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Diagnostics and modeling in nonequilibrium microwave plasmas

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DIAGNOSTICS AND MODELING IN NONEQUILIBRIUM
MICROWAVE PLASMAS
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DIAGNOSTICS AND MODELING IN NONEQUILIBRIUM
MICROWAVE PLASMAS
A dissertation submitted in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
by
UMESH M. KELKAR, B.S.M.E., M .S.M.E.
University of Bombay, India, 1991
University of Arkansas, 1994
December 1997
University of Arkansas
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UMI Number: 9820799
UMI Microform 9820799
Copyright 1998, by UMI Company. All rights reserved.
This microform edition is protected against unauthorized
copying under Title 17, United States Code.
UMI
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This dissertation is approved for
recommendation to the
Graduate Council
DISSERTATION DIRECTOR:
Dr. Matthew Gordon
DISSERTATION COMMITTEE:
Dr. Michael StewartV
Dr. Richard Ulrich
f . Min Xiao
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TABLE OF CONTENTS
Page
List o f ta b le s ................................................................................................................................viii
List o f f ig u r e s ............................................................................................................................... ix
CHAPTER I INTRODUCTION
(1.1) In tro d u c tio n ........................................................................................................
(1.2) Fundamental Plasma P ro c e sse s .......................................................................
(1.3) Forms o f N o n e q u ilib riu m ................................................................................
(1.3.1) Kinetic Nonequilibrium .................................................................
(1.3.2) Ionization N onequilibrium ...........................................................
(1.3.3) Chemical N o n e q u ilib riu m ..............................................................
(1.3.4) Maxwellian N onequilibrium ......................................................
(1.4) Related W o r k ...................................................................................................
(1.4.1) Modeling ........................................................................................
(1.4.2) D ia g n o s tic s .....................................................................................
(1.5) D issertation O v e r v ie w ..................................................................................
CHAPTER II
1
4
7
8
9
9
10
10
11
15
19
EXPERIMENTAL FACILITY
(2.1) M icrowave Plasma R e a c to r ..........................................................................
(2.2) Em ission S y s te m ..............................................................................................
23
25
CHAPTER III DIAGNOSTICS TECHNIQUES
(3.1)
(3.2)
(3.3)
(3.4)
In tro d u c tio n ......................................................................................................
Absolute Line Intensities .............................................................................
Relative Line In te n s itie s ................................................................................
Absolute Continuum Intensity ....................................................................
30
31
35
35
CHAPTER IV MODELING
(4.1) In tro d u c tio n ....................................................................................................
iii
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42
(4.2) Electron Energy D istribution F u n c tio n ...................................................... 42
(4.3) Collisional-Radiative M o d e ls ....................................................................... 44
(4.3.1) Cross-section and Reaction Rate Data for a 21-Level H CRM 45
(4.3.2) Cross-section and Reaction Rate D ata for a 25-Level A r CRM 49
(4.4) Comprehensive H ydrogen M o d e l .............................................................. 49
(4.5) Electron Conservation E q u a tio n ................................................................. 56
(4.6) Power Balance E q u a tio n ............................................................................... 57
(4.7) Methods o f Solution ..................................................................................... 61
CHAPTER V
CHARACTERIZATION OF PLASMAS
(5.1) In tro d u c tio n ......................................................................................................
(5.2) Results and D isc u ssio n ..................................................................................
(5.3) Summary .........................................................................................................
65
66
89
CHAPTER VI ROLE OF EXCITED STATES
(6.1) In tro d u c tio n ...................................................................................................... 90
(6.2) Argon Plasma ................................................................................................ 92
(6.3) Hydrogen Plasma ..............................................................................................103
CHAPTER VII IMPORTANCE OF FREE-FREE CONTINUUM EMISSION
(7.1) In tro d u c tio n .........................................................................................................I l l
(7.2) Results and D isc u ssio n ..................................................................................... 112
CHAPTER VIII ENERGY BALANCE
(8.1)
(8.2)
(8.3)
(8.4)
(8.5)
In tro d u c tio n .........................................................................................................119
Power B a la n c e ................................................................................................... 120
Control Volume Heat Transfer A n a ly sis...................................................... 125
Global Reactor Energy Balance .................................................................... 130
Self-consistent S o lu tio n s .................................................................................. 135
CHAPTER IX DISSOCIATION MODEL RESULTS
(9.1) In tro d u c tio n .........................................................................................................142
(9.2) Parametric S tu d y ................................................................................................ 143
iv
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CHAPTER X CONCLUSIONS AND RECOMMENDATIONS
(10.1) Conclusions..............................................................................................148
(10.2) Recommendations ................................................................................... 150
REFERENCES
............................................................................................................. 152
PUBLICATIONS AND PRESENTATIONS ............................................................ 158
APPENDIX A SPECTROSCOPIC DATA
(A. 1) Atomic hydrogen......................................................................................160
(A.2) Argon ........................................................................................................ 160
APPENDIX B PROGRAM LISTING
(B.l) H ydrogen................................................................................................... 161
ABSTRACT
v
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LIST OF TABLES
Table
Page
3.1
Local Thermodynamic E quilibrium Temperatures (H2/C H 4 P la s m a )
36
3.2
Local Thermodynamic E quilibrium Temperatures
4.1
H ydrogen Model Reactions
5.1
Range o f Parameters Studied in H 2/C H 4 Plasma
.............................
67
5.2
Range o f Parameters Studied in A rgon P l a s m a .................................................
67
6.1
Com parison o f num erical and experimental excited states (A r plasm a) . . . 103
8.1
Experimental Enthalpy D ata in H2/C H 4 P la s m a ....................................................135
8.2
Experimental Enthalpy Data in A r P l a s m a ............................................................ 135
(Ar P la s m a ) .................
36
...................................................................................
50
VI
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LIST OF FIGURES
Figure
Page
1.1
Schematic o f diamond d e p o s itio n ..............................................................................
3
1.2
Fundamental plasm a processes
.................................................................................
5
2.1
Schematic o f W AVEM AT microwave plasma r e a c to r ......................................
24
2.2
Schematic of em ission s y s t e m .................................................................................
27
2.3
Optical emission scan in a hydrogen/methane p l a s m a ......................................
28
2.4
Optical emission scan in an argon plasma
..........................................................
28
3.1
Partial energy level diagram o f atomic h y d ro g e n ...............................................
33
3.2
Partial energy level diagram o f a r g o n ..................................................................
33
3.3
Experimentally measured excited states in the FF/CHj plasma
.....................
34
3.4
Experimentally measured excited states in the Ar plasma
.............................
34
3.5
A Boltzmann plot showing the presence o f nonequilibrium in the H-,/CH4
p la s m a .........................................................................................................................
38
3.6
A Boltzmann plot showing the presence o f nonequilibrium in the Ar plasm a 38
4.1
H olstein's escape factor as a function o f optical depth
4.2
Inelastic energy loss factor for H;
4.3
Iterative scheme used for the parametric study
4.4
Iterative scheme used to obtain self-consistent solutions
...................................
48
........................................................................
60
.................................................
................................
vii
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63
63
4.5
Iterative scheme for a comprehensive hydrogen m o d e l ..................................
5.1
Com parison o f a numerically predicted non-M axwellian EED F and a
M axwellian EED F with the average energy in a H 2/C H 4 p la s m a .................
5.2
70
Com parison o f a numerically predicted non-M axwellian EED F and a
M axwellian EED F w ith the average energy in an A r plasma .......................
5.3
64
70
A Boltzmann plot com paring excited states predicted by using
a non-M axwellian E E D F and those predicted by using a Maxwellian
EED F with the same average energy in a H2/C H 4 plasma
5.4
............................
71
A Boltzmann plot com paring excited states predicted by using
a non-M axwellian E E D F and those predicted by using a Maxwellian
EEDF with the same average energy in an A r p la s m a .............................
71
5.5
The effect o f electron num ber density on the EED F in a H2/C H 4 plasma . . 73
5.6
The effect o f electron num ber density on the EED F in an Ar plasma . . . .
5.7
A Boltzmann plot showing the effect o f electron number density
on the excited states in a H2/C H 4 p l a s m a ...........................................................
5.8
.................................................................
77
A Boltzmann plot showing the effect o f vibrational superelastic
collisions on the excited states in a H2/C H 4 plasm a
5.11
75
The effect o f vibrational superelastic collision on the EEDF in a H:/C H 4
p la s m a ..........................................................................................................................
5.10
75
A Boltzmann plot showing the effect o f electron number density
on the excited states in an Ar plasma
5.9
73
.......................................
The effect o f H2-superelastic electronic collisions on the EEDF in aH2/C H 4
viii
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77
p la s m a ..........................................................................................................................
5.12
79
A Boltzmann plot showing the effect o f H2-superelastic electronic
collisions on the E E D F in a H 2/C H 4 p la s m a ......................................................
79
5.13
The effect of electric field on the EEDF in a H2/C H 4 p l a s m a ........................
81
5.14
The effect of electric field on the EEDF in an A r plasm a
81
5.15
..............................
A Boltzmann plot showing the effect o f electric field on the excited
states in a H2/C H 4 p la s m a ........................................................................................
5.16
82
A Boltzmann plot showing the effect o f electric field on the excited
states in an A r p l a s m a ..............................................................................................
82
5.17
The effect of gas tem perature on the EEDF in a H-./CH, p l a s m a ..................
84
5.18
The effect of gas tem perature on the EEDF in an A r plasma
84
5.19
........................
A Boltzmann plot show ing the effect o f gas tem perature on the excited
states in a H2/C H 4 p la s m a ........................................................................................
5.20
85
A Boltzmann plot showing the effect o f gas tem perature on the excited
states in an Ar p l a s m a ..............................................................................................
85
5.21
The effect of H-mol fraction on the EEDF in a H2/C H 4 p la s m a ......................
87
5.22
Electron energy loss processes in a H2/C H 4 plasma
.........................................
88
5.23
Electron energy loss processes in an Ar p la s m a ..................................................
88
6.1
Electron production and loss rates in the Ar p l a s m a ..........................................
94
6.2
Self-sustaining average electron temperature as a function o f
electron number density with and without two-step ionization
in the Ar plasma
.......................................................................................................
i.\
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94
6.3
A Boltzmann plot comparing the numerical and experimental
excited state number densities in the Ar p la s m a ................................................
6 .4
96
Self-sustaining average electron temperature as a function o f gas
temperature with and without two-step ionization in the Ar p l a s m a
6.5
Two-wise io n iz a tio n ..................................................................................................
6 .6
Self-sustaining average electron temperature as a function o f pressure
96
99
with and without two-step ionization in A r plasmas (Mak et al.. 1997) . . . 100
6 .7
The effect o f non-M axwellian EEDF on the self-sustaining average electron
temperature in low pressure A r plasmas (M ak et al.. 1 9 9 7 ) .......................... 100
6.8
Electron production and loss rates in the H 2/C H 4 p la s m a ....................................104
6.9
Self-sustaining average electron temperature as a function o f
electron number density with and without two-step ionization in
the H 2/CH 4 p l a s m a ...................................................................................................... 104
6.10
A Boltzmann plot comparing the numerical and experimental excited state
number densities in the H2/C H 4 p la s m a ................................................................. 107
6.11
The effect o f collisional quenching on the excited states in H2/C H 4 plasm a
6.12
Self-sustaining average electron temperature as a function o f
107
H-mol fraction in the H :/C H 4 plasma .................................................................... 110
6.13
Self-sustaining average electron temperature as a function o f
gas temperature in the H2/C H 4 p la sm a .................................................................... 110
7.1
Contributions to the total continuum emission as a function o f
average electron tem perature in the H2/C H 4 p l a s m a ...........................................113
x
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7.2
Contributions to the total continuum emission as a function o f
average electron temperature in the Ar plasma ................................................... 113
7.3
Continuum emission as a function o f electron num ber density for various
average electron temperatures in the H2/CH4 plasm a
7.4
....................................... 114
Continuum emission as a function o f electron num ber density for various
average electron temperatures in the A r p la s m a ................................................ 114
7.5
Continuum emission as a function o f electron num ber density for
various gas temperatures in the H2/C H 4 p la s m a ....................................................116
7.6
Continuum emission as a function o f electron num ber density for
various gas temperatures in the A r p la s m a ............................................................ 116
7.7
Continuum emission as a function o f electron num ber density for H-mol
fractions in the H 2/C H 4 plasm a
7.8
............................................................................... 118
Com parison of the numerical and experimental continuum em ission
for two different wavelengths in the H ;/CH 4 p la s m a .......................................... 118
7.9
Com parison of the numerical and experimental continuum emission
with and without inelastic energy loss factor in the H 2/C H 4 p l a s m a
8.1
Pow er deposition as a function o f electron number density for various
average electron temperatures in the H2/CH4 plasma
8.2
120
........................................125
Pow er deposition as a function o f electron number density for various
average electron temperatures in the A r p la sm a ................................................... 125
8.3
Pow er deposition as a function o f electron number density for various gas
tem peratures in the H2/C H 4 p la s m a ......................................................................... 127
xi
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8.4
Power deposition as a function o f electron number density for various gas
temperatures in the A r p la s m a .................................................................................. 127
8.5
Power deposition as a function of electron number density for various
H-mol fractions in the H:/CH 4 p la s m a .................................................................... 128
8.6
Control volume heat transfer a n a ly s is ......................................................................130
8.7
The heat transfer analysis results in the H;/C H 4 p la s m a .......................................133
8.8
The heat transfer analysis results in the A r p la s m a ............................................... 133
8.9
Experimental enthalpy data in the H:/C H 4 plasma ............................................... 136
8.10
Experimental enthalpy data in the A r plasma
8.11
Experimental enthalpy data in H:/C H 4 plasmas with various
pressures and input power
........................................................136
........................................................................................138
8.12
Self-consistent solutions in the H:/C H 4 plasma ..................................................... 140
8.13
Self-consistent solutions in the H:/C H 4 plasma .....................................................141
9.1
Species concentrations as a function of the electron number d e n s ity
9.2
The effect o f gas temperature on the H mol fraction for various
144
electron num ber d e n s itie s ...........................................................................................144
9.3
The effect o f net diffusion loss coefficient on the H mol fraction for
various gas tem peratures
9.4
...........................................................................................146
The effect o f pressure on the H mol fraction for various
gas tem peratures
.........................................................................................................146
xii
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CHAPTER I
INTRODUCTION
(1.1) Introduction
Plasm a technology plays an important role in the miniaturization o f
sem iconductor chips and the processing o f novel m aterials. Plasma assisted materials
processing techniques have widespread applications w hich include plasma-enhanced
chemical vapor deposition (PECVD) and etching. Low pressure microwave plasmas
have distinct advantages for such processing because o f their potential for high charged
particle density, higher gas and energy efficiency, scaling, and minimal contamination
(M oisan et al., 1991). Additionally, low pressure m icrow ave plasmas are used in
downstream electron-cyclotron resonance (ECR) configurations to provide simultaneous
control over ion energy and ion fluxes arriving at the substrate. The ability to
sim ultaneously control the ion energy and ion fluxes facilitates higher etch/deposition
rates with minimal plasma induced damage. Several com panies such as Tokuda.
Plasma-Therm. Astex. Wavemat. Mattson Technology Instruments. Materials Research
Corporation, and Electrotech have recently produced com ponents or entire systems.
1
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specially designed to create plasmas at microwave frequencies (2.45 GHz). Although the
use o f m icrowave plasma assisted reactors is common, a detailed understanding o f the
various physical processes in complex plasm a environments is far from complete. Such
an understanding is difficult to achieve due to inherent nonequilibrium and the difficulty
in accurately probing this harsh environment.
D iam ond thin films have recently become the subject o f significant technological
interest because o f their remarkable electrical, thermal, and optical properties. Some o f
the important properties o f diamond (Yoder. 1991) are: highest therm al conductivity (20
W/cm-K at room temperature) and hardness (9000 kg/m nr). wide optical bandgap (5.45
eV). high electrical resistivity (1015 ohm -cm ). low coefficient o f friction (0.05-0.7). low
dielectric constant (5.5). and chemical inertness. The synthetic diam ond films are
deposited by a m yriad of techniques such as hot-filament. DC arcjet. and plasm a assisted
chemical vapor deposition. Several researchers used microwave reactors for depositing
polycrystalline diamond films (Bou et al.. 1992: Hyman et al.. 1992: Koemtzopoulos et
al.. 1993: Tahara et al.. 1995: Scott et al.. 1996) because microwave plasm as can
produce high quality diamond films.
Figure 1.1 depicts a diamond deposition process in a plasm a reactor. A plasma is
generated by coupling microwave pow er to a gas mixture o f hydrogen and methane.
The electrons activate the gas mixture by producing atoms and radicals in their ground
and excited states. Once this mixture is activated, the products transport to the substrate
by a com bination o f convective and diffusive effects, with reactions continuing to
modify the concentrations o f the various species present. The gas phase reactions are
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Reactants.
H2 + CH4
Activation
H2e- at2H
CH4 + H =* CH3 + H
Flow and Reaction
Diffusion
y
A d so rp tio n
Figure 1.1 Schematic o f Diamond Deposition
3
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coupled to the adsorption and desorption reactions at the surface. Diffusion along the
growth surface leads to diamond crystal nucleation and growth. Diffusion into the bulk
material completes the deposition process. Som e o f the numerous parameters which
govern this complex diamond deposition process include substrate temperature, cham ber
pressure, gas flow rates, species concentrations in the gas phase and their fluxes at the
surface, and energy input. Although reaction m echanisms vary for different activation
techniques, it has been well accepted that atom ic hydrogen plays a significant role in the
process for producing precursor radicals and suppressing the formation o f graphite
(Spear and Frenklach. 1989; Harris. 1989). To meet the demand for depositing diam ond
films over larger areas and at higher efficiency and growth rates, a detailed know ledge o f
the physical processes in the gas-phase o f the microwave discharge is necessary.
(1.2) Fundamental Plasma Processes
Plasma discharges consist o f electrons, ions, and ground and excited state neutral
particles. Figure 1.2 shows various energy transfer processes in a weakly ionized
plasma. Initially, only a few electrons and ions, due to the ionization by cosmic rays or
by ultraviolet photons, are present (Rossnagel et al.. 1994). The input pow'er (DC. RF.
Microwave) accelerates these charged particles under the influence o f the applied
electric field. The accelerated electrons induce high energy reactions such as
dissociation, excitation, and ionization. The ionization reactions produce electrons, and
recombination in the gas-phase or convection and diffusion consume electrons. In
steady state, a balance is established between electron production and loss rates to
4
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Radidon Losses
^ fS S d u c t,'
— Qjovectfo,,
Electric
Energy
Source
Joule Heating
W Electron
Gas
Elastic and
Inelastic Collisions
Discharge
Reactor
Wall
Recombination
of Ians «nH
Electrons
Free Radical
Recombination
Neutral W
Gas
Heat Conduction
and Convection
Radidon Losses
Figure 1.2 Fundamental Plasma Processes
5
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maintain charge neutrality the plasma.
The energy imparted to the electrons is much greater than the energy imparted to
the ions because work done on the charged particles by an electric field between
collisions varies inversely with the particle mass. The weak coupling between the
applied electric field and ionic gas is represented by dashed line in Figure 1.2. The
electron energy distribution depends on the nature o f the gas (atomic or molecular),
operating pressure, and strength o f the applied electric field. In equilibrium, the
electrons follow' a M axwellian electron energy distribution function (EEDF). The
electron energy gain from the applied electric field, the various electron energy loss
processes and the operating conditions determine the nature o f the electron energy
distribution function (EEDF) and the value o f the self-sustaining electric field in the bulk
plasma. Since the electrons and ions follow a non-Maxwellian energy distribution, the
plasmas are described by average electron and ion energy. We will use both electron
volts (eV) and Kelvin (K) to describe the electron energy and electron temperature.
Approximately. 1 eV corresponds to 11.650 K.
The energy from the electrons is transferred to the neutral particles by collisions.
The electron-neutral collisions plav a dominant role and govern the plasma chemistry.
The electron-neutral collisions can either be elastic, in which the internal energy o f the
neutral particles remains the same, or inelastic, in which the electrons dissociate, excite,
or ionize the neutral species. The electron energy transferred in each elastic collision is
proportional to the ratio o f electron and neutral particle masses. Since the electron mass
is about four orders o f m agnitude smaller than the neutral mass, a very small am ount o f
6
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energy is exchanged in elastic electron-neutral collisions. Because o f the weak energy
coupling between the electrons and neutrals, the plasmas are generally characterized as
nonequilibrium plasmas in which the electron temperature is much higher than the
neutral temperature. The inelastic electron-atom processes include electronic excitation
and ionization whereas electron-molecule reactions include rotational, vibrational and
electronic excitation and ionization. The high energy electrons cause these inelastic
reactions and produce the active atoms and radicals which are essential for the plasma
chemistry. Thus, the selective heating o f the electrons by the applied electric fields, the
weak elastic energy transfer between the electrons and neutrals, and the efficient
production o f atoms and radicals by the high energy electrons facilitate low temperature
plasma processing.
(1.3) Forms o f N onequilibrium
To understand plasm a processes, knowledge o f the thermodynamic state o f the
plasma is essential. Plasm as in complete thermodynamic equilibrium (CTE) can be
described solely by tem perature and pressure. W hen all the microscopic processes in the
plasma are in detailed balance, including the radiative field, the plasma is said to be in
CTE. For a plasma to be in CTE. several conditions need to be met. All the plasma
constituents—electrons, ions, atoms, and molecules—must be described by Maxwellian
velocity distribution functions at the same temperature. The plasma should be
homogeneous and optically thick, and the radiation field should follow a Planck
distribution at the same temperature. The internal energy states (rotational, vibrational.
7
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and electronic) o f atoms and m olecules must follow Boltzmann statistics described by
the same temperature. The neutral and charged particles must satisfy Saha equilibrium,
and neutral particles must be in chem ical equilibrium. Nonequilibrium implies any
deviations from the state o f CTE.
Laboratory' plasmas deviate from CTE because finite dim ensions allow for
radiation escape, and the radiation field is not that o f a black body. In addition to
radiation escape, gradients in tem perature, concentration, or velocity give rise to various
kinds o f nonequilibrium. We can still define a plasma in local therm odynamic
equilibrium (LTE) if the collisions are sufficiently numerous to m aintain Maxwellian.
Boltzmann. Saha, and chemical equilibrium although a black body radiation field is not
observed.
(1.3.1) Kinetic Nonequilibrium
Collisions among the plasm a constituents redistribute energy and momentum and
work to restore the plasma towards equilibrium . Thus, some o f the plasm a processes
remain in equilibrium, and others deviate from the state o f equilibrium . For example,
when an electric field is applied to a plasm a, the energy imparted to the lighter electrons
is much greater than that to the ions. However, as mentioned previously, the energy
transfer efficiency from electrons to heavy particles can be sufficiently low such that the
electrons reach a steady state tem perature which is much greater than that for the heavy
particles. In such situations, two separate temperatures are needed (one for heavy
particles and the other for the electrons) to define the state o f the plasm a. This form o f
8
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kinetic nonequilibrium is referred to as a two-temperature plasma. The energy coupling
in ion-neutral collisions is very efficient due to their com parable masses, and thus the
ionic and neutral species are generally at the sam e tem perature.
(1.3.2) Ionization Nonequilibrium
Saha equilibrium between the neutral and charged particles can be disturbed if
the effects o f spatial and tem perature gradients dom inate ionization and recombination
processes. Finite reaction rates lead to an overpopulation or underpopulation o f
electrons at a given local temperature. This form o f ionizational nonequilibrium has a
strong effect on plasm a properties. For example. Owano et al. (1992) showed that, for
50 kW radio frequency inductively coupled plasma (RF-ICP). argon’s radiation source
strength is overpredicted when the plasma is incorrectly assumed to be in LTE.
