will be individually
addressed and simplified in order to simplify and condense the complete differential equation.
Substituting the plasma momentum into the General Transport Equation results in the
following expression given in Eq. (3.19)
∂
(nα < mα v >) +
∂t

· (nα < mα vv >) − nα <
=

F
·
mα

(v) mα v >

∂
(nα < mα v >)
∂t
coll

(3.19)

where the total plasma velocity average value is again represented by v. Once again assuming
the species mass is a scalar quantity whose average is equal to its value, Eq. (3.19) is further
simplified. Finally, recognizing that the species mass is not a spatially-dependent function,
it may be pulled out front of the
∂
(ρα < v >α ) +
∂t

(v) operator resulting in

· (ρα < vv >α ) − nα < F ·

∂
(3.20)
(v) v >α = ∂t (ρα < v >α )
coll

At this point, the total velocity of each species may be expanded in terms of its average and

36

random velocities
v = cα + uα

(3.21)

where cα is the random velocity of species α, and uα is its average velocity. The velocity
gradient operating on the velocity vector results in the identity vector, further simplifying the
third term contained in Eq. (3.20). Recognizing < v >= 0, and substituting the expanded
velocity terms into Eq. (3.20) yields
∂
(ρα uα ) +
∂t

· (ρα uα uα ) +

· (ρα < cc >α ) − nα < F >α =

∂
(ρα uα )
∂t
coll

(3.22)

where the average velocity terms inside the divergence operator may be expanded due to
their linear and independent nature. The random velocity and volumetric mass density
tensor product in the third term of Eq. (3.22) represents the kinetic pressure dyad, Pα [67].
Similar to the EEDF, the velocity distribution is also assumed to be Maxwellian in form. As
a result, the pressure dyad reduces to the scalar pressure, p, further reducing its divergence
operator to a spatial gradient. The volumetric number density multiplying the average force
term may be rewritten as the volumetric mass density multiplying the total acceleration due
to external forces. Assuming local charge neutrality, this external acceleration term is only
due to gravity, and so may be written as ρg · z. Finally, Eq. (3.22) may be summed over all
species in order to solve for the average plasma velocity, rewritten as v and given in
∂
(ρv) +
∂t

· (ρvv) +

p − ρg · z =

∂
(ρv)
∂t
coll

(3.23)

The final alteration to Eq. (3.23) comes in the form of the viscous stress tensor. In fluid mechanics, the viscous stress tensor represents the change of fluid motion due to self-interactions
dependent on the kinetic viscosity, µ, of the fluid. This can be represented by the collisional
term on the right side of Eq. (3.23) and re-written as a divergence operator acting upon the

37

viscosity tensor as
∂ (ρv)
+
∂t

· (ρvv) = − p −

· τ + ρg · z
ˆ
ˆ

(3.24)

where τ is the viscous stress tensor.
ˆ
Although compact, Eq. (3.24) is more conveniently expressed and clearly understood
when written explicitly in its individual variable components. The chain rule may be applied
to the time rate of change in momentum as represented by the first term in Eq. (3.24) yielding
∂ρ
∂v
∂ (ρv)
=v
+ρ
∂t
∂t
∂t

(3.25)

where the time rate of change of volumetric mass density is forced to zero, which is a result of
the time independence assumption from Eq. (3.18). Thus, the total time rate of momentum
change may be calculated
∂ (ρv)
∂v
=ρ
∂t
∂t

(3.26)

In order to set apart only the time rate of change in the plasma velocity, the volumetric mass
density divides Eq. (3.24), resulting in the final form of the momentum transport equation
∂v 1
+
∂t ρ

· (ρvv) = −

1
1
p−
ρ
ρ

·τ +g·v
ˆ
ˆ

(3.27)

where Eq. (3.27) governs the average gas, or plasma, flow velocity throughout the plasma
region.
Note in Eq. (3.27) the

· (ρvv) and

· τs terms form rank-two tensors representing
ˆ

both convective and viscous forces acting in both orthogonal and shear directions. They are
expressed in the following matrix representations for a generic two-dimensional coordinate
frame

 

· (ρvv) =


2 v v
r z 
  vr
· ρ 

2
vz vz vz

38

(3.28)


·τ =
ˆ



 τrr τrz 
·

τzr τzz

(3.29)

where vr and vz are the average plasma velocity components along the r and z unit vectors,
ˆ
ˆ
and τrr , τrz , τzr , and τzz are the viscosity tensor components representing the shear and
bulk forces due to plasma viscosity, respectively. These viscosity tensor terms are calculated
[68]
τrr = −µ 2

∂vr 2
−
∂r
3

·v

(3.30a)

τzz = −µ 2

∂vz
2
−
∂z
3

·v

(3.30b)

τrz = τzr = −µ
·v =

∂vr ∂vz
+
∂z
∂r

∂vz
1 ∂
(rvr ) +
r ∂r
∂z

(3.30c)
(3.30d)

where µ is assumed to have spatial dependence, and so is left inside the divergence operator.
This time dependence present in Eq. (3.27) remains in order to accurately update the gas
momentum in the presence of the non-linear velocity dyad. This choice will become more
clear during the derivation of the numerical framework. However, this is purely due to the
choice of numerical solution method, as the final system solution remains steady-state in the
absence of turbulent flow.
Finally, the left-hand-side of Eq. (3.27) represents a balance between the total time rate of
change in average plasma velocity and the rate of momentum flux across a closed infinitesimal
surface due to convective forces. The right-hand-side terms of Eq. (3.27) represent the net
rate of volumetric momentum change as a consequence of pressure, viscous, and gravitational
forces, respectively [68].

39

3.4.4

Electron Energy Transport

The energy, and thus temperature, of electrons is expected to be much higher than that of
the ions and neutral species at certain operating conditions. This is inferred from their much
smaller mass when compared to ions and neutrals. As a result, their kinetic energy increases
more quickly due to high frequency oscillations in the presence of electromagnetic fields
post collisions and energy transfer before colliding again with heavy species. As mentioned
earlier, the electron energy is treated separately from the ion and neutral energy to maintain
accuracy by following the two-temperature model. The electron energy transport equation
is now derived starting with the Generalized Energy Transport Equation [67].
To begin, the species place holder in the Generalized Energy Transport Equation is
replaced with the electron distinction

· (λe Te ) +

· ρe v e

1 2 3 kb Te
v +
2 e 2 me

+

· (pe ve ) +

· (ˆ · ve )
τ


δ 1 ρe < v 2 >e
2

· [ρe he (v − De xe )] = 
δt
coll
(3.31)


+ ρe g · ve + Pabs +

where λe is the electron thermal conductivity coefficient.
At this point, the collisional term on the right side of Eq. (3.31) must be explicitly treated
for the case of electrons. Numerous physical processes are simultaneously occurring within
the plasma during the nominal operation of MPACVD reactors. Some of these processes
include the transfer of energy between species through collisions. These collisions between
electrons and ions or neutral species represent a kinetic energy loss mechanism and affect the
overall electron temperature. Two distinct families of these collision types exist depending on
the final state of the heavy species involved and include translational-rotational or vibrational
modes, and are denoted Qet and Qev .
At the same time, another electron energy transfer mechanism is associated with electrons
40

involved in chemical reactions. These result in a net change of the electron kinetic energy
(and decreased temperature) through the activation of these chemical reactions. Energy loss
may also be due to the transformation of converted mass or changes in excited energy states.
This generic energy change rate is represented in the electron transport equation by Qec .
Substituting these terms, the time rate of change for the electron kinetic energy as represented by the collisional term on the right side of Eq. (3.31) now becomes

1
δ 2 ρe < v 2 >e


= Qet + Qev + Qec
δt
coll


(3.32)

which may be back substituted into Eq. (3.31) resulting in the final form of the electron
energy transport equation in Eq. (3.33)

· (λe Te ) +

· ρe v e

1 2 3 kb Te
v +
2 e 2 me

+ ρe g · ve + Pabs +

+

· (pe ve ) +

· (ˆ · ve )
τ

· [ρe he (v − De xe )] = Qet + Qev + Qec

(3.33)

which governs the energy and thus temperature of electrons during the simulation process.
The left-hand-side terms in Eq. (3.33) represent the rate of change in the electron energy
due to thermal conduction, kinetic and potential energy related to electron flow, pressure,
gravity, electromagnetic fields, and enthalpy, respectively. The three energy exchange terms
on the right side of Eq. (3.33) represent the time rate of change in electron energy due to
collisions between electrons and heavy species in translational-rotational modes, vibrational
modes, and their chemical reactions, respectively.

3.4.5

Total Energy Transport

The total energy transport equation is arguably the most important of all the transport
equations governing the macroscopic quantities associated with the MPACVD plasma. This
is because it relates all macroscopic solution variables in one equation. In this way, it can be
41

thought of as a pseudo master transport equation. Moreover, the majority of the physical
processes (including chemical reaction rates, energy transfers rates, diffusion coefficients,
etc.) depend on the gas temperature that is governed by this total energy transport equation.
These are thusly present in the gas temperature transport equation and influence the gas
temperature. As with the electron energy transport equation derivation, the generalized
energy transport equation provides the starting point in this derivation.
To begin, a summation over all present species is performed on the individual terms from
the Generalized Transport Equation. These will be treated individually and discussed in the
following paragraphs. The total rate of energy loss via thermal conduction is calculated by
a summation of the three heat flux vectors

qt + qv + qe = − λtr Tg + λv Tg + λe Te

(3.34)

where qt and qv are the heat flux vectors associated with the translational-rotational and
vibrational heavy molecular modes, respectively, and λtr and λv represent their thermal
conductivity coefficients, respectively. Note in Eq. (3.34) the gas temperature, Tg , is used
in both the heat loss due to translational-rotational and vibrational modes. This is due
to the assumed gas temperature and heavy species vibrational mode temperature equality
[16]. These three terms are then substituted back into the total energy transport equation,
replacing the single, leading species-dependent heat flux term.
Next, the kinetic and potential energy terms are addressed. The inclusion of this summation of energy over all present species requires only the change of species-based quantities
to the average or total macroscopic quantity in each case. For example, the transformation

42

can be made

· ρα vα

1 2 3 kb Tα
v +
2 α 2 mα

=

· ρe ve

1 2 3 kb Te
v +
2 e 2 me

Ns
· ρi v i

+
i=1, i=e−

1 2 3 kb Tg
v +
2 i
2 mi

(3.35)

where mi is the mass of the ith heavy species.
The next three terms in the Generalized Energy Transport Equation representing pressure, viscosity, and gravity forces are likewise summed over all species. This again results in
the simple removal of their species dependence, and hence subscript.
The absorbed power density term in the Generalized Energy Transport Equation must be
treated carefully, as it is the sole term through which the electromagnetic fields influence the
plasma properties. The time rate of change in total plasma energy due to the electromagnetic
fields is dependent upon the summation over all changes in species accelerations
Ns
1
ρα aα,E/M · vα
2
α

(3.36)

This summation may be expanded into its dependence on heavy species and electrons. In
particular, the following argument may be made regarding the relative magnitude of influence
of these two families. While the electron acceleration is expected to be much greater than
that of the heavy species, its product with the electron volumetric mass density negates
any large quantity argument. This suggests the summation over all species is necessary to
ensure accuracy. However, the electron velocity is in fact expected to be much larger than
the heavy species velocities due to the contribution of the electron diffusion velocity. This
effect is purely due to its far smaller mass, as is often the case in plasma systems. The large
value argument can thus be made when comparing the summation of all total energy changes
in heavy species with that of electrons. Approximating the summation over all species in

43

Eq. (3.36) with only the influence due to electrons is thus argued as valid, and once more
represented by the absorbed power density as in Eq. (3.33).
The influence of total species enthalpy likewise necessitates the summation over all
species. This may be simplified in the following expression as
Ns

Ns
· ρi hi v − Di xi

=

· (ρhv) −

i=1

· ρi hi Di xi

(3.37)

i=1

where the total enthalpy of all species, h, is calculated
Ns
h=

hi

(3.38)

i
Eq. (3.37) is thus substituted for the species-dependent terms in the total energy transport
equation.
Finally, the collisional term is addressed. In the case of total plasma energy, the net energy
exchange between all species due to collisions is zero. This is due to the physical requirements
of conservation of energy and momentum during any contained physical process. Thus,
this collisional term may be neglected in the case of the total energy transport equation.
However, the energy loss due to charged species radiation must be included. That is, the
de-excitation of electrons radiates energy as phonons. This time rate of energy loss due to
high frequency electric fields is denoted as Qrad , and is substituted in for the total energy
transport equation.
All necessary additions and substitutions to the generalized energy transport equation

44

have been made. The result is the total energy transport equation

· λtr Tg + λv Tg + λe Te +
+

· ρe ve

1 2 3 kb Te
v +
2 e 2 me

· (pv) + ρg · v +
Ns
· ρi v i

+
i=1, i=e−

· (ˆ · v)
τ
1 2 3 kb Tg
v +
2 i
2 mi

Ns
+

· (ρhv) −

· ρi hi Di xi

= −Pabs + Qrad

(3.39)

i=1
where Eq. (3.39) represents the final form of the total energy transport equation.
The left-hand-side terms in Eq. (3.39) represent the time rate of change in the total
plasma energy due to heat conduction, pressure, gravity, viscosity, kinetic and potential
energy, and enthalpy processes across all species, respectively. The right-hand-side terms
represent the gain in electron energy via the electromagnetic heating of the electrons and
their loss due to phonon radiation via de-excitation. The gas temperature, Tg , will become
the solution variable in Eq. (3.39).

3.4.6

Boundary Conditions

The plasma density at any wall or boundary is known to be non-zero for all plasma species,
regardless of mass or charge. As a result, the flux of species mass to the boundaries must
balance the total rate of species production at the wall due to surface chemistry processes.
This is expressed mathematically as

−Ds xs = Wsurf,s + νcat,s Ms
ˆ

(3.40)

where Wsurf,s is the surface rate of net mass density production, and νcat,s is the total
ˆ
surface reaction rate. The two terms on the right side of Eq. (3.40) together represent the
net rate of mass density production from both surface reactions and chemical reactions at
the boundary. These surface rates are calculated as a function of the plasma properties at
45

the boundaries, along with the recombination coefficients
Nr,cat
Wsurf,s =

(2)
νcat,r ηr,s Ms

(3.41a)

r=1
Ns
νcat,r =

ζr xq νχ

(3.41b)

q=1,q=H2 ,q=e−
νcat,r = ζr νχ
ˆ
c
νχ = tot
4

8RTg
πM

(3.41c)
(3.41d)

where νcat,s is the solution-dependent surface reaction rate, ζr is the recombination coefficient of catalytic reaction r, and νχ is the total collision frequency of ions and neutrals at
the boundary. A list of surface reactions that are tracked in the simulation of MPACVD
plasmas, along with their physical constants and assumptions are detailed in the Appendix
and elsewhere [45].
The boundary conditions which apply to the plasma velocity fall into one of two categories: forced and interfacial. Due to the directional nature of the plasma velocity, both
normal and tangential boundary conditions must be addressed. These are discussed in detail
in the following paragraphs.
Hydrogen gas flows into the experimental MPACVD reactor through the inlets toward
the base of the reactor (see Fig. 2.1). It is assumed the gas is flowing at the specified flow
rate at this boundary, and is also laminar. Thus, the normal gas flow velocity component is
assumed to be forced at this boundary. Moreover, the gas flow is essentially oriented normal
to these boundaries.
Unlike the inlet region, the gas flow rate at the outlet region is not forced. However, in
order to avoid momentum and gas accumulation, its flow rate may be calculated dependent
on the inlet flow rate and the ratio of inlet to outlet surface areas. Moreover, the plasma

46

velocity is also assumed laminar at this boundary. The above considerations suggest a finitevalued, or Dirichlet, boundary condition imposed at the outlet.
All other boundaries represent hard material interfaces either with the quartz bell jar
or substrate wall, reactor boundaries, or the substrate surface. The normal component of
plasma flow at these boundaries must be zero. At the same time, the tangential plasma flow
must also go to zero at these boundaries. This is often referred to as the no-slip boundary
condition. That is, it is known at a hard material interface the tangential flow at the surface
drops to zero.
The effects of the plasma sheath are represented through approximations to physical
observables. This includes the assumption that electrons diffusing more rapidly than ions
populate the sheath layer and remain at a constant energy across this layer. Such an approximation results in a zeroed normal derivative in the electron temperature approaching
all boundaries. Mathematically, this is expressed as

n·
ˆ

Te = 0

(3.42)

The gas temperature at the physical boundaries of the MPACVD reactor, and especially
at the diamond substrate, is critical to the diamond deposition process. This is even more
important to the various mass and heat fluxes directed toward the diamond sample during
MPACVD system operation. The relationship between substrate temperature and diamond
deposition rates have been documented in several instances for MPACVD systems [5, 21].
Moreover, this relationship has been demonstrated to be crucial to the diamond deposition
rate. As highlighted earlier, the solution to the total energy transport equation given in
Eq. (3.39) is paramount to the accuracy of the entire solution. Its solution results in an
updated gas temperature distribution throughout the plasma region.
Historically, a uniform substrate temperature of 1200 K is often applied to the substrate surface, while all other boundaries are set to 600 K. These values were chosen to

47

match experimental observations [16]. However, as documented experimentally, the substrate temperature is rarely uniform and fluctuates significantly depending on many reactor
and operating factors [5]. As a result, forcing a uniform substrate temperature is nonphysical,
requiring a more accurate boundary condition be applied. This will be addressed through
the inclusion of a substrate temperature calculation.

3.5

Electrical Conductivity

The complex electrical conductivity distribution is calculated based upon the plasma properties. MPACVD plasmas, and most plasmas in general, represent lossy dielectrics in the
electromagnetic sense. In this way, the complex electrical conductivity represents the ability
of the plasma to absorb electromagnetic power. Due to the steady system assumption and
frequency-domain electromagnetic field solutions, the often time-dependent complex electrical conductivity calculation must once again be translated to the frequency domain. The
resulting derivation is provided in the following paragraphs.
The collisional equation of motion governing the time rate of change in electron momentum is provided below in Eq. (3.43)
∂pe
= −q0 E + ve × B − me νef f ve
∂t

(3.43)

where pe is the electron momentum and νef f is the effective collision frequency between
heavy species and electrons. Again, the magnetic field dependence in Eq. (3.43) may be
neglected. Likewise, the time rate of change in the electron mass is zero, and so this term
may be pulled out of the partial time derivative and divided through. This results in the
following modified equation of motion
q
∂ve
= − 0 E − νef f ve
∂t
me

48

(3.44)

which may now be translated to the frequency domain by replacing the partial temporal
derivative operator with jω, resulting in
∂
→ jω
∂t
q
jωve = − 0 E − νef f ve
me

(3.45a)

where ω is the angular excitation frequency of the incident microwave power. Solving for
the electron velocity results in

ve jω + νef f

q
=− 0E
me

q
E
∴ ve = − 0
me jω + ν
ef f

(3.46a)

Recalling Ohm’s Law, it is known that

Je = qe ne ve = σe E
ˆ

∴ σe =
ˆ

q e ne
E

ve

(3.47a)

(3.47b)

Thus, substituting for the electron velocity from Eq. (3.46) into Eq. (3.44) yields the final
form of the complex electrical conductivity in Eq. (3.48)


2n
νef f − jω
q e

σe = 0 
ˆ
2 + ν2
me
ω
ef f

(3.48)

where the effective collision frequency between electrons and hydrogen species is calculated
as
2
νef f = σx n v 2 + ve
th

(3.49)

with σx being the collision cross section for H2 , vth is the electron thermal velocity and ve
49

is the total electron speed [16]. Thus, in the case of low operating pressures where flow is
neglected, the electron collision frequency is solely dependent on its thermal velocity. This
electrical conductivity is used in the calculation of the absorbed power density from Eq. (3.9).

3.6

Inclusion of Substrate Temperature

The gas temperature at the diamond substrate is critical to the diamond deposition process.
That is, it has a large effect on the diamond deposition rate and resulting sample quality
as documented elsewhere [5, 21]. Its accurate representation at physical boundaries is thus
important not only experimentally, but crucial to the accuracy of the numerical model as a
whole.
The gas temperature in previous MPACVD simulations is often explicitly set along different material boundaries [16, 34]. However, this does not take into account the variations
in plasma position and properties, along with substrate conditions, which experimentally are
known to vary. Thus, the development of a substrate thermal model is required to accurately
capture these changes in substrate temperature. A derivation of the governing equation and
boundary conditions for the substrate model is presented in the following paragraphs.

3.6.1

Heat Equation

The conduction of heat across any system is governed by the heat equation as given by

ρCp

∂T
+
∂t

· (λ T ) = 0

(3.50)

where ρ is the mass density and Cp is the specific heat capacity. Eq. (3.50) relates the time
rate of change in the temperature distribution within any infinitesimal volume element to
the rate of heat flux flowing through its enclosed surface area. The fact that this expression
is set to zero means no sources of heat are present, as is physically relevant to MPACVD
systems.
50

The rate of heat conduction within the substrate is expected to be much slower than that
within the plasma. Combining this fact along with the long experimental run time suggests
a steady-state thermal substrate model is again valid. As a result, the time rate of change
in the substrate model is dropped, leaving the governing equation for the substrate thermal
model as
· (λ T ) = 0

(3.51)

Eq. (3.51) is applied from hereon in order to update the absolute temperature distribution
within the substrate.

3.6.2

Boundary Conditions

The steady-state heat equation given in Eq. (3.51) yields a trivial temperature solution in
the absence of boundary conditions. That is, with no heat sources present, the substrate
temperature would necessarily go to zero. Thus, it is important to set these correctly not
only to avoid a trivial solution, but also to ensure a single coupled solution within the entire
MPACVD system. A derivation of these boundary conditions follows.
The substrate thermal model simulation region extends throughout the substrate pedestal,
the substrate itself, the gas outlet cavity, and the quartz wall at the outer radius of the substrate as seen in Fig. 2.1. Temperature is a scalar quantity whose distribution must be
continuous across material interfaces. This condition avoids the necessity to treat internal
interfaces as boundaries, thus simplifying the boundary considerations to only those lying on
the external domain boundaries. The heat flux must likewise remain continuous throughout
the substrate and at its boundaries.
Experimentally, the pedestal is water-cooled within its inner-most region. It contains
a recirculating flow that is constantly replaced in order to conduct heat away from the
growth surface [4]. The substrate pedestal within the same region is assumed to mimic this
cooling affect by behaving as a thermal sink with a specified temperature. The remaining

51

region at the bottom of the reactor is approximated with a floating boundary, meaning the
temperature is unspecified with zero net heat flow.
The substrate thermal and plasma models are coupled by substituting the total power
flux incident at the top substrate surface and outer wall boundaries for these same boundaries
in the substrate thermal model. This may be expressed in terms of the heat flow at these
boundaries, resulting in the following boundary equation

QH = −λ T

(3.52)

where QH is the heat flow at the substrate pedestal outer boundaries. This heat flow
is a summation of both thermal fluxes and hydrogen energy transfer via bombardment as
calculated from the previously derived plasma properties as in Eq. (3.53)
(x)
(T )
QH = QH + QH

(3.53)

(x)
(T )
where QH is the heat flow due to hydrogen bombardment energy transfer and QH is the
heat flow due to the gas temperature gradient.
This heat flow to the substrate surface due to energy transfer via bombardment of atomic
hydrogen is calculated as the product of the atomic hydrogen recombination coefficient,
the average energy associated with the neutral hydrogen atoms, and the flux rate. As
stated previously, this recombination coefficient of hydrogen at the substrate surface is often
approximated to be 0.1 in similar models [32, 34]. However, a more accurate approximation
which is useful in the present formulation, was proposed by Krasnoperov et al. and is applied
here [48]. The resulting power flux at the substrate surface due to hydrogen bombardment
is given in Eq. (3.54)
(x)
QH = βH H ΓH

(3.54a)

ΓH = −cH DH xH

(3.54b)

52

where βH is the unitless recombination coefficient of atomic hydrogen at the substrate
surface, H is the average exothermic energy deposited by hydrogen atoms involved in the
dominant surface reaction (104 kcal/mol), ΓH is the atomic hydrogen number flux at the
substrate, and cH is the atomic hydrogen molar concentration [45]. This exothermic energy
is associated with the dominant
2H → H2

(3.55)

diamond surface chemical reaction [45]. This flux is calculated at the substrate surface.
Similarly, the heat flow to the substrate surface due to the gas temperature gradient may
be calculated as given above
(T )
QH = − (λt + λv )

Tg

(3.56)

where the summation of the vibration and translation-rotation thermal conductivities is
included. Substituting Eq. (3.56) and Eq. (3.54) into Eq. (3.53) produces the final form of
the heat flux boundary condition calculated within the plasma fluid module.
Once the substrate temperature is calculated, the temperature at the boundaries of the
plasma region is passed to the plasma fluid module and imposed at the corresponding locations. This coupled formulation of passing the heat flux from the plasma to thermal, and
temperature from thermal to plasma ensures not only continuous temperatures, but also continuous fluxes. In this way, the atomic hydrogen flux contribution is not directly influenced
as with the gas temperature at the substrate boundaries, but instead indirectly through its
physical property dependence on this updated temperature.

53

CHAPTER 4
SOLUTION PROCEDURE
With the governing equations outlined, the numerical solution procedure is now developed.
This will include discussions pertaining to domain discretization, the translation of mathematical expressions to matrix equations, the different solution schemes used in updating the
governing equations, and the overall solution scheme.

4.1

Axisymmetric Approximation

The majority of the physical reactor components labeled in Fig. 2.1 exhibit cylindrical symmetry. That is, the cross sectional view provided in Fig. 2.1 may be rotated one full revolution
to trace out the three-dimensional geometry. This cylindrically symmetric geometry makes
a strong case for exploiting axial symmetry during the simulation process.
The domain of the reactor is discretized using a standard rectangular mesh. Per standard
convention, individual nodes are represented by whole integer coordinate pairs (e.g. (m, n)).
Here, m and n represent whole integers colocated at the bottom left node of an arbitrary
element. This results in edges having coordinate pairs represented by (m, n + 1 ) and (m +
2
1 , n) where {m, n} ∈ {Z + } for vertical and horizontal edges, respectively, and (m+ 1 , n+ 1 )
2
2
2
for elements. An example of the index pair labeling of all specified mesh features for a generic
mesh element is provided in Fig. 4.1.
54

(i,j+1)
h

×

1
(i,j+ 2 )
×

(i+ 1 ,j+1)
2

1
(i+ 1 ,j+ 2 )
2

(i,j)
×

(i+1,j+ 1 )
2
×

+

h

(i+1,j+1)
h

(i+ 1 ,j)
2

(i+1,j)
h

Figure 4.1: Generic mesh element with labeled geometry feature index pairs.

These index pairs representing the various mesh geometry features seen in Fig. 4.1 will
also be cited extensively during the matrix representation and stencil derivations.

4.2

Finite Difference

With the numerical domain now discretized into a set of ordered solution sample locations,
the numerical solution of the various equations may be derived. The governing equations
contain differential equations, each requiring the discretization and numerical approximation
of spatial derivatives. Here, the finite-difference (FD) method is chosen for simplicity, familiarity purposes, and maintaining consistency with previous MPACVD numerical simulation
schemes [16, 22, 64].

55

4.3

Electromagnetics Module

The spatial locations of the electromagnetics module numerical solution samples must be
chosen. With the mesh previously defined, this includes the locations of the electromagnetic
fields with respect to this mesh. A common sampling approach for electromagnetic fields
solved via FD methods is the Staggered Yee mesh [69]. In the Staggered Yee method, the
electric field samples are aligned along mesh edges, while orthogonal magnetic fields are
sampled at each mesh face, or element centers in the case of the present two-dimensional
mesh. This is represented in Fig. 4.2.
Er
-

Er
-

H

6

Ez

`
hφ

6

Ez

Er
-

6

Ez

Hφ
`
h

Hφ
`
h

6

Ez

Er
-

6

Ez

Er
-

Hφ
`
h

6

Ez

Er
-

Figure 4.2: Staggered Yee mesh in two-dimensions.

The advantage of this Staggered Yee mesh is its accurate representation of the curl
operators present in Eq. (3.8). For example, the inner curl operator in Eq. (3.8) represents
the spatial derivative of the orthogonal magnetic field component evaluated at the electric
field location. Since the electric field is chosen to lie tangential to the mesh edges and
the magnetic field is directed into the page at the element centers, this curl operation is
calculated exactly without the need for numerical estimation through quadrature, averaging,
or translation. Likewise, the outer curl operator in Eq. (3.8) represents the orthogonal electric
56

field derivative evaluated at the magnetic field sampling location. This involves calculating
both the radially- and vertically-directed derivatives from the mesh edges at the element
centers in the present system. Again, this calculation is performed exactly without any
required modifications or estimations to the field structure or mesh geometry. Moreover,
the Staggered Yee field algorithm maintains the second order accuracy of the curl operator
calculation in the case of uniform mesh spacing. Thus, the Staggered Yee mesh provides a
distribution of electromagnetic field samples that are simple to assign, are well-defined with
respect to the mesh, and result in inherent curl operator calculations.
The choice of the Er , Ez and Hφ electromagnetic field components falls out from the
coordinate frame, symmetry assumption of the reactor geometry, and known incident power
polarization. The electromagnetic field solution is halved to these three components, thus
reducing the number of numerical unknowns.
The accurate simulation of the electromagnetic fields is governed by the sufficient sampling of these fields on the mesh. Thus, an accurate sampling rate within the MPACVD
reactor is crucial to a successful simulation. This numerical mesh must also accurately capture the geometric features of the reactor to similarly ensure successful representation of the
system. A typical electromagnetics domain numerical mesh for the MSU Reactor B (see
Fig. 2.1) is provided for reference in Fig. 4.3. Although dense and somewhat difficult to
distinguish between adjacent elements, a rectangular grid is clearly depicted in Fig. 4.3.
Two different mesh resolutions are clearly visible along z from Fig. 4.3. That is, the
ˆ
solution domain between the top of the bell jar and the antenna probe exhibits much lower
mesh resolution. The relative sparsity of sample points in this region was chosen due to
its distance from any curved surfaces, source regions, or the plasma solution domain itself.
As a result, the electromagnetic fields are expected to remain well-behaved in this region
without large fluctuations in field structure or intensity. This decreased mesh resolution also
decreases the overall number of unknowns, thus reducing the solution time of the simulation
as a whole.

57

Figure 4.3: Typical electromagnetics numerical solution domain mesh.

The sampling rate of any numerical mesh must be sufficient to capture any effects or
changes to the solution variable contained within its domain. In other words, the mesh
discretization size must be small enough to accurately capture the solution itself. A good rule
of thumb for electromagnetic simulations solved via FD is a minimum mesh discretization
size of λ0 /40. This level coincides with the minimum grid spacing observed in Fig. 4.3,
while the higher rate approaches λ0 /100. Although this increased discretization rate is not
necessarily required to accurately capture electromagnetic fields, it will become important in
capturing the plasma solution. Since the numerical mesh is shared between electromagnetics
and plasma solution domains, the minimum discretization rate is given precedence.
Material properties are assigned at element centers and assumed to remain constant over
that entire element. This results in material interfaces being located at mesh edges, which
will coincide with the appropriate boundary conditions for the electric fields. Assigning
materials to entire mesh elements also ensures no fractional steps or approximations in
material transition occur within elements. However, this does result in the need to translate

58

electromagnetic permittivities and electrical conductivities from element centers to edges.
This is done by performing a bilinear interpolation of the material properties at the two
neighboring elements to each edge and computing their weighted average at the edge location.
This ensures accurate translation of material properties in the case of non-uniform mesh
spacing, while also approximating this material interface effect.
In the case of an edge lying on a domain boundary, the materials assigned to the corresponding element are also assigned to the edge. This translation of material properties also
allows for the calculation of the electrical current density, which is concurrently located and
oriented as with the electric fields. The reason for this choice of location and orientation will
be more evident in the proceeding derivations.

4.3.1

Finite-Difference Frequency-Domain

The combination of a frequency domain solution and the FD numerical solution method
for solving Eq. (3.8) is often referred to as the finite-difference frequency-domain (FDFD)
solution method. This is a well-known method in computational electromagnetics, although
not as popular as the time-domain formulation, similarly known as the finite-difference timedomain (FDTD) solution method. Previous MPACVD simulations traditionally have employed these FDTD methods for updating the electromagnetic field solutions [16, 34, 35].
However, the field pattern and wave interactions within MPACVD reactors are known to be
dominated by certain modes at the excitation frequency [5, 21, 22]. This suggests capturing
the lowest order cavity modes is sufficient in representing the majority of the electromagnetic field structure. Moreover, the time-domain solution of these fields is computationally
expensive in terms of both scaling and numerical run time. These two disadvantages will be
discussed in detail in the following paragraphs.
In FDTD, the electric and magnetic fields are updated at half-integer time steps in a
leapfrog manner. The maximum time step (known as the Courant-Fuchs-Levy limit, or CFL
condition) allowed in these updates is limited by the discretization rate of the mesh and the
59

wave propagation speed. It specifies the minimum number of solution steps required during
one wave period. The disadvantage of this CFL condition is that as the mesh discretization
rate increases (and hence mesh spacing decreases), both the number of system unknowns and
the number of update steps to represent one wave cycle increase. Thus, the overall numerical
solution time increases exponentially, or according to a power law using the FDTD method.
The time-averaged absorbed power density which is the only electromagnetic-based quantity present in Eq. (3.33) and Eq. (3.39). Typically, the FDTD simulation is run until this
quantity converges to a steady solution [16]. Several dozen wave cycles are often required
before convergence including those capturing the initial wave propagation into the cavity.
As a result, many calculations are performed unnecessarily making the FDTD method quite
inefficient. Instead, the FDFD method may be employed ensuring a steady-state solution
(including time-averaged absorbed power) and avoiding the need to simulate several wave
cycles.
However, the FDFD method is not without its own disadvantages, which must be addressed. In the matching coaxial structure below and to the outside of the substrate pedestal
from Fig. 2.1 the electromagnetic field is known to support transverse electromagnetic (TEM)
mode structure with respect to z [21]. That is, only the Er and Hφ field components are
ˆ
non-zero, while Ez = 0. The FDFD formulation does not force Ez to zero anywhere in the
solution domain, meaning it can be non-zero in the matching coaxial region. This difference
between experimental and numerical results could affect not only the power density profile,
but possibly the plasma behavior and its physical properties as well. However, these influences are argued as negligible under nominal operating conditions for MPACVD systems, as
the bulk of the power absorption takes place within the plasma body above the substrate
top surface.

60

4.3.2

Equation Discretization

The equation governing the electromagnetic fields as given in Eq. (3.8) may be written in
simplified form as
×

1
ˆ

×−

ω 2
H=0
c

(4.1)

where ˆ now represents all material dependencies explicitly stated in Eq. (3.8). Eq. (4.1)
must now be discretized in terms of the field samples located on the mesh. This will be
done by assigning the solution to a generic mesh geometric feature index pair and relating
its terms to neighboring indices.
To begin, Eq. (4.1) is first expanded into the electromagnetic field components allowed
in the present simulation. Since this includes only the azimuthal magnetic field component,
the result is given by
ω 2
1
H
×−
ˆ
c
∂ 1 ∂
∂ 1 ∂
rHφ +
Hφ
∂r rˆ ∂r
∂z ˆ ∂z

×
=

−

ω 2
Hφ = 0
c

(4.2)

where the curl operators have been expanded in cylindrical coordinates. Eq. (4.1) may now
be represented by the field and material quantities as they were assigned on the numerical
mesh. This requires the discretization of both the azimuthal magnetic field component and
the complex permittivity, ˆ.
Magnetic field samples are located at half-integer element centers and may be represented
1
by generic index pairs of the form (i+ 2 , j + 1 ). Eq. (4.2) may then be rewritten in discretized
2

61

form as
r 3H
− r 1H
1
i+ 2 φ,i+ 3 ,j+ 1
i+ 2 φ,i+ 1 ,j+ 2
2
2
2
r 3 −r 1
ri+1 − ri
i+ 2
i+ 2
− r 1H
r 1H
1
i− 2 φ,i− 2 ,j+ 1
i+ 2 φ,i+ 1 ,j+ 1
2
2
2
−

ˆ
r 1
r,i+1,j+ 1 i+ 2
2

r 1 −r 1
ri+1 − ri
i+ 2
i− 2
−H
H
1
1
φ,i+ 2 ,j+ 1
φ,i+ 2 ,j+ 3
2
2

ˆ
1r
r,i,j+ 2 i+ 1
2
+

ˆ
z 3 −z 1
zj+1 − zj
r,i+ 1 ,j+1 j+ 2
j+ 2
2
H
1
1 − Hφ,i+ 1 ,j− 1
φ,i+ 2 ,j+ 2
2
2
−
ˆ
1
r,i+ 2 ,j

z 1 −z 1
j+ 2
j− 2

zj+1 − zj

−

ω2
H
1
1 =0
c2 φ,i+ 2 ,j+ 2

(4.3)

where once again, whole integer indices represent nodes and edges, while half-integers repre1
sent element centers. Thus Eq. (4.3) relates the magnetic field sample located at (i+ 2 , j + 1 )
2
to the magnetic field samples at its nearest four neighbors adjacent to its four bordering
edges. Moreover, the complex permittivity terms are located at the center of these four
adjacent edge segments, with translated values calculated as discussed earlier. These spatial
dependencies are visually depicted in Fig. 4.4.
Eq. (4.3) and Fig. 4.4 represent the discretized governing equation for the electromagnetic
fields in the bulk region of the MPACVD reactor. That is, Eq. (4.3) is valid for all magnetic
field samples away from any numerical domain boundary. However, in the presence of such
boundaries, special attention must be paid to the numerical sampling and governing equation
in order to ensure an accurate solution. The discretized form of the electromagnetic field
boundary conditions are now presented.

62

j+3
2

h

×
j+1
2

h

h

×

×

h

×
j−1
2

h

i− 1
2

i+ 1
2

i+ 3
2

Figure 4.4: Magnetic field and permittivity sample locations present in the discretized magnetic field equation. Here, ◦ represents the magnetic field sample locations, and × the
permittivities.

4.3.3

Boundary Conditions

The boundary conditions are now numerically discretized and presented in the following
paragraphs. Along with these, a third numerical boundary condition related to the artificial
termination of the reactor microwave power feed will also be discussed.
The first, and by far the simplest boundary condition to numerically implement in the
electromagnetics module is located at the domain boundaries. Electric fields fall to zero
intensity well inside a good conductor. Numerically, these conductors are often approximated
as PECs, resulting in exactly zero electric fields within such regions. The electromagnetic
fields within PECs are thus neglected, and the domain is truncated to include only nonPEC regions, while all domain boundaries (save the axis of symmetry) are assumed to
represent PEC interfaces. Since electromagnetic fields lying tangential to any interface must
be continuous across that interface, this also requires tangential electric fields lying on any

63

domain boundary to be zero. Since the electric field is sampled on mesh edges, this forces all
electric fields lying on domain boundaries to zero. This is enforced in the electromagnetics
module by forcing the following condition

Et |P EC = 0

(4.4)

where Et is the tangential electric field component at the PEC (or domain) boundary.
The discretized governing electromagnetics equation provided in Eq. (4.3) is written in
terms of the magnetic field. This likewise requires the domain boundary representation
to be written for the magnetic field. This is already explicitly included in the discretized
equation, as the contribution from PEC-based magnetic fields, which are already neglected in
the calculation, are necessarily zero. Thus, their contribution along with their corresponding
reference element term, are neglected in Eq. (4.3). This results in an expected zero-tangential
electric field boundary condition when calculated from the magnetic field solution.
Due to the half-plane numerical domain, numerical calculations performed about r = 0
must be carefully treated. In order to ensure accurate simulation of a continuous function,
the radial component of any vector quantity must be zero at r = 0. Likewise, the rate
of change in the radial direction for any vertically-directed vector quantity lying on this
boundary must be constant along r. Mathematically, these conditions are expressed as
ˆ

Ar

=0

(4.5a)

r=0

∂Az
=0
∂r r=0

(4.5b)

where Ar and Az are the radially- and vertically-directed components of some vector quantity, A = Ar r + Az z . These conditions extend to the electric field intensity, which consists
ˆ
ˆ
of vector components. Thus, the above conditions must be enforced during the solution
procedure. However, applying these conditions to Eq. (4.3) results in a singularity at this

64

axis related to the Ez field component. Thus, a new formulation and governing equation
is required in this case. This singularity is avoided by making use of the system symmetry,
realizing the left-directed neighbor of the magnetic field across the r = 0 axis is equal but
opposite the magnetic field value itself. The discrete form of the left-neighboring radial term
from Eq. (4.3) may thus be rewritten at the axis of symmetry as in Eq. (4.6)
r1 H 1
1 − r− 1 Hφ,− 1 ,j+ 1
φ, 2 ,j+ 2
2
2
2
2 =
ˆ
r
r,0,j+ 1 1
2 2

r1 − r 1
−2
2

(r1 − r0 )

2H 1
1
φ, 2 ,j+ 2
ˆ
r (r − r0 )
r,0,j+ 1 1 1
2 2

(4.6)

where i = 0 represents the radial index located at the axis of symmetry and − 1 represents
2
the location at the center of the non-physical element radial image across this axis. Thus, this
simple multiplication by a factor of two accurately represents the magnetic field boundary
condition at the r = 0 axis, as well as the electric field components at this boundary.
A final numerical boundary condition is imposed in order to accurately simulate the electromagnetic field structure within the waveguide feed. The experimental MSU MPACVD
system comprises a long microwave waveguide structure which delivers the power to the
main reactor chamber as seen in Fig. 2.1. It is usually important to capture these fields
during the simulation. However, the fields in this region are not of particular interest and
have little effect on the plasma properties. Moreover, the complete discretization of this
waveguide would unnecessarily increase the size of the solution domain, thus decreasing the
overall numerical efficiency of the solution. In order to avoid simulating these unnecessary
field samples, the solution domain is artificially terminated at an arbitrary location at least
one free space wavelength away from the main reactor chamber. However, this termination
introduces another domain (and hence PEC) boundary, resulting in the non-physical scattering of electromagnetic fields from its surface if left untreated. In order to avoid these
non-physical reflections, an absorbing boundary condition is applied.
Here, the perfectly matched layer (PML) as proposed Berenger is applied via stretch-

65

ing parameters across several layers [70–73]. A detailed derivation of the PML and these
stretching parameters are provided elsewhere and summarized here [72, 73]. This boundary
condition will end up being applied over a series of successive layers approaching the artificially terminated boundary in the numerical domain. These stretching parameters are given
by [72, 73]
p

sr =

r − r0
1+α
r1 − r0

p

sz =

z − z0
1+α
z1 − z0

2

1 − j sin

π r − r0
2 r1 − r0

2

1 − j sin

π z − z0
2 z1 − z0

(4.7a)

(4.7b)

where sr and sz are the stretching parameter functions used in the PML implementation
as a function of r and z, respectively, α is the PML scaling coefficient, z0 is the vertical
position of the first PML layer, z1 is the position of the last PML layer, and p is the PML
scaling order. Here, z0 < z < z1 , while the radial stretching function is one everywhere. The
PML scaling coefficient and corresponding order often take on values between one and ten,
achieving minimal reflections depending on microwave wavelength. Hence, z1 also coincides
with the domain boundary in Eq. (4.7). These stretching parameters modify the material
dependencies in Eq. (4.3), resulting in an augmented governing equation given in discrete

66

form as
r

3H
1 − ri+ 1 Hφ,i+ 1 ,j+ 1
i+ 2 φ,i+ 3 ,j+ 2
2
2
2
2

r 3 −r 1
ri+1 − ri
i+ 2
i+ 2
− r 1H
r 1H
1
1
i− 2 φ,i− 2 ,j+ 1
i+ 2 φ,i+ 2 ,j+ 1
2
2
−

ˆ
r 1
r,i+1,j+ 1 i+ 2
2

ˆ
r 1
r,i,j+ 1 i+ 2
2
+

r

1
i+ 2

−r 1
i− 2

H
−H
1
1
φ,i+ 2 ,j+ 3
φ,i+ 2 ,j+ 1
2
2

s
¯
z,i,j+ 1
2

ri+1 − ri


z
zj+1 − zj
ˆ
−z 1
1
r,i+ 2 ,j+1 j+ 3
j+ 2
2
H
1
1 − Hφ,i+ 1 ,j− 1
φ,i+ 2 ,j+ 2
2
2
−
ˆ
r,i+ 1 ,j
2

s
¯
1
z,i+1,j+ 2





1
s
¯
1
z,i+ 2 ,j+1










1




s
¯
1
z,i+ 2 ,j
z 1 −z 1
zj+1 − zj
j+ 2
j− 2
ω2
s
H
=0
−
z,i+ 1 ,j+ 1
φ,i+ 1 ,j+ 1
c2
2
2
2
2

(4.8)

where sz are the element-centered stretching parameters, and sz are stretching parameters
¯
translated to the mesh edges via a weighted bilinear interpolation scheme. These stretching
parameters must be removed from the material dependence during the electric field calculations that follow.

