close

Вход

Забыли?

вход по аккаунту

?

Numerical and experimental investigation of nonequilibrium microwave argon plasmas

код для вставкиСкачать
INFORMATION TO USERS
This manuscript has been reproduced from the microfilm master. UMI films
the text directly from the original or copy submitted. Thus, some thesis and
dissertation copies are in typewriter face, while others may be from any type of
computer printer.
The quality of this reproduction is dependent upon the quality of the
copy submitted. Broken or indistinct print, colored or poor quality illustrations
and photographs, print bleedthrough, substandard margins, and improper
alignment can adversely affect reproduction.
In the unlikely event that the author did not send UMI a complete manuscript
and there are missing pages, these will be noted.
Also, if unauthorized
copyright material had to be removed, a note will indicate the deletion.
Oversize materials (e.g., maps, drawings, charts) are reproduced by
sectioning the original, beginning at the upper left-hand comer and continuing
from left to right in equal sections with small overlaps.
Photographs included in the original manuscript have been reproduced
xerographically in this copy.
Higher quality 6" x 9" black and white
photographic prints are available for any photographs or illustrations appearing
in this copy for an additional charge. Contact UMI directly to order.
Bell & Howell Information and Learning
300 North Zeeb Road, Ann Arbor, Ml 48106-1346 USA
800-521-0600
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Numerical and Experimental
Investigation of Nonequilibrium
Microwave Argon Plasmas
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Numerical and Experimental Investigation of
Nonequilibrium Microwave Argon Plasm as
A dissertation submitted in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
By
Yunlong Li, B.S.M.E., M.S.M.E.
Xian Jiaotong University, Xian, China, 1992, 1995
May 2000
University of Arkansas, Fayetteville
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
UMI Number 9987250
__
®
UMI
UMI Microform9987250
Copyright 2000 by Bell & Howell Information and Learning Company.
All rights reserved. This microform edition is protected against
unauthorized copying under Title 17, United States Code.
Bell & Howell Information and Learning Company
300 North Zeeb Road
P.O. Box 1346
Ann Arbor, Ml 48106-1346
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
This dissertation is approved for
recommendation to the
Graduate Council
DISSERTATION DIRECTORS
DrTLafry A. Rde
Dr. Matthew H. Gordon
DISSERTATION COMMITTEE
"D r. KnaledTTassouni -
Dr. Ajay Malshe
Dr. R i^ C o u v illio n
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
A C K N O W LED G M EN TS
I would like to thank Dr. Larry Roe and Dr. M att Gordon for their support
throughout my Ph.D. program. Also, thanks to my committee members: Dr.
Khaled Hassouni, Dr. Ajay Malshe, Dr. Min Xiao, and Dr. Rick Couvillion. Special
thanks go to my wife, Yuhong Cai, who has made these years the best ever.
The Mechanical Engineering Department supported my Ph.D. study all
through the past four and half years. I want to thank Dr. Schmidt, the department
head, for his consideration and kindness.
iii
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
TABLE OF CO NTENTS
Page
ACKNOWLEDGEMENTS.................................................................................. iii
TABLE OF CONTENTS.......................................................................................iv
LIST OF SYMBOLS.............................................................................................vi
LIST OF TABLES..................................................................................................x
LIST OF FIGURES............................................................................................. xi
CHAPTER 1 INTRODUCTION............................................................................. 1
1.0 Motivation.................................................................................................................................1
1.1 Introduction.............................................................................................................................. 2
1.2 Diamond CVD Processes...................................................................................................... 4
1.3 Fundamental Plasma Processes and Forms of Nonequilibrium..................................... 9
1.4 Plasma Diagnostics.............................................................................................................. 12
1.5 Plasma Modeling..................................................................................................................13
1.6 Objective............................................................................................................................... 22
CHAPTER 2 EXPERIMENTAL FACILITY......................................................... 24
2.1 Microwave Plasma Reactor................................................................................................ 24
2.2 Emission/Absorption Systems............................................................................................ 26
CHAPTER 3 EMISSION/ABSORPTION DIAGNOSTIC TECHNIQUES............. 30
3.1 Emission Spectroscopy....................................................................................................... 30
3.2 Absorption Spectroscopy.................................................................................................... 37
3.3 Spectral Line Broadening Theories.................................................................................... 40
CHAPTER 4 MODELING................................................................................... 48
4.1 Pseudo-1-D Plasma Model ............................................................................................... 48
4.2 Electromagnetic Model........................................................................................................ 56
iv
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
4.3 Two Dimensional Plasma Fluid Model...............................................................................60
4.4 Two Dimensional Argon Plasma Model.............................................................................72
4.5 Two Dimensional Collisional Radiative Model.................................................................. 85
CHAPTER 5 ENERGY BALANCE STUDY........................................................ 89
5.1 Pseudo-1-D Argon Model Results......................................................................................90
5.2 Experimental Global Reactor Energy Balance................................................................. 96
5.3 Control Volume Heat Transfer Analysis.............................................................................99
5.4 2-D Argon Fluid Model Results......................................................................................... 104
CHAPTER 6 EXCITED STATES OF THE ARGON PLASMAS....................... 114
6.1 The Experimental OES D ata.............................................................................................115
6.2 Absorption Measurement of the Metastable S tate.........................................................116
6.3 Characterization of the Argon Plasma with the 2-D Fluid Model................................. 121
6.4 Characterization of the Argon Plasma with the 2-D C R M ............................................. 140
6.5 Non-uniformity of the Argon Plasmas.............................................................................. 159
CHAPTER 7 CONCLUSIONS AND RECOMMENDATIONS............................162
7.1 Conclusions.........................................................................................................................162
7.2 Recommendations..............................................................................................................165
REFERENCES..................................................................................................167
APPENDIX A .....................................................................................................173
APPENDIX B .....................................................................................................178
v
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
LIST OF SYMBOLS
The following is a list of symbols used frequently in this dissertation. A
number of symbols have been used for several different purposes.
a
a, b
A
x
Voigt parameter
Coefficients in the curve-fitted equation for reaction rates
Area
Einstein coefficient, transition probability from energy level I to j, or at
wavelength X
B
Magnetic flux density
B,|
Stimulated absorption or emission probability
c
Speed of light, = 3e8 m/s
c
Specific heat
C
Constant
d
Stark shift
D
Diameter
D
Diffusion coefficient
Dx The absolute calibration factor
E
Electric field strength
E
Electric field
f
Electron energy/velocity distribution function
f
Gas flow rate
F, Fx, Fy
F,
Flux density vector and its component in x and y directions
External force
g
Relative velocity
g
Degeneracy
h
Planck’s constant, = 6.6262e-34 Js
h
Heat transfer coefficient
H
Magnetic field
I
|(v)
Signal intensity
Lineshape function
J
Current density
k
Boltzmann’s constant, = 1.3806e-23 J/K
k
Spectral absorption coefficient
k
Reaction rate
vi
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
K
Total production or loss
I
Angular momentum quantum number
L
Chord length
m
Mass
m
Mass flow rate
n, N
Number density
Nu
Nusselt number
P
Pressure
P
Power
Pr
Prandtl number
q
Electron charge
Q
Heat rate
Q
Cross-section
Q
Electron partition function
r
R
Re
s, p, d, f
S, P, D, F
t
T
u. v, U, V
Radial coordinate
ratio
Reynolds number
Electronic state
Atomic state
Time
Temperature
Velocity
V
Signal strength
V
Volume
w
Electron impact parameter
W
x
x,y,z
Chemical production or consumption rate
Mole fraction
Cartesian coordinates
Z
Effective nuclear charge
a
Ion-broadening parameter
8
Skin depth
y
Reduced velocity
e
Continuum emission
e
Energy level
e
Emissivity
VII
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
e
permittivity
4> Lineshape factor
X
Wavelength
X
Energy relaxation length, or mean free path
X
Thermal conductivity
n
Dynamic viscosity
po
Magnetic permeability
I*
Ion mobility
v
Collision frequency
v
Frequency
p
Density, or charge density
po
Debye radius
a
Collision cross-section
a
Plasma conductivity
a
Ratio of electron to ion collision frequency
t
Shear stress
t
Radiative lifetime
cd Angular frequency
cDpe
^
Electron plasma frequency
The free-bound Biberman factor
The free-free Biberman factor
(p
Conserved variable
A
Characteristic diffusion length
¥
Conserved variable per unit volume
n
Collision integral
Subscripts
0
1 ,2
aa
Reference value, or at center frequency
At energy le v e P I” or“2"
Atom-atom
abs
Absorbed
amb
Ambipolar
D
Doppler, or based on diameter
e
Electron
ea
Electron-atom
viii
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
ei
g, a
Electron-ion
Gas
i Ion
i, j
At energy level ‘ I* or “j’
m
Medium value, or mean (bulk) value
p Plasma, or perturber
R
Reaction “R’
s Species ‘ s'
sub
t
th
Substrate
Total
Thermal
x, y, z, r In a given direction
X
At a given wavelength
v
At a given frequency
<P
Conserved variable
¥
Conserved variable per unit volume
Superscripts
ea
ei
•
Electron-atom
Electron-ion
Excited state, or the metastable state (4s) for argon, or the mean value
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
LIST OF TABLES
Chapter 3
Table 3.1
Physical classifications, mathematical approximation, and lineshapes of various
line-broadening mechanisms
Chapter 4
Table 4.1
The Conservation Equations
Table 4.2
Species considered in 2-D argon plasma chemistry
Table 4.3
2-D Argon Plasma Chemistry Model
Chapter 5
Table 5.1
The Ranges of Studied Parameters Used for Argon Plasmas at Pressure of 5
Torr
Table 5.2
The Experimental Data for Microwave Argon Plasma Energy Balance
Table 5.3
Calculated Enthalpy Data for the Argon Plasma in Each Cooling Line with an
Argon Flow Rate of 250 seem and an assumed gas exit temperature of 350 K
Table 5.4
Heat Transfer Analysis Data for the Control Volume Encompassing the Argon
Plasma (250 seem argon flow, 5 Torr)
Table 5.5
The Total Heat Predicted from 2-D Model Comparing with the Global Energy
Balance Study and Control-Volume Analysis Results
Chapter 6
Table 6.1
The Excited State Number Densities Calculated from the OES Data Comparing
with the Numerical Predictions At Ne = 5e17 m-3, Tg = 350K
Table 6.2
The calculated argon linewidths (FWHM) with different gas temperatures at
Ne=1e18 rn3, Te=10.000K
Table 6.3
The experimental results of the excited state number densities comparing with
the 2-D CRM predictions for an argon plasma at 5 Torr, 250 seem (Best matched
case at microwave power of 30W)
Table 6.4
Error between peak values comparing the reaction rates calculated from 2-D
CRM and the curve-fitted equations
APPENDIX A
Table A.1
Spectroscopic Notation of Selected Argon Energy Levels
Table A.2
Transition Lines Spectroscopic Data (from NIST Atomic Spectra Data)
x
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
LIST OF FIGURES
Chapter 1
Figure 1.1
Hot Filament CVD Diamond Reactor
Figure 1.2
Combustion Flame Assisted CVD Process
Figure 1.3
RF Plasma Enhanced CVD Process
Figure 1.4
Fundamental Plasma Processes
Chapter 2
Figure 2.1
The WAVEMAT microwave plasma reactor
Figure 2.2
Schematic of the microwave reactor system
Figure 2.3
Schematic of the optical emission spectroscopic measurement system
Figure 2.4
Schematic of the absorption measurement experimental setup
Figure 2.5
Some pictures of the absorption measurement setup
Chapter 3
Figure 3.1
Calibration of the tungsten lamp passing through the belljar
Figure 3.2
Calibration of the tungsten lamp without passing through the belljar
Chapter 4
Figure 4.1
The iterative scheme to obtain a self-consistent solution
Figure 4.2
Schematic of a control volume AxAy for <1>conservation
Figure 4.3
Non-Maxwellian EEDF plots with averaged Te = 1.07e4 K and 300K <Tg<600K
comparing with the Maxwellian EEDF
Figure 4.4
Comparison of the curve-fitted reaction rates (Kea(0,1)) and those calculated
from the CRM as a function of electron temperature
Figure 4.5
Comparison of the curve-fitted reaction rates (Klea(O)) and those calculated from
the CRM as a function of electron temperature
Figure 4.6
Comparison of the curve-fitted reaction rates (Kiea(1)) and those calculated from
the CRM as a function of electron temperature
Figure 4.7
The iterative scheme of the coupled 2-D plasma model and the EM model
Figure 4.8
The simulation domain of the microwave plasma discharge with the WAVEMAT
reactor
XI
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Figure 4.9
The Flow Chart of the Iterative Scheme for the 2-D Collisional-Radiative Model
(CRM)
Chapter 5
Figure 5.1
The comparison of non-Maxwellian and Maxwellian EEDFs At Ne = 7e17 nrf3, Tg
=350 K, and P= 5 Torr
Figure 5.2
The electron energy loss terms with non-Maxwellian EEDF changing with the
averaged electron temperature at Ne = 7e17 m'3, Tg = 350K and P = 5 Torr
Figure 5.3
Eiectron production and loss rates changing with the averaged
temperature at ne of 1e18 m"3 and T0 of 350 K
Figure 5.4
The self-consistent solutions (Ne-Te pairs) with non-Maxwellian EEDF at Tg =
350K, P = 5 Torr
Figure 5.5
Microwave power deposition with non-Maxwellian EEDF changing with the
electron number density while electron number density and energy conservation
are imposed
Figure 5.6
The experimentally measured and theoretically calculated total continuum
emission changing with the electron number density
Figure 5.7
The schematic used for the control-volume heat transfer analysis
Figure 5.8
Electron temperature distributions at the centerline changing with the microwave
power (grid 20 is at substrate)
Figure 5.9
Gas temperature distributions at the centerline changing with the microwave
power
Figure 5.10
The spatial distributions for microwave argon plasma at 5 Torr and 680W
Figure 5.11
A vertical stacked column graph that shows the total heat along the belljar walls
and the substrate surface changing with the microwave power
Figure 5.12
Heat fluxes along the belljar walls changing with the microwave power (5 Torr
and 250 seem)
Figure 5.13
Heat fluxes along the substrate surface changing with the microwave power (5
Torr and 250 seem)
electron
Chapter 6
Figure 6.1
The Boltzmann plots showing that the experimental data can only be matched by
the model predictions with the changed rates
Figure 6.2
The Voigt parameter a changes with gas temperature at an electron number
density of 1e18 m'3, and an electron temperature of 10000K
Figure 6.3
The Boltzmann plot of the numerical predictions with the experimentally
measured 4s state number density and the OES data
Figure 6.4
Spatial distributions of the argon plasma parameters at 5 Torr, 250 seem and
xii
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
10W
Figure 6.5
Spatial distributions of the argon plasma parameters at 5 Torr, 250 seem and
30W
Figure 6.6
Mol fraction of 4s state at the centerline changing with the power
Figure 6.7
Mol fraction of electron number density at the centerline changing with the power
Figure 6.8
The comparison of gas temperature distributions at the centerline with different
plasma pressures
Figure 6.9
The comparison of electron temperature distributions at the centerline with
different plasma pressures
Figure 6.10
The comparison of the electron number density distributions at the centerline with
different plasma pressures
Figure 6.11
The comparison of 4s state number density distributions at the centerline with
different plasma pressures
Figure 6.12
The spatial distribution of the 4s state number density at 5 Torr and 10W
Figure 6.13
The spatial distribution of the 4p state number density at 5 Torr and 10W
Figure 6.14
The spatial distribution of the 5p state number density at 5 Torr and 10W
Figure 6.15
The spatial distribution of the 5d state number density at 5 Torr and 10W
Figure 6.16
The averaged 4s-state number density distribution along the z-direction from the
numerical predictions comparing with the results from the absorption
measurement
Figure 6.17
The averaged 4p-state number density distribution along z-direction from the
numerical predictions comparing with the experimental results
Figure 6.18
The averaged 5p-state number density distribution along z-direction from the
numerical predictions comparing with the experimental results
Figure 6.19
The averaged 5d-state number density distribution along z-direction from the
numerical predictions comparing with the experimental results
Figure 6.20
The wire frame plot of the comparison between MW PD profiles used in 2-D fluid
model and predicted from 2-D CRM at 5 Torr and 10W
Figure 6.21
The wire frame plot of the comparison between the MW PD used in 2-D fluid
model and predicted from 2-D CRM
Figure 6.22
The comparison of the spatial distributions of MWPD used in the fluid model and
predicted from the 2-D CRM at 5 Torr and 10W
Figure 6.23
The comparison of spatial distributions of MWPD used in fluid model and
predicted from the 2-D CRM at 5 Torr and 30W
xiii
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Figure 6.24
The comparison of Kea[0,1] rates calculated from 2-D CRM and from curve-fitted
equation
Figure 6.25
The comparison of Kea[1,0] rates calculated from 2-D CRM and from curve-fitted
equation
Figure 6.26
The comparison of Klea[0] rates calculated from 2-D CRM and from curve-fitted
equation
Figure 6.27
The comparison of Klea[1] rates calculated from 2-D CRM and from curve-fitted
equation
Figure 6.28
4s State number density comparison between the results from the 2-D CRM and
the 2-D fluid model at 5 Torr, 10W
Appendix A
Figure A.1
The Optical Emission Scan of Microwave Argon Plasma at 680W, 5 Torr, and
250 seem flow rate (Data taken on 04/16/98)
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
CHAPTER ONE
INTRODUCTION
1.0
M o tivatio n
Chemical
Vapor
Deposition
(C V D )
plays
an
important role
in the
semiconductor industry. It offers an easy and cost-effective w ay to deposit thin
films on certain substrates by varying input gases. Diamond is an attractive
material that offers many potential uses despite its high cost. The High Density
Electronics C enter (H iD EC ) of University of Arkansas, which was established in
1991, expanded its Thermal Management Program to include synthetic diamond
substrates for Multi-Chip Modules (M CM s) to take advantage of diamond's high
thermal conductivity. In 1992, HiDEC purchased W A V E M A T s (M odel M P D R 3135)
Microwave
Diamond
Deposition
System
for
its
potential
of
high
efficiencies, scaling and minimal contamination. Although high quality thin films
are achievable, the understanding of the gas phase reactions and surface
chemistry is fa r from complete.
To characterize the microwave plasma, optical emission data w ere used in
earlier studies [Gordon, 1996, Kelkar, 1996 and 1997] to verify the predicted
excited state num ber densities from an in-house Collisional Radiative Model
(CRM ). However, it was very difficult to match the measured excited state
number densities with the predicted ones. Further study [Kelkar, 1 99 7 and Li,
1997] showed
a discrepancy between the m etered
input power and the
numerically predicted power. It was thus concluded that the metered pow er could
1
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
not be used as input in the model. Therefore, a 2-D model, which couples the
electromagnetic fields and plasma discharge model, is employed to achieve selfconsistency. For the microwave argon plasma, this is the first self-consistent 2 -D
model.
1.1
Introduction
Plasma
is
the
fourth
state
of
matter
in
the
universe.
From
a
phenomenological point of view, an ionized gas (or plasma) is distinguished from
a room-temperature gas
primarily by its ability to conduct
electricity.
At
temperatures of about 100 ,00 0,000 K, gases are fully ionized. T h e description of
this kind of gas is much sim pler as compared to the general situation. For most
other engineering applications, the gas temperatures are lower than 20,000 K,
and thus such gases are partially ionized [Mitchner and Kruger 1973]. Comprised
of electrons, ions, and neutral species, partially ionized gases open a door of
vast wealth of new physical phenomena and add a lot of complexity to their
characterization.
Natural diamond was first characterized to be of organic origin by Sir Isaac
Newton. In 1955, General Electric first documented synthetic diamond by using a
High-Pressure, High-Temperature (H P H T) process. However, a later study of the
notes of W . G.
Eversole of Union Carbide Corporation would determine that
diamond had actually been deposited using a Chemical Vapor Deposition (C V D )
process
in
1952
[Lettington,
1994].
This
evidence
shows
that the first
2
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
reproducible synthetic diamond process was CVD instead of HPHT.
Diamond,
with
its
attractive
optical
characteristics
and
unbeatable
mechanical hardness, is a perfect material for use in windows, lenses, and
mirrors. Except the electron mobility, diamond’s electrical properties also exceed
virtually those of all other semiconductor materials currently in use [Yodor 1990].
Therefore, the potential use of diamond as a semiconductor material has
attracted a lot of interest.
Among the various diamond CVD techniques, microwave plasma C VD
has gained considerable importance because of its capability to produce high
quality diamond films with reasonable growth rates and areas. However, the
optimization and development of microwave plasma reactors are very difficult
since it is hard to predict the performance of the reactors. The difficulty comes
mainly from the fact that the microwave field and the plasma are highly
interactive. A development based on trial and error is very time-consuming and
costly; thus the numerical simulation of microwave plasma reactors has gained
considerable interests. Moreover, the physics and chemistry involved in the
microwave plasma are far from fully understood. Numerical modeling can thus
add a lot of basic understanding. Before we discuss the microwave plasma C V D
processes in detail, a review of diamond CVD processes is presented for better
understanding of the CVD mechanisms.
3
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
1.2
Diamond CVD Processes
The diamond C V D processes, according to their different activation
techniques, are well summarized in Celii and Butler, 1991 and Palmer, 1992. And
they are discussed below.
1.2.1 Filament-Assisted CVD
The first diamond nucleation on a non-diamond substrate was achieved by
a Hot Filament-assisted CVD process (H FC V D ) [Matsumoto, 1982]. HFCVD is
considered as the simplest of all methods except the combustion flame CVD. In
this method, diamond particles or thin films are deposited on a heated substrate
from a mixture of m ethane and hydrogen activated by a hot tungsten filament. An
experimental laboratory setup can be constructed with a capital investment of
$15,000. Thus it makes HFCVD among the least expensive techniques [Taher
1999]. A typical H FC VD reactor is shown as in Figure 1.1.
Filament
Methane
Substrate
Hydrogen
Pump
Figure 1.1 Hot Filament CVD Diamond Reactor
Diamond deposition using the hot filament method is achievable under a
4
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
wide range of deposition conditions. The deposition param eters should be taken
into consideration for good quality film, such as the filament tem perature, the
filament-substrate distance, the substrate temperature, the reactor pressure and
the m ethane concentration.
1.2.2 Combustion Flames
The growth of diamond at atmospheric pressures using combustion
flames offers the simplicity of the process method and
low cost of the
experimental apparatus, as well as the high growth rates that is already observed
[Matsui 1990]. A schematic diagram of the CVD process using a premixed
oxyacetylene flame [Capelli, 1990] is shown in Figure 1.2. The diamond thin film
is deposited at the second flame boundary, and no diamond deposition occurred
at the central area. This suggests that OH and 0 radicals, which are present in
the second flame layer, are critical to the diamond growth.
Silicon Substrate
Diamond
Thin Film
dnmary
Flame
Diffusion
Nozzle
Figure 1.2 Combustion Flame Assisted CVD Process
5
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Using optical emission, Hirose et al. [1990] described the conditions for
production of optically transparent diamond crystals. Spatial and parametric
variation of C 2 and O H concentration were obtained and indicate equilibrium
concentrations. By correlation with the deposition rate, CH, C 2 , or C H 3 were
considered candidates for growth species.
1.2.3 Low Pressure Plasmas
Unlike the filament CVD process, plasmas offer a ready source for self­
diagnostic information in the form of visible optical emission. Active species in the
plasma can be identified by Optical Emission Spectroscopy (O E S), which could
potentially be used for process control.
1.2.3.1
DC Plasma Enhanced CVD
In the DC plasma enhanced CVD processes, a plasma of m ethane-
hydrogen mixture is excited by applying a DC bias across two parallel plates, one
of which is the substrate. The substrate can be attached to either the anode or
cathode. Another configuration of this method uses a hollow cathode, which is
m ade of refractory metal. This method allows a stable discharge to be
maintained at lower voltages than the planar cathode.
However, there are problems encountered with diamond films by DC
plasmas, including high stress and high concentration of hydrogen. Moreover,
the diamond films may be contaminated by the erosion of the electrodes [Spear,
1993].
The high-pressure DC arcjet plasma is also a DC plasma process but its
6
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
operating conditions are closer to combustion flame techniques. This process
offers an extremely high deposition rate, but this deposition can only occur over a
very small area and may be contaminated by graphite and plasma-generated
materials.
RF Generator
Match
Electrost. Screen
/
6 000
OOOO
Probe
Metal
Figure 1.3 Experimental Setup of the Inductively Coupled RF Plasma
1.2.3.2
RF Plasma Enhanced CVD
Plasma etching and deposition of thin films using low-pressure radio
frequency
(R F)
glow
discharges
is currently
in widespread
use
in
the
microelectronics industry. A lot of experimental and numerical research work has
been
devoted
to this
method.
A
purely
inductively
coupled
rf plasma
experimental setup [Kortshagen, 1995] is shown in Figure 1.3. In this figure, no
substrate is shown, and this structure was used for research work only. A pulsed
7
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
probe technique was used to measure the electron energy distribution function
(E E D F) for the RF plasma discharges.
For RF plasma enhanced CVD processes, the substrate is generally
raised to above 800° C by inductive and plasma heating. High power in the
discharge was reported to be necessary for efficient diamond growth. This is due
to the fact that the average electron energy in a RF discharge at 13.56 M H z at 1
Torr is about 4 eV, which is insufficient to dissociate hydrogen. However, the high
power will result in physical and chemical sputtering from the walls of the silica
tubes, and the diamond films thus were frequently contaminated with silicon
carbide [Spear, 1993],
The driving force to use this method is the availability of the fully
developed equipment capable of depositing large areas. However, uniformity is
still a big concern. It is hard to synthesize high quality coatings at a reasonable
rate using this technique.
1.2.3.3
M icrow ave P lasm a Enhanced CVD
Microwave plasma enhanced CVD processes have been used more
extensively than any other methods for diamond film growth.
Microwave
deposition has two distinctive advantages over other techniques:
•
Contamination can be avoided since it is an electrodeless process,
•
The microwave discharge has a higher plasma density with higher energy
electrons than RF discharges due to higher frequency electric field.
8
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
However, as shown in the later sections, it is very hard to characterize
microwave plasmas both experimentally and numerically. Due to the strong
microwave field inside the cavity, the results of direct-probing measurements are
not reliable. Therefore, the non-intrusive measurements, such as O ES and Laser
Induced Fluorescence (LIF) are desirable. On the modeling side, the composition
and
temperatures
of
the
microwave
plasmas
strongly
depend
on
the
electromagnetic field, which is in turn affected by the discharge itself. Therefore,
a
model of such a microwave-plasma-enhanced
CVD
system
must self-
consistently solve for both the electromagnetic fields and the plasma discharge
[Hassouni, 1998].
1.3
Fundamental Plasma Processes and Forms of Nonequilibrium
In a partially ionized gas, there are generally six "basic" kinds of particles
[Mitchner and Kruger, 1973]: photon, electron, ground-level atom/molecule,
excited atom/molecule, positive ion and negative ion. Figure 1.4 shows the
various energy transfer processes in partially ionized plasma. The input power
accelerates the charged particles under the influence of the applied electric field.
Most of the energy is absorbed by electrons because the energy imparted to the
charged particles between collisions is inversely proportional to the particle
mass.
9
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Heat Conduction
and Convectior
Radiation
Losses
Electric
Energy
Source
^ t ,
Joule
Heating
I
Ion
Gas
Electron
Gas
Reactor
Wall
Recombina­
tion ofIons
and Electrons
Free Radical
Recombina­
tion
Elastic
and
Inelastic
Collisions
Neutral
Gas
Radiatio
n Losses
Heat
Conduction
and
Convection
Figure 1.4 Fundamental Plasma Processes
The energy is transferred from electrons to neutral particles and ions by
collisions, which can be classified as either elastic or inelastic collisions. The
collision is elastic if the internal energy of the neutral particle/ion remains the
sam e after the collision. Otherwise, it is called inelastic, in which the electrons
may induce dissociation, excitation and/or ionization. The ionization reactions will
produce electrons as well as positive or negative ions. Due to recombination in
the gas phase and the convection and diffusion loss of electrons, a balance is
established according to the input energy to maintain the charge neutrality of the
plasma. The electron induced dissociation reactions will produce active atoms
and/or radicals. These particles will also reach a balance by recombination and
diffusion losses. High-energy electrons will also populate the number densities of
10
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
the excited states of either atoms or molecules. At steady states, the number
densities of these excited states are balanced by the radiation losses, in which
the energy is released as photons with certain frequency.