(1.3.3) Chemical Nonequilibrium
Chemical reactions occurring in plasmas are very complicated due to interactions
between electrons, ions, atoms, molecules, and photons. In steady state, all species tend
toward an equilibrium chem ical composition at a given time when forward and reverse
reaction rates for each reaction are in balance. However, plasma reactors involve large
thermal gradients because o f the presence o f cold surfaces (w ater cooled reactor walls,
substrate, etc.). In such situations, kinetic rates for som e o f the chemical reactions may
not be sufficiently fast to m aintain equilibrium chem ical composition across the plasma.
This situation leads to chem ical nonequilibrium in w hich some species are present in
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larger quantities than would be found under equilibrium conditions.
(1.3.4) M axwellian Nonequilibrium
Two-temperature plasmas have been m odeled by assuming the existence o f
Maxwellian energy distribution functions, at different temperatures, for both the
electrons and neutral particles. However, in some cases, the EEDF is non-Maxwellian.
significantly affecting plasm a transport properties, excited state distributions, and the
degree o f ionization. The shape of the EEDF depends on the frequency at which the
electrons gain energy from the electric field and on the frequency at which electrons
interact with the various plasm a constituents. Electron-electron collisions and elastic
electron-neutral collisions drive the EEDF towards Maxwellian. In the presence o f
kinetic nonequilibrium , the electron-neutral energy exchange is very weak, and the
elastic electron-neutral collisions drive the EEDF tow ards Maxwellian. Inelastic
electron-atom or electron-molecule collisions cause dissociation, excitation, and
ionization and thereby deplete the electrons with energy greater than the threshold
energy.
Typically, microwave plasmas have electron densities between IxlO 16 and l x l OIS
m°. average electron temperatures between 0.5 and 4 eV. and neutral temperatures
between 300 and 4000 K. For these ranges o f parameters, the discharges are
characterized as tw o-tem perature plasmas with non-M axwellian EEDFs.
(1.4) Related W ork
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Plasma reactor modelers typically focus on either neutral chemistry and transport
or on glow discharge physics. The neutral transport and reaction models analyze
complex neutral chemistry in multidimensional geometries. Several authors (Harris et
a L 1990: Goodwin and Gavillet. 1990: Matsui et al.. 1990: Spear and Frenklach. 1989)
have investigated the com plicated hydrocarbon chemistry in the diamond deposition
process. A thorough overview o f the gas phase kinetic modeling studies is given by
Celii and Butler (1991). The glow discharge models calculate electron number density.
EEDF. and derived quantities such as dissociation, excitation and ionization rates. The
EEDF plays an important role in the plasma chemistry. In turn, determination o f the
EEDF requires the knowledge o f the plasma composition. Thus models have to be
coupled self-consistentlv to obtain an accurate description o f the plasma.
Several optical and probe diagnostic techniques are used to determine plasma
parameters o f interest (electron num ber density and temperature, gas temperature and
composition, etc.). This thesis concentrates on self-consistent hydrogen and argon
plasma modeling and the use o f optical emission spectroscopy as a diagnostic tool. The
following sections present an overview o f the relevant studies reported in the literature.
(1.4.1) Modeling
Several researchers have investigated the effects o f various electron-neutral
interactions the EEDF through solution o f the Boltzmann equation. Loureiro and
Ferreira (1989) reported that, for H: positive columns at low electric fields, superelastic
vibrational collisions strongly enhance the tail o f the EEDF which significantly increases
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the electron excitation rates. Cacciatore et al. (1978) obtained self-consistent results by
solving the Boltzmann equation with the vibrational m aster equation to investigate the
role o f dissociation kinetics in N : and H: discharges. Their numerical results showed
that the dissociation rates calculated according to jo in t vibro-electronic mechanism are
larger than those obtained by a pure vibrational or pure electronic impact mechanisms.
Colonna et al. (1993) found that electron-electron collisions must be included in EEDF
solvers for post discharge conditions. For nitrogen discharge with electric field o f 5 X10'
17 V-cm2 (or 50 Td). they demonstrated that the electron-electron collisions are effective
in shaping the EEDF when the degree o f ionization exceeds 10~\ The degree o f
ionization is defined as the ratio of electron num ber density to total neutral number
density (n^n,). Capitelli et al. (1994) studied the dependance o f the EEDF on
superelastic electronic collisions by specifying parametrically the concentrations o f the
electronically excited states for given concentrations both o f H: vibrational excited states
and o f atomic hydrogen. They concluded that the EEDF and related properties depend
on electronic superelastic collisions at low electric fields (< 30 Td) when average
electron energies are less than 1.5 eV.
Kune and G undersen (1983) developed a pure hydrogen model for a high
electron number density (>10|l) m'3) plasma by assuming a Maxwellian EEDF.
Additionally, their model included collisional and radiative processes involving excited
states o f atomic hydrogen. They concluded that the step-wise ionization from the atomic
hydrogen excited states plays a dominant role in electron production and the inverse
three-body recom bination in the gas phase is the major electron loss mechanisms.
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Penetrante and Kunhardt (1986) studied the influence o f various collisional processes in
shaping the EEDF for various values o f electric field, degree o f ionization, and
dissociation fraction. Their comprehensive numerical study showed that, for highly
ionized (degree o f ionization>10'4) and highly dissociated (H mol fraction >0.5)
hydrogen plasm as, stepwise ionization from the atomic hydrogen excited states is the
dom inant mechanism o f electron production. Koemtzopoulos et al. (1993) developed a
mathematical model to predict the degree o f dissociation in a H2 discharge. Their
approach involves solving the Boltzmann equation and then solving the mass and energy
conservation equations to determine the self-sustaining electric field, the electron
energy, and the atomic hydrogen density in the discharge. The results suggested that the
gas dissociation can be increased by lowering the total gas density and/or increasing
power. However, neither o f these extensive modeling studies include experimental data
for validation o f their predictions. M ore recently. Scott et al. (1996a) developed a threetem perature thermochemical model for analyzing the chemical com position and energy
states o f a hydrogen microwave plasm a used for studying diamond deposition. Their
zero-dim ensional model showed that H mol fraction increases significantly with
deposited pow er and pressure, whereas the electron number density and temperature
decrease w ith increasing pressure. Their model assumed a M axwellian EEDF and
neglected the excited states.
Bou et al. (1992a. 1992b) perform ed kinetic calculations to investigate the effects
o f plasm a parameters such as EEDF. electron density, and electron temperature on
diamond growth. They concluded that the ion-molecule interactions do not modify the
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plasma composition for electron densities between 1016 and 1018 m '5. Surendra et al.
(1992) used a hybrid particle-fluid approach to study the physical aspects o f DC
discharges used for diam ond deposition. They observed that the system exhibited anode
glow which is an im portant source o f atomic hydrogen. Hyman et al. (1992) are
developing a comprehensive model which includes electromagnetic fields, the EEDF
and electron number density, time dependent hydrocarbon densities, and fluid dynamics.
Their numerically predicted electron number density o f 5x10 18 m '5 in a 1.6 kW. 40 Torr
H:/CH4 plasma was found to be much larger compared to those predicted by the other
researchers (Koemtzopoulos et al.. 1992: Tahara et al.. 1995: Scott et al. 1996). St.
Onge and Moisan (1994) investigated hydrogen atom yield in pure H: radio-frequencv
and microwave discharges. Their work suggests that the hydrogen atom yield can be
increased dramatically by cooling the reactor walls and thus reducing wall
recombination. Tahara et al. (1995) developed a comprehensive hydrocarbon chemistry
model to study the gas phase o f microwave reactor and correlated plasma properties and
film features. They inferred that the density ratio o f C H 5 radical to H atom is closely
related to the hardness o f the deposited diamondlike carbon film. Additionally, they
compared their modeling results with the optical emission spectroscopic (OES)
measurement o f the 4s excited state o f atomic hydrogen. However, their five level
excited state model assum es a Maxwellian EEDF and does not include the effects o f
optical trapping o f resonant radiation.
In addition to the above mentioned studies in diamond deposition plasmas, an
abundance o f literature is available on modeling low pressure plasmas used for etching
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and deposition in semiconductor manufacturing. Park and Economou (1990) used a
continuum model to analyze charged particle transport and potential distribution in low
pressure RF chlorine glow discharges. They found that, for Cl: discharge, the electron
number density was two orders o f magnitude lower than the negative ion number
density. The low electron number density resulted in large potential drop across the bulk
plasma and comparatively high bulk electric field. Sommerer and Kushner (1992) used
a Monte Carlo-fluid hybrid model o f RF discharges used for etching. They found that Cl
atom production efficiency is similar in He/Ch. He/HCl. and He/CCI4 mixtures, while
the sources o f ionization, location o f attachment, and electronegativity dramatically
differ. Lymberopoulos and Economou (1992) developed a coupled Ar glow discharge
neutral transport model. They found that, for a pressure o f 1 Torr. metastable ionization
was the main mechanism for electron production to sustain the discharge despite the fact
that their mol fraction was less than 10‘5. Their results showed that the bulk electric field
and electron energy were lower, and a sm aller fraction o f power was dissipated in the
bulk plasma when compared to the case without metastables.
(1.4.2) Diagnostics
Critical to the understanding o f the diamond CVD process is the ability to
determine the presence, concentration and spatial distribution o f chemical species in the
gas phase. Gas phase diagnostic techniques are utilized to monitor transport o f the
important species, such as atomic hydrogen, and methyl and hydroxyl radicals, to the
diamond growth surface. Quantitative determination o f the composition and temperature
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o f the various species present in the gas phase also facilitates the developm ent o f
chemical, thermal, and mass transport models. The diagnostics techniques used for
monitoring diamond CVD process can be divided into three categories: sample
extraction, physical probe and optical methods. The sample extraction methods (e.g. gas
chromatography and mass spectroscopy) perform ex-situ analysis o f the extracted
portion o f the gas. Physical probe methods (e.g. Langmuir probe) introduce a probe into
the reactor chamber and can m easure potential, electron number density and electron
temperature. Optical methods (e.g. OES and laser induced fluorescence) use
spontaneous or induced photon emission to probe the gas phase environm ent. An
excellent overview o f relevant diagnostics techniques used for diam ond CVD is given by
Thorsheim and Butler (1991).
Optical methods are preferred for detecting stable as well as transient species
because o f their non-intrusive nature and thus their negligible perturbation o f the
diamond growth environment. Spontaneous emission from the excited species can be
easily detected by using OES. Absorption o f laser light and observation o f laser induced
fluorescence (LIF) is used for detecting ground state species (M eier et al.. 1991).
Resonance enhanced m ultiphoton ionization (REMPI) is a com bination o f optical and
physical techniques and requires the introduction o f probes to collect the ions produced
by photon induced ionization (Zachariah and Joklik. 1990). Fourier Transform Infrared
Spectroscopy (FTIR) or infrared diode laser absorption (IR-DLAS) m onitors absorption
o f photons through vibrationally and rotationally excited gaseous environm ents (Celii et
al.. 1990). Coherent anti-Stokes Raman spectroscopy (CARS) and third-harmonic
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generation (THG) use laser induced nonlinear optical conversion through phase
matching to measure absolute species concentrations (Connell et al.. 1995: Chen et al..
1992: Owano et al.. 1992: Green et al.. 1993).
OES is a simple, inexpensive and easy to implement optical method. However.
OES is applicable only to systems containing a sufficient density o f excited state species.
Because o f its in-situ nature. OES can be potentially used as a process control tool.
Many researchers detected various species present in the gas phase and correlated their
emission intensities with the structural quality o f the diamond films. Inspektor et al.
(1989) and Muranka et al. (1991) detected emission from electronically excited atomic
hydrogen (Ha. Hp. H j . methyl radicals (CH) and molecular carbon (C; Swan bands) in a
microwave plasma reactor. Beckmann et al. (1993) observed that the intensity o f CH
correlates with the growth while C: intensity correlates inversely to the film quality.
Chen et al. (1993) monitored H and C : emission and found that the growth rates o f
diamond films in CO-CH4 gas systems are three times that in H:-CH4 gas systems.
Marcus et al. (1991) observed the ratio o f emission intensities o f CH and H and
concluded that, for pressure above 5 Torr. the ratio remains constant and that the
structural quality o f deposited films could not be conclusively correlated to the changes
in the emission spectra.
The amount o f OES signal collected depends on the excited state population and
transition probability. For equilibrium plasmas, ground state densities can be inferred,
by assuming a Bolzmann distribution, from the measurements o f the excited state
densities. However, for nonequilibrium plasmas, the procedure is much more complex
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and requires know ledge o f the EEDF. electron density and temperature, gas temperature
and the reaction mechanisms involved in the production o f the excited states.
Generally, actinometry is used for obtaining relative ground state concentrations
o f some o f the plasm a species. In this m ethod, a noble gas is chosen that has an excited
state energy level close to that o f the species o f interest, and a small amount is added to
the input gas stream . Since the energy levels are close, the excitation efficiencies by
electron impact can be assumed to be the sam e and often show sim ilar dependence on
the plasma param eters. The relative ground state concentrations are determined by
ratioing the target molecule emission intensity to that o f the noble gas. Cobum and
Chen (1980) illustrated this method using F atom ic emission from plasma etching
discharges by adding argon as an actinometer. Muranka et al. (1991) and Beckmann et
al. (1993) determ ined relative H concentration by adding argon to the diamond
depositing plasm a. However, use o f actinom etry is restricted because either an
appropriate actinom eter does not exist or the addition o f the actinometer changes the
deposition environm ent (Thorsheim and Butler. 1991).
In addition to the line and band em ission, spectra include continuum emission.
The continuum em ission results from electron-ion and electron-neutral interactions in
which an electron either recombines (free-bound) or remains free (free-free). When freebound reactions dominate, the dependence o f this continuous radiation can be utilized to
infer electron density and temperature (W ilbers et al.. 1991: Gordon and Kruger. 1993:
Benoy et al.. 1993). If free-free emission produces the majority o f the continuum, the
procedure can not be applied directly to infer electron density or temperature. However.
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we demonstrated here that the continuum em ission intensity can be utilized as a
consistency check for our numerical predictions (Kelkar and Gordon. 1997).
Langmuir probe measurements o f the EEDF and electron density are difficult
because of interfering electric fields. Thus the measurements are typically performed in
the downstream region o f the plasma. Tahara et al. (1995) report that the electron
temperature decreases from 3.5 to 2 eV with increasing ambient gas pressure. Their
electron density indicates a peak at 50-70 Pa and ranges from 5 x l0 l6-2 .5 x l0 17 m '3.
Tachibana et al. (1984) performed heated electric probe measurements in a methane
plasma and found that the EEDF deviates considerably from its Maxwellian form and is
in fact close to Druvvesteynian. OES can also be used to determine temperature.
Virtually every gas kinetic collision changes the rotational quantum number, while
collisions that change the electronic or vibrational quantum number are much less
frequent. Thus, generally, rotational temperature is assumed to represent the gas kinetic
temperature. Stalder and Sharpless (1990) reported excited state temperature
measurements o f C; Swan band emission in a arc jet plasma operating between 100-400
Torr. Reeve and W eimer (1995) concluded that rotational excitation temperatures for C:
and vibrational excitation temperatures for CH closely track the plasma gas temperature.
For DC plasmas. Suzuki et al. (1990) postulated that, above 200 Torr. the plasma was in
thermodynamic equilibrium and gas kinetic temperature can be obtained from the
relative intensities o f the atomic hydrogen lines o f Balmer series. Chen et al. (1991)
measured the gas kinetic temperature o f CH4/H : discharges using OES o f both the H:
and CN bands.
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All the numerical studies reviewed above either assume a M axwellian EEDF or
do not self-consistently account for the excited states in their plasm a models.
Experimentally, several researchers used OES. only qualitatively, to identify species in
the bulk plasma or to correlate the em ission signal with the diamond growth. This work
uses a self-consistent plasma model and OES in tandem to obtain quantitative
information about the plasma parameters o f interest.
(1.5) Thesis Overview
The purpose o f this thesis is to describe the gas-phase discharge physics and
plasma chemistry in a W AVEM AT microwave hydrogen plasma. A theoretical
modeling and diagnostic study is also performed in a nearly chemically inert argon
plasma to facilitate a better understanding o f the discharge physics and to validate the
use o f diagnostic techniques.
Chapter II describes the m icrowave plasma reactor and optical emission setup.
Chapter III describes the diagnostic techniques used in this research to determine
particle densities and temperatures. The techniques allow the determ ination o f excited
state number densities and tem peratures from absolute and relative line intensities,
respectively. The use o f absolute continuum emission intensities to verify the
consistency between theoretical predictions o f the plasma parameters is also described.
The experimental measurements show' that the plasmas under study deviate considerably
from a local thermodynamic equilibrium .
Chapter IV describes the theoretical modeling study in H2/C H 4 and Ar plasmas.
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A detailed description o f the Boltzmann equation, electron num ber density balance
equation, electron energy balance equation, and neutral particle (ground as well as
excited state) balance equations and the required numerical data are given. Finally, an
iterative scheme used to perform a parametric study and to obtain self-consistent
theoretical solutions is presented.
Chapter V presents results from the comprehensive param etric studies in H2/CH4
and Ar plasmas. This chapter evaluates the validity o f assum ptions and identifies the
important physical processes in both H2/C H 4 and Ar plasmas. It will be demonstrated,
for example, that the EEDF is highly non-Maxwellian which greatly influences the
plasma chemistry.
Chapter VI presents self-consistent numerical results obtained by the
simultaneous solutions o f the Boltzmann equation, electron conservation equation,
power balance equation, and neutral species conservation equations. The numerically
predicted excited state num ber densities are compared with the OES measurements. It
will be demonstrated that the excited states play an important role in accurate predictions
o f electron number density, self-sustaining electric field, and average electron
temperature. It will be shown that, for an Ar plasma, the majority o f electrons are
produced by step-wise ionization involving excited states and that neglecting excited
states results in an overprediction o f the self-sustaining electric field and average
electron temperature by 30-50%. However, in our H2/CH4 plasm a, the excited state
ionization is found to be negligible due to collisional quenching (depopulation) o f the
excited states by m olecular hydrogen.
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Chapter VII describes the use o f experimentally measured absolute continuum
emission to validate the theoretical predictions. It will be shown that free-free electronneutral collisions produce the majority o f the continuum emission. For the H;/CH4
plasma, the theoretical predictions will be shown to be inconsistent with the
experimentally measured continuum emission unless the electron-H: interaction crosssection is increased by a factor between 4 and 20 above the momentum cross-section
value.
Chapter VIII examines the power coupling efficiency in microwave reactors. It
will be demonstrated that only 25-100 W out o f 680 W o f input power are absorbed by
the pure argon plasma. An analytical heat transfer model and experimental global
reactor energy balance data will show that the remaining energy bypasses the plasma and
is directly absorbed by the base-plate and applicator walls. Although a larger fraction
(35-50%) o f the input power is absorbed by the H:/C H 4 plasma, it will be shown that a
considerable am ount o f the input power also bypasses the plasma and is dissipated in the
base-plate. and applicator wall cooling water.
Chapter IX presents results from the comprehensive hydrogen model. It will be
shown that gas temperature and diffusion greatly influence the dissociation fraction in
the bulk plasma.
Chapter X presents the conclusions o f this work and discusses possible future
work.
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CHAPTER II
EXPERIMENTAL FACILITY
(2.1) Microwave Plasma Reactor
The research is conducted with a WAVEMAT* microwave plasma reactor
(Model M PD R -3135). The reactor is supplied with 2.45 GHz. 6 kW Sairem*
microwave power supply. The power supply is equipped with incident and reflected
power meters. In case o f mismatch between impedances o f the waveguide and cavity,
the reflected power is absorbed and thermally dissipated in the m atched load. The gas
flows are controlled by MKS type mass flow controllers (MKS Instrum ents11). The
pressure in the chamber is controlled by a throttle valve placed between the chamber
and the mechanical pump (Franklin Electric®'). A Honeywell com puter monitors the
system operating conditions and controls the experimental time and shut down
sequence. Figure 2.1 shows a schematic o f the principal components o f the reactor.
The 7" diam eter cavity microwave plasma disk reactor is designed for large area
uniform diam ond film deposition. The microwave probe assembly and sliding short
form the top portion o f the cavity. The lower section o f the cavity consists o f the
23
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■
Cooling Water
Power coupling probe
Sliding short
Cavity wall
w im m .
Quartz dome
Plasma discharge
Optical access window
Substrate
Base plate
jjj— To vacuum pump
F ig u re 2.1 Schematic of WA VEMAT microwave plasma reactor
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bottom surface, the base plate, and a metal plate. The reactive gases, which are supplied
through the source gas input tunnel, are confined to the lower end o f the cavity by the
quartz bell jar. W ater cooling is incorporated in the sliding short, probe, and the base
plate by annular w ater cooling tunnels. For high input power operations, the bell ja r is
air cooled. The substrate is placed on the substrate holder assembly. The position o f
the substrate in the plasma discharge can be changed by using a vertically movable
mechanical stage. An optical pyrom eter (M ikron Instruments*) m onitors the substrate
temperature. The 3" diameter metal tube acts as a resonance breaker and ensures that
the plasma is created only above the substrate and not in the small cavity underneath the
substrate. Diamond films are deposited on a 3" diameter silicon substrate seeded with 5
pm grain size diamond paste. Typical operating conditions are 40 Torr pressure. 300
seem hydrogen and 3 seem methane flow rates, and 1.6 kW o f input power. Each
deposition run is about 20 hours long, and the growth rate is around 0.25 pm /hr for the
typical operating conditions. Additionally, the reactor was run using nominally pure
argon as a diagnostic gas.
(2.2) Emission System
The primary diagnostic system is a 0.5 m SPEX 500M m onochromator, fiber
optic assembly, and Hamamatsu R-928 phto-multiplier tube (PMT). The miscellaneous
equipment include a dedicated com puter (Gateway 2000) for data acquisition, a tungsten
lamp (GE/18A) for calibration, an optical pyrometer, a power supply (Lambda LLS
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8018) for the lamp, and several optics. A 1-mW He-Ne laser is used for assuring
alignment and for checking spectral response o f the system. The accuracy o f the system
o
e
agrees with the reported value o f ±5 A. and the results are repeatable within ± 1 A.
Figure 2.2 shows the optical setup used for collecting and focusing the em itted
light onto the entrance slit o f the monochromator. A 40 cm focal length (2" diameter)
lens is used to collect the light from the plasm a through a typically 0.5" diameter
aperture stop. A 30 cm (2" diameter) lens images the plasma onto the fiber optic cable.
An x-y translation stage in combination with a labjack which provides z-translation
allows to accurately access various spatial locations across the plasma. However, the
view port only allows optical access across about 1" o f the 3" diameter plasma. Figures
2.3 and 2.4 show typical emission scans in H:/CH4 and Ar plasmas, respectively.
To determine the absolute intensity (W/m:'sr) o f line or continuum emission,
calibration o f the detection system is required. A tungsten lamp was used as a known
source o f emission (3000-8000 A). The temperature o f the lamp was measured using a
Pyro-micro* vanishing filament optical pyrometer. Owano et al. (1991) reported
experimental data for the surface emissivity o f tungsten for several wavelength regions
from 310 nm to 800 nm and temperatures ranging from 1600 K. to 2400 K. The signal
(counts/s) measured from the tungsten lamp is used to determine a wavelength
dependent absolute intensity calibration factor. D; (counts m3/s W sr).