4.3.4

Matrix Representation

Eq. (4.8) is represented in matrix form, allowing for the simultaneous solution of all unknown
magnetic field samples. This is in sharp contrast to the FDTD method where fields are
stepped forward in time and dependent on previous solutions. A derivation of this matrix
formulation is provided below. Eq. (4.8) may be expressed in matrix equation form as

[A0 − DI] H = 0

67

(4.9)

where A is the n × n matrix representing the double spatial curl operators and permittivity
term on the left side of Eq. (4.8) with n being the number of mesh elements contained in
the simulation domain, D is the diagonal matrix containing the PML stretching parameters
and multiplied by the free space wavenumber, I is the identity matrix, H is the n × 1 vector
containing the unknown magnetic field solutions, and 0 is a n × 1 vector containing zeros.
The filling of A is dependent on the indexing scheme used to number the magnetic field
sampling locations. In general, the coefficients multiplying the five magnetic field samples
given in Eq. (4.8) are used to fill the corresponding matrix entries.
The Total-Field/Scattered-Field (TF/SF) formulation is used in simulating the scattering
of electromagnetic fields [73]. Many numerical electromagnetic simulations solve for the total
field solution spanning the entire simulation domain. This total field is the summation of
the incident and scattered electromagnetic fields that resulting from wave reflections and
refractions in the presence of materials. This summation may be represented mathematically
in terms of the present solution magnetic fields as
H (t) = H (i) + H (s)

(4.10)

where H (t) , H (i) , and H (s) represent the total, incident, and scattered magnetic fields
within the computational domain, respectively.
In order to implement the TF/SF formulation, the numerical solution domain must be
separated into regions of total and purely scattered fields. In the present simulation, the
total field (TF) region must extend across the entire plasma solution domain to ensure an
accurate power absorption profile. This translates to all space contained within the quartz
bell jar and that outside the substrate. On the other hand, the scattered field (SF) region
must exist removed from the plasma solution domain, which in this case is located within the
excitation waveguide at the top of the MPACVD reactor. As a result, the TF/SF boundary
is chosen to reside within the waveguide region at the top of the MPACVD reactor, extending

68

Figure 4.5: Location of the TF/SF regions in the electromagnetics solution domain for the
MSU Reactor B configuration. Red is SF, all blue shades represent TF. For interpretation
to the references to color in this and all other figures, the reader is referred to the electronic
version of this dissertation.

across the entire waveguide in r as shown in Fig. 4.5.
ˆ
In this way, both the reflected power within the waveguide region, along with the total
field and resulting power absorption density, may be calculated accurately. The exact location of the TF/SF boundary within the waveguide is completely arbitrary, but is chosen
to reside at least one half wavelength above the main reactor cavity height. This boundary
position avoids the significant contribution of decaying higher order modes.
The derivation of the TF/SF matrix equation formulation begins with the choice of a
numerical solution in the absence of scatterers or materials [73]. This function, denoted here
as F , must be continuous and represent a solution to the governing Maxwell’s Equations in
free space. It is then reshaped into a column vector, fsrc . The masking function, Q, is then
filled taking on values of unity along its diagonal only within SF regions. Q is a square,
69

n × n matrix with zeros everywhere except those diagonal entries stated above. From these
two terms, the scattered and total field source vectors may be calculated as fSF = Qfsrc
and fT F = (I − Q)fsrc . Here, fSF and fT F represent the scattered and total field source
vectors, respectively.
However, the above formulation results in some of the discretized matrix terms representing the SF region being contained in the TF region and vice versa [73]. Since these two
regions are incompatible with each other, these discrepancies are resolved by re-introducing
the source term back to these quantities and moving its calculation to the right side of
Eq. (4.9). The result is a right-hand-side vector in Eq. (4.9) now dependent on the source
field
b = (QA − AQ) fsrc

(4.11)

where A is the square matrix representing both the spatial operator and material property
terms on the left side of Eq. (4.9). The matrix products on the right side of Eq. (4.11) may
be efficiently calculated due to the sparse nature of both A and Q. In fact, since Q is a
diagonal matrix that is sparse even on its diagonal, these computational steps may scale
sub-linearly with the number of numerical solution samples.
With the source vector now formulated, the complete matrix equation may be expressed
as
[A0 − DI] H = [Q (A0 − DI) − (A0 − DI) Q] fsrc

(4.12)

which may be solved for the unknown magnetic field samples contained in H. The choice of
the electromagnetics matrix equation solution method may now be discussed.

4.3.5

Solution Procedure

There are generally several categories of solution methods that may be applied when solving
a matrix equation. These include the direct, iterative, and sparse methods. Each has
advantages and disadvantages associated with its implementation, yet sparse methods are
70

the primary solution scheme employed in the current simulation.
Sparse matrix solution methods solve the resulting matrix equation exactly to machine
precision and often at much faster rates compared to other methods. However, their solution
methods are much more complex and dependent on the particular implementation chosen.
Aside from their somewhat complicated data storage formats, sparse matrix solutions represent the best of both worlds of accuracy and efficiency. For this reason, sparse matrix solution
methods are chosen for any direct matrix inversion required in the simulation including the
solution of Eq. (4.12). Specifically, the MUMPS sparse numerical solver is applied to the
current numerical system [74, 75].
The efficient implementation of a sparse matrix solver requires storing the matrix information in sparse format. Matrices may be stored in sparse format by keeping only their
non-zero entries. Its percentage of non-zero to total number of entries represents the fill
rate of the matrix. For example, the square matrices associated with two-dimensional FD
solutions often exhibit a <1% fill rate. This suggests FD method matrix storage and corresponding calculations performed in sparse format would significantly improve the overall
numerical efficiency of the solution procedure. This would also cut down on both the memory required to store the matrix information, and the number of numerical computations
required during algebraic manipulations.
There are several standard sparse matrix data storage formats. The coordinate list (COO)
sparse matrix storage format is chosen here for simplicity. As a result, the COO-formatted
matrix information is used to solve for the magnetic fields in Eq. (4.12). This same MUMPS
sparse matrix solver will be applied for all required matrix equation solutions throughout
the remainder of the simulation.
Once the magnetic field samples have been calculated by solving Eq. (4.12), the electric
fields must be calculated. These electric fields will be used along with the electrical conductivity to calculate the absorbed power density profile, which will be passed to the plasma fluid
module. The electric fields are calculated via Eq. (3.6) with the PML stretching function

71

dependence now explicitly included as
1
s (ˆe + jω 0 r )
¯ σ

E=

×H

(4.13)

where s denotes the generic form of the edge-based PML stretching functions outlined earlier.
¯
Expanding Eq. (4.13) into its two components yields
∂Hφ
1
sz (ˆe + jω 0 r ) ∂z
¯ σ

(4.14a)

1 ∂ rHφ
sz
¯
(ˆe + jω 0 r ) r
σ
∂r

(4.14b)

Er =

Ez =

where sz is the stretching function dependent on the vertical coordinate. The edge-based
¯
electric fields are thus calculated via Eq. (4.14). Writing in discretized form, these electric
field components located at their respective edge indices are given by

E
1 =
r,i+ 2 ,j

E

1
z,i,j+ 2


H
−H
1
1
φ,i+ 1 ,j− 1 
 φ,i+ 2 ,j+ 2
2
2

z 1 −z 1
j+ 2
j− 2

1
s
¯
1
z,i+ 2 ,j

=
ri

σ
ˆ
+ jω 0
e,i+ 1 ,j
r,i+ 1 ,j
2
2
(4.15a)


s
¯
r 1H
1
1 ,j+ 1 − ri− 1 Hφ,i− 1 ,j+ 1
z,i,j+ 2
 i+ 2 φ,i+ 2
2
2
2
2


r 1 −r 1
i+ 2
i− 2
σ
ˆ
1 + jω 0 r,i,j+ 1
e,i,j+ 2
2
(4.15b)

With both the electric and magnetic fields now calculated at their respective numerical
mesh locations, the Poynting vector may be calculated. This will be useful in calculating
the vector power flux across the simulation domain, and thus the reflected power ratio. Its
general form is
S = E × H∗

(4.16)

where S is the electromagnetic poynting vector. From Eq. (4.16), it is clear the calculation of

72

the Poynting vector depends on the colocation of the electric and magnetic fields. To achieve
colocation, the electric fields located at the edges bordering each element are averaged at
the element center. This results in element-based electric field samples, which may be used
to calculate the Poynting vector along with the element-based azimuthal magnetic field.
As mentioned previously, the reflected power is often experimentally measured and used
as an indicator of the effective power coupling efficiency within the plasma. Thus, a similar
numerical metric is desired in order to gauge the accuracy of the simulation compared to
the experimental system, and to aid in tuning the reactor geometry parameters during the
solution procedure. To calculate the total reflected power, the total flux of electromagnetic
power is integrated across the waveguide feed and compared to the incident power. This
involves integrating the previously-calculated Poynting vector over the surface area of the
coaxial waveguide feed. This integration position is located at the top of the reactor in the
SF numerical region. It is located at least one quarter wavelength away from the TF/SF
boundary to avoid any discontinuities or interpolation errors between these regions. The
reflected power integration is thus calculated
r1

Pref =

Srdrdφ

(4.17)

r0

where Pref is the total reflected power from the MPACVD system, and r0 and r1 represent
the inner and outer radii of the coaxial waveguide feed, respectively. The reflected power
ratio is then calculated as the ratio of the incident power delivered to the system as given
by the scaled source vector, Pinc , and this reflected power via
R=

Pref
Pinc

(4.18)

where R is the reflected power ratio.
Finally, the quality factor of the MSU MPACVD reactor may be calculated, which can
prove useful in comparing against experimentally measured quality factors [22]. The quality
73

factor of any system is a measure of the system bandwidth at resonance. In MPACVD
systems, this value is a measure of how much energy the system can store relative to the
power loss at the microwave excitation frequency. It is often used to gauge the reactor
performance and design, and is calculated from Eq. (4.19) [22]

Q=

E · E ∗ d3 r

ω

(4.19)

Pinc − Pref
where Q is the quality factor.

4.3.6

Reflected Power Minimization

The MSU MPACVD reactor is tuned during the deposition process to minimize the reflected
power. This is done by physically adjusting certain reactor components, namely the main
cavity height and antenna probe length. As a result, the reflected power is minimized ensuring maximum microwave power transfer to the oscillating free electrons and heavy species
in the plasma. Since this represents an important physical operation, it is important the
numerical simulation also mimics this process via a reflected power minimization algorithm
as detailed below.
Each adjustment to the physical reactor settings and corresponding numerical domain
and mesh requires a new electromagnetic field solution. While the sparse solver described
earlier is numerically efficient, this can still take several seconds per solution depending on the
computer architecture, number of processors, and domain size. Thus, it is not advantageous
to minimize the reflected power at every electromagnetic field solution step. Instead, a
certain number of solution steps is skipped before this minimization occurs. Moreover, this
allows for the formation and stabilization of the plasma solution before large changes in the
absorbed power density profile are imposed.
During reflected power minimization, the electromagnetic fields are first solved within the
current reactor configuration and the reflected power saved. The main cavity height is then

74

stepped through a series of evenly spaced positions on either side of the current cavity height,
with the reflected power again saved at each position. This results in a set of cavity heights
and corresponding reflected power values, which are then searched for the minimum value.
The cavity height corresponding to this minimum reflected power is then saved and used as
the new cavity height position. Next, the antenna probe length is varied, again for positions
on either side of the current antenna probe length. The electromagnetic fields are again
solved at each of these positions, along with the reflected power. Finally, the antenna probe
length corresponding to the minimum reflected power is likewise saved. The end result is a
combination of antenna probe length and main cavity height which minimize the reflected
power within the MSU MPACVD reactor. These new probe and cavity settings are then
used as the reference positions during the next reflected power minimization update. Before
updating the plasma fluid module, the electromagnetics solution is once again run at the
minimized cavity height and probe length positions.
Changes in the reflected power calculated during this routine are due solely to adjustments in these physical cavity configuration settings. Although the substrate position is
adjustable, it cannot be tuned during experimental operation. Thus, the exclusion of the
substrate vertical position adjustment during the reflected power minimization algorithm
follows experimental capabilities.
This power minimization routine is repeated throughout the numerical simulation. However, it is often advantageous and results in a more stable final solution when it is applied
only after the formation of a stable plasma within the plasma simulation module.

4.4

Absorbed Power

Once the electromagnetic fields have been calculated, the absorbed power density profile
must be updated. This quantity will be used in the electron and total energy transport
equations. The absorbed power density as derived in Eq. (3.9) will be used here.

75

As with the Poynting vector calculation, the numerical sampling of the absorbed power
density is performed at the element centers. This will ensure colocation with the volume
source terms prescribed in the plasma transport equations given in Eq. (3.33) and Eq. (3.39).
Since the electric fields are sampled along mesh edges, these fields are again be averaged at
the element centers. The original complex electrical conductivity passed from the plasma
module is already defined at the element centers. This avoids the necessity of translating
the already element-based conductivity from the mesh edges back to element centers, which
could result in numerical errors. The result is a discretized absorbed power density profile
for a generic mesh location
1
σ
P
1 ,j+ 1 = 2 {ˆe,i+ 1 ,j+ 1 }
abs,i+ 2
2
2
2


∗
∗
E
1 ,j E
1 ,j + Er,i+ 1 ,j+1 Er,i+ 1 ,j+1
r,i+ 2
 r,i+ 2

2
2
× 

2

∗
∗
1E
1 + Ez,i+1i,j+ 1 Ez,i+1,j+ 1
 z,i,j+ 2 z,i,j+ 2
2
2
+

2


where

E

(4.20)

{ˆe } is the real component of the complex electrical conductivity. This dependence
σ

on only the real component of the electrical conductivity mathematically represents the physical power absorbed by hydrogen ions and neutrals via electron collisions. Again, measurable
energy transfer only occurs in the presence of these collisional processes. Eq. (4.20) is used
to generate a two-dimensional profile of absorbed power density across the entire simulation
domain. However, this power density is known to only be non-zero in the regions where
σe = 0, requiring ne = 0.
ˆ
Once the absorbed power density has been calculated, the total absorbed power level
must be enforced. Experimentally, the incident microwave power supplied to the MPACVD
reactor is set, with the reflected power measured. This means the total absorbed power
is not physically enforced, but rather calculated as the difference between incident and
76

reflected powers. Numerically, this total absorbed power level is explicitly set prior to the
simulation process, with the incident power scaled to result in this total absorbed power.
This is ensured by calculating the total power in the absorbed density profile and scaling
the incident electromagnetic fields accordingly. A more detailed description of this scaling
algorithm follows.
Upon initialization of the complete simulation, the incident electromagnetic field intensity
is set to unity. Once the absorbed power density profile has been calculated, the resultant
absorbed power density profile is numerically integrated for the total absorbed power level
via Eq. (3.10). The ratio between this value and the specified total power level is calculated
as Ptot /P0 where Ptot is the total absorbed power calculated via Eq. (3.10) and P0 is the
desired total absorbed power level. All absorbed power density solutions are then scaled
by this value, resulting in a modified absorbed power density profile corresponding to the
desired total absorbed power. This power ratio is also used to scale the incident source
electromagnetic fields at the next electromagnetics module solution step. As a result, the
absorbed power density passed to the plasma module always corresponds to the desired total
power level. As the coupled solution converges, this scaling coefficient approaches unity.
An advantage of this scaling method and its back substitution in the case of the scaled
incident microwave power is the realization of the absolute electromagnetic field intensities.
That is, the scaling of the incident electromagnetic source fields results in field intensities
which may be directly compared to experimental values.

4.5

Plasma Fluid Module

As with the electromagnetics module, the locations of the various plasma fluid module macroscopic variable samples must first be assigned before proceeding with its numerical solution.
The behavior of these physical quantities is governed through the transport equations. Numerically, these relate the influence of fluxes between mesh elements with source terms inside

77

v

v

6z

v

-r

xs , T
h

6z

v

-r

v

v

xs , T
h

h

v

-r

v

6z

-r

xs , T

6z

v

-r

v

xs , T
h

v

-r

v

6z

6z

Figure 4.6: Locations of plasma module samples on numerical mesh.

each mesh element. As a result, the natural choice for the location of scalar plasma module
variable samples is element centers. However, the vector flow components are involved in the
flux of various quantities between mesh elements. The behavior of these flow components
also depends on gradients of scalar quantities, including pressure and viscosity. This suggests
sampling the vector flow components at the mesh edges, as will be the case here. Unlike
the electromagnetics module, these edge-based vector samples will be oriented orthogonal to
the edge segments themselves. This results in a directed flux quantity based on flow components moving between mesh elements where other scalar samples are located. A graphical
representation of this configuration is provided in Fig. Fig. 4.6.
Note in Fig. 4.6 the flow components are only defined on their respective oriented edges.
That is, the radial flow is only sampled at the vertical edges, while the vertical flow is
only sampled at the horizontal edges. This will lead to non-forced shear boundary flows and
conditions. The advantage of these particular sample locations detailed in Fig. 4.6 is the flux
due to flows between elements is represented exactly with no need to translate or interpolate
flux quantities. Moreover, the flow components may be updated based on the various scalar
gradients, which are also easily calculated.
78

Figure 4.7: Numerical mesh within the plasma region for the plasma fluid module.

As with the electromagnetics solution module, the plasma region must be carefully represented and sampled by the plasma domain mesh. The choice of plasma module sample
points is equally crucial to the accurate modeling of the MPACVD reactor. Since the electromagnetics solution domain extends throughout the plasma solution domain as well, this
mesh has essentially already been produced. However, it is presented here set apart from
the electromagnetics domain as given in Fig. 4.7.
The uniform grid spacing shown in Fig. 4.7 is chosen for simplicity and ease of visualization. However, the present simulation is capable of nonuniform mesh discretization schemes
as are common in similar MPACVD plasma simulations [16, 34, 76]. This grid spacing also
corresponds to roughly λ0 /100, where λ0 is the free space electromagnetic wavelength. No
direct dependence on the electromagnetic wavelength is required for the plasma fluid module
solution quantities. However, familiarity with the simulation and typical operating conditions
suggests this spacing is sufficient to ensure the majority of continuous and smooth solution
distributions. This particular level of mesh spacing is more than sufficient to capture the
important electromagnetic field structure within the MPACVD system.
79

4.5.1

Equation Discretization

The transport equations are now discretized using the sampling scheme presented in Fig. 4.6.
The majority of the transport equations often include many terms and are rather long. Only
a representative selection of one scalar and one flow component transport equation will be
discretized in the following paragraphs. Moreover, only unique samples of representative
scalar and gradient terms will be provided. In this sense, the generic discretization of the
transport equations is presented, with their exact forms derived from these.
The mass transport equation governing the behavior of all chemical species seen in
Eq. (3.18) is restated here for convenience in Eq. (4.21)

· (ρs Ds xs ) +

· (ρs v) = Ws

(4.21)

which may be discretized at any reference generic element-centered index as given in Eq. (4.22)

−x
1
3
s,i+ 1 ,j+ 1 
 s,i+ 2 ,j+ 2
2
2

r 3 −r 1
i+ 2
i+ 2


x
1 ,j+ 1 − xs,i− 1 ,j+ 1
 s,i+ 2
2
2
2


r 1 −r 1
i+ 2
i− 2


ρ
1 ,j+1 Ds,i+ 1 ,j+1 xs,i+ 1 ,j+ 3 − xs,i+ 1 ,j+ 1
s,i+ 2

2
2
2
2
2
+


z 3 −z 1
zj+1 − zj
j+ 2
j+ 2


ρ
1 ,j Ds,i+ 1 ,j xs,i+ 1 ,j+ 1 − xs,i+ 1 ,j− 1
s,i+ 2
2 
2
2
2
2
−


z 1 −z 1
zj+1 − zj
j+ 2
j− 2
ri+1 ρ
v
ri ρ
v
1
s,i+1,j+ 1 r,i+1,j+ 2
s,i,j+ 1 r,i,j+ 1
2
2
2
+
−
r 1 ri+1 − ri
r 1 ri+1 − ri
i+ 2
i+ 2
ρ
v
ρ
v
1
s,i+ 2 ,j+1 z,i+ 1 ,j+1
s,i+ 1 ,j z,i+ 1 ,j
2
2
2 =W
+
−
1
s,i+ 2 ,j+ 1
zj+1 − zj
zj+1 − zj
2

ri+1 ρ
1D
s,i+1,j+ 2 s,i+1,j+ 1
2
r 1 ri+1 − ri
i+ 2
ri ρ
D
s,i,j+ 1 s,i,j+ 1
2
2
−
r 1 ri+1 − ri
i+ 2



x

80

(4.22)

where the first four lines represent the flux of mass due to diffusion and the last two represent
mass flux due to convective gas flow and the production of mass within each mesh element
domain. Eq. (4.22) is representative of all scalar transport equations (mass, electron energy,
total energy) in that each term in these equations can be derived in the form of one of the
terms contained in Eq. (4.21). Thus, the discrete form of all the scalar transport equations
comprising the plasma fluid module have been provided.
The momentum transport equation governs the behavior of the vector plasma flow as
given in Eq. (3.27). It is restated here for convenience in comparison in Eq. (4.23)
∂v 1
+
∂t ρ

· (ρvv) = −

1
1
p−
ρ
ρ

·τ +g·v
ˆ
ˆ

(4.23)

Choosing the vertical flow component for its inclusion of the gravity dependence, Eq. (4.23)
may be expanded into the individual vertical component flow terms as in Eq. (4.24)
2
1 ∂ (rρvr vz ) 1 ∂ ρvz
∂vz
+
+
∂t
rρ
∂r
ρ ∂z

=−

1 ∂p
1 ∂ (rˆzr ) 1 ∂ τzz
τ
ˆ
−
+
− vz g
ρ ∂z rρ ∂r
ρ ∂z

(4.24)

where g is the standard acceleration magnitude. The various terms in Eq. (4.24) must now
be condensed into their generic calculation forms for discretization reporting. The time rate
of change in the vertical flow must remain, as it will be required in the solution scheme. This
leaves three sets of vertical and two radial derivative calculations, each showing the same
form related to the derivative calculation. Thus, these divergence terms may be condensed
and represented by two divergence terms in either direction (ˆ or z ). Finally, the gravitational
r ˆ
term remains as it is unaltered by a spatial operator. Thus, Eq. (4.24) may be written in its
representative operator form as in Eq. (4.25)
∂vz
1 ∂ (rαvz ) 1 ∂ (βvz )
=−
−
− vz g
∂t
rρ ∂r
ρ ∂z

(4.25)

where α and β are generic scalar macroscopic variable sampled at the mesh nodes and element
81

centers, respectively. In this way, Eq. (4.24) may be discretized in a minimal amount of terms.
This is given in Eq. (4.26) for a generic horizontal edge index and scalar quantities defined
above as given in Eq. (4.26)
−v
v
1
z,i+ 1 ,j
z,i+ 2 ,j
2
=−
r
tn − tn−1

ri+1 αi+1,j vz,i+1,j − ri αi,j vz,i,j
¯
¯
1
ri+1 − ri
1ρ
i+1 2 i+ 1 ,j
 2

¯
¯
β 1
1 vz,i+ 1 ,j+ 1 − βi+ 1 ,j− 1 vz,i+ 1 ,j− 1
1
 i+ 2 ,j+ 2
2
2
2
2
2
2
−


ρ 1
z 1 −z 1
j− 2
i+ 2 ,j
j+ 2
−v
1 g
z,i+ 2 ,j

(4.26)

where vz is the vertical flow component at the current time sample, tn , vz is the vertical flow
at the previous time sample, tn−1 , and vz denotes interpolated vertical flow samples at non¯
mesh edges. These interpolated, or translated, samples are required in order to calculate the
divergence terms in Eq. (4.26). However, the vertical flows themselves do not exist at these
locations. Instead, weighted averages of the flow components will be calculated at nodes
and element centers via bilinear interpolation and used in the update of these flow fields.
Eq. (4.26) now represents the discretized form of the two components of the momentum
transport equation (Eq. (3.27)) used to update these quantities within the plasma fluid
module.

4.5.2

Boundary Conditions

The boundary conditions must now be expressed in numerical form. These will be listed for
the scalar solution variables first, then the vector quantities. Just like any other distribution
in cylindrical coordinates, the boundary condition for all variables at the axis of symmetry
must be treated. However, this boundary is explicitly treated in the numerical representation
of scalar distributions by ignoring the flux across this corresponding edge. This assumes

82

h

N−3
2

x

h

N−1
2

N −1

x

N

Figure 4.8: One-dimensional mass transport boundary condition schematic for right-edge
domain boundary.

a zero-net flux, and thus a symmetric distribution, regardless of the physical variable in
question.
There is a single boundary condition governing the termination of the mass transport equation at the domain bounds. It is a finite-valued Neumann condition as given
in Eq. (3.40). The mass transport boundary condition involves a gradient in the mole fraction at the domain boundary. Its exact numerical representation is impossible, since the
elements outside the domain are not available. Instead, this boundary condition is numerically approximated by translating the boundary to the nearest two elements contained in
the domain mesh approaching the boundary. A generic boundary element with a domain
border located along its right edge, denoted by the index pair, N − 1 , j + 1 , is used here
2
2
although its form is easily translated to any boundary orientation. This right vertical edge
boundary condition is represented in the one-dimensional schematic in Fig. 4.8.
The open circles in Fig. 4.8 denote the element centers, while the filled circles are the
nodes. Note the right-most node at the coordinate N is lying completely on the boundary.
In order to exactly represent this boundary, the derivative should be calculated at N . This
is impossible since the calculation of a numerical derivative requires at least two samples on
either side of the derivative evaluation location. Thus, a more reasonable calculation would
be to calculate the derivative approaching the boundary as between N and N − 1. However,
this is also not possible, since the numerical samples are located at the element centers and
not the nodes. This boundary condition may instead be calculated using the two nearest
element centers via a translation algorithm derived below.

83

The original Neumann boundary condition calculation using the node positions would be
calculated according to
A=

AN − AN −1
=γ
xN − xN −1

where A is some scalar variable defined at the nodes, x is a continuous spatial variable,
and γ is some non-zero scalar quantity. As argued earlier, these A values at the nodes are
unavailable, and so must be translated to the boundary. This requires calculating these two
values using the element-based samples via

1 − AN − 3
 N−2
2 x −x
+
AN = A

1
N
x
N−1
N−2
1 − xN − 3
2
N−2
2


A

A
AN −1 =

1 + AN − 3
N−2
2
2

where the average of the adjacent element-based samples results in AN −1 and the slope of
the continuous A variable is used to interpolate its value on the boundary node. Plugging
these equations in to the original Neumann boundary condition equation results in the generic
discretized form of the Neumann boundary condition expression for the scalar transport
equations given in Eq. (4.28)



xN −x
1
1
N−2
N−2
1
−A
1

A
3 2+x
1 2+x
−x
−x
1 N−3
1 N−3
N−2
N−2
N−2
N−2
2
2
=γ
xN − xN −1




xN −x

(4.28)

The above expression is valid for non-uniform mesh spacings. However, in the case of a
uniform mesh, Eq. (4.28) collapses to the standard numerical derivative of the form
A
−A
N−1
N−3
2
2
x
−x
3
N−1
N−2
2

84

(4.29)

where it is now assumed xN − xN −1 = x
−x
. With the general numerically
N−1
N−3
2
2
discretized form of the Neumann boundary condition for scalar transport equations available,
their exact forms may be expressed. This is represented in Eq. (4.30) for the case of a generic
mass transport boundary condition with a domain boundary again at its right edge


x
rN − r
1
1
1
N−2 
s,i+ 2 ,j+ 2 s,N − 1 ,j+ 1  1
2
2  +
−

rN − rN −1
2 r
1 − rN − 3
N−2
2

D
x
rN − r
1
1
s,i+ 1 ,j+ 2 s,N − 3 ,j+ 1  1
N−2 
2
2
2  +

rN − rN −1
2 r
1 − rN − 3
N−2
2
D

=W

ˆ
1
1 + νcat,s,i+ 1 ,j+ 1 Ms,i+ 1 ,j+ 1
surf,s,i+ 2 ,j+ 2
2
2
2
2

(4.30)

where the diffusion coefficient, Ds , has been applied at the element center. This usage agrees
with the above derivation, as the original derivative is calculated across the element and the
diffusion coefficient is necessarily located at the element midpoint. Eq. (4.30) represents the
entirety of the boundary conditions related to the mass transport equations for each species.
The discrete boundary condition representing the electron transport equation results in
the simple condition of setting the electron temperature at the boundary element equal to
its neighboring element as given in Eq. (4.31)

T
=T
3
1
e,N − 2 ,j+ 1
e,N − 2 ,j+ 1
2
2

(4.31)

This falls out from Eq. (4.28) for a zero-condition Neumann boundary.
Finally, total energy transport equation must be discussed. The solution to this equation
results in an updated gas temperature distribution, which is explicitly set at all material
interfaces. This even holds for the substrate thermal model, where the energy fluxes to
the substrate will be used to solve for the temperature at this boundary. However, within
the plasma fluid module, the gas temperature is still set here. Representing a finite-valued

85

Dirichlet boundary condition, the gas temperatures at the node boundary is set via

Tg,N = γ
where γ is a non-zero, finite temperature. However, just like the scalar Neumann boundary
condition derived above, the temperature is not sampled at the desired node location, but
rather the element. Again referring to Fig. 4.8, the scalar quantity at this node may be
interpolated via the following

1 − AN − 3
 N−2
2 x −x
AN = A

1 +x
N
N−1
N−2
1 − xN − 3
2
N−2
2


A

which results in the following equation representing the Dirichlet boundary condition for the
gas temperature at the right edge domain boundary



N−1 
N − 1  Tsub , substrate


2 −T
2 =
T

1 1 + x

−x
g,N − 3 x
g,N − 2
600,
1 − xN − 3
2

N−2
N−1
N−3
else
2
2
2


xN − x





xN − x



(4.32)
where Tsub is the gas temperature profile provided by the substrate thermal model at all
substrate boundaries. The 600 K temperature that is assumed on all boundaries follows
from previous simulations and experimental observations [16].
The momentum transport equation boundary conditions are divided into two categories
of orthogonal and tangential flows. Orthogonal flow boundaries denote those mesh edges
where the flow component is directed orthogonal to the boundary edge. In the case of a
non-inlet or outlet region, which occurs at all physical material interfaces, this flow is forced
to zero.
A small feed gas inlet is present in the physical MSU MPACVD reactor toward the bottom
of the reactor and on the outer coaxial block wall as seen in Fig. 2.1. For simplicity, this inlet
86

is approximated by setting the entire numerical domain boundary resting at the bottom of
the simulation domain to the inlet region. This also ensures laminar flow. This change in
inlet location has little effect on the overall flow profile. With the inlet location specified, the
flow itself must now be calculated. This corresponds to the vertical flow component due to
the inlet region being located on horizontal edges. Experimentally, the inlet flow is set by a
flow controller external to the system to a specified flow rate, Qf . Typical experimental flow
rates are often between 100 to 500 standard cubic centimeters per minute, or sccm. However,
it is first corrected for the pressure difference between standard and operating pressure. This
flow rate must also result in a speed by converting from a volumetric quantity to speed via
the following calculation
vin =

Q f p0
6 × 107 pA

(4.33)

where vin is the inlet region speed, Qf is the specified flow rate at the inlet region, p0 is
standard atmospheric pressure, and A is the total numerical surface area covered by all inlet
edge segments. The numerical factor in Eq. (4.33) is due to the conversion from cm3 /min
to m3 /s. Once this inlet speed has been calculated, the vertical component of the vector
average plasma velocity is explicitly set at the generic inlet edge segment via

v
1 = vin
z,i+ 2 ,j

(4.34)

where i + 1 , j represents the generic mesh indices of an inlet boundary horizontal mesh edge.
2
An outlet region must physically exist to avoid gas and momentum accumulation. The
MSU MPACVD reactor has outlet gas flow regions located toward the outer radius of the
substrate as seen in Fig. 2.1. This location of the outlet region is maintained numerically.
Likewise, the speed of the vertical velocity component must be calculated and forced during
simulation. Unlike the inlet flow region, this outlet flow speed is not set or known. Instead,
the only specification is the avoidance of flow accumulation. To ensure this, the outlet
speed is once again calculated using Eq. (4.33) with the surface area now representing the
87

6

N−3
2

6

N−1
2

N −1

6

N

N+1
2

Figure 4.9: Tangential velocity component sample locations surrounding the domain boundary.

total outlet region surface area. This results in vout , and the final boundary condition for
orthogonal flows
v
1 = vout
z,i+ 2 ,j

(4.35)

where vz is now within the outlet region.
The tangential velocity components lying adjacent to mesh boundaries must also be
considered, but do not physically lie on the domain boundaries. These tangential velocity
boundary components are provided in Fig. 4.9.
The physical tangential velocity at any material interface is necessarily zero. This is
observed experimentally and is referred to as the no-slip or non-slip boundary condition. It
can be deduced logically when considering the bulk flow of a fluid in the following argument.
As a fluid flows across a surface, the frictional forces at the interface counteract the fluid flow
on a microscopic length scale. This prohibits the fluid motion, resulting in zero tangential
flow, and thus no-slip boundary conditions. This effect propagates away from the interface,
resulting in a parabolic speed profile directed orthogonal to this surface.
Numerically, this no-slip boundary condition is not necessarily enforced for all systems.
In fact, the opposite, or driven, system will be used in the validation of the fluid model.
No-slip boundary conditions may be approximated by forcing the flow at the index N to
zero. However, the tangential flow component numerical sample is not located here. Instead,
the no-slip boundary condition will be enforced during the simulation by applying a zeroed
Dirichlet condition at the nearest tangential flow samples.

88

4.5.3

Matrix Representation

The generic spatial operator, flux, and source terms representing the scalar transport equations as given in Eq. (4.22) may now be converted to matrix equations. That is, these
governing equations are now translated to a generic matrix equation form making their
simultaneous solution possible.
The solution variable in Eq. (3.18) is the species molar fraction. This equation is solved
for all non-molecular hydrogen and electron species contained in the hydrogen gas mixture.
A diffusion flux quantity operates on the gradient of these solution samples, translating to
their matrix formulation and representation. As a result, Eq. (4.22) may be written in matrix
form as
As xs = [ws + vs ]

(4.36)

where As is a square matrix representing the combined numerical diffusion flux operator for
species s, xs is the solution vector containing the unknown molar fractions, ws is a vector
containing the net rate production terms, and vs is the vector containing the net rate of
mass flux due to convective forces via the average gas flow. Thus, the diffusion flux operator,
As , represents the calculation of

As ≈

· (ρs Ds )

(4.37)

where the right side of Eq. (4.37) operates on the distribution of mole fraction samples. The
numerical calculation of the entries for the last term, vs , are calculated by summing the
last two lines of Eq. (4.22) related to the convective mass flux dependence. As a result, the
k th row entry in vs representing the net flux of mass density due to convective forces is

89

calculated
ri ρ
ri+1 ρ
v
1v
1
s,i,j+ 1 r,i,j+ 2
s,i+1,j+ 2 r,i+1,j+ 1
2 −
2
vs,k =
r 1 ri+1 − ri
r 1 ri+1 − ri
i+ 2
i+ 2
ρ
ρ
v
1
1 v
1
s,i+ 2 ,j+1 z,i+ 2 ,j+1
s,i+ 2 ,j z,i+ 1 ,j
2
+
−
zj+1 − zj
zj+1 − zj

(4.38)

where vs,k represents the k th row entry corresponding to the generic spatial mesh index
pair i + 1 , j + 1 . In this way, the flux term is not a direct update scheme, but a summation
2
2
over all mesh edges corresponding to each particular element. This is different than the net
rate production of mass vector, which simply requires a direct filling of the vector entries.
Rearranging for the convective flux term on the right side of Eq. (4.36) is possible due to
its interdependence from the solution molar fractions. That is, the physical dependencies
in Eq. (4.38) are purely based on previously updated physical quantities. This allows for
the source term to be calculated ahead of time and added as a vector term to the right side
of Eq. (4.36). Once again, the matrix entries for As are given in Eq. (4.22) as the terms
multiplying the solution molar fractions.
Likewise, the electron and total energy equations may also be represented in matrix
equation form as given in Eq. (4.39) and Eq. (4.39b)

Le te = [ke + ue + pe + τe + ge + P + he + q]

(4.39a)

Lg tg = le + lv + p + g + τ + k + u + h + P + qrad

(4.39b)

where Le and Lg square matrices representing the thermal conductivity matrix operators,
respectively, and te and tg are the solution vectors containing the unknown electron and
gas temperature samples, respectively. The remaining vectors represented on the right side
of Eq. (4.39) represent the various fluxes and source terms listed in order and described previously in their respective energy transport equations. As with the matrix representation of
90

the mass transport equation, some of these right side vectors are precomputed and represent
fluxes across mesh edges. Other vectors on the right side of Eq. (4.39) are source terms that
are simply translated to vector form without any computation required. However, unlike the
mass transport equation matrix representation, the potential energy terms on the right side
of Eq. (4.39) do in fact depend on the solution temperature samples. This would normally
require the flux dependencies of this potential energy term to be represented within the
square, left side matrix operating on the solution vector. However, the influence of these
potential energies on the solution temperatures are known to be insignificant compared to
the fluxes via thermal conduction. Thus, their contribution is removed from the operating
matrix and instead represented in the right side source vectors. This results in a faster numerical matrix calculation and filling, and provides a much simpler formulation. The matrix
equation representations for the scalar transport equations are now available in Eq. (4.36)
and Eq. (4.39).
Since the momentum transport equation is nonlinear, it is not solved via a simple matrix
solution as with the scalar transport equations. Instead, its solution is more complex, requiring intermediate matrix representations. This matrix representation and filling, along with
the momentum transport equation numerical solution, is derived in the following paragraphs.