To
understand the plasma processes,
it is essential to
have the
knowledge of the thermodynamic states of the plasmas. The principle of detailed
balancing states that the different reaction rates for each microscopic process
and for the corresponding inverse process are
thermodynamic
equilibrium
[Mitchner
and
equal under conditions of
Kruger,
1973].
W hen
all
the
microscopic processes in the plasma are in detailed balance, including the
radiative fields, the plasma is said to be in complete thermodynamic equilibrium
(CTE). For plasma in CTE, all the particles in the plasma can be described by
Maxwellian velocity distribution functions at the sam e temperature. The plasma is
homogeneous and optically thick, and the radiation field follows a Planck
distribution at the same temperature. The internal energy states (rotational,
vibrational, and electronic) of each particle will follow Boltzmann statistics at the
same tem perature. Moreover, the neutral and charged particles will satisfy Saha
equilibrium,
and the plasma will be in chemical equilibrium at the same
temperature.
Few plasmas satisfy all the conditions for C TE . Actually, many plasmas
can be described as being in local thermodynamic equilibrium (LTE), where the
collisions are sufficiently numerous to maintain the Maxwellian, Boltzmann, Saha,
and chemical equilibrium although a black body radiation field may not be
11
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
observed. In terms of nonequilibrium, which implies any deviation from the state
of CTE, it can be classified as kinetic nonequilibrium, ionization nonequilibrium,
chemical nonequilibrium, and Maxwellian nonequilibrium.
The collisions among plasma constituents will redistribute the energy and
momentum. Thus the plasma is reaching its equilibrium by collisions. However,
the energy transfer efficiency of the electron-neutral collisions m ay be too low to
reach a steady state temperature due to the much lighter mass of electrons. On
the other hand, the energy coupling efficiency of the ion-neutral collisions is very
high because of their comparable masses. Therefore, the ionic and neutral
species can generally be described with the same tem perature, while the
electron gas has to be described by a much higher electron temperature. This
kind of plasma is referred to as two-temperature plasma. However, even the
assumption of a two-temperature plasma may not be acceptable in some cases.
The electron energy distribution function (EEDF), which is non-Maxwellian, may
significantly affect the plasma transport properties, excited state distributions,
and degree of ionization, etc.
In the presence of cold surfaces, such as the water-cooled reactor walls
and substrate, the kinetic rates of some reactions may not be able to achieve
chemical equilibrium across the plasma. There may exist large gradients of the
gas and electron temperatures as well as the concentrations of certain species.
Therefore, a complicated two-dimensional or three-dimensional model is needed
to treat all these non-equilibrium effects that may significantly affect the plasma
12
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
discharges.
1.4
Plasma Diagnostics
The diagnostic techniques are critical for the understanding of diamond
CVD processes, especially the low-pressure plasma enhanced CVD processes.
Quantitative determination of the composition and tem peratures of various
species present in the gas phase will provide the necessary data for plasma
modeling. In-situ diagnostics may also be used for the plasma process control.
The diagnostic techniques used for diamond CVD processes can be
divided into three categories: sample extraction, physical probe, and optical
methods. Sam ple extraction methods, such as mass spectroscopy, perform an
ex-situ analysis of the extracted portion of the reactive gas. Physical probe
methods, such as those using a Langmuir probe, introduce a probe into the
reactor cham ber to
measure
the
electric potential,
electron
density
and
temperature, etc. However, these two methods may not be available for some
plasma processes. For microwave plasmas, the probe method is not appropriate
since the probe will disturb the microwave electromagnetic field significantly.
Since the microwave plasma is highly nonequilibrium, the ex-situ measurements
cannot provide accurate information. Therefore, optical methods are preferred
because of their non-intrusive nature. The optical methods used in this research
will be discussed in detail in Chapter 2 and 3.
13
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
1.5
Plasma Modeling
The power of computers (in both speed and memory) has increased by
approximately 107 over the last 40 years. As a result, computer modeling has
become a powerful tool for exploring the physics and chemistry of plasmas.
Although most substance in the universe is in the form of plasmas, they present
us with a wealth of complexity not found in ordinary fluids. W hat m akes it even
harder to characterize the plasmas is that plasmas are difficult to experimentally
probe. Thus, to m eet the great challenges associated with plasmas, computer
modeling offers an affordable and effective way to achieve more understanding.
1.5.1 Non-local Approximation
In recent years, the modeling of nonequilibrium plasmas has gained in
importance for the development and understanding of plasma sources. One of
the major challenges is the development of a simple but realistic description of
the spatial dependence of the electron energy distribution function (E E D F ) in
spatially inhomogeneous nonequilibrium plasmas. A number of methods have
been developed to treat this problem, such as the Monte Carlo method, the
particle in cell method (PIC), the converted scheme method, or the direct
numerical integration of the electron Boltzmann equation.
However, these
methods are often very slow if multidimensional systems are considered. Thus to
develop
spatially
two-
or
three-dimensional
discharge
models,
some
approximation methods are considered. For low-pressure, w eakly collisional
14
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
plasmas, the “ nonlocal approximation" is particularly suited [ Kortshagen, 1996].
The nonlocal approximation addresses weakly collisional plasmas, w here the
energy relaxation length (Xc) exceeds the discharge dimensions in the range of
the kinetic energies of interest.
A ( u ) = — -—
Na„(u)
,,
(1.1)
X («) = -----------------------N ( a a (u ) + cTXu ))
where Xm is the m ean free path for elastic momentum transfer of the electrons,
and X is the m ean free path for all kinds of inelastic collision processes; u is the
electron kinetic energy; a is the cross section area.
The Boltzmann solver we used in this research, ELEND IF, relies on the
nonlocal approximation [Morgan, 1994]. Using the assumption that the electron
velocity distribution function (E V D F) is nearly isotropic, it can be replaced by the
first two terms of a spherical harmonic expansion as will be shown in Equation
4.1.
1.5.2 Argon Models
Argon is often used in plasma studies for two reasons. O ne is that argon is
a frequently used rare gas in laboratory studies of plasmas and industrial
applications. Argon can either be a earner gas or the gas introduced before other
gases for safety and better tuning. Another reason is that extensive studies of
15
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
argon collisional radiative models have been conducted, especially the argon
plasmas with the Maxwellian EEDF [Vlcek, 1985].
For simplicity, zero-dimensional models are generally developed first,
which may be modified with diffusion to account for the effects of the geometry.
Vlcek has developed a numerical method for the Boltzmann equation to obtain
the EE D F in a nonequilibrium argon plasma characterized by a set of parameters
(Te, T a, T it ne, etc.) [Vlcek, 1985], Vlcek then developed an argon C R M with 65
discrete effective levels. This model provided information about mechanisms
populating the excited levels under various conditions in non-isothermal plasmas
and about the effects caused by the departure of the actual E E D F from the
corresponding Maxwellian distribution. This model proved that in some region the
assumption of the Maxwellian EEDF was not justified [Vlcek, 1986 and 1989].
Braun and Kune developed a three-level atomic model that was used to
determine the steady-state collisional-radiative coefficients in nonequilibrium
partially ionized argon plasmas. Rate equations for populations of the atomic
levels then were coupled to an electron Boltzmann equation that included
inelastic processes. The solution of these coupled equations yielded an analytical
form of non-Maxwellian EEDF [Braun, 1987]. Ferreira [1985] investigated the
contribution of the ionization from the two metastable and the two resonance
levels of argon to the total ionization rate in a low-pressure argon positive
column. The results showed that the values of the predicted m aintenance field
w ere considerably lower than those with only ground state. A kinetic model that
16
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
included 25 levels was developed by Repetti et al. [1991] to the study of
relaxation of an atmospheric thermal argon plasma from equilibrium condition
near 1 eV. It was shown that resonance radiation trapping was an important
process in such plasmas. This paper also suggested that future work should
include spatially dependent processes such as particle diffusion and plasma
expansion, as well as photoionization modeled as trapping of recombination
radiation. Benoy et al. [1993] discussed the radiative energy loss in a non­
equilibrium
argon
plasma
which
allowed
for
deviation
from
local
Saha
equilibrium.
Kelkar et al. [1997] developed a self-consistent 25-level CRM that could
solve for the excited state number densities as well as the power deposited into
the argon plasma. This model, as outlined in Chapter 4, can be considered as a
pseudo-1 -dimensional model by considering the electron diffusion loss term.
Lymberopoulos et al. [1993 and 1998] developed one-dimensional fluid
models for a 13.56 MHz argon glow discharge and for a pulsed-power inductively
coupled argon plasma. For a pressure of 1 Torr, metastables w ere found to play
a major role in the discharge despite the fact that their mole fraction was less
than 10"5. The studies suggested that neutral transport and reaction must be
considered in a self-consistent manner in glow discharge simulations.
1.5.3 Hydrogen/Hydrocarbon Models for Diamond Deposition
Although the main purpose of this dissertation is to study microwave argon
17
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
plasmas, microwave hydrogen plasmas were also studied in comparison with the
argon plasmas.
Hydrogen and hydrocarbon models are much more complicated than
argon models due to the fact the plasma chemistry will affect the discharge
significantly. Since the hydrocarbon gas only consists 0 .5 -2 % of the gas mixture
in the diamond CVD processes, some studies have devoted to the pure
hydrogen modeling. However, to understand the mechanisms of the diamond
CVD processes, hydrocarbon chemistry has to be included in the model.
Kune and Gundersen [1983a and 1983b] developed a pure hydrogen
model intended to have a detailed understanding of the physics of hydrogen
thyratrons. Their model included collisional and radiative processes involving
excited states of atomic hydrogen. A Maxwellian E E D F was assumed for their
rate calculations. They concluded that the step-wise ionization from atomic
hydrogen excited states played a dominant role and the inverse three-body
recombination in the gas phase was the major electron loss mechanism.
In
Kune’s later work [ Kune 1987], the role of atom-atom inelastic collisions in twotemperature nonequilibrium plasmas was investigated. Their results can be
applied to plasmas that have atomic hydrogen as one component with large
enough amounts so that e-H and H-H inelastic collisions and interaction of these
atoms with radiation dominate the production of electrons and excited hydrogen
atoms. Burshtein et al. [1987] investigated the effects of quenching of excited
hydrogen atoms by H2 molecules. The delayed coincidence method was used to
18
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
measure the quenching of the intensities of the Ha, Hp, and HY Balmer lines in the
afterglow. St-Onge and Moisan [1994] investigated the hydrogen atom yield in
RF
and
microwave
hydrogen
discharges.
A
particle
balance
model was
developed to solve for the H, H2, H+, H2\ and H3+ species. It was observed that
the H-atom concentration decreased when the wall temperature increased, due
to the increased efficiency of atomic recombination on the wall.
Several researchers have investigated the effects of various electronneutral interactions with the EE D F through the solution of the Boltzmann solver.
Loureiro and Ferreira [1989] reported that superelastic vibrational collisions
strongly enhanced the tail of the E E D F that significantly increased the electron
excitation rates. Colonna et al. [1993] found that the electron-electron collisions
must be included in the E E D F solver for post discharge conditions. Capittelli et
al. [1994] showed the dependence of EE D F on superelastic electronic collisions
by specifying parametrically the concentrations of the electronically excited states
both for H2 and atomic hydrogen. They concluded that the EEDF and related
properties depended on electronic superelastic collisions at low electric field (<30
Td) when the average electron energy was less than 1.5 eV.
Kelkar et al. [1997] developed a comprehensive hydrogen model, which
coupled the non-Maxwellian EEDF with the zero-dimensional CRM modified with
the ambipolar diffusion. The predicted excited states number densities were
compared with the optical emission data. T h e microwave power absorbed by the
plasma was also calculated and compared with the experimental data.
19
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
For the modeling of the diamond CVD processes, different approaches
have been adopted for different deposition systems. Surendra et al. [1992]
developed the self-consistent particle-fluid hybrid simulations to study the
structure of hydrogen dc discharges between parallel plates. Dissociation of H 2 in
the anode region contributed significantly to the flux of atomic hydrogen to the
anode, where diamond is typically grown. In this model, a M onte Carlo simulation
was used to describe individual energetic electrons in the cathode sheath, while
the electrons and ions in the low-field region of the discharge were modeled as
fluid.
Koemtzopoulos et al. [1992] developed a model to predict the degree of
gas dissociation in an H 2 microwave discharge in a tubular reactor. Ambipolar
diffusion rates were calculated for the electron number density balance. The
E E D F solved from the Boltzmann equation was not Maxwellian. It was found that
lowering the total gas density and/or increasing the pow er would enhance gas
dissociation as well as atom density. Flow rate had a minor effect on atom
density, and the addition of 1%
CH4 to the hydrogen discharge did not
appreciably affect the EEDF.
Bou et al. [1992] adopted a model that took into account 106 reactions for
kinetic calculations in a microwave plasma of 1% CH4 gas mixture with H * This
model did not consider the energy conservation, and only electro-neutrality was
imposed in the plasma volume. The increase in the electron temperature,
electron density, or in the number of effective electrons raised the amount of H
20
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
atoms and consequently induced an increase in C atom concentration.
T ah ara et al. [1995] used a kinetic model to analyze the particle-species
compositions in the C H 4/H 2 microwave plasmas, in which 14 neutral species and
16 ionic species w ere considered. Several assumptions w ere m ade in particular,
such as the Maxwellian EEDF, chemical equilibrium, and uniform tem perature
and pressure across the cylindrical plasma volume, etc. The diamond-like-carbon
synthesis, plasma diagnostic measurements and kinetic model analyses were
conducted to correlate between plasma properties and the film features.
M icrowave enhanced plasmas are of current interests and have already
demonstrated
good
potential
in moderate
pressure
(1 -1 5 0
Torr)
plasma
processing applications including diamond film deposition. Tan and Grotjohn
[1994 and 1995] have developed a self-consistent model of the electromagnetic
field and plasma discharge in a microwave plasma diamond deposition reactor.
Their modeling is well described in Section 5.2. The understanding obtained from
the electrom agnetic solutions allowed the reactor design to be analyzed for
improvement and optimization of such quantities as the uniformity of microwave
power absorption. Hassouni [1994, 1996, 1997 and 1998], Scott [1996] and
Gicquel [1994, 1996 and 1998] developed a series of microwave hydrogen
models from zero-D to two-D. A lot of experimental data w ere obtained to
facilitate the plasma modeling, such as the ground state and excited state Hatom densities. These data were also used to validate the microwave plasma
model. For the detailed two-dimensional plasma modeling, one can refer to
21
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Section 4.3.
A self-consistent model of the microwave fields and the plasma discharge
in a moderate pressure (2 5 0 0 -1 1 0 0 0 Pa) resonant cavity type reactor was
recently developed
by Hassouni et
al.
[1999].
The
self-consistent model
presented by Hassouni et al. differs from those reported in other papers.
Generally, the models we addressed before require experimental data, such as
the power needed to sustain the plasma and the tem peratures for different
modes (electron, gas or ion temperatures). Moreover, this self-consistent model
addresses the coupled phenomena of chemistry, energy transfer, species and
energy transport.
1.6 Objectives
Based upon the above literature review, one can conclude that more work
is needed to better understand plasma discharges as well as the diamond
deposition processes.
In this thesis,
the major purpose
is to add more
understanding to the nonequilibrium microwave argon and hydrogen plasmas.
Several problems that have occurred in our zero-dimensional argon model
cannot be solved by this model alone, such as the big discrepancy between the
predicted power absorption and the experimental measurement. Developing a
two-dimensional self-consistent argon model is promising. M ore experimental
data can also add more understanding to the study of the microwave plasmas.
The objectives of this dissertation are to:
22
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
•
Develop
a self-consistent
pseudo-1-D
plasm a
model,
which
couples the Boltzman solver (E LE N D IF) with our CRM;
•
Perform the energy balance study on the microwave reactor to gain
a better understanding of the energy transfer mechanisms in the non­
equilibrium plasmas;
•
Demonstrate the potential of OES as a quantitative diagnostic
method for microwave plasmas when used with the numerical model;
•
Perform the absorption measurement for the metastable excited
state of argon plasma for additional experimental data to verify the CRM;
•
Develop
a
self-consistent
two-dimensional
model
of
the
electromagnetic field and the plasma, which will provide more detailed
information about the plasmas and the microwave fields.
23
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
CH A PTER TWO
EXPERIM ENTAL FA CILITY
2.1
Microwave Plasma Reactor
Th e experimental research is conducted on the W A V E M A T microwave
plasma reactor (Model M P D R -3135).
A picture of the W A V EM A T reactor is
shown in Figure 2.1. This 7-inch diameter cavity microwave plasma reactor is
designed for large-area uniform diamond film deposition. The microwave probe
assembly and sliding short form the top portion of the cavity. The lower section of
the reactor consists of the bottom surface, the base plate and a metal plate.
Figure 2.1 The W A V E M A T microwave plasma reactor
T h e microwave is produced by Sairem microwave power supply with a
frequency of 2.45 G H z and maximum power of 6 kW. The microwave energy is
coupled into the cavity through the waveguide and probe. The microwave power
is measured by two power meters, which indicate the incident power and
24
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
reflected power respectively.
The microwave discharge can be created when
excited in a single cavity electromagnetic mode. A ball shaped plasma is formed
inside the quartz bell jar, and the substrate is placed near the plasma. T h e typical
operation conditions for diamond deposition are 30-80 Torr pressure, 300 seem
hydrogen, 3-5 seem m ethane flow rates and 1.6 kW input power. However, for
pure argon plasmas, the chamber pressure is reduced to about 5 Torr to stabilize
the plasma. The pressure in the chamber is controlled by a throttle valve and a
mechanical pump (Franklin Electric). All the gas flows are controlled by MKS
mass flow controllers.
145GHz
j
Microwave
i
Probe Cooling
Short Cooling
Short
Optical
Pyrometer
Probe
Bell Jar
Cooling
Argon plasma
[_
Applicator
Cooling
Substrate
Chamber
Cooling
Substrate Cooling
Figure 2.2 Schematic of the microwave reactor system
25
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Figure 2.2 shows a schematic of the microwave reactor configured for the
energy balance study. The base plate, chamber, probe, short and substrate are
cooled by water, and the belljar is cooled by air. An optical pyrom eter is installed
on the reactor to monitor the substrate surface temperature, which is a critical
param eter for diamond deposition. The belljar temperatures are m easured by the
infrared thermocouples by E X ER G E N . For the purpose of energy balance study,
several K-type thermocouples are installed on the cooling-water and cooling-air
lines to monitor the inlet and outlet temperatures.
2.2
Emission/Absorption Systems
The primary diagnostic system is composed of a 0.5 m S P E X 500M
monochromator, Hamamatsu R -928 PMT, the monochromator driver (M SD), and
fiber optic assembly (Figure 2.3).
2.2.1 Emission system
The
emission
equipment
involves
a
dedicated
com puter for
data
acquisition, a tungsten lamp for calibration, an optical pyrometer, and a set of
collecting optics. A 1-m W H e-N e laser is used for assuring alignment and for
checking spectral response of the system. The accuracy of this system agrees
with the reported value of ± 5 A, and the results are repeatable within ±1 A.
26
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Aperture
Stop
Optical
Fiber
Microwave
Plasma
Monochromator
D ata Acquisition
Computer
Figure 2.3 Schematic of the optical emission spectroscopic measurement
system
Figure 2.3 shows the optical setup used for collecting and focusing the
emitted light onto the entrance slit of the monochromator. A 40-cm focal length
(2” diameter) lens is used to collect the light from the plasma emission. T h e light
then is passed through a typical 0.5” aperture stop. A 30-cm lens focuses the
image of the plasma onto the fiber optic cable. To determine the absolute
intensity of emission, it is necessary to calibrate the emission system. The
tungsten lamp is used as a known source of emission (3 0 0 0 -8 0 0 0 A). T h e
temperature of the lamp stripe is measured by a Pyro-micro® vanishing filam ent
optical pyrometer.
27
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
2.2.2 Absorption system
Figure 2 .4 and Figure 2.5 show the absorption experimental setup. The
emission from the CENC O argon spectrum lamp (Model # 87210) was collimated
with a 10-cm focal length lens through a typical 1-cm aperture stop. Then another
Argon Lamp
r
Microwave
Argon
Plasma
chopper
Oscilloscope
MSD
Lock-in
Amplifier
Monochromator
Figure 2.4 Schematic of the absorption measurement experimental setup
10-cm focal length lens imaged the lamp onto the optical chopper. The beam
was collimated once more by another 10-cm focal length lens. A 40-cm lens
focused the beam through the holes on the reactor cavity window and through
the plasma. Another 40-cm focal length lens collimated the beam again. The
collecting optics was the same as the emission system. The analog signal from
P M T was the input to the MSD controller. W e then connected the analog signal
28
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
to the lock-in amplifier in order to m easure small signals.
The lock-in amplifier is synchronized with the chopper frequency to
maximize the probed signal. Thus the absorption m easurem ent system can read
very w eak signals, especially useful for the signal after absorption through the
plasma volume.
For the wavelength used for absorption measurement, the monochromator
is preset to its position. However, due to the fact this wavelength may be shifted,
it
is
safe
to
scan
the whole
emission
spectrum
before
presetting
the
monochromator to this specific wavelength.
(a)
(b)
(c)
Figure 2.5 Some pictures of the absorption measurement setup: (a) the argon lamp
with the focusing lenses; (b) the Wavemat reactor with the collecting optics; (c) the
monochromator and P M T
29
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
CHAPTER THREE
EM ISSIO N /A BSO RPTIO N DIA G N O STIC TEC H N IQ U ES
3.1
Emission Spectroscopy
Emission diagnostics is the oldest and most commonly used non-intrusive
plasma diagnostic technique. The equipment for emission spectroscopy is
relatively inexpensive and commercially available from a variety of vendors.
Therefore, despite the limitation on the spatial and spectral resolution, emission
diagnostics will continue to be a useful
and reliable
method for plasma
diagnostics.
3.1.1 Absolute Line Intensity
Absolute atomic line emission leads directly to the population (nO of an
excited state through the following relation:
A id
k
", = - t JT L
hcAtJ
( 3 -1 >
w here ly is the emission intensity, Xq is the transition’s wavelength, h is the
Planck’s constant, c is the speed of light, and A,- is the Einstein transition
probability. This measurement is independent of lineshape if the entire line’s
emission is collected. This measurement is also independent of LTE or PLTE if
the emission lines investigated are optically thin over all plasma operating
30
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
conditions. T h e optically thin condition may be expressed as:
f
exp
i
hv
\
« 1
(3.2)
where T ex is the characteristic temperature associated with the excited states,
and v is the frequency of the measured atomic line.
Two measurements are made to obtain n,-. One measurement is m ade at
the transition’s center wavelength
and collects both line and background
continuum emission. Another one is performed 10A away from the center
wavelength and collects only the continuum emission. It is assumed that the
continuum
emission does
not change
appreciably over
10 A. Then
the
subtraction of the latter m easurem ent from the former one will yield the desired
emission intensity.
After the population of a given excited state n\ is determined, one can
define a temperature
T lte
based on the following Maxwell-Boltzmann relation:
n.
n„
g,
Qa
— = —
e.
exp
\
(3.3)
kTLTEi J
In the above equation, the total number density na can be replaced by the
ground number density ni, p/kTg, since for all experimental condition reported
here, the fractional ionization is less than 0.5% . Qa is the electronic partition
function. It can also be replaced by its ground state degeneracy (gi) for the same
reason.
31
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
3.1.2 Relative Line Intensities
By using two or more excited state number densities, the ground state
number density in equation (3.3) can be eliminated and we have the following
Boltzmann relation:
n
s
(e -e
nj
Sj
\
— = — exp —-----^
kkT
1b
(3.4)
y
The slope inferred from a plot of the natural logarithm of n/gj versus upper
state energy 8j is then inversely proportional to the temperature. This relation
does not require thermodynamic equilibrium between the ground and excited
states. This method is valid for PLTE and LTE.
3.1.3 Continuum Radiation
Continuum emission results from interactions between free electrons and
ionized or neutral particles in the plasma. Absolute emission measurement of
continuum radiation can provide information about both absolute electron number
densities and electron temperatures.
For singly
ionized
plasmas,
the
total
continuum
emission
represented as follows:
32
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
can
be
8
—
8 J jj
+
8
g-
+
6 g
(3.5)
C x = 1.63 x 10-43 Wm*Kl/2sr~l C 2 = 1.026 x 1(T34 WrrrK-V2sr-1
where na, ni, ne and X denote the neutral particle, ion, electron number densities
and the wavelength. The needed cross-section information describing the freebound recombination is contained in the so-called free-bound Biberman factor,
£fb.
Similarly,
the
free-free
electron-ion
information in the free-free Biberman factor
term
contains
the
cross-section
The free-free electron-atom term
also requires an appropriate cross-section, Q (Te), which is taken from Deveto
[1973].
3 .1 .4
C alib ratio n
Calibration is a critical
step
for the optical
emission
spectroscopy
measurements. A tungsten lamp was used here acting as the calibration light
source, and the temperature of the tungsten filament was measured by a
pyrometer. T h e absolute calibration factor is determined as:
v
lam p
(3-6)
33
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
w here Viamp is the signal measured, e is the emissivity of the tungsten filament
with a value of approximately 0.45 [Larrabee, 1959], Ci and C 2 are constants
with C 1 = hco2, and C 2 = hco/k, dX/dx is the reciprocal iinear dispersion, and Wex#
is the width of the exit slit of the monochromator.
The emission intensity can be calculated using the following expression:
j _ ^ngnal
(3.7)
"~ ~ d J
w here L is the chord length, and Vsignai is the measured signal strength.
Since the OES signal passes through not only the collecting optics, but
also the belljar, it is interesting to compare the calibration of the tungsten lamp
signals with and without the belljar. As shown in Figure 3.1 and 3.2, the two
calibration curves are different. Thus it will introduce large uncertainty to the
excited state number density measurements if a wrong calibration curve is used.
For the calibration curve of the light passing through the belljar, it is observed
that the emission light with wavelength smaller than 4000 A was not able to pass
through the belljar. Comparing Figure 3.1 with 3.2, not only the m easured signal
strength is lower in Figure 3.1, but also the shape of the calibration curves is
slightly changed. Argon plasma should not have any contamination on the belljar
surface, thus the condition of the belljar remain the same during the calibration
procedure and the O ES measurements. The O E S signal also passes through the
optical windows on the reactor. The windows were tested to show that their
effects on the OES signal were negligible.
34
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Signal Strength ( a.u.)
3500
4000
4500
5000
5500
6000
6500
7000
7500
8000
wavelength (A)
Figure 3.1 Calibration of the tungsten lamp signal passing through the belljar
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
Signal Strength (a. u.)
1.0000E+07
1.00006+04
3500
4000
4500
5000
5500
6000
7000
7500
8000
wavelength (A)
Figure 3.2 Calibration of the tungsten lamp signal without passing through the
belljar
36
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
3.2
Absorption Spectroscopy
T he transmission of light through a linearly absorbing medium can be
described by the Beer-Lam bert relationship:
(3.8)
W here T(v) is defined as the spectral transmittance of the medium, l(v) is
the intensity at frequency v observed after propagation through the absorbing
medium, lo is the incident intensity of the probe beam, and k(v) is the spectral
absorption coefficient.
The two strongly absorbing transitions that can be used to measure 4s
state number density are at 8115 A and 763 5 A. The latter transition was chosen
because the system’s spectral response decreases quickly towards the infrared.
An argon low-pressure discharge lamp is used as our source for this 7635 A
signal. This lamp operates at low pressure, and the gas temperature is close to
room temperature. These conditions are similar to the lamp used in the
measurements m ade by Baer et al [1993]. His m easurements showed that the
FW H M (full width at half of the maximum value) of the lamp’s 8115 A line is
0 .022 A. Both this transition line and the 763 5 A one have the same metastable
lower state and are predominantly Doppler broadening. W ith respect to line shift,
Griem’s [1964 and 1974] theoretical work shows that this shift is negligible and is
also supported by Baer’s experimental work. Therefore, this lamp is used as the
37
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
source for a centerline absorption measurement.