(>-) = e M
Xs \e
2C.
c . .;.r
- 1
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Aperture
Stop
Focussing
Lens (f=30 cm)
Mlcrpwaw^
gg»»—
J
/
Collection
Lens (f=40 cm)
Optical
Fiber
Hamamatsu
R-928 PMT
Datascan
Controller
DS 1000
SPEX-500M
Monochromator
Data Acquisition
Computer
Figure 2.2 Schematic o f Emission System
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35000
Pressure ■=40 Torr
Power = 1.6 IcW
CH< = 3 seem
Hi = 300 seem
30000
25000
20000
15000
10000
5000
0
6500
5500
Wavelength (A)
4500
3500
7500
Figure 2.3 Optical emission scan in a hydrogen/methane plasma
25000
20000
-
~ 15000
d
4
w
-
1| 10000
'
5000
-
0
X40
Pressure “ 5 Torr
Power - 680 W
Ar - 250 seem
^
4000
,
1*
X40
i
5000
6000
Wavelength (A)
*
11
7000
L
8000
Figure 2.4 Optical emission scan in an argon plasma
t
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The spectral intensity for a grev-bodv em itter at a temperature T is given by the
Planck distribution (Incropera and Dewitt. 1990):
where C,=hc2= 0.595x 10'16 W n r. C:=hc/k=0.01438 mK. /, is the wavelength o f radiation,
and e is the surface em issivity. To determine the absolute intensity calibration factor.
Equation 2.1 must be multiplied by the spectral bandpass o f the system. The spectral
bandpass (Owano. 1991) o f the emission system is determined by the w idth o f the
entrance and exit slits o f the monochromator and its reciprocal linear dispersion. dX/dx
(SPEX 500M M anual. 1991). The reciprocal linear dispersion for the SPEX 500M Vz m
monochromator w ith a 1200 grooves/mm grating blazed at 400 nm approximately
remains constant at 16 A/mm. If one were interested in the total system efficiency (i.e.
counts/s per collected W) an accurate measurement o f the collection solid angle and
collection area is required. In our emission diagnostics, we are interested in the
volumetric emission coefficient (W /m3sr) and no such measurements are required.
Because the em ission measurements and intensity calibration are performed with the
exact same m onochromator, collecting optics, and photomultiplier tube, the collection
volume and the solid angle are identical. However, it is important to note that the
tungsten filament is a surface emitter and thus its intensity is given per unit area, while
the plasma is a volum e em itter and its intensity is given per unit volume. We assumed
that the plasma is uniform over entire volume and used the plasma diam eter to convert
the surface emission to the volume emission. The absolute intensity calibration factor
can be written as:
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(counts/s)
D.
'lam p
'
=
d
,
plasm a
2 C.
il.T)
}J le
c.
>.T
dk
- 1
W’
dx
where (counts/s ),amp represents the lamp signal. dplasma represents the plasm a diameter,
and w „lt represents the exit slit width.
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CHAPTER III
DIAGNOSTIC TECHNIQUES
(3.1) Introduction
Microwave plasmas provide a ready source o f diagnostic information via the
emission by various gas phase species. OES is a simple non-intrusive tool applied to
study plasma parameters o f interest. The wavelength at which the spectral lines occur
facilitates the detection o f the species present. The spectral line profiles contain
quantitative information about the emitting species such as temperature and number
densities (Herzberg. 1964). The total energy emitted from each line correlates to the
excited state number density and excitation temperature. In addition to the line
emission, continuous em ission from the plasma can be used to determine the electron
number density and temperature. However, the application o f these techniques needs
careful evaluation o f the nature o f the plasma environment. As mentioned earlier, the
presence o f nonequilibrium in microwave discharges limits the possible inferences from
the OES measurements. However. OES can be used very effectively in tandem with a
self-consistent plasma model to obtain accurate quantitative information.
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(3.2) Absolute Line Intensities
Figures 3.1 and 3.2 show the partial energy level diagrams o f atomic hydrogen
and argon, respectively. Atom s in electronically excited states spontaneously emit
photons while transitioning to lower levels. A bsolute atomic line emission
measurements lead directly to the population nm o f an excited state through the following
relation:
Aidm n Xm n
— .
" • = “i s —
mn
where Imn is the volumetric em ission coefficient.
(31)
is the transition’s wavelength, h is
Planck's constant, c is the speed o f light, and A mn is the Einstein transition probability.
This measurement is independent o f equilibrium assum ptions requiring only that the
emission is optically thin. The required volumetric em ission coefficient was obtained by
dividing the surface emission coefficient from calibration by the imaged path length
assuming that the emission is uniform in the im aged volume. Experimentally two
measurements are made to obtain nm. One m easurem ent is made at the transition's center
wavelength and collects line and continuum em ission: the second measurement is
performed 10 A away from the first wavelength and collects only continuum emission.
If the continuum emission rem ains constant over the 10 A. then the subtraction o f the
latter measurement from the form er yields the desired em ission coefficient.
In a FT/CH., plasma, measurements o f n 3 (12.08-eV). n4 (12.74-eV). and n5
(13.05-eV) are inferred from em ission coefficients at 6563-A ( H j. 4861-A (H„). and
4340-A (H,). respectively. Figure 3.3 shows the Boltzmann plot o f the experimentally
32
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16
15
!T
M
Ionization Limit
JL
&
a5
W
13
12
11
Selected Transition*
4340 A: 5-2 HY
4861 A: 4-2 Hp
6563 A: 3-2 H«
10
Figure 3.1 Partial energy level diagram o f atomic hydrogen
16
Ionization Limit
_Z£_
Sd
15
-ZdL
-3£-
JsL.
14
SB 13
t
a
W
12
11
Selected Transitions
6043 A: 5d-4p
6032 A: 5d-4p
4300 A: 5p-4s
7147 A: 4p'-4s
7272 A: 4p'-4s'
10
Figure 3.2 Partial energy level diagram o f argon
33
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1.6 kW, 40Torr
300 teem H2
3 scan CH.
1011 -
\
10 '°
109
-
1q8 —i i—i i I i i i i I i i i i I i i i i I i i ' i I i i i i
11.0
11.5
1£0
12.5
13.0
13.5
14.0
Energy [eV]
Figure 3.3 Experimentally measured excited states in the H-2/CH4 plasma
10
680 W. 5Torr
250 seem Ar
10'3
10'2
10 '
12
13
14
15
16
Energy [eV]
Figure 3.4 Experimentally measured excited states in the A r plasma
34
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measured excited states in a 1.6 kW input power. 40 Torr pressure H;/CH4 plasma. In an
Ar plasma, excited state number densities for levels 4p (I3.17I-eV ). 5p (14.569-eV). and
5d (15.206-eV) are measured using emission from transitions at 7147.7-A (4p-4s). 4300A (5p-4s). and 6043-A (5d-5p). respectively. Figure 3.4 shows the Boltzmann plot o f the
experimentally measured excited states in a 680 W input power. 5 Torr pressure A r
plasma.
The populations o f the excited state atoms can be utilized to determine the local
thermodynamic equilibrium temperature (Tlte ) by using Maxwell-Boltzmann relation:
/
n
Zm
-E
- n — exp
iq
* kT
LTE
(3.2)
where n, (p/kTg) is the ground state population o f the species under consideration. gm is
the degeneracy o f the excited state. Q is the ground state electronic partition function
(Q =gi) f°r the temperature range considered, k is Boltzm ann’s constant, p is the pressure.
Tg is the gas temperature, and Em is the energy o f the excited state. Table 3.1 show the
LTE temperatures calculated based on the experimentally measured excited states by
assuming 5% dissociation o f H: at a gas temperature o f 2000 K. Table 3.2 shows the
LTE temperatures in the Ar plasma calculated by assum ing a gas temperature o f 350 K.
LTE temperatures represent electronic temperatures only if the ground and
excited states are in equilibrium. However, radiation escape drives plasmas away from
LTE. and thus the plasmas studied in this thesis are not in LTE because collisional
processes are unable to replenish the radiative loss o f excited states. The temperatures
35
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Table 3.1 LTE Temperatures (H:/CH4 plasma)
m
nm/g m (m-3)
T ltEjti (^ )
J
4 .1 2 x l0 10
5501.4
4
1.24x1010
5541.2
5
2.96x109
5386.8
Table 3.2 LTE Temperatures (Ar plasma)
m
nm/g m (m "3)
T lTE. m (K-)
4p
2 .0 x l0 13
6748.4
5p
7.0x10"
6502.5
5d
8.5x10'°
6275.4
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shown in Tables 3.1 and 3.2. then, do not represent the electron temperature.
(3.3) Relative Line Intensities
By using num ber densities o f two or more excited states, we can eliminate the
ground state density from Equation 3.2 to generate the Boltzmann relation:
n
m
------ W.'
n
en
em
kTB j n n
n
(3.3)
t
This analysis does not require thermodynamic equilibrium between the ground and
excited states. For plasm as in local thermodynamic equilibrium , the two temperatures.
T lte and TBmn would be equal. A Boltzmann diagram can be generated by plotting the
natural logarithm o f nm/g m against excited state energy sm. The slope o f the Bolzmann
plot is inversely proportional to temperature. If partial local thermodynamic equilibrium
prevails in which excited states are in thermodynamic equilibrium with electrons, the
Boltzmann temperature represents the electron temperature. Figures 3.5 and 3.6 show
the Boltzmann tem peratures calculated by using relative line intensities in H:/CH4 and
Ar plasmas, respectively. The existence o f several Boltzmann temperatures reveals that
the excited states are not in equilibrium with the electrons.
(3.4) Absolute Continuum Intensity
Continuum em ission results from interactions between free electrons and ionized
or neutral particles in the plasma. When an interacting electron recombines with an ion
the reaction is called free-bound. If the electron remains free the interaction is called
37
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1.6 kW, 40 Torr
300 sc an Hj
3 seem CH.
a>
23
TB34 = 6407K
Tb<£ = 2466 K
W
11.5
=« 15K
12.5
12.0
13.0
13.5
Energy [eV]
Figure 3.5 A Boltzmann plot showing the presence o f nonequilibrium in the
HJCH4plasma
35.0
680 W. 5 Torr
250 seem Ar
3£5
S -30.0
c~ 27.5
T,
25.0
= 5041 K
= 4418 K
= 3266 K
22.5
13
14
15
16
Energy [eV]
Figure 3.6 A Boltzmann plot showing the presence o f nonequilibrium in the
Ar plasma
38
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free-free. The interactions can be represented as follows:
Free-Bound:
Free-Free:
A' - e'
A’ + e
-
A + hv
-
A ' - e ' - hv
(3.4)
- A + e ‘ ^ /tv
A * e
where hv represents the energy o f the photon emitted due to the kinetic energy loss o f
the electron, and A and A* represent the atomic or m olecular neutral and ionic particles,
respectively.
For a singly ionized plasma, the total continuum emission can be represented as
follows: (W ilbers et al.. 1991)
/!>
ei
en
ff
ff
where.
Th
C nn
, -I _ I ">
k T
IT
eh
it
1 - exp
C I ne ni
------------ exp
/.2 r !:
c
— ne nu
Tii20
L ( Te )
h
e
eh
**■>
C, - 1.63x10 43 \Vm*Kl2sr 1
he
}JcT
O
(3.5)
he
UT
he
UT
* 1 exp
he
UT
- 1.026x10 34 WmzK iZsr
C
In the above equation. nh. n,. and X denote the neutral num ber density, the ion number
density, and the wavelength, respectively. The needed cross-section information
describing free-bound recombination is contained in the so-called free-bound Bibermann
39
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factor c,b (Biberman et a l..l9 6 l). Similarly, the free-free electron-ion term contains its
cross-section inform ation in the free-free Bibermann factor clT(Schnapauff R.. 1968).
For the electron-neutral free-free term, the necessary electron-neutral momentum crosssection. Qeh(Tc). is directly incorporated (Cabannes and Chapelle. 1971). For a
multispecies plasma, the above equations were m odified to include the mole fraction
contributions o f various species.
For the FT/CH., plasm a, electron collisions w ith the dom inant species H. FT. and
CH 4 were accounted for to calculate the free-free electron-neutral emission. The
relevant cross-sections for FT and CH 4 were taken from ELENDIF (1990) and those for
H were adapted from K oem tzopoulos et al. (1993). The free-bound and free-free
electron-ion terms account for interactions between the electrons and the FT ions. Both
Bibermann factors for H ' are unity (Biberman et al.. 1961). For highly ionized (n^n^lO *
3) and highly dissociated (X M>0.5) hydrogen plasm as. Flyman et al. (1992) and
Penetrante and K unhardt (1986) showed that the FT ions account for the 99% o f all ions.
For our operating conditions, we found that FT,' and H ' ion concentrations are
comparable. Biberm ann factors for H;, ' are also assum ed to be unity. It will be
demonstrated in C hapter VII that the contribution o f the electron-ion free-free and freebound terms is negligible and thus our assumption does not cause any inaccuracies in the
numerically predicted continuum emission.
For the Ar plasm a. electron-Ar momentum cross-sections reported by Devoto
(1973) are used to calculate the free-free electron-neutral em ission. Bibermann factors
for the free-free electron-ion and tree-bound em issions w ere taken from Gordon (1992).
40
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In general, the total continuum radiation depends on the electron number density,
the electron temperature, the neutral density, and the w avelength (Equation 3.5). If the
continuum emission predominantly results from the free-bound term, then the
dependence on the neutral density and electron temperature, at certain wavelengths, is
weak. Under such conditions, this procedure has been used successfully in several
studies to infer both the electron density and temperature (W ilbers et al.. 1991: Gordon
and Kruger. 1993: Benoy et al.. 1993). For our operating conditions, however, the freefree electron-neutral term dominates the total continuum em ission—because o f an
abundance of neutral species at low gas temperatures (<4000 K.)—and thus there is a
strong dependence on the neutral number density, the electron temperature, and the
electron number density. Equations 3.5 are used to theoretically predict the continuum
emission which is then compared with the experimental data. This procedure is used to
verify the consistency between the plasma parameters.
41
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CHAPTER IV
MODELING
(4.1) Introduction
To accurately model a plasma, it is important to account for all th e physical
processes. This chapter describes a modeling procedure used to simulate H /C H * and Ar
plasmas. Initially, the various reactions used to obtain a non-Maxwellian EED F are
described. The following section describes the development o f collisional-radiative
models (CRMs) along with the cross-section and reaction rate data used in the models.
Then a comprehensive hydrogen model consisting o f two neutral (H and EU) and three
ionic (FT,H2", and H 3") species is described followed by a description o f electron and
energy conservation equations. Finally, the solution methods used to obtain the
parametric and self-consistent results are outlined.
(4.2) Electron Energy Distribution Function
The EED F is inferred from the Boltzmann equation. The steady state, isotropic
form o f the Boltzmann equation can be w ritten as (Koemtzopolous et al., 1993)
42
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where e represents the electron energy, f represents the EEDF, m and M represent the
electron and neutral masses, respectively, k represents the Boltzmann constant, Tg
represents the gas tem perature, ei represents the threshold energy o f process j, E
represents the electric field, co represents the angular frequency o f the excitation field, e
represents the electron charge, and vmrepresents the momentum exchange collision
frequency. The collision frequency for process j is
(4.2)
where Oj(e) is the collision cross-section and N is the neutral species density.
The collision term on the right hand side o f the Boltzmann equation includes
electron-neutral elastic collisions, and vibrational, rotational, and electronic excitation and
ionization o f neutral species. Superelastic vibrational and electronic processes are
accounted for by specifying a vibrational temperature and relative population o f electronic
levels in the Boltzmann solver. It will be shown in Chapter V that, for the operating
conditions studied in this work, the superelastic vibrational and electronic processes can be
neglected. The distribution function is normalized such that /f(e )e 1/2de = 1. The EED F
was found using ELENDEF (1990) which allowed for the specification o f electron num ber
density, electric field strength, and gas composition. The necessary cross sections for Ar,
43
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H 2 and CH 4 w ere supplied with ELENDIF, and those for H w ere obtained as described by
K oemtzopoulos et al. (1993).
(4.3) Collisional-Radiative Models
A n-level atomic collisional-radiative model is developed to predict number
densities o f (n-2) excited states. The ground state and electron number densities are held
constant. The following electron and heavy particle collisional reactions and radiative
reactions w ere used to develop the non-linear system o f equations to be solved:
n>m
Am . M - An . M
n>m
A m » e »• A ’ * e ♦ e
(4.3)
Am . M - A ’ . e . M
rt>m
A’ • e - A
For each unknown excited state number density, a rate equation was generated based on
the above reaction set as follows:
(4.4)
J<tn
j>ni
where n„, represents the number density o f an atomic excited state (m >l),
represents the
electron number density, n* represents the ion number density, and k represents the
appropriate reaction rate coefficient. The major species present in the plasma (Ar, H,
and CH4) are generically represented by the symbol M. The total number density, n,
(p/kTg), is used for calculating heavy particle impact collisional rates. The first tw o terms
44
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account for electron-impact and heavy particle-impact excitation and de-excitation, the
third term accounts for the electron and heavy particle induced ionization, the fourth term
accounts for the respective recombination reactions, the fifth and sixth terms account for
radiative excited state transitions, and the last term accounts for radiative recombination.
The evaluation o f the needed electron-atom collisional rate coefficients requires
integrating the appropriate cross-sections (a ) over the chosen EEDF. The Boltzmann
equation solver described in the Section (4.2) provides the EEDF. The rate coefficient for
an electron-atom excitation reaction, for example, would be determined by
* .7 nO) •
— jXe)ea^"(e)</e
N M« o
(4.5)
(4.3.1) Cross-section and Reaction Rate Data for a 21-Level H CRM
The electron-atom ionization and excitation cross-sections were taken in their
analytical form from M itchner and Kruger (1973 pp. 26-28). Since the lower-energy
portion o f the distribution function will be shown to be nearly Maxwellian, the inverse
reaction rates can be obtained through the principle o f detailed balancing using forward
reaction rates given by a Maxwellian EEDF (Mitchner and Kruger, 1973 pp. 80-88). This
procedure was implemented to determine the three-body electron-atom recombination rate
coefficients. For electron-atom deexcitation reaction rate coefficients, the principle o f
detailed balancing was used to relate directly the deexcitation cross sections to the
excitation cross sections (see M itchner and Kruger, 1973 pp. 80-82). W ith these cross
sections, the rate coefficients are calculated using Equation 4.5.
45
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The heavy particles are assumed to follow a Maxwellian energy distribution
function at the gas temperature. Accordingly, excitation and ionization rates were taken
from Kune (1987), and the inverse reaction rates were evaluated through detailed
balancing. The deexcitation rates for collisions with molecular hydrogen are considerably
larger than those predicted by analytical expressions in Kune (1987). Several researchers
investigated the depopulation o f H-excited states by collisions w ith H 2 (Burshtein et al.,
1986; Meier et al., 1986). The collisional quenching rates for the first four H-excited
states are adapted from Burshtein et al. (1986).
The radiative recombination cross sections were calculated from the analytical
expressions given by Mitchner and K ruger (1973, pp. 83-85). F o r radiative transitions
between bound-excited states, the rate constants do not require integration over the EEDF
and are simply the Einstein transition probabilities which are taken from Wiese et al.
(1966). The direct use o f these radiative rates reflects the assumption that the emission is
optically thin and escapes the plasma. However, in some cases the radiation is reabsorbed
in the inverse absorption process before reaching the plasma's edge. Holstein (1951)
introduced the idea o f an escape factor which directly multiplies the optically thin emission
rate to account for this trapping. An escape factor o f zero indicates an optically thick or
fully absorbed transition whereas an escape factor o f unity indicates an optically thin or
fully escaping transition. For a cylindrically symmetric plasma o f radius r and o f infinite
length, Holstein developed the following expressions for a radiative absorption coefficient,
a 0, and an escape factor, gef:
46
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1.6
(4.6)
«</V* log (*(/■)
where A.21 is the transition's wavelength, nt is the absorbing state's population, g 2 and g t
are the emitting and absorbing state's degeneracies, respectively, A21 is the Einstein
transition probability, and v0 is the absorbing atom's most probable speed and equal to
(2kTg/M ) 1/2where M is the mass o f the absorbing species. Holstein (1951) defined the
non-dimensional product a 0r as the optical depth. Figure 4.1 shows the variation o f the
escape factor with optical depth. The plasma is optically thin for optical depths larger than
1000 and is optically thick for optical depths smaller than 1. For the experimental
conditions in this work, the optical depth for the Lyman alpha transition is around 6,500
for the range o f parameters studied. However, it is not uncommon to find that the inverse
o f the absorption coefficient ( l / a 0) is used as the characteristic length for optical trapping
and thus, defined as the optical depth. By this definition, the Lyman-a transition in our
HVCH 4 plasma has an optical depth o f 0.0007 cm. Thus, our 3.81 cm radius plasma is still
optically thick for this transition as expected. Equation 4.6 shows that the escape factor
approaches an impossible value o f infinity as either the optical depth approaches zero.
Thus, once optical depth becomes less than about 3, an escape factor must be calculated
differently. W e follow the procedure o f Jolly and Touzeau (1963). Validation o f the
general operation o f this 21-level CRM was accomplished by comparing its output using a
Maxwellian EED F with the output o f another hydrogen CRM which also uses a
Maxwellian E ED F (Bates and Kingston, 1963). Excellent agreement among the results
was observed.
47
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10'1
102
1 03
1 04
Optical Depth
Figure 4.1 Holstein's escape factor as a function of optical depth
48
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(4.3.2) Cross-section and Reaction R ate Data for a 25-Level A r CRM
The analytical expressions reported by Repetti et al. (1991) were used for the
electron-atom excitation and ionization cross-sections. These analytical expressions use
scaling factors to modify Drawin's cross-sections. The required scaling factors for the
excited state transitions were obtained from Kimura et al. (1985) and those for the ground
state transitions were obtained from Vlcek (1988). The deexcitation cross-sections were
found by direct relation with the excitation cross-sections based on the principle o f
detailed balancing (Mitchner and Kruger, 1973). The deexcitation rate coefficients were
obtained by using Equation (4.6). The three-body recombination rates were obtained from
the ionization rates by using the principle o f detailed balancing. The analytical expressions
reported by Bacri and Gomes (1976, 1978) were used for heavy particle impact excitation,
deexcitation, ionization and three-body recombination. The spontaneous emission
coefficients for transitions between bound-excited states were adapted from Kimura et al.
(1984). The optical trapping o f the resonance radiation was accounted for by using
Holstein's escape factors. The reaction rates given by Bacri and Gomes (1978) were used
for radiative recombination to excited states. The model's predictions were validated by
observing a good agreement, for the same input parameters, with those from Repetti et
al.'s (1991) code.
(4.4) Comprehensive Hydrogen Model
A pure hydrogen plasma model consists o f two neutral (H, H 2) and four charged
species (e, IT , H / , H3"). The model is based on the methodology o f Fukamasa et al.
49
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T able 4.1 P u re H ydrogen M odel R eactions
No.
R eactions
R ate CoefTs.
References
M olecular Dissociation
1
H 2+ e « H + H + e
Phelps and Englehardt, 1965
2
h 2+ m « h + h + m
Harris et al., 1990
M olecular Ionization
3
H , + e - H 2" + e + e
k3
Phelps and Englehardt, 1965
4
H 2* + e - H2" + e + e
K
Kune and Gundersen, 1984
ks
Peart and Dolder, 1974
Dissociative Recombination
5
H 2" + e - H + H
6
H3" + e ~ H 2 + H
7
H 3* + e - H + H + H
k7
Kune and Gundersen, 1984
8
H2* + e - H + W + e
k8
Peart and Dolder, 1974
kg
Kune and Gundersen, 1984
M ath u re ta l., 1978
H-," Ion Formation
9
H ,” + H 2 - H3* + H
Three-Bodv Reaction
10
W + H 2 + H2 - H i + H2
k 10
M artin and McDaniel, 1968
11
Diffusion Loss o f H,
^w> ^dl>
McDaniel, 1964
IT , H2’, H f
kj2>kd3
Convection Loss o f H
^cH
12
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(1985) and Kune and Gundersen (1983, 1984). Table 4.1 shows the reactions accounted
for in the model.