4.5.4

Solution Procedure

Separating the average flow velocity into its two radial and vertical components results in
a set of five separate transport equations governing the plasma fluid module. Furthermore,
the mass transport equation must be solved for all non-molecular hydrogen and non-electron
species included in the simulation. Instead, these masses are updated via the charge and
species conservation equations. With the resulting seven hydrogen species included in the
present simulation, this results in a set of eleven governing equations which must be simultaneously updated. The choice of numerical solution procedure for this set of equations is
discussed and derived in detail in the following paragraphs.
91

The most important criterion for choosing a solution method is accuracy of the overall
simulation. Second, the numerical procedure should be generalizable to the physical structure
of the numerical mesh and its representative MPACVD system. Finally, the solution method
should be numerically efficient and scale reasonably with numerical size. This suggests a near
O(N ) scaling metric for the solution method.
At this point, a distinction should be made between the two classes of governing transport equations in the plasma fluid module. These classes are divided according to their
linear or nonlinear representations. In particular, the scalar transport equations form linear
expressions, while the momentum transport equation is characterized as a nonlinear equation. This suggests these two classes of transport equations may be solved differently, as is
in fact the case here. Further discussion related to the reasoning behind this decision and
the particular numerical solution methods are provided in the following paragraphs.
The linear, scalar transport equations will be discussed first. As noted above, these
equations are characterized as linear, first order differential equations. This suggests their
numerical solution is attainable through simple matrix manipulation and linear algebra.
However, the system as a whole must first be considered before choosing a suitable numerical
solution method. That is, the complete set of eleven governing equations forms a highly
nonlinear system of simultaneous solutions. Even though individual equations are linear,
their solutions are interdependent and expected to exhibit nonlinear behavior. Moreover,
the physical constants, by which these equations are influenced, are often exponentially
dependent on several transport variables. This last fact also suggests nonlinear system
dynamics with regards to the overall solution behavior.
As a result, the sparse direct solver is not chosen as the numerical solution method for
updating the scalar transport equations. Its superior numerical precision is not advantageous
for a nonlinear system with rapidly changing and interdependent solutions. Moreover, its
computation time is still slightly more than that required by an iterative method. Instead,
an iterative numerical method is chosen to update the scalar transport equations. The choice

92

of which iterative solution method to use must now be made.
One promising nonlinear solution method is Broyden’s method, which guarantees convergence for a set of nonlinear equations. However, this similarly requires a large number
of numerical calculations, often resulting in an unreasonable computation time. Coupling
this with the need to update physical constants, the guaranteed convergence of the particular matrix equation does not guarantee overall system convergence or even a reasonable
solution. Thus, Broyden’s method was not employed for the solution to the scalar transport
equations.
Traditionally, Gauss-Seidel iterative solvers have been applied to solve steady MPACVD
systems [16, 32]. The Gauss-Seidel family of numerical solvers are based upon the guess and
check methodology, where an approximate solution is calculated and used as the guess in the
next iteration. Several implementations and variations of this method have been proposed
and developed over the years. One particularly popular implementation for the simulation of MPACVD systems is the Gauss-Seidel Line Relaxation method [16]. This numerical
solution method was first applied to a set of coupled velocity moment equations by MacCormack in the late 1980s [77]. The Gauss-Seidel Line Relaxation method essentially applies the
Gauss-Seidel method successively along each mesh traversal direction. It has proven highly
successful in solving the types of nonlinear systems of equations present when simulating
MPACVD systems. However, a rectangular, uninhibited grid is required for accurate numerical evaluation, meaning irregular or non-rectangular-shaped meshes are not allowed. This is
unacceptable in the current simulation, as irregularly shaped meshes are required in order to
accurately capture system geometry, and improve numerical efficiency and solution accuracy.
Instead, the Gauss-Seidel Successive Over-Relaxation (SOR) numerical solution method is
employed. SOR is a variation of the Gauss-Seidel solution method, the only difference being
the insertion of solution relaxation during the final update step. It was originally developed
as a numerical solution method in 1950 by Young [78]. Its advantage over Gauss-Seidel Line
Relaxation is its independence of the mesh structure required for solution. SOR often boasts

93

faster convergence rates than the standard Gauss-Seidel method depending on the choice of
this relaxation factor. Thus, SOR is chosen as the numerical solution method applied for
the scalar transport equations.
However, there are several disadvantages to SOR that must be pointed out. Firstly, even
though the convergence rate is often maximized during over-relaxation, under-relaxation is
utilized here to avoid divergence of the overall solution. This drastically reduces the effective
convergence rate of the numerical solution, but can often avoid its divergence depending
on the solution tolerance. That is, an accurate and stable multiphysical model is currently
preferred over a faster solution time. Improvements in the numerical solution time could
be addressed in future implementations of the present simulation by applying various other
iterative solution methods.
A second discussion point is the uniqueness of the overall numerical solution. That is,
with the set of coupled transport equations forming a nonlinear numerical system, the final
solution is not guaranteed to be unique. Instead, this may represent a local solution that
depends on the initial numerical and physical operating conditions. If these conditions, or
the solution procedure itself, are changed, a completely new solution may arise. Although
unlikely due to the multiphysics nature of the coupled solution between various physical
models and the aggressive under-relaxation imposed, large differences in the final solution
depending on these initial conditions could arise. However, little change in the final solution
is observed when altering the initial conditions under the same physical operating conditions.
Before the SOR solution algorithm is provided, the resulting matrix structure given
the choice of sampling scheme must be addressed. Using the finite-difference method, the
resulting matrix is penta-diagonal, or two-dimensional tridiagonal in structure. That is, the
main diagonal is always nonzero with two nonzero entries for each row on either side of
this main diagonal. This results from the second spatial derivatives in both directions, with
each element-based scalar solution quantity being related to its nearest four neighboring
scalar variables centered at their respective element centers. As a result, each matrix row

94

has at most five nonzero entries. These entries may be split into the r- and z -going terms
ˆ
ˆ
by separating out their specific terms and keeping track of the individual main diagonal
contributions. In this way, the SOR method as derived below may be applied twice over,
once in each direction.
The SOR method is derived in many texts and works, and so is only provided here in
final form for convenience. The numerical solution for any matrix equation containing a
tridiagonal structure may be calculated via the original, iterative Gauss-Seidel method via
forward substitution as given by

xn+1 =
i

1 
bi −
Ai,i

i−1

n

Ai,j xn+1 −
j


Ai,j xn 
j

(4.40)

j=i+1

j=1

where xn+1 is the ith solution vector entry at the (n + 1)th iteration step, Ai,j is the lefti
side matrix entry in the ith row and j th column, bi is the ith entry of the source vector,
and xn is the j th row entry of the previous solution vector. In this way, the unknown
j
solution vector is successively updated in terms of a previous solution. Upon starting the
solution procedure, typically a non-zero solution vector is assumed. The closer this initial
guess is to the final solution, typically the faster the overall solution is likely to converge.
The right-most term in Eq. (4.40) is then evaluated for the given solution vector row. Once
this backward substitution is calculated, the forward calculation of the second summation
term is evaluated.
The numerical update scheme presented in Eq. (4.40) forms the basis of the SOR method,
with only a slight modification. That is, the numerical solution via SOR is still updated as
given in Eq. (4.40), but now the solution is relaxed. A combination of the new and old
solution vectors now form the final updated solution vector to be used in the next iteration.
This may be expressed mathematically as
xn+1 = (1 − ω) xn + ωxi
i
i
95

(4.41)

where ω is the relaxation coefficient, and xi is the new solution calculated via the original
Gauss-Seidel method given in Eq. (4.40). Relaxation coefficients are typically in the range
0 < ω < 2, and heavily influence the convergence rate with these factors depending on the
properties of A. For example, setting ω = 1 clearly results in the original Gauss-Seidel
numerical solution algorithm. Moreover, setting ω > 1 denotes over-relaxation, while forcing
ω < 1 represents under-relaxation. Due to the stiffness of the present numerical model,
under-relaxation is utilized here. The end result is a new solution that is a combination of
the previous and current solutions. Thus, the full numerical form of the SOR method is
expressed


i−1

n



ω 
bi −
Ai,j xn+1 −
Ai,j xn 
xn+1 = (1 − ω) xn +
j
i
j
i
Ai,i
j=1
j=i+1

(4.42)

where xn+1 is the new solution.
i
The determination of these relaxation coefficients is important to the stability, efficiency,
and accuracy of the scalar transport equations, and the simulation as a whole. As stated
earlier, due to the nonlinear nature of the numerical system, these relaxation constants often
approach zero, again representing under-relaxation. They are updated during the solution
process itself, with increasing value if the current solution is converging and decreasing value
otherwise.
To begin, an initial relaxation coefficient on the order of 0.1% is applied to all scalar solution variables. During the solution process, a maximum scaling parameter is chosen to either
increase or decrease each individual relaxation coefficient. This value is typically around
10%, but can vary depending on the physical operating and initial numerical conditions.
The determination of whether the individual solution variable is converging or diverging is
calculated by comparing the difference between previous and current matrix equation residuals. If the current residual is less than the previous residual, the corresponding relaxation
coefficient is increased, otherwise, it is decreased. In this way, individual determination of
96

the relaxation parameters is made and typically avoids divergence of the complete solution.
At the start of the next complete iteration step, the same initial relaxation coefficient again
resets across all solution variables. A maximum relaxation limit is also specified, typically
less than one. This may be reached in the case of a steadily converging solution variable with
a large number of allowed iterations. Such a limit diminishes the possibility of a stable solution variable becoming unstable, or the multiphysics solution diverging if the electromagnetic
module is not promptly updated.
Finally, since the set of nine scalar transport equations are solved independently, each
has its corresponding relaxation coefficients. These individual relaxation coefficients are thus
tracked and adjusted according to convergence of the individual solutions. That is, the total
error associated with any one of these scalar transport equations may be calculated as

n+1 =
err

N |xn+1 − xn |
i=1 i
i
N |xn |
i=1 i

(4.43)

where n+1 is the total residual error associated with the present scalar transport equation
err
where xn+1 and xn are the current and previous solution vectors, respectively. The result is
an estimation of the convergence of the solution. That is, if n+1 < n , then the solution is
err
err
assumed to be converging, and its relaxation coefficient is increased. Instead, if n+1 > n ,
err
err
then its relaxation coefficient is increased, resulting in a greater dependence on the previous
solution. Limits must be imposed on these relaxation coefficients to avoid both convergence
that is too slow or too fast. The relaxation coefficient limits imposed here include a minimum
(and initial) of 0.005 and a maximum of 0.5. Such limits have been found previously to result
in an acceptable balance between efficient convergence and a stable overall solution [16].
Of course, since SOR is an iterative process, many iterations are required before the
solution is converged enough to consider it a final solution. Thus, the number of these iterations that are allowed, and also necessary, must be discussed. At this point, the numerical
solution to the linear matrix equations expressed in Eq. (4.36) via the SOR method have

97

been presented. Thus, the nonlinear transport equation numerical solution procedure must
now be treated.
The momentum transport equation contains nonlinear dependencies on the flow components related to convection influences. As a result, its numerical solution procedure is not
expected to remain efficient or even stable under the SOR numerical method applied in the
case of the nonlinear transport equations. In fact, even though SOR could be applied in this
case, its solution is not expected to converge efficiently, and may even diverge depending
on the relaxation parameter. Thus, the SOR method is not applied to the solution of the
momentum transport equation. Instead, the Chorin Projection method is used to solve the
momentum transport equation.
The Projection method was first proposed and developed simultaneously, yet independent
of each other, by Chorin and Temam in the 1967 [79]. It is an implicit update scheme
that splits the nonlinear equation solution into intermediate implicit and explicit solution
components tracking individual physical forces. These intermediate expressions must be
solved in succession for each iteration, and require either matrix solutions or explicit solution
projections. The result is a stable solution to the nonlinear momentum transport equation
in a reduced number of iterations compared to purely explicit methods. A more detailed
derivation of its solution procedure is presented below.
The momentum transport equation derived in Eq. (3.27) is once again provided for convenience in Eq. (4.44)
∂v 1
+
∂t ρ

· (ρvv) = −

1
1
p−
ρ
ρ

·τ +g·v
ˆ
ˆ

(4.44)

which may be split into its linear and nonlinear dependencies as in Eq. (4.45)
v − vn
1
=−
δt
ρ

98

· (ρvv)

(4.45a)

1
1
v n+1 − v
=−
p−
δt
ρ
ρ

·τ +g·v
ˆ
ˆ

(4.45b)

where v n+1 , v , and v n are the new, intermediate, and previous vector average plasma flow
distribution solutions, respectively, and δt is the discrete time step. These two expressions
in Eq. (4.45a) and Eq. (4.45b) are sometimes referred to as the predictor and corrector
steps, respectively. Clearly, Eq. (4.45a) contains the nonlinear dependencies and Eq. (4.45b)
contains the linear dependencies. The gravitational dependence in Eq. (4.45b) is moved to
Eq. (4.45a) to group explicit calculation terms as will be derived below. Finally, the viscous
and pressure forces are split once more, resulting in a series of three intermediate solution
steps representing the standard Projection method algorithm in Eq. (4.46)
1
v − vn
=−
δt
ρ

· ρv n v n + g · v n
ˆ

1
v −v
=−
δt
ρ
v n+1 − v
δt
where v

=−

(4.46a)

·τ
ˆ

(4.46b)

1 n+1
p
ρ

(4.46c)

is a second intermediate flow solution vector. Thus, moving forward linearly in

time results in the following relationship between vector flow solutions
v n → v → v → v n+1 ,

tn → tn+1

(4.47)

where tn and tn+1 are the times at the numerical solution samples corresponding to the
previous and current solutions, respectively. Note in Eq. (4.46) the time-dependence and
solution time associated with all important spatial quantities and physical constants have
been explicitly provided. This will be important when deriving their respective numerical
solution procedures. Also note the density term in all expressions in Eq. (4.46) are calculated
at the previous time step, n, and are not expected to change significantly over one numerical
time step. As stated previously, Eq. (4.46) represents the set of three intermediate expressions
99

that must be solved at each successive iteration during the Projection method solution.
Explicit time stepping is used to step forward the numerical solution. As a result, the
CFL condition must again be satisfied, with the time step now dependent on the magnitude
of the flow speed.
Eq. (4.46a) contains the nonlinear flow dependence term represented by the vv tensor
product. An explicit numerical solution is applied to this expression, resulting in an updated
intermediate flow quantity, v . The gravitational force is also included in the explicit velocity
update, as its flow dependence is due to the previous solution at time n. Thus, the solution
to the explicit equation given in Eq. (4.46a) may be translated to discretized form as given
in Eq. (4.48) for the generic radial and vertical flow components
n
v
1 = vr,i,j+ 1
r,i,j+ 2
2



v2 1
v2 1
−ρ 1
1
1
1
i− 2 ,j+ 1 r,i− ,j+ 1 
∆t  i+ 2 ,j+ 2 r,i+ 2 ,j+ 2
2
2
2

−


ρ
r 1 −r 1
i,j+ 1
i+ 2
i− 2
2

−

∆t
ρ

i,j+ 1
2

n
v
1 ,j = vz,i+ 1 ,j
z,i+ 2
2 

ρ

ρi,j+1 vr,i,j+1 vz,i,j+1 − ρi,j vr,i,j vz,i,j
zj+1 − zj


v2 1
−ρ 1
v2 1
1
1
1
i+ 2 ,j− 1 z,i+ ,j− 1 
∆t  i+ 2 ,j+ 2 z,i+ 2 ,j+ 2
2
2
2

−

ρ 1 
z 1 −z 1
j− 2
i+ 2 ,j
j+ 2
ρi+1,j vr,i+1,j vz,i+1,j − ρi,j vr,i,j vz,i,j
∆t
−
− gv
ρ 1
ri+1 − ri
z,i+ 1 ,j
2
i+ 2 ,j

(4.48a)

ρ

(4.48b)

where the discretized forms of the average plasma flow vector components within the parentheses are all assumed to represent the previous solutions at time step n. Note, the normal

100

and shear terms, along with the densities, are thusly represented at the element centers and
nodes, respectively for the modifying terms in Eq. (4.48). This suggests the interpolation
of average flow quantities to these non-standard mesh locations. As with previous translation operations, this is performed using bilinear interpolation to nodes, whereas a simple
averaging scheme suffices for element-based quantities. Eq. (4.48) represents the first intermediate numerical solution update for the Chroin Projection method for the two velocity
components.
The second step is to update the next intermediate flow solution taking into account
the viscous forces as given in Eq. (4.46b). To begin, Eq. (4.46b) is expanded into its two
orthogonal velocity components, its viscosity tensor expanded as in Eq. (3.30) in terms of
these components, and rearranged as given in Eq. (4.49)
∂vz
∂vr 2 1 ∂
δt 1 ∂
−
(rvr ) +
r −µ 2
vr = vr −
ρ r ∂r
∂r
3 r ∂r
∂z
δt ∂
∂vr ∂vz
+
−µ
+
ρ ∂z
∂z
∂r

∂vr
δt 1 ∂
+
vz = vz −
r −µ
ρ r ∂r
∂z
δt ∂
2
∂vz
+
−
−µ 2
ρ ∂z
∂z
3

(4.49a)

∂vz
∂r
∂vz
1 ∂
(rvr ) +
r ∂r
∂z

(4.49b)

where all velocity components are assumed to be sampled at the t intermediate time step
unless otherwise noted. The discrete form of Eq. (4.49) is complex and rather lengthy,
and is derived separately in Appendix B. It is clear these expressions form a set of coupled
equations, suggesting their simultaneous solution in a block matrix form, which may be
formulated in terms of generic matrix expressions of the form






 I 0 

−
0 I

∆t
ρ

 







 Arr Arz   vr   vr 

 
=

Azr Azz
vz
vz

101

(4.50)

where I is the identity matrix, 0 is a square matrix of size n × n containing all zeros, and
A represent the block matrices corresponding to the spatial operators containing both the
gradient and divergence operators as well as the viscosity dependence.
Thus, the matrix equation in Eq. (4.50) must be solved in terms of the intermediate
velocity component sample vectors, vr and vz . The A matrices in Eq. (4.50) are rather
sparse, with a maximum of nine non-zero entries per matrix row. This makes the use of
the sparse direct solver attractive. Thus, Eq. (4.50) is solved for the second intermediate
velocity components via the sparse direct matrix solver.
Finally, the pressure correction must be applied to arrive at the full time step and final
solution. This is done by first rearranging Eq. (4.46c) in terms of velocity solution at the
full time step
v n+1 = v −

δt n+1
p
ρ

(4.51)

At this point, both the velocity and pressure samples at the full time step are unknown,
resulting in an under-defined equation. To avoid this, the divergence of all terms may be
applied, resulting in Eq. (4.52)
· v n+1 =

· v − δt

·

1 n+1
p
ρ

(4.52)

where notice the time step has been removed from the spatial divergence operator, but the
density dependence remains within this operator. In the case of the incompressible flow
assumption applied here, momentum cannot be created nor destroyed. Thus, this left side
term must vanish since
·v =0

(4.53)

leaving the following implicit expression governing the pressure correction
1
δt

·v =

102

·

1 n+1
p
ρ

(4.54)

Eq. (4.54) is first expanded into its two velocity and spatial divergence components as given
in Eq. (4.55)


1  1 ∂ rvr
δt r ∂r


∂ vz
n+1
∂ 1 ∂pn+1
 = 1 ∂ r ∂p
+
+
∂z
r ∂r ρ ∂r
∂z ρ ∂z

(4.55)

and then expressed in discrete form at a generic index pair i + 1 , j + 1 via
2
2

1 

δt 

ri+1 v




− ri v
v
−v
1
1
r,i+1,j+ 1
r,i,j+ 2  1  z,i+ 2 ,j+1
z,i+ 1 ,j 
2
2 
+ 
 δt 

ri+1 − ri
zj+1 − zj



=



n+1
ri+1 pn+1
3 ,j+ 1 − pi+ 1 ,j+ 1
i+ 2
2
2
2







r 3 −r 1
i+ 2
i+ 2


n+1
ri pn+1


1 ,j+ 1 − pi− 1 ,j+ 1


i+ 2
2
2
2


−



ρ
1 ri+1 − ri ri+ 1 − ri− 1
i,j+ 2
2
2

n+1
pn+1

1 ,j+ 3 − pi+ 1 ,j+ 1

i+ 2
2
2
2
+


ρ 1
z
− zj
z 3 −z 1
j+ 2
i+ 2 ,j+1 j+1
j+ 2




−



ρ

1
i+1,j+ 2

ri+1 − ri



n+1
pn+1
1 ,j+ 1 − pi+ 1 ,j− 1
i+ 2
2
2
2

ρ

i+ 1 ,j
2

zj+1 − zj

z 1 −z 1
j+ 2
j− 2










 (4.56)



Once again, the density terms must be translated to the mesh edges as seen from the nomenclature and sample locations in Eq. (4.56). Finally, Eq. (4.56) may be expressed as a matrix

103

equation of the form








1  vr 
 Ar  n+1
=


p

∆t
vz
Az

(4.57)

where Ar and Az represent the spatial second derivative operators in the r and z directions,
ˆ
ˆ
respectively, and ∆t is the full numerical time step. These A matrices also contain the
density dependence as well. Eq. (4.57) is now solved for the pressure at the full time step
once again using the sparse direct solver.
However, the boundary conditions related to the pressure must first be addressed. The
velocity dependence on the pressure is related to the gradient of the pressure distribution.
Thus, to remain consistent with the zero normal flow at material and domain boundaries,
the normal gradient of the pressure must also remain zero at these boundaries. This results
in a zeroed Neumann boundary condition at all material and domain boundaries, which is
mathematically represented as
pn+1 · n|bc = 0
ˆ

(4.58)

where ·ˆ represents the unit vector directed orthogonal to all boundaries.
n
Finally, once the full time step pressure has been updated, the full time velocity must
also be updated. This is done by reapplying Eq. (4.46c) and updating all non-boundary
velocity components using the current pressure solution and the previous velocity solution.
The end result is an updated average vector plasma velocity solution at a full time step.
At this point, the momentum transport equation error is calculated using Eq. (4.43), and
the maximum allowed time step calculated. The previous solution algorithm represents
one complete solution cycle over a single time step. In practice, this algorithm is repeated
until a given tolerance is reached. This tolerance can vary on the size of the mesh and the
nonlinearity of the system, but a maximum error limit of 0.001 typically results in a good
balance between final solution accuracy and numerical efficiency. This convergence level is
set by the user in the current simulation. The momentum transport equation update may
also be terminated after a maximum number of full time step solutions, which is also set by
104

the user.

4.5.5

Solution Algorithm

The plasma fluid module solution algorithm is now discussed and presented. Its major
components, the order of the various solutions, and a discussion on these considerations
follows.
Before the plasma fluid module begins, the absorbed power density is passed from the
electromagnetics solution module. Initialization of the plasma fluid module requires the
passing of the various scalar and vector transport equation solutions, along with the corresponding plasma physical properties, from the previous solution.
If this is the first complete iteration of the numerical simulation, then the plasma fluid
module has not been run before. These solution variables must then be chosen to reflect
some non-zero distributions in order to avoid trivial or non-physical numerical solutions.
For example, all mole fractions are chosen as uniform, low-value approaching machine precision. This typically results in all mole fractions of constant 10−14 everywhere, except for
the molecular hydrogen, which is nearly one. The plasma flow fields are chosen to be zero
everywhere except at the inlet and outlet regions. Finally, the gas temperature distribution
is solved via the steady heat flow equation given the substrate and quartz boundary temperatures outlined previously. Since it is known to generally result in higher temperatures, the
electron temperature is set as uniformly five times this gas temperature distribution. If this
is not the first complete solution iteration, then the plasma fluid module has been previously
updated, and the previous solutions that were stored at the end of the previous update are
read from file.
Next, the physical constants are updated according to these initial distributions. This
includes the calculation of the corresponding physical quantities related to plasma composition such as molar, mass, and number concentrations. Once the physical constants have
been updated, the transport equations must be solved.
105

The momentum transport equation is solved first, and is expected to exhibit less coupling
dependence on the other transport equations. This is primarily due to its dependency on
only volumetric mass density and viscosity and not the scalar transport variables (including
electron temperature). That is, the electron temperature is known to be tightly coupled
to the other transport quantities and is often the underlying influence of nonlinear effects.
Moreover, its implicit numerical solution procedure typically results in a more stable solution
compared to the iterative processes for the scalar transport equations. Thus, its solution
is only updated along with the physical constants. However, the momentum transport
equation solution procedure does require convergence to a desired tolerance. Thus, it is
solved concurrently until the tolerance level is reached, resulting in a converged average
vector plasma velocity distribution. More on this choice of solution order and number will
be discussed below.
After the average vector plasma velocity is updated by solving the momentum transport
equation, the scalar transport equations must be solved. These equations are solved in the
following order. First, the mass transport equations for all non-molecular hydrogen and
electron species are solved. Then, the electron energy equation is solved, and finally the
total energy transport equation. However, all molar fraction and temperature dependencies
in all of these transport equations are based upon the previous solutions. That is, the
molar fraction dependence in the energy transport equations related to enthalpy diffusion
are taken as the previous mole fraction solutions, and not those current. This ensures
the energy transport equations are solved using the physical constants associated with the
previous transport quantities, and also helps to avoid a divergent solution.
These scalar transport equations are numerically updated using the SOR method described earlier. Their solutions are then relaxed according to their individual relaxation
rates, and their residuals along with the total system error, are calculated. This completes
one iterative numerical solution cycle, which is repeated for a set number of iterations as
set by the user. After the maximum number of iterations has been reached, the physical

106

constants are updated and the momentum transport equation is once again solved. These solution iterations are often referred to as the inner iteration loop structure within the plasma
fluid module. Change in the electrical conductivity profile is also tracked as a total error
metric when comparing to the distribution upon initialization to the electromagnetics module. This updating of physical constants and transport variables continues until a maximum
number of iterations, or the maximum error allowed between successive electrical conductivity distributions, have been reached. These iterative updates and solutions are collectively
referred to as the outer iteration loop. At this point, the physical constants and electrical
conductivity are once again updated and passed to the electromagnetics module.
The choice of number of allowed iterations related to both the scalar transport equation
solutions, and the maximum physical constants and transport equation resets is crucial to
the stability of the numerical simulation. It essentially represents a balance between allowing
the numerical solution to converge toward a solution in an efficient manner, with maintaining
a simultaneous solution between different physical modules. That is, the choice of maximum
number of inner plasma iterations provides a balance between convergence in the numerical
SOR solution with the divergence in the transport quantities before updating their related
physical constants. Moreover, the choice of maximum outer iterations relates the need for efficient solution convergence with the divergence of the electrical conductivity. Large changes
in the electrical conductivity will inevitably influence the absorbed power density profile and
thus, the overall solution. These outer and inner iteration settings will depend on not only
the numerical mesh and its resolution, but also the initial plasma solution conditions, and
the physical system operating conditions. Typical maximum iteration numbers which have
proven in the past to strike a good balance between numerical efficiency and the avoidance
of divergent numerical solutions are in the range of 5 to 20 for both settings. The maximum
error or residual between successive electrical conductivities should also be set accordingly,
with typical relative errors between 1% to 10%. Of course, again, this will depend on the
particular system to be simulated. A visual depiction of the plasma fluid module solution

107

algorithm in flowchart form is provided in Fig. 4.10.
Pabs
no
Momentum
Transport

Physical
Constants

Initialize

Converge?

no

σe
ˆ

yes

yes

Converge?

yes

Converge?

Scalar
Transports
no

E/M Module
Figure 4.10: Plasma fluid module numerical solution flowchart .

The final step in the plasma fluid module solution algorithm is to calculate the complex electrical conductivity distribution. This is the main coupling mechanism going from
the plasma fluid module to the electromagnetics solution module. From Eq. (3.48), it is
clearly dependent on primarily scalar transport and related quantities. That is, the vector
flow dependence in the electrical conductivity is only present due to the effective collision
frequency given in Eq. (3.49). Thus, the electrical conductivity is calculated at the element
centers with little interpolation required. As a result, the edge-based plasma velocities are
averaged at the element centers given the values at the four corresponding edges. Once the
conductivity is calculated, it is passed to the electromagnetics module and used in both the
material dependence and the absorbed power density calculation. This concludes the plasma
fluid module solution algorithm.

108

4.6

Substrate Thermal Module

The numerical solution and update of the substrate thermal model is finally visited. This solution results in a physical temperature distribution across and within the substrate pedestal.
It is coupled to the gas temperature at the substrate surface from the plasma fluid module.
In this way, these two modules form a coupled, multiphysical numerical solution resulting in
a more accurate representation of the reactor operation and characteristics.
The numerical sample locations with respect to the mesh must first be defined. The
relationship between the temperature and the material properties of the substrate suggest
sampling these offset from each other. That is, it is numerically advantageous to sample the
temperatures within the substrate one-half mesh cell removed from the physical material
assignments. Keeping with the standard of sampling scalar solution quantities at mesh
centers as seen in the plasma fluid module, the substrate temperature is also located at the
mesh element centers. This leads to the assignment of material characteristics at the mesh
edges. Although material assignments are optimally located at the element centers, these
are readily interpolated to the mesh edges.
The substrate temperature module is solved completely within the substrate region. This
includes the substrate itself, the outflow gas region, and the outer substrate pedestal quartz
wall as seen in Fig. 2.1. Once again, the numerical approximation and discretization of the
solution domain is critical to the accuracy of this module in representing the physical system.
In this case, the choice of mesh is rather straight forward with very little approximation due to
the absence of curved material interfaces. As such, the FD sampling scheme is aptly suited
to represent this rectangular domain. An example of the substrate temperature module
solution domain is provided in Fig. 4.11 for reference.
Although completely nondescript, the numerical sampling mesh shown in Fig. 4.11 is
bounded on its top and outer right edges by the plasma domain from 4.7. Moreover, the
uniform mesh discretization rate is again chosen for simplicity, with nonuniform mesh spacing

109

(a)

(b)

Figure 4.11: Substrate temperature module solution domain (a) sampling mesh and (b)
material mapping for the MSU MPACVD Reactor B.

110

being equally applicable to the solution method.

4.6.1

Equation Discretization

The steady heat equation given in Eq. (3.51) is now discretized given the material and
temperature sample locations. This is given for a generic element index pair, i + 1 , j + 1 ,
2
2
and written in terms of thermal diffusivity in Eq. (4.59)

ri+1 α
1
T,i+1,j+ 2

T 3
1 − Ti+ 1 ,j+ 1
i+ 2 ,j+ 2
2
2

r 1 ri+1 − ri
i+ 2
ri α
−

1
T,i,j+ 2

r

i+ 3
2

T 1
−T 1
i+ 2 ,j+ 1
i− 2 ,j+ 1
2
2

r 1 ri+1 − ri
i+ 2
α
+

1
T,i+ 2 ,j+1

−

T,i+ 1 ,j
2

r 1 −r 1
i+ 2
i− 2

T 1
3 − Ti+ 1 ,j+ 1
i+ 2 ,j+ 2
2
2

zj+1 − zj
α

−r 1
i+ 2

z 3 −z 1
j+ 2
j+ 2

T 1
−T 1
i+ 2 ,j+ 1
i+ 2 ,j− 1
2
2

zj+1 − zj

=0

(4.59)

z 1 −z 1
j+ 2
j− 2

where T is the absolute substrate temperature in Kelvin and αT is the thermal diffusivity.

4.6.2

Boundary Conditions

The boundary conditions must also be treated, and are now given in discrete form. These can
be broken into two categories of Neumann boundary conditions: zeroed and finite-valued.
At the axis of symmetry, the radial derivative of the substrate temperature must be zero.

111

As a result, the substrate temperature is numerically calculated via

T1
= T3
1
,j+ 1
2
2
2 ,j+ 2

(4.60)

where 1 , j + 1 represents the element centers lying one-half mesh cell to the right of the
2
2
axis of symmetry. Likewise, at the bottom of the pedestal corresponding to the domain
boundary, a similar condition is satisfied

T 1 1 =T 1 3
i+ 2 , 2
i+ 2 , 2

(4.61)

1
where again, i + 2 , 1 represents all element centers lying one-half mesh cell above the nu2
merical domain boundary bottom edge within the substrate pedestal.
At the top and right pedestal edges, the total energy flux passed from the plasma module
is again applied to the substrate temperature module domain edges. This is a combination of
the atomic hydrogen bombardment and energy transfer and the thermal flux from the plasma
at these boundaries. It is denoted here as the term QH denoting the total heat flow due to
the plasma. Thus, for a generic right-edge substrate thermal model boundary representing
the boundaries of the plasma and thermal modules located at the one-dimensional element
1
index N − 2 , the discrete boundary condition is expressed
T
−T
N−3
N−1
2
2
QH,N = λN
r
−r
N−1
N−3
2
2

(4.62)

where λN is the translated thermal conductivity of the substrate material associated with
the boundary index N . Again, QH,N has been supplied via calculations from the plasma
fluid module at the adjacent and opposite element across the current edge. Indices and
corresponding relationships between adjacent samples are the same as shown in Fig. 4.8.

112

4.6.3

Matrix Representation

Eq. (4.59) is now expressed in matrix form given in Eq. (4.63) as

At = 0

(4.63)

where A is the square n × n matrix containing the spatial divergence and gradient operators, and the thermal diffusivity dependence at the mesh edges, n is the number of elements
contained in the substrate mesh, t is the vector containing the unknown substrate temperature solutions, and 0 is a vector containing zeros. The solution to Eq. (4.63) is trivial in
the absence of non-zero source terms on its right side. These non-zero values correspond
to the substrate right and top boundaries where the thermal flux due to plasma processes
are coupled to the thermal model. Eq. (4.63) is solved for the substrate temperature vector
using the same sparse direct matrix solver from the electromagnetics module and momentum
transport equation in the plasma fluid module.
Finally, the solution substrate temperature at the top and right substrate mesh boundaries must be interpolated to the respective edges and passed to the plasma fluid module.
Since the same parent mesh is used to generate all solution module domains, this passing is
one-to-one between elements, requiring no interpolation or averaging transverse to the mesh
interface. The substrate temperatures are interpolated to the domain boundaries via the
following generic solution located at a right boundary
T
−T
N − 1 ,j+ 1
N − 3 ,j+ 1
2
2
2
2
T
1 = TN − 1 ,j+ 1 +
r
−r
N,j+ 2
2
2
N−3
N−1
2
2

rN − r
1
N−2

(4.64)

where TN is the substrate temperature that is passed to the plasma fluid module resulting
in the gas temperature solution being forced to this value at all substrate boundaries. This
completes the substrate thermal module solution procedure. Its location and order within
the entire simulation algorithm is discussed in the following paragraphs.
113

4.7

Overall Solution Algorithm

A detailed description and discussion of the complete numerical solution algorithm governing
the entire simulation follows. This will cover both the choice of solution method, the order
of submodule solutions, and an explanation of advantages and shortcomings to this method.
The overall structure of the simulation is essentially a compartmentalized iterative solver.
That is, the electromagnetics, plasma fluid, and substrate thermal modules are all updated
simultaneously by passing related field distributions, yet solved individually. This iterative
solver ensures simultaneous solutions which approach a single, coupled solution. That is,
since the plasma fluid module is comprised of primarily iterative numerical solution methods, their solutions approach steady conditions upon system convergence. However, at any
one time during the simulation, especially during the initial phase, these solutions are not
necessarily in agreement. This is more so the case with the plasma fluid module and its dependence on both the electromagnetics and substrate thermal modules than the other way
around.
Unfortunately, there are several disadvantages associated with this overall solution procedure. Firstly, the iterative procedure does not guarantee a specified convergence rate, or
even convergence at all. The complete solution may diverge under certain conditions related
to the physical system settings, the mesh resolution, or the numerical solution procedure. It
may diverge depending on the initial conditions or the maximum plasma module iterations
allowed during the transport update procedures. The simulation also does not guarantee
the uniqueness of the final solution, which is a byproduct of the nonlinear system behavior.
However, even with these admitted disadvantages, the successful and accurate simulation of
MPACVD reactor plasmas will be presented.
To begin the complete simulation, a complex conductivity profile must be assumed and
supplied to the electromagnetics module. This ensures a non-trivial absorbed power density
profile which is then passed to the plasma fluid module. If this absorbed power density

114

profile contained all zeros, the energy transport equations would not be driven by the power
density. Physically, this would result in zero net energy transferred to the electrons, leading
to no species production or other energy transfer. So, in the case of zero initial conductivity, no plasma would form and the simulation would terminate. Thus, an initial complex
conductivity profile is required for the simulation to begin.
The choice of this initial conductivity profile is not particularly important to the final
solution. However, its distribution will influence the rate of solution convergence and, in
the extreme case, whether the simulation diverges before a solution is reached. Thus, it is
typically recommended to begin with a conductivity profile which is a reasonable estimate
to the expected profile. This estimate should be within one order of magnitude in both size,
profile, and absolute values (both real and imaginary components) of this expected profile.
As a result, this initial conductivity profile is often centered at the axis of symmetry just
above the substrate top surface with gaussian profiles in both r and z . After this initial
ˆ
ˆ
conductivity has been supplied to the electromagnetics module, the updated solution from
the plasma module is used in all successive conductivity dependent calculations.
A schematic of the complex solution process is provided in Fig. 4.12.
The electromagnetics solution module is outlined in the dashed black box in Fig. 4.12,
while the plasma fluid module solution components are outlined in the red dashed box.
After the initial conductivity profile has been chosen, the electromagnetic fields are updated and the absorbed power density profile calculated via the electromagnetics module.
This absorbed power density profile is passed to the plasma fluid module, where the various
transport quantities are initialized, physical constants calculated, and transport quantities
updated. Once these transport equations have converged to the desired error or tolerance,
the complex conductivity profile is once again calculated in terms of the updated transport
quantities. At this point, the substrate temperature is updated via the substrate thermal
module. This temperature distribution is solved using the previously updated gas temperature from the plasma fluid module, with the substrate temperature being passed to the

115

σe,init
ˆ

E/M
Module

Pabs

no

no

Converge?
Tsub

Thermal

Momentum
Transport

Physical
Constants

Initialize

Converge?

no

σe
ˆ

yes

Converge?

yes

yes

Converge?

Scalar
Transports
no

yes
Stop
Figure 4.12: Complete simulation flowchart.

future plasma solution.
As a final step, the total system error is estimated by summing over all transport matrix
equation residuals via
Nt
tot =
t=1

|At xn+1 − bn+1 |
t
t
n+1 |
|bt

(4.65)

where Nt is the total number of transport equations solved, At is the left side, square matrix
associated with the tth transport equation, xn+1 is the transport equation solution at the
t
n+1 is the current source vector associated with transport equacurrent iteration step, and bt
tion. In this case, Ntrans = 11 representing the seven mass, two momentum components,

116

and two energy transport equations. The above represents one complete numerical iteration
step. This process repeats until either tot has dropped below the specified maximum system residual error, or the maximum number of allowed iterations, have been reached, the
various numerical solution distributions are plotted, and the simulation terminates. Typically, several hundred to thousand such iterations are required for a finely discretized domain
simulating higher pressure operating conditions to converge to a total residual of 10%.

117

CHAPTER 5
VERIFICATION AND
VALIDATION
An important step in gauging the accuracy and success of any numerical simulation is its
comparison against other known solutions and systems. These systems may represent experimental data, analytic functions or solutions, or previous numerical simulations. This
comparison of numerical solutions against other systems is commonly referred to as the
verification and validation process.
The current Microwave PACVD system numerical simulation has numerous interconnected and coupled components, referred to as modules. Its three solution modules are
coupled through different physical variables in order to arrive at a single global solution.
These modules must be verified and validated individually, along with the entire numerical
solution as a whole. These results are presented in the following sections.

5.1

Verification

The verification of individual solution modules is first discussed and presented. This involves verifying the different numerical solution procedures via their related matrix equations. Eigenvalue analysis will be performed on the corresponding matrices. As is standard
118

practice, examples of lowest eigenvalue convergence with increasing mesh resolution are presented.

5.1.1

Electromagnetics Module

The solution to the electromagnetics module is calculated from Eq. (3.8) and Eq. (4.12)
in terms of the azimuthal magnetic field. Eigenvalue analysis may performed on the mass
matrix in Eq. (4.12) by comparing the lowest mass matrix eigenvalues to the corresponding
wavenumbers of an analytic geometry. A cylindrical cavity lined with PEC serves as the
analytic geometry as shown in Fig. 5.1.
r=0
6

2λ0

0

z
ˆ
6
-

r
ˆ

?


λ0

-

Figure 5.1: Simple cylinder schematic for verifying Helmholtz matrix equation solution in
electromagnetics module.

Supported wavenumbers for the cavity in Fig. 5.1 may be analytically derived [80]. Assuming TMz polarization, these wavenumbers are given by
χ
kr = 0n ,
λ0

n ∈ Z+

(5.1a)

nπ
,
2λ0

n ∈ {0, Z + }

(5.1b)

kz =

119

Table 5.1: Lowest 20 analytic and calculated eigenvalues for a regular PEC cylinder waveguide with varying mesh refinement for the electromagnetics module.
Mode No.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20

Analytic
386.24058868
551.03045703
1045.40006209
1869.34940386
2035.07865405
2199.86852241
2694.23812746
3022.87848233
3518.18746923
4505.98729751
4671.71654770
5001.46489722
5166.25476558
5660.62437063
6154.82536288
6318.67584939
6484.57371240
7638.10279087
7967.51391476
8460.94413798

λ0 /20
385.69556448
550.40074519
1043.50082514
1861.95567835
2021.79786182
2186.50304254
2679.60312249
3000.71925848
3498.05797569
4452.77070940
4636.82155583
4923.14226854
5087.84744926
5580.94752921
6088.87300674
6209.15765104
6399.40238241
7538.16596255
7845.25994838
8259.05137385

λ0 /30
385.99816625
550.75039349
1044.55550081
1866.06000274
2029.16077897
2193.91300621
2687.71811353
3013.01221253
3509.22261546
4482.26841391
4656.17482525
4966.47051486
5131.22274210
5625.02784943
6125.43102663
6269.80147776
6446.53235136
7593.48456115
7912.96409048
8370.71190019

λ0 /40
386.10419169
550.87288775
1044.92491456
1867.49847970
2031.74684394
2196.51554000
2690.56756681
3017.32523434
3513.14113194
4492.63222899
4662.96788659
4981.74409267
5146.51278874
5640.56481554
6138.27488123
6291.14464717
6463.13838068
7612.96513533
7936.78729942
8410.08931311

2
2
kr + kz

(5.1c)

k = kr r + kz z
ˆ
ˆ

(5.1d)

k0 =

where kr , kz , k0 , and k are the radial, vertical, magnitude, and total wavenumbers, respectively, χ0n is the nth zero crossing of the zeroeth order Bessel function of the first kind,
and Z + is the set of all positive integers. The eigenvalues of the corresponding mass matrix should match these eigenvalues given in Eq. (5.1). All numerical eigenvalues presented
in this work were calculated using the MATLAB R eigenvalue function, eig. The five lowest
zero crossings of the zeroeth order bessel function were also calculated to sixteen digits via
MATLAB R during the radial component calculations. The results for different uniform mesh
spacings are provided in Table 5.1.
120

Clearly, the numerical eigenvalues presented in Table 5.1 are converging to the analytic
wavenumbers as the mesh resolution increases. This verifies the correct mass matrix filling
within the electromagnetics module.
The material dependence of the matrix solution within the electromagnetics module
must also be verified. This is achieved by filling the entire PEC cavity with a homogeneous
dielectric and again performing eigenvalue analysis. As a result, the same geometry from
Fig. 5.1 is again employed, but now filled with a perfect dielectric. The eigenvalues are
calculated in much the same way as in Eq. (5.1), but now with a material dependence as
given in Eq. (5.2)
χ0n
χ
kr = 0n = √ ,
λ
λ0 r

n ∈ Z+

(5.2a)

nπ
nπ
=
√ ,
2λ
2λ0 r

n ∈ {0, Z + }

(5.2b)

kz =

where the total and vector wavenumbers are the same as given in Eq. (5.1). For numerical
verification purposes, an arbitrary relative permittivity of r = 2 was chosen. The corresponding analytical and calculated numerical eigenvalues for the electromagnetics module
matrix equation solution are provided in Table 5.2.
As with the empty cavity results presented above, the calculated mass matrix eigenvalues
provided in Table 5.2 converge toward their corresponding analytic values when mesh refinement is increased. Again, this verifies the correct perfect dielectric dependence of the mass
matrix filling in the electromagnetics solution module. This is important, as the presence
of dielectrics in the MPACVD reactor is vital to the coupling of microwave power to the
plasma discharge.
Finally, the accuracy of the mass matrix generation in the presence of a lossy dielectric
must be verified. Once again, the same homogeneously filled cylindrical PEC cavity is once
again employed. However, the material is now assumed to have a total permittivity matching
that of free space, while its complex permittivity is set to an arbitrary value. Eigenvalue
analysis is again performed to confirm solution convergence with mesh refinement.

121

Table 5.2: Lowest 20 analytic and calculated eigenvalues for filled cylindrical waveguide with
varying mesh resolutions and homogeneous permittivity of r = 2.
Mode No.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20

Analytic
193.12029434
275.51522851
522.70003104
934.67470193
1017.53932702
1099.93426120
1347.11906373
1511.43924116
1759.09373461
2252.99364875
2335.85827385
2500.73244861
2583.12738279
2830.31218531
3077.41268144
3159.33792469
3242.28685620
3819.05139543
3983.75695738
4230.47206899

λ0 /20
192.84778224
275.20037259
521.75041257
930.97783917
1010.89893091
1093.25152127
1339.80156124
1500.35962924
1749.02898784
2226.38535470
2318.41077791
2461.57113427
2543.92372463
2790.47376460
3044.43650337
3104.57882552
3199.70119120
3769.08298127
3922.62997419
4129.52568692

122

λ0 /30
192.99908312
275.37519674
522.27775040
933.03000137
1014.58038948
1096.95650310
1343.85905676
1506.50610626
1754.61130773
2241.13420695
2328.08741262
2483.23525743
2565.61137105
2812.51392471
3062.71551331
3134.90073888
3223.26617568
3796.74228057
3956.48204524
4185.35595009

λ0 /40
193.05209584
275.43644387
522.46245728
933.74923985
1015.87342197
1098.25777000
1345.28378340
1508.66261717
1756.57056597
2246.31611449
2331.48394329
2490.87204633
2573.25639437
2820.28240777
3069.13744061
3145.57232358
3231.56919034
3806.48256766
3968.39364971
4205.04465655

Table 5.3: Lowest 20 analytic and calculated eigenvalues for a homogeneously filled cylindrical waveguide with r = 1 and σe = 1 + j0 with varying mesh refinements.
ˆ
Mode No.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20

Analytic
7.04454840
10.05011083
19.06679814
34.09461031
37.11730590
40.12286834
49.13955564
55.13354735
64.16736782
82.18360927
85.20630486
91.22050502
94.22606745
103.24275476
112.25636677
115.24479605
118.27056693
139.30950397
145.31755355
154.31710770

λ0 /20
7.03460786
10.03862571
19.03215849
33.95975808
36.87508090
39.87909875
48.87263153
54.72939086
63.80023112
81.21300514
84.56986390
89.79199794
92.79601579
101.78954857
111.05347818
113.24732064
116.71714816
137.48678094
143.08779368
150.63483514

λ0 /30
7.04012694
10.04500286
19.05139446
34.03461586
37.00937136
40.01424728
49.02063888
54.95359907
64.00386027
81.75100662
84.92284349
90.58225132
93.58712724
102.59351884
111.72025104
114.35338869
117.57674023
138.49572344
144.32263310
152.67138440

λ0 /40
7.04206071
10.04723701
19.05813211
34.06085189
37.05653797
40.06171426
49.07260937
55.03226323
64.07532914
81.94002973
85.04674048
90.86082240
93.86599869
102.87689380
111.95450699
114.74266158
117.87961357
138.85102491
144.75713884
153.38957948

The analytic eigenvalues are now dependent on both the permittivity and the conductivity
as given in Eq. (5.3).
χ
χ
√
kr = 0n = 0n ,
λ
λ0 ˆ
nπ
nπ
√ ,
kz =
=
2λ
2λ0 ˆ
ˆ= 1+

n ∈ Z+

(5.3a)

n ∈ {0, Z + }

(5.3b)

σe ∈

(5.3c)

−1

1
(σe ω)2

,

where ˆ is the total real component of the permittivity, σe is the electrical conductivity, and
is the set of all real numbers. Eigenvalue analysis was performed with results provided in
Table 5.3. The lossy dielectric was assumed to have the properties r = 1 and σe = 1 + j0.
ˆ
Once again, the calculated mass matrix eigenvalues provided in Table 5.3 converge toward
123

their respective analytic wavenumbers with mesh refinement. The mass matrix filling in the
case of lossy dielectrics in the electromagnetics module is thusly verified as accurate.