The spectral absorption coefficient near a spectral line may be expressed
as [Com ey 1977]:
C
I
g ^ \J;
g2”
(3.9)
g A z = SzB 21
B2l = c 3A 21/ ( S t B i v 3 )
Substituting Equation 3.7 into Equation 3.6, w e get:
T(v) s I ( v ) I I 0 = exp|(n2B2]- , hBX2)<Kv)—
c
(3.10)
w here B 12 and B 21 are the stimulated absorption and emission probabilities,
respectively, and 4>(v) is the associated line shape. Here, ni and n2 denote the
lower and upper state number densities. The num ber density of the upper state,
n2, can be obtained through emission. As shown in Equation 3.9, Bi2 and B2i are
converted from the Einstein coefficient A2i [W iese et al., 1966],
If w e take the natural logarithm of Beer's law, w e obtain the following
equation:
In ——
T h ree
measurements
= (n ,5 21 - n xBn )(j){v)
are
required to
hvx
c
(3.11)
experimentally determine
the
transmission of a probe beam. The first one is taken with just the lamp in
operation (L). This gives the reference signal lo. A second measurement is taken
38
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
with just the plasma in operation (P). As shown in Figure 3.4, the optical chopper
is located between the discharge lamp and the plasma, allowing filtering out the
plasma’s emission. However, the chopping wheel alternates between equally
spaced open and solid segments so that a fraction of the plasma emission can
be reflected off of the solid segments and thus a signal will be generated also at
the chopping frequency. This extraneous signal is phase shifted exactly 180’
relative to the lamp’s signal. T h e phase sensitive detector will display the
difference of the lamp’s signal and the plasma signal (the lamp’s signal is zero in
this particular measurement). W e have set the detector's phase such that the
lamp produces a positive signal. The plasma alone thus will produce a negative
signal which represents an offset. Finally, the last m easurem ent is taken with
both the lamp and plasma in operation (LP). The desired transmission is then:
n v ) = l ( , v ) / I a = ( L P ~ P)
39
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(3.12)
3.3
Spectral Line Broadening Theories
A spectral line m ay be broadened and shifted by a variety of processes that
depend
on
certain
environmental
parameters.
Therefore,
accurate
measurements of a spectral lineshape should be combined with an adequate
theory for the determination of important plasma parameters. For a more detailed
description of broadening theory, one should refer to Griem, Breene, Shore, or
Sobelman [Griem, 1964 and 1974][Breene, 1961 [[Shore, 1968][Sobelman, 1981]
3.3.1
Natural Broadening
Natural broadening is due to the finite lifetimes of the states involved in the
probed transitions. A Lorentzian lineshape is described as:
r, .
A v v /2;r
K v ) = - ----------------------(v '-v 'o )
tttt
+ (A v V 2 )
(3.13)
where vo is the linecenter frequency and Avn (sec*1) is the full width of the line at
the half-maximum intensity values (FW HM ). The integrated area under the
lineshape is normalized to unity. The width is proportional to the sum of the
inverse radiative lifetimes of the two states and m ay be expressed as:
a ,.--!-'
'
where ti and
12
1
—
1
i- —
In vr>
(3.14)
Tu
are the total radiative lifetimes of states 1 and 2, respectively.
The total radiative lifetime may be determined from the sum of all radiative decay
40
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
rates from that level.
(3.15)
where A] is the spontaneous emission coefficient for the transition i-> j w here the
index j runs all over the possible lower states.
3.3.2 Doppler Broadening
Doppler broadening is caused by the Doppler effect and is described as
an inhomogeneous broadening mechanism since it does not affect all particles
equally. Specifically, for an emitter (absorber) approaching the observer (light
source) with a relative velocity Vo, the effective emission (absorption) frequency
is shifted as shown in the following equations:
(3.16)
or equivalently,
w here Av and AX represent the Doppler shift in frequency and wavelength, vo and
Xo represent the frequency and wavelength in the stationary reference frame.
For a system in thermal equilibrium, the number density of particles in
state j moving with a velocity between Vo and V q + dVo m ay be described by
41
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Maxwellian distribution:
N,
n tfo V n j =
7tU1V.
-exp
V
VD V dVn
V r,j
f
(3.18)
where N* is the total number density of particles at state j. V p= (2kT/m )1/2 is the
most probable veiocity. Substituting Equation 3.15 into Equation 3.16, we obtain
an expression that describes the number of particles with linecenter frequencies
shifted from vo to vo + dv:
NjCc/vo)
nj {v)dv =
exp
c v-v n
V
dv
(3.19)
Since the emitted or absorber power from an optically thin spectral line is
proportional to the number density
0 j(v)dv,
the resulting Doppler broadened
profile (Gaussian shaped) can be expressed as:
f
I(v) =
me 2
f
1/2
exp
2nkTv o
v-vn
me
2kT
(3.20)
The Doppler FW HM is given by:
f8ln2 fcT
A vD = v 0
n
I/2
(3.21)
me'
In terms of the Doppler width, the profile can be rewritten in the following
expression that is more convenient:
f
f 41n2y
/(v ) =
V «■ J
f M—1/ \ 2>
v -v 0
1
- — exp -41n 2
Avd
I VD J y
V.
42
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(3.22)
3.3.3 Pressure Broadening
In addition to natural and Doppler broadening, a spectral line may be
affected by interactions with neighboring particles (i.e., molecules, atoms, ions,
and/or electrons). Pressure broadening effects can be classified either by the
mathematical approximations m ade in treating the perturbation or by the types of
the perturber. The following sections will review the types of interactions
classified by the type of perturber.
3.3.3.1
van d erW aals Broadening
Interactions
broadening,
result
with
neutral
from
the
particles,
also
dipole-induced
known
dipole
as
van
forces.
der W aals
The
impact
approximation is used and the width (Avc) and shift (Avs) in the ideal case are
given by Breene [1961]:
Avc = 2.71N C l ' 5V V5
(3.23)
A v, =0.98 N pC ; ,5V
where Np is the perturber num ber density and V is the mean relative velocity
between the atom and the perturber.
3.3.3.2
Resonance Broadening
Resonance interactions occur between identical particles and are often
significant for transitions that involve the ground state through an allowed
43
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
transition. Classically, the broadening may be caused by the efficient energy
exchange between the two identical oscillators and effectively results in a
reduction of the level’s lifetime [Breene, 1961]. Due to symmetry, resonance
broadening of the transition (the transition involving the ground and first-excited
states, denoted 1 and 2, respectively) will result in an unshifted Lorentzian profile
with a width (FW H M ), Avres, which is given by [Griem, 1964]:
f _ V '2
= - gx
"
2\g i j
(3.24)
e0m,arl2
where so is the permittivity of free space, Ni (rrf3) is the population number
density in state 1, g is the degeneracy of a given state, and fo. coi2 are the
oscillator
strength
and
angular
frequency
of
the
resonance
transition,
respectively. Avres reflects the energy uncertainty of state 2 as a result of the
lifetime reduction due to the resonance interaction since the lifetime of state 1
(the ground state) is effectively infinite.
3.3.3.3
Stark Broadening
Stark broadening, or pressure broadening caused by charged particles,
can be treated with a classical electrodynamics approach that considers the
effect of an external electric field on the energy levels of an atomic system and
the resulting influence on a spectral line. An external electric field F applied to an
atomic system effectively distorts the electron distribution and induces an electric
dipole aF , where a is the atomic polarizability. The interaction between the
44
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
induced-dipole moment and the electric field will shift the atomic energy levels by
an amount proportional to the interaction energy given by a F 2. Therefore, the
Stark shift varies quadratically with the electric field strength.
The motion of charged species within a Debye sphere gives rise to varying
electric fields and results in the broadening of a spectral line. The lineshape may
be determined from a combination of both the ion and electron broadening
effects. Thus the Stark-broadened profile of an isolated neutral non-hydrogenic
atomic line can be considered as a convolution of the electron-impact profile with
the quasi-static profile for ion broadening.
Tabulated theoretical Stark parameters relate the broadening and shift of
spectral lines to electron number density and temperature. For predominantly
singly-ionized plasmas, the total theoretical width (F W H M ) due to the quadratic
Stark effect, wth (A), and the corresponding total theoretical Stark shift, dm (A), are
given by the relations [Griem, 1964 and Baer, 1992]:
ajA * 2[1 +1.75 x 1O'4ntx'4a (l - 0.068/;,1'6Tt' x' 1)]1 (T16wnt
dth * 2[c//w ± 2.00 x 10'J«r,/4a (l - 0.068«t l/<T e~l/: )]10"16wnt
w here w(A) is the electron impact parameter, a is the ion-broadening parameter,
and (d/w) is the relative electron-impact shift. The Stark shift is towards longer
wavelengths except for the negative values of (d/w). Equation (3.24) includes the
param etric dependencies on ne and T e so that the state-specific parameters may
be used directly from the table. The restrictions on the applicability are given by
the following relations (in cgs units) [Griem, 1974]:
45
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
0.05 < a
r n. \' 1/4
V1016 j
< 0 .5
^3 2 6 )
R = ^ - = 6 u l K V 6 { e 2 / k T ) xn- n \ ' 6 < 0 .8
a
3x ft.tin *2Y
=
-------------
v.v,
m.Z
"1^
>1
/3
n.
where a is a measure of the relative importance of ion broadening; R is the ratio
of the mean distance between electrons, pm, where 47tpm3ne/3=1, and the Debye
radius, po, where pD2 = kT/47cnee2; ve,vi are the mean relative electron and ion
velocities, respectively; Z is the effective nuclear charge; n’ is the effective
principal quantum number of the upper state; and a is the ratio of electron to ion
collision frequency and indicates the relative significance of the time-varying
fields generated by the respective charged species.
For parameters outside the ranges specified by Equation 3.26, the Stark
broadening and shift equations should be appropriately modified or the entire
theoretical profiles should be used [Griem, 1974].
Table 3.1
mathematical
presents a brief discussion of the physical classifications,
approximation,
and
lineshapes
of
various
line-broadening
mechanisms [Baer, 1993]:
46
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Table 3.1 Physical classifications, mathematical approximation, and lineshapes of
various line-broadening mechanisms
Hom ogeneous Broadening
Inhom ogeneous
Broadening
Stark
(charged
perturbers)
Resonance
(like
perturbers)
VanderW aals
(neutral
perturbers)
Physical
Classification
Other
Doppler
(relative
velocity)
Mathematical
Approximation
Maxwellian
Velocity
Distribution
Impact
Quasi-static
Other
(Avc tp^ « 1)
(Avc tp >> 1)
Lineshape
Other
Gaulsian
Lorentzian
Modified Lorentzian
47
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
CHAPTER FOUR
MODELING
4.1
Pseudo-1-D Plasma Model
This section describes the Collisional-Radiative-Model (C R M ) used in our
research to predict the excited state number densities as well as the electron
number density, electron temperature, the deposited microwave power, etc. The
reason to label the CRM as a pseudo-1-D plasma model is that CR M itself is a
P, Tg, ne, R, Lj y, etc.
/
E/N /
A
EEDF
No
D n e /d t= 0
Yes
Figure 4.1 The iterative scheme to obtain a self-consistent solution
48
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
zero-D model. However, the electron diffusion loss is taken into consideration so
that one can estimate the effects of the electron diffusion and the cavity
geometry.
Figure 4.1 shows the iterative scheme to obtain a self-consistent solution.
First, the program reads in the geometric data as well as the operation
conditions, such as the chamber pressure (P). Some estimated values are also
input, such as the excited state number densities, electron number density, the
electron and gas temperatures, and the electric field strength (E/N is used where
E stands for the electric field strength and N stands for total density).
Both argon and hydrogen models have been developed. One can refer to
Kelkaris thesis for more detailed information [Kelkar, 1997]. Here only the argon
model is discussed for the interests of this research.
4.1.1
E lectro n E nergy D istribu tion Function
T h e general form of the Boltzmann equation can be found in any statistical
mechanics text, such as [Mitchner and Kruger, 1973]:
at
V n .J .
mt
= 2
X
(4 ' 1)
W here fe is the electron velocity distribution function (EVD F). c is the
electron velocity. Fe is the external force that can be simplified as qE. ne is the
electron number density. me is the mass of electron. Rer represents collisional
rates between electrons in a particular velocity class and species r.
Further simplification is needed in the general case of non-equilibrium. W e
49
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
assume that the EVDF is nearly isotropic so that it can be replaced by the first
two terms of a spherical harmonic expansion:
/( W
.( n + /,( i') ~
(4'2)
Often it is desirable to work with Electron Energy Distribution Function
(E E D F) rather than EVDF. Using the relation E=mV2/2, one can convert between
the EV D F and EEDF:
/ . w
=
^
8 7T
/ , w
< 4 ' 3 )
The distribution function is normalized such that:
\af ( s ) s V2d s = 1
0
(44)
A commercially available software, ELENDIF, is used as the Boltzmann
solver. It includes the terms for ionization, attachment and recombination,
photon-electron processes, as well as an external source of electrons such as
an electron beam. The diffusion effect can also be taken into consideration by
ELENDIF.
4.1.2
25-Level Argon Model
An n-level atomic collisional-radiative model was developed to predict the
excited state number densities. The ground state and electron num ber densities
are held constant during calculation. The following electron and heavy particle
collisional reactions and radiative reactions w ere used to develop the non-linear
50
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
system of equations:
A m+ e
An + e
n> m
A„m + M <-> A„ft + M
n> m
A m + e <-» A ' + e + e
(4.5)
Am
+ M <-> A~ + e + M
Iff
A„ —> A m + h v
n> m
A * + e - > Am
m+ h v
For each excited state number density, a rate equation was generated
based on the above reactions as follows:
+
«,*r')+
(4.6)
j<m
j>m
where nm represents the number density of an atomic excited state (m>1); ne, nj
are the electron and ion number densities, respectively; k is the appropriate
reaction rate coefficient; nt is the total number density, which can be calculated
as (P/kTg); the symbol M represents the major species present in the plasma.
In Equation 4.6, the first two terms account for the electron-impact and
heavy particle-impact excitation and de-excitation. The third term is for the
electron and heavy particle induced ionization. The fourth term represents the
respective recombination reactions. The fifth and sixth terms account for
radiative excited state transitions. And finally, the last term is for radiative
recombination.
51
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
4.1.3 Electron Conservation Equation
Electrons are produced inside the bulk plasma once the discharge is
formed. They are lost either by recombination in the bulk plasma or by diffusion
and convection to the cavity wall and substrate, or to the after-glow region. The
electron number density balance can be expressed in the following equation:
dne _ kr
— —
— A ,
„
A
at
r
TS
r^diff
_ kr
cairv
(4.7)
w here Kj, Kr, ( W , Kconv represent the total production of electrons by ionization,
the total loss of electrons by recombination, the electron loss rate due to
diffusion and the electron loss rate due to convection, respectively. For pure
argon plasmas, the electron number density balance can be written as the
follows, according to the set of reactions present in Equation 4.5:
tin
24
-
/
\
£ »„ ( "AT"+
-
24
I «.», ( » . * - " + ) -
m=l
^
>
ix
n,nk‘
Da
* e / x 4 x l 0 17
~ n , ------------ --- ------------------------------------------A
nt
Vp
( ‘ )
In the above equation, the terms of diffusion (neD a/A 2) and convection
(the last term) need more discussion since they can affect the solution of CRM
significantly.
Diffusion is due to the existence of the gradient of concentration of the
spatially distributed species. For charged particles in plasmas, such as electrons
and ions, one must account for ambipolar diffusion, which occurs in moderate
52
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
pressure plasmas and ensure the electrical neutrality for a length scale greater
than the Debye length [Liebermann, 1994]. The electron free diffusion rates are
much higher than those of ions because of their lighter masses. Therefore, the
electrons diffuse faster than the ions, disturbing the quasi-neutrality of the
plasmas. However, since more electrons diffuse outside the bulk plasma and
accumulate at the boundary, the so-called sheath region will be formed. A
significant voltage drop will exist across the sheath, with the voltage drop
varying from several volts to hundreds of volts. This resulting space charge field
will retard the electron diffusion and increase the ion diffusion so that the quasineutrality can be restored at all points inside the bulk plasma. For argon, only
one ion (Ar+) is considered here, the ambipolar diffusion coefficient is calculated
as follows [Cherrington, 1973]:
(4.9)
11 J
v
w here Dj represents the ionic diffusion coefficient; T e, Tj represent electron and
ion temperatures, respectively. To calculate the ionic diffusion rate, Einstein’s
relation is applied here [McDaniel, 1973]:
(4.10)
where p* is the ionic mobility, which is given by the following relation:
Mi =
Mio
53
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
(4.11)
where
1*0
is the reduced ionic mobility, NL is the Loschmidt number (2.689x1025
nrf3), and nt = P/kTg represents the total gas density. The reduced ion mobility
for Ar+ (in argon) is 1.535±0.007 cm2/Vs [M cDaniel, 1973]. The characteristic
diffusion length (A) for a cylindrical geometry is given by [Chem'ngton, 1979]:
where
L and
R represent the length of radius
of the plasma volume,
respectively. For a plasma in contact with a cold surface (reactor walls,
substrate, etc.), the electrons and ions diffusing out are lost by recombination.
Thus the surface acts as a third body and absorbs the energy released in the
recombination reactions. For plasmas not in contact with a surface, the diffusing
electrons and ions may either combine in the surrounding gas or be pumped
out.
The convection rate coefficient for the charged and neutral particles is
given by:
f
4 x l 0 17
(4 .1 3 )
conv
where f represents the gas flow rate in seem (4x10 17 is a conversion factor from
seem to particles/s).
4.1.4 Pow er Balance Equation
The input microwave energy accelerates the charged particles present in
54
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
the plasma. As a result of ions’ heavier mass, the energy absorbed by the ions
in the bulk plasma is almost negligible. However, ions can be accelerated in the
DC sheath at the plasma boundaries and absorb a small part of microwave
energy. The total power balance can be written as:
(4.14)
The electron energy
(Pabs.e)
balance can be written as:
(4.15)
Pabv
'E -Q la s + Q in c la s + Q ra d + Q c o n v + Q d ijr
The above equation shows that the electron energy is gained by the
electrical (ohmic) heating. Most of the electron energy is lost by the elastic and
inelastic energy exchange in electron-neutral collisions, and the radiative losses
that comprise free-free and free-bound collisions between electrons and ions or
neutrals. The third and fourth terms in equation 4.15 represent the kinetic
energy loss by electron diffusion and convection, respectively.
The free-free
and free-bound
radiative losses were calculated
by
integrating the continuum emission over a wavelength range of 10 nm to 100
pm. The expressions for other terms in equation 4.15 can be written as follows:
n
(4.16)
= n. ' Z kj£1nj
j
55
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
The kinetic energy loss due to ionic diffusion and convection is given as
[Kelkar, 1997]:
Pa b s j =
4.2
( n,D.
w. y x 4 x ! 0 ITY
»,
V,
J
(4.17)
'
Electrom agnetic Field Model
The pseudo-1-D model discussed in the above section is not self-
consistent. One should specify several important parameters such as the
electron number density, gas temperature, and the electrical field to get a
satisfactory solution. Another important parameter, the microwave input power,
cannot be used in the model. As shown in the later sections, w e found a big
discrepancy between the reading from the pow er meters and the prediction from
the CR M . Therefore, a 2 -D plasma model that couples the electrom agnetic (EM )
model is desirable for further investigation.
4.2.1 Governing equations of the EM m odel
The behavior of the electromagnetic energy and how it couples to and
excites the plasma discharge depend on the geometry of the structure, the input
power coupling structure, and the parameters of the plasma discharge (size,
density, pressure, composition, etc.). The finite difference time-domain method
(F D T D ) is used to solve the microwave electromagnetic fields in the plasma
reactor [Tan, 1994 & 1995]. Since the microwave wavelength (X=c/v=0.122m ) is
sm aller than the dimension of the electromagnetic field confinement region, a
56
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
full w ave solution of the Maxwell equation is necessary. The Maxwell equations
are given by:
V x E = - m0 —
V x / 7 = e— + J
(4.18)
dt
V-£ = —
e
V-B = 0
where E is the electric field, H is the magnetic field, B is the magnetic flux
density, J is the current density, s is the permittivity, po is the permeability, and p
is the charge density. The plasma discharge contributes to these four equations
through the current density and the charge density terms.
T h e current density is determined by solving the momentum transport
equation for electrons in the microwave frequency range simultaneously with the
Maxwell equations 4.18. The momentum transport equation can be written as:
m. ^
— = -q E -m ,vtjrv
(4 -1 9 )
dt
And the current density is:
7
J = -qnev
where
v
isthe
-
(4.20)
average electron velocity,qis the electron
electron mass, veff is the effective collisionfrequency,
charge, me
is the
and ne is the electron
density. The expression for the effective collision frequency can be written as:
,V e f f-^ a
v th nt
°
57
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
<4 -2 1 >
w here a is the momentum transfer cross-section for electron-heavy particle
collisions, vtt, is the electron thermal velocity, and nt is the total neutral gas
density.
The microwave power density absorbed by the plasma, M W P D ,
is
calculated for each grid point with the following expression:
(4.22)
MWPD - J ■E
4.2.2 Electrom agnetic Properties of Plasma Discharges
For the collisional, non-magnetized plasma discharges, such as the one
used for diamond CVD deposition, the frequency of excitation is generally less
than the plasma frequency. Therefore, the microwave electromagnetic fields do
not propagate through the plasma discharge. In this case, the electromagnetic
w ave will penetrate into the plasma discharge a distance on the order of the skin
depth (defined in Equation 4.23). Then for the region of the discharge within
approximately a skin depth of the plasma surface, the electron gas absorbs the
energy from the microwave fields.
The
general
expression
for
the
skin
depth
5
of
a
transverse
electromagnetic wave penetrating into plasma can be written as [Bittencourt
1986]:
r
(4.23)
— = -Lm ag — <a
8
c
58
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
where c is the speed of light, v is the electron collision frequency, o is the
microwave frequency, and £ope is the electron plasm a frequency, which is given
by [Mitchner and Kruger 1973]:
r
cop* =
t \ 1/2
ne2 ]
(4.24)
For example, if the electron density is given as n=5e17 nrf3, the electron
plasma frequency cope will be equal to about 40 G Hz. For a typical argon plasma
discharge, with the collision frequency of v= 5 x 1 0 10 H z and a microwave
frequency of 2 .45 GHz, the skin depth is approximately 5 cm. If the collision
frequency is reduced to 5 GHz, the skin depth will be about 1.2 cm.
59
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
4.3
Two Dimensional Plasm a Fluid Model
In the moderate pressure plasma flows, the assumption of a continuum
description of the flow is valid and the plasma species and energy transport are
governed by the fluid dynamic equations, which express the conservation of
mass, momentum and energy in the plasma flow [Prelas et al., 1998].
In this section, details of the modeling of the nonequilibrium plasma will
be discussed. For argon plasmas, a two-temperature flow assumption is m ade
for the expression of the transport equations. For previous hydrogen plasmas
studies, a three-temperature flow assumption was m ade due to the existence of
the vibrational modes of the molecules [Hassouni, 1997],
4.3.1
Plasma Flow Descn'ption
For thermo-chemically nonequilibrium plasma flow, one has to write
conservation equations for each
species,
each
energy
mode and
each
momentum component. G enerally we use mass density (ps), the momentum per
unit volume (pU) and energy per unit volume (Emode) to denote the conserved
variables.
60
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
COcD
7ET
Figure 4.2 Schematic of a control volume AxAy for O conservation
Considering a control volume element, as shown in Figure 4.2, the
conservation equation for each conserved variable
0
(mass,
momentum,
energy, etc) can be written as:
= Fxmx (AyAz) + Fymx (AxAr)
A/
(4.25)
- Fxoul x (AyAx) - FyMt x (AxAz) +
where AV=Ax*Ay with Az=1 for a control volume element AxAy and the sourceterm can be denoted as coa>. T h e conservation equation can be written in the
following differential format:
^ _ = dFx_Sfy_+ w
dt
dx
Ay
*
61
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(4.26)
where ¥ = —
dV
, which is the conserved variable per unit volume.
The following table shows the formulation for each conserved variable in
Cartesian coordinate systems. For cylindrical system, the equations should be
changed for axial and radial directions.
Table 4.1 The Conservation Equations
Continuity
Equation
ps for
species
"s"
Momentum
Component
Equation
pu (p the
plasma
total mass
density)
Electron
Energy
Equation
Ee
Total
Energy
Equation
Equation
Conservation Equation
Conserved
Variables
#
v (p su + p%
u ,) = W,
where u is the plasma
averaged velocity, and us is the diffusion velocity of
species "s" relative to the averaged velocity
x-direction: v((/3«)u + P -
r*)=
pgx
y-direction: v((pv)u + P - r y)=
pgy
(4.27)
(4.28)
V f t . v r , - J , h , ) + M W PD -Q ,_ , - 0 , _ , = 0
where Xe, Je, he denote the thermal
conductivity, the mass diffusion flux and electron
enthalpy, respectively
(4.29)
v T a ,V 7 ; + x y T ' - Y , J 1h y + M W P D - Q raJ = 0
E
*
J
where X( is the thermal conductivity of heavy
species translational mode, Qrad is the power lost
from the plasma by radiation
V
(4.30)
The continuity equation can also be written as the following equation for
simplicity:
62
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
where Ms, DSl Xs, and W s denote the molar mass, diffusion coefficient, mole
fraction, and chemical production rate of species "s", respectively. M and p are
the averaged molar mass and total mass density of the plasma.
4.3.2
•
Flux and Source Term Expressions for the Transport Equations
Species Diffusion Velocity
To write expressions for species diffusion velocities in a multi-component
flow is very difficult. Thus an approximation similar to Fick’s Law is used, which
uses an effective species diffusion coefficient in the plasma mixture,
[Curtiss et ai., 1949 and Lee, 1984]:
(4.32)
P,
•
Shear Stress Tensor Components
For the Newtonian fluids, the shear stress tensor components are
expressed as following [Bird 1960]:
at
2
at + av
(4.33)
(4.34)
where p is the averaged plasma viscosity.
63
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
•
E lectron-Translatio nal Energy T ra n s fe r S ource T erm
This term can be derived from classical kinetic theory. The expression is
as following [Vincenti and Kruger, 1967]:
(4.35)
w here M is the species molar mass. Subscript e and s denote electron and
species "s" respectively. Oe-s is the electron-heavy particle momentum transfer
collision cross-section, and n is the total plasma number density.
•
Electron Energy Loss by C h em ical A ctivatio n
This term is determined by the reaction rate
(A E
r
).
(V
r
)
and activation energy
The expression is as following:
Q .-c
(4.36)
=
Re actions
W here
4 .3.3
cxr*
is the electron stochiometric coefficient in reaction "R".
T ra n s p o rt C oefficients
Yos [1963] derived the rigorous expressions for the transport coefficients
of multi-temperature gas mixture flow in thermal equilibrium (all distribution
functions are close to Maxwellian about their inherent temperatures). Lee
adapted these expressions for thermally nonequilibrium flows [Lee 1984].
•
V is c o s ity
The viscosity of a nonequilibrium multi-component flow depends on the
second order collision integrals (H s/2,2*). The expression of the calculation is as
64
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
following [Lee, 1984]:
mexe
msxs
(4.37)
™<Tm)
£ x X ^ ) + * eA < 2(7;)
\ r
where xs is the mole fraction of species "s", and Asr(2)( T ) is defined as:
16
/
\ 1/2
2 msmr
n£l
Krd<T{ms + m r )
( 2. 2 )
(4.38)
In the above equations, ms denotes the mass of species "s".
For a plasma only having one major species, the viscosity calculation can
be simplified from the Wilke Equation as following [Wilke, 1950]:
r
\
(4.39)
* - Z
k *s
where xs is the mole fraction of species "s", (is is the viscosity of the pure gas
"s", and cbsk is given as:
-
1/ 2
(
1+ E l
V
' '
i+^
Mk
Y jy
' M kV
'4'
(4.40)
\ Mkj
The calculation of ps is given by Bird [1960] and W ilke [1950]:
(4.41)
M s
•
=
1
Therm al Conductivity
The thermal conductivity consists of two terms, one for the translational
thermal conductivity and the other for the rotational energy mode if molecules
65
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
are present in the gas mixture.