The electron impact molecular dissociation and ionization cross-sections were
obtained from Phelps and Englehardt (1965). The heavy particle molecular dissociation
rate was obtained from Harris et al. (1990). The inverse reaction rates were calculated
using the principle o f detailed balancing (M itchner and Kruger, 1973). The number
densities for electronically excited H 2 were estimated by assuming a Boltzmann
distribution corresponding to the average electron energy, and the H 2-excited state
ionization rates were obtained from Kune and Gundersen (1984). The experimentally
determined cross-section curve-fits by Peart and Dolder (1974) and M athur et al. (1978)
were used for the dissociative recombination Reactions 5 and 6 , respectively. The
dissociative recombination rate for Reaction 7 is found to be two times that for Reaction 6
(Kune and Gundersen, 1984). The cross-sections reported by Peart and Dolder (1974)
were used for dissociative recombination Reaction 8 . The reaction rates for the
dissociative recombination reactions leading to H excited states are found to be negligible.
A constant rate o f 2><10' 15 m3/s was used (Kune and Gundersen, 1984) for the ion
formation Reaction 9. Several authors (St. Onge and Moisan, 1994; Fukamasa et al.,
1985) neglect the low-tem perature three-body Reaction 10. Martin and McDaniel (1968)
reported that the reaction rate at room temperature is 3X10"*1 m6/s and slowly decreases
with temperature. Since no reaction rate data were found for our range o f gas
temperature (1000-4000 K), we used the room temperature rate for all temperatures.
Diffusion is a spatial process and causes non-uniformity in plasmas. To better
51
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understand the role o f diffusion in a zero-dimensional model, some discussion is necessary.
The charged particle diffusion in plasmas is extensively studied by several researchers
(McDaniel, 1973; Cherrington, 1979; Liebermann and Litchenberg, 1994). The electron
free diffusion rates are much higher than those o f ions because o f their lighter mass (at
least 4 orders o f magnitude). The electrons tend to diffuse more rapidly than the ions,
disturbing the quasi-neutrality o f the plasma. The resulting space charge field retards the
electron diffusion and increases the ion diffusion so that the quasi-neutrality is restored at
all points in the bulk plasma. Under these conditions the electrons and ions diffuse at the
same rate as determined by the ambipolar diffusion coefficient (McDaniel, 1973). The
ambipolar diffusion coefficient is calculated as follows (Cherrington, 1973):
w here D; represent ionic diffusion coefficient and T ; represent ion tem perature (we assume
T^Tg). Einstein's relation gives the ionic diffusion coefficient as follows (McDaniel,
1973):
D,
kT,
- ■—
n,
e
(4.8)
w here p; is ionic mobility and can be calculated as follows:
f,
•
nl
—
n.
(4.9)
w here p i0 represents the reduced ionic mobility, NL represents the Loschmidt number
52
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(2.689* 102 m ), and n, (p/kTg) represents total gas density. The reduced ionic mobilities
for IT , H2", H3", and Ar" are 15.9 cm 2V 'Is'1, 12.6 cm 2V 'Is'1, 11.1 cm2V"1s, '1and 9 cm 2^ ^ ' 1
(McDaniel, 1973), respectively. The characteristic diffusion length (A) for a cylindrical
geometry is given by (Cherrington, 1979) as:
'
A2
1C
Si
' 2.405 ) 2
(4.10)
1)
where L and R represent the length and radius o f the plasma, respectively. For a plasma in
contact with a cold surface (reactor walls, substrate etc.), the ions and electrons diffusing
out are lost by recombination. The surface acts as a third body and absorbs the energy
released in electron-ion recombination. For plasmas not touching the surfaces, the
diffusing electrons and ions either recombine in the gas surrounding the plasma or are
pumped out. For a mixture o f ions, the ambipolar diffusion rate depends on the individual
ionic diffusion coefficients (Cherrington, 1979). In the present work, it is assumed that
H 3" is the dominant ion in the plasma and governs the net diffusion rate o f electrons. The
net loss o f electrons, or equivalently, H3" due to ambipolar diffusion is calculated as
follows:
C4 - H )
The diffusion rates for the other ions (H" and H2") were calculated from that o f H3" by
assuming that the diffusion rates are proportional to (Mri) I/2,
being the reduced mass o f
the H;7H 2 system (St. Onge and Moisan, 1994).
The neutral particle diffusion in H 2/CH 4 requires further discussion. Atomic
53
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hydrogen diffuses out o f the bulk plasma, and molecular hydrogen diffuses into the bulk
plasma from the surrounding cold gas. The radially diffusing hydrogen atom s either
recombine in the surrounding cold gas or are pumped out, whereas axially diffusing
hydrogen atoms either recombine on the substrate or are reflected back into the bulk
plasma. The axial diffusion from the top o f the cylindrical plasma volume is accounted for
by increasing the surface area o f radial diffusion. The incoming hydrogen molecules
participate in the various reactions shown in Table 4.1 and produce H, IT , H2*, H,*, and
Hj~. Knowledge o f spatial concentration gradients and surface recombination probabilities
is essential to accurately account for diffusion. In the present work, a parametric
approach is followed. The inward diffusion o f FL, and the outward diffusion o f H are
lumped together in a net H loss coefficient y. For a homogeneous plasma, the diffusion
loss rate coefficient (#/s) is given by
kw - y
c As
4
Vp
y
—
4-VJ
H
^
Vp
(4.12)
In this work, the influence o f y on H concentration was investigated by parametrically
varying it between lxlO 'M . A net loss coefficient o f 1 indicates that all the H atoms
diffusing out o f the plasma are lost whereas a net loss coefficient o f 0 indicates that all the
H atoms diffusing out o f the plasma either bounce back into the bulk plasma o r are
replenished by the dissociation o f incoming H , molecules. The ratio A /V p converts the
surface reaction rate to volume reaction rate.
The convection rate coefficient (#/s) for the charged and neutral particles is given
54
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where f represents the gas flow rate in seem (4 x l0 17 is a conversion factor from seem to
particles/s).
In addition to the reactions shown in Table 4.1, the collisional and radiative
reactions responsible for the production and loss o f H were calculated as described in
Section 4.2. Based on the above reactions, the particle number density balance equations
can be written as follows:
dnH
^
i
*^■k <&f,H1n t . 2
*k g iHJ , *i k 7n [i^ i t .
j
♦k j t H:n H^
'* H
- k J ' H * k cf/t H '
CR&l
dnH£
'
k lP H .? t ’ k f tH f
<
'
k in H f *
■
*8 n H ( l *
■
k ^ lH f lH1 ' k £ p H {
(4.14)
dn
*5 tr *
. k, lcp2HfiH^ - kgi Hn t - kyijfji # - kj /i ff .
nt ’ nH• *
* nH;
The rate coefficients in the set o f Equations (4.14) were calculated by integrating
the relevant cross-sections over a non-Maxwellian EEDF obtained from ELENDIF. The
set o f Equations (4.14) w ere solved simultaneously by using ne, Tg, and E/N as parameters
55
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to obtain steady-state concentrations o f H, H2, IT , H 2", and H 3+.
(4.5) Electron Conservation Equation
The electron number density balance can be w ritten as follows:
dn
■ r , - Kr - Kj, - K cr
(4.15)
where Kj represents the total rate o f production o f electrons by ionization in the bulk
plasma and K,. represents the total rate o f loss o f electrons by recombination in the bulk
plasma. The electron loss rates by diffusion and convection are represented by
and
Kcc, respectively. For a hydrogen plasma, the electron conservation equation can be
written as follows:
. k ^ iH n t .
- k 5nf f . - k j i Hn f - k y i ^ n f
i
j
1
:
" , / x 4 x 1017
A
Similarly, for an argon plasma electron number density balance can be given by
(4.17)
56
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Equations (4.16) and (4.17) are functions o f average electron temperature, electron
number density, pressure, reactor geometry and gas flow rate. The electron temperature
dependence is implicit in the reaction rate coefficients (k) and the particle number densities
(nj). Several researchers (Hyman et al., 1992; Koemtzopolous et al, 1992; M ak et al.,
1997) neglected the role o f excited states in electron production. However, in argon
plasmas, it is well accepted that neglecting the role o f the excited states in electron
production leads to an overprediction o f average electron energy (Vriens, 1978; Ferreira
et al., 1985; Lymberopolous and Economou, 1993; Kelkar et al., 1997). It will be
demonstrated in Chapter VI that, for our operating conditions, the H-excited state
ionization in hydrogen plasmas is insignificant. However, the experimental and numerical
data for the excited states can be utilized to validate the numerically predicted average
electron temperature and electron number density.
(4.6) Pow er B alance E quatio n
The input microwave energy accelerates the electrons and ions present in the
plasma. The energy gained by ions is much less than that gained by electrons because the
ionic mass is at least 104 times greater than the electron mass. The ions are accelerated in
the DC sheaths at the plasma-surface interface. The total power balance can be written as
follows:
57
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where
represents the power deposited in the electrons and
represents the power
deposited in the ions. The electron energy balance can be written as follows:
(4.19)
The left hand side represents the electrical (ohmic) heating o f electrons. The energy gain
from the electric field is partly lost by the inelastic and elastic energy exchange in electronneutral collisions. The radiative losses comprise free-free and free-bound collisions
between electrons and either ions or neutrals. The third and fourth terms represent the
kinetic energy loss by electron diffusion and convection, respectively. Generally, it is
assumed that the absorbed power equals the amount o f energy deposited into the cavity.
However, it will be shown in Chapter 8 that, for argon plasmas, only 10% o f the pow er
coupled into the cavity is absorbed by the plasma.
The elastic energy exchange between the electrons and the plasma constituents is
given by
7 * (r. h
2
mh
(4.20)
where the heavy particle mass and temperature are represented by mh and Tp respectively.
The average collision frequency between electrons and species h is given by v eh; k is the
Boltzmann constant. The summation is performed over all the important species present
in the plasma (H, H,, and CH 4 for a H 2/CH 4 plasma and A r for a pure Ar plasma)
The inelastic electron energy loss comprises the vibrational, rotational, and
electronic excitation, dissociation, and ionization and is given by
58
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QmsLx ■ " « £ kjePj
j
(4.21)
where kj, ej; and n, represent rate coefficient, energy threshold, and species number density
involved in an inelastic process j, respectively. Sutton and Sherman (1967) reported the
inelastic energy loss factors for molecular species, 6 ^ as a function o f average electron
energy. These experimentally measured energy loss factors are multiplication factors and
are to be directly applied to the elastic energy loss rates to account for the enhanced
energy exchange in electron-molecule collisions. For a H 2/C H 4 plasma, it was found that
the electron collisions with H 2 and H produce the majority o f the inelastic losses and those
due to the electron-CH 4 collisions are negligible. Figure 4.2 shows the inelastic energy
loss factor for H 2. For an argon plasma, the energy losses produced by the excitation and
ionization were obtained from the CRM.
The free-free and free-bound radiative losses were calculated by integrating
Equations (3.4) over a wavelength range o f 100 nm-100 pm (Wilbers et al., 1991; Benoy
et al., 1993). N ote that the electron energy loss produced by the bound-bound line
emissions are already accounted for in the excitation losses.
The pow er deposited in the electrons is partly carried away as a kinetic energy by
the electrons via diffusion and convection. For a Maxwellian EED F, the average kinetic
energy o f electrons escaping the plasma volume is 2 kTe (Liebermann and Litchenberg,
1994). Thus the energy loss due to diffusion and convection o f electrons can be written
as:
59
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e-H
10
1
2500
5000
10000
7500
12500
15000
T e [K]
Figure 4.2 Inelastic energy loss factor fo r H2
*
60
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(
Da
Qdflr * @c
n t f x 4 x l 0 17
n . * — ------------
a2
«t
yp
(*«*.)
(4-22)
The power absorbed by the ions was calculated as shown in Liebermann and
Litchenberg (pp. 305-307, 1994). The kinetic energy gained by the ions entering the
sheaths depends on the sheath potential. The total kinetic energy o f ions leaving the
plasma depends on the initial kinetic energy o f ions entering the sheaths and the additional
kinetic energy gained by the ions because o f the acceleration in the sheaths. The ion
energy in the low sheath voltage microwave plasmas is approximately 5.2Te (Liebermann
and Litchenberg, 1994 pp. 300-315). Thus, the net kinetic energy loss due to ionic
diffusion and convection is given by:
„
Pabsj •
nPa»
n i
A2
«,
/*
4*10 17
\
(5.2 kTt)
(4.23)
r.p
(4.6) M ethods of Solution
Figure 4.3 depicts the modeling procedure used for the parametric study in the
H 2/CH 4 and Ar plasmas. This modeling scheme is used in Chapter V to assess the
influence o f the various plasma parameters. The electron number density, electric field,
and gas temperature and composition are specified in ELENDIF. The resulting EEDF is
used on the CRM to obtain the excited state number densities. The model also calculates
the power deposited and continuum emission for the specified plasma parameters.
Although this procedure does not provide self-consistent solutions it can be used to study
the relative importance o f various plasma processes.
61
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Figure 4.4 depicts the iterative scheme used to obtain self-consistent solutions in
hydrogen and argon plasmas. For an assumed electron number density, and gas
temperature and composition, the electric field is varied until electron conservation is
satisfied. Once a self-consistent solution is obtained the numerically predicted excited
states and continuum emission are compared with the experimental data (see Chapters VI
and VII). This modeling procedure does not predict the H 2 dissociation fraction in the
hydrogen plasma. However, it simplifies the computational efforts required to obtain a
self-consistent solutions.
Figure 4.5 depicts the iterative scheme used to self-consistently predict the
dissociation fraction in hydrogen plasmas. For an assumed electron number density and
gas temperature, the H-mol fraction and electric field are varied until H and electron
conservation equations are satisfied. Chapter IX describes the influence o f diffusion and
gas temperature on the H2-dissociation fraction.
Some discussion is necessary about the plasma power deposition. An exact
amount o f power deposition is necessary for valid predictions. Although the total amount
o f power deposited in the cavity is known, the exact amount o f pow er absorbed by the
plasma is unknown. We used power deposition as an additional parameter. Our
numerical models assume that the power is deposited uniformly in the bulk plasma
volume, Vp. The electron and ion energy losses are calculated based the electron number
density, average electron energy, gas temperature and composition. The numerically
calculated power deposition is then compared with the experimentally measured power
forwarded into the cavity.
62
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BO LTZM ANN
SOLVER
n-LEVEL
CRM
y ...................
l l j 's ,
Figure 4.3
Te, Pdep
Modeling scheme used fo r the parametric study
( j K ' s , p , T g, n e , R , L , 0
BOLTZMANN
E /N
SOLVER
I
—
n-LEVEL
CRM
N O
d n e /d t= 0
Y E S
n j 's ,
F ig u re 4.4
T e} E
/N j
Pdep
Iterative scheme to obtain self-consistent solutions
63
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#
BOLTZM ANN
E /N
Xh
SO LVER
I
EEDF
2 1 -L E V E L
H CRM
Z
H2
NO
t Z
d is s o c ia t io n
d n e /d t= 0
YES
X
NO
h ,assum.—XH.precIT
YES
X h, iij's, T e , E /N , Pdep
Figure 4.5
Iterative scheme for a comprehensive hydrogen model
64
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CHAPTER V
CHARACTERIZATION OF PLASMAS
(5.1) In tro d u c tio n
A plasma model must accurately account for all the physical processes for valid
predictions. Several researchers (Kune and Gundersen. 1984: Tahara et al. 1985: Repetti
et al.. 1990: St Onge and Moisan. 1993: Scott et al.. 1996: Mak et al.. 1997) assume a
Maxwellian EEDF to calculate reaction rates. However, low pressure weakly ionized
plasmas often have a highly non-Maxwellian EEDF w hich greatly influences the plasma
chemistry. Some o f the models reported in the literature (Koemtzopoulos et al.. 1992:
Penetrante and Kundhardt. 1986) neglect the electron-electron and superelastic reactions
in the solution o f Boltzmann equation, and do not account for the effect o f neutral
particle impact reactions on the excited state number densities. This chapter evaluates
the validity o f these assumptions and investigates the influence o f various plasma
parameters on the excited state populations. A comprehensive study is performed byusing electron number density (ne). electric field (E/N). gas com position (XH:. X H. XCM4.
and Ar) and temperature (T J as parameters.
65
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For H;/CH 4 plasmas, the EEDF deviates considerably from its Maxwellian form.
For our operating conditions, the superelastic vibrational and electronic collisions can be
neglected. For degrees o f ionization less than 10"1. the electron-electron collisions do not
influence the shape o f the EEDF. Inelastic processes cause the majority o f the electron
energy losses.
In Ar plasmas, the EEDF is highly non-M axwellian. Even for the lower degrees
o f ionization (< 10”‘). the electron-electron collisions are found to dramatically change
the shape o f EEDF as well as affect the average electron energy. Electron-electron
collisions drive the EEDF towards Maxwellian. Superelastic electronic collisions are
found to be insignificant for our operating conditions.
(5.2) R esults a n d D iscussion
Table 5.1 shows the range o f parameters studied and their base values in the
H:/CH 4 plasma. The parameters are held at their base values unless mentioned. The gas
temperature and H mol fraction ranges were selected based on experimental data
published in the literature (M cMaster et al.. 1994; St Onge and Moisan. 1994: Scott et
al.. 1996). M cM aster et al. reported a H: dissociation fraction o f 0.001 near the
substrate. This 1 part per 1000 value, then, represents a minimum; the amount o f H:
dissociation in the gas phase away from the substrate would be larger. The upper bound
on the H mol fraction is based on the measurements by Scott et al. (1996) in a similar
microwave reactor. St Onge and Moisan (1994) measured gas temperature as a function
o f pressure and pow er density in a pure H: microwave plasm a using Doppler broadening
66
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Table 5.1 Range of Parameters Studied in H:/CH4 Plasma
Pressure=40 Torn Flow rates=300 seem H2. 3 seem CH 4
Parameter
Range
Symbol
Base
Value
(Units)
E/N (Td)
20-60
50
H Mol Fraction
XH
0.001-0.3
0.05
Gas temperature
TS(K)
1000-4000
2000
Electron num ber density
ne (n r3)
l x i o 16- l x i o ig
1 * 1017
Plasma Radius
R (cm)
—
3.8
Plasma height
L (cm)
—
2.5
Electric field/Total gas num ber density
Table 5.2 Range of Parameters Studied in Ar Plasma
Pressure=5 Torr. Flow rate=250 seem
Parameter
Symbol
Range
Base
Value
(Units)
E/N (Td)
0.01-0.5
0.15
Gas temperature
Tg (K)
300-1000
350
Electron num ber density
ne (n r3)
3* 10l6- l x 10|g
5*10 17
Plasma radius
R (cm)
—
6.35
Plasma length
L (cm)
—
10
Electric field/Total gas num ber density
67
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o f H0. Scott et al. (1996) measured the vibrational and gas temperature using TALIF and
CARS in a hydrogen microwave plasma (20 Torr and 1000 W). Their results are
extrapolated to infer gas temperature in our slightly higher pressure plasma (40 Torr).
The direct probe measurements o f the electron num ber density are not feasible in
microwave plasmas because oscillating high frequency electric fields interfere with the
measurements. Our electron num ber density range is based on the theoretical (M oisan et
al.. 1992) and numerical results reported in the literature (Koemtzopoulos et al.. 1992:
Hyman et al.. 1992: Scott et al.. 1996).
Table 5.2 shows the parameter range and their base values in the Ar plasma. The
electron number density and electric field ranges were selected based on the typical
estimates for microwave discharges (Mak et al.. 1997). The gas temperature range was
selected based on the experimentally measured belljar temperature. For a 5 Torr
pressure. 680 W input pow er Ar plasma, the belljar temperature was 19 C. The gas
temperature is estimated using simple conduction heat transfer calculations and
assuming that the plasma is in close contact with the belljar. The plasma radius and
length were determined based on visual inspection. The parameters are held at their base
values unless mentioned.
Figure 5.1 compares a numerically predicted EEDF in a H:/CH 4 plasma (40 Torr
pressure. 1.6 kW input power) with a Maxw'ellian EEDF with the same average energy.
The numerically generated EEDF deviates considerably from its Maxwellian form for
electron energies above 9 eV. These high energy electrons are depleted in electronimpact excitations o f the H: and H ground states to the various electronic levels. The
68
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non-Maxwellian EEDF also shows a small depression for electron energies less than 2
eV. This depression is caused by the vibrational excitation o f H: to various levels with
energies between 0.5-2.0 eV. Figure 5.2 compares the EEDFs in an Ar plasm a (5 Torr
pressure. 680 W input power). The numerically predicted EEDF deviates from
Maxwellian for electron energies above 10 eV. This large depletion o f electrons is
caused by the electronic excitation o f Ar ground state to the 4s excited state with a
threshold energy o f 11.65 eV.
The depression o f the high energy tail (e>10 eV) in the EEDF greatly influences
the plasma chem istry. Figures 5.3 and 5.4 show Boltzm ann distributions o f excited
states generated from the EEDFs shown in Figures 5.1 and 5.2. respectively. The
excited state distributions clearly show the presence o f non-equilibrium. An equilibrium
excited state distribution will be represented by a straight line on a Boltzmann plot. The
escaping plasma radiation depletes the excited states. More importantly, note that the
excited state num ber densities predicted with the M axwellian EEDF exceed by orders o f
magnitude those predicted with the non-M axwellian EEDF. The non-Maxwellian
EEDFs have orders o f magnitude fewer high energy electrons compared to the
Maxwellian EEDFs. The threshold energies for the H and Ar ground state excitation to
their first excited states are 10.19 and 11.65 eV. respectively. Only electrons with
energies greater than the threshold energy can produce the electronically excited states.
Thus, the models which assume a Maxwellian EEDF will over-predict the excited state
number densities com pared to those which correctly account for a non-M axwellian
EEDF with the sam e average electron energy. M any important inelastic reactions, such
69
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plasma
40 Torr
>
■— MaxwslEan
— Numerically Predcted
10-
0
5
10
IS
20
Energy [eV]
Figure 5.1
Comparison o f a numerically predicted non-Maxwellian EEDF
and a Maxwellian EEDF with the average energy in a HJCH4
plasma
Arptaama
5 Torr
n# = 5x10,7 m
B N =0.15Td
e= 1.6 aV
T. = 350 K
Maxwelian
Numartcally Prsdictsd
Energy [eV]
Figure 5.2
Comparison o f a numerically predicted non-Maxwellian EEDF
and a Maxwellian EEDF with the average energy in an Ar plasma
70
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Hj /CH4 plasma
40 Torr
n , = 1xi017m-3
E/N = 50 Td
e = 3.0 eV
T -2 0 0 0 K
^ = 0.05
c
MaxwelSan
Numerically Predicted
10s
0
2.5
7.5
5.0
12.5
10.0
15.0
Energy [eV]
Figure 5.3 A Boltzmann plot comparing excited states predicted by using a
non-Maxwellian EEDF and those predicted by using a Maxwellian
EEDF with the same average energy in a Fl-JCH4plasma
i
i |
i i
i
10" . 1
1022 r
^
1021
1020 r
101S r
1018 r
1017 r
......... Maxwellian
1016 p
------- Numerically Predicted
10,s r
10" r
10'3 r
io ,z 1011 i 1 i 1 1
lO'O 1 • *
i
1
i
'Ar plasma
5 Ton
n ^ S x I O 17™-3
E/N = 0.15 Td
e= 1.6 eV
Tg = 350 K
J
i
-1
4
j
I
■
.
i
i
I
Energy [eV]
Figure 5.4 A Boltzmann plot comparing excited states predicted by using a
non-Maxwellian EEDF and those predicted by using a Maxwellian
EEDF with the same average energy in an Ar plasma
71
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as dissociation and ionization, have very high threshold energies. For example. H:. H.
and Ar ionization threshold energies are 15.4. 13.6 eV. and 15.755 eV . respectively.