5.1.2

Transport Equations

The mass matrix filling for the transport matrix equations within the plasma fluid module
must also be verified similarly. These matrix equations will be broken into two categories:
vector transport and scalar transport. Since the momentum transport equation contains
both explicit and implicit numerical solutions, its verification is unavailable. Instead, the
numerical solution to this equation is validated against canonical solutions for simplified flow
systems.
Eigenvalue analysis may be performed on the scalar transport equations exactly the
same way as seen in the electromagnetics module. The same empty cavity geometry from
Fig. 5.1 and corresponding wavenumbers from Eq. (5.1) are utilized and compared against
the numerical eigenvalues calculated from the scalar transport mass matrix as provided in
Table 5.4.
Once again, the first twenty eigenvalues are captured with convergence during mesh
refinement. The scalar transport mass matrix filling is thus verified as accurate.
Of course, the scalar transport matrix solutions are never fully incremented during the
plasma fluid module solution procedure. As a result, the SOR numerical solution procedure
used in updating the scalar transport equations must also be verified. Since it is an iterative
solution method, it is verified against the exact numerical solution of a diagonal matrix
equation, with agreement to machine precision expected. Specifically, a positive definite,
penta-diagonal square matrix of size 1600 × 1600 was generated with its entries filled by
random numbers between 0 and 1. The resulting matrix equation was first solved using the
MUMPS sparse direct solver and then solved via the SOR solution method. An arbitrary

124

Table 5.4: Lowest 20 analytic and calculated eigenvalues for an empty cylindrical cavity with
varying mesh refinement.
Mode No.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20

Analytic
164.78986835
659.15947341
980.71259656
1145.50246492
1483.10881517
1639.87206998
2463.82141174
2636.63789364
3287.53142937
3452.32129772
3617.35049021
3946.69090278
4119.74670882
4770.64024455
5100.45930539
5924.16932302
5932.43526070
6911.76305866
6913.14785727
7076.55292701

λ0 /20
164.70518071
657.80526066
977.56612986
1142.27131058
1476.26011387
1635.37139053
2453.82624373
2615.02369400
3253.57205610
3418.27723682
3592.58982387
3911.37731676
4067.07514491
4729.83216997
5044.64127478
5823.46208656
5868.59575010
6764.61276082
6801.02821643
6929.31794153

125

λ0 /30
164.75222724
658.55733456
979.22975848
1143.98198572
1480.06183649
1637.78709305
2459.29159498
2627.01404628
3272.19023028
3436.94245753
3606.24380477
3930.74756485
4096.27024766
4752.25206678
5075.50000615
5883.80331151
5899.20427657
6846.40563292
6863.03306999
7011.15786017

λ0 /40
164.76869606
658.82072287
979.81250866
1144.58120472
1481.39428800
1638.63323153
2461.20679666
2631.22104265
3278.72607008
3443.49476614
3611.03355131
3937.54679295
4106.52803729
4760.12035809
5086.34054595
5905.04045548
5909.94711273
6875.21649374
6884.85296414
7039.98518981

relaxation coefficient of 0.1 was specified. SOR numerical solution error was calculated as
N

2
(0)
bi − bi

=

(5.4)

i=1
where

is the total solution error, N is the number of solution unknowns (1600 in this

case), b(0) is the solution vector calculated via the MUMPS sparse solver, and b is the
solution calculated via the SOR routine. The solution procedure terminated once the SOR
error dropped below the double precision tolerance of the computer, or 10−16 . Thus, SOR
solution procedure was verified as accurate.

5.1.3

Substrate Thermal Model

Finally, the correct mass matrix filling within the substrate thermal module must also be
verified. Again, this was carried out via eigenvalue analysis. The standard empty cylindrical
cavity from Fig. 5.1 is once again assumed with supported wavenumbers given by Eq. (5.1).
The resulting eigenvalues calculated from the substrate thermal module mass matrix are
given in Table 5.5.
Again, the first twenty eigenvalues are captured in Table 5.5 with clear convergence
toward the analytic eigenvalues during mesh refinement observed. This verifies the mass
matrix filling within the substrate thermal module.

5.1.4

Complete Simulation

The verification of the complete, coupled simulation is warranted and provides confidence in
the model as a whole. However, due to its complexity, nonlinear behavior, and dependence
on numerous physical and numerical variables, the verification of the complete simulation is
unavailable. Validating the complete simulation is a more reasonable undertaking and will
be discussed in the following section.

126

Table 5.5: Lowest 20 analytic and calculated eigenvalues for an empty cylindrical waveguide
with varying mesh refinements for the substrate thermal module.
Mode No.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20

Analytic
164.78986835
659.15947341
980.71259656
1145.50246492
1483.10881517
1639.87206998
2463.82141174
2636.63789364
3287.53142937
3452.32129772
3617.35049021
3946.69090278
4119.74670882
4770.64024455
5100.45930539
5924.16932302
5932.43526070
6911.76305866
6913.14785727
7076.55292701

λ0 /20
164.70518071
657.80526066
977.56612986
1142.27131058
1476.26011387
1635.37139053
2453.82624373
2615.02369400
3253.57205610
3418.27723682
3592.58982387
3911.37731676
4067.07514491
4729.83216997
5044.64127478
5823.46208656
5868.59575010
6764.61276082
6801.02821643
6929.31794154

127

λ0 /30
164.75222724
658.55733456
979.22975848
1143.98198572
1480.06183649
1637.78709305
2459.29159498
2627.01404628
3272.19023028
3436.94245753
3606.24380477
3930.74756485
4096.27024766
4752.25206678
5075.50000615
5883.80331151
5899.20427657
6846.40563292
6863.03306999
7011.15786017

λ0 /40
164.76869606
658.82072287
979.81250866
1144.58120472
1481.39428800
1638.63323153
2461.20679666
2631.22104265
3278.72607008
3443.49476615
3611.03355131
3937.54679295
4106.52803729
4760.12035809
5086.34054595
5905.04045548
5909.94711273
6875.21649374
6884.85296414
7039.98518981

5.2

Validation

With the accurate filling of the matrix equations in the various modules verified, their resulting numerical solutions must be validated. This includes comparing these solutions with
known physical phenomena and tabulated experimental data. As with verification process,
the validation of the overall simulation will initially be broken into its individual modules.
The validation of the complete simulation is performed by numerical trends and data to
physical trends and experimental data under similar operating conditions.

5.2.1

Electromagnetics Module

The electromagnetics module is validated by comparing the numerical field solutions against
analytical fields for canonical system geometries. This canonical geometry is most easily
realized as an empty cylindrical waveguide as presented in Fig. 5.2.
E
-

k ?
z
ˆ
6
-

r
ˆ

0
-



r=0

λ0

Figure 5.2: Cylindrical waveguide with TM(z) polarized incident field for validating electromagnetics module solution.

A TM(z) -polarized electromagnetic wave is assumed traveling downward from the top of
the waveguide. The analytic electromagnetic field solutions are given in Eq. (5.5) [80].
µ
Hφ = 0 J1 (kr r) exp (jkz z)
0
128

(5.5a)

Er = J1 (kr r) exp (jkz z)

(5.5b)

Ez = J0 (kr r) exp (jkz z)

(5.5c)

These solutions provided in Eq. (5.5) may be used for direct comparison with the numerical
solutions.
An infinite cylindrical waveguide as shown in Fig. 5.2 is numerically unattainable. Instead, it is truncated and lined with PMLs on either end as presented in Fig. 5.3 [70, 73].
r=0
PML
-

6

E

?
k

SF

4λ0

TF
0

z
ˆ
6
-

PML

r
ˆ

?


λ0

-

Figure 5.3: Cylindrical waveguide for quantifying reflections from PMLs and validating the
electromagnetics module numerical solution.

The incorporation of the PMLs introduces a numerical reflection artifact, which must
be quantified. In order to do this, the resulting electromagnetic field solutions must be
calculated purely in the presence of the PMLs. These solutions are all provided in Fig. 5.4,
along with the initial traveling wave.
In order to quantify the reflection from the PMLs, the magnetic field magnitude may be

129

(a)

(b)

(c)

(d)

Figure 5.4: Real components of the (a) TF/SF source vector, (b) azimuthal magnetic (H/m),
and (c) radial and (d) vertical electric field (V /m) intensities for the canonical waveguide
solution.

130

compared between the scattered and total field regions as
(s)
|H |2
φ
RP M L =
(i)
|H |2
φ

(5.6)

(s)
(i)
where RP M L is the reflection coefficient from the PMLs, and |H |2 and |H |2 represent
φ
φ
the maximum scattered and incident magnetic field intensities, respectively. The incident
magnetic field intensity is equal to the maximum of the first-order Bessel function of the first
kind (approximately 0.575) divided by the free space impedance as given by

η0 =

µ0

≈ 1.526 × 10−3

(5.7)

0

where η0 is the free space impedance. The maximum scattered magnetic field intensity
within the scattered field region is extracted as 6.168 ×10−6 H/m. Thus, the reflected
power ratio may be calculated
|6.168 × 10−6 |2
RP M L =
= 1.633 × 10−5
−3 |2
|1.526 × 10

(5.8)

resulting in a total reflected power percentage due to the numerical domain truncation via
PMLs of approximately 0.16%. This result is very small, and translates to a maximum
absorbed power error of roughly 3 Watts for a MPACVD system operating at 2.0 kW. This
validates the incorporation of the PMLs in the electromagnetics solution procedure. The
accurate incorporation of both the PEC and axis boundaries were similarly validated within
the electromagnetics module. Finally, the numerical results provided in Fig. 5.4 validate the
electromagnetic field solutions for the electromagnetics module for an empty cavity.
The electromagnetic field solutions must also be validated in the presence of both lossless
and lossy dielectrics. In order to validate the electromagnetics module numerical solution in
the case of lossless materials, a dielectric material fills the lower half of the numerical domain.

131

A schematic of this partially filled lossless dielectric waveguide is provided in Fig. 5.5.
r=0
PML
-

6

E

?
k

SF 4λ
0
TF

0

z
ˆ
6
-

PML

r
ˆ

?


λ0

-

Figure 5.5: Cylindrical waveguide with lossless dielectric region for validating the electromagnetics module numerical solution.

The supported analytic electromagnetic field distributions may be calculated and used for
comparison [80]. The resulting numerical electromagnetic solutions for TM(z) polarization
and r = 4 are provided in Fig. 5.6.
As expected, the wavelength of the electromagnetic fields in the lossless dielectric region
of Fig. 5.6 is one half that of the free space region. In order to validate the lossless material
response of the numerical solution, both the wavelength and reflected and transmitted field
intensities are quantified and compared to their corresponding analytic values. Here, the
wavelengths were determined to be within 2% of their analytically calculated values. These
relationships remained valid for both magnetic and electric fields in the scattered and total
field regions. Likewise, the magnitude of both the electric and magnetic fields were observed
to agree with the analytical amplitudes to within 2% for both regions. These results were
reported for a relatively coarse mesh discretization rate of twenty sample points per free
space wavelength, and validate the electromagnetics solution in the presence of a lossless
dielectric.
132

(a)

(b)

(c)

(d)

Figure 5.6: Real components of the (a) TF/SF source vector, (b) azimuthal magnetic (H/m),
and (c) radial and (d) vertical electric field (V /m) intensities for the lossless material ( r = 2)
validation.

133

Table 5.6: Comparison of analytic and measured skin depth values for wave propagation into
a lossy material with r = 1 − j0.1 for mesh discretization of ∆ = λ0 /20.
Resolution
δ(m)
Analytic 6.158497226941845E-002
Simulation 6.118213428571451E-002

The electromagnetic field solutions in the presence of a lossy dielectric must also be
validated. The same geometry from the lossless dielectric system is employed, but now with
a complex permittivity, in this case chosen to be ˆr = 1−j0.1. The resulting electromagnetic
field intensities exponentially decay as the wave propagates into the lossy material with a
skin depth given by
δ=

c
1
= √
≈ 0.062m
α
ω 0.1

(5.9)

This decay of electromagnetic field strength is observed in Fig. 5.7.
The skin depth was calculated by measuring the distance from the lossy dielectric surface
to the point where the electric field magnitude dropped to 1/e of its surface value. For
comparison, both the analytic and numerically extracted distances are listed in Table 5.6.
The numerical and analytic skin depth data provided in Table 5.6 agree to within 0.65%
error. Thus, the accurate solution of the electromagnetic fields in the presence of a lossy
dielectric are validated for the electromagnetics module.
As a final validation of the electromagnetics module, the field solutions and power distribution within the MSU Microwave PACVD reactor may be analyzed. This will include a
qualitative analysis of the electromagnetic fields in the absence of a lossy dielectric (representing the plasma). Numerical electromagnetic field solutions in the presence of a plasma
discharge will be presented later.
Electromagnetic fields within the MSU MPACVD reactor are known to be dominated by
(z)
TM013 -like modes in the main cavity [5, 21, 22]. Simulated electromagnetic field solutions
in the absence of a plasma discharge are provided in Fig. 5.8.

134

(a)

(b)

(c)

(d)

Figure 5.7: Real components of the (a) TF/SF source vector, (b) azimuthal magnetic (H/m),
and (c) radial and (d) vertical electric field (V /m) intensities for the lossy material ( r =
1 − j0.1) validation.

135

(a)

(b)

(c)

(d)

Figure 5.8: (a) Discretized MSU MPACVD schematic with magnitudes of the (b) azimuthal
magnetic (H/m), and (c) radial and (d) vertical electric field (V /m) intensities for the
validation fo the unfilled reactor. In (a), red denotes the TF/SF interface, green the silica
bell jar and subsrate walls, and blue the plasma and reactor regions. White denotes metal
(PEC). The color scales for the electric fields are the same.

136

Within the main cavity, three electric field maxima are present at the outer reactor
(z)
wall in the |Er | solution from Fig. 5.8. This confirms the TM013 -like mode dominance.
Similarly, the corresponding two maxima along the reactor axis in the |Ez | solution also
(z)
suggest the TM013 mode dominates. The combination of all these factors suggest the total
electromagnetic field solution in the absence of a plasma discharge within an approximated
MSU MPACVD reactor is validated.

5.2.2

Plasma Fluid Module

The numerical solutions within the plasma fluid module are now validated. Unlike the
electromagnetics module, this includes both the physical constants and matrix equation
solution variables. These physical constants are addressed and validated first.
Some of the physical constants present in Eq. (3.18) may be compared directly against
reported data under similar conditions [16, 35, 45, 64, 81]. Two such constants include the
species mass diffusion and the chemical reaction coefficients, the exact calculation of which
are provided in the appendix.
The species mass diffusion coefficient is based upon Kinetic Theory and gas dynamics,
and as a result is dependent on many separate plasma properties [81]. Several approximations have been previously substituted for these calculations and are often implemented
in MPACVD simulations [35, 64]. These simplified diffusion coefficient expressions require
much less information and computation time, but are not nearly as accurate. The latter can
lead to large errors in the final system solution. Two simplified diffusion coefficient models
have been reported by Koldanov, et al. and Mankelevich, et al. [35, 64]. As a comparison between these simplified diffusion expressions and that calculated in the current simulation, the
ratio between these values are recorded over the range of typical operating conditions. The
resulting ratios of current simulation to both the Koldanov and Mankelevich approximated
mass diffusion coefficients are shown in Fig. 5.9.
Clear differences between the present and simplified diffusion models are apparent from
137

(a)

(b)

Figure 5.9: Comparison of present to approximated mass diffusion coefficients as used by (a)
Koldanov, et al. and (b) Mankelevich, et al. [35, 64].

138

Fig. 5.9. That is, the wide range of ratios between present and the approximated diffusion
models suggest large differences in the resulting calculations. One possible reason for this discrepancy is that the species diffusion coefficient is not completely–or even well–characterized
within microwave discharges in MPACVD systems. This makes it difficult to definitively
choose one model over another. However, the two simplified diffusion models and that used
in the present work are of the right order of magnitude over a large range of expected operating temperatures and pressures. Their ratios do not deviate by more than one order of
magnitude for the majority of the typical operating conditions range, as well. Thus, this
qualitative agreement is taken as a validation of the species diffusion coefficient calculation.
Calculated chemical reaction rates associated with MPACVD hydrogen-based plasmas
may also be validated against experimental data [16, 34, 45–47, 64, 82–85]. Several reactions
at various locations throughout the plasma solution domain were compared to those stated in
the literature under the same temperature, pressure, and density conditions [16, 45]. Agreement to within machine precision was observed, validating the chemical reaction calculations.
The momentum transport equation in Eq. (3.27) resembles the Navier-Stokes equation.
Thus, this equation may be validated against canonical Navier-Stokes solutions for simple
geometries. One of the most common canonical problems is the driven cavity system as
shown in Fig. 5.10.
The coordinate frame chosen in Fig. 5.10 is the Cartesian frame, as is typically assumed.
Thus, the momentum transport equation solution will first be validated against the Cartesian
coordinate frame solution, and later the cylindrical solution will be supplied. Also, the
gravitational force present in Eq. (3.27) is turned off, and the volumetric mass density and
viscosity distributions are assumed constant. The resulting numerical vector velocity solution
to the momentum transport equation is presented in Fig. 5.11.
As expected, a circulating flow is supported with the fluid circulating about a center
vortex. Two turbulent regions are observed to exist in the two lower corners as expected.
This solution may be compared directly with numerical results reported elsewhere [86] by

139

vx = −1, vy = 0

vx = 0,
vy = 0

vx = 0,
vy = 0

y
ˆ
6
-

x
ˆ

vx = 0, vy = 0
Figure 5.10: The canonical driven cavity system used to validate the momentum transport
equation.

Figure 5.11: Numerical vector velocity solution in units of m/s to the momentum transport
equation for the driven cavity canonical system in Cartesian coordinates.

140

Figure 5.12: Validation of driven cavity numerical solution for two Reynolds numbers against
reported data. [86]

tracking the horizontal flow component at the horizontal midpoint across the vertical domain
as provided in Fig. 5.12.
The numerical vector flow solutions at the two given Reynolds numbers are in excellent
agreement with the previously reported results as shown in Fig. 5.12 [86]. With 40 numerical
samples in each direction, agreement to within 1% is observed. This validates the momentum
transport equation in Cartesian coordinates.
However, the MSU MPACVD reactor exhibits cylindrical symmetry, and so the numerical
simulation must also be validated in the cylindrical coordinate frame. Since a similar driven
cavity solution does not exist in cylindrical coordinates, flow through a cylindrical pipe is
used for validation.
Schematics of laminar flow through both square and cylindrical pipe canonical systems
are provided in Fig. 5.13.
The resulting numerical solutions for both square and cylindrical systems are provided
in Fig. 5.14.

141

vx = 0, vy = 1

vx = 0,
vy = 0

vr = 0, vz = 1

vx = 0,
vy = 0

vr = 0

vr = 0,
vz = 0

y
ˆ
6

z
ˆ
6

-

x
ˆ

vx = 0, vy = 1

-

r
ˆ

vr = 0, vz = 1

(a)

(b)

Figure 5.13: The laminar flow through a pipe in the (a) Cartesian and (b) cylindrical coordinate frames.

As expected, the flow speed at the center of the Cartesian coordinate frame system is
greater than that present at the axis of symmetry in the cylindrical coordinate system. This
is due to the cylindrical pipe having roughly twice as much area for the fluid to flow through
compared to the square pipe. These results validate the solution of the momentum transport
equation under uniform fluid properties in cylindrical coordinates.
Finally, the mass transport equation solution must also be validated inhomogeneous
viscosity and mass density. To do so, a Gaussian distribution for both density and viscosity is
assumed as shown in Fig. 5.15. The relative scaling between the density and viscosity mimics
typical MPACVD plasma values. The resulting vector flow solution was again calculated for
matching inflows and outflows of 1 m/s as given in Fig. 5.16.
The solution in Fig. 5.16 is very similar to that given for the constant density solution from
Fig. 5.14. However, a slight difference in flow speed and direction is observed toward the right
half of Fig. 5.16 as the gas flow speed is slightly greater than that from the constant density
and viscosity plot. This is undoubtedly due to the variable density and viscosity distributions.
These results validate the numerical solution to the momentum transfer equation in the

142

(a)

(b)

Figure 5.14: Numerical vector velocity solutions in units of m/s to the momentum transport
equation for the laminar pipe flow canonical systems in (a) Cartesian and (b) cylindrical
coordinates.

143

(a)

(b)

Figure 5.15: (a) Mass density (kg/m3 ) and (b) viscosity (P a s) distributions for the validation of the variable density momentum transport solution.

144

(a)

(b)

Figure 5.16: Vector velocity solutions in the presence of (a) constant and (b) variable density
and viscosity distributions to the momentum transport equation. Units are given in m/s.

145

-

0.2 m

1m
dT = 0
dz

QH = −1000

T = 100

z
ˆ
λ=1

6
-

r
ˆ

dT = 0
dz
Figure 5.17: Right toroid used for validating the substrate thermal module and related
boundary conditions.

presence of variable density and viscosity.

5.2.3

Substrate Thermal Model

Finally, both the internal temperature solution flux boundary conditions were validated
utilizing the unit cylinder shown in Fig. 5.17.
The analytic temperature at the right side of the numerical domain may be calculated
in terms of the heat flux boundary conditions as
∆x
ln
T2 = T1 − QH
λ

r2
r1

(5.10)

where T1 and T2 are the temperatures at the left and ride side of the numerical domain, QH
is the heat flux, ∆x is the mesh spacing, and r1 and r2 are the inner and outer radii of the
forced temperature and boundary flux on either side. The resultant analytic temperature at
the right side of the domain becomes ≈1387 K. The corresponding numerical solution to
this system is provided in Fig. 5.18.

146

Figure 5.18: Numerical solution for cylindrical toroid of thickness 1 m used to validate the
substrate thermal model.

The numerical solution at the right side of Fig. 5.18 agrees with the analytic temperature to within 20 K. This discrepancy is simply due to interpolation error when plotting.
When comparing colocated solutions, these two values agree to machine precision. Thus,
the substrate temperature module is validated in the case of heat flux boundary conditions
in cylindrical coordinates.

5.2.4

Complete Simulation

The complete numerical simulation may finally be validated, but first the reproducibility of
the geometry configurations is addressed. These reported data can be imprecise at times,
and error bars will be reported when appropriate and available [40]. Similarly, the exact
reactor configurations used during data extraction are often partially reported. This makes
the simulation of the MPACVD reactor under the exact same conditions difficult or even
impossible. Of course, this may lead to a inaccurate simulation, which makes the validation
between the experimental data and the numerical results appear inaccurate. Keeping these

147

points in mind, the validation of several important simulated quantities against experimental
data is described below, and will include:
• Radial electric field intensity at the reactor wall
• Quality factor of the reactor at a lower pressure
• Average gas temperature and electron density in the plasma
• Average vector plasma flow solutions in the presence of the plasma
• Maximum substrate temperature at the growth surface
The electric field intensity at the reactor wall was previously measured by Tan and
Grotjohn and is used for comparison here [22]. Measurements varying along z were recorded
ˆ
using a field probe during plasma operation under lower pressure operating conditions. The
MSU MPACVD reactor geometry used for the |Er | measurements by Tan and Grotjohn was
approximated and discretized as presented in Fig. 5.19 [22].
The original electric field data was recorded at an absorbed power level of 1.5 kW and a
pressure of 50 Torr which were again enforced during simulation [22]. The substrate thermal
model was not included. The resulting |Er | numerical solution along the wall, compared
against the measured data, is provided in Fig. 5.20.
Very good agreement between the simulated and experimental data is observed in Fig. 5.20.
In particular, three maxima are captured in both the experimentally measured data and in
(z)
the numerical solution. This confirms the TM013 -like mode dominates in the main cavity
of the reactor. Furthermore, agreement of maxima and minima locations along z suggests
ˆ
wavelength and good reactor configuration match. Finally, the magnitudes of the simulated
and experimental electric field intensities also agree. Also, the simulated quality factor of
119.3 agrees with the experimental value as reported by Tan and Grotjohn of ≈100 [22].
These characteristics from Fig. 5.20 suggest the complete numerical simulation is validated
with regards to the electromagnetic module.
148

Figure 5.19: Schematic of MSU MPACVD reactor used for Er |wall measurements. [22]

Figure 5.20: Comparison of measured and simulated |Er | component along the outer wall
of the MSU MPACVD reactor.

149

Figure 5.21: Schematic of the modified MPACVD reactor used during the measurement of
Tg and ne at low pressures.[40]

Plasma fluid module based variables may also be used to validate the complete simulation.
Here, the gas temperature and electron volumetric number density are used as measured by
Grotjohn, et al. [40]. Line-of-sight averages of these two quantities were recorded over a
range of pressures at a constant absorbed power level. The modified MPACVD reactor
provided in the original work was approximated and discretized as shown in Fig. 5.21 [40].
Replication of the line-of-sight experimental data was achieved by integrating across the
plasma within a cylinder. A visual depiction of this beam configuration with respect to
the plasma is provided in Fig. 5.21 with corresponding mathematical representation of this
integration calculation within the beam across the plasma region is given by
Rb 2π rb
0 0 X(r , z )r dr dφ dz
χ= 0
Rb 2π rb
0 0 r dr dφ dz
0

150

(5.11)

where χ is the scalar average value of the chosen solution variable, X, rb is the radius of the
microwave beam, and Rb is the maximum outer radius marking the boundary of integration
inclusion in the weighted average sum. These general coordinates may be translated to the
standard cylindrical coordinate frame via
z0 +rb
z0 −rb
χ=
z0 +rb
z0 −rb

Rb
2
−1
0 X(r, z) π cos
Rb
2
−1
0 X(r, z) π cos

|z−z0 |
rb
|z−z0 |
rb

rb
tan−1 r rdrdz
rb
tan−1 r rdrdz

(5.12)

Finally, the discrete form of Eq. (5.12) becomes

χ(i) =

2
X(r, z) π cos−1
2
−1
π cos

|zm −z0 |
rb

|zm −z0 |
rb

rb
tan−1 rm

rb
tan−1 rm

2
2
r2 − r1 |z2 − z1 |
(5.13)
2
2
r2 − r1 |z2 − z1 |

where r2 , r1 , z2 , and z1 are the radial and vertical coordinates at the right/top and
left/bottom edges of the ith mesh element, respectively, and rm and zm are the radial
and vertical coordinates at the element midpoints, respectively.
The approximate beam radius and vertical position are given in the original work by
Grotjohn, et al. and so are used here, while a radial beam extent of 1 cm was imposed [40].
Plots of both gas temperature and electron density experimental and averaged numerical
data at a total absorbed power level of 400 Watts are provided in Fig. 5.22. Experimental
error bars are included in Fig. 5.22 to represent the approximately 20-30% uncertainty in
the physical data [40].
Several important observations are made when comparing the experimental data and
numerical results in Fig. 5.22. First, both gas temperature and electron density increase
for increasing pressure as expected. The simulated gas temperature results agree with the
experimental data to within the reported uncertainty for pressures below 30 Torr. Above 30
Torr, the simulated gas temperature exceeds the experimental data. Similarly, the simulated
electron densities are generally greater than their experimental counterparts. These discrep151

(a)

(b)

Figure 5.22: Comparison between experimental and averaged numerical simulation data of
(a) gas temperature (K) and (b) electron density (cm−3 ) for the modified MSU MPACVD
reactor at 400 W. Experimental data taken from [40].

152

Figure 5.23: Validation of substrate temperature module simulation versus experimental
data at low and high pressures in the MPACVD reactor.

ancies may in part be due to several factors including differences in reactor configuration and
geometry; inaccurate substrate position; and the simplification of the numerical approximation to the experimental measurement technique. With these acknowledged differences, the
observed trends of increasing gas temperature and electron density with increasing pressure
suggest the complete simulation is validated with respect to the plasma fluid module solution
quantities.
Finally, the substrate temperature solutions must be validated within the complete, coupled simulation. This is done by comparing the simulated substrate temperature results at
the substrate top surface to experimental data as reported in the literature [5]. Such results
are presented for various operating conditions in Fig. 5.23.
The maximum simulated substrate temperature results given in Fig. 5.23 are slightly
lower than the experimental data reported by Hemawan, et al. at lower pressures, yet the
upward trend and rate of increase agree well [5]. At 200 Torr, the simulated substrate temperature increases much more rapidly than the measured data at similar absorbed power levels.
These differences could be due to several factors including high calculated reflected power
(denoting a detuned geometry); the over-simplification of the internal substrate structure
153

and introduction of a spacer layer; and the omission of a diamond sample and recessed sample holder. These would all undoubtedly affect the simulation results and thus could explain
the small differences observed between the numerical and experimental data in Fig. 5.23.
In light of these small differences, the general agreement between the simulated results and
experimental data validates the complete simulation with respect to the substrate temperature.

154

CHAPTER 6
RESULTS
This chapter presents data on the simulation of MPACVD reactors at moderate pressures.
These results will be presented and discussed in order of the numerical solution algorithm
and its corresponding modules. All results presented from hereon are assumed to be taken
directly from fully coupled multiphysics simulations unless otherwise noted.

6.1

High-pressure example

The multiphysics numerical solution of the MSU MPACVD Reactor B configuration is first
solved for power and pressure combinations of 180 Torr and 2.2 kW. An inlet gas flow rate of
400 sccm is specified. These operating conditions represent typical values during MPACVD
diamond deposition experiments. A minimum numerical mesh element width within the
plasma region of λ0 /150 (roughly 0.8 mm) is used. The numerical results presented include
those related to the electromagnetic, plasma fluid, and substrate temperature solutions.

6.1.1

Electromagnetics module

The magnitude of the complex electrical conductivity solution given in Fig. 6.1 is confined to
the immediate region surrounding the reactor axis of symmetry. Its maximum conductivity

155

Figure 6.1: Magnitude of electrical conductivity numerical solution for the MSU MPACVD
”Reactor B” configuration operating at 2.2 kW and 180 Torr. Units are given in S/m.

amplitude is slightly less than 3.0 S/m suggesting a much high rate of spatial electromagnetic
absorption and scattering. However, the relatively narrow conductivity profile in Fig. 6.1
means the electromagnetic waves are still able to penetrate the discharge before decaying
significantly. This narrow conductivity profile is more narrow than those typically observed
at lower pressures [16].
The resulting magnitudes of the various electric and magnetic fields are provided in
Fig. 6.2 with interesting features readily apparent. In general, the electromagnetic field
intensities for the higher pressure results are greater than those observed at lower pressures
(z)
[22]. The radial electric field pattern presented in Fig. 6.2 exhibits a TM103 -like structure
with three clear maxima located at the outer reactor wall. This radial component intensity
is significantly higher in the immediate region just outside the plasma ball surrounding the
reactor axis of symmetry. This is indicated by the lighter blue false coloring extending from
the substrate surface toward the top inner surface of the quartz bell jar at a radius of roughly

156

(a)

(b)

(c)

(d)

Figure 6.2: Plots of (a) radial and (b) vertical electric (V /m), (c), azimuthal magnetic
(H/m), and (d) total electric field (V /m) components from the electromagnetics module.
Shown for MSU MPACVD Reactor B at a pressure of 180 Torr and absorbed power of 2.2
kW.
157

5 mm in r. A small increase in radial electric field intensity is also witnessed just above the
ˆ
substrate surface.
The vertical component of the electric field exhibits high field strength in the immediate
vicinity of the substrate surface with its overall solution distribution agreeing with that of
the lower pressure result. That is, this vertical component intensity varies between roughly
25,000 V /m at the substrate surface, drops to below 10,000 V /m within the main plasma ball
which coincides with the maximum electrical conductivity magnitude, and then increases
again approaching the top wall of the quartz bell jar when moving along z . Both the
ˆ
radial and vertical electric field component intensity solutions presented in Fig. 6.2 remain
nonuniform within the plasma region. This suggests the plasma has a strong influence on
the electromagnetic fields, which in turn effects the absorbed power profile.
All of these observations and characteristics come about from the combination of the
electrical conductivity and electromagnetic field numerical solutions, and result from the
increased operating power and pressure. As a result of the increased operating pressure, the
plasma is more tightly confined to the reactor axis. Moreover, the increased absorbed power
level results in increased electron and hydrogen species dissociation within the plasma. This,
in turn, results in an increased electrical conductivity magnitude and altered electromagnetic
field structure.
Finally, the volumetric absorbed power density numerical solution is presented in Fig. 6.3
where its maximum reaches slightly more than 2.3×108 W/m3 . This maximum absorbed
power density is on the same order as values observed at similar operating conditions and
reactor configuration reported elsewhere [5]. The absorbed power density profile is clearly
concentrated near the substrate surface surrounding the reactor axis. This profile extends
upward toward the inner surface of the top quartz bell jar. It even increases moving away
from the plasma ball toward this location. This suggests the electromagnetic energy is
primarily being transferred to the electrons near the substrate surface, which may in turn
ultimately increase the rate of diamond deposition through the production of other ions and

158

(a)

(b)

Figure 6.3: Plots of absorbed power density (a) across the full MPACVD reactor and (b) a
detail of the plasma region. Shown for MSU MPACVD Reactor B at a pressure of 180 Torr
and absorbed power of 2.2 kW. The color scales are the same for both plots. Units are given
in W/m3 .

159

Figure 6.4: Molecular hydrogen mole fraction solution for the MSU MPACVD Reactor B
operating at 180 Torr and 2.2 kW.

radicals.

6.1.2

Plasma fluid module

The nine mole fraction, vector average gas flow, and electron and gas temperature distributions are presented in Fig. 6.4 through Fig. 6.15. These profiles were extracted from
the same coupled, multiphysics simulation presented in the previous section detailing the
electromagnetic field solutions.
The molecular hydrogen mole fraction drops to a little less than 40% at the center of
the plasma ball as given in Fig. 6.4. Conversely, the atomic hydrogen mole fraction reaches
slightly greater than 60% as seen in Fig. 6.5. The atomic hydrogen clearly dominates the
gas mixture when comparing produced species. Thus, the molecular hydrogen feed gas is
predominantly dissociated into atomic hydrogen at moderate pressures. This is in agreement
with previous experimental observations and numerical results at lower pressures [4, 16, 35,
160

Figure 6.5: Atomic hydrogen mole fraction solution for the MSU MPACVD Reactor B
operating at 180 Torr and 2.2 kW.

40]. The numerical solution for both the molecular and atomic hydrogen mole fractions
are tightly confined to the reactor axis. This is clear from the rapid decrease in molecular
hydrogen, and rapid increase in atomic hydrogen mole fractions extending outward along r.
ˆ
These mole fractions reach roughly 10% of their respective minimum/maximum values at
roughly 2 cm from the axis, respectively. This is in sharp contrast to previous simulation
results at lower pressures and powers where these mole fractions have been observed to extend
much further toward, and even past, the outer radius of the substrate pedestal [16, 33]. The
combination of increased operating pressure and reactor configuration settings are attributed
to affecting this change in the solution profile.
Both the first and second excited atomic hydrogen mole fraction numerical solutions
seen in Fig. 6.6 and Fig. 6.7 exhibit maxima of approximately 6.5 × 10−9 and 8.4 × 10−11 .
As with absorbed power density and the previously discussed hydrogen mole fractions, these
excited atomic hydrogen mole fractions are tightly confined to the reactor axis. Moreover, the
vertical position of the maxima reported for these excited hydrogen mole fractions is shifted
161

Figure 6.6: First excited atomic hydrogen mole fraction solution for the MSU MPACVD
Reactor B operating at 180 Torr and 2.2 kW.

Figure 6.7: Second excited atomic hydrogen mole fraction solution for the MSU MPACVD
Reactor B operating at 180 Torr and 2.2 kW.

162

Figure 6.8: H + ion mole fraction solution for the MSU MPACVD Reactor B operating at
180 Torr and 2.2 kW.

upwards, away from the substrate surface. This suggests the density of these particular
hydrogen species are present in limited quantities near the deposition surface at moderate
pressures.
The proton mole fraction solution shown in Fig. 6.8 is also tightly confined to the reactor
axis with a maximum mole fraction reaching more than 3 ×10−8 . The vertical location
of this maximum mole fraction is shifted upwards along z and away from the substrate
ˆ
+
surface. The positive molecular hydrogen ion (H2 ) solution increases to 1.565 ×10−11
as seen in Fig. 6.9. It shares some similarities with the proton mole fraction in that it is
now concentrated along the reactor axis, and even begins to exhibit significant concentration
roughly midway between the substrate and bell jar surfaces. However, unlike the previous ion
+
solution, the H2 ion remains in significant concentration just above the substrate surface.
+
The location of the maximum value of the H3 ion is shifted toward the center of the
plasma ball, and its overall distribution is more concentrated at the r = 0 axis as seen in
Fig. 6.10. Similarly, its concentration remains significant just above the substrate surface.
163

+
Figure 6.9: H2 ion mole fraction solution for the MSU MPACVD Reactor B operating at
180 Torr and 2.2 kW.

+
Figure 6.10: H3 ion mole fraction solution for the MSU MPACVD Reactor B operating at
180 Torr and 2.2 kW.

164

Figure 6.11: H − ion mole fraction solution for the MSU MPACVD Reactor B operating at
180 Torr and 2.2 kW.
The maximum value of the H − ion reaches more than 7.7 ×10−9 . Similar to other mole
fraction solutions, its overall distribution and shape is elongated along z .
ˆ
+
Finally, the electron mole fraction solution closely follows the H3 trends given the dominance of that ion with respect to the total density of all ions in the plasma. The vertical
location of this maximum mole fraction is removed from the substrate surface several millimeters, and its overall distribution is now more closely confined to the reactor axis.
As quantitative convective flow data have not been previously reported for MPACVD
reactors at these moderate pressure operating conditions, considerable attention will be
paid to describing these data in detail. The vector average gas flow at the gas inlet region
extending along the bottom edge of the solution domain in Fig. 6.13 contains an upward
direction component. The flow just above the inlet region is already downward directed
suggesting the gravitational forces acting on the gas mixture away from the plasma ball
dominate. Such gravitational dominance is expected, as the gas is significantly cooler in this
region outside the substrate pedestal and below the substrate top surface as is shown in the
165

Figure 6.12: Electron mole fraction solution for the MSU MPACVD Reactor B operating at
180 Torr and 2.2 kW.

Figure 6.13: Vector average gas flow solution for the MSU MPACVD Reactor B operating
at 180 Torr and 2.2 kW. Color denotes speed while the arrow denotes flow direction. Units
are in m/s.

166

gas temperature plot in Fig. 6.15. Moreover, with an inlet flow of ∼2 mm/s, the gas flow
throughout the plasma region is primarily free convection-driven.
The hydrogen feed gas is strongly upward directed along both the outer surface of the
substrate pedestal, and the inner surface of the quartz bell jar walls. These flow streams
represent the primary method for momentum and feed gas transport to the plasma region.
A vortex is present in this lower and outer substrate region just above the inlet as evident
from the swirling vector pattern and low overall gas flow speed located at a radius of roughly
4 cm. The flow begins to turn back upward after the cooling gas initially falls under the
force of gravity with moderate speed close to the substrate outer wall.
The flow picks up speed at the top and outer substrate radius in order to conserve
momentum in its flow around this corner. Some of the gas mixture then flows through the
outlet region toward the outer radius of the substrate top surface and evacuates the reactor
system. The hydrogen gas mixture which does not exit the reactor flows tangential to the
substrate surface toward the reactor axis. At the r = 0 axis, the average gas flow is heated
from the increase in gas temperature, causing it to decrease in density, become buoyant, rise,
and flow upward toward the quartz bell jar inner surface. Once heated within the plasma
ball, the gas mixture begins to cool and starts to exits the plasma region. Flowing back along
the top quartz bell jar surface in the positive r-direction, the gas mixture cools and starts
ˆ
to succumb to the gravitational force resulting in the downward direction. This influence of
both buoyant and gravitational forces leads to an overall circulating flow pattern, creating
a vortex outside the plasma ball and substrate.
The electron temperature solution provided in Fig. 6.14 shows a maximum temperature
of roughly 13,770 K. Due to the increased operating pressure and total microwave power,
the energy transfer rate from electrons to heavy species is increased. This in turn results in
a decreased average electron energy (and therefore temperature) within the plasma, as the
electrons will transfer their kinetic energy to the heavy species more rapidly before gaining
significant microwave energy.

167

Figure 6.14: Electron temperature solution for the MSU MPACVD Reactor B operating at
180 Torr and 2.2 kW. Units are given in K.

The electron energy solution distribution again reaches its maximum value at the top
surface of the substrate and decreases from this value along the positive z direction. However,
ˆ
this higher pressure electron temperature solution does not decrease nearly as quickly as
lower pressure distributions reported elsewhere [16, 33]. Instead, the electron temperature
remains close to its maximum value within a very short length from the top quartz bell
jar surface. Although the electron temperature remains close to the maximum temperature
moving toward the outer radius of the substrate along r, it first decreases before increasing
ˆ
again.
The gas temperature solution shown in Fig. 6.15 reaches a maximum of 3654 K. This
value is consistent with the expected trend of increasing gas temperature with absorbed
power and operating pressure often reported at lower pressures [16, 34, 35, 40]. It is due
to the increased energy transfer from the electrons to the translational, rotational, and
vibrational modes of the hydrogen molecules in the plasma. The gas temperature distribution

168

Figure 6.15: Neutral species and ion temperature solution for the MSU MPACVD Reactor
B operating at 180 Torr and 2.2 kW. Units are given in K.

is slightly more elongated along z than its lower pressure counterpart [16, 34]. However, the
ˆ
gas temperature does remain at significant temperature levels extending outward in r past
ˆ
the substrate radius. This elongated gas temperature shape is not commonly reported in
the literature at lower pressures, but may be commonplace at higher pressures.
A very small change in relative local pressure is shown in the pressure difference plot in
Fig. 6.16. In this case, the operating pressure is 2.4×104 P a, or approximately 180 Torr
with a maximum deviation from the baseline operating pressure is less than 1.5 × 10−5 P a.
This suggests very low flow speeds present during system operation.
The volumetric number density distributions associated with the atomic hydrogen and
electron species, along with the total gas density, are provided in Fig. 6.17 to Fig. 6.19. The
maximum gas density shown in Fig. 6.17 reaches almost 3 × 1024 m−3 . This total gas
number density is more confined near the reactor axis, while at the same time more removed
from the immediate vicinity of the substrate surface. The number density remains high even
169

Figure 6.16: Absolute pressure difference solution from 180 Torr (∼2.4 × 104 P a) for the
MSU MPACVD Reactor B operating at 180 Torr and 2.2 kW. Units are given in P a.

Figure 6.17: Volumetric total number density solution for the MSU MPACVD Reactor B
operating at 180 Torr and 2.2 kW. Units are given in m−3 .

170

Figure 6.18: Volumetric atomic hydrogen number density solution for the MSU MPACVD
Reactor B operating at 180 Torr and 2.2 kW. Units are given in m−3 .

below the vertical position of the substrate and to its outside, although the plasma density
is significantly lower in this region.
Many differences are observed when comparing the atomic hydrogen volumetric number
density from Fig. 6.18 to the previous total gas number density. This is clear from the change
in maximum number densities, where a more than ten times decrease going from the total
to atomic hydrogen solutions is present. Moreover, the solution distribution itself is more
tightly packed at the reactor axis. These effects are echoed by the atomic hydrogen mole
fraction solution discussed in preceding paragraphs and shown in Fig. 6.5.
The electron volumetric number density numerical solution as shown in Fig. 6.19 also
depicts a narrow distribution. Similar to the atomic hydrogen number density solution discussed in the preceding paragraph, this behavior is also found in the electron mole fraction
solution given in Fig. 6.12. Furthermore, the decrease of roughly five orders of magnitude
between the atomic hydrogen and electron number density solutions echoes their corresponding mole fraction solutions. These solution variables and related physical quantities provide

171

Figure 6.19: Volumetric electron number density solution for the MSU MPACVD Reactor
B operating at 180 Torr and 2.2 kW. Units are given in m−3 .

significant insight into the behavior and characteristics of plasmas within MPACVD systems
when operating at higher pressure and power conditions.
Finally, the internal substrate temperature distribution is shown in Fig. 6.20. Its maximum temperature is located at the substrate top surface with almost uniform temperature,
and then gradually decreases traveling down its outside radius. Similarly, a sharp transition
from high to low temperature is observed across the spacer region at approximately 5 cm in
the vertical direction. The maximum substrate temperature of 1438 K is much greater than
values reported in the literature [5]. Differences are attributed to the over-simplified nature
of the present simulation in terms of the actual physical structure and material properties.
The calculated reflected power also remained relatively high for this simulation, at 37%,
which may suggest a detuned MPACVD reactor, and thus less than optimal power transfer.

172

Figure 6.20: Substrate temperature solution for the MSU MPACVD Reactor B operating at
180 Torr and 2.2 kW. Units are given in K.

6.2

Extreme high-pressure example

Numerical solutions at 180 Torr and 2.2 kW were presented in the previous section, and
represent typical MSU MPACVD system operating conditions [5, 21]. This operating pressure is only slightly above the generally accepted lower pressure/higher pressure boundary
of 150 Torr. Moreover, MPACVD experiments have recently been conducted at pressures
approaching 300 Torr with total absorbed powers approaching 3 kW [5, 21]. Diamond deposition rates often change significantly at these operating conditions, which suggest the plasma
properties are also altered [5]. Thus, results of a second simulation run at the upper limit of
these operating conditions (300 Torr and 2.5 kW) are also presented. The numerical mesh
spacing was decreased to roughly 500 µm in the plasma region and the resulting numerical
solutions are presented and described in the following paragraphs.

173

Figure 6.21: Magnitude of electrical conductivity numerical solution for the MSU MPACVD
”Reactor B” configuration operating at 2.5 kW and 300 Torr. Units are given in S/m.