The translational thermal conductivity of heavy particles is given as:
(4.42)
4 r = - * £
4 r
£ a „ * , A ™ ( r f ) + 3 .5 4 * .A ™ ( r ,)
V
r
where asr is defined as:
\
1
a„= I +
\
-0 1 l
0 .4 5 -2 .5 4 —
mr
mrJ
f
(4.43)
\2
1+ ^
V
mrJ
If all the rotational energy modes are assumed to fully excited, the overall
rotational thermal conductivity of a mixture is given as:
x.
s -M o le c u le
(4.44)
Xx,Al"(rs)+xXUrj
\ r
where Asr(1)(T) is defined as:
1/2
2 msmr
A“ ’ ( n = |
(4.45)
nQ!(i.D
jdcT{ms + m r )
If vibration energy mode
is also involved,
the vibrational
thermal
conductivity should also be calculated. The equation is the same as Equation
4.45 with the vibrational temperature.
The electron thermal conductivity is given as follows:
66
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
As in the case for viscosity, it is time-consuming to calculate the thermal
conductivity according to the above equations. For the plasmas with only one
major species, the thermal conductivity may be well estimated from the one of
the pure gas component. The conductivity can be calculated as following by
Eucken relations [Bird, 1960]:
(4.47)
Ira n i
(4.48)
where Ctrans, Crot, and Cv are the specific heat for the translational, rotational and
vibrational modes.
The electron thermal conductivity can also be simplified as follows
[Jaffrin, 1965]:
K
15k
nkTe
6 4 (1 + V 2 )OeJ
me
(4.49)
where Q ee is the electron-electron collision cross section.
•
Diffusion Coefficients
The binary diffusion coefficient Dsr of a species "s" in another species "r"
is given by [Yos, 1963]:
(4.50)
Dsr
PA{H (T )
67
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
T h e diffusion coefficient of a neutral species "s" in a gas mixture can be
expressed as follows (Curtiss, 1949]:
_ M,
M
For
charged
species,
2 > r/D J
ambipolar
diffusion
has
to
be
taken
into
consideration to ensure the electrical neutrality for the length scale that is larger
than the Debye length. The calculation is the same as discussed in the pseudo1-D model:
f
T \
(4.52)
1+ ^
Damb - A
T
\
&J
T h e electron diffusion coefficient is derived from the assumption of
electrical neutrality and zero electric current. It is given as [Delcroix, 1958]:
' Z ( zsM sxsD s.amb)
—_£________________
n
(4-53)
e.am b
1
- - ^ X,
s
where zs is the electrical charge of the ion species "s".
•
Comments on the Calculation of Transport Coefficients
In the calculations for most transport coefficients, the collision integral
jS d e fin e as the weighted average of a collision cross section in the
form as [Lee, 1984]:
=f r
smx d x d r
e_rV 2m+3(l - c°smx) sinxdxdy
68
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(4.54)
where csr = asr(x,g) is the collision cross section for the collision pair s-r, x is the
scattering angle in the center of mass system, g is the relative velocity of the
colliding particles, and y = [msmr/2(ms+mr)kT]1/2g is the reduced velocity.
From the above equation, one can see that the accuracy of the
calculated transport properties is greatly dependent on the collision integrals.
The collision cross sections for ion-neutral and electron-neutral interactions still
need more investigation.
4.3.4
Boundary Conditions
The boundary conditions are critical to the numerical modeling. Different
boundary conditions for the computation domain will generate different results.
The boundary conditions may be provided by experimental measurements, or
extracted from the conservation laws.
For the modeling of the cylindrically symmetric deposition reactor, the
computation domain consists of the belljar walls, the substrate surface, and the
symmetric axes.
•
At the Sym metric Axis
Considering the axi-symmetric computation domain, at the symmetric
axis, zero gradient boundary conditions are imposed to all the parameters:
*®-0
dr " °
=
xs,u,v,Ts^andTt
69
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(4 -55>
•
At the Belljar W all
At the belljar wall and the substrate surfaces, the velocity components
should vanish. The gradient of pressure will approach zero. Without a detailed
sheath model, one can assume that the gas temperature is in equilibrium with
the wall temperature and that the gradient of electron tem perature is zero due to
the fact that the accommodation of electron energy at the wall is very weak
[Scott. 1993]. The resulting boundary conditions are as follows:
11 - =v
0
=
dP
-------
=0
(4.56)
(4.57)
d r wall
T g.wall - T wall
dT.
=0
(4.58)
(4.59)
d r wall
•
At the Substrate Surface
The resulting boundary conditions at substrate surfaces are as following:
W= V = 0
dP
0
=
(4.60)
(4.61)
dz sub
Tg.sub = Twall
57;
=
0
dz sub
70
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(4.62)
(4.63)
•
Surface Chemical Kinetics
The boundary conditions for the chemical species concentrations at the
substrate surfaces and walls can be derived from the modeling of the surface
chemistry kinetics. Some models are available for diamond deposition plasmas.
The resulting boundary conditions can be expressed as [Scott, 1993 and
McMaster, 1994]:
-D
w here
s%
rrux
W Si surface
dx
= W
(T k c r = 1 N species J\
VY s.surfaceK1 s ->K ->C r ’ r
(4.64)
surface
is the production or consumption rate of the species "s" by
surface reactions. As shown in the above equation, it depends on the surface
temperature, the rate constants of the surface reactions, and the species mole
fractions at the surface.
71
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
4.4
Two Dimensional Argon Plasma Model
The 2-D plasma model was originally designed for hydrogen plasma
discharges in France. Characteristics of the hydrogen plasmas have been
studied extensively by applying this model [Hassouni, 1999]. However, no study
of the 2-D collisional-radiative-model was done according to the literature review.
Moreover, no such 2-D model for the microwave argon plasma is available.
Some studies of 2-D EED F have been reported [Kortshagan, 1999], but their
models are very crude thus cannot provide useful information such as excited
state
number
densities,
power
densities,
and
gas/electron
temperatures.
Therefore, we constructed a 2 -D argon plasma model. This 2-D model adds
more understanding to the microwave non-equilibrium plasmas since it provides
more detailed spatially resolved and self-consistent data.
4.4.1
Argon Plasma Chem istry Model
The 2-D model requires more calculation time than the 0-D models.
Although the time issue may not be critical, constructing a model that includes all
the excited states is not an efficient way for research and development.
Therefore, four species are considered for the argon plasma chemistry. This
approach has been widely adopted by other researchers [Lymberopolous, 1998],
Tab le 4.2 and 4.3 show the species and reactions considered in the argon
plasma chemistry.
72
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
Table 4.2 Species considered in 2-D argon plasma chemistry
Species
Argon ground level
Symbol
Ar
Argon 4s excited state (metastable)
Ar*
Argon ions
Ar*
Electrons
e
Table 4.3 2-D Argon Plasma Chemistry Model
Number
Reaction
Description
1
Ar + e -> Ar* + e
Electron-atom excitation reaction
2
Ar* + e -» Ar + e
Electron-atom de-excitation reaction
3
M + A r-> M + Ar*
Atom-atom excitation reaction
4
M + Ar* -> M + Ar
Atom-atom de-excitation reaction
5
Ar + e -> Ar* + 2e
Electron-atom ionization reaction
6
Ar* + 2e -» A r+ e
Three body electron recombination
Atom-atom ionization reaction
7
M + Ar
M + Ar* + e
8
M + Ar* + e -> M + Ar
Three body atom recombination
9
Ar* + e -» Ar* + 2e
Electron-atom ionization reaction
10
Ar* + 2e -» Ar* + e
Three body electron recombination
11
M + Ar* -> M + Ar* + e
Atom-atom ionization reaction
12
M + Ar* + e -> M + Ar*
Three body atom recombination
13
Ar* -> Ar + hv
Radiative de-excitation reaction
14
Ar* + e -> Ar + hv
Radiative electron recombination
15
Ar* + e -» A r + hv
Radiative electron recombination
In Table 4.2, the m etastable excited state (4s) is considered along with the
argon ground state, ion and electron as the four species in the argon plasma
chemistry. Although only the 4s excited state is included, good results have been
73
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
obtained because excited state ionization is dominant from the 4s state, and 2 step ionization will be discussed in Chapter 5. Neglecting the excited state
ionization will cause large error in characterizing the argon plasmas.
In Table 4.3, the third body atoms, denoted as "M", are simply argon
atoms in the pure argon plasma discharges. All the reaction rates can be
obtained in the CRM by specifying the values of E/N, electron number density,
pressure, and gas temperature. The curve-fitted rate expressions w ere obtained
by curve-fitting the rates with the averaged electron temperature and electron
number density.
Surface chemistry for the 2-D model is necessary to simulate the effects of
the existence of the physical walls. In this argon model, it was assumed that
100%
of the
argon excited state particles and
ions were
de-excited
or
recombined to the ground state at the wall.
In the argon plasmas, since no molecules are available, only the atom
translational and electron energy modes are considered. As discussed in the
Section 4.3.1, for each energy mode, an energy conservation equation is
needed. Due to the fact that the EEDFs of the argon plasmas w e investigated are
highly non-equilibrium, the energy transfer terms should also be curve-fitted.
74
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
4.4.2 Curve-Fitted Reaction Rates
In this section, some of the reaction rates are discussed. T h e pseudo-1-D
model was used to generate the rates at first. Then they w ere curve-fitted to
some function
of electron number density,
electron temperature,
or gas
temperature if needed. Since the Boltzmann solver is no longer coupled inside
the 2-D model, these curve-fitted reaction rates are used instead of the EEDF to
calculate them directly.
10' 1
Li.
N, = 5e16 m'1
Nt = 5e17 m'1
N, = 5618 m'1
Maxwellian Distribution
0
5
10
15
20
Energy [eV]
Figure 4.3 Non-Maxwellian EEDF plots with averaged T e — 1.07e4 K and
300K < T g < 600K comparing with the Mawellian EEDF
Figure 4 .3 shows three EEDF plots with the same averaged electron
75
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
temperature at 1.07e4 K. From this figure, the effect of electron number density
to the EE D F shape is very clear. At a high electron number density (5 e 1 8 nrT3),
the Boltzmann plot is close to the Maxwellian distribution, which is a straight line.
With lower electron number densities, the high-energy tail of EEDF will deviate
from the Maxwellian distribution much further.
T h e parameters changed in the pseudo-1-D model are electron number
density (Ne), gas temperature (Tg), and E/N, which corresponds to an averaged
electron temperature (Te). To curve-fit the reaction rates, w e use:
K r^ , = c { \ . 0 - e - ^ ) ' T e ‘’
(4 '65)
Microsoft Excel’s Solver was used to optimize the parameters for the
curve-fitted equations. The exponent “a” is introduced to account for the degree
of the non-Maxwellian effects of EEDF. Figures 4 .4 -4 .6 show the comparison of
the original calculated reaction rates with the curve-fitted ones. The curve-fitting
range for electron temperature is from 5000K to 30000K . This is the sam e range
that is specified inside the 2-D model. Larger error is present when both the
electron number density and electron temperature are low (ne < 1e17 nrf3 and T e
< 10000 K). Fortunately, most of the ne- Te pairs are not in this region. Thus the
numerical uncertainty introduced by the curve-fitted rates is still reasonable
(<30% ) as discussed in Chapter 6.
76
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
.-16
0
0
c
,-17
o
o
CO
K e a(0 ,1 )
0
c
o
eg
o
X
0
E
o
i-2 0
Ne=5e17 m
Ne=1e18 m-3
,-21
Ne=5e18 m"3
0I
c
o
Curvefitted Rate
cS
0
0
8000 10000 12000 14000 16000 18000 20000 22000 24000 26000
T e (K)
Figure 4.4 Comparison of the curve-fitted reaction rates (Kea(0,l))
and those calculated from the C R M as a function of electron
temperature.
77
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
rate
ionization
Klea
(0)
reaction
10
.-20
• ••
10"
electron-atom
Ne=5e17 m
,-22
N9 =1e18 m 3
N9 =5e18 m 3
,-23
Curve-fitted Rate
.‘2*
8000 10000 12000 14000 16000 18000 20000 22000 24000 26000
T e (K)
Figure 4.5 Comparison of the curve-fitted reaction rates
(Klea(0)) and those calculated from the C R M as a function of
electron temperature.
78
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
0
2
c
o
K lea(1)
o
0
0
,-14
c
o
ra
N
'c
o
E
0
<1
3
c
Ne=5e17 m
o
N,=5e17 m'3
o
Curve-fitted Rate
Ne=5e17 m'3
0
0
.-16
8000 10000 12000 14000 16000 18000 20000 22000 24000 26000
T.(K)
Figure 4.6 Comparison of the curve-fitted reaction rates
(K lea(l)) and those calculated from the C R M as a function of
electron temperature.
79
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
4.4.3
Iterative Scheme between the Plasm a Model and the EM Model
The 2-D argon plasma model is fully self-consistent. Once the plasma
chemistry and boundary conditions are specified, the only information necessary
to
start
the
calculation
is
the
plasma
pressure
and
microwave
power.
Unfortunately, as we will discuss in the energy balance study, the microwave
power measured from the power meters on the reactor could not be used. Thus a
parametric study of the effects of microwave power on the argon plasma
discharges is necessary. In this section, the iterative scheme between the
plasma model and the EM model is discussed as following.
Figure 4.7 is an iterative scheme showing the coupling between the 2-D
plasma fluid model and the EM model. The self-consistent solution is obtained by
iterating between these two models until a converged solution is reached. Due to
the different grid structures in the two models, grid interpolation is needed for the
exchange of information between these two models. The microwave power
density (M W PD ) is obtained from the EM model and serves as the input to the
plasma fluid model. On the other hand, the fluid model provides the information
about the plasma discharge, such as the electron number density, electron
temperature, gas temperature, etc.
The major difference between the two grid structures is that w e have to
consider the whole cavity volume for the EM model, while w e only consider the
region inside the belljar for the plasma fluid model. The two grid structures will be
80
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
shown in the next section.
Te;>Tg, lie a —- ■
^1
V
Grid Interpolation
|
Argon Plasma
2-Temperature
Model
(300-400
Iterations)
I
1
Electromagnetic
Model
E, H , J, M W P D
(15-25 Microwave periods)
Grid Interpolation
MW PD
Figure 4.7 The Iterative Scheme of the Coupled 2-D Plasma Model
and the E M Model
W e will show that the argon plasma is highly non-uniform. Therefore, we
do need a 2-D model instead of the 0-D model that assumes a uniform plasma
discharge. Also due to the high non-uniformity, more than 15 iterations will be
needed to achieve a converged solution for the microwave argon plasma.
81
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
4.4.4 Grid Structures and Boundary Conditions
As discussed in the above section, the grid structures for the plasma fluid
model and the EM model are different. Therefore, for a specific reactor design,
two sets of grids are generated from the reactor geometry.
Figure 4.8 shows the reactor geometry considered for the models. One
can refer to the schematic of the reactor in Figure 5.8 to have a better idea about
the reactor structure.
Due to axi-symmetry, we only need to consider the simulation domains as
shown in the Figure 4.9. The grid structure is half of the cross-section of the
reactor cavity in the r and z directions. For the FDTD EM model, the maximum
number of grids used is 100x130 in r and z directions, respectively. The grid
spacing in the plasma discharge region is set to be denser than in other regions,
both in r and z directions. For the plasma model, the num ber of grids is 40x72 in r
and z directions, respectively. The plasma grid is defined only inside the belljar
where the plasma discharge is actually confined. The grid spacing is varied to be
denser in the region of the plasma discharge ball.
The base plates, substrate holder, probe, short, and the cavity walls form
the boundaries for the EM
model simulation region.
By assuming these
boundaries are made of perfect conductors, the boundary conditions for the
electric fields on these surface are that only the normal components of the
82
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Microwave
Simulation
Boundary
Plasma
Simulation
Boundary
Belljar
| Substrate i
Holder
^
X w x w w w w W iw w w w w w w ^
777777,
I
Figure 4.8 The Simulation Domain of the Microwave Plasma Discharge
with the W A V E M A T Reactor
electric fields exist, while the tangential components on these surfaces are set to
zero. The probe is actually a coaxial antenna. At its input end is an open
boundary where the electric field computed is unbounded. At this boundary, a
truncation method was used to prevent any artificial reflection of outgoing waves
[Tan and Grotjohn, 1994, 1995], The coordinate system sets z= 0 at the top of the
substrate holder.
The technique used to excite the electromagnetic field is to select the grid
83
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
points on the cross section plane of the coaxial probe as the source points. The
time-varying electric field components are assigned to these points based on the
theoretical transverse
electromagnetic
(TE M ) w ave
solutions in a
coaxial
structure. The electromagnetic w ave then propagates downward into the cavity
where the microwave power is absorbed by the discharge. The reflected
electromagnetic w ave will propagate to the end of the open boundary of the input
power antenna and be terminated by the non-reflecting boundary.
84
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
4.5
Tw o Dimensional Collisional-Radiative Model
For the highly non-equilibrium plasmas we investigate, one should
consider the non-Maxwellian effects of the EEDF. This requires the coupling of
a Boltzmann solver for the derivation of the EEDF. Moreover, for our research
interests, the excited state num ber densities are used for comparison with the
experimental data. Therefore,
a 25-level
CRM
is necessary to gain the
information of all the excited states. If we couple this C R M with the 2-D fluid
model, the computation load will make this approach infeasible. Another
problem is that our Boltzmann solver is not a 2-D code, thus the E/N data
cannot be imported from the electromagnetic model directly.
Therefore, the 2-D fluid model for pure argon only has four species: Ar,
Ar' (4s), A r+, and electron, while the 2-D CRM still keeps 25 levels. Inside the 2 D CRM,
all input parameters come from 2-D fluid model.
Recalling the
numerical schematic for the pseudo-1-D model (Figure 4.1), electron number
density balance is achieved by changing the E/N value. In the 2-D CRM , the
electron number density balance equation is no longer implemented because in
the 2-D fluid model, the electron number density balance has been realized by
the continuity equation for electrons. However, the E/N value is again changed
inside the 2-D CR M to ensure the resulting averaged electron temperature from
the Boltzmann solver is equal to the one from 2-D fluid model.
85
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Gas Temperature (Tg)
Electron Temperature (Te)
Electron Number density (Ne)
Ion Number Density (N i)
Electrical field => E/N
SaSSSliEJfi SdSSSaSEfii ’
W r-
'
<
Excited State Number
Densities
(File: excite.dat)
Microwave Power
Density, E/N
(File: Pabs2d.dat)
Reaction Rates
(File: rate.dat)
Other data if needed
(EEDF.etc.)
2-D CRM
Figure 4.9 The Flow Chart of the Iterative Scheme
for the 2-D Collisional-Radiative Model (C R M )
86
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Figure 4 .9 shows the iterative scheme of the 2-D CRM . T h e outputs from
2-D fluid model are used as inputs to the 2-D CRM. The excited state number
densities from the outputs of the 2-D CR M are of primary interest since w e can
compare the results with our experimental measurements. Other outputs include
the reaction rates and the microwave power density. The gas temperature
remains unchanged in the 2-D CRM, while the value of E/N is changed inside
the 2-D CR M to match the averaged electron temperature with the one from the
2-D fluid model.
Another iteration between the 2-D CRM and 2-D fluid model can be
formed to achieve higher accuracy. In this case, the reaction rates can be first
written into a data file inside the 2-D CRM . Then in the 2-D fluid model, all these
reaction can be read directly from this data file instead of calculating the rates
from the curve-fitted equations, which could produce some uncertainty. Thus
there will be two major iterations to get a converged solution (Fluid M odel/EM
Model, and Fluid M odel/CRM ).
The tradeoff between the 2-D fluid model and the 2-D C R M is that the 2 D fluid model sacrifices the accuracy of the plasma chemistry model to gain
efficiency in numerical calculations. And the 2-D CRM can provide more
detailed data of excited states, but is not able to consider the 2-D effects within
the model itself.
More detailed structures of the plasma fluid model, the EM models and the
2-D C R M can be found in the Appendix B. The codes for the fluid model and the
87
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
EM model were written in the W A T C O M FO R TR A N format. The 2 -D C R M codes
were written in Fortran 90 and C languages.
88
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
CHAPTER FIVE
E N ER G Y BALANCE STUDY
For the study of the nonequilibrium microwave argon plasmas, both
numerical and experimental approaches have been used to gain a better
understanding. In this chapter, numerical results will be presented to show that
the argon plasmas absorbed only a small portion (2-5% ) of the metered
microwave energy (800W forwarded, and 120W reflected) under the studied
conditions (5 Torr and 2 5 0 seem argon flow).
An energy balance study was then conducted to finalize this issue. A
global reactor energy balance was performed on the microwave CVD reactor to
observe how the microwave energy is dissipated into the cooling lines. After we
understood the global reactor energy balance, a control-volume heat transfer
model was constructed to perform the heat transfer analysis with a control
volume encompassing the plasma discharge region directly.
The 2-D argon model provides more detailed data about the heat fluxes
around the plasma simulation boundaries. These data were used to check the
results of the energy balance study. Good agreem ent was achieved between the
argon 2-D model and the energy balance study results.
To begin the discussion with the energy balance study, the pseudo-1-D
model results should be introduced first.
89
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
5.1
P seudo-1-D A rgon M odel R esults
For the non-Maxwellian EEDF, the Boltzmann solver was coupled with the
argon CR M to solve for the excited state number densities and the predicted
power deposition. In this case, the input parameter E/N, which represents the
ratio of the electric field strength and the total number density, was varied instead
of electron temperature. In this coupled CRM, the electron number density
balance was imposed in the model. The ranges of the parameters studied are as
follows:
T ab le 5.1 T h e Ranges o f S tudied P aram eters Used fo r A rg o n
P lasm as at Pressure o f 5 T o rr
E/N (Td)
0.005 - 2 . 0
T g (K )
300 - 600
ne (m*3)
8e16 - 3e19
As shown in Figure 5.1, the non-Maxwellian EEDF deviates from the
Maxwellian distribution dramatically at high energies. This phenomenon could
impact the plasma chemistry significantly. Therefore, non-Maxwellian E E D F was
used for all the models used in this research work.
Figure 5.2 shows the relative electron-energy loss terms changing with the
averaged electron temperature. For the conditions studied, at lower electron
temperature, the elastic energy loss term is dominant. However, at higher
electron temperature, the inelastic energy loss term becomes dominant. For
90
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
argon plasmas, the elastic energy transfer between the electrons and the neutral
particles is the major source for gas heating at low electron temperatures (Te <
14000 K). However, at higher electron temperatures, the reaction rates of the
inelastic collisions become much higher. Thus the inelastic energy loss term
becomes dominant at higher electron temperatures.
The electron production and loss mechanisms are demonstrated in Figure
5.3 as a function of the averaged electron temperature for a particular ne and T g
(1e18 nrf3 and 350 K, respectively). Under the studied conditions, the dominant
production and loss mechanisms are excited state ionization and ambipolar
diffusion terms. Therefore, the electron number density balance w as mainly
determined by these two terms. In Figure 5.3, the point where the excited state
ionization curve crosses the ambipolar diffusion curve actually means that the
electron number density balance is satisfied. Thus for certain ne and T g, the
electron temperature can be determined. In Figure 5.3, the electron temperature
that satisfies the electron conservation is determined to be around 9000 K.
Figure 5.4 shows the solutions (N e-T e pairs) generated by varying E/N at
different electron number densities. As discussed in Figure 5.3, the dominant
production mechanism was the excited state ionization. Therefore, the solutions
change dramatically if one neglects two-step ionization. This is critical in
simplifying the 2-D fluid model since one had to include two-step ionization.
Figure 5.5 shows the microwave power predicted from the C R M changing
with the electron number density while the electron number density and energy
balances
were
imposed.
From
this
microwave power deposition curve, it is
91
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
clear that if 6 8 0 W is absorbed by the argon plasma, the electron number density
has to be higher than 2e19m '3. For the argon plasma at the conditions as
studied, such a high electron number density is not feasible for the microwave to
penetrate the argon plasma, although it is desirable. From the 2-D model results
discussed in Chapter 6, it will show high electron number density may be
possible on the edge of discharge region. However, the microwave can only
penetrate within the skin depth. Thus a uniform argon discharge with electron
number density higher than 2e19m '3 is impossible. Such a high electron number
density also exceeds the reported experimental measurements performed in the
downstream region of microwave plasma by 1-2 orders of magnitude [M ak 1996].
Moreover, for the electron number higher than 1x1019 m'3, it is impossible to
numerically match the measured excited state number densities within a factor of
10. In Chapter 6, the excited states will be studied to characterize the argon
plasmas.
Such a large electron number density (> 1e19 m*3) also conflicts with the
continuum
emission
data.
Since
large
uncertainty is associated with
the
continuum emission measurements, the continuum emission data was only used
to determine the upper limit for electron number density. Figure 5.6 shows the
calculated continuum emission curve with the experimental measurement data.
The upper limit of electron number density determined from this plot is about
2 e 1 8 rn3, which corresponds to a much lower microwave power (< 100W ) as
determined in Figure 5.5.
92
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
P = 5 Torr
f = 250 seem
10-‘
n0 = 7e17 m
T„ = 350 K
%
>
<D
• Non_Maxwsliian
— Maxwellian
1 -1 0
10 ‘
1-13
IQ-
0
5
10
20
15
e [eV]
Figure 5.1 The comparison of non-Maxwellian and Maxwellian EE D F At
Ne = 7el7 m'3, Tg =350 K , and P= 5 T o rr
100
P = 5 Torr
f = 250 seem
nfl = 7e17 m
T„ = 350 K
S
o
_l
>.
E>
Inelasbc
Elastic
Diffusion and Convection
®
c
UJ
40
u
UJ
0
10000
5000
15000
20000
Te [K]
Figure 5.2 The electron energy loss terms w ith non-Maxwellian EE D F
changing with the averaged electron temperature at Ne = 7el7 m"3, Tg =
I S n K an ri P = 5 T n r r
93
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(0
a:
10
"I
Argon Plasma
5 Torr
10 "
10 * ■*
■Ground State Ionization
- Excited State Ionization
■Total Recombination
- Ambipolar Diffusion
nt = 1e18 m'3
T0= 350 K
1o’*!
10"!
10"
—
6000
8000
—
i—
1----
i—
12000
10000
14000
----1
16000
T [K]
Figure 5.3 Electron production and loss rates changing with the
averaged electron temperature at ne of le!8 nT5 and T t of 350 K
30000
28000
P = 5 Torr
Argon Flow 250 seem
T = 350 K
26000
24000
22000
with 2-step ionization
without 2-step ionization
20000
£
18000
l - “ 16000
14000
12000
10000
8000
6000
1E18
1E17
1E19
ne [ m
l
Figure 5.4 The self-consistent solutions (Ne-Te pairs) with nonMaxwellian EEDF at T s =350K and P =5 To rr
94
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
250 seem
P = 5 Torr
T a = 350 K
10 '
10'
10
n. [m3]
Figure 5.5 Microwave power deposition with non-Maxwellian
EEDF changing with the electron number density while electron
number density and energy conservation are imoosed
250 seem Argon
P = 5 Torr
T9 = 350 K
Upper Limit Determined from Experiment
Total Continuum Emission
”
10
' -
10* ■
T
"
I
I
1
T
I T
"i
I |
1E18
1E17
i
r i t i p1
‘i
111
1E19
ne[nr3]
Figure 5.6 The experimentally measured and theoretically calculated
total continuum emission changing with the electron number density
95
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
5.2
Experim ental Global Reactor Energy Balance
A global reactor energy balance was performed to check the accuracy of
the power meters. The schematic of the reactor with the cooling lines installed for
the global energy balance study is shown in Figure 2.2. The temperatures of the
cooling lines w ere measured by the K-type thermocouples installed specifically
for this study. The cooiing w ater and air were pre-cooled (13.3°C for water and
14.0°C for air respectively), then flowed through the base-plate/applicator walls,
chamber, substrate, probe, short, and belljar cooling lines separately. The inlet
and outlet temperatures were measured for each cooling line. Thus the actual
absorbed energy by the cooling lines can be obtained by calculating the enthalpy
increase.