Thus, the m odels which assume a Maxwellian EEDF with the sam e average electron
energy as a non-M axwellian EEDF will incorrectly predict the dissociation and
ionization fractions.
Electron-electron and electron-neutral elastic collisions replenish the high energy
electrons and drive the EEDF tow'ards its Maxwellian form. G inzberg and Gurevich
used a param eter P (^v^ydv^) to compare relative importance o f electron-electron to
electron-neutral collisions. The electron-electron and electron-neutral collision
frequencies are given by va and ven. respectively. The electron-neutral collision
frequency includes elastic as well as inelastic collisions. The electron mass (me) to
heavy particle m ass (M) ratio is given by 5 (2m,/M). Their study shows that the EEDF
is Maxwellian for P » 5 . For base values o f parameters in our H:/C H 4 and Ar plasmas. P
varies between 0.1 and 0.2 and is well below the M axwellain limit.
Figures 5.5 and 5.6 show the influence o f the degree o f ionization on the shape o f
the EEDF in FT/CH., and Ar plasmas, respectively. The electron-electron collisions
redistribute the energy among the electrons and thus drive the EED F towards
equilibrium. For our H:/C H 4 plasmas, the electron-electron collisions do not affect the
shape o f EEDF.
For Ar plasmas, the electron-electron collisions significantly influence
the shape o f the EEDF. The average electron energy increases w ith increasing electron
number density due to enhanced coupling between the electric fields and the electrons.
The EEDF shape depends on the electric field strength, and the electron-electron and
72
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Hj /CH4 plasma
40 Torr
E/N = 50Td
T *2000K
Xu = 0.0S
n =1x101S m43, c=3.0 eV.
na=1x10 m , e=3.0 eV,
n,=1x10,a m-3, e=3.0 eV.
n_=1x1019 in'3, &=3.0 *V,
P=0.008
P-O.076
P=0.68
P-6.14
Energy fevl
Figure 5.5
The effect o f electron number density on the EEDF in a HfCH4
plasma
10
T
h
10-’ F
io-3 F
S . —■s.
v. ^
io-5 F
>
io'7 F
io9 F
io’11 F
> io'13 F
m io-15 F
N
io*17 F
i o 19 F __ — na=3x1016 m' , e=1.18eV, P=1.43
io 21 F
— ne=1x10’7 m'
io 23 F .... .... ne=1x1O10 m'
io-^ F n •=1x1019 m' \e=2.18eV, P=1026
io^ F
io » F
IO*31 f c j— i
■
i
I
i
Arplasma
5 Torr
E/N = 0.15 Td
-
i
10
Energy [eV]
Figure 5.6 The effect o f electron number density on the EEDF in an Ar
plasma
73
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electron-neutral collision frequencies. In H:/CH 4 plasm as, the presence o f relatively
high electric fields dom inates the collisional effects. A dditionally, the inelastic electronneutral collisions dom inate the elastic electron-neutral and electron-electron collisions in
predominantly molecular H 2/C H 4 plasmas. A com bination o f these effects leads to an
insignificant role o f electron-electron collisions in H:/C H 4 plasmas. A numerical study
using ELENDIF indicates that, for our H:/CH 4 plasma, electron number densities in
excess o f 1*10:: m '3 (P=3908) are necessary to obtain a M axwellian EEDF. However,
for the electron num ber density range in our Ar plasma, the electron-electron collisions
are extremely important because o f the relatively low electric field strengths. A
numerical study using ELENDIF indicates that, for our A r plasma, electron number
densities in excess o f 1 * 10:o m '3 (P=8894) are necessary to obtain a M axwellian EEDF.
Colonna et al. (1993) observed a similar phenomenon in N : discharges.
Figures 5.7 and 5.8 show the influence o f the degree o f ionization on excited
state production in H;/C H 4 and Ar plasmas, respectively. The electron impact excitation
rates increase with increasing electron number density and average electron temperature.
Although the average electron energy remains constant in H 2/C H 4 plasmas, higher
excited state number densities are produced with increasing electron number density.
For all other parameters at their base values, a comparison o f the numerical and
experimental data in our H:/CH 4 indicates that the electron num ber density is around
I * 10 17 m'3. For Ar plasm as, the excited state number densities dramatically increase
with increasing electron number densities. This significant increase in the excited state
number densities is attributed to the considerable increase in the average electron energy
74
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Hj /CHj plasma
40 Torr
E/N = 50Td
T =2000 K
Xh-0.06
0
5
10
IS
Energy [eV]
Figure 5.7 A Boltzmann plot showing the effect o f electron number density on
the excited states in a H JCH 4plasma
Ar plasma
SToir
r
10! h
n8=3x10 b m , e = 1.18 aV
na=1x10l 7 m'3>e = 1.30 aV
n,=1 x10l 8 m‘3, e = 1.75 eV
nB=1x10’9 m -3 , E = 2-18eV
r
103 5-
o
5
10
15
Energy [eV]
Figure 5.8 A Boltzmann plot showing the effect o f electron number density on
the excited states in an A r plasma
75
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with increasing electron num ber densities. For all other parameters at their base values,
a comparison o f the numerical and experimental data in our Ar plasm a indicates that the
electron number density is around 5* 10 17 m \
In addition to electron-electron collisions, superelastic collisions, in which the
low energy electrons exchange energy with excited states, can replenish the high energy
electrons lost in vibrational and electronic excitations. Knowledge o f excited state
number densities is essential to investigate the effect o f superelastic collisions. In this
work, the effect o f vibrational and electronic superelastic collisions on the EEDF was
parametrically investigated. The vibrational superelastic collisions were studied by
assuming that the H: vibrational temperature varies between the estimated gas
temperature (2000 K) and the average electron temperature (22.500 K). Figure 5.9
shows a comparison o f EEDFs generated by specifying various values o f vibrational
temperature in ELENDIF. It is observed that the fractional populations o f the low
energy electrons (e<2 eV) decrease with increasing vibrational temperature. This result
is counter-intuitive because one expects that the superelastic collisions will replenish the
low energy electrons lost in vibrational excitations. The reason for this seemingly
counter-intuitive observation is the presence o f electron-electron collisions in our model.
The energy gained through superelastic vibrational collisions is redistributed by the
electron-electron collisions. The electrons follow an EEDF corresponding to a higher
average electron temperature and thus, effectively, the fractional populations o f the lowenergy electrons are reduced. Figure 5.10 shows the Boltzmann distributions for various
values o f vibrational temperatures. A small increase in excited state number densities is
76
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Hj /CH4 piasma
40 Torr
CJ
IO-*
Tv = 0 K e = 3.0 aV
Tv-5 0 0 0 K .e« 3.36 aV
Tv = 10.000 Kt = 3.72 «V
IO"6
0
5
15
10
20
Energy [eV]
Figure 5.9
The effect o f vibrational superelastic collision on the EEDF in a
HJCH4plasma
10
—Tv = 0 K ,e= 3 .0 eV
- Tv = 5000 K, e = 3.36 *V
— Tv = 10.000 K, c = 3.72 aV
5
10
15
Energy [eV]
Figure 5.10 A Boltzmann plot showing the effect o f vibrational superelastic
collisions on the excited states in a H fCH 4plasma
77
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attributed to the increase in average electron energy. In the presence o f superelastic
reactions, a slightly sm aller E/N is required to maintain the same average energy when
superelastic reactions are neglected. Recently. Scott et al. (1996) m easured vibrational
and gas temperatures using TALIF and CARS in microwave hydrogen discharges. They
found that the vibrational and gas temperatures differ by less than 5%. Figures 5.9 and
5.10 show that, for vibrational tem peratures around 2000 K. superelastic collisions can
be neglected. Because o f the negligible effect on the results, all o f the subsequent data
neglect superelastic vibrational collisions.
The influence o f superelastic electronic collisions is investigated by specifying
fractional populations o f H 2 electronically excited states in ELENDIF. Figure 5.11
shows a comparison o f EEDFs for various fractional populations o f H 2 electronically
excited states. A significant increase in the number o f high energy electrons and
average electron energy is only observed for the fractional populations o f H 2 electronic
levels in excess o f IO-4. The H 2 electronic excited states lie between 8.9 and 13 eV. The
observed increase in the num ber o f electrons with energies greater than 8 eV is due to
collisions between low energy electrons and H 2 excited states. Some o f these electrons
then collide with other H 2 excited states to yield an abundance o f electrons above 16 eV.
Figure 5.12 shows a com parison o f Boltzmann distributions for various fractional
populations o f H 2 electronic excited states. A significant increase in the excited state
number densities is seen only when the fractional populations o f H 2 electronic states
exceed 10"4. This observed increase is due to an increase in the average electron
temperature and a subsequent increase in the electron impact excitation rates. We
78
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\
n7n=10-®. e = 3.0aV
n"/n = 10'5, e = 3.0 aV
n / n = 10'2. t = 3.34 eV
0
5
15
10
20
Energy [eV]
Figure 5.11 The effect o f H2-superelastic electronic collisions on the EEDF in
a HJCH4plasma
Hj /CHj plasma
40 Toit
E/N = 50 Td
T = 2000 K
Xh = 0-05
109 g-
— n’/n = 10'9. s = 3.0 *V
- n7n = 10's.e = 3.0eV
— n'/n = 10*2, e = 3.34 «V
107 f
0
5
10
15
Energy [eV]
Figure 5.12 A Boltzmann plot showing the effect o f H2-superelastic electronic
collisions on the EEDF in a H fCH 4plasma
79
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estimated the H 2 electronically excited state populations using a com bination o f
numerical and experimental results. For our operating conditions, a LTE temperature
(5500 K.) was calculated using the experimentally measured 3s excited state number
density. Since the excitation threshold energies for the first excited states o f H 2 (8.9 eV)
and H (10.19 eV) are approxim ately the same, the H2 electronic states are also assumed
to follow a Boltzmann distribution at an excitation temperature o f 5500 K. The
fractional populations for the H 2 electronically excited states were found to be between
1O'8- 10*13. Such low populations o f H: electronically excited states do not affect the
EEDF and excited state distributions. A similar study in our Ar plasma showed that the
superelastic electronic collisions can be neglected in the solution o f Botlzmann equation.
Because o f the negligible effect on the results, all o f the subsequent data neglect
superelastic electronic collisions.
Figures 5.13 and 5.14 show the effect o f electric field strength (E/N) on the
EEDF in H 2/CFI4 and Ar plasm as, respectively. The average electron energy increases
with increasing E/N. Equation 4.1 shows that an increase in the electric field results in
the acceleration o f electrons if other conditions remain the same. In an Ar plasma, a
much lower E/N is required to accelerate the electrons to the same average electron
energy compared to that in a H 2/CH 4 plasma. In H 2/CH 4 plasmas, a 50% increase in E/N
results in a 50% increase in the average electron temperature whereas, in A r plasmas, a
factor o f 3 increase in E/N is necessary to increase the average electron temperature by
50%. This behavior can be explained by comparing the electron-neutral momentum
exchange collision frequencies for the FI2/CH 4 and Ar plasmas. The electron-neutral
80
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IO-1
E/N = 40 Td, e = 2.40 eV
E/N = 50 Td, e= 3.0 eV
E/N = 6 0 Td, e= 3.56 eV
5
10
15
20
Energy [eV]
Figure 5.13 The effect o f electric field on the EEDF in a HJCH4plasma
Arplasma
5 Torr
ngsSxIO ^m -3 1
T =350 K
n
>
ffl
w
E/N = 0.05 Td. e = 1.08 eV
E/N = 0.15 Td. e = 1.60 eV
E/N = 0.35 Td, 6 = 2.08 eV
IO*1
,-13
0
5
10
15
20
Energy [eV]
Figure 5.14 The effect o f electric field on the EEDF in an Ar plasma
81
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Hj /CHj plasma
40Torr
ne = 1x10l7 m'3
T . = 2000 K
Xh = 0-0S
3" 10
cf* 10
c IO1
10? i
10® r
E/N = 40T d,e=2-40eV
E/N = SOTd, e = 3.0 «V
E/N = 60Td, e = 3.56eV
5
10
15
Energy [eV]
Figure 5.15 A Boltzmann plot showing the effect o f electric field on the
excited states in a HJCH4plasma
Ar plasma
-j
5 Torr
n# = 5x1017m'3-j
T = 350 K
-i
E/N = 0.05 Td. e= 1.08 eV
E/N =0.15 Td. e= 1.60 aV
E/N = 0.35 Td. e = 2.08 aV
10®
0
5
10
IS
Energy [eV]
Figure 5.16 A Boltzmann plot showing the effect o f electric field on the
excited states in an Ar plasma
82
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momentum collision frequency in the Ar plasma increases three times faster than that in
the H-,/CH4 plasm a w ith increasing electric field strength. Thus, a smaller increase in the
collisional loss freqencv in the Ar compared to the H;/C H 4 plasm a leads to a smaller
increases in the average electron energy. Figures 5.15 and 5.16 show Boltzmann
distributions for various values o f E/N (or average electron energies) in HyCH 4 and Ar
plasmas, respectively. An increase in the average electron energy increases the reaction
rate coefficients and results in higher excited state num ber densities. For the base value
parameters, the com parison o f the numerical and experim ental data indicates that the
electric field strengths in our H:/CH 4 and Ar plasmas are approximately 50 Td and 0.15
Td. respectively.
Total gas num ber density depends on the operating pressure and the gas
temperature and can influence the EEDF and average electron energy. The electronneutral collision frequency increases with increasing gas density and influences the
EEDF and the average electron energy. Figures 5.17 and 5.18 show a comparison o f
EEDFs for various gas temperatures at constant pressure in H:/CH 4 (40 Torr) and Ar (5
Torr) plasm as, respectively. An increase in the gas tem perature rarefies the gas and thus
reduces the electron-neutral collision frequency. The average electron energy increases
because the electrons suffer fewer collisions with increasing gas temperature. Figures
5.19 and 5.20 show the Boltzmann distributions for various values o f gas temperature in
H;/CH 4 and A r plasm as, respectively. Note that an increase in gas temperature decreases
the ground state num ber density. The excited state num ber densities remain almost the
same because o f the counteracting effects o f the increased average electron energy and
83
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HJGM i plasma
40T
o «t
n#= IxlO^m-3
E/N = 50Td
Xh= °05
T_ = 1000 K, e= 2.99 «V
T =2000 K , « - 3.00 «V
T =3000 K,e= 3.15 «V
T =4000 K. 8=3.2$ *V
10
Energy [eV]
Figure 5.17 The effect o f gas temperature on the EEDF in a HfCH4plasma
Arplacma
5 Tonn8 = SxlO^m-3
T„ = 350 K
=
!
1
1
r
H
>
r
T = 3 5 0 K ,a = I.6 0 « V
T = 500 K, e = 1.63 #V
T = 1 0 0 0 K ,e = 1 .6 7 * V
v
ris
0
5
15
10
20
Energy [eV]
Figure 5.18 The effect o f gas temperature on the EEDF in an Ar plasma
84
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Hj/CH^ plasma
40 Tonn, = 1x10 Tm
E/N = 50Td
Z« = 0.05
T =1000 K,e = 2.99 aV
T® = 2000 K.a = 3.00 aV
T. = 3000 K. e = 3.15 aV
T = 4000 K, e = 3.26 aV
Energy [eV]
Figure 5.19 A Boltzmann plot showing the effect o f gas temperature on the
excited states in a HJCH4plasma
T = 350 K, a = 1.60 eV
T =500 K.e= 1.63 aV
T =1000 K,e= 1.67aV
0
5
10
15
Energy [«V]
Figure 5.20 A Boltzmann plot showing the effect o f gas temperature on the
excited states in an Ar plasma
85
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decreased ground state num ber density. The distribution o f the higher excited states
(n> 6 ) changes considerably w ith the gas temperature. Interestingly, it is observed that
these higher excited states follow a Boltzmann distribution at the input gas temperature.
The threshold energies for transitions between these states is small (c<0.3 eV).
Although the gas temperature (2000 K) is much smaller than the electron temperature
(22.500 K). the heavy particle collisional rates are sufficient to govern the excited state
populations because o f their larger number densities (1 * 10:5 m °) com pared to the
electron number density (1 * 1 0 17 m '3).
Gas composition affects the shape o f the EEDF and the average electron energy.
In turn, the EEDF determines the reaction rates and thus the plasm a composition. A
self-consistent procedure is necessary to determine the gas com position. O ur diamond
deposition plasma primarily consists o f H:. H. and small am ounts o f CH., (<1%). The
influence o f CH 4 at such low concentrations is found to be insignificant. The importance
o f atomic hydrogen for diam ond deposition is well established in the literature (Spear
and Frenklach. 1989; Harris. 1989). Therefore, it is imperative to study the influence o f
H mol fraction in shaping the EEDF. Figure 5.21 compares EEDFs for various H mol
fractions. A small increase in average electron energy is observed with an increase in Hmol fraction. Molecular hydrogen can exchange energy with electrons much more
effectively than atomic hydrogen because o f rotational and vibrational excitation modes.
An increase in H number density at constant pressure reduces H; num ber density and
results in higher average electron energy.
Figures 5.22 and 5.23 show the relative contributions o f electron energy loss
86
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H JC H a plasma
40 T o i t
%
>
o
cj
- X M= 0.001.* = 2.78 eV
• XH = 0.01,6 = 2.80 »V
— XH = 0.1,6=3.05 eV
- XH = 0.2, « = 3.35 *V
0
5
10
15
20
Energy [eV]
Figure 5.21 The effect ofH-mol fraction on the EEDF in a H^/CH4plasma
87
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100
s«
3o
—i
Elastic
Rotational*Vibrational
Electronic
Kinetic Energy
e
UJ
40
c
o
u
_o
UJ
20
~ f—
10000
T
I— i
r— r —
1— i— i —
25000
20000
15000
T 9 [K]
Figure 5.22 Electron energy loss processes in a H-JCH4plasma
100
Ar plasma
5Torr
ne = 5x10’7m
T„= 350 K
3O
—
J
>»
E>
UJ
c
o
uo
60
Inelastic
Elastic
40
Kinetic Energy
h_
111
8000
10000
12000
14000
16000
18000
20000
t8[ki
Figure 5.23 Electron energy loss processes in an Ar plasma
88
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processes as a function o f average electron energy in H;/C H 4 and A r plasmas,
respectively. For H2/CH 4 plasmas, vibrational excitation o f H 2 is responsible for the
majority o f electron energy loss over a wide range o f average electron energies. The
energy exchanged between the vibrationallv excited states and neutral gas particles
results in gas heating. The contribution o f electronic excitation loss increases rapidly
w ith increasing average electron energy. Although elastic energy losses due to electronneutral collisions increase slowly with increasing average electron energy, their
fractional contribution to the total loss decreases with increasing average electron
energy. For Ar plasmas, elastic electron-neutral collisions are responsible for the
m ajority o f electron energy loss at lower average electron temperatures (Te< 14.000 K).
However, inelastic losses due to electronic excitation and ionization increase
considerably with an increase in average electron temperature and become dominant
above 14.000 K.
(5.3) S u m m ary
A parametric study is performed in H 2/CH 4 (1 .6 kW . 40 Torr) and Ar (680 W. 5
Torr) plasm as to assess the importance o f various physical processes. The study shows
that the EEDF deviates considerably from its Maxwellian form for both plasmas. The
electron-electron collisions influence the shape o f the EEDF and the average electron
energy in Ar plasmas even for small degrees o f ionization (10'6) whereas, for H 2/CH 4
plasm as, their influence is negligible. For our operating conditions, superelastic
collisions can be neglected in both plasmas.
89
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CHAPTER VI
ROLE OF EXCITED STATES
(6.1) Introduction
Generally, the fractional population o f electronically excited particles in low
pressure, weakly ionized plasmas ranges between 10‘14- 10 '€. Researchers often assume
that excited states do not influence the glow discharge physics and neutral chemistry, and
neglect excited state reactions— including ionization— in their plasma models (Hyman et
al., 1992; Koemtzopolous et al., 1994; St. Onge and M oisan, 1994; Scott et al., 1996;
M ak et al., 1997). Research on positive direct current (DC) discharges, however, has
established the importance o f two-step ionization and its effect on the required electric
field to maintain a discharge. Ferreira et al. (1985) developed a self-consistent discharge
model for low pressure argon positive columns by coupling the balance between the
electron production and loss rates to the steady-state rate balance equations for the
excited states. By comparing numerical and experimental data, they demonstrated the
excited states' dominant role in electron production. Ferreira et al. also experimentally
verified that the strength o f the sustaining electric field is considerably lower (by about
90
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50%) than that required for sustaining the discharge if only ground state ionization w ere
modeled. Vriens (1978) also observed the same effect in N a-N e discharges. According to
Vriens' research, tw o-step ionization lowered the values o f the self-sustaining electric field
by 25-50%. Lymberopoulos and Economou (1992) discovered a similar phenomenon in
an argon DC discharge which was due to step-wise ionization from metastable excited
states.
In this chapter, the optical emission spectroscopic measurements o f Ar and H
excited state number densities are compared to those predicted by our CRMs described in
Chapter IV. A comparison in a 680 W input power, 5 T orr pressure Ar plasma reveals
that all the numerical predictions require electron number densities between 3 x 1017 and
1 x 1018 m'3, electric field strengths between 0.05 and 0.35 Td, and average electron
temperatures between 9000 and 14,000 K. It is dem onstrated that, over the entire range
o f solutions, step-wise ionization from Ar-excited states is the dominant mechanism for
electron production and that neglecting it results in an over-prediction o f the selfsustaining electric field and average electron tem perature by 30-60%. Ambipolar difiusion
and three-body volume recombination are the major electron loss mechanisms.
F or a 1.6 kW input power, 40 Torr pressure H->/CH4 plasma, it is shown that all the
theoretical predictions require electron number densities between 6*10 16 and 3* 1017 m*3,
electric field strengths between 40 and 55 Td, and average electron temperatures between
21,000 and 24,000 K. Interestingly, it is found that the step-wise ionization is insignificant
and that the direct ionization from H, and H ground states produce the majority o f the
electrons. The number densities o f the first four excited states are dramatically reduced
91
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because o f collisional quenching with H ,. In the presence o f quenching, very large values
o f E/N and average Te are required to sustain the plasma solely through direct ionization.
Molecular and atomic ion recombination in the bulk plasma represent the major electron
loss mechanisms.
(6.2) Argon Plasma
The discharge physics in an Ar plasma is simpler then in a H 2 plasma because o f
the absence o f molecular species. Interestingly, it was difficult to stabilize the argon
plasma. The plausible explanation is that the microwave cavity used in this w ork was
specially designed to operate with H 2 containing small amounts o f CH4. The short, probe
and substrate heights define the electric field distribution inside the cavity. After several
tuning adjustments a stable Ar plasma was obtained at 5 Torr pressure, 250 seem flow rate
and 680 W input power. The plasma filled the entire belljar and was in contact with the
belljar and the substrate.