6.2.1

Electromagnetics module

As expected, the complex electrical conductivity shown in Fig. 6.21 is roughly 70% greater
than the electrical conductivity depicted from the 180 Torr simulation, peaking at slightly
under 5 S/m. This increased complex conductivity is due to the increase in total microwave
absorbed power leading to an increased ionization rate within the plasma, and an increased
electron density. As a result, the complex conductivity increases and effects the electromagnetic field distribution. Comparing this extreme high pressure result to the 180 Torr results
from Fig. 6.1, the present electrical conductivity amplitude is even more tightly confined
to the reactor axis. The maximum absolute electrical conductivity translates to a relative
permittivity of more than 25, suggesting significant electromagnetic wave scattering and
influence.
The numerical electromagnetic field solutions at 300 Torr depict many similar traits to

174

(a)

(b)

(c)

(d)

Figure 6.22: Plots of (a) radial and (b) vertical electric (V /m), (c), azimuthal magnetic
(H/m), and (d) total electric (V /m) field components from the electromagnetics module.
Shown for MSU MPACVD Reactor B at a pressure of 300 Torr and absorbed power of 2.5
kW.
175

those given previously at 180 Torr. For example, the incident electromagnetic wave is indeed
(z)
TM(z) -polarized, a T M103 -like cavity mode dominates the solution in Fig. 6.22, and the
electric field is primarily vertically directed just above the substrate. At a maximum electrical
conductivity of almost 5 S/m, the electric field strength within the bulk of the plasma ball
is significantly reduced from previous results. The vertical electric field intensity drops to
roughly 20% of its maximum value within this region. This represents a large decrease in
field strength over a very short length extending from the top of the substrate to the center
of the plasma ball, which then increases once more at the top bell jar wall. The radial electric
field intensity is confined to the outer region of the plasma, suggesting it penetrates only
slightly into the plasma. Similarly, the radial electric field is nonzero just outside the reactor
axis just above the substrate top surface. This coincides with a very high vertical electric
field intensity at this location, leading to an increased total electric field intensity. This, in
turn, results in a very high absorbed power density just above the substrate surface, as will
be discussed in the following paragraphs.
The vertical component of the electric field intensity falls off significantly traveling along
the top substrate surface along r. Moreover, this elevated vertically-directed field intensity
ˆ
extends upward several millimeters into the plasma region. Once again, this suggests increased microwave power absorption extending to regions away from the substrate. Finally,
the vertical component of the electric field is also elevated at the top bell jar inner surface.
This is unexpected and not common during experimental operation, as this leads to an
increased absorbed power density and therefore electron temperature at this location. Experimentally, this can lead to the unwanted heating of the bell jar, which may even damage
the bell jar and entire MPACVD system.
The resulting volumetric absorbed power density profile is presented in Fig. 6.23. Similar
to the previous numerical results, the absorbed power density is highest several millimeters
above the substrate top surface. Again, this is expected as the maximum microwave absorbed
power within MPACVD systems is typically just above the substrate surface where the

176

(a)

(b)

Figure 6.23: (a) Volumetric absorbed power density numerical solution (b) with detail for
the MSU MPACVD ”Reactor B” configuration operating at 2.5 kW and 300 Torr. The same
color scale applies to both. Units are given in W/m3 .

production of ions and electrons leads to the most efficient deposition. Moreover, the power
density has risen to a maximum of roughly 3.4 × 108 W/m3 , which exhibits good agreement
with reported experimental values [5].
The absorbed power density decreases moving along z as is expected. However, unlike
ˆ
the previous volumetric absorbed power solution, the power density observed in Fig. 6.23
increases on the other side of the plasma ball toward another maximum. Again, this signals
the unwanted heating of the electrons away from the substrate surface which could damage
the physical reactor. As a result, the reflected system power is also expected to be very high,
and suggests sub-optimal tuning of the MPACVD reactor. Finally, the power density profile
in Fig. 6.23 shows a discontinuous distribution with finger-like regions of increased absorbed
power observed. This is a numerical artifact due to the rate of mesh discretization stemming
177

Figure 6.24: Molecular hydrogen mole fraction numerical solution for the MSU MPACVD
”Reactor B” configuration operating at 2.5 kW and 300 Torr.

from the balance of production and diffusion fluxes within the plasma transport equations.

6.2.2

Plasma fluid module

As with the previous solution, the molecular hydrogen mole fraction solution presented in
Fig. 6.24 reaches a minimum value at the center of the plasma ball, while it is maximum outside the plasma region. Likewise, the atomic hydrogen mole fraction solution from Fig. 6.25
is maximum within the plasma, and decreases to negligible amounts away from the substrate.
Notice at this operating pressure and total absorbed power level that the minimum molecular hydrogen mole fraction is less than 5% while the atomic hydrogen conversely reaches
a maximum mole fraction of more than 95% at the same location. This confirms the vast
majority of the molecular hydrogen feed gas is being dissociated into atomic hydrogen, a
trend that was also observed for the 180 Torr operating conditions. As was seen in the 180

178

Figure 6.25: Atomic hydrogen mole fraction numerical solution for the MSU MPACVD
”Reactor B” configuration operating at 2.5 kW and 300 Torr.

Torr numerical solutions, the atomic and molecular hydrogen mole fraction distributions are
again elongated along the reactor axis. This is again a result of the increased operating
pressure resulting in confined plasma located at the reactor center.
Unlike the previous excited atomic hydrogen solutions, the maxima of the first excited
atomic hydrogen mole fraction distribution presented in Fig. 6.26 is almost 20 times greater,
and the second excited solution in Fig. 6.27 more than 10 times greater than the corresponding high order results. This confirms increased electron energy promotion within the
plasma, which is expected due to the increased microwave absorbed power. That is, increased absorbed power translates to an increase of energetic electrons in the plasma, which
in turn leads to a greater number of excited hydrogen atoms via chemical reactions. Unlike
the previous solution, both excited hydrogen mole fractions are very tightly confined to the
reactor axis. As a result, the excited mole fraction solutions are greater in magnitude and
narrower. Of interest is the increased distance between the region of greatest excited atomic

179

Figure 6.26: First excited atomic hydrogen mole fraction numerical solution for the MSU
MPACVD ”Reactor B” configuration operating at 2.5 kW and 300 Torr.

Figure 6.27: Second excited atomic hydrogen mole fraction numerical solution for the MSU
MPACVD ”Reactor B” configuration operating at 2.5 kW and 300 Torr.

180

Figure 6.28: Proton ion mole fraction numerical solution for the MSU MPACVD ”Reactor
B” configuration operating at 2.5 kW and 300 Torr.

hydrogen density and the substrate surface. The excited atomic hydrogen mole fraction at
this surface is also higher than the previously presented results. A similar trend is observed
in many other mole fraction solutions, suggesting increased diamond deposition rates at such
operating conditions.
The hydrogen ion mole fraction solutions presented in Fig. 6.28 through Fig. 6.31 all
reach greater maximum values than their predecessors, which is again expected. However,
as with the excited atomic hydrogen mole fractions, their solution distributions are significantly altered from their corresponding lower pressure results being more concentrated
+
toward the center of the reactor axis. Of notable significance is the much greater H3 ion
mole fraction distribution, with almost a ten-fold increase in maximum value over its 180
Torr counterpart. This suggests the electron concentration, and thus energy transfer from
the electromagnetic power to the electrons and ions, are much greater. However, even at
these very high absorbed power levels, this ion and corresponding electron mole fractions

181

+
Figure 6.29: H2 ion mole fraction numerical solution for the MSU MPACVD ”Reactor B”
configuration operating at 2.5 kW and 300 Torr.

+
Figure 6.30: H3 ion mole fraction numerical solution for the MSU MPACVD ”Reactor B”
configuration operating at 2.5 kW and 300 Torr.

182

Figure 6.31: H − ion mole fraction numerical solution for the MSU MPACVD ”Reactor B”
configuration operating at 2.5 kW and 300 Torr.

remain much lower than the atomic hydrogen mole fraction, which is found to dominate
+
species concentrations independent of the physical operating conditions. Again, the H3 ion
mole fraction distribution is also observed to extend upward to the inner surface of the top
quartz bell jar. Its presence in this region confirms its direct dependence on the absorbed
power and resulting production via chemical reactions and recombination processes away
from the substrate.
+
Once again, even though the H3 ion mole fraction solution drops off significantly at the
substrate surface, its number density remains high at approximately 10% of this maximum
value at the surface. The H − ion also exhibits an unusual mole fraction distribution numerical solution in that it appears to separate into upper and lower concentration regions. That
is, the H − ion mole fraction is greatest just above the substrate surface and just below the
upper quartz bell jar surface.
However, with the tightly confined plasma densities observed in these figures, uniform

183

Figure 6.32: Electron mole fraction numerical solution for the MSU MPACVD ”Reactor B”
configuration operating at 2.5 kW and 300 Torr.

diamond deposition is not guaranteed, and may not even be expected. At this point, the
tuning of the MPACVD system becomes important as the adjustment of the physical reactor
geometry will significantly affect the plasma shape and properties. As stated previously,
these adjustments are commonplace and often necessary to maintain uniform deposition
across the substrate surface during the deposition process. These in situ adjustments are
available within the simulation by minimizing the reflected microwave power during the
electromagnetics module update by altering the reactor cavity height and antenna probe
length.
Finally, as is typical with both previously depicted electron mole fraction solutions, the
+
300 Torr electron mole fraction solution given in Fig. 6.32 closely matches the H3 ion
distribution in shape and value. Moreover, the increased maximum value results in an
increased overall volumetric number density. This maximum value remains low even at this
higher operating pressure and absorbed power level and confirms that even at the upper

184

Figure 6.33: Vector average gas flow numerical solution for the MSU MPACVD ”Reactor
B” configuration operating at 2.5 kW and 300 Torr. Units given in m/s.

limit of current operating conditions, the plasma remains very weakly ionized as is typical
of MPACVD systems [16, 34, 35, 40].
The vector average gas flow numerical solution provided in Fig. 6.33 is very similar in
characteristics and behavior to its corresponding results at 180 Torr. For instance, the inlet
gas flow once again has little influence on the overall gas flow throughout the plasma region
at the bottom of the reactor. Moreover, the gas flow is once again upward directed just
outside the substrate pedestal and outer quartz bell jar walls, while it is downward directed
between these regions. Above the vertical position of the substrate surface, the gas flow is
circulatory about a vortex located above the outer substrate radius with upwardly flow at
the reactor axis. At 300 Torr and 2.5 kW, the maximum flow speed occurs not at the outer
quartz bell jar wall, but at the center of the reactor. Thus, the upward flow of momentum
and gas species lies directly above the diamond sample where it can arguably influence
the plasma properties most significantly. Since this upward flow is obviously not driven
185

Figure 6.34: Electron temperature numerical solution for the MSU MPACVD ”Reactor B”
configuration operating at 2.5 kW and 300 Torr. Units given in K.

by the forced gas flow at the inlet region, it is an example of free convection and buoyant
forces. The operating conditions of 300 Torr and 2.5 kW absorbed power represent the upper
limit of currently reported MPACVD system operating conditions. Thus, the maximum gas
flow of roughly 7 cm/s represents the expected upper limit of gas flow speeds in MPACVD
reactors. Of course, this upper limit of average gas flow speed depends heavily on the reactor
geometry and operating mode. However, for the MSU MPACVD systems considered, this
assumption is expected to remain valid for the foreseeable future as operating pressures and
total absorbed power levels continue to increase.
The electron temperature solution provided in Fig. 6.34 reaches a minimum temperature
of roughly 11,600 K, which is much less than the minimum temperature reported in each
of the previous higher pressure example solution. Once again, the electron temperature is
expected to decrease with increasing operating pressure and total absorbed power level due
to the increased transfer of energy from the electrons to the heavy hydrogen species via

186

Figure 6.35: Gas temperature numerical solution for the MSU MPACVD ”Reactor B” configuration operating at 2.5 kW and 300 Torr. Units given in K.

collision processes. However, unlike the previous electron temperature results, the current
distribution remains almost constant along the entire reactor axis line extending from the
substrate surface to the inner surface of the top quartz bell jar. This suggests the electrons
remain significantly energetic even far removed from the substrate, and thus deposition,
surface which would lead to a decreased deposition rate in an experimental system. Thus,
while unwanted and even avoidable, this solution is once again provided for comparison and
reference purposes as an example of an non-tuned reactor and the effects thereof.
The gas temperature distribution from Fig. 6.35 are similar to those reported for the
180 Torr simulation results seen in Fig. 6.15. That is, the distribution shape, location of
maximum temperature, rate of temperature decrease outward along r, and temperature at
ˆ
the plasma region boundaries are all very similar. One key difference is the maximum gas
temperature, which is now 4176 K. Although not experimentally reported elsewhere thus

187

Figure 6.36: Difference in local pressure numerical solution for the MSU MPACVD ”Reactor
B” configuration operating at 2.5 kW and 300 Torr. Units given in P a.

making validation difficult, the trend of steadily increasing maximum and average gas temperature with operating pressure and total absorbed microwave power is expected. Again,
this is due to the increased energy transfer from the high frequency electron oscillations to
the translational, rotational, and vibrational modes of the molecular hydrogen atoms. This
is a direct result of the combination of both increased operating pressure and total absorbed
power.
Once again, the maximum variation observed in Fig. 6.36 is incredibly small, thus confirming low average gas flow speeds. For instance, the maximum difference in the local
pressure solution never reaches more than 10−5 P a, which represents a deviation on the
order of several parts-per-billion. Thus, even at these extreme operating conditions close to
the upper limit in both pressure and total absorbed power, the difference in local pressure
within the plasma region during operation remains incredibly low. This confirms the previous conclusion accompanying the average gas flow solution that at even higher pressures

188

Figure 6.37: Total gas volumetric number density numerical solution for the MSU MPACVD
”Reactor B” configuration operating at 2.5 kW and 300 Torr. Units given in m−3 .

and powers the gas flow would expected to remain at relatively low speeds.
Both the total and atomic hydrogen volumetric number densities from Fig. 6.37 and
Fig. 6.38, respectively are similar to their corresponding previously presented solutions at
slightly lower pressure. These include the overall shape of the solution distributions and
location of corresponding maximum values. Once again, the increased operating pressure
and total absorbed microwave power results in increased maximum total and atomic hydrogen number densities, with likewise an increased minimum total volumetric number density.
These are due to the combination of both the increased pressure translating to higher densities, as well as indirectly via the increased absorbed power level which in turn leads to increased gas temperatures and thus densities. The result of only slightly increased maximum
densities over those reported in the 180 Torr results again confirm the trend of increasing
density with absorbed power and pressure reported elsewhere [16, 40].
The maximum electron volumetric number density as shown in Fig. 6.39 reaches more

189

Figure 6.38: Atomic hydrogen volumetric number density numerical solution for the MSU
MPACVD ”Reactor B” configuration operating at 2.5 kW and 300 Torr. Units given in
m−3 .

Figure 6.39: Electron volumetric number density numerical solution for the MSU MPACVD
”Reactor B” configuration operating at 2.5 kW and 300 Torr. Units given in m−3 .

190

Figure 6.40: Substrate temperature solution for the MSU MPACVD ”Reactor B” configuration operating at 2.5 kW and 300 Torr. Units given in K.

than 1.25 ×1019 m−3 , which is more than double the maximum electron density reported
in the 180 Torr case. This mimics the electron mole fraction trend of increasing maximum
value with pressure and absorbed power levels. Moreover, this electron number density is
again tightly confined to the reactor axis with a maximum value occurring just off the r = 0
axis and decreasing rapidly along r. Its extension of non-negligible levels from the substrate
ˆ
surface to the inner surface of the quartz bell jar along the z -axis also agrees with the
ˆ
electron mole fraction solution. Once again, this suggests the detuned nature of the reactor
and undesirable expected decreased diamond deposition rate.
The substrate temperature solution is provided in Fig. 6.40 depicting a maximum of 2060
K. Once again, the substrate temperature is maximum at the top surface, which is in close
proximity to the plasma. This temperature also decreases along the outer substrate pedestal
wall, approaching roughly 700 K at the bottom of the reactor.

191

6.3

Power and pressure combinations

The plasma characteristics at different power and pressure combinations are expected change
with these operating conditions. For example, the combination of both increasing power and
pressure should result in essentially a consistent plasma size or volume. This is often observed
and enforced experimentally (along with other reactor configuration settings) in order to
maintain a consistent plasma volume and uniform deposition [5, 21]. At the same time, the
plasma intensity and luminescence is expected to change significantly due to higher power
densities. This in turn is expected to result in increased gas temperatures, and electron and
atomic hydrogen mole fractions.
The corresponding numerical results at these increasing power and pressure conditions are
presented in the following paragraphs. Important physical variable solutions are presented.
The specific operating conditions for each are specified, but all reside in the higher pressure
operating regime, ranging from 180 to 240 Torr. The operating pressure is increased in 20
Torr increments, resulting in four such sets of numerical solutions. Along with increasing
pressure, the total absorbed power is increased from 2 kW to 2.3 kW in 100 Watt increments.
The MSU MPACVD Reactor B configuration was again utilized for all results.
The radial and vertical electric field intensities shown in Fig. 6.41 and Fig. 6.42 do not
vary significantly with increased power and pressure. This is expected, as the shape and
location of the plasma also do not change significantly over these power and pressure ranges
suggesting minimal affect on the electromagnetic fields. Moreover, although the conductivity
itself does increase at these conditions, this does not translate to a large change in electric field
(z)
structure or waveform. Again, a TM103 -like mode is observed to dominate the field structure
in Fig. 6.41. Many other similar characteristics are also observed at these conditions. Finally,
the maximum field intensities within the plasma region again reach 25 kV /m as was observed
in the case of the corresponding high pressure results presented earlier.
A marked and consistent change in the simulated absorbed power density solutions is

192

(a)

(b)

(c)

(d)

Figure 6.41: Radial electric field intensity solutions for (a) 180 Torr, 2.2 kW, (b) 200 Torr,
2.1 kW, (c) 220 Torr, 2.2 kW, and (d) 240 Torr, 2.3 kW. Shown for MSU MPACVD Reactor
B. Units are given in V /m.
193

(a)

(b)

(c)

(d)

Figure 6.42: Vertical electric field intensity solutions for (a) 180 Torr, 2.2 kW, (b) 200 Torr,
2.1 kW, (c) 220 Torr, 2.2 kW, and (d) 240 Torr, 2.3 kW. Shown for MSU MPACVD Reactor
B. Units are given in V /m.
194

(a)

(b)

(c)

(d)

Figure 6.43: Absorbed power densities for (a) 180 Torr, 2.2 kW, (b) 200 Torr, 2.1 kW, (c)
220 Torr, 2.2 kW, and (d) 240 Torr, 2.3 kW. Shown for MSU MPACVD Reactor B. Units
are given in W/m3 .
195

observed in Fig. 6.43. That is, the formation and increase in absorbed power at the top of
the plasma region just inside the top surface of the quartz bell jar is evident. An increased
electron density and thus electrical conductivity also corresponds to this location. This suggests the reactor may be detuned as the plasma pulling away from the substrate surface
and forming at the top of the quartz bell jar. Experimentally, this would be avoided by
adjusting the reactor configuration as the increased power residing at this top surface would
draw from that at the substrate surface. This would would likely result in a decreased diamond deposition rate. Moreover, the increased power absorption at the quartz dome surface
may cause overheating of the quartz dome, which could ultimately cause it to melt. Thus,
this operating condition and resulting plasma mode is not recommended for experimental
observation. Again, this is confirmed by the calculated reflected powers of 57%, 65%, 84%,
and 56%, respectively.
The atomic hydrogen mole fraction solutions remain quite high at these operating conditions, indicating very high dissociation of the hydrogen feed gas. This is evident in the
maximum atomic hydrogen mole fraction reaching to almost 88% at 2.3 kW and 240 Torr in
Fig. 6.45. An increased atomic hydrogen mole fraction is a result of the increased absorbed
power density as described previously. As a result, the atomic hydrogen flux to the substrate
surface would also be expected to increase, and possibly lead to an increased diamond deposition rate. However, its narrow distribution above the substrate suggests uniform deposition
may be difficult for this particular reactor tuning.
The electron densities in Fig. 6.46 and Fig. 6.47 exhibit very narrow distributions tightly
confined to the axis of symmetry. Moreover, these solutions steadily increase in maximum
value along with increased absorbed power and pressures. This is expected, as the increased
power density leads to increased ionization rates, which also increases the plasma density.
The maximum electron density approaching 1019 m−3 in Fig. 6.47 is reasonable compared
to electron densities reported elsewhere at lower pressures [4, 35, 40]. Moreover, the position
of the maximum electron density is also observed to rise with increasing power and pressure.

196

(a)

(b)

Figure 6.44: Atomic hydrogen mole fraction solutions at (a) 180 Torr, 2 kW and (b) 200
Torr, 2.1 kW from the plasma module. Shown for MSU MPACVD Reactor B.

197

(a)

(b)

Figure 6.45: Atomic hydrogen mole fraction solutions at (a) 220 Torr, 2.2 kW and (b) 240
Torr, 2.3 kW from the plasma module. Shown for MSU MPACVD Reactor B.

198

(a)

(b)

Figure 6.46: Electron densities at (a) 180 Torr, 2 kW and (b) 200 Torr, 2.1 kW from the
plasma module. Shown for MSU MPACVD Reactor B. Units are in m−3 .

199

(a)

(b)

Figure 6.47: Electron densities at (a) 220 Torr, 2.2 kW and (b) 240 Torr, 2.3 kW from the
plasma module. Shown for MSU MPACVD Reactor B. Units are in m−3 .

200

This follows the absorbed power density shown in Fig. 6.43. Finally, the electron density
solution becomes more discontinuous moving up in pressure. This again is a numerical consequence due to the choice of mesh discretization and is thus not physical. This artificial
limit was imposed due to the serial structure of the simulation, and the choice and availability of processing hardware. Future implementations and versions are suggested to avoid
these shortcomings through the use of parallel processing and other methods as suggested
in Chapter 8.
Marked changes are observed across the vector average gas flow solutions presented in
Fig. 6.48 and Fig. 6.49. For instance, the flow speeds in different regions surrounding the
plasma change significantly with operating conditions. That is, as the absorbed power density
and pressure increase, the flow speeds similarly increase above the substrate surface. This is
particularly noticeable along the axis, at the outer substrate corner, and just inside the curved
quartz dome corner. At these three locations, the flow speed steadily increases along with
operating pressure. Of course, these are physically due to the increase in gas temperature,
which in turn results in a decrease in gas density within the plasma, and ultimately increases
the flow speed. That is, while this increased flow due to decreased gas density is observed
at the reactor axis, the flow outside the expected plasma region is due to cooling gas and
gravitational effects. As a result, the recirculating pattern observed throughout the solution
domain intensifies with increasing speeds.
Of course, several characteristics and regions remain unaffected by this change in absorbed
power and pressure as well. These include the inflow and outflow regions, along with the
entire spatial region below the vertical substrate surface.
This increased upward gas flow at the reactor axis of symmetry within the expected
plasma region was observed and during the high pressure results. Again, the maximum gas
flow speed remains low, not rising above 5 cm/s within the expected plasma region.
The maximum electron temperature solutions presented in Fig. 6.50 and Fig. 6.51 generally decrease with increasing pressure. Again, this is expected as the increased pressure

201

(a)

(b)

Figure 6.48: Vector average gas flows at (a) 180 Torr, 2 kW and (b) 200 Torr, 2.1 kW from
the plasma module. Shown for MSU MPACVD Reactor B. Units are in m/s.
202

(a)

(b)

Figure 6.49: Vector average gas flows at (a) 220 Torr, 2.2 kW and (b) 240 Torr, 2.3 kW from
the plasma module. Shown for MSU MPACVD Reactor B. Units are in m/s.
203

(a)

(b)

Figure 6.50: Electron temperature solutions at (a) 180 Torr, 2 kW and (b) 200 Torr, 2.1 kW
from the plasma module. Shown for MSU MPACVD Reactor B. Units are in K.

204

(a)

(b)

Figure 6.51: Electron temperature solutions at (a) 220 Torr, 2.2 kW and (b) 240 Torr, 2.3
kW from the plasma module. Shown for MSU MPACVD Reactor B. Units are in K.

205

results in increased plasma density, thus increasing the electron collision frequency. This
means the electrons transfer their energy to neutrals and ions more quickly and thus the
electrons have less time to gain energy before their next collision event. As a result, their
most probable energy is decreased, resulting in a lower temperature. The electron temperature distribution changes only slightly across these high pressure operating conditions, with
the majority of the energy confined to the reactor axis. A slight increase in electron temperature is observed toward the outer radius of the substrate surface, but remains lower than
that at the reactor center.
Finally, the maximum gas temperature observed in the gas temperature solutions from
Fig. 6.52 and Fig. 6.53 consistently increase with operating pressure and absorbed power.
From the previous discussion, this is expected, as the increasing absorbed power translates
to an increased rate of dissociation in the atomic hydrogen and thus plasma energy. The
gas temperature distribution does not change significantly across this range. Moreover, this
maximum gas temperature generally agrees with a slowly increasing temperature typically
observed in MPACVD systems at lower pressure conditions [16, 35, 40].

6.4

Effects of reactor geometry

Up to this point, several numerical solutions have been presented at individual operating
conditions. These results have served to present and exemplify the typical results associated
with this wide range of physical conditions imposed during the simulation. However, no
such results have yet been provided demonstrating the various effects of the system geometry on the plasma behavior. To accomplish this, several series of simulation results are now
presented to demonstrate the important influence these geometric features have over the
plasma properties. Two lesser, yet still important, plasma influences related to the reactor
geometry are the substrate vertical position and the reactor cavity height. Several multiphysical simulation results are presented detailing the effects on the plasma properties with

206

(a)

(b)

Figure 6.52: Gas temperature solutions at (a) 180 Torr, 2 kW and (b) 200 Torr, 2.1 kW
from the plasma module. Shown for MSU MPACVD Reactor B. Units are in K.
207

(a)

(b)

Figure 6.53: Gas temperature solutions at (a) 220 Torr, 2.2 kW and (b) 240 Torr, 2.3 kW
from the plasma module. Shown for MSU MPACVD Reactor B. Units are in K.
208

regards to these features. The substrate radius, and finally a complete reactor configuration
alteration are presented. In particular, these effects will be presented within the higher pressure regime, as this is important to future experiments with lower pressure results available
elsewhere [16, 22, 35].

6.4.1

Substrate position

The vertical position of the substrate within the MSU MPACVD class of reactors is known to
significantly affect the plasma properties and power coupling [5]. It is physically adjustable
prior to deposition, but not during. This is also the case in the numerical simulation with
the vertical position being set prior to the simulation run. Its numerical adjustment is
not available due to the unsatisfactory interpolation required that would significantly affect
the simulation results, and possibly lead to divergent solutions. This adjustment is often
necessary to ensure efficient microwave power coupling to the plasma, to maintain contact
between the plasma and the substrate, and to maximize deposition uniformity during the
system operation.
Experimentally, the substrate vertical position is often not quoted as an absolute distance
from the bottom of the reactor, but instead related to the vertical position of the top-most
outer coaxial block. Thus, the difference between these two characterizes this setting, with
the outer coaxial block being the reference height. For example, a setting of -5 mm, which is
often typical during an experimental run, would translate to the substrate vertical position
being 5 mm lower than the outer coaxial block vertical position. This same nomenclature is
used here in the following paragraphs.
In the following series of plots, the substrate position is varied from null to -6 mm in 2 mm
increments. All of these results were extracted from a single multiphysics solution for a MSU
MPACVD Reactor B configuration operating at 180 Torr and 2.2 kW. The substrate thermal
model was not included in the results, improving numerical efficiency and singling out the
effects due to substrate position alone. For comparison, the radial and vertical electric field
209

distributions at these substrate positions are provided in Fig. 6.54 and Fig. 6.55.
As seen from Fig. 6.54 and Fig. 6.55, significant changes to the electric field intensity
component solutions are not readily apparent. That is, these electric field solutions are
very similar in structure, value, and distribution almost completely throughout the entire
simulation domain. However, several small differences between solutions at the various
substrate positions are apparent. For instance, the maximum radial electric field intensity at
the outer substrate pedestal wall below the top substrate surface decreases significantly with
decreasing substrate position. With very little change in the vertical electric field solution
in this region, this suggests the power coupling into the plasma is more efficient at the lower
substrate positions. Likewise, the vertical component electric field intensity both at the
substrate and bell jar surfaces along the reactor axis decrease in magnitude with decreasing
substrate position. This is attributed to the increased distance between these two surfaces
allowing for a less compact plasma distribution, and thus power absorption and plasma
density.
Subtle changes to the absorbed power distribution are also apparent as shown in Fig. 6.56.
As the substrate position is lowered with respect to the outer coaxial block level, the maximum absorbed power density decreases more than 10%. This is significant to the plasma
properties and overall behavior of the reactor as will be discussed related to the plasma
solution results presented in the following paragraphs. Moreover, this decrease in maximum
power density suggests a more distributed power absorption profile accompanying the lower
substrate positions. Again, this will be presented along with the plasma fluid module solution results. With regards to the overall absorbed power density distribution, very little
change in this structure is observed regardless of substrate position.
The most significant changes to the electromagnetic solutions with regards to substrate
position were related to the absorbed volumetric power density distributions, with a more
distributed density corresponding to the lower substrate positions. However, the overall
effect of this substrate position was not readily apparent aside from the maximum power

210

(a)

(b)

(c)

(d)

Figure 6.54: Plots of radial electric field intensities for substrate positions of (a) 0 mm, (b)
-2 mm, (c) -4 mm, and (d) -6 mm from the electromagnetics module. Shown for MSU
MPACVD Reactor B at a pressure of 180 Torr and absorbed power of 2.2 kW. Units are
given in V /m.
211

(a)

(b)

(c)

(d)

Figure 6.55: Plots of vertical electric field intensities for substrate positions of (a) 0 mm,
(b) -2 mm, (c) -4 mm, and (d) -6 mm from the electromagnetics module. Shown for MSU
MPACVD Reactor B at a pressure of 180 Torr and absorbed power of 2.2 kW. Units are
given in V /m.
212

(a)

(b)

(c)

(d)

Figure 6.56: Plots of absorbed power densities for substrate positions of (a) 0 mm, (b)
-2 mm, (c) -4 mm, and (d) -6 mm from the electromagnetics module. Shown for MSU
MPACVD Reactor B at a pressure of 180 Torr and absorbed power of 2.2 kW. Units are
given in W/m−3 .
213

density. To demonstrate the effects of substrate position on MPACVD reactor operation
even further, several plasma-related solution quantities are now presented. These include
several solution mole fractions, the average gas flow, temperatures, and densities.
Similar to the microwave absorbed power density distribution plots shown in Fig. 6.56,
very little structural or spatial change to the atomic hydrogen mole fraction solutions is
observed in Fig. 6.57 and Fig. 6.58. However, the maximum solution values as denoted by
the top color scales suggest an increase of more than 10% from the null to -6 mm substrate
positions. This is again similar to the percent relative change in the maximum absorbed
power density solution, albeit the inverse relationship, which is expected due to the strong
dependence of the rate of hydrogen dissociation on the absorbed power density. Nevertheless,
this again represents a significant change in the plasma properties suggesting that adjusting
a single reactor component by as little as 6 mm can affect the maximum plasma atomic
hydrogen mole fraction by more than 10%.
The effect of substrate vertical position is also observed in the change in the total power
deposited to the top substrate surface as shown in Fig. 6.59. It is clear a nearly linear
relationship is present in this total power flux measurement depending on the substrate
position. This suggests the power delivered to the substrate via hydrogen bombardment
increases as the substrate is lowered. However, this must be balanced with the measured
reflected power in order to maximize the overall diamond deposition rate.
Unlike the atomic hydrogen mole fraction solution, the electron mole fraction solutions
presented in Fig. 6.60 and Fig. 6.61 exhibit no such clear trend in maximum value. Moreover,
their distributions are all narrowly confined around the reactor axis, yet are slightly different
in terms of maximum value positions. Again, this is due to the discontinuous solution
distribution, which is a numerical artifact of the relaxation process. Thus, the comparison
of maximum values here is not necessarily warranted or valid. This same behavior is also
present in the electron density distribution.
Again, the average gas flow velocity solutions presented in Fig. 6.62 and Fig. 6.63 show

214

(a)

(b)

Figure 6.57: Atomic hydrogen mole fraction numerical solutions for substrate positions of
(a) 0 mm and (b) -2 mm for the MSU MPACVD ”Reactor B” configuration operating at
180 Torr and 2.2 kW.

215

(a)

(b)

Figure 6.58: Atomic hydrogen mole fraction numerical solutions for substrate positions of
(a) -4 mm and (b) -6 mm for the MSU MPACVD ”Reactor B” configuration operating at
180 Torr and 2.2 kW.

216

Figure 6.59: Total power deposited on top substrate surface due to hydrogen bombardment
for various vertical substrate positions in the MSU MPACVD Reactor ”B” configuration
operating at 180 Torr and 2.2 kW.

very little change in the overall flow distribution as a whole. Notice in these plots the color
scales have all been normalized to the same 5 cm/s maximum for easy comparison. However,
both the upward flow at the reactor axis within the plasma ball, and the downward directed
flow outside the substrate radius both increase slightly with lowered substrate position. This
is a direct result of the increased gas temperature and decreased gas density in the upward
directed region, and decreased gas temperature in the downward flow case. As a result, the
substrate position appears to have very little effect on the overall, circulating flow pattern
during operation. However, as this position is either raised or lowered, the gas flow within
the plasma region is also affected due to changes in the gas temperature and mass density.
As with all numerical solution quantities in which the diffusion and conduction fluxes are
dominated by source terms, the electron temperatures from Fig. 6.64 and Fig. 6.65 are again
discontinuous and uneven. As a result, their maximum temperatures present from their color
scales tend to provide an inaccurate depiction of the affect of substrate position on electron
temperature. In spite of this, little change is observed between numerical solutions for the

217

(a)

(b)

Figure 6.60: Electron mole fraction numerical solutions for substrate positions of (a) 0 mm
and (b) -2 mm for the MSU MPACVD ”Reactor B” configuration operating at 180 Torr and
2.2 kW.

218

(a)

(b)

Figure 6.61: Electron mole fraction numerical solutions for substrate positions of (a) -4 mm
and (b) -6 mm for the MSU MPACVD ”Reactor B” configuration operating at 180 Torr and
2.2 kW.

219

(a)

(b)

Figure 6.62: Average gas flow velocity numerical solutions for substrate positions of (a) 0
mm and (b) -2 mm for the MSU MPACVD ”Reactor B” configuration operating at 180 Torr
and 2.2 kW. Units are given in m/s.

220

(a)

(b)

Figure 6.63: Average gas flow velocity numerical solutions for substrate positions of (a) -4
mm and (b) -6 mm for the MSU MPACVD ”Reactor B” configuration operating at 180 Torr
and 2.2 kW. Units are given in m/s.

221

(a)

(b)

Figure 6.64: Electron temperature numerical solutions for substrate positions of (a) 0 mm
and (b) -2 mm for the MSU MPACVD ”Reactor B” configuration operating at 180 Torr and
2.2 kW. Units are given in K.

222

(a)

(b)

Figure 6.65: Electron temperature numerical solutions for substrate positions of (a) -4 mm
and (b) -6 mm for the MSU MPACVD ”Reactor B” configuration operating at 180 Torr and
2.2 kW. Units are given in K.

223

various substrate positions, and so is assumed to have little effect on this quantity.
Finally, a clear trend in increasing gas temperature with lowering substrate position is
deduced from Fig. 6.66 and Fig. 6.67. This increase in maximum gas temperature from
3680 K to 3696 K represents a relative change of less than 0.5%. However, this difference
is significant enough to correspond to the previously discussed maximum atomic hydrogen
mole fraction increase of more than 10%. Likewise, this temperature increase does affect the
average gas flow, which from previous paragraphs was observed to have a small change in
behavior with respect to substrate position.
Finally, the reflected power coefficient for the various substrate settings is presented in
Fig. 6.68. It is clear the substrate position has a large effect on the relative percentage of
microwave power reflected from the reactor. As discussed in the preceding paragraphs, the
result of this increased reflected power is a decrease in coupling efficiency and maximum
absorbed microwave power density. Thus, the substrate position clearly affects the reflected
power coefficient. These reflected power ratios of more than 40% are far greater than typically
observed during MPACVD operation, and thus suggest sub-optimal tuning at the specified
operating conditions.
From these numerical solution data, plots, and discussion, the vertical positioning of the
substrate clearly affects the system behavior and operation. This, in turn, affects the plasma
properties and as a result is expected to significantly affect the diamond deposition rate and
perhaps even the quality of the synthetic diamond sample. This dependence on the substrate
position and the resultant plasma properties under such conditions presented in the previous
numerical results are generally in agreement with previously reported experimental data at
similar conditions [5].

6.4.2

Reactor cavity height

It may seem obvious that the vertical positioning, and therefore size, of the plasma region is
significant to the overall behavior and properties of the plasma during operation. However,
224

(a)

(b)

Figure 6.66: Gas temperature numerical solutions for substrate positions of (a) 0 mm and
(b) -2 mm for the MSU MPACVD ”Reactor B” configuration operating at 180 Torr and 2.2
kW. Units are given in K.
225

(a)

(b)

Figure 6.67: Gas temperature numerical solutions for substrate positions of (a) -4 mm and
(b) -6 mm for the MSU MPACVD ”Reactor B” configuration operating at 180 Torr and 2.2
kW. Units are given in K.
226

Figure 6.68: Power reflection coefficient for several substrate positions for the MSU
MPACVD ”Reactor B” configuration operating at 180 Torr and 2.2 kW.

even minute changes to the system geometry configuration significantly removed from the
plasma region can also affect the system operation. The most common of these is the
reactor cavity height. Changing the reactor cavity height is known to significantly affect the
electromagnetic field structure, and thus the absorbed power and plasma properties. As an
example to this effect, many similar numerical solutions are now presented for support of
this statement. These numerical solutions are presented in a similar pattern compared to
the substrate position discussed previously.
As can be seen in Fig. 6.69 and Fig. 6.70, there is very little change in both components
of the electric field solution. Any discernible differences are further masked by the imposed
equal color scales in an effort to demonstrate the fields within the main reactor cavity.
These fields would otherwise be overshadowed by the much more intense electromagnetic
fields within the coaxial waveguide at the top of the figures. Thus, from this brief, albeit
mundane conclusion, the effect of the reactor cavity height on the electromagnetic fields,
and therefore absorbed power and plasma properties, may be naively accepted as negligible.
However, this is not the case as clear differences in these numerical solutions are observed
as given in the absorbed power density solutions in Fig. 6.71.
The maximum volumetric absorbed power density as given in the color scales associated
227

(a)

(b)

(c)

(d)

Figure 6.69: Plots of radial electric field intensities for reactor heights of of (a) 20.0 cm, (b)
20.2 cm, (c) 20.4 cm, and (d) 20.6 cm from the electromagnetics module. Shown for MSU
MPACVD Reactor B at a pressure of 180 Torr and absorbed power of 2.2 kW. Units are
given in V /m.
228

(a)

(b)

(c)

(d)

Figure 6.70: Plots of vertical electric field intensities for reactor heights of of (a) 20.0 cm,
(b) 20.2 cm, (c) 20.4 cm, and (d) 20.6 cm from the electromagnetics module. Shown for
MSU MPACVD Reactor B at a pressure of 180 Torr and absorbed power of 2.2 kW. Units
are given in V /m.
229

(a)

(b)

(c)

(d)

Figure 6.71: Plots of absorbed power density distributions for reactor heights of of (a) 20.0
cm, (b) 20.2 cm, (c) 20.4 cm, and (d) 20.6 cm from the electromagnetics module. Shown for
MSU MPACVD Reactor B at a pressure of 180 Torr and absorbed power of 2.2 kW. Units
are given in W/m3 .
230

with Fig. 6.71 generally decrease with increasing cavity height. Moreover, this decrease
represents a change in maximum power density of slightly more than 10%. This relative
ratio between maximum absorbed power densities between the two most extreme cavity
heights is approximately equal to the relative ratio of absorbed power densities observed
during similar changes in the substrate vertical positioning. The similarity between these
two numbers with regards to the very different and significantly separated reactor geometry
configuration surfaces suggests perturbations of several millimeters of metallic horizontal
surfaces can affect the power distribution, and ultimately plasma properties, more than
10%.
The fact that the maximum absorbed power density is decreasing with increasing cavity
height suggests this distribution is becoming more spread out as the cavity size is increased.
Since the reactor geometry within the plasma region remains constant, this redistribution
of power is expected to result in a decrease of species production and densities. Similarly,
the gas temperature will also be expected to decrease as will be discussed in the following
paragraphs.
This significant change in the maximum absorbed power density profile once again suggests similar effects are expected to the plasma fluid module solution variables. As a result,
these variables are once again presented and discussed in terms of the effect the reactor
cavity height has on the plasma properties. Unlike the previous substrate position results,
the adjustment to the reactor cavity height is removed from the plasma region, and thus any
changes to the plasma properties are completely due to an external reactor configuration
parameter.
The maximum atomic hydrogen mole fractions observed in Fig. 6.72 and Fig. 6.73 increase
with corresponding decreases in maximum absorbed power densities. This is unexpected,
yet the observation of a decreasing maximum absorbed power density may not be correct.
That is, although the maximum absorbed power density does in fact continuously decrease
with increased cavity height, the power density within the center of the plasma ball is not

231

(a)

(b)

Figure 6.72: Plots of atomic hydrogen mole fraction solutions for reactor heights of of (a)
20.0 cm and (b) 20.2 cm within the MSU MPACVD Reactor B at a pressure of 180 Torr
and absorbed power of 2.2 kW.

232

(a)

(b)

Figure 6.73: Plots of atomic hydrogen mole fraction solutions for reactor heights of of (a)
20.4 cm and (b) 20.6 cm within the MSU MPACVD Reactor B at a pressure of 180 Torr
and absorbed power of 2.2 kW.

233

able to be precisely extracted from Fig. 6.71. Instead, a second set of plots providing the
same solution over only the plasma region would aid in this distinction and analysis, and is
presented in Fig. 6.74 and Fig. 6.75.
The maximum absorbed power density from Fig. 6.74 and Fig. 6.75 is located just above
the substrate surface. With the maximum absorbed power decreasing here, the microwave
power is instead being absorbed more within the plasma ball than elsewhere. As a result,
the absorbed power density distribution indeed relates to an increase in atomic hydrogen
mole fraction.
The electron mole fraction solutions again do not exhibit significant changes in their
distributions as seen in Fig. 6.76 and Fig. 6.77. That is, all plots are narrow in distribution about the reactor axis with maximum mole fractions that are approximately similar.
Again, these slight differences in the electron mole fraction scales can be attributed to their
oscillatory and discontinuous numerical solutions. Thus, any change to the electron mole
fraction, temperature, and number density due to an adjustment in the reactor cavity height
is difficult to discern.
As can be seen from Fig. 6.78 and Fig. 6.79, a decrease in average gas flow outside the
plasma and about the outer substrate corner increases with the decrease in absorbed power
density. This is expected, not as a result of the absorbed power density, but instead related
to the increase in gas temperature and corresponding decrease in gas density in these regions.
As a result, the buoyant and gravitational forces are more prominent, and affect the average
flow.
A clearly increasing maximum gas temperature with rising cavity height is observed
across Fig. 6.80 and Fig. 6.81. This again suggests the decrease in maximum absorbed
power density results in an increased power density within the bulk plasma region, leading
to increased hydrogen dissociation and energy transference to the hydrogen kinetic modes,
thus raising the gas temperature. The reactor cavity height only slightly alters the maximum
gas temperature value by much less than 0.5%. Moreover, the numerical distributions are

234

(a)

(b)

Figure 6.74: Plots of absorbed power density distributions for reactor heights of of (a) 20.0
cm and (b) 20.2 cm within the plasma region. Shown for MSU MPACVD Reactor B at a
pressure of 180 Torr and absorbed power of 2.2 kW. Units are given in W/m3 .
235

(a)

(b)

Figure 6.75: Plots of absorbed power density distributions for reactor heights of of (c) 20.4
cm and (d) 20.6 cm within the plasma region. Shown for MSU MPACVD Reactor B at a
pressure of 180 Torr and absorbed power of 2.2 kW. Units are given in W/m3 .
236

(a)

(b)

Figure 6.76: Plots of electron mole fraction solutions for reactor heights of of (a) 20.0 cm and
(b) 20.2 cm within the MSU MPACVD Reactor B at a pressure of 180 Torr and absorbed
power of 2.2 kW.

237

(a)

(b)

Figure 6.77: Plots of atomic hydrogen mole fraction solutions for reactor heights of of (a)
20.4 cm and (b) 20.6 cm within the MSU MPACVD Reactor B at a pressure of 180 Torr
and absorbed power of 2.2 kW.