Table
5.2
and 5.3
show the
experimental
measurements
and the
calculated enthalpy data for the argon plasma:
Table 5.2 The Experimental Data for Microwave Argon Plasma Energy Balance
Cooling Lines
Inlet Temperature
(°C)
Flow Rate (LPM)
Outlet Temperature
(°C)
m l (prob)
1.55
14.0
m2 (short)
2.88
14.0
m3 (base­
plate/applicator)
5.73
14.6
m4 (not used)
N/A
N/A
m5 (MW generator)
5.77
N/A
m6 (chamber)
1.02
14.4
m7 (substrate)
0.72
14.0
13.8
Belljar air cooling
10 scft3/min
14.0
21.5
96
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Table 5.3 Calculated Heat Transfer Rates for the Argon Plasma in Each Cooling
Line with an Argon Flow Rate of 250 seem and an assumed gas exit temperature of
350 K
Flow Rate
(kg/s)
Cp (J/kg K)
A T(K )
P,bs(W )
Experimental
Uncertainty (W)
prob
0.0258
4182
0.2
21.6
21.6
short
0.048
4182
0.2
40.1
40.1
Base-plate/
Applicator
0.0955
4182
0.8
320
80
chamber
0.017
4182
0.6
42.7
14.2
Substrate
0.012
4182
0.2
10.0
10.0
Belljar air
cooling
0.00548 (air)
1003.5
7.5
42.6
1.1
6.81 e-6
(argon)
520.3
77
0.1842
0.6
477
168
Cooling
Lines
Argon gas
convection
Total
In Table 5.3, a total of 477±168 W microwave-energy can be accounted
for in the cooling lines. The bulk of the microwave energy (320±80 W ) is
dissipated into the base-plate/applicator cooling line. The energy carried away by
the argon gas convection is negligible (0.2W ). Therefore, although the argon exit
temperature was not measured in the experiments, the assumption on this
temperature will not affect the final results significantly.
The large uncertainty associated with the enthalpy data was mainly due to
the poor resolution of the
K-type thermocouples and the
low inlet-outlet
temperature differences. The uncertainty caused by the thermocouple reading
97
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
alone can be added up to 167.6W , about 25% of the metered microwave energy
(680±40 W ). Comparing the total energy absorbed in the cooling-lines (477±168
W ) with the metered microwave energy, a 30% difference is calculated. That
indicates the enthalpy data calculated from the cooling lines can match the
metered microwave energy within a conservatively estim ated
experimental
uncertainty (about 30% ).
M ak et al. [1996 and 1997] calculated the power absorbed at the reactor
walls by probe measurement of electric fields in a similar microwave reactor.
Their measurements were performed in a different operating regime (5 mTorr,
277 W input power) and in the absence of the substrate.
They concluded that
the wall losses amount to only 1% of the forward power and assumed that 99%
of the input power is absorbed by the plasma. However, they did not measure
the power absorbed by the base-plate and the short. Also, their Langmuir probe
measurements of electron number density in the bulk plasm a are a factor of 4
lower than those predicted from a simple global reactor m odel.
Since electron
density and power follow almost a linear relationship, a factor of 4 lower electron
density reflects a factor of 4 lower absorbed power. Thus, our analysis of their
reported data suggests only about 65 W is absorbed by their plasma.
Som e discussion is necessary on the validity o f assumptions and
uncertainties involved in the numerical predictions and experim ental data.
O ur
pseudo-1-D plasma model with ambipolar diffusion correction assumes uniform
bulk plasma and neglects the multidimensional effects.
model of a 13.56
M H z argon glow
discharge
by
A one-dimensional
Lymberopoulos
98
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
and
Economou [1993 and 1998] showed that a greater fraction of power was
dissipated into the sheaths near the plasma-surface interface.
Therefore, a
control-volume heat transfer model was constructed, which was independent of
the numerical modeling. As discussed in the next section, it will demonstrate that
it is impossible to remove 60 W microwave energy out of the control volume that
encompasses the plasma discharge region.
5.3
Control Volume Heat Transfer Analysis
Figure 5.7 shows a control volume encompassing the plasma, belljar,
base-plate, and substrate-holder assembly.
Microwave energy absorbed by the
Microwave Power
-r
Q nui
Qbj by belljar
Cooling air
Argon Plasma
Base-plate
Substrate
Holder
Y
Qcj by substrate
Figure 5.7 The schematic used for the
control-volume heat transfer analysis
99
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
plasma represents energy flowing into the control volume.
conduction through the
substrate,
gas convection,
Plasm a radiation,
and forced
air-cooling
comprise the energy loss terms for the control volume.
Plasma radiation consists of line and continuum emission and was
calculated
based
on
the
radiative
rates
in
the
CR M
and
amounts
to
approximately 0.65W , mainly due to the line radiation. The free convection heat
transfer between the plasma and the substrate is calculated as follows:
Q ,.,= h „A ,{T l - T , )
<5 ' 1>
where hps represents the average heat transfer coefficient between the plasma
and the substrate and is calculated as follows [Incropera and Dewitt, 1990]:
_ NiipA = 0.664Rep2 Pr1' 3 A
ps~
D
~
(5.2)
D
where A is the thermal conductivity of the argon plasma, and D is the diameter of
the substrate. A value of 20 W /m 2 K was calculated for hps. The energy transfer
between the plasma and the substrate is found be less than 6 W based on the
gas and substrate temperatures of 350 K and 287 K, respectively. The gas
temperature was assumed to be 350K since the measured belljar temperatures
were close to room temperature (about 296K). The substrate temperature was
actually measured by the reactor. If the gas temperature is assumed to be at
500K, the heat transfer rate is found to be 19.5W . This result is consistent with
the global energy balance study that found only a small amount of energy (about
15W ) was dissipated into the substrate cooling line.
The energy carried by the gases leaving the control volume is calculated
100
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
as follows:
Q.=«.Cn<-T. - V
(5 '3)
where mg and Cpg represent mass flow rate and specific heat of the gas,
respectively. The inlet and outlet gas temperatures are represented by T g,i and
T g, respectively. The gas transport heat loss was found to be negligible (about
0.2W ).
The heat transfer between the belljar and cooling air is calculated as
follows:
a,
where
Abj,
T bj,
temperature,
and
and
Ta
mean
(54)
represent the belljar surface area, measured belljar
air temperature,
respectively.
The
heat transfer
coefficient for the forced air cooling of belljar is represented by hb| (150 W / m2 K).
It is also calculated according to Equation 5.2 since the gas flow is laminar, but
with the thermal conductivity and Prandtl number of air. The belljar and cooling
air mean temperatures were 294K and 2 9 1 K, respectively. If the belljar surface
area above the substrate is used, the amount of heat dissipated in the cooling air
was found to be 13.7 W . If the whole surface area of the belljar is used, and also
accounting for the uncertainties associated with the temperature and flow
measurements, a maximum of 35 W heat transfer rate is obtained.
Based on an estimated contact area between the base-plate and the
belljar
and
experimentally
measured
base-plate
temperature
(287K),
the
conduction through the base-plate was found to be insignificant. As shown in
Figure
5.8,
the
base-plate
barely
touches the outside of the belljar while
101
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
the applicator walls are not encompassed in the control-volume at all. To
maintain vacuum conditions inside the belljar during operation, an O-ring is
designed to be between the belljar and the base-plate as a sealing device. Since
the O-ring is made of rubber, it should insulate the belljar and base-plate.
The theoretical calculations in the control-volume heat transfer analysis
are very straightforward. Recalling the results in the global energy balance study,
the bulk of microwave energy (320W ) is absorbed by the base-plate/applicator
w ater cooling line. If this amount of energy comes from the argon plasma, it has
to go through the belljar and dissipate into the cooling air, and then dissipate into
the applicator walls. Therefore, a total of 3 6 2 .6 W microwave energy has to be
transferred through the belljar (320W into the applicator wall and 4 2 .6 W into the
belljar air cooling line). For this heat transfer rate, a belljar tem perature of 370K is
needed.
That
is
76K
higher
than
the
experimentally
measured
belljar
temperature, far exceeding the claimed uncertainty of the infrared thermocouples
(about 2% accuracy).
Therefore, w e can conclude that the bulk of the microwave energy actually
bypassed the argon plasma and dissipated into the water cooling lines. Table 5.4
is a review of the results obtained in this section. The maximum heat transfer
rates are obtained by accounting for the uncertainties in the tem perature and flow
measurements. Also considering the possible minimum value for the heat
transfer rates, we conclude that only 10-60 W can be removed from the control
volume through thermal processes. These results support the conclusion in the
pseudo-1-D model.
102
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Table 5.4 Heat Transfer Analysis Data for the Control Volume Encompassing the
Argon Plasma (250 seem argon flow, 5 Torr)
AT(K)
Heat
Transfer
Rate (W)
Maximum
Heat
Transfer
Rate (W)
/
/
680±40
720
/
/
/
0.65
1.0
Convection
(plasma-substrate)
20
4.56e-3
63
5.76
19.5
Convection (belljarcooling air)
150
3.04e-2
3
13.7
35
/
/
/
0.2
0.8
20.3
56.3
Heat Transfer
Terms
Heat Transfer
Coefficient
(W/m2 K)
Area (m2)
Metered Microwave
Energy
/
Plasma Radiation
Convective Gas
Flow
Total Loss
103
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
5.4
2-D Argon Fluid Model Results
The control-volume heat transfer analysis and the pseudo-1-D model
results suggest that only 10-60W microwave power could be deposited in the
argon plasmas under the studied conditions. T h e 2-D argon fluid model will
provide more detailed information on the spatial temperature distributions,
microwave power density in the discharge region, and heat fluxes along the
boundaries.
Except for the boundary conditions and the plasma chemistry, the only two
parameters varied in the 2-D fluid model are pressure and microwave power.
Since the metered microwave power cannot be directly used, a parametric study
by varying the microwave power is helpful. The heat fluxes along the substrate
surface and the belljar walls will be a good check with the experimental data
obtained in the global energy balance.
5.4.1
Characteristics of Plasma Discharge by Varying Microwave Power
It is clear in the above discussions that the microwave power is significant
in characterizing the microwave argon plasmas. Therefore, a parametric study by
varying the microwave power was conducted to see the characteristics of the
plasm a discharges. Although w e have concluded that it was impossible to
deposit 6 8 0 W into the argon plasmas at our running condition, one case with
input microwave power at 6 8 0 W was also tried for comparison.
T h e microwave powers used in this parametric study are 2W , 5W, 10W,
104
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
30W , 50W , 10OW, and 680W . No self-consistent solution was available for both
the 2W and 5 W cases due to the curve-fitted rates. Figures 5.8 and 5 .9 show the
electron and gas temperature distributions. Four power levels (10W , 30W , 100W,
and 680W ) w ere used for the simulation. For simplicity, only the distributions at
the centerline w ere shown in the figures. However, the peak values of the plasma
parameters may not be on the centerline at all. Thus, these figures can only
show the general trends of the variation of these parameters changing with the
microwave plasma power.
The gas temperature distributions can be affected by the microwave
power significantly, as shown in Figure 5.9. From the energy balance study, we
stated that all the microwave power absorbed by the plasma has to be dissipated
either through the belljar and substrate, or by radiation. To dissipate more power
from the plasma, a higher gas temperature is expected.
105
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
18000
* — 10W
-•— 30W
^ -- 1 0 0 W
■r— 680W
16000
14000
12000
t-
10000
8000
6000
4000
20
30
40
60
50
70
z direction grid
Figure 5.8 Electron temperature distributions at the centerline
changing with the microwave power (grid 20 is at substrate)
1400
1200
—
10W
—• — 30W
1100
—A— 100w
1300
—▼— 680W
1000
900
*
800
05
I-
700
600
500
400
300-
200
20
30
40
50
60
70
80
z direction grid
Figure 5.9 Gas temperature distributions at the centerline
changing with the microwave power
106
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
It is interesting to see the spatial distributions with microwave pow er of
680W . Figure 5 .10 shows the spatial distributions of the gas and electron
temperatures, the microwave power density, and the electron number density.
Although w e can see much higher microwave power density and gas
temperature
distributions,
there
is
almost
no
change
with
the
electron
temperatures. This is consistent with our previous discussion. From Figure
5.10(d), it is clear that the lowest electron number density is about 1e19 m"3,
while the highest electron
number density surpasses 7e19
m"3. Previous
investigation with the continuum emissions showed that the averaged electron
number density could not go above 1e19 m*3. Therefore, the 6 8 0 W case is
impossible for our discharge conditions. However, the continuum emission has a
large uncertainty and it does not account for the non-Maxwellian effects. Thus it
may not be able to rule out all the high-energy cases. Thus we should find
another method to determine it quantitatively.
Heat fluxes are obtained in the 2-D fluid model. Combining this information
with the experimental global energy balance study and the control-volume heat
transfer analysis, it offers a better way to determine the microwave power as
discussed in the following section.
107
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
^
N
1.1E4
~N)1.2E4
9.6E3
9.6E3
2E4-1E4-
'sV \
10-
.3E4 I
8.3E3
E
o
1.1 E4
\
\
■1,2E4\
N
4-
7E3
-1.3E4
2-
-6
(b)
-4
0
-2
2
4
6
r(cm)
Figure 5.10 The spatial distributions for microwave argon
plasma at 5 Torr and 680W: (a) gas temperature distribution
(K), (b) the electron temperature distribution (K).
108
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
-6
-4
-2
(c)
0
2
4
6
r(cm)
1.9E19
1.9E19
10H
E19
E
u_
nT
8-
1E19
1.9E19
1.9E19
:.8E19
I
1E19.
S II1 ^ 4 .6 E 1 9
-6
(d)
-4
0
■2
2
4
6
r(cm)
Figure 5.10 (cont.) (c) the absorbed microwave power density
(W/m3), (d) the electron number density distribution (in'3)
109
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
5.4.2 Heat Fluxes in the 2-D Fluid Model
The control-volume heat transfer analysis showed that about 10-60 W
microwave power could be absorbed by the argon plasma. Moreover, the heat
transfer rate from the plasma to substrate has already been calculated in the
global energy balance study. The control-volume heat transfer model also
provides detailed information on the heat transfer rates both from the belljar to
cooling air and from plasma to substrate surface. Since the 2-D fluid model has
detailed information about the temperature distributions, it is interesting to
compare the heat flux calculated from the 2-D model with the results from the
control-volume heat transfer model.
Table 5.5 lists the heat transfer rates calculated from the 2-D model by
integrating the heat fluxes along the belljar walls or substrate surface, from the
control-volume heat transfer analysis, and from the global energy balance study.
It should be noted that although the heat flux along the top of the belljar was not
calculated, the heat transfer rates along the belljar walls and substrate surface
have almost reached the specified microwave power for 10W , 30W , 5 0 W and
even 10 0 W case. For the 6 8 0 W case, more heat is dissipated through the top of
the belljar. For the 10W case, the total heat along the belljar walls and substrate
surface (10.6W ) has exceeded the specified 10W. This error could be introduced
during the grid interpolation and the iteration between the EM mode! and fluid
model.
Combining the results from both the global energy balance study and
control-volume heat transfer analysis,
the total heat along the belljar walls
110
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
should be lower than 35W , and the total heat along the substrate surface should
be lower than 19.5W . As shown in Table 5.5, the cases with microwave power
higher than or equal to 5 0 W cannot m eet these criteria. Then it is concluded that
the possible microwave pow er absorbed in the plasma should be lower than
50W . For better determination of the microwave power, the uncertainty of the
control-volume heat transfer analysis and the global energy balance study should
be reduced.
Since most of the
uncertainty comes from
the temperature
measurements, better instrumentation of the temperature measurements is
needed.
Table 5.5 The Total Heat Predicted from 2-D Model Comparing with the Global
Energy Balance Study and Control-Volume Analysis Results
Total Heat along
Belljar Walls (W)
Total Heat along
Substrate Surface
(W)
10 W
9.72
0.87
30 W
23.2
1.91
50 W
39
5.36
100 W
63.7
18.3
680 W
337
35.6
Nominal
13.7
5.76
Maximum
35
19.5
Nominal
/
10
Maximum
/
20
Items
Microwave Power
Specified in the 2-D
Model
Heat Transfer Rate
from Control-Volume
Analysis
Heat Transfer Rate
from Global Energy
Balance
111
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Figure 5.11 shows the plot of the total heat changing with the microwave
power, which is the same result as in Table 5.5. Figure 5.12 and 5.13 show the
heat fluxes along the belljar walls and the substrate surface changing with the
microwave power, which are calculated from the 2-D model results. T h e total
heat transfer rates are
obtained
by integrating the
heat fluxes with
the
corresponding surface area. In Figure 5.15, the heat fluxes increase with the
microwave power due to the increase in the gas temperature. However, Figure
5.16 shows no significant change between the heat fluxes of the 1 0 0 W and
680 W cases. This is caused by the high non-uniformity of the plasma discharge,
as shown in Figure 5.9, 5.10 and 5.13)
1000 •
l = j Total Heat along Belljar Walls
llllllll Total Heat along Substrate Surface
100 -
(0
0)
X
0067
10W
30W
50W
100W
680W
Microwave Power
Figure 5.11 A vertical stacked column graph that shows the total
heat along the belljar walls and the substrate surface changing with
the microwave power
112
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
(0
ra
a3
m
<u
B "g
§ I
(0
X
_2
- A
Li­
• — 30W
— 5 0W
—T— 1oow
ra
• — 680W
0)
X
0.12
Figure 5.12 Heat fluxes along the belljar walls changing with the
microwave power (5 Torr and 250 seem)
03
0
■C
10s i
3
CO
10W
— a — 50W
03
— 0— 680W
1«
— • — 30W
—T — 100W
10*1
C=O ~E
|
I
101
X
3
ra
<D
X
102 1
—
0.00
I—
0.02
0.01
0.03
0.04
0.05
r (m)
Figure 5.13 Heat fluxes along the substrate surface changing with the
microwave power (5 Torr and 250 seem)
113
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
CHAPTER SIX
EXCITED STATES OF THE AR G O N PLASM AS
Optical emission spectroscopy provides an inexpensive and non-intrusive
diagnostic method to probe the plasma parameters, especially for the microwave
plasmas we studied where the Langmuir probe cannot be used in the reactor
cavity. Excited state number densities can be calculated directly from the O ES
data. Then how sensitive the measured excited state number densities are to the
plasma
parameters,
such
as
electron number densities
and temperature,
becomes an interesting topic.
W e used the OES measurements in tandem with the pseudo-1-D model to
investigate the microwave argon plasmas [Kelkar 1999]. However, the model
predictions could not match the experimental results within the experimental
uncertainties. As discussed in Chapter 5, w e also could not match the predicted
microwave power from pseudo-1-D model with the metered power (680W ).
These two topics motivated us to have further investigation of microwave argon
plasmas both experimentally and numerically.
The absorption measurement was conducted to obtain 4s state number
density, which is the metastable state of the argon plasma. These data will be
helpful in characterizing the argon plasma.
Both the 2-D fluid model and 2-D C R M were constructed to model the
114
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
plasma more accurately. The 2-D results show high non-uniformity of the argon
plasma discharges, in contrast to the uniform assumption implicit in the pseudo1 -D model.
6.1
The Experimental OES Data
For the microwave argon plasma with param eters of 6 8 0 W input power
(8 0 0 W forward power and 120W reflected power), 5 Torr pressure and 250 seem
argon flow, the measured excited state number densities from O ES are listed in
Table 6.1. The lines used in the O ES measurements are listed in Appendix A.
O ur CRM results indicate that some of the 4p-4s and 5p-4s transitions are selfabsorbed. Therefore, the experiments were m ade by choosing optically thin
transitions.
Table 6.1 The Excited State Number Densities Calculated from the OES Data
Comparing with the Numerical Predictions At Ne = 5el7 nT3, Tg = 350K
Numerical Prediction with Changed Coefficients
(n/gj)
Experimental
Uncertainty
Level
Data (n/gj)
A4p-4s =
Alp-41 “
3.34x107 s'1
1.0x10® s‘1
QjjM* X 10
4p
2.29e13
40%
1.1 x1013
2.48 x1013
2.22 x1013
Sp
6.97e11
56%
3.15x10”
5.53 X1011
5.74x1011
Sd
1.60e11
43%
9.67 x101°
1.24 x1011
1.22x10”
115
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Table 6.1
excited state
also lists the experimental uncertainty associated with the
number densities. The
uncertainties are
mainly due
to the
calibration factor and the transition coefficients. The 5p state has the highest
experimental uncertainty (56% ) because of its wavelength.
In T ab le 6.1, the numerical predictions with changed coefficients are
listed, which are the results of trying to match the experimental data within the
experimental uncertainties. Kimura et al. [1985] claimed that some of the
transition probabilities had a large uncertainty of one magnitude. Thus some of
coefficients w ere changed to check the sensitivity of the excited state number
densities to these coefficients. For detailed information, one should refer to the
discussion in Chapter 6 of [Kelkar, 1999].
6.2
A bsorption Measurement o f the Metastable State
The
motivation
for
the
absorption
measurements
came
from
the
Boltzmann plots for the predictions from our pseudo-1-D model to match the
measured excited state number densities from O ES measurements. Figure 6.1
was generated to show that the measured excited state number densities could
be matched by the model predictions only with changed rates. The experimental
data points w ere drawn with the experimental uncertainties.
However, it was observed that the 4s state (metastable state) number
density varied with different electron number density and temperature pairs,
which cam e from the pseudo-1-D model. The measured 4s state number density
116
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
will then provide another method to verify our models.
250 seem Argon
P = 5 Torr
T „ = 350 K
■
0
n8=1x1017 n r3 ; T b=1 6855 K V
n9=1x1018 m-3 ; T 9=9160 K
n9=1x1019 m"3 ; T 9=8118 K
n9=1x1018 m"3 with changed rates
Experimental
5
10
15
Energy [eV]
Figure 6.1 The Boltzmann plots showing that the experimental data can only
be matched by the model predictions with the changed rates
W e m easured the 4s population through absorption measurement at 7635
A. A typical optical emission scan of the argon plasma running at 5 Torr and
6 8 0 W condition can be found in the Appendix A. The line we used for absorption
measurement, 7635 A, actually is the strongest line.
The calculation of the 4s state number density needs the lineshape factor.
The lineshape can be calculated from the line-broadening theories as described
in Chapter 3. It turned out that the Doppler broadening was the dominant
mechanism [Li, 1999], This actually eased the necessity for us to get a measured
lineshape that requires a tunable laser system. The calculated argon linewidths
117
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
(F W H M ) are listed in Table 6.2. The data in this table w ere generated at the
following conditions: electron number density, N e = 1 e18 rrf3, T e= 10 ,00 0 K.
Table 6.2 The calculated argon linewidths (F W H M ) with different gas temperatures
at N e=lel8 m'3, Te=l0,000K
M ech an ism
S ym b ol
Linew idth (FW H M , unit in Hz)
Tg = 350k
450k
Center
Frequency
Natural
Broadening
V a n der
W aals
Vo
3.93E + 1 4
Avn
5.33E + 0 6
Resonance
Avr
0.00E +0 0
Avs
4 .95E + 0 5
Stark
Broadening
Doppler
Broadening
A vc
A vq
2.87E +05
8.29E+08
2.41 E +05
9.40E +08
550k
2.10E +05
1.04E +09
To investigate the effects of gas temperature, the Voigt param eter “a”,
defined as (ln 2 )1/2AvH/Avo, was used. For N e=1e18 nV3, and Te=10,000K , Figure
6 .2 is generated by the theoretical calculations.
The Voigt param eter "a" actually indicates the significance of either the
homogeneous or inhomogeneous broadening mechanisms. Figure 6.2 shows
that the Voigt param eter a «
1 with the gas tem perature changing from 3 5 0 K to
550K . As the gas temperature increases, the Voigt param eter “a” decreases.
Therefore, the Doppler broadening mechanism
is dominant for the argon
plasm as at the conditions w e studied.
118
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
6 . 506-03
6006-03
5. 506-03
4506-03
4. 006-03
'
300
350
400
450
500
550
600
Gas Temperature (K)
Figure 6.2 The Voigt parameter “a” as a function of gas temperature at an electron
number density of le !8 m'3, and an electron temperature of 10000K
For a Doppler broadening dominant lineshape, the lineshape function can
be expressed as:
7(v) =
( 41n2
V
nW2
1
-exp -41n2
(6-1)
/
With this lineshape, the 4s-state number density was calculated from the
experimental results, which is 6.50e1 6 nrf3. The experimental uncertainty for the
119
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
4s state measurement is about 42% , which was estimated conservatively from
the uncertainties in the absorption measurements and the uncertainty associated
with the transition line.
Figure 6.3 is the Boltzmann plot of the numerical predictions with the
experimentally measured 4s state number density and the OES data. The 4s
state is drawn with an error bar for its 42% uncertainty. It shows that although the
numerical prediction with the changed coefficients can match the experimental
OES data, it cannot match the measured 4s state number density within the
experimental uncertainty. Therefore, more modeling work is needed in this study.
n#=1e17 m'3, Te=16855K
n#=1e18 m’3, T e=9160K
—
na= 1e19m '3, T #=8118K
—
ne=1e18 m‘3 with changed rates
•
Experimental OES/absorption Data
s(eV)
Figure 6.3 The Boltzmann plot of the numerical predictions with the
experimentally measured 4s state number density and the OES data
120
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
6.3
Characterization of the Argon Plasma with the 2-D Fluid Model
From the energy balance study, it was concluded that not all the
microwave power could be absorbed by the argon plasmas at our studied
conditions.
To investigate the role of excited states in the argon plasmas, the study
with the base set in the 2-D fluid model will be discussed first. The microwave
power and plasma pressure were varied to study the sensitivity of the excited
states to these two parameters. Then the 2 -D CR M was used to generate the
detailed excited state number densities. Since the higher microwave power
cases
(>
50W )
have
been
ruled out in the energy
balance
study, the
experimental results were compared to the 2 -D predictions only from the lower
power cases (10W , 30E, and 50W ). Good agreem ent was found between the
experimental and numerical results.
The 2-D results show high non-uniformity with the argon plasmas at the
studied conditions. And the excited state number densities can be used as the
indicator of the non-uniformity.
6.3.1
A rgo n P lasm a a t 5 T o rr and 10W /30W
After much study with the 2-D model, the base sets we chose are at the
*
pressure of 5 Torr and the microwave power of 10W or 3 0 W at a frequency of
2.45 GHz.
121
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
T h e results of the simulation data for the plasma discharge at 5 T o rr and
10 W are shown in Figure 6.4, which indicates high non-uniformity in the plasma
discharge region. Figure 6.4 (a) shows the gas temperature distribution in the
simulation region. The shadowed area represents the substrate holder. The
quartz belljar forms most of the walls. The dimensions for the full plasma
simulation region are from -7cm to 7cm and from Ocm to 12 cm, in the r and z
directions, respectively. The spatial distribution shows a peak value of the gas
temperature of 472 K at the centerline about 4 cm above the substrate.
Figure 6.4 (b) shows the spatial distribution of the electron temperature.
Not like the gas temperature distribution, the electron temperature has peak
values at different places. This is caused by the non-uniformity of the microwave
power density that in turn is affected by the electric field.
Figure 6.4 (c) and (d) are the distributions of the mol fractions of the 4s
excited state number density and the electron/ion number density. In the pure
argon plasma discharge, the electron and ion number densities are the sam e if
we neglect the second ionization from the Ar+ ions. It is clear that the excited
state num ber density and the electron number density are highly non-uniform.
Figure 6 .4 (e) and (f) show the spatial distributions of the microwave
power density and the electric field. The microwave power density distribution
varies a little from the electrical field strength distribution. This could be caused
by the fact that both the electric field strength and the electron flux affect the
actual pow er deposition.
Due
to the
high
non-uniformity of the
122
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
electron
distribution, the electron flux should be expected to be highly non-uniform
spatially.
Figure 6.4 (g) is a gray scale map for the electric field strength. In this
figure, it is much easier to see the non-uniformity of the field strength.
Figure 6.5 shows the distributions for the 3 0 W case. Figure 6.5 (a) and (b)
show the gas temperature and electron temperature distributions. T h e gas
temperature in the 30W case is higher than in the 10 W case, which is caused by
the increase in the microwave power density. However, the electron temperature
is lower than in the 10W case. To understand this phenomenon, one can refer
back to Figure 5.4, where the electron number densities and temperatures were
in pairs for a self-consistent solution. Actually at higher electron number
densities, lower electron temperature is needed to sustain the plasma.
From
Figure 6.5 (d),
it can be shown that in the 3 0 W
case, the
electrons/ions number densities are much higher than in the 10W case. The
highest electron mol fraction in the 3 0 W case is about 4.3e-5, while in the 10W
case, the highest electron mol fraction is about 4.0e-6. Although the total particle
number density should be lower in the 3 0 W case due to higher gas temperature,
there still is a difference about one magnitude between the 10 W and 3 0 W cases,
which is significant for an increase in the microwave power with only a factor o f 3.