A parametric study was performed by varying the electron number density, selfsustaining electric field, absorbed power, and gas temperature. The plasma power
deposition is allowed to vary between 5-900 W because the exact amount o f power
absorbed by the plasma is unknown. Although the power coupled into the cavity is 680
W, a maximum plasma power absorption o f 900 W is used to account for the uncertainties
in the numerical data. A self-consistent solution is obtained when the electron
conservation is satisfied. The numerically predicted excited state number densities are
compared with those experimentally measured. The numerical and experimental excited
92
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states are matched within a factor o f 2 to account for the uncertainties in the numerical
data used in the models. The experimentally measured excited states are found to be
uncertain within ±45%. Once the electron conservation is satisfied and the experimental
and numerical excited states are matched the model inputs represent the self-sustaining
plasma parameters.
Figure 6 .1 shows the variation o f electron production and loss rates with average
electron temperature. The reaction rates were calculated by varying E/N and holding all
other parameters at their base values. It is observed that the excited state ionization rates
exceed the ground state ionization rates over the entire range o f average electron
temperatures by 2-4 orders o f magnitude. Ambipolar diffusion increases with increasing
average electron temperature and represents the dominant electron loss mechanism. In the
bulk plasma, the electrons and ions are lost by three-body collisional recombination and
radiative recombination. The collisional recombination require a third body to absorb the
energy released in recombination. This third collisional partner can be either an electron
or a neutral particle. However, in the presence o f a surface the recombination energy is
absorbed by the surface resulting in surface heating. Thus, the electrons and ions diffusing
out of the bulk plasma recombine on the belljar and the substrate at much faster rates than
the volume recombination rates. Under steady state conditions, a self-sustaining plasma is
obtained when electron loss and electron production rates are equal. Figure 6 .1 reveals
that, in the presence o f excited states, an average electron tem perature o f approximately
12,500 K is required to sustain the plasma. However, an electron temperature in excess o f
20,000 K is necessary to sustain the plasma when only ground state ionization is allowed.
93
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i
i
|
i
i
I
i
|
i— i ■ i
i
|
I— I
I
I— i”
i — i— i— r -
Ar plasma
5 Torr
n0 = Sx1O17m-3
Tfl = 350K
- Gr. S t Ion.
Ex. S t km.
Tot Raoomb.
Amb. Dilf.
J I I L.
10000
_l
15000
I
I
I
I
I
I
I
25000
20000
30000
T8 [K]
Figure 6.1 Electron production and loss rates in the Ar plasma
Ar plasma
5 Torr
40000
T = 350 K
Wifi stapvwaa ion. _
Without stepwiso ion-
30000
20000
10000
1O10
tm'3!
Figure 6.2 Self-sustaining average electron temperature as a Junction of
electron number density with and without two-step ionization in
the Ar plasma
94
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Figure 6.2 shows the variation o f average electron tem perature required to
theoretically sustain a plasma as a function o f electron number density. An increase in
electron number density drives the EEDF tow ards its Maxwellian form (Fig 5.6). The
ionization rates increase with increasing electron number density due to the increase in the
number o f high energy electrons. Thus, a smaller average electron tem perature is required
with increasing electron number density to replenish the electrons lost by ambipolar
diffusion and recombination and to sustain the plasma. The role o f excited states was
investigated by numerically neglecting ionization and recombination involving excited
states. N ote that the average electron temperature in the absence o f the excited states
exceeds that in their presence by 30-50%.
Corresponding to the self-consistent solutions, Figure 6.3 shows a few plausible
solutions which have a reasonable agreement between the numerically predicted and
experimentally measured excited state number densities. The parametric study revealed
that all the solutions require electron number densities between 3*10 17 and 1x 1018 m'3 and
average electron temperature between 9000 and 14,000 K. Correspondingly, the power
absorbed by the plasma varies between 15 and 70 W (Chapter VIII).
Figure 6.4 shows the influence o f gas temperature on the self-sustaining average
electron temperature. An increase in the gas temperature reduces the total gas number
density and thus the electron-neutral collision frequency. The gas tem perature does not
affect the diffusion loss o f electrons. Thus, to maintain the same ionization rate, a higher
average electron temperature is required. N ote that the self-sustaining average electron
tem perature in the presence o f the excited states is lower by 20-50% than that in their
95
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Ar plasma
5 Torr
T . = 350 K
ne=3x1017 m-3 ; Te=14000 K
n,=5x1017 m '3 ; T#=12500 K
n,=2x1018m-3 ;T#=9300K
•
0
Experimental
5
10
15
Energy [eV]
Figure 6.3
A Boltzmann plot comparing the numerical and experimental
excited state number densities in the Ar plasma
25000
Ar plasma
5 Torr
n . = 5x1017 m
22500
20000
—With ttopvusa ion.
Without stepwisa ion.
17500
15000
12500
10000
200
400
600
800
1000
1200
T„[K]
Figure 6.4 Self-sustaining average electron temperature as a function o f gas
temperature with and without two-step ionization in the Ar plasma
96
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absence. Figures 6 .2 and 6.4 demonstrate the strong influence o f excited states on the
average electron temperature. Thus, the electron conservation equation must include
reactions involving Ar excited states to ensure accurate predictions.
Previous researchers (Liebermann and Litchenberg, 1994; Mak et al., 1997)
neglected the presence o f excited states in their models. Implicit in their neglect is the
assumption that the number densities o f the excited states are sufficiently low such that all
reactions involving these states— including ionization— can be neglected. However, the
CRM predictions demonstrate that ionization from the Ar-excited states produces more
than 99% o f the free electrons (Figure 6 .1). This important step-wise production o f
electrons is explained in Figure 6.5. Although there are orders o f magnitude fewer atoms
in the excited states than in the ground state (Figure 6.3), ionization from the excited
states dominates because there are orders o f magnitude more electrons w ith energy
sufficient to ionize from the excited states (< 4 .1 eV) than there are electrons with
sufficient energy to ionize directly from the ground state (15.755 eV). O f course, to
create the excited states from the ground state, electrons with energy at least 11.65 eV are
required. Therefore, if only ionization from the Ar-ground state is modeled, to maintain a
specified electron number density, a sufficient number o f electrons with energy greater
than 15.755 eV (argon's ground state ionization energy) are required. In contrast, if twostep ionization is included, then only a sufficient number of electrons with energy greater
than 11.65 eV (argon's ground state excitation energy) are required. Theoretically, these
high energy electrons are produced through interactions with the applied electric field.
Thus, when tw o-step ionization is neglected, a larger electric field and subsequently a
97
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larger average electron tem perature (30-50%) is required to generate, theoretically, the
needed number o f electrons w ith energy greater than 15.755 eV.
Recently, Mak et al. (1997) measured the electron num ber density in the
downstream region o f a microwave plasma (1-10 mTorr, 277 W). Recently, the use o f
low-pressure plasmas (<100 m Torr) has become widespread in etching o f metal or
dielectric thin films used in semiconductor manufacturing (Liebermann and Litchenberg,
1994). Although this pressure regime differs considerably from that (5 Torr) studied in
the present work, it is interesting to evaluate the viability o f conclusions reached in the
present w ork under different operating conditions. Figure 6 .6 shows the variation o f
average electron temperature with pressure. Experimental data for electron number
density (2* 1017 m‘3), plasma geom etry (R=6.2 cm, L=8.5 cm) reported by Mak et al.
(1997) were used. Even under low-pressure conditions, the average electron temperature
required to theoretically sustain a plasma in the presence o f the excited states is lower by
30-50% compared to that in the absence o f the excited states. Thus, the models used for
low-pressure plasma processing must include step-wise ionization involving excited states.
Figure 6.7 evaluates the validity o f assuming a Maxwellian EEDF. Recall that a
non-Maxwellian EEDF has orders o f magnitude fewer high energy electrons (e> 15.75 eV)
compared to a Maxwellian EED F with the same average electron temperature (Figure
5.2). Thus, the models which assume a Maxwellian EEDF underpredict the self-sustaining
average electron temperature by approximately 5% compared to those which correctly
account for a non-Maxwellian EEDF. It is observed that accounting for the nonMaxwellian EEDF becomes more important at higher pressures. For our 5 Torr pressure
98
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Argon Energy Level Diagram
ip s is
Figure 6.5
Two-step ionization
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100000
T JK ]
aoooo
With stepwise ion.
Without stepwise ion.
60000
40000
0
8
6
2
10
12
^
p [mTorr]
Self-sustaining average electron temperature as a junction of
pressure with and without two-step ionization in Ar plasmas (Mak
et al., 1997)
60000
Ar plasma
STorr
ne = 2x1017m
T„ = 350 K
T0 (K]
50000
Numericaly Predicted
Maxwellian
40000
30000 -
0
2
4
a
6
10
12
p [mTorr]
Figure 6.7 The effect of non-Maxwellian EEDF on the self-sustaining average
electron temperature in low pressure Ar plasmas (Mak et al.,
1997)
100
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Ar plasma, a 25% larger average electron temperature is required to sustain the plasma
when non-Maxwllian EED F is used.
It is observed that the self-consistent solutions can not simultaneously match all
three excited states within experimental uncertainties. The uncertainties in the
experimental data are caused by the uncertainties involved in measuring the plasma signal,
calibration lamp temperature, and path length. Also, the uncertainties on the Einstein's
transition probabilities can influence accuracy o f the experimental measurements.
Conservatively accounting for these uncertainties the excited state number densities were
found to be accurate within ±45%.
Some discussion is necessary about the uncertainties in the numerical results. O ur
model used cross-sections and Einstein transition probabilities from Kimura et al. (1985).
They reported that some o f the collisional excitation cross-sections and Einstein transition
probabilities are accurate within a factor o f 10. Thus, a complete sensitivity analysis is
essential to determine the influence o f the uncertainties on the excited state predictions.
O ur CRM is based on Repetti et al.'s (1990) model which lumps the Ar excited states into
25 levels. This lumped approach calculates the effective rates for the collisional and
radiative reactions between the various levels. A direct comparison o f the excited state
number densities predicted by this lumped model with the experimental data is not
feasible. We compared the numerical (iv ) and the experimental data (n^J using the
following relation (Gordon, 1992):
exp
(e« - eft)
kT
101
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(6. 1)
where e and g represent the energy threshold and degeneracy. The numerical excited state
densities obtained using Equation 6.1 differ by less than 10% from the experimental data.
O ur C RM results indicate that the 4p-4s and 5p-4s transitions are self-absorbed.
The experimental measurements were made by carefully selecting optically thin transitions.
Repetti et al. (1990) used Einstein probability for the optically trapped transition in their
lumped model. W e investigated the influence o f Einstein's coefficient on the model
predictions by parametrically varying it between that for the optically thin (6.3 * 10s) and
optically thick (3.35* 107) transition.
Table 6 .1 compares the numerical and experimental excited state num ber densities.
It is observed that all three o f the experimentally measured excited states can be matched
within their experimental uncertainty (±45%) when Einstein probability for the 4p to 4s
transition equals 1* 106. Our numerical study also demonstrated that the numerical and
experimental data can be matched when the electron impact cross-section for excitation
from the ground to the 4s excited state is increased by a factor o f 10. This factor o f 10
increase is within the reported uncertainty on the cross-section data (Kimura, 1985). It is
onbserved that the excited state distribution is more sensitive to the radiative rates than the
collisional excitation rates. Additionally, the reduction in the radiative rates is theoretically
supported by the optical trapping theory. The 4s excited state should be measured to
determine the amount o f trapping, and to justify the reduction in the radiative rates.
Although a reasonable agreement between the numerical and experimental d ata can be
obtained by modifying a variety o f reaction rates within their numerical uncertainty, the
goal was to dem onstrate that all three o f the experimentally measured excited states can be
102
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simultaneously matched within their experimental uncertainties.
T able 6 .1 C o m p ariso n of N um erical a n d E xperim ental E xcited States (A r Plasm a)
n =5xl017 mJ, Tg=350 K
Level
E xpt.
N um erical (n/gj)
N um erical (nj/gj)
N um erical (nj/g,)
(nj/gj)
A 4M = 3.34xl0 7 s 1
A 4M = 1 .0 x l0 6 s' 1
Q3p-ux1°
4p
2 .0 * 1013
l.lx lO 13
2.48x10“
2 .2 2 x l 0 13
5p
7.0x10“
3.15x10"
5.53x10"
5.74x10"
5d
8 .5 x l0 10
9.6 7 x l0 10
1.24x10"
1.2 2 x 10 "
(6.3) H y d ro g en Plasm a
A systematic study was performed by using the electron number density, electric
field, H mol fraction, and gas temperature as parameters. The absorbed power is varied
between 400 to 2400 W to account for uncertainties on numerical data and power
coupling efficiency. The numerical and experimental excited state number densities are
matched w ithin a factor o f 2 , and the solutions are verified for electron conservation.
Figure 6.8 shows electron production and loss rates as a function o f average
electron tem perature for an electron number density o f 1 x 1017 m‘3, gas temperature o f
2000 K, and H mol-fraction of 0.05. For average electron temperatures less than 10,000
K, the majority o f electrons are produced from the H excited states. The H 2 and H ground
state ionization rates are comparable. M olecular ion recombination rates exceed atomic
recombination rates by a factor o f 10-20. Ambipolar diffusion is the major electron loss
mechanism and exceeds all production rates. Thus, in this region the plasma will not
103
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:HXplasma
n, = 1a17 m'
T =2000 K
Xh=0.05
H Gr. S t Ion.
H Ex. S t Ion.
H Raccmb.
£
H2 Ion.
— - Mol. Ion. Racomb.
Amb. Diff.
10’
10000
30000
20000
40000
T„[K]
Figure 6.8 Electron production and loss rates in the H-JCH4plasma
24000
23000 22000
T = 2000 K
X u = 0.05
I-a
21000
20000
— Wilh sWpwisa ion.
— Without stopwica ion.
19000
18000
1 0 '9
Figure 6.9 Self-sustaining average electron temperature as a function of
electron number density with and without two-step ionization in
the HJCH4plasma
104
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sustain. For average electron temperatures above 15,000 K, the majority o f electrons are
produced from the H 2 and H ground states, and H excited state ionization is negligible.
Molecular ion recombination and ambipolar diffusion represent the important electron loss
mechanisms. Note that the H 2 ionization and molecular ion recombination rates are almost
equal for average electron temperatures greater than 15,000 K. The electron conservation
is satisfied when the ambipolar diffusion and the H ground state ionization rates are equal.
It is interesting to note that the ambipolar diffusion and th e H ionization rates play a
crucial role in determining the self-sustaining average electron temperature even though
they are not the dominant rates. A self-sustaining average electron temperature o f 22,500
K was obtained.
Figure 6.9 shows the variation o f average electron temperature required to sustain
a plasma as a function o f electron number density. N ote that the average electron
temperature is independent o f electron number density. The electron number density is
eliminated from the electron conservation equation (Equation 4.16) because the H 2 and H
ground state ionization and ambipolar diffusion rates are directly proportional to the
electron number density. Also, Figure 5.5 shows that the shape o f the EEDF and average
electron energy are independent o f electron number density. Thus, the average electron
temperature almost remains constant with increasing electron number densities. The
electron number increases linearly with plasma power deposition. Thus, we conclude that
the average electron temperature does not directly depend on the plasma power
deposition.
The results in the absence o f the excited states are generated by numerically
105
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neglecting ionization and recombination reactions involving excited states. Unlike the
conclusions in Ar plasmas, it is seen that the excited states do not influence the selfsustaining value o f the average electron temperature. Because the majority o f electrons
are produced from H 2 and H ground states, neglecting H excited states does not influence
the self-sustaining average electron tem perature.
Figure 6.10 shows a Boltzmann plot for H excited states generated by using selfconsistent electron number densities and average electron temperatures as shown in Figure
6.9. The solutions which match the numerical and experimental data within a factor o f 2
require electron number densities between 6 * 10l6-2* 1017 m’3, E/N's between 40-55 Td,
and average electron temperatures between 21,000-24,000 K (corresponding average
electron energies between 2.7-3.0 eV). It is observed that, for an electron number density
o f l x 1017 m ‘3 and an average electron tem perature o f 22,500 K, the experimentally
measured excited states are matched within their uncertainty (±45%). A direct
comparison o f these predictions is not feasible due to lack o f reported measurements.
Tahara et al. (1995) did report electrostatic double probe measurements o f electron
temperature and ion number density but under low pressure conditions (<15 Torr).
However, extrapolations from their work to the 40 Torr H 2/CH 4 plasma studied in the
present w ork yield electron number density predictions in reasonable agreement (40%)
with those predicted in this work. Also, our numerical predictions o f electron number
density and average electron temperature are in reasonable agreement with those from
Scott et al.'s (1996) numerical model.
The numerical predictions in Ar plasmas demonstrated the important role o f
106
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'h./CH^ piaJma
'
40 Torr
T_ = 2000 K
XL = 0.05
J
<? 101
n a= 6 x 1 0 16 m - 3 ; T e= 2 2 .S O O K
- n •.^ _x lO
^ m -_<3* ;T.=22.500K
. .40
• -_ ______ _
n„=2x1018 in'3 ; T,=22.500 K
Experimental
•
i
I
I
I
I
2.5
I
I 1—1 I
5.0
I I
I
I
I
I I
7.5
I
I
I
10.0
I
I I
I
I
I I
1ZS
L
15.0
Energy [eV]
Figure 6.10 A Boltzmann plot comparing the numerical and experimental
excited states in the H /C H 4plasma
Hj/CHj plasma
40 Torr
ne»1x10l7 m-3
e = 3.0 »V
T =2000 K
Xu = 0.05
•
0
2.5
Without quenching
With quenching
Experimental
5.0
7.5
10.0
12.5
15.0
Energy [eV]
Figure 6.11 The effect o f collisional quenching on the excited states in
HJCH4plasma
107
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excited states. However, the results in H /C H 4 plasmas revealed that the excited states
can be neglected. Thus, it is o f interest to investigate reasons for an insignificant role o f
excited states in HVCH, plasma. The insignificant role o f excited states is attributed to the
depopulation o f H excited states through H 2 collisions. This so called collisional
quenching mechanism has been studied extensively in literature (Meier et al., 1986;
Burshtein et al., 1987).
We used experimentally measured H 2 collisional quenching rates
by Burshtein et al. (1987) for the first four H excited states. Figure 6.11 demonstrates the
important role o f quenching. The excited state number densities in the presence o f
quenching are 2-7 orders o f magnitude lower than those in the absence o f quenching. A
much larger average electron temperature is required to satisfy the experimentally
measured excited states. For such large average electron temperatures, a sufficient
number o f high energy electrons are available to directly ionize H , and H from their
ground states. Because the majority o f electrons are produced from H and H , ground
states, neglecting excitation and ionization reactions involving excited states does not
affect the self-sustaining average electron temperature. However, it is important to
account for this strong quenching mechanism when using OES and numerical data to
obtain quantitative information. For example, the results presented in our previous
publication (Gordon and Kelkar, 1996) assumed an incorrect electron number density o f
4* 10lg m ‘3 and did not self-consistently account for the electron conservation equation.
Our assumption o f electron number density was based on the numerical predictions by
Hyman et al. (1992). For an electron number density o f 4* 10l* m'3, it was found that an
average electron temperature o f around 10,000 K satisfies the experimentally measured
108
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excited states within their experimental uncertainties. Also, it was observed that the step­
wise ionization from H excited states produced the majority o f electrons. However, our
model neglected the quenching o f H excited states through H , collisions and led to
incorrect conclusions.
Figure 6 .12 shows the influence o f H mol fraction on the self-sustaining average
electron temperature. For H mol fractions less than 0.1, the average electron temperature
decreases with increasing H mol fraction. For H mol fractions greater than 0.1, the
average electron temperature then increases with increasing H mol fractions. The ground
state ionization energies for H and H 2 are 13.6 eV and 15.6 eV, respectively. It is easier
to produce electrons from H ground states compared to H 2 ground states. An increase in
H mol fraction increases the H ground state number density which results in higher
ionization rates. The H 2 ionization and molecular ion recombination rates decrease
proportionally and do not influence the electron conservation. Thus, the plasma can be
sustained at lower average electron temperatures with increasing H mol fractions.
H owever, for H mol fractions greater than 0.1, although H ionization rates increase with
increasing H mol fraction, the H 2 ionization rates decrease more rapidly than do the
molecular recombination rates. Effectively, a slightly larger average electron temperature
is required to sustain the plasma.
Figure 6.13 shows the influence o f gas temperature on the average electron
temperature. An increase in the gas temperature decreases the electron-neutral collision
frequency and thus the ionization rates. A significantly larger average electron
tem perature is required to maintain the ionization rates and to thus to sustain the plasma.
109
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28000
Hj /CHj plasma
40 Ton
26000
n^lxlO17!!!"3
T = 2000 K
“
24000
22000
20000
0
0.05
0.10
0.15
0.20
0.30
Figure 6.12 Self-sustaining average electron temperature as a Junction ofH mol fraction in the HJCH4plasma
28000
26000
Hj /CHj plasma
40 Ton-
24000
22000
20000
18000
1000
3000
2000
4000
Tf l[K]
Figure 6.13 Self-sustaining average electron temperature as a function o f gas
temperature in the HfCH4 plasma
110
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CHAPTER VII
IMPORTANCE OF FREE-FREE CONTINUUM EMISSION
(7.1) Introduction
In this chapter, the continuum emission is studied in tw o-tem perature
nonequilibrium plasmas characterized by electron number densities between 6 >: 10 161x 1018 m '5 and average electron temperatures between 9000-24.000 K. The emission is
monitored in both a molecular plasm a (H:/C H J and a sim pler atom ic plasm a (Ar).
Absolute continuum em ission collected from the plasma is used to verify the consistency
between various plasma parameters. For the operating conditions studied here, free-free
electron-neutral interactions produce the majority o f the continuum emission. For a
H:/CH 4 plasma, the experimental continuum emission is shown to be inconsistent with
the theoretical results unless the electron-H: interaction cross-section is increased by a
factor between 10 and 20 above the momentum cross-section value. For an Ar plasma,
the experimental continuum em ission is used only to obtain an upper bound on the
electron number density (3 x 10 18 m '3).
Ill
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(7.2) R esults a n d Discussion
The set o f Equations 3.4 is used to calculate the total continuum em ission based
on the numerical predictions given in the previous chapter. The numerical and
experimental continuum emission are compared to verify the consistency between the
predicted plasma parameters. Figures 7.1 and 7.2 show the relative contributions o f the
various terms as a function o f average electron temperature for H:/CH 4 and pure Ar
plasmas, respectively. All other parameters are held constant at their base values (Tables
5.1 and 5.2). For our H:/CH 4 plasma, an experimental continuum em ission collected at a
wavelength o f 4285 A is shown in Figure 7.1. in our A r plasma, a weak continuum
emission (signal to noise ratio o f 1.05) was observed. It is noted that the free-free
electron-neutral term produces the majority o f the continuum emission over the entire
average electron temperature range. Although cross-sections for free-bound and freefree electron-ion collisions are about 100 times larger than that for free-free electronneutral collisions, for weakly ionized plasmas (< 10"5). the emission from these terms is
insignificant. Over the entire range o f average electron temperatures in the H:/C H 4
plasma, the numerically calculated continuum emission is a factor o f 10-20 lower than
that measured experimentally. For average electron temperatures between 11.00014.000. the numerical and experimental continuum emission agree well in the Ar
plasma.
Figure 7.3 and 7.4 show the variation o f total continuum emission as a function
o f electron number density for various average electron temperatures in H:/CH 4 and Ar
plasmas, respectively. The electron temperatures are selected based on the self112
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Experimental
Hj /CHj plasma
n,=1x1017m-3
T =2000K
XH = 0.05
'k.