238

(a)

(b)

Figure 6.78: Plots of average gas flow velocity solutions for reactor heights of of (a) 20.0
cm and (b) 20.2 cm within the MSU MPACVD Reactor B at a pressure of 180 Torr and
absorbed power of 2.2 kW. Units are given in m/s.

239

(a)

(b)

Figure 6.79: Plots of average gas flow velocity solutions for reactor heights of of (a) 20.4
cm and (b) 20.6 cm within the MSU MPACVD Reactor B at a pressure of 180 Torr and
absorbed power of 2.2 kW. Units are given in m/s.

240

(a)

(b)

Figure 6.80: Plots of gas temperature solutions for reactor heights of of (a) 20.0 cm and (b)
20.2 cm within the MSU MPACVD Reactor B at a pressure of 180 Torr and absorbed power
of 2.2 kW. Units are given in K.

241

(a)

(b)

Figure 6.81: Plots of gas temperature solutions for reactor heights of of (a) 20.4 cm and (b)
20.6 cm within the MSU MPACVD Reactor B at a pressure of 180 Torr and absorbed power
of 2.2 kW. Units are given in K.

242

Figure 6.82: Plot of the reflected power ratio for various reactor heights within the MSU
MPACVD Reactor B at a pressure of 180 Torr and absorbed power of 2.2 kW.

very similar across the different reactor height settings.
Finally, the microwave reflected power given in Fig. 6.82 confirms the trend of increasing
absorbed power density within the plasma ball by presenting the opposite trend of decreasing
reflected power. That is, this decrease in reflected power conversely suggests an increase in
power transferred to the plasma via the electrons and their collisions with hydrogen species.
The reflected power values themselves are quite high being between 47% to 65%. Again, this
is not typical of experimental reflected power coefficients, but is reasonable in a numerical
simulation where the reactor has not been tuned to minimize this reflected power.
From the results presented in the previous paragraphs and corresponding figures, adjusting the reactor cavity height clearly affects the plasma properties. This is seen not only in the
power coupling from the microwave source into the plasma, but also in the plasma properties
itself; most notably the atomic hydrogen and gas temperatures. More specifically, increasing
the cavity height by a mere 6 mm may affect the atomic hydrogen molar fraction by more
than 10%. Since atomic hydrogen plays an important role in the diamond deposition, this
alteration is significant to the total operation of the MPACVD reactor.
243

6.4.3

Reactor Configuration

The strong, nonlinear dependence of important plasma properties in the physical reactor
geometry was observed to be dependent even on millimeter changes in either the substrate
position or reactor height. It would thus be expected that more drastic changes to the
reactor configuration, such as the alteration of the entire MPACVD system configuration
and physical layout would affect the overall solution even more. These differences have
been previously reported over a wide range of reactor configurations [5, 21, 76]. However,
the majority of these data were reported while operating at lower pressures. Similar higher
pressure results are now briefly outlined in the following paragraphs for comparison and
reference. The MSU MPACVD reactors will be primarily used due to familiarity and the
existence of experimental data, consisting of configurations ”A”, ”B”, and ”C”. Results
could similarly be obtained for other reactor configurations.
Although ”A” and ”B” are clearly different reactors as noted by their associated letters,
only the substrate pedestal wall radius is altered. More specifically, this pedestal radius is
decreased in Reactor ”B” compared to ”A” by almost a factor of two. As a result, numerical
simulation data are not expected to vary significantly between these two configurations. On
the other hand, Reactor ”C” is strikingly dissimilar from Reactors ”A” and ”B”, suggesting
the plasma solutions are also dissimilar. Results are given for various simulated physical
quantities across all three reactor configurations for comparison.
The two electric field component solutions are provided in Fig. 6.83 and Fig. 6.84.
As expected, the electric field solutions presented in Fig. 6.83 and Fig. 6.84 are quite
similar between reactors ”A” and ”B”, while that of ”C” is markedly different. For instance,
the largest change seen going from Reactor ”A” to ”B” is the increase in radial field intensity
below the substrate position, and the increase in vertical electric field intensity within this
same region. This means the electromagnetic energy below the substrate vertical position
changes from being directed radially to vertically. However, as it is very far away from

244

(a)

(b)

(c)

Figure 6.83: Plots of radial electric field intensities for MSU MPACVD reactor configurations
(a) A, (b) B, and (c) C from the electromagnetics module. Shown for a pressure of 150 Torr
and absorbed power of 2 kW. Units are given in V /m.
245

(a)

(b)

(c)

Figure 6.84: Plots of vertical electric field intensities for MSU MPACVD reactor configurations (a) A, (b) B, and (c) C from the electromagnetics module. Shown for a pressure of 150
Torr and absorbed power of 2 kW. Units are given in V /m.
246

the plasma ball position, the absorbed power density here is extremely low and thus is not
expected to significantly affect the overall power distribution.
Both radial and vertical electric field intensities seen in Fig. 6.83 and Fig. 6.84 for the
MSU MPACVD Reactor ”C” are very different than their counterparts. The radial electric
(z)
field appears to exhibit a TM202 -like mode structure with two maxima residing above the
substrate along z . Likewise, the presence of a removed minimum in r from the axis toward
ˆ
ˆ
the top of the reactor suggests this second radial mode excitation. Within the plasma region,
the radial electric field is more uniform than the previous configurations, while this intensity
drops to almost zero within the plasma at the reactor axis. The radial electric field once
again drops off roughly as 1/r below the substrate at the outer pedestal. The verticallydirected electric field once again reaches very high levels at the substrate surface and the
top of the quartz bell jar. However, outside the substrate radius, this field component drops
significantly, possibly resulting in a decreased absorbed power density here. Finally, Ez
is much lower within the plasma region, which is a result of increased conductivity and
thus increased electron density. More on these quantities will be provided in upcoming
paragraphs.
Little change is observed between the power density results presented in Fig. 6.85 for
MSU MPACVD Reactors ”A” and ”B”. However, those results corresponding to Reactor
”C” show very large differences to the other two solutions. That is, the same power and
pressure combination used for the previous two configurations results in greater power density present along the top surface of the bell jar. However, it does provide an example of
a combined unstable system and operating conditions. Moreover, this power density is also
even more heavily concentrated at the reactor axis, which is likely due to a combination of
a reduced substrate radius, increased pressure, and increased plasma region volume. Thus,
the absorbed power density observed for Reactor ”C” is clearly altered from the other configurations. As a result, the various plasma properties are also expected to vary significantly
as discussed below.

247

(a)

(b)

(c)

Figure 6.85: Plots of absorbed power density for MSU MPACVD reactor configurations (a)
A, (b) B, and (c) C from the electromagnetics module. Shown for a pressure of 150 Torr
and absorbed power of 2 kW. Units are given in W/m3 .
248

(a)

(b)

Figure 6.86: Plots of atomic hydrogen mole fraction for MSU MPACVD reactor configurations (a) B and (b) C from the plasma fluid module. Shown for a pressure of 150 Torr and
absorbed power of 2 kW.

249

The plasma quantities calculated using reactor configurations ”A” and ”B” were very
similar, and thus ”A” results are omitted here in order to allow for easy comparison with
the ”C” results. As seen in Fig. 6.86, the distributions, location, and maximum values of the
simulation results for the atomic hydrogen mole fraction are very different between Reactors
”B” and ”C”. For instance, while the ”B” atomic hydrogen mole fraction is more round and
spherical in shape, the ”C” distribution is more narrow being tightly confined to the reactor
axis. Moreover, its maximum mole fraction is almost 40% greater than that of Reactor ”B”.
Finally, these changes in distribution and maximum value confirm the previously reported
findings of decreased operating microwave power required for Reactor ”C” in order to achieve
the same plasma volume and density of that in Reactor ”A” or ”B” [21].
As with the atomic hydrogen mole fraction results, the electron mole fraction numerical
results in Fig. 6.87 depict a much more narrow electron mole fraction with slightly increased
maximum value for Reactor ”C” compared with its counterpart in ”B”. Similar to the previous results for altered substrate and reactor heights, the electron mole fraction and density
is not exponentially-dependent on the absorbed power density as is the atomic hydrogen.
Thus, the small increase in electron density follows as expected.
Marked differences in the vector average gas flow solutions are shown in Fig. 6.88 and
Fig. 6.89 as noted by both changes in maximum gas speed and the overal solutions. For
instance, the maximum gas flow within the plasma increases moving from MSU MPACVD
Reactors A to B, and to C, respectively. Moreover, a more complex flow pattern is observed
for Reactor C with increased gas flow located at the reactor axis, outer substrate corner,
and just inside the top quartz dome surface. Similarly, the vertical flow below the top
substrate surface also changes between these configurations. The overall circulating pattern
of flow with upward direction within the plasma region remains throughout. As these results
presented in Fig. 6.88 and Fig. 6.89 all correspond to the same pressure and absorbed power
levels, the reactor configurations clearly have a marked affect on the average gas flows within
the MPACVD system.

250

(a)

(b)

Figure 6.87: Plots of electron mole fraction for MSU MPACVD reactor configurations (a)
B and (b) C from the plasma fluid module. Shown for a pressure of 150 Torr and absorbed
power of 2 kW.

251

(a)

(b)

Figure 6.88: Plots of vector average gas flow for the MSU MPACVD reactor configurations
(a) A and (b) B from the plasma fluid module. Shown for a pressure of 150 Torr and absorbed
power of 2 kW. Units are given in m/s.

252

Figure 6.89: Plot of vector average gas flow for the MSU MPACVD reactor configurations
C from the plasma fluid module. Shown for a pressure of 150 Torr and absorbed power of 2
kW. Units are given in m/s.

The electron temperature simulation results corresponding to Reactor ”C” as given in
Fig. 6.90 is more confined to the reactor axis with much cooler temperatures approaching
the substrate radius along its top surface. Moreover, this temperature is much cooler below
the substrate vertical position compared to the Reactor ”B” results. Again, this is a result
of the decreased electromagnetic field presence, and thus absorbed power density within this
region and is dependent on the reactor geometry itself. Interestingly enough, the maximum
electron temperatures, and thus the temperatures at the substrate surfaces, agree to within
2%, denoting very little difference in the electron energy here.
Finally, the increased absorbed power density also results in increased gas temperature
comparing ”B” and ”C” results in Fig. 6.91. This agrees with the much larger relative increase in the atomic hydrogen mole fraction presented previously. The elongated distribution
with higher temperatures residing higher within the plasma region also confirm the changes
in absorbed power density and electron mole fraction.
Clearly the reactor configuration, geometry settings, and parameters present during the

253

(a)

(b)

Figure 6.90: Plots of electron temperature for MSU MPACVD reactor configurations (a) B
and (b) C from the plasma fluid module. Shown for a pressure of 150 Torr and absorbed
power of 2 kW. Units are given in K.

254

(a)

(b)

Figure 6.91: Plots of gas temperature for MSU MPACVD reactor configurations (a) B and
(b) C from the plasma fluid module. Shown for a pressure of 150 Torr and absorbed power
of 2 kW. Units are given in K.
255

simulation can significantly affect the plasma properties, and thus diamond deposition rate.
As a result, the reactor design must carefully be chosen, analyzed, and vetted to ensure
efficient and hight quality diamond deposition.

6.5

Power Minimization

As described earlier, several of the important physical reactor components are often adjusted
during the deposition process in order to minimize the measured reflected power. This
reflected power is an indication of the power absorption efficiency into the plasma and the
resultant matching within the reactor. It also can be used as a diagnostic for indicating an
unstable plasma or reactor configuration. In this way, the effect of the minimization of power
plays an important role in ensuring the successful and efficient deposition of diamond.
The accurate capture of this power minimization and its related effect on the plasma
properties and location is also important. This is demonstrated by simulating similar reactor configurations under the same operating conditions, but only allowing one to actively
minimize the reflected power during the solution procedure. That is, in one case, the reactor
cavity height and probe antenna length are chosen to be 21.0 cm and 2.5 cm respectively.
These settings represent a typical set of configuration parameters for the MSU MPACVD
systems. During the power minimization simulation, the cavity height and antenna probe
length are adjusted in order to achieve a minimum reflected power. As a result, the minimized values are 20.7 cm and 3 cm for the cavity height and antenna length, respectively.
Clearly, these settings do not change much, but result in large differences in calculated reflected power. The resulting differences between the plasma solutions are attributed solely
to the differences in reactor configurations. This test was performed at the higher pressure
operating conditions of 180 Torr and 2.2 kW for the MSU MPACVD Reactor B. Plots of
several physical constants are provided and described in the following paragraphs. Both
simulations are run at operating conditions of 180 Torr and 2.2 kW.

256

(a)

(b)

Figure 6.92: Plots of radial electric field intensities for MSU MPACVD Reactor B with (a)
minimized and (b) non-minimized reflected powers from the electromagnetics module. Units
are given in V /m.

(a)

(b)

Figure 6.93: Plots of vertical electric field intensities for MSU MPACVD Reactor B with (a)
minimized and (b) non-minimized reflected powers from the electromagnetics module. Units
are given in V /m.

257

The radial and vertical component electric field intensities shown in Fig. 6.92 and Fig. 6.93
exhibit many differences between field solutions. For the radial electric field solution, the
field intensity is much greater within the waveguide region in the case of the non-minimized
power solution. This increased field intensity confirms a greater reflected power, which in this
case is roughly 30% higher than that of the minimized power solution. Similarly, this field
intensity at the outer top region of the reactor is much greater for the non-minimized power
solution. Again, this suggests more energy is remaining within the cavity and not being
transferred to the plasma and aiding in the deposition process. For the vertical electric field
intensity, the electric field is again much greater in the immediate region surrounding the
antenna probe in the case of the non-minimized power solution compared to that of the
minimized power.
Similar traits shared between both of these solutions include relatively unchanged field
structures within the plasma region. Moreover, the intensities or amplitudes of these fields
within the plasma region do not change significantly between these solutions. This suggests
the plasma solution and related quantities may also exhibit little change between the two
solutions. Although unexpected from an experimental standpoint, the fact that these solutions remain unchanged could be due to the relatively high reflected power (∼40%) even for
the minimized power solution. More on these plasma solution quantities will be discussed in
the following paragraphs.
The absorbed power density solutions shown in Fig. 6.94 exhibit small changes in both
maximum power density and location of this maximum. That is, the maximum absorbed
power density is slightly higher in the case of the non-minimized reflected power. Moreover,
the location of this maximum is closer to the substrate surface than in the actively minimized
reflected power solution. This indicates that more power is being transferred to the electrons,
and thus plasma ions and radicals as well, closer to the substrate and at a higher rate.
However, the overall absorbed power density profiles are very similar, which again suggests
the plasma solutions will also remain similar.

258

(a)

(b)

Figure 6.94: Plots of absorbed power densities for MSU MPACVD Reactor B with (a)
minimized and (b) non-minimized reflected powers from the electromagnetics module. Units
are given in W/m3 .

Again, very little difference between both the maximum and solution distribution for
the atomic hydrogen mole fractions are observed in Fig. 6.95 when comparing the reflected
power solutions. The only clear difference is the slight change of maximum atomic hydrogen
solution with the non-minimized reflected power value being slightly greater. This represents
only a 0.62% relative difference between the two maximum values again confirming very little
difference between the two solutions.
Very similar numerical solutions are again encountered comparing both the electron and
gas temperatures, and vector average gas flow solutions. However, the electron mole fraction
and densities exhibit stark contrasts in maximum value when comparing their solutions.
These distributions are provided and discussed in the following pages.
Similar to the electron mole fraction, the electron density solutions shown in Fig. 6.96
exhibit very dissimilar maximum values between the minimized and non-reflected power
minimized numerical solutions. In this case, the adjusted reactor configuration resulting in

259

(a)

(b)

Figure 6.95: Plots of atomic hydrogen mole fraction for MSU MPACVD Reactor B with (a)
minimized and (b) non-minimized reflected powers from the plasma fluid module.

260

(a)

(b)

Figure 6.96: Plots of electron number density for MSU MPACVD Reactor B with (a) minimized and (b) non-minimized reflected powers from the plasma fluid module. Units are
given in m−3 .

261

a minimized reflected power leads to a sharp increase in maximum electron number density.
However, this is in contrast to the absorbed power density solution where the two solutions
were very similar. Since the absorbed power density is closely related to the electron number
density, this represents an oscillating solution with insufficient mesh discretization. That is,
although the solution maxima do not agree, their overall shape and absolute density values
are expected to remain in good agreement to within the uncertainty associated with one
solution iteration.
As a final observation on reactor geometry configuration adjustment and its affect on the
plasma properties, the energy flux to the substrate may be calculated and compared between
the two cases. These solutions are presented below.
The energy fluxes shown in Fig. 6.97 exhibit small changes between solutions, particularly at the axis of symmetry. For instance, while the energy flux due to atomic hydrogen
bombardment is slightly greater at this location in the case of a non-minimized reflected
power reactor geometry configuration, it is slightly lower for the thermal conduction energy
flux. Since the energy due to thermal conduction is roughly five times that due to the atomic
hydrogen bombardment, the total energy flux to the substrate is greater when the reactor
is tuned to minimize reflected power. This confirms the experimental practice of actively
minimizing reflected power during diamond deposition. Of course, since hydrocarbons are
not included in the present simulation, these may also affect the flux to the substrate and
may even further confirm the minimization of reflected power practice. However, the above
discussion has confirmed that even minute changes of reactor geometry configuration (3 mm
in this case) can have observable affects on the plasma properties, the resulting energy flux
to the substrate, and likely the deposition rate.

262

(a)

(b)

Figure 6.97: Plots of energy flux from (a) thermal conduction and (b) atomic hydrogen
bombardment at the substrate surface. Units are given in W/m2 .

263

6.6

Convergence

The convergence of the complete numerical solution is vital to both the accuracy and efficiency of the simulation as a whole. This includes not only the rate of convergence within
the various modules consisting of numerous equations, but also the minimum convergence
levels attainable. The simulation only terminates once satisfactory convergence has been
achieved. However, this level of satisfaction is determined by the user with numerous conditions spanning several solution variables within the plasma fluid module. The intricacies
of the simulation convergence, examples of its calculation and levels during simulation, and
physical interpretation will be presented below.
The passing of physical quantities between solution modules after successive updates
results in an overall guess-and-check-style iterative solution procedure. As a result, the
system is expected to slowly converge toward a single solution, but may require many updates to achieve this. The overall simulation convergence toward a single, stable solution is
not guaranteed. However, with the correct choice of relaxation coefficients applied to the
most unstable numerical equations, reasonably stable convergence is often achieved. Various
plasma fluid module transport equation residuals, representing the system convergence, are
provided in Fig. 6.98 and Fig. 6.99. These convergence rates are typical of those achieved
for the stated relaxation quantities with null initial solution and higher pressure operating
conditions.
The most important numerical residuals calculated from the plasma fluid module transport equations are presented in Fig. 6.98 and Fig. 6.99. These data represent the scaled,
absolute difference between left and right hand sides of their respective matrix equations.
That is, the absolute value of these residuals closely mimics their maximum solution values
+
within their numerical distributions. Thus, the numerical error associated with the H3
ion mole fraction is much lower than the atomic hydrogen mole fraction simply due to the
large difference between their maximum solution values. In order to somewhat normalize

264

Figure 6.98: Numerical transport equation residuals plotted against total iteration number
for the MSU MPACVD Reactor B operating at 180 Torr and 2 kW.

Figure 6.99: Numerical transport equation residuals plotted against total iteration number
for the MSU MPACVD Reactor B operating at 300 Torr and 3 kW.

265

the error calculations across all variables, the temperature errors were divided by the upper
temperature range of 30,000 K.
Note in Fig. 6.98 and Fig. 6.99, the error associated with many of the variables changes
rapidly over the first several iterations. Starting from the numerical condition of null initial
solution, the various mole fractions and temperatures are undergoing large changes in both
their distribution shapes and absolute values. This eccentric convergence continues for several iterations until a stable plasma begins to form above the substrate as characterized by
a gently increasing error across solution variables until 500 to 750 iterations. Once a stable plasma has formed, the solution converges toward a single solution by undergoing small
changes over the next several hundred to thousand iterations. The simulation terminates
once all specified convergence limits are met for the solution variables.
The errors associated with the higher pressure and power simulations are generally greater
than those associated with the lower pressure and power conditions. This is due simply to
the increase in ion and species production, and an increase in gas temperature at higher
pressure and power conditions. As a result, the error calculated for these simulations is
necessarily greater.
The atomic hydrogen and gas temperature numerical errors are relatively well-behaved
+
and continue to decrease after plasma formation. This is in contrast to the H3 ion, electron
temperature, and total errors which plateau once a plasma forms. Again, this is a result of
their local oscillations within the spatial distributions and discontinuous solutions. It is a
direct result of insufficient mesh discretization coming from a source term dominating over
that species or temperature flux.
Finally, the total plasma residual is dominated by the electron temperature regardless
of operating conditions as indicated by a relatively constant and increased Te residual in
Fig. 6.99. This suggests the numerical solution, and thus physical MPACVD system, is
heavily influenced by the EEDF within the plasma.

266

CHAPTER 7
ANALYSIS AND DISCUSSION
The following analysis is presented to provide a more detailed description, and underlying
reasoning, behind the numerical results provided in Chapter 6. Significant attention is paid
to the role of convective flows within MPACVD systems at moderate operating pressures and
the substrate temperature during the deposition process. Both quantitative and qualitative
analysis is provided and compared at typical operating conditions within the MSU MPACVD
reactors. Comparisons with previous experimental and numerical simulations are also made.

7.1

Electromagnetics

Many interesting characteristics of the electromagnetic fields within the plasma reactor during operation have been outlined in Chapter 6. These include the overarching themes (regardless of operating conditions or minute tuning of the reactor configuration) of maximum
electric field intensity residing just above the substrate surface, this maximum electric field
being primarily vertically directed at the substrate surface, and the electric fields below the
substrate are significantly diminished from those above the substrate. On top of these, the
(z)
MSU MPACVD Reactors A and B were indeed found to support TM103 -like field structure as observed at the outer reactor walls. Overall, the electromagnetic fields (and thus
absorbed power density, and ultimately plasma characteristics) were clearly sensitive to the
267

reactor configuration parameters during operation. As expected, the reactor configuration
and geometry played an important role in these characteristics. Even small changes of a
few millimeters made to reactor surfaces (even those removed from the plasma region) were
observed to significantly affect the plasma properties. For instance, the lowering and raising
of the reactor cavity height 6 mm resulted in a 10% change in maximum atomic hydrogen
mole fraction within the plasma as given in Fig. 6.72 and Fig. 6.73. Since atomic hydrogen is
a precursor to the carbon deposition process, this increase is expected to significantly affect
the rate of diamond deposition during the experimental run.

7.2

Absorbed Power

The maximum absorbed power level often resides just above the substrate surface in the case
of stable plasmas. However, for those plasmas elongated vertically along the reactor axis,
or formed at the top of the bell jar, this was not necessarily observed. Instead, for unstable
or detuned solutions, the absorbed power density often reached a maximum at the top bell
jar surface and centered at the axis of symmetry. Moreover, the microwave absorbed power
density was observed to closely influence the electron temperature solution. This is expected
and often observed, as the microwave power is primarily transferred to the electrons via their
interactions with the electromagnetic fields, and then to the rest of the plasma via collisions
with heavy species [16]. Finally, the microwave power is absorbed at non-negligible levels
outside and below the substrate surface. However, these absorbed power densities are still
very low compared to the maximum absorbed power density residing just above the substrate
surface. In turn, this results in very little hydrogen dissociation or plasma ionization below
the substrate, resulting in a relatively confined and stable plasma above the substrate.

268

7.3

Plasma Fluid Module

A more detailed analysis of the plasma fluid module solution and related quantities is now
presented. This will primarily focus on the two important contributions of the present work:
the influence of convective flows and the substrate temperature at moderate pressures.

7.3.1

Validation

The numerical average gas flow results were never validated in the case of the fully coupled,
multiphysics simulation. Since no known experimental data on these flows exists, these flows
at moderate pressures will be validated against previous simulations. The average gas flow
velocity component profiles just above the substrate surface were simulated and presented
previously by Yamada, et al. [36]. Although these results utilized a much different reactor
configuration, they provide a basis for comparison in both spatial distribution, sign, and
magnitude.
Comparing the numerical results presented in Chapter 6 with the results of Yamada, et
al., similar magnitudes and orientation of the flow are observed [36]. These results provide a
validation of very low average flow components near the substrate. Furthermore, the radial
component is observed to be negative in sign just above the substrate top surface, signaling
a flow back toward the reactor axis. Finally, the vertical flow is consistently upward-directed
within the plasma. The agreement between these several characteristics is taken to validate
the momentum transport solution, and average convective gas flow results, in the case of
moderate pressure MPACVD systems.

7.3.2

Effect of average gas flow

The effect of the average gas flow on the overall plasma properties, stability, and location
within MPACVD reactors at pressures exceeding 150 Torr is of significant interest and is
one of the primary thrusts of this work. Thus, a detailed analysis of this effect via numer269

ical simulations is warranted. This is done by quantifying the three gas flow-related forces
affecting the plasma; convection, buoyancy, and gravitation. These are analyzed in the following paragraphs over a range of operating pressures, absorbed power levels, and for several
solution variables. They are analyzed for their influence in the respective scalar transport
equations for both mole fractions and temperatures. Finally, a direct comparison between
the relative size of these forces is provided.
Unlike buoyant or gravitational forces, the force of convection may be oriented along any
direction. In order to quantify the force of convection and its role within the plasma discharge
during MPACVD system operation, the related flux across mesh edges is calculated. Here,
both mass and energy are tracked and compared with their respective maximum fluxes,
which are calculated as
Γm,s = ρs v

(7.1a)

1
E
˜
Γe = Ev = v = ρv 2 v
V
2

(7.1b)

˜
where Γm,s and Γe represent the mass and energy fluxes, respectively, E is the plasma
volumetric energy density, and V is the volume associated with individual mesh elements.
Since many terms contained in Eq. (7.1) are element-centered, they are translated to their
respective edges via a bilinear interpolation scheme. In this way, both the mass and energy
fluxes may be directly compared to their diffusive flux counterparts as given in the scalar
transport equations. Moreover, these fluxes provided in Eq. (7.1) follow directly from their
respective transport equations from Eq. (3.18) and Eq. (3.39). Vector plots of both the mass
and energy fluxes across the entire plasma domain are provided in Fig. 7.1 and Fig. 7.2.
The convection flux vector plots provided in Fig. 7.1 and Fig. 7.2 clearly follow closely
to the average vector gas flow solution. That is, both energy and mass convection fluxes
exhibit vortices centered at the outer region of the plasma with upward-directed flux within
the plasma and downward going outside. The slight differences between the local values of
these convective fluxes are due to changes between mass and kinetic energy density within

270

(a)

(b)

Figure 7.1: Vector energy (a) convection and (b) conduction fluxes within the plasma region
for the MSU MPACVD Reactor B operating at 180 Torr and 2.2 kW. Units are given in
kg/s3 .
271

(a)

(b)

Figure 7.2: Vector mass (a) convection and (b) diffusion fluxes within the plasma region
for the MSU MPACVD Reactor B operating at 180 Torr and 2.2 kW. Units are given in
kg/m2 /s.
272

the plasma region. Moreover, convective fluxes are present throughout the entire plasma
region, albeit at much smaller levels far away from the plasma.
Comparing both the maximum and average energy conduction and mass diffusion fluxes
with their respective convection fluxes further exemplifies the small effect the average gas
flow has on the transport of these quantities. As a result, the gas flow influence on the
overall plasma properties and characteristics is very small. The absolute difference between
the maximum conduction and convection energy fluxes spans more than twelve orders of
magnitude. Likewise, the separation between the maximum convection and diffusion mass
fluxes is more than eight orders of magnitude within the plasma, as the diffusion color scale
is truncated in Fig. 7.2. Since both the mass diffusion (in the case of atomic hydrogen)
and energy conduction terms are known to dominate over their respective source transport
equation terms, the role of the convection fluxes due to the average gas flow are confirmed
to be insignificant.
Reviewing these convective forces specifically in terms of their vertical components results
in the specific treatment of gravitational and buoyant forces. Using the same convection flux
plots provided in Fig. 7.1 and Fig. 7.2, the effect of these gravitational and buoyant forces
on the plasma properties may be discussed as follows.
Simulations suggest the average gas flow within the plasma changes direction from downward to upwardly-directed depending on operating pressure and absorbed power level. This
is a result of an increased gas temperature, and thus decreased gas density, yielding a much
larger buoyant force (or likewise smaller gravitational force) at increased operating pressures.
Moreover, the gas flow outside this plasma region can either be downwards or upwards directed, as is the case of lower pressure operation.
In the case of lower pressure operation with a decreased gas temperature, the average
gas flow is downward directed within the plasma. This results in the further transport of
mass toward the substrate surface, as the diffusion flux is necessarily downward directed
toward this surface in the presence of a plasma. As a result, the mass flux to the substrate

273

is expected to be greater than in the case of no gas flow. When operating at higher pressure
conditions, the gas flow is upward directed within the plasma region, which may naively be
interpreted as reducing the mass flux to the substrate compared to lower pressure operation.
However, due to the much greater atomic hydrogen mole fraction at these conditions, the
flux to the substrate is much greater than that at lower pressures. Thus, even with a slightly
reduced mass flux toward the substrate due to the upward-directed average gas flow, this
flux is still much greater than that observed at lower pressures.
It is clear from this analysis that convective fluxes, buoyant, and gravitational forces are
all present throughout the plasma region regardless of operating conditions. This results in
the transport of both mass and energy throughout the solution domain. Due to their relatively small influence on the physical quantities when compared to their respective diffusion
or conduction fluxes, the gas flow within MPACVD reactors is not expected to contribute significantly to the properties or position of the microwave discharges even at higher pressures
approaching 300 Torr.
A final recommendation may be made relating to the inclusion of the average gas flow
in the multiphysics simulation of MPACVD reactors. Due to the insignificant amount this
average gas flow affects the overall plasma behavior and characteristics at the given conditions, it is beneficial to estimate its relevance during numerical simulation and question its
inclusion in the solution process. The employed matrix inversion solution requires a significant amount of processing power and physical memory, especially for dense meshes. On top
of this, its inclusion also results in a slightly less efficient rate of convergence for the overall
solution, and can contribute numerical instability and noise. As a result, the numerical solution of the average gas flow (and thus momentum transport equation) within the present
simulation is not recommended for operating conditions where a plasma is known to remain
experimentally stable.
It should be mentioned that the above statements are only valid for the combination
of operating conditions and reactor geometries included in the present work. In fact, these

274

effects may become significant under the right conditions that have not yet been simulated.

7.4

Atomic Hydrogen Diffusion Coefficient

As stated earlier, the atomic hydrogen mole fraction and resulting volumetric number density
are important to the diamond deposition process. Thus, the atomic hydrogen mole fraction
solution and distribution is vital to accurately simulating the system behavior and characteristics. Since the diffusion flux dominates the convective flux and steady rate production
of species in the mass transport equation, the accurate modeling and calculation of this
diffusion coefficient is paramount to the accurate atomic hydrogen mole fraction solution.
Two previously referenced and analyzed atomic hydrogen diffusion coefficients were presented in Chapter 6 [35, 64]. However, simulation results using these approximated diffusion
coefficients were not presented, but are now included in Fig. 7.3 through Fig. 7.5 for atomic
hydrogen mole fraction and gas temperature.
Comparing Fig. 7.3 through Fig. 7.5, it is clear the atomic hydrogen diffusion coefficient
is effected by the choice of diffusion coefficient representation. Moreover, since this mole
fraction is closely related to the gas temperature, it too is changed slightly from its nominal
value. More specifically, the maximum atomic hydrogen mole fraction is observed to decrease
by more than 20% when going from the Hirschfelder atomic hydrogen diffusion coefficient
to the Koldanov approximation. This also results in a roughly 0.25% change in maximum
gas temperature. The atomic hydrogen distributions are also slightly altered when comparing those using different diffusion coefficients. For example, the solution associated with
the Mankelevich approximation is more confined to the immediate region surrounding the
reactor axis, while that using the Koldanov approximation is more diffuse. This suggests
the approximated Koldanov diffusion coefficient is greater than that of both the Hirschfelder
and Mankelevich coefficients.
It is also clear the choice of diffusion coefficient plays a large role in the final solution

275

(a)

(b)

Figure 7.3: (a) Atomic hydrogen mole fraction and (b) gas temperature (K) within the
plasma region for the MSU MPACVD Reactor B operating at 180 Torr and 2.2 kW.

276

(a)

(b)

Figure 7.4: (a) Atomic hydrogen mole fraction and (b) gas temperature (K) calculated using
the Mankelevich approximated atomic hydrogen diffusion coefficient within the plasma region
for the MSU MPACVD Reactor B operating at 180 Torr and 2.2 kW.
277

(a)

(b)

Figure 7.5: (a) Atomic hydrogen mole fraction and (b) gas temperature (K) calculated using
the Koldanov approximated atomic hydrogen diffusion coefficient within the plasma region
for the MSU MPACVD Reactor B operating at 180 Torr and 2.2 kW.
278

for the atomic hydrogen mole fraction, and can even effect other physical quantities. While
the stated approximations are often much easier to calculate, they are not suggested for
future use, as they remain approximations to the much more rigorous Hirschfelder diffusion
calculation. However, the future analysis of the present diffusion coefficient is suggested and
even posited for experimental validation.

7.5

Conductivity

The electrical conductivity solution is important to analyze in terms of the different operating
powers and pressures. For example, its maximum absolute value is observed to range between
0.02 S/m and 5 S/m in the given power and pressure ranges typically used in MPACVD
systems.
The lower conductivity value of 0.02 S/m results in complete electromagnetic wave propagation through the entire plasma with little attenuation. Moreover, when realizing this
maximum conductivity is only present at the very center of the plasma ball with diminishing values moving away, the electromagnetic wave attenuation is expected to decrease even
further.
On the other hand, the maximum conductivity absolute value approaching 5 S/m represents a marked increase in electromagnetic wave attenuation at the particular frequencies
often associated with MPACVD systems. For example, the attenuation coefficient corresponding to this conductivity value may be calculated in the following derivation. While the
absolute conductivity value approaches 5 S/m, its imaginary component is often around 1
S/m and the real component likewise approaches 5 S/m. Taking these values, the attenua-

279

tion coefficient is calculated by first calculating the corresponding complex wavenumber

k = k − jk
−j

= k0




−

= k0 

where

and

σe
−j
ω 0

+

σe 
ω 0

(7.2)

denote the real and imaginary components of the complex permittivity and

conductivity, respectively. Assuming the permittivity in the plasma region in Eq. (7.2) is
completely real (which is valid in this case assuming vacuum), the imaginary wavenumber
within the plasma may be calculated as
σe
ω 0

k = k0

2πf
σe
c
ω 0
√
2πf σe
=
c
0

=

(7.3)

where f is the frequency of the electromagnetic excitation. Finally, the attenuation coefficient
may be calculated by approximating a 2 cm thick plasma region via

α = exp −

√


2πf
c

σe 
rp
0

(7.4)

where rp is the thickness of the plasma through which the plane wave propagates. Plugging
in the values for this expression given in Eq. (7.4) results in an attenuation coefficient of
∼2 × 10−3 . This, of course, is a very small coefficient, suggesting the majority of the electromagnetic wave is attenuated by the time it reaches the center of the plasma. However,
comparing this value to the numerical results at this extreme high pressure condition, the

280

electromagnetic field only drops to slightly more than 10% of its intensity just outside the
plasma region. The discrepancy between these two values is attributed to the unrealistic
assumption of consistent conductivity within the plasma when calculating this attenuation.
That is, the electrical conductivity magnitude within the plasma region is not homogeneous,
and more closely resembles a Gaussian distribution. Due to this spatial variation, the attenuation of the electromagnetic fields is only maximized at the very center of the plasma, with
little attenuation toward its outer region. As a result, the effective attenuation of the electromagnetic fields is much smaller than that given in the above derviation in Eq. (7.2) through
Eq. (7.4). Moreover, with a maximum conductivity magnitude approaching 5 S/m even
at the most extreme operating conditions, the electromagnetic fields, and thus microwave
power, is expected to keep penetrating completely into the plasma within MPACVD reactors.

7.6

Stability

Stability refers to both the physical system in terms of the location of the plasma and
its physical formation, and the numerical solution and convergence. These two different
meanings of stability will be discussed in the following section.
The physical stability of the system is characterized by the consistent formation of the
plasma above the substrate within the plasma region. Moreover, the plasma is considered
stable if its location does not change significantly over several solution iterations. The
combination of these two definitions excludes even a consistent plasma forming at the bell
jar corner, its top surface, or below the substrate from being classified as stable. This
definition agrees with the experimental observation of nominal system operation, where
these conditions would be classified as unstable.
The physical stability of the system depends on several factors, including the reactor
configuration, positions of the reactor height and substrate surface, and the operating conditions. For example, the operation of the MPACVD reactor at high power levels but low

281

pressures often results in the formation of a plasma at the top bell jar surface. The very
high electron and gas temperatures in close proximity to the top bell jar surface can cause
its overheating and even lead to its mechanical failure.
Numerical stability is important to the accuracy and numerical efficiency of the simulation. This is characterized by the convergence toward a single numerical solution, often
coinciding with the physical stability of the plasma at the same time. In this case, the
divergence of any one solution variable would signal numerical instability.
The numerical stability of the simulation depends on many factors including the initial
solution distributions, the system operation parameters, mesh resolution, and choice of solution procedures. The maximum number of updates to the physical parameters within the
plasma fluid module SOR routine are of critical importance. The choice of these numerical
parameters is important to an efficient numerical solution and thus must be chosen carefully
to avoid long run times or inefficient solution methods.
The numerical stability of the complete simulation is not fully understood at this time and
has not been rigorously characterized. Several factors influence this stability, including mesh
resolution, numerical initial and physical operating conditions, and choice of SOR relaxation
coefficients within the transport equations update. In light of this, the combination of
reasonable physical operating conditions, the incorporation of a previous solution as the
initial solution guess, and fewer inner and outer SOR iterations often leads to the most
stable simulations. These choices are recommended for future simulations and the numerical
stability analysis in future work.

7.7

Convergence

The numerical convergence of the solution within the coupled, multiphysics simulation is
necessary to gauge its progress and finality. That is, without the tracking of solution convergence, it would never be known whether the simulation has finished. This trait is not

282

typically reported along with results from previous simulations, and thus is difficult to gauge
an appropriate or similar tolerance level for the solution. However, residual tolerance levels
may be approximated and set according to visual observation of changes to the solution
across iterations, and even inspection of the relative convergence rates themselves. These
limits are typically chosen to correspond to levels of non-changing distributions or plots and
range from 10−8 to 10−1 .
The convergence rate of the numerical solution depends on many factors within the simulation, and is often different between solution variables. This is evident from the error plots
given in, for example Fig. 6.98. The smooth convergence of certain solution variables, such
as the atomic hydrogen mole fraction and gas temperature, is attributed to the dominance of
the diffusion and conduction fluxes over their source terms, respectively. On the other hand,
+
the numerical residuals associated with the H3 ion and the electron temperature appear
to plateau after several hundred complete iteration cycles. This is expected, as the electron
temperature is used in many physical parameter calculations, the set of which result in a
nonlinear dependence on this variable.
The solution convergence depends primarily on the solution procedure associated with the
plasma module update scheme. For the SOR scheme employed in the present simulation, the
choice of inner and outer iteration limits is most important. In order to analyze these choices
of iteration limits, the same reactor configuration and operating conditions are applied to
the MSU MPACVD Reactor B configuration. Maintaining consistency with previous results,
the operating pressure was set to 180 Torr with 2.2 kW of total absorbed power. Various
combinations of iteration limits were chosen representing likely choices for these values are
presented in Table 7.7 with resulting total residual error plot in Fig. 7.6.
The total summed plasma matrix equation residuals shown in Fig. 7.6 exhibit very different behavior from one another. Most importantly, the only combination of iteration limits
that results in the actual convergence of the simulation toward a single solution is that corresponding to the smallest numbers of five inner and outer iterations. As a result, the total

283

Table 7.1: Maximum iteration limits imposed in plasma module solution to demonstrate
convergence.
No.
1
2
3
4

Inner Outer
5
5
10
5
20
5
10
10

Figure 7.6: Total residual in plasma matrix equations for differing allowed maximum iteration
limits during Gauss-Seidel update.

plasma matrix equation residual in this case begins at roughly 1% and approaches 0.01%.
On the other hand, the remaining combination of iteration limits correspond to divergent
solutions, as interpreted by their seemingly stochastic behavior indicating no convergence.
This interpretation is confirmed after observing the resultant plasma quantity data with
either discontinuous or unreasonable solutions.
These data in Fig. 7.6 confirm the necessity to update the plasma physical quantities
often and limit the allowed convergence of the related solution. Although these particular
iteration limits are not necessarily required in all cases, at these higher pressures and with
zero initial conditions, they appear sufficient and are recommended for future use.
284

The convergence of the numerical solution is also heavily dependent on the initial conditions and choice of initial solution. This initial solution can either take on the form of
an applied previous solution at slightly different operating conditions, or a cold or zero condition start. In the latter case, only the electrical conductivity profile is supplied to the
electromagnetics solution module, with all related plasma solution variables remaining close
to their lower limits. As a result, all non-molecular hydrogen mole fractions are set to their
lower limits of 10−14 and the gas flow is set to zero everywhere outside the in- and outflow regions. The gas temperature resembles a parabolic function across the entire solution
domain with a value of 1200 K at the substrate top surface and 600 K everywhere else.
Likewise, the electron temperature is simply set to five times the gas temperature at every
sample location. This choice of initial temperature distributions ensures solutions ensures
temperature gradients which are important to future plasma updates. These initial cold
temperature distributions are provided in Fig. 7.7.
This choice of electron temperature being five times that of the gas temperature is an
approximation to the typical relationship between these two quantities over a wide range of
operating conditions. The solution itself is an approximation to the temperature solution in
the absence of any foreknowledge of the final solution. In this way, the cold initial solutions
are used in the case of an unknown final solution distribution arising from the absence of a
previous solution. This may be due to either changes in the physical reactor configuration,
or to the mesh resolution of the solution domain. The numerical solution convergence rates
associated with these two initial conditions are drastically different as shown in the total
plasma residual data given in Fig. 7.8.
As can be seen from Fig. 7.8, the application of a previous solution at slightly different
operating conditions to the initial solution of the present simulation results in a lower overall
residual limit. This also leads to a much faster convergence rate at the onset of the solution
procedure as seen in decrease of the residual in the first roughly 500 iterations. On the other
hand, the choice of a cold initial solution results in a solution with an increasing plasma

285

(a)

(b)

Figure 7.7: Initial (a) electron and (b) gas temperature solutions during cold start conditions
for plasma solution module. Units are given in K.