As shown in Equation 4.15, the absorbed microwave power should have roughly
a linear relation with the electron number density. Therefore, it is concluded that
the non-uniformity is even worse in the 30 W case than in the 10 W case.
123
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
However, the 4s number densities in the 10W and 30W cases do not have
as large difference. Comparing Figure 6 .4 (c) and 6.5 (c), roughly a factor of 3
increase in the p eak values of the 4s state number density is observed. This is
caused by lower electron temperatures in the 30W case. As discussed in
Chapter 5, higher microwave power will generate higher electron num ber density.
However, lower electron temperature will go with the higher electron number
densities, then reduce the sensitivity of the excited state number densities to the
plasma parameters.
Figure 6 .5 (e) and (f) show the absorbed microwave power density and
the electric field distributions. Unlike the 10W case, here the higher microwave
power densities and the higher electric fields are much closer to either the belljar
walls or the substrate surface. Therefore, the non-uniformity is even worse in this
case. As shown in Figure 6.5 (e), the electric field is higher close to either the
belljar walls or the substrate surface. Since the surfaces are acting as catalysts
to the recombination or de-excitation reactions, it is possible that the plasma
absorbs more energy in the regions closer to the walls and then maintains the
electron number density balance in these regions by recombination at the
surfaces. Figure 6 .5 (g) is a wire frame plot of the electric field inside the EM
simulation region. This plot shows that the microwave is unable to penetrate the
whole plasma discharge. High electric field exists along the belljar walls and
substrate surface, while the bulk of the plasma discharge has relatively low
electric field.
124
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
- 6 - 4
-2
0
2
4
6
4
6
r(cm)
'1.1 E4
5E4
8- .3E4
_ \
9.7E3
1.1E4'
■1.3E4
E «N*
4-
9.7E3'
2-
-6
-4
0
•2
2
r(cm)
Figure 6.4 Spatial distributions of the argon plasma discharge
characteristics at 5 Torr, 10W: (a) Gas Temperature (K ), (b)
Electron Temperature (K). (Note: the shadowed area is the
substrate holder.)
125
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
s
2.5E4E-6
10-
4E-6
E
N
o_
IE-6
;.5E-7
I.3E-7-
4-
2
-
-6
-4
0
•2
2
4
6
r(cm)
Figure 6.4 (cont.) (c) the mol fraction of 4s state number
density, (d) the mol fraction of electron/ion number density
126
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
12-
10-
8
-
E* 6 *
'Zt
4-
2-
0
-6
-4
-2
0
2
4
6
2
4
6
r(cm)
12
10
8
"g
6
4
2
0
-6
-4
-2
0
r(cm)
Figure 6.4 (cont.) (e) the absorbed microwave power density
(W /m3), (f) the electrical field strength (V/m)
127
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
-6
-4
-2
0
2
4
6
r(cm)
Figure 6.4 (cont.) (g) the gray scale map o f the electric field strength (V/m )
128
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
-6
-4
-2
0
2
4
6
r(cm)
Figure 6.5 Spatial distributions of the argon plasma
discharge characteristics at 5 Torr, 30W: (a) Gas
Temperature (K), (b) Electron Temperature (K). (Note: the
shadowed area is the substrate holder.)
129
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
-6
-4
-2
0
2
4
6
r(cm)
r(cm)
Figure 6.5 (cont.) (c) the moi fraction of 4s state number
density, (d) the mol fraction of electron/ion number density
130
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
1.4E5
5.2E4
5.2E4
7 2E3
7.2E3
12-
•1,6E:
10-
8
-
6.3E2
Figure 6.5 (cont.) (e) the absorbed microwave power density
(W/m3), (0 the electrical field strength (V/m)
131
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
probe position
1.4x10
belliar walls
o 1 .2 x 1 0
©
substrate holder
® 1 .0 x 1 0 s
position
o 8 .0 x10 *
•c
£ 6 .0 x 1 0 *
4 .0 x 1 0
2 .0 x 1 0
(g)
in ner m etal chuck
position
Figure 6.5 (cont.) (g) the wire frame plot of the distribution for the
electric field inside the EM simulation region (unit in W/m)
132
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
From the above discussion, excited states are not very sensitive to plasma
parameters due to the fact that when the electron number density increases, the
electron temperature decreases in a self-consistent solution. To explain this
phenomenon, the electron conservation equation should be revisited. The
electron conservation equations in the pseudo-1-D and 2-D models should be
sufficient for the purpose here. For simplicity, Equation 4.8 in the pseudo-1-D
model is used. This equation is copied here for convenience:
(6 .2 )
Also as shown in Figure 5.3, the dominant electron production and loss
terms are excited state ionization and ambipolar diffusion loss, which correspond
to the first and fourth terms in the right-hand-side of Equation 6.2, respectively.
The electron production term actually is a function of electron number
density and temperature since the reaction rates are mainly functions of the
electron temperature as shown in Equation 4.65, which is also copied here for
convenience:
K
re a c tio n
= wc(l\ *w
.0 -
e~Ne/Ne" yJ Teb
(6.3)
The parameter “a” reflects the EEDF’s deviation from Maxwellian. For
higher electron number densities, the EEDF will be Maxwellian, and “a" will
approach zero. Therefore, the reaction rate is a function of electron temperature
133
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
at higher electron number density.
Thus at higher electron number densities, Equation 6.2 can be simplified
to the following if only the dominant production and loss terms are included and
the steady state solution is considered:
For a pure argon plasma, electron and ion number densities are the same.
Then further simplification of Equation 6 .4 will lead to an equation with only one
unknown variable, that is, the electron temperature. Therefore, for high electron
number densities, the electron temperature is actually a constant determined by
the pressure, cavity geometry and ambipolar diffusion rate. It is no longer
affected by the electron number density.
However, for low electron number densities, the non-Maxwellian effect will
reduce the reaction rate. Thus a higher electron temperature is needed to
maintain electron conservation.
134
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
6.3.2 Effects of the Microwave Power
In Chapter 5, the effects of the microwave power have been discussed
related to the energy balance. In this section, the effects of microwave power on
the electron and excited state number densities are discussed. Figure 6.6 and
6.7 show the 4s state and electron mol fractions changing with the microwave
power.
Figure 6.7 shows that the electron number density can be affected by the
microwave power significantly. From the review of the plasma processes, the
electrons play a very important role here. Almost all the microwave power is
absorbed by electrons first, then the energy is transferred to other particles in the
plasma discharges through electron-heavy particle reactions. In Figure 5.11, it is
observed that the electron temperature distributions did not change significantly
with different power levels, which is consistent with our pseudo-1-D model
solutions.
Figure 6 .6 shows that the 4s state number density will change with
microwave power. However, the change is not so significant. This leads to a fact
that the excited state number densities are not very sensitive to the microwave
power.
135
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
1E-4
as
ra
55
T3
a>
o
iS
<0
IE-5
1E-6
*♦—
O
— ■—
— *—
— a—
—
c
o
o
as
Li­
as
o
10W
301/7
100W
680W
5
1E-9
20
30
40
50
60
70
z direction grid
Figure 6.6 Mol fraction of 4s state at the centerline changing with the
power
w
c
o
w
o
as
lii
o
c
o
—
—
10W
30W
— a — 100W
1E-7-;
— ▼— 680W
IE-8
20
30
40
50
60
70
z direction grid
Figure 6.7 Mol fraction of electron number density at the centerline
changing with the power
136
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
6.3.3 Effects of the Plasma Pressure
It was very hard to get a stable argon discharge experimentally. Our
W A V E M A T reactor was designed for diamond CVD processes. Therefore, it was
optimized for the hydrogen plasmas at pressure around 4 0 Torn However, at
such pressures, the argon discharge was not stable. Thus the pressure was
reduced to 5 Torn. Argon discharges at lower pressure (at the magnitude of
mTorr) were studied previously. However, most of the reactors were parallel
plates or other simpler structures.
Three pressure conditions (3, 5, and 8 Torr) were simulated for the
discharge with microwave power of 10W . Figures 6.8 -
6.11
compare the
distributions of the gas, electron temperatures, and the electron, 4s state number
densities. No significant changes can be observed in these figures. Comparing
the results of 3 Torr and 8 Torr, the pressure is increased roughly three times.
However, the peak values of gas temperature, electron temperature, electron
number density and 4s state only changes about 30K (<10% ), 1000K (<10% ),
6e17 rrf3 (within the same order), and 5e17 m*3 (within the same order),
respectively. Thus the plasma parameters are not very sensitive to pressure over
the range investigated.
137
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
— ■— 3 Torr
XL
480
<u
c
TT
ffl
C
a>
a
460-
®
400-
<U
(U
W
3
W
0)
380-
E
a)
320
5 Torr
— a — 8 Torr
440 420-
360
340-
Q.
I</>
(0
O
300
280
~r~
20
T -
~r~
—r~
—r~
—r~
30
40
50
60
70
-l
80
z direction grid (20 at the substrate)
Figure 6.8 The comparison of gas temperature distributions at the
centerline with different plasma pressures
18000
17000-
<
D
u.
3
2
<5
Q.
E
<D
H
C
o
L.
o
0)
UJ
— ■— 3 To rr
m•
16000150001400013000-
— • — 5 Torr
— *■— 8 To rr
1 2 0 00 -
A
I
1 1 0 00 10000-
'
i
9000
8000-
A
J
7000-
A
£
6000
5000
4000
3000
•
A
A
—r~
20
- 1-
l
30
40
I50
- T"
- r-
60
70
~1
80
z direction grid
Figure 6.9 The comparison of electron temperature distributions at the
centerline with different plasma pressures
138
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
0)
•*r
E
3
II
J3
Z
C
10 "
3 Torr
— • — 5 Torr
-
— a — 8 Torr
s
o
_©
UJ
10
i
T"
20
30
—
r~
40
—T~
~r~
50
60
T
“
“I
80
70
z direction grid (20 at the substrate)
Figure 6.10 The comparison of the electron number density
distributions at the centerline with different plasma pressures
10 '*
1017
'£
10 "
</>
c
<D
Q
k.
0)
E
3
14
10
— ■— 3 Torr
Z
— • — 5 Torr
©
— a — 8 Torr
13
co
in
rr
10
"
20
30
40
50
60
70
80
z direction grid
Figure 6.11 The comparison of 4s state number density distributions
at the centerline with different plasma pressures
139
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
6.4
Characterization of the Argon Plasma with the 2-D CRM
The 2-D fluid model provided the spatial distributions of gas and electron
temperatures, as well as other parameters. The 2 -D CR M will then use these
parameters to calculate all the excited state num ber densities, which will enable
us to compare them with the experimental data. Since w e follow the similar
plasma chemistry for argon discharges in both the 2-D fluid model and the 2-D
CRM,
similar results of the 4s-state
number densities,
microwave power
densities, and other parameters are expected.
6.4.1
Predicted Excited-State Number Densities Distribution
Figures 6 . 1 2 - 6 . 1 5 show the spatial distributions of the 4s, 4p, 5p, and 5d
state number densities at a pressure of 5 Torr and the microwave power of 10W .
The high non-uniformity is clearly shown in these figures. The spatial distributions
of the excited state number densities follow the sam e distribution profile, which
follows the profile of the electron number density distribution. Due to the fact that
there is not a large change with the electron temperature in the discharge region,
the electron number density affects the excite-state number densities more
significantly.
For the microwave power level at 30W , similar results were obtained. It is
of interest to compare the experimental results with the predictions from the
models, recalling that w e could not match all the three excited states (4p, 5p, and
140
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
5d) simultaneously without changing any rates in the pseudo-1 -D model.
One assumption of our pseudo-1-D
model is the uniformity of the
discharge. However, as shown in Figures 6.12-6.15, this assumption can no
longer be held after the study of 2-D modeling. To compare the numerical data
with the experimental data, the predicted excited state number densities must be
reduced to the O ES signal along the probing path. If axi-symmetric distribution is
assumed, the general expression for the averaged optical emission signal along
the r-direction is:
I,iavs=
2 \J l , Iij .rr d r / D = — —
All// J\ n i . r dr I D
y.tfvs
^
(6 -5 )
where Xy is the wavelength of the used transition line, D is the diameter of the
belljar, r is the radius from the centerline of the belljar, ni,r is the excited state
number density at radius r, Aj is the transition probability.
For
the
absorption
measurements,
the
Beer-Lambert
relationship
(Equation 3.7 and 3.8) also relate the spatially distributed excited state number
density to the measured transmission of a probe beam of light.
For the uniform mesh spacing, the above expression can be simplified
further. Although the non-uniformity along the radius direction cannot be shown
after this averaging procedure, it still can be clearly shown by the non-uniform
distribution along the z-axis.
141
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
1 .4 x10
1.2x10
£ • 1.0x10
°
o
8 .0 x 1 0 "
6 .0 x 1 0
m
4 .0 x 1 0
2.0 x10
0 -7
Figure 6.12 The spatial distribution of the 4s state number density at 5
Torr and 10W
5x 10
£ 4X10
X
"tn
Q
3x10*'
s
a
E
3
2
©
2x 10
1x10
Figure 6.13 The spatial distribution of the 4p state number density at 5
Torr and 10W
142
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
1 .5x10
0
-7
Figure 6.14 The spatial distribution of the 5p state number density at 5
T o rr and 10W
1.4x10
1.2x10
£
10
c
ffl
Q
o
•O
1.0x10
8.0 x10
E
a
Z
6 .0 x 1 0
3
a
in
4.0x10
2.0x10
0
-7
Figure 6.15 The spatial distribution of 5d state number density at 5
T o rr and 10W
143
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
During the study of the 2-D CRM, the plasma pressure is set at 5 Torr
while the microwave power is changed from 10W to 50W . As a result of the
previous sections, microwave powers higher than 100W are excluded. The 5 0 W
case was added later on trying to get a prefect match between the experimental
and numerical results, which turned out to be not successful.
Figures 6.16-6.19 show the excited-state number density distributions
along the z-direction compared with the experimental results. The experimental
uncertainty was shown in the figures for each excited state. It is interesting to
observe that the 50W case does not increase the excited-state number densities
at all. Therefore, excited-state number densities have only a w eak dependence
on the microwave power.
Figure 6 .1 7 shows that none of the cases studied provides a good match
between
the
numerical
and
experimental uncertainty (40% ).
experimental
results for
4p-state
within
the
However, all other figures show that the 3 0 W
case can match the numerically predicted 4s, 5p and 5d state number densities
with the experimental ones within the experimental uncertainty. Even for the 4p
state, the error between the numerical and experimental results is about 64% ,
which is much better than the results we obtained from the pseudo-1-D model.
For the larger error associated with 4p state, a lot of factors may play their
roles here. First of all, the top of the rector (short) actually is adjustable for better
tuning during the experiments. A few millimeters adjustment was always
144
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
reasonable to obtain a stable plasma discharge. Second, the excited state
number density distribution is significantly non-uniform. The values on the grids
next to the position of the probe beam may introduce an error larger than 15% .
The simulation region for the reactor was based on the design values. Thus a
small deviation of the actual short position may introduce much larger error than
expected.
The uncertainty caused by the model itself should also be considered. In
the next two sections, the microwave power density and the curve-fitted rates will
be studied to estimate the uncertainty caused by these parameters.
It needs to be pointed out that although the excited-state number densities
are not good indicators for the plasma parameters (power, electron number
density, electron temperature, etc.), they do provide valuable information about
the non-uniformity of the discharge. With the scale-up of the plasma reactors, the
uniformity becomes a critical problem for the designer. Therefore, the optical
emission spectroscopic data may be an easy way to monitor the uniformity inside
the plasma discharge region. In summary, the experimental and numerical data
are listed in Table 6.3
145
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Table 6.3 The experimental results of the excited state number densities
comparing with the 2-D C R M predictions for an argon plasma at 5 Torr, 250 seem
(Best matched case at microwave power of 30W)
Best Matched 2-D CRM Data
at 30W
Experimental Data
Excited
States
nj/gj (m-3)
nj (m*3)
Uncertainty
nj (mJ )
Uncertainty
4s
5.42e15
6.5e16
42%
8.09e16
+24.5%
4p
2.29e13
8.24e14
40%
2.99e14
-63.7%
5p
6.97e11
2.51e13
56%
1.31e13
-47.8%
5d
1.600e11
9.60e12
43%
6.59e12
-31.4%
146
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
3.0x10" -
2.5 x 1 0 "-
U)
(D
Q
2.0x10 17-
xi
1.5x10"-
c
experimental
E
3
Z
ffl
re
O
)
■
in
1.0X10” - 1
5.0 x 1 0 "0.0 -
I
10
20
30
—I—
T -
I
40
50
60
T ”
70
-1
80
z-Direction Grid (0 at the bottom)
Figure 6.16 The averaged 4s-state number density distribution along
the z-direction from the numerical predictions comparing with the
results from the absorption measurement
1.4x10'* ”
1.2x10'* ”
%
>»
expenmental
1 .0 x 1 0 "-
c
0)
Q
l.
0)
8 .0 x10'*-
X) 6.0x1014 E
3
z 4 .0 x10'*0)
2
0)
i
Q.
2 .0 x10'*0 .0 -2.0x10"
i
- I-
I
- i—
10
20
30
40
i—
|—
50
i—
|—
i—
60
r -
70
I
80
z-Direction Grid (0 at the bottom)
Figure 6.17 The averaged 4p-state number density distribution along zdirection from the numerical predictions comparing with the
experimental results
147
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
10W
8x10
30W
7x10'
— — 50W
>» 6x10'
experimental
a) 5x10’
<D 4x10
5 3x1013"
2x10
A 1x10
m
-1x10
0
30
20
10
40
50
60
80
70
z-Direction Grid (0 at the bottom)
Figure 6.18 The averaged 5p-state number density distribution along
z-direction from the numerical predictions comparing with the
experimental results
'E
>.
'55
c
a>
Q
w
(U
XI
e
b
3
Z
©
ra
4.5x10 13-
10W
4.0x1 O'3-
30W
50W
3.5x1 O'3-
- — experimental
3.0x10132.5x10132.0x10,J -
/VV
1.5x10’31.0x10 11"
35
5.0x10tJ-
•6
in
0 .0 -5.0x101J-
- 1-
~I-
l
I
~ r-
i
I
- 1
10
20
30
40
50
60
70
80
z-Direction Grid (0 at the bottom)
Figure 6.19 The averaged 5d-state number density distribution along zdirection from the numerical predictions comparing with the
experimental results
148
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
6.4.2 Effects of the Microwave Power Density
Microwave
power density
(M W PD )
distribution
is one
of the most
important parameters used in our 2-D fluid model. The total microwave power is
specified in the EM model, then the M W P D distribution is calculated and serves
as the input to the 2-D fluid model for the next iteration. During the iterations, the
M W PD distributions may introduce some error due to the interpolation. The
highly non-uniform argon plasma discharge also introduces more uncertainty with
M W PD.
T h e 2-D CRM generates the M W P D results by calculating the electron
energy balance. Therefore, comparing the M W PD distributions used in the fluid
model and the CRM will be a quantitative method to evaluate the uncertainty with
these models, which could attribute to the uncertainties of microwave power and
the excited state number densities.
Figure 6 .2 0 shows the comparison of two spatial distributions of the
microwave power densities at 5 Torr and 10W; one was used for the 2-D fluid
model and the other was predicted from the 2-D CRM . To show the difference
graphically, the 2-D fluid model results are shown in the right half of the base
plane, with the 2-D CRM results shown in the left half. T h e two distributions have
similar profiles as shown in Figure 6.20. However, the predicted peak value of
the M W P D in the 2-D CRM (about 1.2e5 W /m 3) is higher than the one used for 2 D fluid model (about 6e4 W /m 3). Therefore, in terms of the peak values, roughly a
149
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
50% uncertainty is introduced with the M W P D distributions.
In our 2-D CRM , the electron tem perature distribution was taken from the
2-D fluid model. Then the E/N value was iterated inside the C R M to get the
predicted averaged electron temperature converged to the value that is provided
by the 2-D fluid model. The microwave power absorbed by the plasma was
calculated from the electron energy balance inside the 2-D C R M . Since we
followed much smoother profiles of electron and gas temperature distributions,
the predicted absorbed M W PD was also much smoother than the one generated
from the 2-D electromagnetic model.
Figure 6.21 compares the M W PD distributions used in the 2-D fluid model
to those predicted from the 2-D CR M at 5 Torr and 30W . In this figure, the
difference between these two distributions is much smaller. Comparing the peak
values, only an 18% uncertainty is calculated for this case.
Figure 6 .2 2 and 6.23 are the contour plots for the same distributions in
Figure 6.20 and 6.21, respectively. These plots provide a quantitative method to
compare the two profiles. Also the discrepancy between the two distributions is
shown more clearly in these two figures.
The large uncertainty with the M W P D distribution in the 10W case could
be reduced if an iteration scheme between the 2-D CR M and 2-D fluid model is
employed. However, for the 3 0 W case, the 18% uncertainty is acceptable.
150
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
1.4x10* ■"
1.2x10* '
M W P D Used
in the 2-D
Fluid Model
1.0x10* "
E
8.0x104 ‘
6.0x104 ‘
4.0x104
2.0x104
0 ft
MW PD
Predicted
from 2-D
CRM
Figure 6.20 The wire frame plot of the comparison between M W PD
profiles used in 2-D fluid model and predicted from 2-D C R M at 5 Torr
and 10W
151
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Figure 6.21 The wire frame plot of the comparison between the
M W PD used in 2-D fluid model and predicted from 2-D C R M
152
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
,4E'
4-
MWPD used in Fluid Model
2
1.9E4
-
2 E 2.
E
o
-
2-
MVyPD P re d ic te d from 2 - D C R M
I.7E4 1.9E4
-4 -
-6-
z(cm)
Figure 6.22 The comparison of the spatial distributions of
M W PD used in the fluid model and predicted from the 2-D
C RM at 5 Torr and 10W
1E4 MWPD Predicted
I
from 2-D CRM
,6E!
1E4
\
_L5T864rJ._.
E
u
-
22E5
-4 -
.6E5
hi u
1
— --------
^1.1E5
-6
-
8E4'
12
MWPD Used in
Fluid Model
S.8E4
5.8E4.
10
1.1E5
m
5.8E4
5.8E4^—^
6
8
4
2
0
z(cm)
Figure 6.23 The comparison of spatial distributions of M W P D
used in fluid model and predicted from the 2-D C R M at 5 To rr
and 30W
15 3
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
6.4.3 Effects of the Reaction Rates
As discussed in the modeling part, the reaction rates w ere curve-fitted to
avoid the troubles in the calculation time and programming techniques. Large
errors between the curve-fitted rates and the exact ones w ere observed,
especially when the electron number densities and the electron temperatures
were both low. Therefore, it would be better if this can be avoided in the actual
cases. Figure 6.24 - 6.27 show the comparisons between the rates calculated
frum the 2-D CRM and the curve-fitted rates for the 5 Tonr and 1 0 W input power
case. Other curve-fitted rates are not presented here because they are not as
sensitive to the non-Maxwellian EEDF as are the four rates shown in these plots.
The rates for Kea[0,1] and Kea[1,0] have larger error than the other two
rates (Kiea[0] and Kiea[1]) between the 2-D CRM calculations and the curvefitted calculations. Comparing the peak values of the two profiles shown in the
same plot, the errors caused by curve-fitting are listed in Table 6.4.
The problem associated with the low electron number density and/or low
electron temperature did not show up here.
This actually is good for our
analysis. Even in Figure 5.53 where the biggest discrepancy occurs, the peak
value calculated from the 2-D CRM is about 2 times higher than the peak value in
the curve-fitted calculation. Checking the electron number density distribution for
5 Torr and 10W case, the range is from 1e17 to 1e18 m*3, which is a much
sm aller range than the one used in the curve-fitting process. Therefore, the low
154
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
electron number density region actually has been avoided. To have an accurate
curve-fitted equation, a smaller electron number density range should be used
that will not introduce too much error.
Table 6.4 E rror between peak values comparing the reaction rates calculated from
2-D C R M and the curve-fitted equations
Error between Peak Values
Rates
Kea[0,1], electron-atom excitation
48.8%
Kea[1,0], electron atom de-excitation
44.4%
Kiea[0], electron-atom ground state
ionization
< 5%
Kiea[1], electron-atom excited state
ionization
< 5%
In the above discussion, Kea[0,1] and Kea[1,0] rates have the largest
errors. It is interesting to see how much uncertainty this error will introduce
between the results from both the 2-D CRM and 2-D fluid models. Figure 6.28
shows the 4s-state number densities obtained from both the 2-D C R M and fluid
models. From this figure, a 30% uncertainty has been observed between the two
distributions if peak values are again used to evaluate the uncertainty. That
means the error from the reaction rates is actually lessened in the final results.
Comparing Figures 6.2 4 and 6.25, Kea[0,1] and Kea[1,0] rates both have larger
values
in the 2-D
CRM.
During the calculation,
the combination
of the
uncertainties from these two rates may generate a lower uncertainty for 4s state
155
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
number density. For accurate results, more accurate curve-fitted reaction rates
are desired. However, there is always a tradeoff between a more efficient
working model and more accurate results.
:4.1E-20
1.1E-2I
Kea[0,1] calculated
4 -
from 2-D CRM
'E - o -
2-
Kea[0,1] calculated from
1E-21
curve-fitted equation
12
10
6
8
4
2
0
z(cm)
Figure 6.24 The comparison of Kea[0,l] rates calculated from 2-D
CRM and from curve-fitted equation
156
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
1.9E-176.3E-17
4-
Kea[1,0] calculated
from the 2-D CRM
5.6E-17-
3.5E-17
4.2E-17.
2.1 E-1
E-TS'
1.4E-17
Kea[1,0] calculated from
the curve-fitted equation
2.1 E-17
3.5E-17
-
6
1.8E-17-
12
10
6
8
4
2
0
z(cm)
Figure 6.25 The comparison of Kea[l,0| rates calculated from 2-D
C R M and from curve-fitted equation
6-
’6.7E-1 E-21
4-
Klea[0] calculated
from the 2-D CRM
2-
E
o
-
21E-23.
Klea[0] calculated from
the curve-fitted equation
-4 -
-
6-
12
10
8
6
4
2
0
z(cm )
Figure 6.26 The comparison of Klea[0] rates calculated from 2-D
C R M and from curve-fitted equation
157
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
.6E-1
L7E-1
4-
Klea[1] calculated
from the 2-D CRM
2E-1&
‘3.3E-15
-
2
-
Klea[1] calculated from
-4 -
-
the curve-fitted equation
612
10
6
8
4
2
0
z(cm)
Figure 6.27 The comparison of K Ien[l] rates calculated from 2-D
C RM and from curve-fitted equation
6
-
/
6.4El7r
I.7E17-
4s number density
4-
in 2-D CRM
2-
-
2-
4s number density
1E17
-4 -
in 2-D fluid model
3.7E17
-
6
V6.4E17
-
12
10
6
8
4
2
0
z(cm)
Figure 6.28 4s State number density comparison between the
results from the 2-D C R M and the 2-D fluid model at 5 Torr, 10W
158
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
6.5
Non-uniformity of the Argon Plasmas
The 2-D
model results show that high non-uniformity exists in the
microwave argon plasmas under the studied conditions. It should be noted that
the W A V E M A T reactor studied was originally designed for diamond CVD
process. That could explain why major problems with the hydrogen modeling
were not previously encountered. For the extensive microwave argon plasma
study conducted in this research, a lot of problems have been solved, and the
conclusions drawn from previous studies are also confirmed by this study.
The applications of the 2-D fluid model and the 2-D C R M could be far
beyond the areas shown in this study. In this study, although w e failed to relate
the OES data to the plasma parameters directly, it is clear that the O E S data can
be used to indicate the uniformity inside the discharge region. This should be an
inexpensive and easy technique to implement for most reactors if spatial OES
measurements are possible.