5000
20000
15000
10000
25000
30000
T„[K]
Figure 7.1
Contributions to the total continuum emission as a Junction of
average electron temperature in the H fCH 4 plasma
Experimental
106
_
1°5| ’
i
£
u
c
10 * '
5000
15000
10000
20000
T JK ]
Figure 7.2 Contributions to the total continuum emission as a function o f
average electron temperature in the Ar plasma
113
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T---- 1--- 1—li l 11
10s
Hj/CH, plasma
40 Ton
T =2000K
XH= 0.05
Experimental
108
.7
Tg = 20.000 K
T . -22,500 K
T =25,000 K
10®
io16
na ['n'3]
Figure 7.3 Continuum emission as a Junction o f electron number density for
various average electron temperatures in the H fCH 4plasma
108
Ar plasma
5 Ton
T„ = 350 K
Experimental
s>
Y
E
U
T9 =9000 K
T , = 12.500 K
T = 14,000 K
104
Figure 7.4 Continuum emission as a Junction o f electron number density for
various average electron temperatures in the Ar plasma
114
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consistent solutions presented in Chapter VI. The total continuum em ission increases
linearly with increasing electron number density because free-free electron-neutral
interactions produce the majority o f emission. The electron-neutral collision frequency
increases linearly with increasing electron num ber density leading to an increase in the
total emission. Recall that the self-consistent solutions in H;/CH 4 plasmas have electron
num ber densities between 6 * 10 16 and 2* 10 17 m '3. The numerical continuum emission is
a factor o f 10-20 lower than the experim ental measurement. For Ar plasmas, the selfconsistent solutions have electron num ber densities between 3* 1017 and 1* I0 IS m '3. The
experimental uncertainty on the total em ission is considerably large (± 2 0 0 %) due to the
very weak signal. However, it is seen that several numerical solutions are consistent
with the experimental data. Note that the electron number density must remain below
3* 10 18 m '3 to ensure reasonable agreem ent betw een the theoretical and experimental
data.
Figures 7.5 and 7.6 compare the total continuum emission for various gas
temperatures for H:/CH 4 and Ar plasmas, respectively. An increase in the gas
temperature rarefies the gas and reduces the electron-neutral collision frequency. The
total continuum emission decreases with this increase in gas temperature because o f less
frequent collisions. Note that, for the H:/C H 4 plasm a, the numerical predictions are
lower than the experimental data by a factor o f 5-30. For the Ar plasma, the figure can
be used to obtain an upper bound on the electron num ber density for various gas
temperatures.
Figure 7.7 shows the variation o f total continuum emission as a function o f the
115
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tt---- 1—r i-rrrr
10s
Wj/CH^ plasma
40Torr
T, = 22,500 K
Xu = 0.05
T =1000
T®= 2000
T# =3000
T .4 0 0 0
K
K
K
K
10®
Figure 7.5 Continuum emission as a function of electron number density fo r
various gas temperatures in the HfCH4plasma
II II!]---- 1-- 1
10s
Ar plasma
5Torr
T, = 12,500 K
L_
CO
T
E
U
T =350 K
T , = 500 K
T. = 1000 K
104
10'6
Figure 7.6 Continuum emission as a function o f electron number density for
various gas temperatures in the Ar plasma
116
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electron number density for various H mol fractions. The results were generated by
using a self-consistent average electron temperature o f 22.500 K and an assumed gas
temperature o f 2000 K. The total continuum emission is found to increase slightly with
increasing H mol fraction. The electron-H: elastic collision cross-section is larger than
electron-H and electron-CH 4 cross-sections by a factor o f 10. For our operating
conditions. H mol fraction varies between 0.001-0.3. Thus. FT comprises more than
80% o f plasma and produces the majority o f free-free electron-neutral continuum
emission. An increase in H mol fraction, holding the gas temperature constant, then
results in a decrease in H: num ber density and consequently a decrease in emission.
Note that, for the self-consistent solutions, the numerical predictions are lower than the
experimental em ission by a factor o f 10- 2 0 .
The parametric results in the Ar plasma suggest that, for average electron
temperatures between 9000-14.000 K.. the electron number density can not exceed
3* 1018 m T h i s information is valuable for the energy balance calculations performed
in Chapter VIII. However, for the H:/CH 4 plasma, all the theoretical solutions underpredict continuum emission by a factor o f 5-30. To produce the experimental amount o f
continuum emission, either higher electron number densities (>1 * 10 IS m '3) or average
electron temperatures (>30.000 K.) are needed. However, such values lead to an over­
prediction. by orders o f magnitude, o f the excited state num ber densities (Chapter VI)
and contradict the power balance (Chapter VIII).
Such a discrepancy between the theoretical and the experim ental emission data
greatly exceeds that allowed by experimental uncertainty. Further, the numerical models
117
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'W2/6H4 pli*mW,Tl
4 0 T o it
Experimental
10®
0>
T
E
£
CJ
—
10s
x H= 0.001
XH= 0.01
XH= 0.1
1016
Figure 7.7
Continuum emission as a junction o f electron number density for
H-mol fractions in the HJCH4 plasma
•
10s
2000
Te = 20.000 K
T# = 22.500 K
Te = 25.000 K
Experimental
4000
6000
6000
X[nm]
Figure 7.8
Comparison o f the numerical and experimental continuum
emission for two different wavelengths in the HfCH4plasma
118
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which were used have been well validated by comparing their results with those from the
other models available in the literature. The cause o f this discrepancy is the electronneutral free-free emission term , which produces most o f the signal (refer to Figure 7.1).
To confirm that this m easurem ent was not in error because o f the chosen wavelength, the
measurement was repeated at several other wavelengths. Figure 7.8 compares the
theoretical continuum em ission with the experimental data at two different wavelengths
(4285 A and 6550 A). For both wavelengths, the experimental emission exceeds the
theoretical emission by the same factor o f 10-20. This under-prediction o f continuum
emission greatly exceeds the experimental uncertainty (±45%). Thus, the theoretical
method requires further investigation.
The electron-neutral free-free term which produces the majority o f emission
depends on the electron num ber density, average electron temperature, electron-neutral
momentum cross-sections, and gas temperature and com position. A possible source o f
error could be an incorrect electron-neutral momentum cross-section because all other
parameters are varied parametrically. As described earlier, the electron-neutral
momentum cross-section used by previous researchers characterizes only elastic energy
exchange between an electron and a neutral particle. However, it is well known that an
electron can exchange energy much more effectively w ith a molecule than an atom due
to vibrational and rotational modes. As already described in Section 3.5. Sutton and
Sherman (1967) reported a temperature dependent energy loss factor to account for this
enhanced energy exchange between an electron and a m olecule (Figure 3.2 for H:). For
our range o f energies studied in this work, this factor varies between 10 and 20. Figure
119
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plasma"'*
«
E
I------ 1
I
I
: I I II
40Torr
T , = 22,500 K
T = 2000 K
= 0.05
10i o L
"
“ I -------- r T T T T T T I —
t
Experimental
f
-------------
With inelasic loss factor
Without inelasic loes factor
1qS I i m l
1016
i
l x i .u u
I
I
i i ' i ' i 'I
101S
1017
i
i i i i 11 il
i
ii
1019
n8 [™'3l
Figure 7.9
Comparison of the numerical and experimental continuum
emission with and without inelastic energy loss factor in the
HJCH4plasma
120
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7.9 shows excellent agreement between theoretical and experim ental continuum
emission when the electron-H; cross-section is enhanced by this inelastic energy loss
factor.
Although this loss factor was not originally meant to include continuum
emission, an enhanced electron-molecule energy exchange will plausibly lead to an
enhanced em issive output. As an additional check. OES results were also obtained in a
3.2 kW power. 70 Torr pressure H:/C H 4 plasma. It is found that the self-consistent
electron num ber density increases by approximately 20 % and average electron
temperature decreases by approximately 10%. The decrease in self-sustaining average
electron temperature is caused by an increase in the ionization rates with increasing
pressure. The experimental emission increases by 20%. Here. too. a self-consistent
solution required an enhancement o f the electron-H; m om entum cross-section by a factor
between 10 and 2 0 .
This relatively large cross-section discrepancy has not been noted before for the
following reasons. The electron-neutral free-free emission dominates the total
continuum em ission for the conditions previously studied. In other sim ilar studies
(W ilbers et al.. 1991: Gordon and FCruger. 1993: Galley and Hieftje. 1993: Yang and
Barnes. 1989: Megy et al.. 1995). the free-bound emission dominates, obscuring this
effect. Further, these other studies involve either noble gases or very high electron and
gas tem peratures such that molecular species comprise a sm all portion o f the plasma.
The continuum method does require a Maxwellian EEDF for validity. The presence o f a
non-M axwellian EEDF would discourage the application o f this method. However, we
121
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claim the method is still valid as described below.
Figure 5.1 com pared a Maxwellian EEDF and a predicted EEDF with the same
average electron energy o f 2.9 eV. A significant deviation from the Maxwellian EEDF
is only noted for electrons with energies greater than 10 eV. and these electrons
represent only about .005% o f all electrons. The continuum emission measured at a
wavelength o f 4285 A arises because o f interactions involving electrons with energies
greater than 2.9 eV. and there are 1000 times more electrons with energies between 2.9
eV and 10 eV than electrons with energies greater than 10 eV. Thus, a very small
fraction ( 0 . 1%) o f the total em ission is assumed to originate from the highly nonMaxwellian electrons. Com paring the fraction o f predicted electrons between 2.9 eV
and 10 eV to the equivalent fraction o f M axwellian ones, it is found that there is only a
20% difference. Thus, this technique can be confidently applied.
The electron-ion free-free and free-bound continuum emission were calculated
by assuming Bibermann factors o f 1 for all the ions ( H \ H: ". and H f). The molecular
ion recombination rates are approximately 100 times m ore than that for the atomic
hydrogen ions. Only some o f the molecular recom binations lead to emission. As an
upper bound, we assumed that all o f the molecular recom binations lead to emission, and
accounted for the radiative m olecular recombinations by increasing the electronmolecule free-bound Biberm ann factor by 100. However. Figure 7.1 shows that, for our
operating conditions, the free-free electron-neutral collisions produce the majority o f
continuum emission, and thus, even a factor o f 100 increase in the electron-ion emission
does not affect our conclusions.
122
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CHAPTER VIII
ENERGY BALANCE
(8.1) Introduction
Plasma pow er deposition greatly impacts the discharge physics and plasma
chemistry. M icrowave plasma reactors use an adjustable excitation probe to couple the
power across a dielectric window. Generally, a load matching network is utilized to
reduce the coupling losses. M ak et al. (1997) showed that cavity tuning is extremely
important to ensure good coupling efficiency. The arithmetic difference between forward
and reflected pow er equals the amount o f pow er coupled into the cavity. However, the
net power forwarded into the cavity does not necessarily equal the power absorbed by the
plasma. The plasma power coupling efficiency depends on the microwave coupling probe
and discharge vessel wall losses. Most researchers (Hyman et al., 1992; Koemtzopolous
et al., 1994) assume that these parasitic losses are negligible and that the difference
between forward and reflected power represents pow er absorbed by the plasma.
In this chapter, a comprehensive energy balance is performed to determine the
power absorbed by the H 2/CH 4 (40 Torr, 1.6 kW ) and Ar (5 Torr, 680 W) plasmas. The
123
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numerical and experimental results presented in previous chapters are verified for
consistency with the pow er balance. For the HVCH* plasma, several numerical solutions
have plasma power deposition between 400 and 2400 W whereas, for the A r plasma, all
the numerical solutions have power deposition between 20 and 100 W. A control volume
heat transfer and global reactor energy balance are performed to estimate the plasma
power deposition. The results reveal that approximately 600 to 1000 W are deposited in
the H 2/C H 4 plasma. The results in the Ar plasma indicate that approximately 50 to 100 W
are deposited. The experimental enthalpy data and analytical heat transfer calculations
indicate that the remaining energy bypasses the plasma and is directly absorbed by the
base-plate and applicator walls. Finally, this chapter presents the best possible solutions in
H 2/CH 4 and Ar plasmas.
(8.2) P ow er B alance
Figure 8 .1 shows the variation o f the power deposition as a function o f electron
number density for various average electron temperatures in the H^CH* plasma. The
power deposition increases linearly with increasing electron number density. For the selfconsistent solutions (nc between 6 * 1016 and 3* 1017 m‘3, and Te between 21,000 and
24,000 K) presented in Chapter VI, the pow er deposition varies between 500 and 2300
W. Figure 8.2 shows the variation o f the power deposition as a function o f electron
number density for various average electron temperatures in the Ar plasma. It is observed
that the pow er deposition varies between 15 and 100 W for the numerical predictions
presented in Chapter VI.
124
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1111
i
i i r i i
fcj/CH* P*a6ma
40 Torr
T = 2 000K
XH=0.0S
Te = 20.000 K
T « 22,500 K
T = 25,000 K
10 '
Figure 8.1 Power deposition as a function o f electron number density fo r
various average electron temperatures in the HJCH4plasma
Ar plasma
5 Tort
T = 350 K
T , = 11,000 K
T , = 12,500 K
T , = 14,000 K
10 -
Figure 8.2
Power deposition as a function o f electron number density fo r
various average electron temperatures in the Ar plasma
125
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Figures 8.3 and 8.4 compare the power deposition for various gas temperatures in
H j/C H , and Ar plasmas, respectively. The electron-neutral collision frequency determines
the rate o f energy exchange between electrons and neutrals. An increase in the gas
tem perature reduces the neutral particle gas density and the electron-neutral collision
frequency. Thus, for the same electric field strength, the pow er deposition decreases w ith
an increase in the gas temperature.
Figure 8.5 shows the influence o f H mol fraction on the plasma power deposition.
The electron-H 2 energy transfer is much more effective than the electron-H energy transfer
due to vibrational and rotational modes. Thus, a fewer energy loss channels are available
to the electrons with an increase in H-mol fraction, and this loss results in a small increase
in the pow er deposition.
The theoretical results in the HVCH, plasma show that several combinations o f
electron number density, average electron temperature, gas tem perature, and H-mol
fraction satisfy the pow er balance. Knowledge o f the exact am ount o f power absorbed by
the plasma is essential to obtain a unique solution. The results in the Ar plasma reveal that
all the solutions have plasma pow er deposition between 15 and 100 W. The incident and
reflected pow er meters indicate that 680 W are deposited in the cavity. To match the
input pow er deposition o f 680 W, electron number densities greater than 1*10 19 m‘3 o r
average electron temperatures greater than 15,000 K are required. However, for such
larger electron number densities and temperatures, it is impossible to simultaneously
satisfy the electron conservation equation, the experimental excited states (Chapter VI),
and the continuum emission (Chapter VII). Also, such large electron number densities
126
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
H2/CH4 plasma
40 Torr
T# = 22.500 K
XM= 0.05
104
a.
T = 1000 K
T®= 2000 K
T =3000 K
Figure 8.3 Power deposition as a junction o f electron number density fo r
various gas temperatures in the HJCH4plasma
Ar plasma
5 Torr
E/N =0.15 Td
o.
o
■o
0.
T =350 K
T =500 K
T„ = 1000 K
10'6
Figure 8.4 Power deposition as a function o f electron number density fo r
various gas temperatures in the Ar plasma
127
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Hj /CH4 plasm;
40 Toit
Te = 22.S00 K
T = 2000 K
10 *
XH= 0.001
XH=0.01
101
Figure 8.5 Power deposition as a Junction o f electron number density fo r
various H-mol fractions in the HfCHt plasma
128
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exceed the reported experimental measurements perform ed in the downstream region o f
microwave plasma (Tahara et al., 1995; M ak et al., 1997) by 1-2 orders o f magnitude.
Thus, the spectroscopic data and modeling results suggest that a maximum o f 100 W out
o f 680 W o f input pow er deposited in the Ar plasma.
(8.3) Control Volume Heat Transfer Analysis
Figure 8.6 shows a control volume including the plasma, belljar, base-plate, and
substrate-holder assembly. Microwave energy absorbed by the plasma represents energy
flowing into the control volume. The conduction through the substrate, forced air
cooling, gas convection, and plasma radiation represent the energy loss terms for the
control volume.
The energy conducted through the substrate equals sum o f energy exchanged
between the plasma and the substrate and energy released in H recombinations on the
substrate. The energy exchanged between the plasma and the substrate can be described
by free convection heat transfer because o f very small gas flow rates (300 seem o f Hj).
The free convection between the plasma and the substrate is calculated as follows:
Q »'
( 8 - 1)
where hp, represents the heat transfer coefficient between the plasma and the substrate and
is calculated as shown in Incroperra and Dewitt (1990 p. 345). The hydrogen atoms from
the bulk plasma recombine on the substrate. The am ount o f heat released in the Hrecombination can be calculated as follows:
129
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Microwave Energy
1“
'rad
Q a ir
; Base Plate
Qbp
777
Substrate
Holder
S ta g e
Qgas
Figure 8.6
Q snb
Control volume heat transfer analysis
130
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(8.2)
where n and C represent number density and average velocity o f particles in the gas phase.
The H recombination probability and energy released in H recombinations on the substrate
are represented by y r and en respectively. The H recombination probability depends on
the type and conditions o f the surface. For H recombinations on the silicon substrate at
820 C, the recombination probability was assumed to be 0.01 (St Onge and Moisan,
1994). The characteristic dissociation energy for H 2 (8.9 eV) is released when two H
atoms combine on the substrate.
The energy carried by the gases leaving the control volume is calculated as
follows:
(8.3)
where nig and Cpg represent the mass flow rate and specific heat o f the gas, respectively.
The inlet and outlet gas temperatures are represented by T&j and Tg, respectively.
The heat transfer between the belljar and cooling air is calculated as follows:
e* • W
* * r «)
(8.4)
where A^, Tbj, and T, represent the belljar surface area, belljar temperature, and mean air
temperature, respectively. The heat transfer coefficient for the forced air cooling o f belljar
is represented by hbJ and is calculated as shown in Incroperra and Dewitt (1990 p. 345).
Plasma radiation consists o f line and continuum emission and was calculated based on the
131
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radiative rates in the CRMs.
Figure 8.7 shows the control volume heat transfer data for the H /C H * plasma. A
belljar temperature o f 48 C was measured by infrared sensors, and the mean cooling air
temperature was 27 C. The infrared sensor calibration was checked by comparing
temperatures o f a spare belljar measured using infrared sensors and a thermocouple. The
temperatures matched within ± 2 C. Based on the theoretically calculated forced
convection heat transfer coefficient (140 W/m 2FC), the energy dissipated in the belljar
cooling air is found to be 130 W. A substrate temperature o f 820 C was measured by an
optical pyrometer, and the plasma gas temperature was assumed to be 2000 K (Scott et
al., 1996). The substrate heat dissipation was the dominant heat loss mechanism (525 W)
and is mainly due to the energy released in H recombinations. Some hydrogen atoms are
carried away by the gases pumped out o f the chamber. The subsequent H recombinations
on the chamber walls release a considerable amount o f energy. The gas convection and
plasma radiation losses were 10 W and 15 W, respectively. Based on the experimentally
measured base-plate temperature and contact area between the belljar and the base-plate,
the base-plate conduction was found to be negligible. A total o f 700 W can be removed
from the control volume.
For a 5 Torr A r plasma, a microwave pow er o f 680 W was coupled into the cavity.
Figure 8.8 shows the control volume heat transfer calculations for the Ar plasma. The
belljar and cooling air mean temperatures were 21 C and 18 C, respectively. Based on the
theoretically calculated forced convective heat transfer coefficient (140 W/m 2 K) between
the belljar and air, the amount o f heat dissipated in the cooling air was found to be 18 W.
132
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m
■
oO
<J
.o
3
CO
o
a
i/i
CO
0
CO
"O
CD
a:
CO
CO
a>
E
</)
CD
jo
CL
lb s
™o2
m
Figure 8.7
77»e heat transfer analysis results in the HfCH4plasma
Figure 8.8
The heat transfer analysis results in the Ar plasma
133
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The plasma radiation is approximately less than 20 W, and the gas convection loss was
found to be negligible. The minimum temperature measured by the optical pyrom eter is
500 C. For an assumed substrate temperature o f 293 K, the energy conducted through the
substrate was found be less than 5 W. Based on an estimated contact area between the
base-plate and the belljar and experimentally measured base-plate tem perature (14 C), the
conduction through the base-plate was found to be insignificant. Thus, a total o f only 50
W can be removed from the control volume.
(8.4) G lobal R eactor E n erg y B alance
Tables 8 .1 and 8 .2 show the experimental enthalpy data for the H /C H * and A r
plasmas, respectively. The corresponding global reactor energy balance results are shown
in Figures 8.9 (H 2/CH4) and 8 .10 (Ar). For both plasmas, the bulk o f the input energy is
dissipated in the base-plate and applicator wall cooling water. For the Hj/CH* plasma, the
substrate heat dissipation was found to be around 400 W and is in reasonable agreement
with the theoretical heat transfer calculations (Figure 8.7). The belljar air, short, probe,
and chamber water account for the remaining energy. The enthalpy data are highly
uncertain in the Ar plasma due to very small temperature differences (0.1 C). The
measurements indicate that at least 400 W are dissipated in the cooling lines.
The heat transfer analysis indicates that the base-plate and applicator walls do not
gain a significant amount o f energy from the plasma. However, the global reactor energy
balance data indicate that the bulk o f the input energy is dissipated in the base-plate and
applicator cooling water. Thus, the results suggest that a significant portion o f the input
134
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Table 8.1 Experimental Enthalpy Data in a H2/CH4 Plasma
Cooling Channel
f (kg/s)
Cp (J/kg-K)
Tin (C)
Tout (C)
P * (W )
Base-pIate/Applicator walls
0.1077
4182
13.8
15.8
900.80
Substrate
0.012
4182
13.8
2 2 .2
421.54
Chamber
0.0158
4182
13.8
15.2
92.50
Belljar
0.0055
1003
14.2
40.2
143.40
4.08x1 O' 7
14270
15
2200
11.55
Short
0.0475
4182
13.8
14.1
59.6
Probe
0.0259
4182
13.8
14.3
53.95
Gas convection
Table 8.2 Experimental Enthalpy Data in a Ar Plasma
Cooling Channel
f (kg/s)
Cp (J/kg-K)
Tjn (C)
T0ut (C)
Pd-(W )
Base-plate/Applicator walls
0.0955
4182
13.8
14.6
319.50
Substrate
0.012
4182
13.8
14
10.04
Chamber
0.017
4182
13.8
14.2
28.44
Belljar
0.0055
1003
14
21.5
41.37
6.81X10-6
520.3
15
77
0 .2
Short
0.0475
4182
13.8
14
39.72
Probe
0.0259
4182
13.8
14
21.66
Gas convection
135
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Figure 8.9 Experimental enthalpy data in the H/CH4plasma
lt
I
Figure 8.10 Experimental enthalpy data in the Ar plasma
136
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microwave energy bypasses the plasma and is directly absorbed by the base-plate and
applicator walls. Recently, M ak et al. (1997) investigated the coupling efficiency in E C R
microwave discharges. They calculated pow er absorbed by the reactor walls by probe
measurement o f electric fields in a similar microwave reactor. Their measurements w ere
performed in a different operating regime (5 mTorr, 277 W input power) and in the
absence o f the substrate. They concluded that the wall losses amount to only 1% o f the
forward power and assumed that 99% o f the input pow er is absorbed by the discharge.
However, they did not measure the power absorbed by the base-plate, short, and probe.