286

Figure 7.8: Total residual in plasma matrix equations for differing initial solutions provided
to the plasma fluid module.

matrix equation residual over this same time, which only then stabilizes and approaches its
residual limit. From Fig. 7.8, it is clear the application of a similar, previous solution as the
choice of initial plasma solution is preferable over a cold start when available. However, even
in the case of this cold start, convergence to a reasonable residual limit is achievable with
the valid choice of iteration limits.
Finally, the choice of mesh resolution, or grid spacing, is addressed. The mesh resolution
of any spatially sampled numerical simulation is important to the accuracy and numerical
efficiency of the particular model. This is equally important in the present MPACVD simulation as discussed below. In order to highlight the affect mesh spacing has on the overall
solution accuracy and numerical efficiency, the same reactor geometry configuration and
physical operating conditions were supplied to simulations of varying mesh resolution within
the plasma region. These grid spacings were defined as the width between opposite facing
edges within each element, and ranged from approximately 0.5 mm to 1.2 mm. Four different
grid spacings were used, and are calculated in terms of the electromagnetic free space wavelength as λ0 /100, λ0 /150, λ0 /200, and λ0 /250, corresponding to approximately 1.2 mm to
287

0.5 mm, respectively. These calculations assume the standard MSU MPACVD microwave
excitation frequency of 2.45 GHz and are provided in the remaining chapter figures.
Little difference is observed between the electromagnetic field intensity solutions throughout the MPACVD reactor cavity as shown in Fig. 7.9. This is expected, as the mesh resolution
outside the plasma region remained fixed at ∆x = λ0 /40, while the change in electrical conductivity was not significant. As a result, the electromagnetic fields were captured sufficiently
at the current grid spacing.
In contrast to the electric field intensities shown in Fig. 7.9, the absorbed power densities
presented in Fig. 7.10 and Fig. 7.11 exhibit significant changes in both maximum values and
overall distributions. For example, the maximum absorbed power density increases more
than two fold when decreasing the mesh spacing from 1.2 mm to 0.5 mm. This significant
increase confirms the region of maximum absorbed power is very small and resides close to the
substrate surface. Thus, a fine mesh spacing must be employed in order to accurately capture
the power absorption profile close to the substrate. At the same time, the absorbed power
toward the top of the quartz dome becomes less significant in the overall power distribution.
As a result, the total absorbed power profile is shifted more toward the substrate, suggesting
the unstable plasma and reflected power conditions are not as significant as indicated at the
lower mesh resolution. The major observation associated with these results suggest a mesh
spacing of less than 0.5 mm is required in order to accurately resolve the absorbed power
density profile at higher pressures, perhaps approaching 250 µm.
The atomic hydrogen mole fraction solutions presented in Fig. 7.12 and Fig. 7.13 suggest
a mesh spacing of 0.5 mm is significant in capturing its physical solution. This is indicated
by the decrease in maximum atomic hydrogen mole fraction solution at the smallest mesh
spacing from its previous solution. Moreover, the fact that these solutions do not change
significantly in shape or position also confirm a sufficient grid spacing. Similar results were
also observed for the gas temperature solution.
Due to its close relationship with the absorbed power density profile, the electron mole

288

(a)

(b)

(c)

(d)

Figure 7.9: Plots of electric field intensities for (a) radial component at coarse grid spacing,
(b) radial component at fine grid spacing, (c) vertical component at coarse grid spacing, and
(d) vertical component at find grid spacing throughout the MSU MPACVD Reactor B at
180 Torr and 2.2 kW. All units are given in V /m.

289

(a)

(b)

Figure 7.10: Plots of absorbed power density in the plasma region of MSU MPACVD Reactor
B at 180 Torr and 2.2 kW for mesh spacings of approximately (a) 1.2 and (b) 0.8 mm. Units
are given in W/m3 .

290

(a)

(b)

Figure 7.11: Plots of absorbed power density in the plasma region of MSU MPACVD Reactor
B at 180 Torr and 2.2 kW for mesh spacings of approximately (a) 0.6 and (b) 0.5 mm. Units
are given in W/m3 .

291

(a)

(b)

Figure 7.12: Plots of atomic hydrogen mole fraction in the plasma region of MSU MPACVD
Reactor B at 180 Torr and 2.2 kW for mesh spacings of approximately (a) 1.2 and (b) 0.8
mm.

292

(a)

(b)

Figure 7.13: Plots of atomic hydrogen mole fraction in the plasma region of MSU MPACVD
Reactor B at 180 Torr and 2.2 kW for mesh spacings of approximately (a) 0.6 and (b) 0.5
mm.

293

(a)

(b)

Figure 7.14: Plots of electron mole fraction in the plasma region of MSU MPACVD Reactor
B at 180 Torr and 2.2 kW for mesh spacings of approximately (a) 1.2 and (b) 0.8 mm.

294

(a)

(b)

Figure 7.15: Plots of electron mole fraction in the plasma region of MSU MPACVD Reactor
B at 180 Torr and 2.2 kW for mesh spacings of approximately (a) 0.6 and (b) 0.5 mm.

295

fraction solutions given in Fig. 7.14 and Fig. 7.15 exhibit similarly large differences in both
profiles and maximum values. The discontinuous solution at the lowest mesh resolution is
quickly replaced by continuous solutions that steadily increase in maximum electron mole
fraction. This suggests a grid spacing of less than approximately 1 mm is necessary in
capturing a continuous electron mole fraction solution at the given operating conditions.
Similar results to the electron mole fraction solutions were also observed in the electron
number density solutions.
Like the electron mole fraction solutions, the electron temperature solutions provided
in Fig. 7.16 and Fig. 7.17 move from a discontinuous solution at the largest mesh spacing
toward continuous distributions as this numerical sampling rate is increased. Moreover, the
maximum electron temperature also steadily increases. Finally, the electron temperature
becomes more concentrated and elevated at the substrate surface as the mesh resolution is
increased. This corresponds to the increased mesh resolution required to accurately capture
the absorbed power density profile discussed in earlier paragraphs. Thus, the microwave
power is more strongly absorbed and localized about the substrate top surface at higher
pressures. This is likely an important indicator for the increased radical production and
thus deposition rate at the substrate surface at higher pressures. The electron temperature
solution is still changing significantly between the 0.5 and 0.6 mm solutions, thus suggesting a
mesh resolution of at least 0.5 mm is required to accurately capture the electron temperature
solution at higher pressures.
Finally, the effect of mesh resolution on the overall convergence rate of the plasma fluid
module solution can be quantified through the related matrix equation residuals as given in
Fig. 7.18.
The total plasma fluid module transport solution matrix residuals observed in Fig. 7.18
confirm better convergence as the mesh resolution is increased. This is expected, as smaller
changes in the overall solution variable distributions are achieved with this increasing mesh
resolution. This leads to a smaller matrix equation residual error between successive update

296

(a)

(b)

Figure 7.16: Plots of electron temperature in the plasma region of MSU MPACVD Reactor
B at 180 Torr and 2.2 kW for mesh spacings of approximately (a) 1.2 and (b) 0.8 mm. Units
are given in K.

297

(a)

(b)

Figure 7.17: Plots of electron temperature in the plasma region of MSU MPACVD Reactor
B at 180 Torr and 2.2 kW for mesh spacings of approximately (a) 0.6 and (b) 0.5 mm. Units
are given in K.

298

Figure 7.18: Total plasma fluid module transport equation matrix residuals for four mesh
resolutions corresponding to the MSU MPACVD Reactor B configuration at 180 Torr and
2.2 kW.

schemes, and thus better final convergence.
This numerical error or uncertainty can be quantified in the overall solution as well. That
is, the final total residual computed between all iterative matrix equation solutions may be
applied to the physical variable distributions. As a result, uncertainties may be associated
with these values, with particular interest in their maximum values. For example, since the
electron temperature typically comprises the bulk of the residual error in the solution of the
plasma transport equations, its numerical uncertainty is expected to be the largest. This
may be calculated from the minimum total electron temperature residual given at the most
coarse grid spacing in Fig. 7.18. With a minimum residual of approximately 30% over 3600
total samples, this results in a estimated numerical uncertainty of roughly 0.0083%. This
translates to an expected accuracy to the sixth decimal place.
However, the electron temperature rarely converges to the sixth decimal place. Instead,
it will often oscillate about mean values in the fourth or even third decimal place. This
numerical uncertainty and instability is attributed to the solution procedure required by the
stiffness of the overall set of governing equations. The total numerical uncertainty associated
with specifically the electron temperature, and thus the maximum uncertainty applied to the
299

overall simulation, is thusly estimated at roughly 0.1% to 1%. This estimation is based upon
numerical observation and experience with simulation results, and the resulting data plots.
It should be noted that a trade off of this lower convergence limit achieved with increasing mesh resolution also requires significantly greater simulation run time as indicated by
Fig. 7.19.

Figure 7.19: Total simulation run times (in seconds) for four mesh resolutions corresponding
to the MSU MPACVD Reactor B configuration at 180 Torr and 2.2 kW.

Given the results of Fig. 7.19, the choice of mesh resolution clearly has a significant
impact on the numerical efficiency of the complete simulation. That is, although roughly an
order of magnitude better total residual is achieved when decreasing the mesh spacing from
1.2 to 0.5 mm, the effective run time increases more than six fold. Moreover, the relationship
between grid spacing and run time scales worse than linear, meaning a linear decrease in
mesh spacing results in an exponential increase in run time. As a result, a sufficient balance
between accuracy and run time must be achieved.
Combining these observations, a mesh resolution of 0.5 mm or less is suggested within the
plasma region in order to accurately simulate hydrogen-based microwave discharges within
MPACVD reactors at moderate pressures. Similar mesh spacings are typically used in similar

300

moderate pressure simulations, and may contain even smaller grid spacings in the substrate
region [34]. Thus, such grid spacings are recommended for use at these higher pressure
operating conditions as well.

301

CHAPTER 8
SUMMARY
This work details the theory behind, solution of, and results produced from a multiphysics
numerical simulation of hydrogen plasmas in MPACVD systems at moderate pressures. The
simulation is composed of three primary physics modules which are coupled together to
arrive at a single, multiphysics solution. These modules simulate the microwave energy
propagation and absorption within the MPACVD reactor; the chemical, kinetic, and temperature behavior and properties of the hydrogen plasma; and the temperature distribution
within the internal substrate structure. The electromagnetic fields are simulated within the
MPACVD reactor using a finite-difference frequency domain (FDFD) formulation. The microwave power density is extracted and supplied to the plasma fluid module where it is used
in the electron and total energy transport equations. The mass and momentum transport
equations are also solved within the plasma fluid module via a Gauss-Seidel Successive OverRelaxation scheme. The energy flux to the MPACVD reactor substrate is extracted from
the plasma fluid module and passed to the substrate temperature module. The steady heat
equation is solved within the substrate structure and coupled back to the plasma fluid module by forcing the gas temperature at the correct boundary. Finally, the complex electrical
conductivity of the plasma is inserted into the electromagnetics module and used to recalculate the electromagnetic fields. This iterative solution process is strongly under-relaxed and

302

continues until a specified tolerance in the overall system residual is reached.
This multiphysics MPACVD plasma model improves upon previous lower pressure models by including both the average gas flow and substrate temperature in the coupled solution.
The incorporation of the average gas flow via the momentum transport equation update offers
a more accurate solution at higher pressures where convective transport has been hypothesized to affect the plasma properties. The inclusion of the substrate temperature results in
more accurate fluxes and capturing of the critical plasma behavior in the region immediately
surrounding the substrate surface. On top of these new physical processes, a more efficient
electromagnetic solution method was developed and included, significantly improving the
numerical run time of the overall simulation. Finally, the simulation framework and nomenclature were carefully chosen and organized in an effort to make it easier to model a wider
range of MPACVD reactor configurations, operating conditions, and physical processes.
Many insights into the behavior and properties related to hydrogen plasmas in MPACVD
reactors at moderate pressure operating conditions were achieved. Most notable were the
effective role of convective forces within the plasma at higher operating pressures, and the
influence of the substrate temperature on fluxes at the deposition surface. The average gas
flow throughout the plasma region at typical operating conditions within MPACVD reactors
remains quite low, never rising above 15 cm/s at any point in the simulations. As a result,
the related forces due to the gas flow acting upon the plasma remain too small to significantly
influence its properties or location. This was evident regardless of operating conditions and
reactor configuration.
The substrate temperature varies significantly across this structure, and is greatly affected
by both the plasma properties and its internal material makeup. Simulation results suggest
the heat flux to the substrate due to gas temperature gradient in the plasma dominates over
the energy flux due to hydrogen collisions and recombination. These flux contributions are
themselves significantly affected by the plasma properties in the immediate vicinity of the
substrate. As a result, the substrate vertical position and even reactor configuration also

303

significantly impact the steady substrate temperature distribution. Changes to the substrate
position within the reactor of as little as 5 mm can alter the resulting substrate temperature
by as much as 200 K. A pseudo-gas layer was inserted and its material properties adjusted
in order to approximate imperfect heat conduction across substrate materials and match
substrate temperature experimental data [5]. The diamond deposition rate may be estimated
from the hydrogen flux to the substrate surface, but is left for future implementations of the
simulation.
The coupled, multiphysics simulation of MPACVD systems and their hydrogen plasmas
presented and discussed in this work is suggested for use in the future design and prediction
of MPACVD reactors. It is the first known simulation that incorporates numerical solutions
of the electromagnetic fields, plasma behavior, gas flow, and substrate temperature within
MPACVD systems at moderate pressures into a single, coupled, multiphysical manner. As
such, it is expected to be more accurate than previous multiphysics simulations including only
some of these physical processes at lower pressures [16, 34, 35, 64, 76]. The code structure also
makes it easy to analyze and predict the behavior of a wide range of MPACVD reactors at
various operating conditions, and including numerous physical processes. Thus, the present
simulation is recommended for the future prediction of, and insight into, the behavior of
hydrogen plasmas in MPACVD systems.

8.1

Future work

Further improvements of the presented simulation are recommended in several areas as described in the paragraphs below. The majority of these improvements add or ensure numerical accuracy, or capture further physical processes within the MPACVD system.

304

8.1.1

Further physics modules

The incorporation of more physics modules would ensure a more accurate overall simulation
while also providing further insight into the behavior and properties of MPACVD system
plasmas. Some of these described below have been included in other MPACVD or separate
simulations.
The decision to exclude hydrocarbon species in the present simulation is due in part to
the resulting simplicity related to the reduced number of species and chemical reactions to
track. Their inclusion is not uncommon in similar simulations and results in the capture of
many more important chemical reactions [33, 34, 42, 64, 76]. Thus, the inclusion of hydrocarbon species and their related chemical reactions is recommended in future versions of the
multiphysics simulation.
Another suggested improvement on the present simulation is the explicit inclusion of a
plasma sheath model. This plasma sheath is currently approximated through the numerical boundary conditions and surface interaction rates. However, a separate model for this
behavior would undoubtedly result in more accurate fluxes at the substrate surface, and
perhaps the simulation as a whole. Due to its extremely narrow interaction zone at the
plasma domain boundaries, a one-dimensional model is recommended.
Finally, a diamond deposition model is recommended for inclusion in future versions of
the simulation. Plasma discharges within experimental MPACVD systems are currently used
as more diagnostic tools for estimating the deposition rates associated with these systems.
The development of a diamond deposition model would instead provide a direct prediction
of the deposition rate prior to the experiment. It would likely remain a one-dimensional
model, and could take on the form of previously developed models [87].
The above mentioned and briefly discussed physical modules are by no means the only
physical processes that could be added to it to improve its accuracy and the capturing of
more physical phenomena within MPACVD systems. Instead, these are recommended as

305

the next foreseeable steps in future implementations of the present simulation.
Several improvements are also recommended with regards to the numerical solution
scheme. Moving to more advanced relaxation coefficient update schemes in the plasma
transport equations could improve the overall numerical efficiency., A new plasma transport
equations solution method could also improve the numerical run time. Similarly, the parallelization of the numerical code would significantly improve the numerical efficiency and
run time if parallel computer architectures are available. Parallelization of the code could be
implemented in several areas, including the sparse solver architecture and method, the SOR
routine over solution variables within the plasma fluid module, and the updating of physical
constants and rate coefficients also in the plasma fluid module. Thus, several opportunities
for this parallelization exist, which suggests the simulation may be capable of running much
faster.

8.2

Recommendations for experiments

As a final remark, several recommendations are made for the gathering of experimental data.
Experimental data served as the primary method for validating the present simulation. As
such, much of this experimental data was only available at lower pressures and for several
specific physical variables including substrate temperature, electron density, and gas temperature. However, important physical values including average gas speeds and diffusion
coefficients were unavailable. Moreover, all of these variables except substrate temperature
were unavailable at pressures exceeding 200 Torr. Thus, it is recommended that future experiments measure these data within the higher pressure regime to assist in the validation
of future simulations.

306

APPENDICES

307

Appendix A: Collision Integrals
The calculation of the collision integral associated with the collision of species and various
chemical reactions is vital to the accuracy of the self-consistent simulation. That is because
this data is used to not only calculate the total effective mass diffusion coefficients, but also
the thermal conductivities associated with each species and mode. The collision integrals
must also be calculated in an efficient manner, as they are updated along with the plasma
physical constants. To this end, they are often approximated by logarithmic fits to experimental data. The data associated with these fits is provided in the following section. Direct
comparisons with experimental data are also provided where available.

Coefficients
The collision integrals in the present simulation take on the form
(ii)
Ω
= T C1 exp C (ii,Np ,j,k)
jk
Np
C1 =

Cii,i,j,k [ln (T )]Np −i−1

(1a)

(1b)

i=1
where the indices j and k represent the two species involved in the collision, ii is denotes
the collision integral of either the first or second kind, T is the temperature associated with
the collision, Np is the number of coefficients associated with the numerical fit, and C is a
numerical fit coefficient array. This constant is provided below in Table A.1.

308

Table A.1: Collision integral polynomial fit coefficients.
Species 1 Species 2

Type C1

C2

C3

C4

1

1

0.0

-5.0949e-2

0.47409

0.80454

2

0.0

-5.889e-2

0.64142

0.18926

1

0.0

-3.3403e-2

0.13563

2.0844

2

0.0

-3.1267e-2

0.12675

2.2555

1

0.0

-3.3403e-2

0.13563

2.0844

2

0.0

-3.1267e-2

0.12675

2.2555

1

0.0

-3.3403e-2

0.13563

2.0844

2

0.0

-3.1267e-2

0.12675

2.2555

1

0.0

0.0

-0.4

5.6096

2

0.0

0.0

-0.4

5.6313

1

0.0

0.0

-0.4

5.6096

2

0.0

0.0

-0.4

5.6313

1

0.0

0.0

-0.4

5.6096

2

0.0

0.0

-0.4

5.6313

1

0.0

0.0

-0.4

5.6096

2

0.0

0.0

-0.4

5.6313

1

-3.4122e-2

0.79562

-5.9754

15.8856

2

-3.4122e-2

0.79562

-5.9754

15.8856

1

0.0

5.8478e-3

-0.42247

4.2854

2

0.0

-8.1062e-3 -0.20755

3.6016

1

0.0

5.8478e-3

-0.42247

4.2854

2

0.0

-8.1062e-3 -0.20755

3.6016

1

0.0

5.8478e-3

-0.42247

4.2854

2

0.0

-8.1062e-3 -0.20755

3.6016

1

1

1

1

1

1

1

1

2

2

2

1

2

3

4

5

6

7

8

9

2

3

4

309

Table A.1 (contd)
Species 1 Species 2

Type C1

C2

C3

C4

2

1

-7.372e-6

5.9999e-4

-2.1169e-2

3.4623

2

0.0

0.0

-0.4

5.6313

1

-7.372e-6

5.9999e-4

-2.1169e-2

3.4623

2

0.0

0.0

-0.4

5.6313

1

-7.372e-6

5.9999e-4

-2.1169e-2

3.4623

2

0.0

0.0

-0.4

5.6313

1

-7.372e-6

5.9999e-4

-2.1169e-2

3.4623

2

0.0

0.0

-0.4

5.6313

1

1.7253e-2

-0.47276

3.9085

-7.6063

2

1.0911e-2

-0.32361

2.7619

-4.8733

1

0.0

5.8478e-3

-0.42247

4.2854

2

0.0

-8.1062e-3

-0.20755

3.6016

1

0.0

5.8478e-3

-0.42247

4.2854

2

0.0

-8.1062e-3

-0.20755

3.6016

1

-7.372e-6

5.9999e-4

-2.1169e-2

3.4623

2

0.0

0.0

-0.4

5.6313

1

-7.372e-6

5.9999e-4

-2.1169e-2

3.4623

2

0.0

0.0

-0.4

5.6313

1

-7.372e-6

5.9999e-4

-2.1169e-2

3.4623

2

0.0

0.0

-0.4

5.6313

1

-7.372e-6

5.9999e-4

-2.1169e-2

3.4623

2

0.0

0.0

-0.4

5.6313

1

1.7253e-2

-0.47276

3.9085

-7.6063

2

1.0911e-2

-0.32361

2.7619

-4.8733

1

0.0

5.8478e-3

-0.42247

4.2854

2

2

2

2

3

3

3

3

3

3

3

4

5

6

7

8

9

3

4

5

6

7

8

9

4

310

Table A.1 (contd)
Species 1 Species 2

6

6

7

3.4623

0.0

0.0

-0.4

5.6313

1

-7.372e-6

5.9999e-4

-2.1169e-2

3.4623

0.0

0.0

-0.4

5.6313

1

-7.372e-6

5.9999e-4

-2.1169e-2

3.4623

0.0

0.0

-0.4

5.6313

1

-7.372e-6

5.9999e-4

-2.1169e-2

3.4623

0.0

0.0

-0.4

5.6313

1

1.7253e-2

-0.47276

3.9085

-7.6063

1.0911e-2

-0.32361

2.7619

-4.8733

1

0.0

0.0

-2.0

22.6789

0.0

0.0

-2.0

23.2154

1

0.0

0.0

-2.0

22.6789

0.0

0.0

-2.0

23.2154

1

0.0

0.0

-2.0

22.6789

0.0

0.0

-2.0

23.2154

1

0.0

0.0

-2.0

22.6789

0.0

0.0

-2.0

23.2154

1

0.0

0.0

-2.0

22.6789

0.0

0.0

-2.0

23.2154

1

0.0

0.0

-2.0

22.6789

2

6

9

-2.1169e-2

2

5

8

5.9999e-4

2

5

7

-7.372e-6

2

5

6

1

2

5

5

3.6016

2

5

9

-0.20755

2

4

8

-8.1062e-3

2

4

7

0.0

2

4

6

C4

2

4

C3

2

5

C2

2
4

Type C1

0.0

0.0

-2.0

23.2154

1

0.0

0.0

-2.0

22.6789

2

0.0

0.0

-2.0

23.2154

311

Table A.1 (contd)
Species 1 Species 2

Type C1

C2

C3

C4

6

1

0.0

0.0

-2.0

22.6789

2

0.0

0.0

-2.0

23.2154

1

0.0

0.0

-2.0

22.6789

2

0.0

0.0

-2.0

23.2154

1

0.0

0.0

-2.0

22.6789

2

0.0

0.0

-2.0

23.2154

1

0.0

0.0

-2.0

22.6789

2

0.0

0.0

-2.0

23.2154

1

0.0

0.0

-2.0

22.6789

2

0.0

0.0

-2.0

23.2154

1

0.0

0.0

-2.0

22.6789

2

0.0

0.0

-2.0

23.2154

1

0.0

0.0

-2.0

22.6789

2

0.0

0.0

-2.0

23.2154

1

0.0

0.0

-2.0

22.6789

2

0.0

0.0

-2.0

23.1613

6

7

7

7

8

8

9

8

9

7

8

9

8

9

9

Validation
Since these data provided in Table A.1 only represent a numerical fit to experimental data,
some differences between these two data sets are expected. For reference, the calculated collision integrals based upon the above data for several interactions of hydrogen-based species
are plotted against their corresponding experimental data in Fig. A.1 through Fig. A.11.
The experimental data were extracted from [88].
312

Figure A.1: Experimental data and corresponding numerical approximation for the collision
integral of type 1 in the H − H+ interaction.

Figure A.2: Experimental data and corresponding numerical approximation for the collision
integral of type 2 in the H − H+ interaction.

313

Figure A.3: Experimental data and corresponding numerical approximation for the collision
integral of type 1 in the H − e interaction.

Figure A.4: Experimental data and corresponding numerical approximation for the collision
integral of type 2 in the H − e interaction.

314

Figure A.5: Experimental data and corresponding numerical approximation for the collision
integral of type 1 in the H2 − e interaction.

Figure A.6: Experimental data and corresponding numerical approximation for the collision
integral of type 1 in the H2 self interaction.

315

Figure A.7: Experimental data and corresponding numerical approximation for the collision
integral of type 2 in the H2 self interaction.

Figure A.8: Experimental data and corresponding numerical approximation for the collision
integral of type 1 in the H self interaction.

316

Figure A.9: Experimental data and corresponding numerical approximation for the collision
integral of type 2 in the H self interaction.

Figure A.10: Experimental data and corresponding numerical approximation for the collision
integral of type 1 in the H − H2 interaction.

317

Figure A.11: Experimental data and corresponding numerical approximation for the collision
integral of type 2 in the H − H2 interaction.

As can be seen from this series of data presented in Fig. A.1 through Fig. A.11, the
approximated logarithmic to the experimental data is sufficient for most interactions. An
exception is the collision integral of the first kind for the H − H+ interaction, which is
significantly inaccurate for a majority of given temperatures. However, this interaction is
not as significant as many of the others included in the simulation, and thus represents an
acceptable fit.

318

Appendix B: List of Chemical
Reactions
The list of chemical reactions used in the simulation are provided, with all falling into one
of two categories. The first includes those whose rates follow the Arrhenius equation form,
while the second is composed of those reactions which do not follow the Arrhenius equation.
In order to accurately simulate the chemical reaction rates of the second category, an exponential polynomial fit is applied. The various coefficients corresponding to the chemical
reaction rate calculations for each category are listed below in Table B.1 and Table B.2.
The recombination of chemical species at material surfaces are governed by seven surface reactions listed in Table B.3. These are given in qualitative form, with only species,
stoichiometric, and recombination coefficient data given.

319

Table B.1: List of Arrhenius chemical reactions.
Reaction
H+ + 2e− → H + e−
H+ + e − → H
H2 + 2H → 2H2
3H → H2 + H
H + H+ → H2 + H+
2
H2 + H+ → H + H+
3
2
2H2 → H2 + 2H
H2 + H → 3H
H + H− → 2H + e−
H + H− → H2 + e−
H2 + H+ → H + H+
2
H+ + H− → 2H
H+ + 2e− → 2H + e−
2
H+ + 2e− → H2 + H + e−
3
2H2 + H+ → H2 + H+
3
H+ + H − → H2 + H
2
H+ + H− → 2H2
3
H2 + H(n=2) → H+ + e−
3
H2 + H(n=3) → H+ + e−
3
H(n=3) → H(n=2) + γ

k0 (m3 /s mol)
η
3.63 × 1025
-4.0
8
1.46 × 10
-0.699
5
1.0 × 10
-0.6
3
3.2 × 10
0.0
8
3.85 × 10
0.0
9
1.27 × 10
0.0
8.61 × 1011
-0.7
10
2.7 × 10
-0.1
2
4.5 × 10
1.5
9
1.43 × 10
-0.15
8
1.94 × 10
0.0
11
1.78 × 10
0.0
9
3.17 × 10
-4.5
3.17 × 109
-4.5
8
1.95 × 10
-4.5
2.08 × 1012
-0.5
12
2.08 × 10
-0.5
7
1.675 × 10
0.5
1.675 × 107
0.5
7
6.58 × 10
0.0

320

Ta (K)

Tg , Te

0.0
0.0
0.0
0.0
0.0
0.0
5.253 × 104
5.253 × 104
6.98 × 102
8.15 × 102
2.1856 × 104
1.768 × 103
0.0
0.0
0.0
0.0
0.0
0.0
0.0
0.0

Te
Te
Tg
Tg
Tg
Tg
Tg
Tg
Tg
Tg
Tg
Tg
Te
Tg
Tg
Tg
Tg
Tg
Tg
Tg

Table B.2: List of non-Arrhenius chemical reactions. All reactions depend on the electron
temperature, Te .
Reaction

C1

C2

H2 + e− → H+ + 2e−
2
H + e− → H+ + 2e−
H2 + e− → 2H + e−
H+ + e− → 3H
3
H+ + e− → H2 + H
3
H+ + e− → 2H + H+ + e−
3
H2 + e− → H + H−
H+ + e− → H + H+ + e−
2
H+ + e− → 2H
2
H− + e− → H + 2e−
H2 + e− → H + H(n=2) + e−
H2 + e− → H + H(n=3) + e−
H + e− → H(n=2) + e−
H + e− → H(n=3) + e−

-2.37e3
-1.84e3
-1.15e3
-4.24e1
3.30e1
-2.42e3
-1.70e2
-2.93
-2.20
3.13e1
-2.13e3
-2.57e3
-1.40e3
-1.60e3

3.95e2
3.04e2
1.91e2
1.25e1
-1.25
4.09e2
3.32e1
2.25
4.61
-0.963
3.65e2
4.43e2
2.44e2
2.77e2

C3

C4

-1.64e1 1.88e6 -6.78e9
-1.25e1 1.54e6 -5.76e9
-7.77
1.03e6 -4.08e9
-6.1
2.58e4 -4.26e7
3.06e-2 -1.53e4 4.03e7
-1.72e1 1.81e6 -6.28e9
-1.48
1.29e5 -5.07e8
2.13e-2 8.02e4 -5.74e8
-0.214 6.91e3 -9.08e6
4.70e-2 -5.05e4 1.25e8
-1.56e1 1.28e6 -3.16e9
-1.90e1 1.54e6 -3.81e9
-1.05e1 8.09e5 -1.78e9
-1.19e1 9.50e5 -2.21e9

Table B.3: List of surface chemical reactions.
Surface Reaction

β

2H → H2
H(n=2) → H
H(n=3) → H
H+ → H
H+ → H2
2
H+ → H + H2
3
H− → H

0.1
1.0
1.0
1.0
1.0
1.0
1.0

321

C5

C6
6.74e12
5.78e12
4.22e12
3.15e10
-3.50e10
6.20e12
5.04e12
6.44e11
5.97e9
-1.25e11
-2.41e12
-2.23e12
-2.87e12
-2.71e12

Appendix C: Physical Constants
Calculations
The derivations of the numerous physical constants present in Eq. (3.18), Eq. (3.27), Eq. (3.33),
and Eq. (3.39) and related terms are derived in the following paragraphs. Some of these
constants are based upon empirical observations and data, while others are derived from
quantum mechanical or classical kinetic theories. These are available with further details
and citations in other works [63, 81].

Collision Integral
The interactions between the various chemical species within the plasma gas mixture are
important to the plasma behavior and characteristics. In fact, these inter-atomic forces and
interactions are the primary source of new species production and internal energy transfer
within the plasma. This suggests the accurate capture of these interactions is important to
the overall accuracy of the numerical model. Not surprisingly, collisions and their related
physical quantities appear in several of the physical constants listed in the previously derived
transport equations. Their characteristics are well documented and have been developed
over several decades of research [81, 88]. The various forces and considerations required to
accurately resolve these interactions at the molecular level mark the starting point for the
derivation of said physical constants. The majority of these derivations are available in other
works, and are summarized here for convenience [63, 81, 88].

322

Collisions between electrons, ions, and neutrals may be calculated by estimating the
collision integral between each species. Consider two particles moving along trajectories in
the absence of external forces. If their trajectories are close enough, they will both influence
the position and velocity of the other, resulting in an altered ending trajectory. This crude
example is often referred to as a binary system, due to only two particles being present.
Many physical constants may be extrapolated and calculated from this simple interaction
system. It is used as the basis for the following derivations, and is an estimation of the
weighted average of the cross sectional area of the target mass experienced by the projectile
in plasma particle interactions. It has been shown that these collision integrals may be used
in combinations to represent transport coefficients [81]. These coefficients are only referenced
here, with more detailed descriptions available elsewhere [81]. The collision integral may be
calculated analytically by performing the following integrations [63]
∞ π
2 2l+3 1 − cosk χ σ sin χdχdγ
ij
0 0 exp (−γ )γ
(k,l)
¯
π Ωij = 4π
∞ π
2 2l+3 1 − cosk χ sin χdχdγ
0 0 exp (−γ )γ

(2)

¯ (k,l) represents the collision integral with units of area, σ is the collision cross
where π Ωij
ij
section for the pair of interaction species, i and j, χ is the scattering angle between the two
interacting particles, and γ is their reduced velocity as calculated [63]

γ = v

mi mj
2 mi + mj kb T

1/2


(3)

where v is the relative velocity of the projectile particle taken with respect to the target
¯ (k,l) follows the later works by Yos and Lee [63, 88]. It
particle at rest. This notation of π Ωij
also represents the same collision integral derived by Hirschfelder, et al., where it is instead
represented by simply Ωkl with no leading π symbol [81]. This explicit notation used here
ij
is meant to avoid the confusion over this representation which has appeared in previous

323

simulations.
The integral calculation from Eq. (2) can be quite difficult to calculate and requires
prior knowledge of numerous physical properties including initial relative velocity between
species and their respective trajectories. With collision frequencies in MPACVD systems
often within the GHz to THz range, calculating these integrals for every collision is not
feasible. Instead, experimental observations are used to estimate these collisions in the
plasma under certain physical conditions related to pressure, species temperature, and its
chemical makeup. These collision approximations were observed, with their data tabulated
and summarized, for the case of hydrogen ions and electron collisions by Yos [88]. In this
sense, the basic collision integral data used in the calculation of numerous physical transport
coefficients is taken from experimental measurements. Polynomial fits to these data have
been tabulated and are used here to approximate these experimental collision integral data
[16]. These polynomial fits take the form
βγ
¯ (γ)
π Ωsr = Tg exp α4,γ
3
βγ =

αi,γ log Tg 3−i

(4a)

(4b)

i=1
where αi are coefficients used to approximate the polynomial fit to the empirical collision
integral data, and γ represents the data associated with, and collision integral of, either the
first or second kind. The later notation assumes that the final collision state as denoted by
k, l in Eq. (2) requires k = l. This polynomial fit is essentially a log-polynomial fit, as is
seen in Eq. (4). The result is a set of relatively accurate and continuous collision integral
functions spanning the entire expected temperature range and for all unique interactions.
The modified collision integrals may then be tabulated once the collision integral functions
themselves are known. These are also associated with collisions of the first and second kind,

324

as given in Eq. (5) [63]
8 ¯ (1,1)
(1)
∆sr (T ) = π Ωsr
3

2ms mr
πkb T (ms + mr )

(5a)

16 ¯ (2,2)
(2)
∆sr (T ) = π Ωsr
5

2ms mr
πkb T (ms + mr )

(5b)

having units of m s.

Binary Diffusion Coefficient
The diffusion of particles from regions of high particle concentrations to lower concentration
is a well-known phenomenon governed by Fick’s Law. Although simple in the case of a single
gas, this process can become complex and difficult to mathematically represent in the case
of multicomponent plasmas. In order to better understand this process, a binary system of
two types of interacting particles is assumed. Here, the binary diffusion coefficient governing
the rate of particle flux through an arbitrary plane may be calculated. It is expressed for the
two species, s and r, and is based upon the modified collision integral derived the previous
paragraph as given in [63]
Dsr =

kb Tg
(1)
p∆sr (Tg )

(6)

(1)
where Dsr is the binary diffusion coefficient and ∆sr (Tg ) is the modified collision integral
of the first kind in calculated in terms of the gas temperature. The units of the binary
diffusion coefficient are m2 /s, which may be thought of as representing the rate of flux of a
unitless parameter through a fixed area.

Mass Density Diffusion
At this point, the binary diffusion coefficient has been stated for any two-component system.
However, it is not the term present in the transport equations, which must be calculated for

325

a multicomponent plasma. Instead, the diffusion coefficient for individual species must be
stated, as given in Eq. (7) [16, 63]
ctot Ms 1 − Ms xs
M
Ds =
Ns
xr
Dsr
r=1
r=s

(7)

where ctot is the total molar concentration of the plasma, Ms is the molar mass of species s,
and M is the total molar mass of the plasma. The diffusion coefficient in Eq. (7) represents
the diffusion flux of the volumetric mass density. It may be expressed in terms of the standard
diffusion coefficient as
Ds = ρs Ds,0

(8)

where Ds is the mass diffusion coefficient for species s present in the transport equations
previously derived, and Ds,0 is the standard diffusion coefficient representing the flux of a
unitless quantity related to species s. This mass density diffusion is substituted here in order
to take advantage of its otherwise mass density product calculation present during each of
its uses in the scalar transport equations. That is, the species diffusion coefficient present
in the transport equations in each case is multiplied by the species mass density. This is
common in MPACVD simulations [16].

Ambipolar Diffusion
Due to the presence of oppositely charged species, the ambipolar diffusion coefficient must
be introduced. As was outlined earlier, the behavior of charged particles is influenced in
the presence of electromagnetic fields. This behavior can be calculated through the Lorentz
force law given in Eq. (3.11), resulting in positively charged particles accelerating along the
electric field lines, while negatively charged particles move in the opposite direction.
MPACVD plasmas are known to remain electrically neutral over distances greater than

326

the deBye length suggesting the ion and electron densities are equal, with ne = ni . While
the electron diffusion velocity in the presence of an external electric field is much greater
than that of the ions, this does not lead to drastic changes in overall charge distribution.
Instead, a static electric field forms between the rapidly diffusing electrons and more slowly
diffusing ions. This electrostatic field acts to slow the diffusion velocity of electrons, while
accelerating the ions. The effective rate of diffusion for these charged species is thus altered,
and becomes the ambipolar diffusion coefficient. A detailed derivation of this ambipolar
diffusion coefficient is provided elsewhere and is restated here for convenience as [67]

Da = Ds 1 +

Te qs
Tg

(9)

It is clear from Eq. (9) that the modifying factor multiplying the volumetric mass density
diffusion coefficient is always greater than two, since Te > Tg for MPACVD system plasmas.
In the present system, |qs /q0 | = 1, meaning only singly-charged ions are included. This leads
to a modified ion diffusion velocity that is at least twice the original ion diffusion velocity
simply due to the presence of the more quickly diffusing electrons. The above ambipolar
mass diffusion coefficient replaces all instances of the mass density diffusion coefficient for
ion species in the transport equations. The electron mass density diffusion coefficient is
calculated using an analytic function dependent on empirical data for hydrogen plasmas
[16].

Viscosity
The viscosity of a multicomponent gas mixture is derived elsewhere and stated here for
convenience [63]. It is used as the scalar kinematic viscosity term in Eq. (3.27) and given by

µ=

Ns
m s xs
me xe
+
(2)
(2)
Ns x ∆(2) (T )
s=1 xe ∆se (Te ) +
r=1 xr ∆sr (Tg )
r=1 r er e
s=e−
r=e−

327

(10)

(2)
where µ is the viscosity of the plasma, and ∆sr is the modified collision integral of the
second kind between species s and r. Note the separate treatment of electrons in Eq. (10)
as referenced from the e− terms.

Thermal Conductivities
The thermal conductivity terms present in both Eq. (3.33) and Eq. (3.39) are important for
calculating the rate of energy transfer between species and the cavity surroundings. That
is, their presence is related to the heat flux throughout the plasma region and is important
for the accurate capturing of physical processes. Likewise, these conductivities will become
even more important during the numerical solution procedure, where they will be used in
updating of temperatures via matrix inversion. Thus, their derivation is important to the
accuracy of the electron and total energy transport equations.
Heat conduction within the plasma occurs through both collisional processes and the
diffusion of reacting chemical species [63]. The latter is explicitly treated and already present
in the energy transport equations in Eq. (3.33) and Eq. (3.39). Thermal conduction via
collisional processes is included by treating the contribution of the three kinematic hydrogen
energy modes (translational, rotational, and vibrational) and electrons. They are functions
of the collisional processes resulting in the excitation and filling of the various modes. These
expressions become [63]

λt =

15
k
4 b

Ns
s=1,s=e

xs
(2)
Ns
asr xr ∆sr (Tg )
r=1,r=e
xs

λr = kb
s=mol
λv = kb
s=mol

Ns x ∆(1) (T )
r=1 r sr g
xs
Ns x ∆(1) (T )
r=1 r sr g

328

(11a)

(11b)

(11c)

15
λe = kb
4

Ns
s=1

xs
Ns x ∆(2) (T )
r=1 r er e

(11d)

where
asr = 1 +

(1 − (Ms /Mr )) (0.45 − 2.54(Ms /Mr ))
(1 + (Ms /Mr ))2

(11e)

and λt , λr , and λv represent the thermal conductivities of the molecular hydrogen translational, rotational, and vibrational modes, respectively, λe is the thermal conductivity of
electrons, and s = mol denotes the summation over only molecular species. It is clear from
Eq. (11) that the rotational and vibrational thermal conductivity coefficients are identical
[63].
Finally, the translational and rotational thermal conductivity terms are often combined
to form the translational-rotational thermal conductivity of the hydrogen species, λtr via
λtr = λt + λr

(12)

This term is used in the updating of the neutral and ion species temperature in the total
energy transport equation from Eq. (3.39).

Enthalpy
Before any plasma species can react, there must be sufficient energy for activation, as well
as the energy to produce the space for the new species, with the latter being a measure
of enthalpy. The enthalpy calculation is based upon the difference in stoichiometric coefficients both prior to and following the chemical reaction, and is extrapolated by a fifth-order
polynomial fit with respect to species temperature [63]
(0) (i)
(i)
hs
η1 − η2

hs =
i

329

(13a)

(0)
hs =

5
i=1

αi
iT i

(13b)

where η1 and η2 are stoichiometric coefficients representing the species present prior to and
following the chemical reaction, αi is the smoothing coefficient for each species, and T is the
governing temperature depending on the reaction.

Chemical Reactions
The chemical reactions ongoing between species represent the source of mass production
and consumption within the plasma. Their rates are important to many other physical
processes, as mass and density of the various species significantly influence many other
plasma properties. The majority of the reaction rates for hydrogen-based plasmas within the
temperature ranges common in MPACVD reactors are well known. These specific reactions
are tabulated elsewhere [16, 33].
Chemical reaction rates often take the form of the modified Arrhenius equation as given
by [45]
k = AT η e−Ea /RT

(14)

where k is the chemical reaction rate, A is the pre-exponential factor (or frequency factor), T
is the inherent plasma temperature, η is the Stoichiometric coefficient, R is the gas constant,
and Ea is the activation energy of the chemical reaction. Many of the chemical reactions
tabulated for MPACVD plasmas take this form. However, some chemical reactions do not
follow the Arrhenius form from Eq. (14). In these particular cases, a hybrid logarithmicpolynomial fit to experimental data is employed [16, 45]

k = exp (kr0 )
3
kr0 =

κi logi−1 (RT ) +

i=1

(15a)
6
i=4

330

κi
(RT )i−3

(15b)

where κi are the chemical reaction empirical fit coefficients. These chemical reaction rates
will be used in the following paragraphs to calculate the net rate of mass production and
consumption for each chemical species.

Rate of Mass Production
The volumetric rate of mass production represents the net rate of change in the mass of each
species, and is the source term observed on the right side of Eq. (3.18). This volumetric
rate of mass production may be calculated based on the net rate of species production
via chemical reactions, its mass, and the stoichiometric coefficients associated with its final
reaction products. Its general form is given as [45]


Nr

Ws = Ms 
r=1


(1)
ηrq
(2)
(1)
kr ΠNs cq
ηsr − ηsr 
q=1

(16)

where Nr is the total number of chemical reactions included in the plasma simulation, cq
is the molar concentration of species q, and kr is the reaction rate constant for reaction r.
Once the chemical reaction rates are calculated from the above derived expressions and fits
to empirical data, the rate production of species in Eq. (16) becomes available. This rate of
mass production is tabulated for all chemical species contained in the plasma.

Energy Exchange Rates
The final group of terms from the plasma transport equations left to calculate are the various
volumetric rates of energy exchange found in the two energy transport equations, Eq. (3.33)
and Eq. (3.39). These include the exchange of energy between the various molecular modes,
electrons, chemical reactions, and loss due to electromagnetic radiation. They have been
derived elsewhere in detail, and are listed here for convenience [63].