The spatial distributions for the electron number density (Figure 6.4(d) and
6.5(d)) show that electrons are swarming towards either the belljar walls or the
substrate surface. This is unique for the argon discharge. Hydrogen plasmas will
form a perfect ball above the substrate if finely tuned. During the experiments, it
was observed that the argon plasma filled the whole belljar volume. To
understand this phenomenon, the introduction of the plasma conductivity should
be helpful. T h e plasma conductivity is defined as [Lieberman 1994]:
159
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
where cope is the plasma frequency, a is the microwave frequency, and vm is the
electron-neutral collision frequency. For a «
vm, the plasma conductivity can be
simplified as the dc plasma conductivity, which is defined as:
e \
G dc
—
(6.5)
~
V *
™ 'V m
In the above equation, the electron-neutral collision frequency is related to
the neutral particle number density. A plasma with lower pressure will have lower
neutral particle number density. Then the electron-neutral collision frequency will
be lower, and the plasma conductivity will become higher. Since the belljar walls
or the substrate surface provide more efficient electron-ion recombination
reactions, electrons will always try to escape the discharge volume. Thus a
sheath is formed to confine the electrons inside the discharge region, which we
have not put into the models yet.
Another factor to consider for the non-uniformity inside the plasma
discharge region is that the simulation region is axi-symmetric. That means,
although the grid points on the r-direction may be uniform, the actual volume
each grid represents is totally different (V=2nrZ). The grids that are closer to the
wall will have larger volume. At the start point of the 2-D model calculations, a
ball-like microwave power density is assumed. However, if the electrons in this
assumed discharge region cannot absorb all the input power, the discharge
160
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
region has to be expanded to reduce the power density. On the other hand, even
if more electrons are generated in the discharge region to absorb the input
power, when the calculation of the electromagnetic field begins, the electrons will
escape the original discharge region.
161
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
CH APTER SEVEN
C O NC LU SIO NS AND REC O M M EN DATIO N S
7.1
Conclusions
The major focus of this dissertation is to investigate the nonequilibrium
microwave argon plasmas. The motivation for the argon study originated from
our previous research work [Kelkar 1999 and Li 1997]. Two major objectives of
this dissertation are to solve the puzzle with the microwave power absorbed in
the microwave argon plasmas, and to match the numerically predicted excited
state number densities with the experimentally measured results within the
experimental uncertainties. A better understanding of the microwave argon
plasmas has been achieved. The major achievements and conclusions from this
study are listed as follows:
❖ Microwave power is the most important parameter used in the plasma
modeling,
especially for the 2-D
models.
It affects
discharges under the studied conditions significantly.
the argon
plasma
However, a large
discrepancy exists between the predicted microwave power from the pseudo1-D model (<1 00 W ) and the metered power during the experiment (8 0 0 W
forward power and 12 0 W reflected).
❖ An energy balance study was conducted by performing a global energy
balance study on the W A V E M A T reactor, constructing a control-volume heat
162
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
transfer model, and a self-consistent argon 2-D fluid model with the EM
model. T h e global energy balance study and the control-volume analysis
provided solid evidence that only a small amount of microwave energy (106 0W ) is absorbed by the argon plasma. The 2-D fluid model generates the
heat fluxes along the belljar walls and the substrate surface, which confirm
that upper limit. By combining all the conclusions from experimental and
numerical results, a microwave power less than 50W w as confirmed for the
argon plasmas under the studied conditions (5 Torr, 250 seem argon flow).
Optical emission spectroscopic (O E S ) measurements w ere conducted to
m easure the excited-state number densities. The three excited states used in
this study w ere 4p, 5p, and 5d levels. The experimental uncertainties
associated with the excited-state number densities are 40% , 56% and 43% ,
respectively. T h e absorption measurements w ere conducted to measure the
4s
state
number
density.
These
data
are
independent
of the
OES
m easurem ent and provide additional experimental results to verify our
models.
T he numerical models used to predict the excited states are the pseudo-1-D
model, the 2-D fluid model and the 2-D Collisional-Radiative Model. The
pseudo-1 -D model assumes a uniform microwave power density. However, it
was
impossible
to
match
the
excited
state
number
densities
within
experimental uncertainties. The 2-D model, which is the first 2-D microwave
argon model, provides the detailed spatial distributions for the excited state
163
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
number densities. It is observed from the 2-D results that the argon plasmas
under studied conditions are highly non-uniform. The excited state number
densities predicted from the 2-D models are more accurate than the psedo-1 D model. T h ree of the four predicted excited states number densities (4s, 5p,
and
5d)
can
match
uncertainties. The
the
experimental
results
within
the
experimental
prediction of 4p state number density matches the
experimental results with an uncertainty of 64% . The best matching case from
the 2-D model has a microwave power of 30W , which is consistent with the
energy balance study.
Large uncertainty with the numerical predictions could also be introduced
from the uncertainties of the microwave power densities caused by the
iterations between the 2-D fluid model and 2-D CRM and from the curve-fitted
reaction rates used in the 2-D fluid model.
The high non-uniformity of the argon plasmas is rather unique and should be
avoided if this kind of plasma is going to be used. At the pressure studied (5
Torr), the electron-neutral collision frequency is not high enough to reduce the
plasma conductivity. Thus the recombination reactions are not able to keep
electrons from swarming towards the walls, where the surface recombination
reactions are more efficient.
With
the
extensive
experimental
and
numerical
investigation
of
the
nonequilibrium argon plasmas, a better understanding of the argon plasma
has been achieved. It was confirmed that only about 3 0 W microwave-power
164
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
was absorbed in the argon plasma. The OES and absorption data were used
to verify our pseudo-1-D and 2-D models. However, it has shown that no
simple relations could be found to relate the spectroscopic data to plasma
parameters such as electron number density or tem perature directly. They
have to be used carefully in the models.
6.2
Recommendations
This study was a continuation of previous research at the University of
Arkansas. After this study, there are still some
investigation.
topics that need further
Due to the limitation of the research facilities and funding
resources, it is highly recommended that more cooperative work should be
conducted in the future that will enable us to attack more sophisticated problems.
❖
First of all, spatial measurements of the OES and absorption data are highly
recommended. Although the techniques are straightforward, it does require a
reactor that allows spatial measurements.
❖ Although absorption measurements can generate the 4s number density
using a theoretically calculated lineshape factor, an accurate lineshape is still
desirable
by using
a
tunable
diode
laser.
Since
the
line-broadening
mechanisms are dominated by the Doppler broadening, the actual measured
lineshape could help determine the gas temperatures in the plasma discharge
region.
❖ The 2-D model showed that the plasma discharge could be very sensitive to
165
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
the geometry. The study of the 2-D model with changing geometry should be
very helpful to the reactor designers. If a certain reactor needs to be scaledup to larger w afer size, the 2 -D model should also be good at checking the
effects of the changed geometry.
The curve-fitted reaction rates introduce numerical uncertainty to the 2-D
models. For better curve-fitting results, the gas temperature should be
considered in the curve-fitting process. Dividing the electron number density
range into some small sub-ranges will also reduce the error.
For the experimental study of the argon plasmas, instability is one issue that
we could not tackle at this time. Some researchers have begun to consider
the instabilities in low-pressure discharges [Lieberman 1999], This could be
an interesting topic for the plasm a processing industry.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
REFERENCES:
Baer, D. S., 1993, "Plasma diagnostics with semiconductor lasers using fluorescence and
absorption spectroscopy," Ph.D. Dissertation, Stanford University.
Benoy, D.A., van der Mullen, J.A., and Schram, D.C., 1993, “Radiative energy loss in a
nonequilibrium argon plasma," J o u r n a l o f P h y s ic s D : A p p l ie d P h y s ic s , Vol. 26, pp.14081413.
Bird, R.B., Stewart, W.E., and Lightfoot, E.N., 1960, T r a n s p o r t P h e n o m e n a , John Wiley & Sons
Inc.
Bittencourt, J.A., 1986, F u n d a m e n t a ls o f P l a s m a P h y s ic s , P e r g a m o n P r e s s , Oxford,pp. 425-427.
Bou, P., Boettner, J.C., Harima, H.. and Vandenbulche, L., 1992a, “Kinetic calculations in
plasmas used for diamond deposition," J a p a n e s e J o u r n a l o f A p p li e d P h y s ic s , Vol. 31, pp.
1505-1513.
Bou, P., Boettner, J.C., Harima, H., and Vandenbulche, L„ 1992b, “Kinetics of chemical
reactions in hydrocarbon-hydrogen plasmas for diamond deposition," J a p a n e s e J o u r n a l
o f A p p lie d P h y s ic s , Vol. 31, pp. 2931-2936.
Braun, C.G., and Kune, J.A., 1987, “Collisional-radiative coefficients from a three-level atomic
model in nonequilibrium argon plasmas," P h y s . F lu id s , Vol. 30, pp. 499-509.
Breene, R.G., 1961, T h e S h if t a n d S h a p e o f S p e c t r a l L in e s , Permagon Press, Oxford.
Burshtein, M.L., Lavrov, B.P., and Yakovlev V.N., 1987, “Quenching of excited hydrogen atoms
by H2 molecules," O p t ic a l S p e c t r o s c o p y ( U S S R ) , Vol. 62, No. 6, pp. 729-731.
Capelli, M.A., and Paul, P.H., 1990, ‘An investigation of diamond film deposition in a premixed
oxyacetylene flame," J .A p p l. P h y s ., Vol. 67, No. 5, pp. 2596-2602.
Celii, F.G., and Butler, J.E., 1991, ‘Diamond chemical vapor deposition,” A n n u a l R e v i e w P h y s ic s
C h e m is t r y , Vol. 42, pp. 643-683.
Cherrington, B.E., 1979, G a s e o u s E le c t r o n ic a n d G a s L a s e r s , Pergamon Press Inc., Oxford,
England.
Comey, A., 1977, A t o m ic a n d L a s e r S p e c t r o s c o p y , Oxford University Press, Oxford.
Curtiss, C.F., and Hirchfelder, J.O., 1949, “Transport properties of multi-component gas mixture,"
J . C h e m . P h y s ., Vol. 17, No. 6, pp. 550-555.
Delcroix, J.L., 1958, In t r o d u c t io n a la t h e o r ie d e s g a z io n is e s ( E n g lis h : In t r o d u c t io n t o t h e t h e o r y
o f i o n iz e d g a s e s ) , Translated from the French by Melville Clark, Jr., David J. BenDaniei
and Judith M. BenDaniei, Interscience tracts on physics and astronomy, no.8.
167
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
Devoto, R.S., 1973, Transport coefficients of ionized argon," P h y s . F lu id s , Vol. 16, pp. 616.
Ferreira, C.M., Loureiro, J. and Ricard, A., 1985, ‘ Populations in the metastable and the
resonance levels of argon and stepwise ionization effects in a low-pressure argon
positive column," J . o f A p p l. P h y s ., Vol. 57, pp. 82-90.
Gicquel, A., Chenevier, M„ Kassouni, Kh., Tserepi, A., and Dubus, M., 1998, ‘Validation of
actinometry for estimating relative hydrohen atom densities and electron energy evolution
in plasma assisted diamond deposition reactors," J . o f A p p l . P h y s ., Vol. 83, No. 12, pp.
7504-7521.
Gicquel, A., Chenevier,M„ Breton, Y., Petiau, M„ Booth,J.P., and Hassouni, K.. 1996, ‘Ground
state and excited state H-atom temperatures in a microwave plasma diamond deposition
reactor," J . P h y s . I l l F r a n c e , Vol. 6, pp. 1167-1180.
Gicquel, A., Hassouni, K., Farhat, S., and Breton, Y., 1994, ‘ Spectroscopic analysis and chemical
kinetics modeling of a diamond deposition plasma reactor," D ia m o n d a n d R e l a t e d
M a t e r i a ls , Vol. 3, pp. 581-586.
Gordon, Matthew H., 1992, "Non-equilibrium effects in a thermal plasma," Ph.D. Dissertation,
Stanford University.
Gordon, M.H., and Kelkar, U., 1996, T h e role of two-step ionization in numerical predictions of
electron energy distribution functions," P h y s ic s o f P l a s m a s , Vol. 3, pp. 407-413.
Griem, H., 1964, P l a s m a S p e c t r o s c o p y , McGraw Hill, New York.
Griem, H., 1974, S p e c t r a l L in e B r o a d e n in g b y P la s m a s , Academic Press, New York.
Grotjohn, T.A., Tan, W., Gopinath, V., Srivastava, A.K., and Asmussen, J., 1994, "Modeling the
electromagnetic excitation of a compact ECR ion/free radical source," R e v . S c i., In s t r u m .,
Vol. 65, No. 5, pp.1761-1765.
Grotjohn, T.A., 1998, "Electromagnetic field modeling of diamond CVD reactors," H a n d b o o k o f
In d u s t r ia l D ia m o n d a n d D ia m o n d F ilm s , Prelas, M.A., Popovici, G., and Bigelow, L.K.
Eds, Marcel Dekker, Inc., New York, pp. 673.
Hassouni, K., Farhat, S., Scott, C.D., and Gicquel, A., 1994, "Contribution to the modeling of a
microwave PACVD diamond deposition reactor Chemical kinetics and diffusion
transport," A d v a n c e s in N e w D ia m o n d S c ie n c e a n d T e c h n o lo g y , Saito, S., Fujimori,N.,
Fukunaga, O., Kamo, M., Kobashi, K, and Yoshikawa, M. (Editors), MYU, Tokyo, pp.
131-134.
Hassouni, K., Farhat, S., Scott, C.D., and Gicquel, A., 1996, ‘ Diamond species and energy
transport in moderate pressure diamond deposition H2 plasmas," J . P h y s . I l l F r a n c e , Vol.
6, pp. 1229-1243.
Hassouni, K., Scott, C.D., Farhat, S., Gicquel, A., and Capitelli, M., 1997, "Non-Maxwellian effect
on species and energy transport in moderate pressure H2 plasmas," S u r f a c e a n d
C o a t in g s T e c h n o lo g y 9 7 , pp. 391-403.
Hassouni, K., Leroy, O., Farhat, S., and Gicquel, A., 1998a, ‘ Modeling of H2 and H2/CH 4
moderate-pressure microwave plasma used for diamond deposition," P l a s m a C h e m is t r y
a n d P l a s m a P r o c e s s in g , Vol. 18, No. 3, pp. 325-362.
168
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Hassouni, K„ Scott, C.D., and Farhat, S., 1998b, “Flow modeling fora plasma assisted diamond
deposition reactor,’ H a n d b o o k o f In d u s tr ia l D i a m o n d a n d D i a m o n d F ilm s , Prelas, M.A.,
Popovici, G., and Bigelow, L.K. (Editors), Marcel Dekker, Inc. New York, pp. 653.
Hassouni, K„ Scott, C.D., and Farhat, S., 1998c, ‘Modeling of the diffusion transport of an H2
plasma obtained under diamond deposition discharge conditions," H a n d b o o k o f in d u s t r ia l
D i a m o n d a n d D i a m o n d F ilm s , Prelas, M.A., Popovici, G., and Bigelow, L.K. Eds, Marcel
Dekker, Inc., New York, pp. 697.
Hassouni, K„ Grotjohn, T.A., Gicquel, A., 1999, ‘ Self-consistent microwave field and plasma
discharge simulations fora moderate pressure hydrogen discharge reactor,* J . o f A p p lie d
P h y s ic s , Voi. 86, No. 1, pp. 134-151.
Hirose, Y., Amanuma, S., and Komaki, K„ 1990, T h e synthesis of high-quality diamond in
combustion flames,’ J o u r n a l o f A p p lie d P h y s ic s , Vol. 68, pp. 6401-6405.
Incropera, F.P., and Dewitt, D.P., 1990, F u n d a m e n t a ls o f H e a t a n d M a s s T r a n s f e r , John Wiley &
Sons.
Jaffrin, H.J., 1965, “Shock structure in partially ionized gases," P h y s . F lu id s , Vol. 8, No. 4, pp.
605-625.
Katsonis, K., and Drawin, H.W., 1980, Transition probabilities for argon (I)," J . Q u a n t . S p e c t r o s .
R a d ia t . T r a n s f e r , Vol. 23, pp. 1-55.
Kelkar, U., and Gordon, M.H., 1996, ‘Determination of electron density and temperature using
optical emission spectroscopy and self-consistent modeling in a nonequilibrium
microwave plasma," 43'° N a t io n a l S y m p o s iu m
o f A m e r ic a n V a c u u m
S o c ie t y ,
Philadelphia, Pennsylvania, October 14-18.
Kelkar, U„ Gordon, M.H., Roe, L A , and Li, Y., 1997, ‘ Diagnostic and modeling in a pure argon
plasma: Energy balance study," 28"1 A IA A P la s m a D y n a m i c s a n d L a s e r s C o n f e r e n c e ,
Atlanta, Georgia, June 23-25.
Kelkar, Umesh, 1997, "Diagnostics and modeling in nonequilibrium microwave plasmas,’ Ph.D.
Dissertation, University of Arkansas. Fayetteville.
Kelkar, U„ Gordon, M.H., Roe, L.A., and Li, Y„ 1999, “Diagnostics and modeling in a pure argon
plasma: Energy balance study," J. V a c . S c i. T e c h n o l. A , Voi. 17, No. 1, pp. 125-132.
Kimura, A.. Kobayashi, H., Nishida, M., and Valentin, P., 1985, "Calculation of collisional and
radiative transition probabilities between excited argon levels," J . Q u a n t . S p e c t r o s c .
T r a n s f e r , Vol. 34, No. 2, pp. 189-215.
Koemtzopoulos, C.R., Economous, D.J., and Pollard, R., 1993, “Hydrogen dissociation in a
microwave discharge for diamond deposition,” D i a m o n d a n d R e l a t e d M a t e r ia ls , Voi. 2,
pp. 25-35.
Kortshagan, U., Heil, B., 1999, Two-dimensional experimental characterization of an inductive
discharge and comparison to a kinetic plasma model using parallel computing," AT2.4,
T h e 5 2 r G a s e o u s E le c t r o n ic C o n fe r e n c e , Norfolk, VA.
Kortshagen, U. Pukropski, I., and Tsendin, L.D., 1995, ‘ Experimental investigation and fast twodimensional self-consistent kinetic modeling of a low-pressure inductively coupled rf
169
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
discharge," P h y s ic a l R e v i e w E , Vol. 51, No. 6, pp. 6063-6078.
Kortshagen, U., Parker, G.J., and Lawler, J.E., 1996, ‘Comparison of Monte Carlo simulations
and nonlocal calculations of the electron distribution function in a positive column
plasma," P h y s ic a l R e v i e w E , Vol. 54, No. 6, pp. 6746-6761.
Kune, J A , 1987, “Role of atom-atom inelastic collisions in two-temperature non-equilibrium
plasmas," P h y s ic s o f F lu id s , Vol. 30, pp. 2255-2263.
Kune, J A , and Gundersen, M A , 1983a, ‘A fundamental theory of high power thyratrons II: The
production of atomic hydrogen and positive ions," L a s e r a n d P a r t ic l e B e a m s , Vol. 1, part
4, pp. 407-425.
Kune, J A , Guha, S., and Gundersen, M.A., 1983b, “A fundamental theory of high power
thyratrons I: The electron temperature," L a s e r a n d P a r t ic le B e a m s , Vol. 1, part 4, pp. 395405.
Larrabee, R.D., 1959, Spectral Emissivity of Tungsten, J . O p t ic a l S o c . A m e r ic a , Vol. 49, No. 6,
pp.619-625.
Lee, Jong-Hun, 1984, ‘Basic governing equations for the flight regimes of aeroassisted orbital
transfer vehicles," AIAA-84-1729, A I A A 1 $ th T h e r m o p h y s ic s C o n f e r e n c e , Snowmass,
Colorado.
Lettington, A., Steeds, J., 1994, T h in F ilm D ia m o n d , The Royal Society.
Li, G.P., Takayanagi, T., Wakiya, K., and Suzuki, H., 1988, “Determination of cross sections and
oscillator strengths for argon by electron-energy-loss spectroscopy," P h y s ic s R e v i e w A ,
Vol. 38, pp. 1240-1247.
Li, Y., Kelkar, U., Gordon, M.H., and Roe, L.A., 1997, ‘A global reactor model for pure argon
microwave discharges," T h e a n n u a l 5C fn G a s e o u s E le c t r o n ic s C o n f e r e n c e , Madison,
Wisconsin, October 6-9.
Li, Y., Gordon, M.H., Roe, L A , and Hassouni, K., 1999, ‘Absorption measurements of 4s state
number density for a microwave argon plasma," DT1.5, T h e a n n u a l 5 2 f G a s e o u s
E le c t r o n ic C o n f e r e n c e , Norfolk, VA.
Liebermann, M.A., and Lichtenberg, A.J., 1994, P r in c ip le s o f P l a s m a D is c h a r g e s a n d M a t e r i a l s
P r o c e s s in g , Wiley-lnterscience, John Wiley & Sons Inc., New York. N.Y.
Liebermann, M.A., 1999, “Pulsed power excitation and instabilities in low pressure
electronegative discharges," JW2.1, T h e a n n u a l 52nd G a s e o u s E le c t r o n ic C o n f e r e n c e ,
Norfolk, VA.
Lymberopoulos, D.P., and Economou, D.J, 1993, “Fluid simuiations of glow discharges: Effect of
metastable stoms in argon," J. A p p l. P h y s ., Vol. 73, No. 8, pp.3668-3679.
Lymberopoulos, D.P., Kolobov, V.I., and Economou, D.J., 1998, "Fluid simulation of a pulsedpower inductively coupled argon plasma," J. V a c . S c i. T e c h n o l. A , Vol. 16, No. 2, pp. 564571.
Mak, P., and Asmussen, J., 1997, ‘Experimental investigation of the matching and impressed
electric field of a multipolar electron cyclotron resonance discharge, " J . V a c . S c i. T e c h n o l.
170
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
A , Vol. 15, No. 1, pp. 154-168.
Mak, P., Tsai, M.H., Natarajan, J., Wright, B.L., Grotjohn, T.A., Salam, F.M.A., Siegel, M., and
Asmussen, J., 1996, “Investigation of multipolar electron cyclotron resonance plasma
source sensors and models for plasma control," J . V a c . S c i. T e c h n o l. A , Vol. 14, No. 3,
pp. 1894-1900.
Matsui, Y., Yabe, H., and Hirose, Y., 1990, “The growth mechanism of diamond crystals in
acetylene flames," J a p a n e s e J o u r n a l o f A p p lie d P h y s ic s , Vol. 29, pp. 1552-1560.
Matsumoto, S., Sato, Y., Tsutsumi, M., and Setaka, N., 1982, J p n . I. A p p l. P h y s . , Vol. 21, L18385.
McDaniel, E.W., and Mason, E.A., 1973, T h e M o b ilit y a n d D iff u s io n o f Io n s in G a s e s , John Wiley
& Sons, New York.
McMaster, M.C., Hsu, W.L., Coltrin, M.E., and Dandy, D.S., 1994, “Experimental measurements
and numerical simulation of the gas composition in a hot filament assisted diamond
chemical vapor deposition reactor," J . A p p l. P h y s ., Vol. 76, No. 11.
Mitchner, M., and Kruger, C.H., 1973, P a r t i a lly I o n iz e d G a s e s , Wiley-lnterscience, New York,
N.Y.
Morgan. W.L., 1994, E L E N D I F 9 3 V e r s io n 1 . 7 U S E R 'S G U I D E , Kinema Software, P.O. Box 1147,
Monument, CO 80132.
Palmer, J.D., 1992, “Optimization and scale-up of the hot-filament diamond process by modeling
reaction kinetics and hydrodynamics," Ph.D. Research Proposal, Department of
Chemical Engineering, University of Arkansas, Fayetteville.
Repetti, R.E., Fincke, J.R., and Newman, W.A., 1991, “Relaxation kinetics of argon thermal
plasmas," A S M E H e a t T r a n s f e r in T h e r m a l P l a s m a P r o c e s s in g , Vol. 161, pp. 167-182.
Scott, C.D., 1993," A non-equilibrium model for a moderate pressure hydrogen microwave
discharge plasma," N A S A T e c h n ic a l M e m o r a n d u m 1 0 4 7 6 5 .
Scott, C.D., Fahat, S., Gicquel, A., Hassouni, K., and Lefebvre, M., 1996, “Determination of
electron temperature and density in a hydrogen microwave plasma," J o u r n a l o f
T h e r m o p h y s ic s a n d H e a t T r a n s f e r , Vol. 10, pp. 426-435.
Shore, B.W., and Menzel, D.H., 1968, P r in c ip le s o f A t o m ic S p e c t r a , John Wiley & Sons, New
York.
Sobelman, I.I., Vainshtein, LA., and Yukov, E A ., 1981, E x c it a t io n o f A t o m s a n d B r o a d e n in g o f
S p e c t r a l L in e s , Springer-Verlag, Berlin.
Spear, K., and Dismukes, J., 1993, S y n t h e t i c D i a m o n d : E m e r g in g C V D S c i e n c e a n d T e c h n o lo g y ,
The Electrochemical Society, Inc.
St-Onge, L., and Moisan, M., 1994, “Hydrogen atom yield in RF and microwave hydrogen
discharges," P l a s m a C h e m is t r y a n d P l a s m a P r o c e s s in g , Vol. 14, pp. 87-116.
Surendra, M„ Graves, D.B., and Plano, L.S., 1992, “Self-consistent DC glow discharge
simulations applied to diamond film deposition reactor," J o u r n a l o f A p p l i e d P h y s ic s , Vol.
171
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
71, pp. 5189-5198.
Tahara, H., Minami, K., Murai, A., Yasui, T., and Yoshikawa, T., 1995, “Diagnostic experiment
and kinetic model analysis of microwave ChVH2 plasmas for deposition of diamond-like
carbon films,’ J a p a n e s e J o u r n a l o f A p p l ie d P h y s ic s , Vol. 34, pp. 1972-1979.
Taher, M.A., 1999, ‘An investigation of the mechanical and physical issues of the adhesion of
chemically vapor deposited (CVD) diamond coating on cemented carbide cutting tool
inserts,’ Ph.D. Dissertation, University of Arkansas, Fayetteville.
Tan, W., and Grotjohn, T.A., 1994, ‘ Modeling the electromagnetic excitation of a microwave
cavity plasma reactor," J. V a c . S c i. T e c h n o l. A , Vol. 12, No.4, pp. 1216-1220.
Tan, W., and Grotjohn, T.A., 1995, ‘Modeling the electromagnetic field and plasma discharge in a
microwave plasma diamond deposition reactor," D i a m o n d a n d R e l a t e d M a t e r i a ls , Voi 4,
pp. 1145-1154.
Vincenti, W.G., and Kruger, C.H., 1967, In t r o d u c t io n to P h y s ic a l G a s D y n a m i c s , Kreiger
Publishing Company, Malabar, Florida.
Vlcek, J., and Pelikan.V., 1985, ‘Electron energy distribution function in the collisional-radiative
model of an argon plasma," J . P h y s . D .'A p p l. P h y s ., Vol. 18, pp. 347-358.
Vlcek, J., and Pelikan.V., 1986, “Excited level populations of argon atoms in a non-isothermal
plasma," J. P h y s . D . 'A p p l. P h y s ., Vol. 19, pp. 1879-1888.
Vlcek, J., 1989, "A collisional-radiative model applicable to argon discharges over a wide range of
conditions. I: Formulation and basic data," J . P h y s ic s . D , A p p lie d P h y s ic s , Vol. 22, pp.
623-631.
White. H.E., 1934, In t r o d u c t io n to A t o m ic S p e c t r a , McGraw-Hill, New York.
Wiese, W.L., Smith, M.W., and Glenn, B.M., 1966, A t o m ic T r a n s itio n P r o b a b ilit ie s , U.S. National
Bureau of Standards, National Reference Series 4, U.S. Government Printing Office,
Washington D.C., Vol. 1.
Wilke, C.R., 1950, "A viscosity equation for gas mixtures," J . C h e m . P h y s ., Vol. 18, No. 4.
Yodor, M.N., 1990, ‘The vision of diamond as an engineered material," S y n t h e t ic D i a m o n d E m e r g in g C V D S c ie n c e a n d T e c h n o lo g y , Spear, K.E. and Dismukes, J.P. Eds., John
Wiley & Sons Inc., New York, N.Y.
Yos, J.M., 1963, "Transport properties of Nitrogen, Hydrogen, Oxygen and air to 30000 K,"
T e c h n ic a l M e m o r a n d u m R A D T M - 6 3 - 7 , AVCO-RAD, Wilmington, Mass.
172
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
A PPE N D IX A
ARG O N SPEC TR O SC O PY
Following
is
a
simple
introduction
to
the
notation
of
the
atomic
spectroscopy. For electrons, when the angular momentum quantum number,
£=0,1,2,3, the electronic state is denoted as s, p, d, f, etc. If w e put the main
quantum number before these letters, w e get the symbol for the electronic states.
The capital letters S, P, D, F are used to denote the atomic states. At the upper
left corner, a number such as 2 denotes that it has a double structure. And at the
lower right comer, the j quantum number is put there. Although S states only
have single energy level structure, we still use 2S symbol.
However, the calculation of argon transition strengths between the excited
states requires the use of jl-coupling schemes due to the complex interactions
between the atomic-core electrons and the valence electrons [White, 1934;
Katsonis & Drawin, 1980]. Therefore, the notation describing the electronic states
with the conventional LS-coupling may not be appropriate.
Argon is the fortieth elem ent in the periodic table and has a nominal
atomic weight of 39.95. In the ground state, argon has a closed 3p shell with a
total spin S=0. Singly excited atomic states have the configurations as 3s23p5nls,
where n, I, s are the principle, angular and spin quantum numbers of the excited
electron, respectively.
173
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
T h e energy level structure of argon is typical of all the noble gases. It is
clear to see that there is a large energy gap separating the first excited state from
the ground state. As a result, the electrons in the atomic core will effectively
couple their respective angular momentum and spin vectors before interacting
with the outer electron. For example, the electrons in the atomic core first form
the angular and spin vectors, denoted lc and sc, which then couple together to
yield the total core angular momentum vector j c, where j c = lc + sc. Since sc = 1/2,
the two lowest energy level of the argon ion Ar(ll) will have the two possible
quantum number jc = 1/2 or 3/2. T h e two states are denoted as 2P i /2 and 2Pzt2,
where the 2P ^
is 1431
cm*1 above the 2Pi /2 ground state. The angular
momentum vector I of the excited (outer) electron then couples with the total core
angular momentum vector j c to form the intermediate vector K, w here K = jc + I.
The intermediate vector K then couples with the electron spin vector s to form
the total angular momentum vector for the entire atom, J, where J = K + s. The
splitting of the atomic core forms two term systems. O ne is referred to as the
"primed system" with jc = 1/2, and the other is called as the "unprimed system"
with jc = 3/2. These two term systems may be described as the following
nomenclature [Katsonis and Drawin, 1980]:
[ 2P i /2 ] nl [ K]j
for jc = 1/2
[ 2P 3/2 ] n l [ K ] j
for jc = 3/2
Table A.1 shows the spectroscopic notation for selected argon energy
levels [Li, 1988],
174
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
jl
LS
Paschen
E, (c m 1)
3p6 'So
‘So
Ipo
0
3p5 (2P3/2°)4s [3/2]2°
3p5 4s 3P2°
Is5
93143.800
3p5 (2P3n
4s [3/2],°
3p5 4s 3P,°
Is4
93750.639
3p5 (2P i /2°) 4 s ' [l/2 ]0
3p5 4s 3P0°
is3
94553.707
3p5 (2P i /2°) 4 s' [l/2 ]0
3p5 4s ‘P,0
Is2
95399.870
3p5 (2P3/2°) 4p [5/2]3
3pJ 4 p 3D 3
2p9
105462.804
3p5 (2P3/2°) 4p [5/2]2
3p5 4p 3D 2
•a00
to
Table A .l Spectroscopic Notation of Selected Argon Energy Levels
105617.315
3p5 (2P3/2°)4p [3/2],
3p5 4p 3D ,
2p7
106087.305
3p5 (2P./2°) 4p' [3/2],
3p5 4p *P,
2p4
107131.755
3p5 (2Pi/2°) 4p‘ [3/2]2
3p5 4p 3P2
2p3
107289.747
3p5 (2P i/2°) 4p' [1/2],
3p5 4 p 3P,
2p2
107496.463
)
Figure A.1 shows a typical the optical emission spectroscopic scan from
3500 A to 8000 A. The strongest emission lines are located between 7000 A to
8 00 0 A. Table A .2 shows the spectroscopic data for the used transition lines.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Table A.2 Transition Lines Spectroscopic Data (from N IS T Atomic Spectra Data)
Wavelength
E,
Aul
Eu
Ji - Ju
Uncertainty
of Au,
9i “ 9u
(10s s'1)
A ir (A)
(cm'1)
(cm'1)
4300.101
93750.5978
116999.3259
1 -2
3 -5
3.77E-03
25%
6032.127
105462.760
122036.0704
3 -4
7 -9
2.46E-02
25%
6043.223
105617.270
122160.1502
2 -3
5 -7
1.47E-02
25%
6965.431
93143.7600
107496.4166
2 -1
5 -3
6.39E-02
25%
7067.218
93143.7600
107289.7001
2 -2
5 -5
3.80E-02
25%
7147.042
93143.7600
107131.7086
2 -1
5 -3
6.25E-03
25%
7272.936
93750.5978
107496.4166
1 -1
3 -3
1.83E-02
25%
7503.869
95399.8276
108722.6194
1 -0
3 -1
4.45E-01
25%
7514.652
93750.5978
107054.2720
1 -0
3 -1
4.02E-01
25%
7635.106
93143.7600
106237.5518
2 -2
5 -5
2.45E-01
25%
176
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
1.0000B06
Emission Strength (a.u.)
1.0000EMM
1.0000EMM
1.0000EMJ3
1.0000EMJ2 ----------------------------------------------------------------------------------------------------------------------------------------------3000
4000
5000
6000
7000
8000
Wavelength (A)
Figure A .l A Typical Optical Emission Scan of Microwave Argon Plasma at 680W, 5
Torr, and 250 seem flow rate (Data taken on 04/16/98)
177
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
APPENDIX B
PROGRAM LISTING
B.1
2-D Argon Fluid Model
The codes for the 2-D argon fluid model were written in the W A TC O M
FO R TR A N format. A short description of the source codes is as follows.
B.1.1 Description of the Source Codes
ecrver.for:
This file contains all the routines used for generating output files for either checking or
saving data.
extpow.for:
This file initializes the MWPD for the fluid model at the beginning of solution. After the first
iteration with the EM model, it reads the MWPD from the output file generated in the EM
model,
interpol.for:
This file interpolates the results in the 2-D fluid model between the meshes of the fluid
model and EM model.
plasana.for:
This file is not routinely used in the code during the solution of the equations. It analyzes
different source and fluxes terms in the conservation equations. The heat flux terms are
generated by this subroutine,
plasbloc.for:
This files treats the boundary conditions in the blocked off region situated in the
computation domain.
plascalc.for:
Main driver for the computation of the coefficients of the quasi-linear algebraic system
resulting from the descretization of the transport equation and their boundary conditions.
It contains the calling sequences for the computation of the Peclet numbers, Fluxes term
coefficients, sources terms, and so on.
plaschem.for:
This file calculates the chemistry source terms and their Jacobian if needed.
plasclax.fon
178
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
This file computes the coefficients resulting from the boundary conditions at the reactor
axis.
plasclin.for:
This file computes the coefficients resulting from the boundary conditions at the reactor
inlet.
plasclw.for:
This file computes the coefficients resulting from the boundary conditions at the reactor
walls.
plasco.for:
Secondary driver for the computation of the coefficients resulting from the discretization
of the flux terms in the transport equations.
plascoO.for:
Routine which computes the coefficients resulting from the discretization of the flux terms
of the transport equations
plascomp.for:
This file calculates some plasma physical characteristics needed in the computation:
plasma molar mass, species mass fraction, species molar concentration, species and
total density, etc.
plascons.for:
This file contains the routines that force the electrical neutrality by correcting the electron
mole fraction, and maintain pressure to be constant in the reactor by correcting H2 mole
fraction,
plascvhs.fon
This file splits the fluxes terms of enthalpy into two terms: the temperature dependant
part of the enthalpy is treated as a pseudo-convective term while the one related to the
formation enthalpy at a reference temperature T0 is treated as a source term. The
resulting discretization coefficients are then calculated in the same file,
plasgsri.for & plasgsrj.for:
These routines perform the Gauss-Siedel Line relaxation for all the conservation
equation, plasgsri.for relaxes in the Oi (or Oz) direction while plasgslrj.for relaxes in the
Oj (or Or) direction,
plasinitfon
This is the initialization routine,
plasmain.fon
179
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Main driver of the plasma module.
plasnrex.for:
This routine computes the energy exchange source terms between the different plasma
energy modes and during chemical reactions.
plaspecl.fon
This file contains the routine for the computation of the Peclet number which is important
for setting up the numerical scheme for the differentiation of the conservation equations,
plasphys.fon
Secondary driver for the computation of all the physical properties of the plasma at each
grid point: enthalpy, energy exchange rates, chemistry rate, etc.
plasprpt.for:
Main driver for the computation of all the physical properties and the transport coefficients
of the plasma at each grid point.
plasread.for:
This routine reads all the data of related plasma physical constants and the plasma
chemistry.
plasreru.for:
This routine enables the program to restart the computation starting from previously
relaxed fields.
plasres.for:
This file contains the main relaxation loop where all other routines are called. Thus it is
the most important file among ail the fifes used in the fluid model.
plasrest.for:
This file contains the routines for the calculation of the residues of the conservation
equations and the rates of the changes with the temperatures and specie concentrations
during the relaxation,
plassour.for:
In this routine, the final form of the source terms is calculated. These source terms are
180
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
linearized by splitting the positive part from the negative part,
plastran.fon
This routine calculates the transport coefficients for the
two/three
temperature
nonequiiibrium flow.
rddat.for:
This routine calculates all the parameters related to the reactor geometry.
B.1.2 Input Data Files
Practically all the input data files are read in the routine 'plasread.for1.
T he data files are of three types:
- Those dealing with the plasma chemistry and physics;
- Those dealing with geometry and input power spatial distribution;
- Those dealing with the numerical procedures.
(Note: All the data are in MKS units, that is, kg, m, sec, J, etc.)
A short description of the input data files used in the 2-D fluid model is as
follows:
•
Plasma Chem istry and Physics
plasspec.dat:
This file gives the chemical species that may be present in the plasma. Each data line is
preceded by some comments in French. The data deals with the nature of species
(atoms, molecules, ions, etc.), molar mass, charges, specific heats ratio
(Cp/Cv),
thermodynamic data, transport coefficient data for collision integrals calculation, etc.
plasreac.dat:
This file contains the plasma chemistry model: total number of reactions, the number and
the indices of the reactions where electrons significantly dissipate their energy (line 1),
181
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
and the energy threshold of these reactions (lines 2 and 3). Then the following
information is present for each reaction (as shown in the example below):
1. The reaction equation;
2. Three integers indicating if the reaction involves a third body, the reaction is reversible
and it follows Arrhenius law.
3. The activation mode of the reaction: gas temperature, electron kinetics mode,
vibrational mode, or the mixed vibration-gas mode, (the last mode does not play any role
for Ar and H2 plasmas. It was done for N2 plasma).
4. Curve-fitted coefficients for rate constant calculations. For electron mode activated
reactions, the curve-fitted equation comes from the solution of the electron Boltzmann
equation. Otherwise, the coefficients correspond to the Arrhenius form.
5. LHS Stochiometric coefficient for all the species.
6. RHS Stochiometric coefficient for all the species.
Exam ple:
T1
e- + H2 = 2e- + H2+
000
0 100
-.236880D+04 ,394732D-K)3 .187693D+07 -677875D+10 .673533D+13 -.163752D+02
10000000 1
0 0 0 0 0 1002
plassurf.dat:
This file gives:
•
The number of catalytic reactions;
•
Recombination coefficients on the reactor wall;
•
Recombination coefficient on the substrate surface;
•
Then for each recombination reaction we have three lines:
182
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
•
The species;
•
LHS Stochiometric coefficient of the recombination reaction;
•
RHS Stochiometric coefficient of the recombination reaction.
plasnrje.dat:
This file gives curve-fitted coefficients for the elastic cross-section for calculations of the
energy transfer between the translation modes of electron and atoms/molecules;
•
Geometrical Data
maillage.dat:
This files contains the computational grid information:
1. Number of grids in the z direction;
2. Number of grids in the r direction;
3. An integer indicating if the cavity has a cylindrical (1) or Cartesian (0) geometry;
4. An angle for axisymetrical computation;
5. The z positions of the grid points;
6. The r positions of the grid points;
7. Integers indicating blocked off region (solid region). This is done to treat the
complicated geometry or reactors with substrate in stagnation point configuration without
using generalized grid system (Note that the grid may be, and is in fact, not regular).
•
MWPD Input
powtim.out:
This file gives the microwave power density absorbed at each grid point.
•
Num erical Data
plasmain.dat:
183
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
The significance of different symbols is given in the comments of the file. Some of the
important things to note are the following:
The first line contains general information on the discharge conditions: pressure, wall and
substrate temperatures. It also contains the values of temperatures (electron gas and
vibration) which have to be used during the initialization procedure. These values,
especially for Te, are of prime importance for the convergence of the calculation. The
general requirement for the electron temperature initial values is to put a high value in the
discharge region (where the absorbed power density is high) and a low value elsewhere.
The next nine lines contain the initial values inside and outside the discharge region as
well as the inlet values for species molar fraction. Here it is also required to initialize the
electron molar fraction with a high value (10'7-10"5) in the discharge.
Note:
Set IECOU to 0 because the present version of the code does not treat the convection.
numecou.dat:
Most of the numerical data contained in this file should not be changed except in the case
with serious convergence problems. The only exception is the integer IREP that has to be
set to 0 for a new calculation and to 1 to continue the calculation for better convergence.
IREP is the sixth data (integer) in the second line. Generally IREP is changed routinely
from one simulation to another.
plasnum.dat:
This file is one of the most important files. It contains the relaxation parameters. The
values chosen for these parameters will make the computation successful or not.
The first line contains:
NITPM: the number of total iterations (between 400 and 2000, depending on the
discharge conditions, is sufficient to achieve the convergence);
NITPD: the interval of iterations for the calculated field and other flow characteristics to
be saved;
184
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
NITPE: the calculated field will be saved each NITPD iterations;
NRED: put NRED equal to NITPM;
XRED: not important for the moment;
RPLM & TCPLM: Total residue and convergence rate for all the plasma transport
equations (not used);
The remaining lines give the relaxation factors used during the numerical solution of the
total energy equation, the electron energy equation, the vibration energy equation and
the species equations. For example, the relaxation factors for the total energy equation
are:
RTTM, TCTTM, NGSTTM, FRTTO, FMFRTT, FRTTM, FRGTTU, FRGTTO
The most important parameters that will have to be changed are:
TCTTM: maximum allowed rate of change of the gas temperature, which is the unknown
of the total energy equation;
FRTTO: Initial relaxation factor for the total energy equation;
FRTTM: Maximum relaxation factor for the total energy equation.
Important Notes:
The convergence of the code depend of the choice of these three parameters (for each
equation):
- The initial relaxation factor FRTTO has to be quite low to prevent divergence of the code
in the early stage of the relaxation. The initial fields in general are far from the converged
one and the conservation equation may show a lot of stress and high instability during the
first 100 iterations.
- The maximum relaxation factor must not exceed 1.0. In practice a value of 0.5-0.8 is
OK. Instability may appear above these values.
- The maximum allowed rate of change is very important. This parameter has two main
tasks:
•
To prevent the instability during the first stage of the relaxation;
185
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
•
When the convergence starts to be achieved, small oscillations of the plasma
temperatures and species mole fraction will take place if the relaxation
factors are maintained high (0.5 for example). The residues of the equation
will also slightly oscillate, due to the strong non-linearity and stiffness of the
system. A method to check the convergence validity is to adopt a smaller
value for TCTTM and to restart the calculation from the approximately
converged field. If the convergence is OK, a strong decrease of the residues
will be observed.
numnrj.dat:
Not important (used during the development phase).
B.1.3 Output Files
Practically almost all the output files are generated by the subroutines in
the file 'ecrver.for*. The routinely used output files are: 'champs.out1, 'champs.in1,
'plasnum.out', 'therplas.out' and 'speplas.out'. The other files w ere used only
during the development phase or when changing the grid meshes,
champs.out & champs.in
These files are used to save the currently calculated fields and to restart the calculation,
respectively.
Suppose that a calculation has been completed for 400 iteration and the convergence
was not completely achieved. The currently calculated filed for 400 iteration is saved in
’champs.out'. To continue the iteration:
- Copy 'champs.out' to 'champs.in'
- Change the number of iteration NITPM in 'plasnum.dat' (for example if NITPM is
changed from 400 to 800, the code will run until 800 starting from 400.
- Change IREP from 0 to 1 in the file 'numecou.dat'. This tells the code to read the initial
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
data in the file 'champs.in'.
plasnum.out:
This file is very important for tracking and checking the convergence. For each
temperature and each species, it reports the rate of change, the residue of the
corresponding equation, and the currently used relaxation coefficients,
plasemf.in:
This file provides detailed information about the plasma discharge, which is the input file
for the EM model,
therplas.out & speplas.out
These two files report the temperature distribution (therplas.out) and species mole
fraction fields (speplas.out), respectively,
interpolmp.out, interpolmu.out & interpolmv.out
The solution of the plasma equations makes use of staggered grid systems (insuring the
treatment of convective or quasi-convective terms). These files were used to check the
accuracy of the interpolation between the three grid systems: main, Oz staggered and Or
staggered grid systems.
Jacmp.out, Jacmu.out & Jacmv.out
Check the accuracy the Jacobian of the geometrical transformations from non-regular
(physical) to regular (indices) grid (for spatial derivative computation) for the main and the
staggered grids.
mailp.out, mailu.out & mailv.out
Check the behavior of the grid meshes for the main and the staggered grids,
plassouhs.out
Report the quasi-source term which appears after the discretization of the total energy
equation and which corresponds to the part of the diffusion of enthalpic species involving
only the species formation enthalpy,
plasco.out
187
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Report the values of the coefficients of the quasi-linear system resulting from the
discretization of the transport equations.
B.1.4 How to Run the 2-D Fluid Model
To run the 2-D fluid model with W A T C O M IDE (Integrated Development
Environment), the procedure listed beiow should be followed:
I.
Make sure all the source codes and input data files are in the same directory. For
the source codes, the common block files should also be included in the same
directory.
II.
Open the WATCOM IDE.
III.
If the project has already created, go to the T ile ' menu and choose ‘open project’.
Then browse the directories and choose the project name.
IV.
If this is a new project, go to the tile’ menu and choose ‘new project’. The system
will prompt you for the project name. The default target environment' is 'Win32
(NT/95/32s)’. Set the ‘Image Type’ to ‘Windows Executable (.exe)’. Then click
’OK’. A window for the new project will pop out.
V.
In the blank space for the source file, click the right button of the mouse, a menu
will pop up. Then go to ‘new source’. Choose the source codes needed for this
project.
VI.
Right click the mouse again and go to ‘source options’. Choose 'Fortran Compiler
Switches’. Then click on the ' » ’ button until '9 Application Type Switches’. Set
this switch to ‘Default Windowed Application’. In the source file window, the ‘[sw]’
sign will be shown with the files.
VII.
Go to Target’ menu and select ‘make’. Or click ‘F4’ on the keyboard to compile
the source codes. If ail the data files are set correctly and the compilation is
passed successfully, you then can run the program by clicking ‘ctri-R’.
188
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
VIII.
To start a new calculation, make sure the parameter IREP in ‘numecou.dat’ is set
to 'O'. Otherwise, set the value to ‘1 ’ to read in the MWPD profile from the EM
model.
IX.
The codes will be coupled with the EM model. For better convergence, ten or
more iterations between these two models may be needed. For each iteration,
make sure the 'champs.out’ file has been copied to ‘champs.in’.
189
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
B.2
2-D Electromagnetic Model
The codes for the 2-D EM model were also written in the W A TC O M
FO RTR AN format. Following is a short description of the codes used in the 2-D
EM Model.
interfield.for:
This file interpolates the electrical field information from EM model simulation grids to the
fluid model simulation grids.
interflu.for:
This file interpolates the electron number density, electron temperature, gas temperature
and other information from the 2-D fluid model to the EM model.
interpwd.for:
This file interpolates the absorbed microwave power density distribution from the EM
model to the fluid model.
main2.for:
This is the main control program for the 2-D EM model. The fluid model can be called in
this program if needed.
max.for:
This routine uses the FDTD method to solve for the electrical field in the simulation
domain. It is the most important file in the EM model.
readem.for:
This file reads in the EM simulation information from ‘EMGEOM.IN’.
The input files for the EM model are:
plasemf.in:
This file contains all the plasma discharge results calculated from the 2-D fluid model,
which are needed to start the EM model simulation.
EMGEOM.IN:
This file provides the geometrical information for the EM simulation region.
190
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
The output files are:
powtim.out:
This file contains the MWPD information that will be used in the 2-D fluid model,
eltim.out:
This file contains the electrical field information,
geo.flu:
This file provides the geometric data for the fluid model,
geo.max:
This file provides the geometric data for the EM model.
ver_fd.out:
This file contains the fluid model data. The purpose of this file is to verify the 2-D fluid
results used in the EM model are consistent with the original results.
maxpwd.save:
This file saves the MWPD information that is in Origin format,
maxfield.save:
This file saves the electric field data that is in Origin format.
Other data files used in the EM model are either
for the debugging
purpose or to save the intermediate results during the calculations.
To run the 2-D EM model, the same instruction as in the 2-D fluid model
should be followed. The microwave power can be changed in the ‘main2.for’ file.
The param eter in ,main2.for’ representing microwave pow er
example, to set the microwave power to 30 W, the line should
is 'watt'.
be like:
watt = 30.0
191
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
For
The iteration number for the EM model simulation can be changed in
‘m ax.for\ The parameter to change is ‘niter1. M ore iteration will have better
results for the EM simulations. However, the EM model is coupled with the fluid
model. To achieve a converged solution, more iteration between these two
models is needed. Thus more calculation time with only one simulation may turn
out to be less effective.
192
Reproduced with permission o f the copyright owner. Further reproduction prohibited without permission.
B.3
2-D CRM
Due to the fact that E LE N D IF (the Boltzmann Solver) was written in
Fortran 90 format, the 2-D CRM codes were written in either Fortran 90 format or
C format. The interfaces between the Fortran 90 codes and C codes w ere then
implemented to ensure the correct data flows.
The source codes of the 2-D CRM are as follows:
arcrm.c:
This file contains all the routines for the argon collisional-radiative model. It reads EEDF
from the Boltzmann solver. The excited state number densities and the absorbed
microwave power densities are calculated in this file. This file can also provide the
reaction rates directly from the CRM, instead of from the curve-fitted rates,
am ovalue.c:
Due to the fact that the Boltzmann solver cannot handle extremely low electron number
densities (<1e16 m'3) and E/N values (<0.01 Td), this routine is added to avoid calling the
Boltzmann solver. Then the excited state number densities are set to zeroes. No reaction
rates are calculated.
crm2d.f:
This file is the main driver for the 2-D CRM. The plasma parameters are read in this
routine.
ebyn_0.f:
This is the Boltzmann solver used in 2-D CRM. The source code is from ELENDIF. In this
file, the CRM routine is called after the EEDF is calculated.
The input files for the 2-D CR M are:
ar2.inp:
193
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
This is the input file for argon CRM. In this file, some switches are specified to check the
physical meanings. Some of the data in this file may be overwrite inside the routines.
cvdlib.dat:
This file is a library file for the CVD processes. Most species used in CVD processes can
be found in this file,
elencom.inc & in_outinc:
These two files are used for ELENDIF. Some common blocks are specified here,
eltim.out:
This file contains the electric field information from the 2-D EM model. The data is not
directly used in the 2-D CRM.
Input file for ELENDIF, the Boltzmann solver. This files specifies some default values for
the electron number density, electron temperature, etc. This file also specifies time step
and other parameters for ELENDIF to achieve a converged solution. An example is as
follows:
Arcase
names of chemical species, fractional concentrations. & ionization fractions:
Ar 1.0 0.0 /
End_Species /
End_Pops /
Press_Torr
5
/ gas pressure in Torrs
T_Electron 1.2
/ initial electron temperature in eV
N_Electron lel2
/ in cm-3
d Max_Tsteps 100
/ maximum number of time steps
d N_Cycles
1
/ number of AC cycles to run the calculation
Max_Tsteps 100000
/ maximum number of time steps
c/Time_Step
l.e-11 / time step in seconds (AC case)
Time_Step l.e-7
/ time step in seconds
N_Print
50
/ print every 5 time steps
DC
10.0 0.0 0.0 / 50 Td DC electric field
c/AC
le-1 2.54e9
0.0 / 50 Td DC electric field
Du_Umax 0.20 20.0
/ 100 grid points between 0 and 20 eV
End Data
/
sigma.dat:
Another library file containing plasma physics and chemistry data for most species used
194
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
in plasma processes,
speplas.out:
Input file of 2-D CRM. This actually is the output file from 2-D fluid model, which contains
the spatial species distributions.
thermplas.out:
Input file of 2-D CRM. This is the output file from 2-D fluid model, which contains the
spatial temperature distributions.
The output files are:
amew.out:
Output file for argon CRM for debugging and tracing the program.
excite.dat:
This file contains the spatial excited state number densities,
fek.out:
This file contains the information of EEDF calculated by the Boltzmann solver.
NU.out:
Used in the development phase. This file contains the calculated collision frequency used
for curve-fitting of the electron-neutral energy exchange term.
Out:
Output file of ELENDIF. This file contains almost all the information from the Boltzmann
%
solver.
Pabs2d.dat:
This file contains the predicted absorbed microwave power from 2-D CRM.
rate.dat:
This file contains the information of the rates used in 2-D fluid model, which are
calculated directly in 2-D CRM and can be used in the fluid model to reduce uncertainties
caused by curve-fitting.
\ v e r files:
195
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
These files are for verification.
To run the 2-D CRM, the following procedure should be followed:
1.
Make sure all the input files and library files are copied to the working directory.
2.
Compile the source code as follows:
f90 -c crm2d.f
f90 -c ebyn_0.f
cc -c arcrm.c
cc -c amovaIue.c
f90 -o YOUR_PROGRAM crm2d.o ebyn_0.o arcrm.o amovalue.o
3.
Run YOUR_PROGRAM (the executable program).
196
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Numerical and Experimental Investigation of
Nonequilibrium Microwave Argon Plasmas
Abstract of dissertation submitted in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
By
Yunlong Li, B.S.M.E., M.S.M.E.
Xian Jiaotong University, Xian, China, 1992, 1995
May 2000
University of Arkansas, Fayetteville
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
This abstract is approved by:
DISSERTATION DIRECTORS:
Dr. l^ rry Roe
-----------Dr. Matthew Gordon
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
ABSTRACT
Optical emission spectroscopy (O E S) and absorption m easurements have
been used in tandem with numerical models to characterize microwave argon
plasmas. A W A V E M A T (model M P D R -3135) microwave diamond deposition
system was used to generate an argon plasma at 5 Torn. Three excited state
number densities (4p, 5p, and 5d) were obtained from the O E S measurements.
However, the numerical predictions match the experimental data only with
changed coefficients. The numerical results also showed that only a small
amount (2-5% ) of the metered microwave energy (680±40 W ) w as absorbed by
the argon plasma.
An energy balance study showed that the energy absorbed by the argon
plasma was far less than the metered power, in agreement with the pseudo-1-D
predictions. A global energy balance study showed that the bulk of the energy
(3 2 0 ± 8 0 W ) was dissipated in the base-plate/applicator water-cooling line. Only a
small amount of energy (about 10W ) was dissipated into the substrate. A controlvolume
heat transfer model
was constructed for a
better understanding.
Combining results from the global energy balance study and the control-volume
heat-transfer model, only 10-60W of the microwave energy can be absorbed by
the argon plasma.
1
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
A 2-D self-consistent fluid model coupled with a 2-D electromagnetic
model was constructed for the microwave argon plasmas. High non-uniformity is
observed in the microwave argon plasma. The 2-D fluid model also provides
information of the heat fluxes along the belljar walls and substrate surface.
Comparing with results of the energy balance study, only the cases with
microwave power less than 50 W can have the predicted heat transfer rates
consistent with the energy balance study.
Further experimental work includes absorption m easurements of the
m etastable state (4s) number density. A 2-D Collisional-Radiative Model was
developed along with the 2-D fluid model to predict the spatial excited state
number densities. Good agreement was achieved between the 2-D
predictions and the OES/absorption measurements.
9
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
CRM
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Документ
Категория
Без категории
Просмотров
0
Размер файла
7 099 Кб
Теги
sdewsdweddes
1/--страниц
Пожаловаться на содержимое документа