331

Electron-Vibrational Energy Exchange
The electron-vibrational energy exchange rate represents the rate of energy passed from electrons to the molecular hydrogen vibration mode. Molecular hydrogen is a diatomic molecule
with allowed vibrational modes predicted by the quantum oscillator model. This means the
theoretical number of possible vibration modes is infinite. However, this number is essentially
limited by the energy of the system. Thus, only the first fourteen modes are allowed here,
as keeping with previous simulations [16]. This number is expected to completely capture
all vibrational modes based upon the energies typically present in MPACVD plasmas.
Energy passing from electrons to the molecular hydrogen vibration modes may occur
along one of two pathways [32]. One method includes energy transformation from electron
kinetic energy to the hydrogen vibrational mode [32]
H2 (v) + e− → H2 (w) + e−

(17)

where H2 (v) and H2 (w) represent the prior and final molecular hydrogen vibration modes,
with a net gain of energy being assumed. This energy rate may be calculated in generic form
via the Landau-Teller relaxation equation as [32, 63]
(2)
∂ev
ev (Te ) − ev
=
∂t
τev
1
τev = ne (1 − exp (−θv /Te ))2
2

(18a)

k0j j 2 dj

−1
(18b)

(2)
where ev is the vibrational energy per unit mass, ev (Te ) is the equilibrium vibrational energy at the electron temperature, τev is the relaxation time for the electron-vibration energy
exchange process, θv is the characteristic temperature of vibration, and k0j (j) is the rate
coefficient for the vibrational mode to transition from the ground to the j th vibrational state
[63]. The rate coefficient for the ground state vibrational model transition may be calculated

332

by integrating over the Maxwellian EEDF for each transition [45]. However, this rate coefficient and thus electron-vibration relaxation time would be prohibitively costly to calculate
at each update step. Instead, a polynomial fit to the impacting electron temperature is used
to numerically approximate this term, as given by
1
τev = ne (1 − exp (−θv /Te ))2
2
N
1 ev
αi [log (Te )]i−1
≈
ne
i=1

k0j (j)2 dj

−1

(19)

where αi are the polynomial fit coefficients associated with each polynomial order. As a
result, the final expression for the numerical calculation of the energy exchange rate between
the electrons and the molecular hydrogen vibrational modes due to direct excitation becomes
(2)
cs Ms ev,s − ev,s

(1)
Qev = s=mol
N
1 ev α [log (T )]i−1
e
i
ne
i=1

(20)

where the denominator in Eq. (20) is substituted from Eq. (19).
Energy transfer from the electrons to the molecular hydrogen vibrational modes may also
occur through excitation of hydrogen singlets, which then radiate and return to previous or
lower vibrational modes. The rate coefficients of this energy transfer are calculated dependent
on hydrogen cross sections and the transition probabilities [45]. This results in the following
exchange rate equation [45]
B
C
Q1v = nH Rθv kev (Te )SB (Tv ) + kev (Te )SC (Tv )
ev
2

(21)

where Q1v is the volumetric energy exchange rate density of energy transfer from electrons
ev
to the hydrogen vibration modes via an intermediate energy state, θv is the characteristic
vibration energy, kev are the rate coefficients for electron excitation of the hydrogen singlets,
333

and SB and SC are dimensionless probability functions [45]. These S functions may be
calculated assuming an anharmonic oscillator model, yielding analytic vibrational energy
levels [45]. Higher order polynomial fits of experimental data are used to approximate these
functions taking on the form
2
SB,C (Tv ) =

i
ai Tv

(22)

i=0
where ai represent polynomial fitting coefficients to experimental data, and are provided
B,C
elsewhere [45]. The electron-vibrational rate constants, kev , are similarly calculated from
curve fits of cross sectional data from Buckman and Phelps[89].
The resultant rate of energy lost by electrons through their excitation of singlet vibrational modes may be calculated as
B
C
Q1e = ne nH kev B + kev C
ev
2

(23)

where B and C are the dimensionless energies levels of the vibrational modes associated
with the hydrogen B and C states.
From these three modes of energy exchange between electrons and vibrational modes,
the net rate of energy transfer from the electron to molecular hydrogen vibrational modes
may be calculated through their summation via
(1)
Qev = Qev + Q1v − Q1e
ev
ev

(24)

where Qev is the net rate of energy exchange between electrons and hydrogen vibrational
modes as seen in Eq. (3.33). The net rate of energy radiated by electrons may also be
computed as simply the difference between Q1v and Q1e becoming
ev
ev
Qrad = Q1e − Q1v
ev
ev

334

(25)

where Qrad is the rate of electron energy loss due to radiative processes present in Eq. (3.39).
Translational/Rotational-Electron Energy Exchange
The energy transfer rate between electrons and the translational-rotational kinetic states of
the heavy particles is derived by Hassouni, et al. elsewhere [45]. However, it is restated here
for convenience as
8RTe
πme

Qet = 3Rne Te − Tg

σse
s=e−

ns n
m2
s

(26)

where σse is the momentum transfer collision cross section, and n is the total gas density
for all present species.

Translational/Rotational-Vibrational Energy Exchange
Due to the relatively high pressure of operation for most MPACVD systems, the rotational,
translation, and vibrational modes are all assumed to be in thermal equilibrium. This suggests that the vibrational temperature is equivalent to the gas temperature. As such the
energy exchange rate between the translational-rotational and vibrational modes in molecular hydrogen is nominally zero in most MPACVD systems. Thus, its derivation and final
expression are neglected.

Electron-Chemical Energy Exchange
Electrons also exchange energy through the activation of chemical reactions and the resultant
mass production. This general process is again often derived from empirical data, but can
be expressed in general form following external works as [45]
Nr
Qec =

ηre vr act,r
r=1

335

(27)

where ηre is the electron stoichiometric coefficient for reaction r, vr is the reaction rate,
and act,r the chemical reaction activation energy. This activation energy is tabulated from
empirical observations of the included chemical reactions. This concludes the derivation and
statement of all transport coefficients.

Macroscopic Observables
One further necessary derivation step is to define the remaining macroscopic observable
calculations. These macroscopic variables are often just as important as the macroscopic
solution variables in their influence of the plasma properties. Moreover, their forms may
vary depending on definition and source. As such, many of the important macroscopic
plasma quantities not already derived are provided in detail in the following paragraphs.

Molar Mass and Concentration
The plasma molar mass is calculated as a summation of all mole fractions and molar mass
products
Ns
M=

xs Ms

(28)

i=1
while the total and species molar concentrations are calculated

ctot =

p
R Tg (1 − xe ) + Te xe
cs = xs ctot

(29a)

(29b)

where the pressure, p, is the local pressure distribution as calculated during the momentum
transport equation.

336

Number Densities
Since both the pressure distribution and temperatures are known, the species and total
volumetric number densities may be calculated as

ns = Nav ctot xs

(30a)

n = Nav ctot

(30b)

where n is the total volumetric number density of the plasma.

Mass Densities
Finally, the species and total volumetric mass densities are calculated as

ρs = cs Ms

(31a)

ρ = ctot M

(31b)

where ρ is the total volumetric mass density of the plasma.
The governing transport equations, transport coefficients, and macroscopic observable
calculations have all been derived. The transport equations were derived from the distribution function and Boltzmann equation. This set of expressions, coefficients, and calculations
represents a closed set of equations that may be solved numerically to update the plasma
properties and behavior.

337

Appendix D: Finite Difference Stencil
for Momentum Transport Equation
The finite-difference (FD) numerical stencil representing the discretized form of the viscous
flow update to the momentum transport equation as stated in Eq. (4.49) is now derived. This
represents the first implicit update step in the Chorin projection method. An implicit update
is required since the time rate of change in the momentum depends on the velocity itself.
Moreover, the two vector velocity components similarly depend on not only their own values,
but also those with orthogonal orientation. As such, the numerical stencil will include both
radial and vertical velocity components using the standard whole and half integer indexing
scheme.
To begin, the point form of this viscous force implicit projection step is again stated for
convenience in Eq. (32)
∂vr 2 1 ∂
∂vz
δt 1 ∂
r −µ 2
−
(rvr ) +
vr = vr −
ρ r ∂r
∂r
3 r ∂r
∂z
δt ∂
∂vr ∂vz
+
−µ
+
ρ ∂z
∂z
∂r

δt 1 ∂
∂vr
vz = vz −
r −µ
+
ρ r ∂r
∂z
δt ∂
∂vz
2
+
−µ 2
−
ρ ∂z
∂z
3

338

∂vz
∂r
1 ∂
∂vz
(rvr ) +
r ∂r
∂z

(32a)

(32b)

Both compressive and shear forces are present in both the radial and vertical components
of the velocity update equations provided in Eq. (32) as evident from the presence of radial
and vertical derivatives in both terms. The compressive force calculations (as represented
by derivatives lying parallel to the flow component direction) are easily calculated from the
neighboring two sample points. However, the shear forces in both cases are not naturally
represented at the given sampling locations as defined by the numerical mesh. Instead,
their values must be approximated from likewise neighboring edges. Similarly, due to the
location of the velocity components along the edges of each mesh element, the gas density
must likewise be translated to the element edge. Finally, this also requires the gas kinetic
viscosity to be calculated at the mesh nodes, which is calculated prior to the iterative update
scheme and based upon a two-dimensional bilinear interpolation scheme.
In order to present the complete discrete form of the viscous force projection step update
of the momentum transport equation, it is decomposed into its four primary terms. These
represent both the compressive and shear force terms present in both velocity components.
As a result, a simplified viscous force projection step update is provided in Eq. (33).
δt
vr = vr + [−τrr + τrz ]
ρ

(33a)

δt
vz = vz + [+τzr − τzz ]
ρ

(33b)

The four viscosity tensor terms in Eq. (33) are now presented in discrete form. Thus, these
equations do not represent complete equations themselves, but instead simply discrete forms
of numerical terms to be substituted back into Eq. (33). This decreases confusion with
regards to the lengthy derivation, and also mimics the numerical code structure, with each
viscosity tensor term being calculated and filled in a separate subroutine. As usual, these
terms are derived and centered about an arbitrary mesh index location.

339

τrr
The compressive tensor term corresponding to the radial velocity component update may be
expressed in point form as

τrr =

1 ∂
r ∂r

r −µ 2

∂vr 2 1 ∂
∂vz
−
(rvr ) +
∂r
3 r ∂r
∂z

(34)

which may be expanded and simplified as

τrr =

2 ∂
r ∂r

µr

∂vr
∂r

+

2 ∂
3r ∂r

µ

340

∂
2 ∂
(rvr ) +
∂r
3r ∂r

µr

∂vz
∂z

(35)

1
which is now discretized about the generic vertical edge with corresponding index i, j + 2 as

µ 1
1 r 1 vr,i+1,j+ 1 − vr,i,j+ 1
2 i+ 2 ,j+ 2 i+ 2
2
2
=
τ
ri
rr,i,j+ 1
2
ri+1 − ri r 1 − r 1
i+ 2
i− 2
µ 1
1 r 1 vr,i,j+ 1 − vr,i−1,j+ 1
2 i− 2 ,j+ 2 i− 2
2
2
+
ri
ri − ri−1 r 1 − r 1
i+ 2
i− 2
µ 1
1 ri+1 vr,i+1,j+ 1 − ri vr,i,j+ 1
2 i+ 2 ,j+ 2
2
2
+
3ri
ri+1 − ri r 1 − r 1
i− 2
i+ 2
µ 1
1 ri vr,i,j+ 1 − ri−1 vr,i−1,j+ 1
2 i− 2 ,j+ 2
2
2
+
3ri
ri − ri−1 r 1 − r 1
i− 2
i+ 2
µ 1
1 r 1 vz,i+ 1 ,j+1 − vz,i+ 1 ,j
2 i+ 2 ,j+ 2 i+ 2
2
2
+
3ri
zj+1 − zj
r 1 −r 1
i− 2
i+ 2
µ 1
1 r 1 vz,i− 1 ,j+1 − vz,i− 1 ,j
2 i− 2 ,j+ 2 i− 2
2
2
+
3ri
zj+1 − zj
r 1 −r 1
i+ 2
i− 2

341

(36)

τzz
The compressive tensor term corresponding to the vertical flow vector component may be
expanded and simplified as given in the following

τzz =

4 ∂
3 ∂z

µ

∂vz
∂z

−

2 ∂
3 ∂z

µ ∂
(rvr )
r ∂r

(37)

1
In discretized form, centered about the generic horizontal mesh edge with index pair i + 2 , j,
this becomes

µ 1
1 vz,i+ 1 ,j+1 − vz,i+ 1 ,j
4 i+ 2 ,j+ 2
2
2
τ
1 ,j = 3
zz,i+ 2
zj+1 − zj
z 1 −z 1
j+ 2
j− 2
µ 1
1 vz,i+ 1 ,j − vz,i+ 1 ,j−1
4 i+ 2 ,j− 2
2
2
−
3
zj − zj−1
z 1 −z 1
j− 2
j+ 2
µ 1
1 vr,i+1,j+ 1 − vr,i,j+ 1
2 i+ 2 ,j+ 2
2
2
−
3
r 1 ri+1 − ri z 1 − z 1
i+ 2
j− 2
j+ 2
µ 1
1 vr,i,j+ 1 − vr,i−1,j+ 1
2 i+ 2 ,j− 2
2
2
+
3
r 1 ri+1 − ri z 1 − z 1
i+ 2
j+ 2
j− 2

(38)

τrz
The radial gas flow component shear force term is significantly simpler in form than the
previous compression force expressions given in the preceding sections. However, due to
the orthogonal orientation terms, an averaging of velocity components must be used. Once
342

again, the τrz term is expanded from that given in Eq. (32) to become

τrz = −

∂
∂z

µ

∂vr
∂z

−

∂
∂z

µ

∂vz
∂r

(39)

which may be written in discrete form for the index pair i, j + 1 as
2

µi,j+1 v
=−
τ
rz,i,j+ 1
2

−

r,i,j+ 3
2

z 3 −z 1
j+ 2
j+ 2

−v
r,i,j+ 1
2

µi,j
+

zj+1 − zj

z 1 −z 1
j+ 2
j− 2

µi,j+1 v
−v
1
z,i+ 2 ,j+1
z,i− 1 ,j+1
2
r 1 −r 1
i+ 2
i− 2

v
1 − vr,i,j− 1
r,i,j+ 2
2

zj+1 − zj

µi,j
+

zj+1 − zj

v
1 − vz,i− 1 ,j
z,i+ 2 ,j
2

r 1 −r 1
i+ 2
i− 2

(40)

zj+1 − zj

τzr
Finally, the shear force associated with the vertical gas flow vector component follows is
a similar form as that of τrz . Only now, instead of the reference edge being vertical in
orientation, it is now horizontal. Moreover, a more complicated radial derivative is present
due to the cylindrical coordinate frame. As a result, the expanded τzr term becomes

τzr = −

1 ∂
µr
r ∂r

∂vr
∂z

343

−

1 ∂
µr
r ∂r

∂vz
∂r

(41)

Once again, in discrete form the above expression centered at the generic horizontal edge
with index pair i + 1 , j becomes
2

τ

zr,i+ 1 ,j
2

=−

+

−

+

r

i+ 1
2

1
r

i+ 1
2

1
r

1
i+ 2

1
r

−v
µi+1,j ri+1 v
r,i+1,j− 1
r,i+1,j+ 1
2
2

1

i+ 1
2

ri+1 − ri

µi,j ri

z 1 −z 1
j+ 2
j− 2

−v
v
r,i−1,j− 1
r,i,j+ 1
2
2

ri − ri−1

z 1 −z 1
j+ 2
j− 2

−v
µi+1,j ri+1 v
1
z,i+ 2 ,j
z,i+ 3 ,j
2
r 3 −r 1
i+ 2
i+ 2
µi,j ri

ri+1 − ri

v
−v
z,i+ 1 ,j
z,i− 1 ,j
2
2

r 1 −r 1
i+ 2
i− 2

(42)

ri+1 − ri

Substituting Eq. (36) through Eq. (42) into Eq. (33) results in a Ne × Ne matrix structure,
where Ne is the total number of edges in the simulation domain (horizontal plus vertical).
Moreover, the structure of this matrix is block-diagonal in a 2×2 fashion. Solving the resulting implicit matrix equation with the previous solution provided as the generic right hand
side solution vector results in the first implicit projection method update to the momentum
transport equation.

344

Appendix E: FORTRAN files
master.f90

program master
! master program file for e/m-plasma simulation
use
use
use
use
use
use
use
use
use
use

indices
constants
modtimer
plasmadata
mesh
geometry
em
geometrydeallocate
plasma
thermal

implicit none
save
character*100 :: time
double precision :: time1, time2
! write starting time out to file
call systime(time);
open(unit=8, file=’../status/time.out’, status=’unknown’)
write(8,*) ’Starting time:’; write(8,*) trim(time);
! first, initialize physical constants
if (verbose.ge.2) then
print*, ’Calling [initialize_constants] module...’
345

end if
call initialize_constants
! if necessary, allocate timer values
if (Ctimer.gt.0) then
call initialize_timer
end if
! start multi-physics solution algorithm
if (Csolveplasma.eq.1) then
! initialize global counter
itt=0
! read in plasma data
if (verbose.ge.2) then
print*, ’Calling [plasma_data] module...’
end if
call plasma_data
! generate mesh (if applicable)
if (Cgenmesh.gt.0) then
call generate_mesh
end if
! next, process simulation geometry
if (verbose.ge.2) then
print*, ’Calling [process_geometry] module...’
end if
call process_geometry
! open status files (to be appended during simulation)
call master_status_open
! ** MAIN SOLUTION LOOP **
rplas=2.d0*Cmaxerr
do while ((itt.lt.Cnitermax+1).and.(Cresmaxflag.eq.0))
! step global counter forward
itt=itt+1
! initialize power minimization on/off flag
Crefpowminon=0; Crefpowminfinish=0;
! print status
346

write(*,’(a10,i4,a8,i4,a)’), ’Iteration ’,itt,’ of max ’,Cnitermax,’...’
! write global counter to file
open(unit=10,file=’../status/itt.out’, status=’unknown’)
write(10,*) itt; close(10);
! update e/m solution (if necessary)
if (Csolveem.eq.1) then
if (verbose.ge.2) then
print*, ’Calling [solve_em] module...’
end if
call cpu_time(time1)
if ((Crefpowminflag.gt.0).and.(mod(itt,Cnrefpowmin).eq.0)) then
Crefpowminon=1
call geometry_deallocate
call refpowmin
Crefpowminon=0
else
call solve_em
end if
call cpu_time(time2)
if (Ctimer.gt.0) then
timer(1,1)=timer(1,1)+time2-time1
end if
end if
! solve plasma flow module
if (verbose.ge.2) then
print*, ’Calling [solve_plasma] module...’
end if
! ** MAIN PLASMA LOOP **
itp=0; call cpu_time(time1)
do while (itp.lt.Cniterpmax)
itp=itp+1
call solve_plasma
end do
call cpu_time(time2)
if (Ctimer.gt.0) then
timer(1,2)=timer(1,2)+time2-time1
end if
! update substrate temperature model
if (Ctempsubsolve.eq.1) then
call cpu_time(time1)
347

call solve_thermal
call cpu_time(time2)
if (Ctimer.gt.0) then
timer(1,3)=timer(1,3)+time2-time1
end if
end if
! write to timer file for comparison
call master_timer_write(itt,timer)
end do
! close status files
call master_status_close
else if (Csolveplasma.eq.0) then
! if ONLY e/m solution is desired
if (Csolveem.eq.1) then
! initialize reflected power minimization flag
Crefpowminon=0
! write first iteration to file
open(unit=10, file=’../status/itt.out’, status=’unknown’)
itt=1; write(10,*) itt; close(10);
! reflected power minimization loop
if (Crefpowminflag.gt.0) then
Crefpowminon=1
call refpowmin
Crefpowminon=0
else
if (Cgenmesh.gt.0) then
call generate_mesh
end if
call process_geometry
call solve_em
call geometry_deallocate
end if
end if
! if ONLY substrate thermal solution is desired
if (Ctempsubsolve.eq.1) then

348

! write first iteration to file
open(unit=10, file=’../status/itt.out’, status=’unknown’)
itt=1; write(10,*) itt; close(10);
! check for mesh generation
if (Cgenmesh.gt.0) then
call generate_mesh
end if
call process_geometry
call solve_thermal
call geometry_deallocate
end if
end if
! write starting time out to file
call systime(time);
write(8,*) ’Ending time:’; write(8,*) trim(time); close(8);
end program master
subroutine master_status_open
! open status files for appending during simulation
open(unit=1001, file=’../status/residual.out’, status=’unknown’)
open(unit=1002, file=’../status/powref.out’, status=’unknown’)
if (Ctempsubsolve.gt.0) then
open(unit=1003, file=’../status/tsuberror.out’, status=’unknown’)
end if
open(unit=3333, file=’../status/l2errtg.out’, status=’unknown’)
end subroutine master_status_open
subroutine master_status_close
! close status files
close(1001)
close(1002)
if (Ctempsubsolve.gt.0) then
close(1003)
end if
close(3333)
end subroutine master_status_close

349

subroutine master_timer_write(itt,timer)
! write master timings to file
integer :: itt
double precision :: timesum, timer(1,3)
! calculate total time
timesum=0.d0
do i=1,3
timesum=timesum+timer(1,i)
end do
! write timings to file
if (Ctimer.gt.0) then
open(unit=10, file=’../status/timer.out’, status=’unknown’)
write(10,*) ’Time required for E/M calculations:’
write(10,*) timer(1,1)/float(itt), timer(1,1)/timesum
write(10,*) ’Time required for plasma calculations:’
write(10,*) timer(1,2)/float(itt), timer(1,2)/timesum
write(10,*) ’Time required for thermal calculations:’
write(10,*) timer(1,3)/float(itt), timer(1,3)/timesum
close(10)
end if
end subroutine master_timer_write

em.f90

module em
! solve electromagnetics model
use
use
use
use
use
use
use
use

indices
constants
modtimer
geometry
emconductivity
emsolve
empower
empoynting
350

use
use
use
use
use

emreflected
emqvalue
emsubstrate
emplot
emdeallocate

implicit none
save
double precision :: timeem1, timeem2
contains
subroutine solve_em
if (verbose.ge.3) then
print*, ’Running [solve_em] module...’
end if
if (verbose.ge.4) then
print*, ’Calling [em_conductivity]...’
end if
call em_conductivity
if (verbose.ge.4) then
print*, ’Calling [em_pml]...’
end if
call em_pml
if (verbose.ge.4) then
print*, ’Calling [em_curlcurl]...’
end if
call cpu_time(timeem1)
call em_curlcurl
call cpu_time(timeem2)
if (Ctimer.gt.0) then
timer(2,1)=timer(2,1)+timeem2-timeem1
end if
if (verbose.ge.4) then
print*, ’Calling [em_source]...’
end if
call em_source
if (verbose.ge.4) then
351

print*, ’Calling [em_solve]...’
end if
call cpu_time(timeem1)
call em_solve
call cpu_time(timeem2)
if (Ctimer.gt.0) then
timer(2,2)=timer(2,2)+timeem2-timeem1
end if
if (Cpoyntsub.gt.0) then
if (verbose.ge.4) then
print*, ’Calling [em_poynting]...’
end if
call em_poynting
end if
if (Crefpow.gt.0) then
if (verbose.ge.4) then
print*, ’Calling [em_reflected]...’
end if
call em_reflected
end if
if (Cemqvalue.gt.0) then
if (verbose.ge.4) then
print*, ’Calling [em_qvalue]...’
end if
call em_qvalue
end if
if (Cemsubstrate.gt.0) then
if (verbose.ge.4) then
print*, ’Calling [em_substrate]...’
end if
call em_substrate
end if
if (verbose.ge.4) then
print*, ’Calling [em_power]...’
end if
call em_power
if ((itt.eq.1).or.(mod(itt,Cplotem).eq.0)) then
if (verbose.ge.4) then
print*, ’Calling [em_plot]...’
352

end if
call cpu_time(timeem1)
call em_plot
call cpu_time(timeem2)
if (Ctimer.gt.0) then
timer(2,3)=timer(2,3)+timeem2-timeem1
end if
end if
if (verbose.ge.4) then
print*, ’Calling [em_deallocate]...’
end if
call em_deallocate
if (verbose.ge.4) then
print*, ’Calling [em_solve_write]...’
end if
call solve_em_write
end subroutine solve_em
subroutine solve_em_write
! write em timer values to file
if (Ctimer.gt.0) then
open(unit=10, file=’../status/timerem.out’, status=’unknown’)
write(10,*) ’Time required for curl x curl calculation:’
write(10,*) timer(2,1)/itt, timer(2,1)/sum(timer(2,1:3))
write(10,*) ’Time required for e/m solve:’
write(10,*) timer(2,2)/itt, timer(2,2)/sum(timer(2,1:3))
write(10,*) ’Time required for e/m plot:’
write(10,*) timer(2,3)/itt, timer(2,3)/sum(timer(2,1:3))
close(10)
end if
end subroutine solve_em_write
end module em

353

plasma.f90

module plasma
! update plasma model
use
use
use
use
use
use
use
use
use
use
use
use
use
use
use
use
use

indices
constants
modtimer
geometry
plasmaallocate
plasmainitialize
plasmaphysics
plasmatransport
plasmaupdate
plasmarelax
plasmaconductivity
plasmapass
plasmasize
plasmabeam
plasmafluxsub
plasmaplot
plasmadeallocate

implicit none
save
double precision :: timeplas1, timeplas2, rcompare
contains
subroutine solve_plasma
if (verbose.ge.3) then
print*, ’Running [solve_plasma] module...’
end if
! read in global iteration number from file
open(unit=10, file=’../status/itt.out’, status=’old’)
read(10,*) itt; close(10)
if (verbose.ge.4) then
354

print*, ’Calling [plasma_allocate] module...’
end if
call plasma_allocate
if (verbose.ge.4) then
print*, ’Calling [plasma_initialize] module...’
end if
call plasma_initialize
if (Ctimer.gt.0) then
timer(3,1:2)=0.d0
end if
! update flows via momentum transport
if (Cvel.gt.0) then
! update physical quantities
if (verbose.ge.4) then
print*, ’Calling [plasma_physics] module...’
end if
call plasma_physics
if (verbose.ge.4) then
print*, ’Calling [plasma_momentum] module...’
end if
call cpu_time(timeplas1)
call plasma_momentum
call cpu_time(timeplas2)
if (Ctimer.gt.0) then
timer(3,1)=timer(3,1)+timeplas2-timeplas1
end if
end if
! outer plasma loop
iouter=0
do while (((iouter.lt.Cniteromin).or.(plaserrcond.lt.Cmaxerrcond)) &
1 .and.(iouter.lt.Cniteromax))
! step outer counter forward
iouter=iouter+1
! update physical quantities
if (verbose.ge.4) then
print*, ’Calling [plasma_physics] module...’
end if
call plasma_physics

355

if ((itt.eq.1).and.(iouter.eq.1)) then
iinmax=Cniterimax0
else
iinmax=Cniterimax
end if
! step total counter forward
itotal=itotal+1
! reset maxerrplas
maxerrplas=1.d-1*Cmaxerrplas
! print starting iteration
call solve_plasma_iouter
! inner plasma loop
iinner=0; call cpu_time(timeplas1)
do while (iinner.lt.iinmax)
! step inner counter forward
iinner=iinner+1
if (verbose.ge.4) then
print*, ’Calling [plasma_relax_coefficients]...’
end if
call plasma_relax_coefficients
if (verbose.ge.4) then
print*, ’Calling [plasma_transport] module...’
end if
call plasma_transport
if (verbose.ge.4) then
print*, ’Calling [plasma_update] module...’
end if
call plasma_update
if (verbose.ge.4) then
print*, ’Calling [plasma_relax_convergence]...’
end if
call plasma_relax_convergence
if (verbose.ge.4) then
print*, ’Calling [plasma_relax_error]...’
end if
356

call plasma_relax_error
! print relaxation rates
write(*,100) frxs(2:nspec-1), frte, frtg
end do
call cpu_time(timeplas2)
if (Ctimer.gt.0) then
timer(3,2)=timer(3,2)+timeplas2-timeplas1
end if
if (verbose.ge.4) then
print*, ’Calling [plasma_conductivity] module...’
end if
call plasma_conductivity
! calculate residuals
call plasma_relax_residuals
call plasma_relax_convergence_total
print*, ’Residual summation = ’,rplas
end do
! check for maximum residual condition
call solve_plasma_resmax
! write residuals to file
if (itt.gt.1) then
write(1001,101) itt, rplas, rxs(2), rxs(7), rte, rtg, velerr
end if
if (verbose.ge.4) then
print*, ’Calling [plasma_pass] module...’
end if
call plasma_pass
if (verbose.ge.4) then
print*, ’Calling [plasma_size] module...’
end if
call plasma_size
if (Cpbeam.gt.0) then
if (verbose.ge.4) then
print*, ’Calling [plasma_beam] module...’
end if
357

call plasma_beam
end if
if (Cpfluxsub.gt.0) then
if (verbose.ge.4) then
print*, ’Calling [plamsa_fluxsub] module...’
end if
call plasma_fluxsub
end if
if ((itt.eq.1).or.(mod(itt,Cplotplasma).eq.0)) then
if (verbose.ge.4) then
print*, ’Calling [plasma_plot] module...’
end if
call cpu_time(timeplas1)
call plasma_plot
call cpu_time(timeplas2)
if (Ctimer.gt.0) then
timer(3,3)=timer(3,3)+timeplas2-timeplas1
end if
end if
if (verbose.ge.4) then
print*, ’Calling [plasma_deallocate] module...’
end if
call plasma_deallocate
! write timings to file
call solve_plasma_timer
100 format(8e9.2,e8.1)
101 format(i4,6e12.4)
end subroutine solve_plasma
subroutine solve_plasma_iouter
print*,
print*,
print*,
print*,

’------------------------’
’------------------------’
’Starting outer iteration ’,iouter,’ ...’
’------------------------’

end subroutine solve_plasma_iouter
subroutine solve_plasma_resmax
358

! check for maximum residual error reached
! if so, set flag for simulation termination
! ensure residual error flag is zero
Cresmaxflag=0
! check for condition
do i=1,Cnresmax
if (Ciresmax(i).le.nspec) then
rcompare=rxs(Ciresmax(i))
else if (Ciresmax(i).eq.nspec+2) then
rcompare=rte
else if (Ciresmax(i).eq.nspec+3) then
rcompare=rtg
end if
if (rcompare.ge.Cresmax(i)) then
Cresmaxflag=Cresmaxflag+1
end if
end do
if (rplas.ge.Cresmaxtot) then
Cresmaxflag=Cresmaxflag+1
end if
! change sign on flag
if (Cresmaxflag.eq.0) then
Cresmaxflag=1
else
Cresmaxflag=0
end if
end subroutine solve_plasma_resmax
subroutine solve_plasma_timer
! write plasma timings to file
if (Ctimer.gt.0) then
open(unit=10, file=’../status/timerplas.out’, status=’unknown’)
write(10,*) ’Time required for momentum update:’
write(10,*) timer(3,1)/iouter, timer(3,1)/sum(timer(3,1:3))
write(10,*) ’Time required for transport updates:’
write(10,*) timer(3,2)/iouter, timer(3,2)/sum(timer(3,1:3))
write(10,*) ’Time required for plasma plot:’
write(10,*) timer(3,3)/itt, timer(3,3)/sum(timer(3,1:3))
close(10)
359

end if
end subroutine solve_plasma_timer
end module plasma

thermal.f90

module thermal
! module for updating substrate thermal model
use indices
use constants
use geometry
implicit none
save
integer :: ntlhss, ntsub
double precision, dimension(:), allocatable :: fluxxh0, fluxxh
double precision, dimension(:), allocatable :: fluxtg0, fluxtg
double precision, dimension(:), allocatable :: fluxsub
double precision, dimension(:), allocatable :: trhs, tsoln
double precision, dimension(:), allocatable :: tsoln0, alphat
double precision, dimension(:,:), allocatable :: tlhss
contains
subroutine solve_thermal
if (verbose.ge.3) then
print*, ’Running [thermal] module...’
end if
if (verbose.ge.4) then
print*, ’Calling [thermal_allocate]...’
end if
call thermal_allocate
if (verbose.ge.4) then
360

print*, ’Calling [thermal_diffusivity]...’
end if
call thermal_diffusivity
if (verbose.ge.4) then
print*, ’Calling [thermal_left]...’
end if
call thermal_left
if (verbose.ge.4) then
print*, ’Calling [thermal_read]...’
end if
call thermal_read
if (verbose.ge.4) then
print*, ’Calling [thermal_right]...’
end if
call thermal_right
if (verbose.ge.4) then
print*, ’Calling [thermal_solve]...’
end if
call thermal_solve
call thermal_relax
if (verbose.ge.4) then
print*, ’Calling [thermal_error]...’
end if
call thermal_error
if (verbose.ge.4) then
print*, ’Calling [thermal_pass]...’
end if
call thermal_pass
if ((itt.eq.1).or.(mod(itt,Cplotthermal).eq.0)) then
if (verbose.ge.4) then
print*, ’Calling [thermal_plot]...’
end if
call thermal_plot
end if
if (verbose.ge.4) then
print*, ’Calling [thermal_deallocate]...’
361

end if
call thermal_deallocate
end subroutine solve_thermal
subroutine thermal_allocate
! allocate substrate thermal model arrays
allocate(tlhss(3,5*ntel))
allocate(trhs(ntel))
allocate(tsoln0(ntel))
allocate(tsoln(ntel))
allocate(alphat(ntel))
tlhss=0.d0
trhs=0.d0
tsoln0=0.d0
tsoln=0.d0
alphat=0.d0
end subroutine thermal_allocate
subroutine thermal_diffusivity
! update thermal diffusivity array
integer :: nlamtype
double precision :: cpt(ntel), rhot(ntel)
double precision :: lamtype(10)
! set specific heats and mass densities at
! each element in substrate
lamtype=0.d0; lamtype(1)=lambdasub(1); nlamtype=1;
do i=2,ntel
flag=0
do j=1,nlamtype
if (lambdasub(i).eq.lamtype(j)) then
flag=flag+1
end if
end do
if (flag.eq.0) then
nlamtype=nlamtype+1
lamtype(nlamtype)=lambdasub(i)
end if
end do

362

! fill specific heat and mass density vectors
do i=1,ntel
! lambda > 10 (METAL)
if (lambdasub(i).gt.1.d1) then
cpt(i)=5.d2
rhot(i)=1.d4
! lambda < 0.1 (GAS)
else if (lambdasub(i).lt.1.d-1) then
cpt(i)=1.43d4
rhot(i)=1.d-2
! else (SILICA)
else
cpt(i)=7.d2
rhot(i)=2.d3
end if
alphat(i)=lambdasub(i)/(cpt(i)*rhot(i))
end do
end subroutine thermal_diffusivity
subroutine thermal_left
! fill sparse left-hand-side array
double
double
double
double

precision :: r0, rm1, r1, rp1, r2, dr0, dr1, dr2
precision :: z0, zm1, z1, zp1, z2, dz0, dz1, dz2
precision, dimension(:,:), allocatable :: tlhss0
precision :: drdzterm, alphatavg, coef, tlhssd

! fill sparse left-hand-side array
k=0
do i=1,ntel
! check for otherwise bordering elements
flag=0
do j=1,4
if (tel4tel(j,i).eq.0) then
flag=flag+1
end if
end do
if ((tforced(i).gt.0).and.(flag.eq.0)) then
! inner forced (water-cooled)
k=k+1
tlhss(1,k)=i
tlhss(2,k)=i
363

tlhss(3,k)=1.d0
else if ((flag.gt.0).and.(tforced(i).eq.0)) then
! domain borders
! main diagonal
k=k+1
tlhss(1,k)=i
tlhss(2,k)=i
if ((tel4tel(1,i).eq.0).or.(tel4tel(2,i).eq.0)) then
tlhss(3,k)=1.d0
end if
if (tel4tel(3,i).eq.0) then
tlhss(3,k)=alphat(i)/abs(telCoords(1,i)-telCoords(1,tel4tel(2,i)))
end if
if (tel4tel(4,i).eq.0) then
tlhss(3,k)=alphat(i)/abs(telCoords(2,i)-telCoords(2,tel4tel(1,i)))
end if
! off-diagonal
k=k+1
tlhss(1,k)=i
if (tel4tel(1,i).eq.0) then
tlhss(2,k)=tel4tel(4,i)
end if
if ((tel4tel(2,i).eq.0).and.(tel4tel(3,i).ne.0)) then
tlhss(2,k)=tel4tel(3,i)
end if
if (tel4tel(3,i).eq.0) then
tlhss(2,k)=tel4tel(2,i)
end if
if (tel4tel(4,i).eq.0) then
tlhss(2,k)=tel4tel(1,i)
end if
if ((tel4tel(1,i).eq.0).or.(tel4tel(2,i).eq.0)) then
tlhss(3,k)=-1.d0
end if
if (tel4tel(3,i).eq.0) then
tlhss(3,k)=-alphat(i)/abs(telCoords(1,i)-telCoords(1,tel4tel(2,i)))
end if
if (tel4tel(4,i).eq.0) then
tlhss(3,k)=-alphat(i)/abs(telCoords(2,i)-telCoords(2,tel4tel(1,i)))
end if
else if ((tforced(i).gt.0).and.(flag.ne.0)) then
! bordering, but still forced
k=k+1
tlhss(1,k)=i
tlhss(2,k)=i
364

tlhss(3,k)=1.d0
else if ((flag.eq.0).and.(tforced(i).eq.0)) then
! all other non-boundary locations
! spatial terms
r0=telCoords(1,i)
rm1=tnodeCoords(1,tel2tnode(1,i))
r1=telCoords(1,tel4tel(2,i))
rp1=tnodeCoords(1,tel2tnode(2,i))
r2=telCoords(1,tel4tel(3,i))
dr0=abs(rp1-rm1)
dr1=abs(r0-r1)
dr2=abs(r0-r2)
z0=telCoords(2,i)
zm1=tnodeCoords(2,tel2tnode(1,i))
z1=telCoords(2,tel4tel(1,i))
zp1=tnodeCoords(2,tel2tnode(4,i))
z2=telCoords(2,tel4tel(4,i))
dz0=abs(zp1-zm1)
dz1=abs(z0-z1)
dz2=abs(z0-z2)
! fill sparse array
tlhssd=0.d0
do j=1,4
ii=tel4tel(j,i)
if (j.eq.1) then
drdzterm=1.d0/(dz0*dz1)
else if (j.eq.2) then
drdzterm=rm1/(r0*dr0*dr1)
else if (j.eq.3) then
drdzterm=rp1/(r0*dr0*dr2)
else if (j.eq.4) then
drdzterm=1.d0/(dz0*dz2)
end if
! off diagonal terms first
if (ii.ne.0) then
alphatavg=(alphat(i)+alphat(ii))/2.d0
coef=alphatavg*drdzterm
k=k+1; tlhss(1,k)=i; tlhss(2,k)=ii
tlhss(3,k)=coef; tlhssd=tlhssd-coef
end if
end do
! main diagonal
k=k+1; tlhss(1,k)=i; tlhss(2,k)=i; tlhss(3,k)=tlhssd
end if
end do
365

ntlhss=k
! contract sparse lhs array
allocate(tlhss0(3,5*ntel))
tlhss0=tlhss
deallocate(tlhss)
allocate(tlhss(3,ntlhss))
tlhss=tlhss0(:,1:ntlhss)
deallocate(tlhss0)
end subroutine thermal_left
subroutine thermal_read
! read in flux data from file
! count number of total substrate bounds
ntsub=0
do i=1,ntel
if (tel4tel(3,i).eq.0) then
ntsub=ntsub+1
end if
end do
do i=1,ntel
if (tel4tel(4,i).eq.0) then
ntsub=ntsub+1
end if
end do
! allocate data arrays
allocate(fluxxh0(ntsub)); fluxxh0=0.d0
allocate(fluxtg0(ntsub)); fluxtg0=0.d0
allocate(fluxxh(ntsub)); fluxxh=0.d0
allocate(fluxtg(ntsub)); fluxtg=0.d0
allocate(fluxsub(ntsub)); fluxsub=0.d0
!
!
!
!
!
!
!
!
!

read in temperature flux at substrate
boundaries from files as calculated
in plasma module
if (itt.eq.1) then
fluxxh0=0.d0; fluxtg0=0.d0
else
open(unit=10, file=’../plasma/fluxxhtot0.out’, status=’unknown’)
open(unit=11, file=’../plasma/fluxtgtot0.out’, status=’unknown’)
do i=1,ntsub
366

! read(10,*) fluxxh0(i)
! read(11,*) fluxtg0(i)
! end do
! close(10); close(11)
! end if
open(unit=12, file=’../plasma/fluxxhtot.out’, status=’old’)
open(unit=13, file=’../plasma/fluxtgtot.out’, status=’old’)
do i=1,ntsub
read(12,*) fluxxh(i)
read(13,*) fluxtg(i)
end do
close(12)
close(13)
! calculate total flux at boundary elements
do i=1,ntsub
! if (itt.eq.1) then
fluxsub(i)=(fluxxh(i)+fluxtg(i))
! else
! fluxsub(i)=(1.d0-Ctempsubrelax)*(fluxxh0(i)+fluxtg0(i)) &
! 1 +Ctempsubrelax*(fluxxh(i)+fluxtg(i))
! end if
end do
end subroutine thermal_read
subroutine thermal_right
! fill right-hand-side vector
! check for r=0 or forced region
do i=1,ntel
! zeroed neumann on r=0
if (tel4tel(2,i).eq.0) then
trhs(i)=0.d0
end if
! force room temperature (300 K) in bulk
if (tforced(i).gt.0) then
trhs(i)=3.d2
end if
end do
! initialize flux edge counter
j=0

367

! vertical edges
do i=1,ntel
if (tel4tel(3,i).eq.0) then
j=j+1
trhs(i)=fluxsub(j)
end if
end do
! horizontal edges
do i=1,ntel
if (tel4tel(4,i).eq.0) then
j=j+1
trhs(i)=fluxsub(j)
end if
end do
end subroutine thermal_right
subroutine thermal_solve
! solve substrate thermal model
call axbdmumps(ntlhss,tlhss,ntel,trhs)
tsoln=trhs
! check for NaN
do i=1,ntel
if (tsoln(i).ne.tsoln(i)) then
tsoln(i)=0.d0
end if
end do
end subroutine thermal_solve
subroutine thermal_relax
! relax substrate temperature update
! update solution
do i=1,ntel
if (tforced(i).eq.0) then
tsoln(i)=(1.d0-Ctempsubrelax)*tsoln0(i)+Ctempsubrelax*tsoln(i)
end if
end do
end subroutine thermal_relax
368

subroutine thermal_error
! calculate error between current solution
! and previous temperatures at substrate
double precision :: tsuberror
! retrieve old temperature values
if (itt.eq.1) then
! initialize temperature to zero
tsoln0=0.d0
else
! read in tsoln0 from file
open(unit=10, file=’../plasma/tgsub.out’, status=’old’)
do i=1,ntel
if (tel4tel(3,i).eq.0) then
read(10,*) tsoln0(i)
end if
end do
do i=1,ntel
if (tel4tel(4,i).eq.0) then
read(10,*) tsoln0(i)
end if
end do
close(10)
end if
! calculate L2Norm between current
! gas temperatures on boundaries
! and new solution
tsuberror=0.d0
do i=1,ntel
if (tel4tel(3,i).eq.0) then
tsuberror=tsuberror+(tsoln(i)-tsoln0(i))**2
end if
end do
do i=1,ntel
if (tel4tel(4,i).eq.0) then
tsuberror=tsuberror+(tsoln(i)-tsoln0(i))**2
end if
end do
tsuberror=sqrt(tsuberror)
! write out error to file
369

write(1003,*) itt, tsuberror
end subroutine thermal_error
subroutine thermal_pass
! pass substrate temperature solution
! to plasma module (gas temperature)
! open file
open(unit=10, file=’../plasma/tgsub.out’, status=’unknown’)
! write substrate outer wall to file
do i=1,ntel
! check for element bounding plasma domain
if (tel4tel(3,i).eq.0) then
write(10,*) tsoln(i)
end if
end do
! write substrate top surface to file
do i=1,ntel
if (tel4tel(4,i).eq.0) then
write(10,*) tsoln(i)
end if
end do
! close file
close(10)
!
!
!
!
!
!
!
!

! pass current flux to old files
open(unit=10, file=’../plasma/fluxxhtot0.out’, status=’unknown’)
open(unit=11, file=’../plasma/fluxtgtot0.out’, status=’unknown’)
do i=1,ntsub
write(10,*) fluxxh(i)
write(11,*) fluxtg(i)
end do
close(10); close(11)

end subroutine thermal_pass
subroutine thermal_plot
! plot substrate thermal solution
if (Cplotplasma.gt.0) then
370

! call plotvtk(ntel,lambdasub,’lambdasub’)
! call plotvtk(ntel,tsoln,’tsub’)
!
!
!
!
!
!

open(unit=10, file=’../plot/lambdasub.3d’, status=’unknown’)
write(10,*) ’r z phi lambdasub’
do i=1,ntel
write(10,’(2e16.8,i4,e16.8)’) telCoords(:,i), 0, lambdasub(i)
end do
close(10)

open(unit=10, file=’../plot/alphat.3d’, status=’unknown’)
write(10,*) ’r z phi alphat’
do i=1,ntel
write(10,’(2e16.8,i4,e16.8)’) telCoords(:,i), 0, alphat(i)
end do
close(10)
open(unit=10, file=’../plot/tsub.3d’, status=’unknown’)
write(10,*) ’r z phi tsub’
do i=1,ntel
write(10,’(2e16.8,i4,e16.8)’) telCoords(:,i), 0, tsoln(i)
end do
close(10)
end if
end subroutine thermal_plot
subroutine thermal_deallocate
! deallocate substrate thermal model arrays
deallocate(fluxxh0,fluxxh,fluxtg0,fluxtg)
deallocate(fluxsub)
deallocate(tlhss)
deallocate(trhs)
deallocate(tsoln0)
deallocate(tsoln)
deallocate(alphat)
end subroutine thermal_deallocate
end module thermal

371

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