Also, their Langmuir probe measurements o f electron number density in the downstream
region o f the plasma are a factor of 4 lower than those predicted from a simple global
reactor model. Since electron density and pow er follow almost a linear relationship, a
factor o f 4 lower electron density corresponds to a factor o f 4 lower absorbed power.
Thus, we believe that only 70 W (25%) out o f 277 W o f input power are deposited in their
plasma, which is in general agreement with our findings.
Figure 8 .1 1 shows the global reactor energy balance data in our H 2/CH 4 plasma for
a variety o f operating conditions. An increase in input power increases the dissociation o f
H 2 in the bulk plasma. For the same pressure, the net H flux arriving at the substrate
increases. Assuming that substrate conditions and thus y r remain the same, the total
energy released in H recombination increases and results in higher heat dissipation in the
substrate cooling water. The belljar air cooling also shows a proportional increase in the
heat dissipation rate. The heat removed by the base-plate and applicator wall cooling
w ater increases slightly with increasing input pow er and pressure. The data indicate that
137
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1
1200
--
1000
-
■ 1 5 1 5 W , 40 T orr
8 0 0 --
■ 2 3 5 8 W , 48 T orr
6 0 0 --
□ 3 3 8 0 W . 67 T orr
400 -
I
200
-
a
tr
o
.C
w
£
s
Q-
___ 1
§
=5
=5
o:“
& s
■§
2
=
fi
9 - 3CO
o.
=
£
E0
<
«a
O
o
CD
Figure 8.11 Experimental enthalpy data in H /CH 4plasmas with
various pressures and input power
138
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the increase in input pow er from 1600 W to 3700 W results in an increase in the net
power coupling efficiency from 39% to 50%. However, a considerable amount o f the
input power still bypasses the plasma. A. possible explanation is that the brass base-plate
and applicator walls directly absorb the microwave energy from the electric fields.
Although a mechanism by which a substantial part o f the input power is absorbed by the
base-plate and the applicator walls is unknown, the results shown in this chapter reveal
that the input power is only partially (5-15% in the Ar and 40-50% in the H^CH,,)
deposited in the plasma.
(8.5) Self-consistent Solutions
Figures 8.12 and 8.13 summarize the best self-consistent numerical solutions for
our H 2/CH 4 and Ar plasmas, respectively. These solutions satisfy the electron
conservation (Figures 8 . 12a and 8 .13a), the experimentally measured excited states
(Figures 8.12b and Figure 8.13b), the experimentally measured continuum emission
(Figures 8 . 12 c and 8.13c), and the numerical and experimental energy balance (Figures
8.12d and 8.13d).
139
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1
1022
t o 2'
to20
10“
3
Raw|m'V,J
-1
10 “
a
3
i
4
I0 ‘*
I0IS
3
i
i
I0‘»
l0«L
15000
10000
20000
0
2S O O O
5.0
T#[K1
Figure 8.12a Electron conservation
15.0
F igure 8.12b Comparison o f numerical and
experimental excited states
10*
,sH J C H 4 p la a n ia l" '
4 0 T o rr
T # • 2 2 .5 0 0 K
T -2 0 0 0 K
toM
75
Eh«igy[#Vl
1
H jC H t
4
10»
(ium a
4 0 T o rr
3
x ; * o.o5
r>t-W10,7m“*
T , a 2 2 .5 0 0 K
T «2000K
i
3
x * -a o s
13
10'
tooow
1
r
j
.4 .
10"
10"
", t«i
n ,[m J
Figure 8.12c Comparison o f numerical Figure 8 . 12d Numerical plasma power
(curve) and experimental (band) continuum
deposition
emission
Figure 8.12 Self-consistent solutions in the HJCH4plasma
140
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I 0 24
to23
10=
1021
10»
10"
10"
10 =
f
10®
Numerical? Pradcled
1 ,0”
T o t S e c t P r o a R o le
T o t B e e t Lose R u e
10 "
'
|
10"
7500
10000
12500
15000
17500
I0»
20000
r,[K ]
F ig u re 8.13a Electron conservation
Energy [eV]
Figure 8.13b Comparison o f numerical and
experimental excited states
N u m e n c f tly P f « 4 ic M
Figure 8.13c Comparison o f numerical (curve) Figure 8.13d Numerical plasma power
and experimental (band) continuum emission
deposition
F ig u re 8.13 Self-consistent solutions in the HJCH4plasma
141
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CHAPTER IX
DISSOCIATION MODEL RESULTS
(9.1) Introduction
It has been well established that atomic hydrogen influences diamond deposition by
producing precursor radicals and by etching graphitic carbon (Spear and Frenklach, 1989;
Harris, 19989). The net atomic hydrogen flux at the substrate depends on the atomic
hydrogen concentration in the gas phase. Microwave discharges efficiently produce high
concentrations o f atomic hydrogen, and thereby facilitate the deposition o f good quality
diamond films. Several researchers (Koemtzopoulos et al, 1992; St. Onge and Moisan,
1994; Scott et al., 1996) developed a zero-dimensional model to determine hydrogen
atom yield in microwave plasmas. These studies either use a Maxwellian EEDF (St. Onge
and Moisan, 1994; Scott et al., 1996) or do not validate their numerical predictions with
the experimental data.
This chapter presents preliminary results from our comprehensive hydrogen model
(Chapter IV). The hydrogen atom yield is predicted as a function o f electron number
density, effective diffusion loss coefficient, gas temperature, and pressure. It is
142
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demonstrated that the atomic hydrogen concentration in the bulk plasma increases with
increasing electron number density and gas temperature, whereas th e atomic hydrogen
concentration decreases with increasing diffusion loss coefficient.
(9.2) Parametric Study
Figure 9.1 shows the numerically predicted species concentrations as a function o f
electron number density for a gas temperature o f 2000 K. The atom ic H concentration
increases with increasing electron number density mainly due to increasing electron-impact
H 2 dissociation rates. It is seen that H2" concentration increases linearly with increasing
electron number density. The H2" ions are mainly produced by electron-impact ionization
o f ground state H 2. However, H," ions are rapidly depleted through collisions with
abundantly available H 2 forming H3~ ions. The H 3~ ion concentration increases slowly
because H3" recombination rates increase with increasing electron num ber density. The H"
ions are produced by the direct and step-wise electron-impact ionization o f H. For our
weakly ionized (n^n < 10's) and weakly dissociated (XH<0.2) plasma, direct ionization
from H ground state dominates. The 3 s excited state number density is also plotted for
comparison. A linear increase in the 3 s number density with increasing electron number
density is seen because o f increasing electron-impact excitation rates and H number
density.
Figure 9.2 shows the effect o f gas temperature on the hydrogen atom yield for
various electron number densities. The atomic hydrogen mol fraction increases with
increasing gas temperature. The heavy particle impact H 2 dissociation rates increase
143
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HyCH4 pfasmaJ
40 Torr
:
T =2000 K 1
7= 0.01
|11
-j
r
10'6
F igure 9.1 Species concentrations as a function o f the electron number
density
1
Hj/CH^ plasma
40 Torr
7 = 0.01
1000
3000
2000
4000
T„[K]
F ig u re 9.2 The effect o f gas temperature on the H mol fraction fo r various
electron number densities
144
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
considerably with increasing gas tem perature and lead to larger H mol fractions. N o te that
for larger gas temperatures (Tg>3000 K) the H 2 dissociation is due to heavy particle
collisions. Thus, the H mol fraction does not increase significantly with increasing
electron number density.
Figure 9.3 shows the effect o f the loss coefficient on H mol fraction fo r various gas
temperatures. The diffusion significantly affects the H mol fraction in the bulk plasma.
The H mol fraction decreases with increasing diffusion loss coefficient. The sam e trend is
observed for all gas temperatures. Recall that, our model uses a net H loss coefficient to
calculate the atomic hydrogen loss for the bulk plasma. Our goal is to obtain a qualitative
understanding o f the influence o f various plasma parameters on H mol fraction. However,
a multidimensional model is essential for accurate predictions.
Figure 9.4 shows the variation o f H mol fraction as a function o f pressure for
various gas temperatures. The self-sustaining average electron temperature increases
dramatically w ith decreasing pressure due to fewer electron-neutral collisions. The heavy
particle impact H, dissociation rates decrease with decreasing pressure. The electronneutral collisions cause the majority o f dissociation at lower pressures (<1 Torr, not
shown in the figure), and, thus, the H mol fraction depends weakly on the gas
temperature. In general, the H mol fraction decreases with increasing pressure unless the
neutral-neutral collisions cause the majority o f dissociation. It is observed that, for gas
temperatures less than 2000 K, the electron impact dissociation dominates. A decrease in
self-sustaining average electron tem perature leads to a decrease in the H mol fraction with
increasing pressure. For gas temperatures greater than 4000 K, the heavy particle
145
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T— |-r iT T II|------ 1—
Hj/CH^ plasma
40 Torr
n „*1x1017m'3
1
T = 1000 K
T = 2000 K
T =3000 K
T, = 4000 K
Figure 9.3
The effect o f net diffusion loss coefficient on the H mol fraction
fo r various gas temperatures
H J C H . plasma
T =1000 K
T =2000K
T = 3000 K
T_ = 4000 K
m 10
1
T o [x 104K]
ne = 1*017mJ
T=0.01
10
p [Torr]
F igure 9.4
The effect o f pressure on the H mol fraction fo r various gas
temperatures
146
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reactions cause the majority o f dissociation. Thus, the H mol fraction increases with
increasing pressure. However, for gas temperatures between 2000 and 3000 K, the
electron impact and heavy particle impact dissociation are comparable. For a gas
temperature o f 3000 K, the H mol fraction initially decreases with increasing pressure and
then starts increasing w ith increasing pressure.
Figures 9.1 to 9.4 indicate that H mole fraction can be increased by increasing the
electron number density and gas temperature or by decreasing the diffusion loss
coefficient. The electron number density can be increased by increasing the power
deposition at constant pressure. The gas temperature can be increased by simultaneously
increasing pressure and pow er deposition. The diffusion loss coefficient is a function o f
properties and tem perature o f the surfaces in contact w ith the plasma. St. Onge and
Moisan (1994) suggested that hydrogen atom diffusion can be reduced by cooling the bell
jar walls.
147
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CHAPTER X
CONCLUSIONS AND RECOMMENDATIONS
(10.1) C onclusions
Self-consistent zero-dimensional modeling and optical emission spectroscopic
(OES) measurements were performed in hydrogen/methane (1600-3700 W. 40-70 Torr.
300 seem hydrogen. 3 seem methane) and argon (680 W. 5 Torr. 250 seem argon)
microwave plasmas. The experiments were conducted in a 2.45 GHz. 6 kW
WAVEMAT (Model MPDR-3135) microwave plasma reactor. The use o f numerical
and experimental data in tandem has led to several interesting conclusions concerning
the relative importance o f various physical processes in the bulk plasmas, the application
o f the standard diagnostics techniques under two-temperature nonequilibrium
conditions, the plasma power deposition efficiency, and quantitative predictions o f the
plasma parameters o f interest.
Both plasmas are characterized as weakly ionized plasm as (degree o f ionization
less than 10'5) in which the electron energy distribution function (EEDF) deviates
considerably from Maxwellian. It is demonstrated that the models which assum e a
148
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Maxwellian EEDF lead to inaccurate predictions o f excited states, average electron
energy, and self-sustaining electric field. In argon plasm as, step-wise ionization from
the excited states is the major electron production mechanism. Although there are orders
o f magnitude fewer excited states compared to ground states, this so called tw o-step
ionization is dominant because its threshold (11.65 eV) is lower than that for direct
ionization (15.755 eV). Consequently, a low er electric field and a lower average
electron energy (30-60%) are required to num erically sustain the plasma when tw o-step
ionization is included. Interestingly, in the hydrogen/m ethane plasmas, it is found that
the direct ionization from molecular and atom ic hydrogen ground states produce the
majority o f electrons and that the step-wise ionization from atomic hydrogen excited
states is insignificant. The excited state populations are found to be dramaticallyreduced by collisional quenching with m olecular hydrogen.
For our operating conditions, the free-free electron-neutral collisions produce the
majority o f continuum emission. In argon plasm as, the numerical and experimental data
were used to verify- the consistency between the predicted plasma parameters. H ow ever,
in hvdrogen/methane plasmas, the numerical predictions are found to be inconsistent
with the experimentally measured continuum em ission. It is demonstrated that
consistent results are obtained if the electron-m olecular hydrogen cross-section, used for
calculating free-free electron-neutral em ission, is increased by a factor between 10 and
20 above the momentum cross-section value. This increase is theoretically supported by
the enhanced energy exchange for electron-molecule interactions.
Our numerical and experimental data indicate that only 25-100 W (5-15%) out o f
149
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680 W o f input pow er are deposited in the argon plasma. The numerical results in the
hydrogen/methane plasm a indicate power deposition between 400-2000 W. However,
the experimental enthalpy data in the hydrogen/methane plasma reveal that 600-1000 W
(40-60%) out o f 1600 W o f input power are deposited in the plasma. For both plasmas,
our heat transfer analysis and enthalpy data indicate that a substantial part o f the input
power bypasses the plasm a is directly absorbed by the base-plate and applicator walls.
In the hvdrogen/m ethane plasma, an electron number density o f 1 * 10 17 m '3.
average electron temperature o f 22.500 K (or average electron energy o f 2.9 eV). electric
field strength o f 50 Td. gas temperature o f 2000 K.. hydrogen dissociation fraction o f
0.05. and plasma power deposition o f 1000 W provide the best agreem ent between the
numerical and experimental data. In the argon plasma, an electron num ber density o f
5*10 17 m‘-’. average electron temperature o f 12.500 K (or average electron energy o f 1.6
eV). electric field strength o f 0.15 Td. gas temperature o f 350 K. and plasm a power
deposition o f 50 W provide the best agreement between the numerical and experimental
data. Our procedure demonstrates the possible use o f OES as a quantitative diagnostics
tool.
(10.2) R ecom m endations
The current understanding of our nonquilibirum microwave plasm as can be
enhanced by further numerical and experimental work. Numerically, the zero­
dimensional models can be used to determine the variation o f electron num ber density,
average electron temperature, and hydrogen dissociation fraction as a function o f the
150
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input power, pressure, and gas flow rates. One-dimensional or possibly two-dimensional
models could be developed to investigate spatial effects. The pure hydrogen model
should be modified to include gas phase hydrocarbon chem istry which is crucial to
understanding diamond deposition. The bulk plasma models should be coupled with
surface chemistry model to investigate the diamond growth.
Experimentally, the reactor needs to be modified to obtain better optical access
for the spatially resolved measurements. Abel inversion can be used to access the radial
variation o f the plasma parameters o f interest from the lateral OES data. A systematic
study is required to determine correlations between the operating conditions and the
emission signal. Absorption experiments can be performed to measure 2s and 4s excited
state number densities in hydrogen and argon plasmas, respectively. Our preliminary
analysis indicate that 4s excited state absorbs around 50% o f the input light in the argon
plasma whereas, in the hydrogen plasma. 2s excited state absorbs less than 0.5% o f the
input light. The first excited state number density measurem ent will improve the
accuracy o f our predictions.
151
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157
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PUBLICATIONS AND PRESENTATIONS
Kelkar. U.M .. Gordon. M.H.. Roe. L.A.. and Li. Y.. 1997. "Self-Consistent Modeling o f
Microwave Plasm a Chemical Vapor Deposition Reactor." to be subm itted. Diamond and
Related Materials.
Kelkar. U.M.. Gordon. M.H.. Roe. L.A.. and Li. Y.. 1997. "Energy Balance Study in a
Pure Argon M icrowave Plasma" to be published. Journal o f Vacuum Science and
Technology.
Kelkar. U.M .. and Gordon. M.H.. 1997." The Importance o f Electron-M olecule
Interactions in Free-Free Continuum Em ission for Microwave Discharge CVD." Plasma
Chemistry- and Plasma Processing. Vol. 17. No. 3. pp. 315-329.
Gordon. M.H.. and Kelkar. U.M.. 1996." The Role o f Two-Step Ionization in Numerical
Predictions o f Electron Energy Distribution Functions." Physics o f Plasmas. Vol.3. pp.
407-413.
Gordon. M .H.. Roe. L.A.. Kelkar. U.M .. and Li. Y:. 1997. "Diamond Deposition and
Istotope Effects." to be presneted at a conference in JapanA
Kelkar. U.M.. Gordon. M.H.. Roe. L.A.. and Li. Y.. 1997. "Diagnostic and Modeling in
a Pure Argon Plasma: Energy Balance Study" Invited Paper. 28th AIAA
Plasmadvnamics and Lasers Conference. Atlanta. Georgia. June 23-25.
Kelkar. U.M .. Gordon. M.H.. and Roe. L.A.. 1996. "A Self-Consistent ZeroDimensional Description o f a N onequilibrium Microwave Plasma." Bulletin o f
American Physical Society. Vol. 41(6). p. 1337.
Kelkar. U.M.and Gordon. M.H.. 1996. "Determination o f Electron Density and
Temperature Using Optical Emission Spectroscopy and Self-Consistent Modeling in a
Nonequilibrium M icrowave Plasma." 43rd National Symposium o f the American
Vacuum Society. Philadelphia. Pennsylvania. October 14-18.
Kelkar. U.M. and Gordon. M.H.. 1996. "Self-Consistent Modeling o f Electron Energy
Distribution Functions for Microwave CVD Reactors." Invited Paper. CVD-XIII,
Electrochemical Society Proceedings. 96-5. pp. 75-82.
Gordon. M.H. and Kelkar. U.M.. 1995. "Non-equilibrium Effects in Microwave
Plasmas." Electrochem ical Society Proceedings. Vol. 95-4. pp. 291-295.
158
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Kelkar. U.M. and Gordon. M.H.. 1995. "Practical Applications o f Optical Emission
Spectroscopy in a Nonequilibrium Microwave Plasma." Bulletin o f American Physical
Society. Vol. 40(9). p. 1548.
Gordon. M.H. and Kelkar. U.M.. 1995. "Non-Maxwellian EEDFs." presented at Los
Alamos National Laboratory. Los Alamos. NM.
159
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APPENDIX A
S P E C T R O S C O P IC D ATA
(A .l) Atom ic H ydrogen
Spectroscopic constants (Wiese et al., 1966) for the first three lines o f the atomic
hydrogen B aimer series are listed in Table A. 1.
Aji(A)
E, (eV)
Ej(eV )
&
gj
A ji (s l)
fj.
6562.80
10.19
12.09
8
18
4.41 xlO 7
0.6407
4861.32
10.19
12.75
8
32
8.42x 106
0.1193
4340.46
10.19
13.05
8
50
2.53 *10 6
0.0447
(A.2) Argon
Spectroscopic constants (Wiese et al., 1966) for several argon lines are listed in
Table A.2.
*j. (A)
E i(eV )
Ej (eV)
&
gj
Aji ( s l)
fj.
7272.93
11.62
13.28
5
3
2 .0 0 x 10 s
0.0159
7147.04
11.55
13.28
5
3
6.50* 10s
0.00299
7067.22
11.55
13.30
5
5
3.95x10s
0.0296
6965.43
11.55
13.33
5
3
6.70x10s
0.0292
6043.22
13.09
15.14
5
7
1.53xl0 6
0.0117
6032.13
13.07
15.13
7
9
2.46x10s
0.0173
4300.10
11.62
14.50
3
5
3.94x10s
0.00182
160
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APPENDIX B
PROGRAM LISTING
(B .l) Hydrogen
The hydrogen program listing is as follows. A similar code is deveoped for argon.
Changes from version 6
The CRM and Dissociation m odels are coupled. Difiusion o f
excited is taken into account. Ground state diffusion and flow loss are also included
This program calculates the excited state number densities o f atomic hydrogen under
non-equilibrium conditions. The user can input the Non-maxwellian E E D F through the
data file eedf.out
Input:
hyd.inp, eedf.out (only when Non-M axw ellian distribution is used)
Output: hyd.out
Variable definitions:
Numlev
Te
Ta
Ne
MaxEE
iter
N
dist
=
=
=
=
=
=
=
=
=
NP
L
gamm
=
=
Q
=
Number o f excited states (M ax=20)
Electron temperature (K)
Gas temperature (K)
Number density o f electrons (#/m /'3)
Maximun electron energy (eV )
Number o f iterations for matrix solver
Number o f data points in the electron energy distribution
Choice o f the distribution
(0=M axw ellian, 1=N on-M axw ellian from eedf.out)
Total number density (#/m /'3)
Height o f plasma (in)
Wall recombinationprobability
Volume flow rate (seem )
***************************************************************************************/
161
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#include <stdio.h>
#include <math.h>
# d efin ep i 3.14159
#define size 1000
/****************************************************************************************
Function prototypes
***************************************************************************************/
void sub_inputO;
void sub_convertO;
void sub_avgEE0;
void sub_KEea();
void sub_BCDeaO;
void sub_KIeaO;
void sub_KTeO;
void sub_KReO;
void sub_KSa();
void sub_KaaO;
void sub_KITaaO;
void sub_outputO',
void sub_disso0;
void MatsolverO;
void eliminateO;
void substituteO;
void pivotO;
void messageO;
/****************************************************************************************
Global variable declaration
***************************************************************************************/
int dist, N , Numlev, iter, switcha, switchr, switchi, rout, count,efcount;
double Te, Ta, N e, CO, P2, normfac, guessef, norm, Pret, NP, EN, XH, L,gamin, Q, fudge, GrDif;
double avgEE, area, MaxEE, alpha 1, alpha2, radius, vzero, dndt,check, iea, r3e, rei, iaa, r3a;
FILE ^pointer,
double EEfsize], chiC[size], escfac[20], fefsize], fc[size];
double Kea[20][20], KIea[20], KTe[20], KRe[20], K Sa[20][20], Kaa[20][20], NH, Kde, KdH2, KdH, Kre, KrH2,
KrH, Kw;
double KIaa[20], KTa[20], lnnog[20], lam bda[20][20], kzero[20];
double dum[200];
double K2, K3, K4, K5, K7, K8a, K8b, K9, K13, dl, tau l, taue, EbyN;
double vol, N IC , dhdt, NHp, N H 2p, NH3p, dhdti, dnedt, deltah;
double lhs[20][20], rhs[20], H [20], X [20], IonL[2][2], IonR[2], Y[2];
double E[20]
=
{0, 10.19571, 12.08379, 12.74467,13.05057, 13.21666, 13.31681, 13.38188,
13.4265, 13.45835, 13.4819, 13.49988, 13.51388, 13.52491, 13.53383,
13.54115, 13.54722, 13.55231, 13.55664, 13.56036};
162
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intg[20]
double f[20][20]
double A [20][20] =
=
{ 2 ,8 ,1 8 ,3 2 ,5 0 ,7 2 ,9 8 , 1 2 8 ,1 6 2 ,2 0 0 ,2 4 2 ,2 8 8 ,3 3 8 ,3 9 2 ,4 5 0 , 512, 5 7 8 ,6 4 8 ,7 2 2 ,
800};
{ {1 , 0.4162, 0.0 7 9 1 , 0 .0 2 8 9 9 ,0 .0 1 3 9 4 , 7 .7 9 9 e-3 , 4.814e-3, 3.183e-3, 2.216e-3,
1.605e-3, 1.201e-3, 9.2 1 4 e-4 , 7 .2 2 7 e-4 , 5.774e-4, 4.6 8 6 e-4 , 3.856e-4,
3.21 le -4 , 2 .7 0 2 e-4 , 2.2 9 6 e-4 , 1.967e-4},
{1, 1 ,0 .6 4 0 7 ,0 .1 1 9 3 ,4 .4 6 7 e-2 , 2.209e-2, 1.270e-2, 8.036e-3, 5.429e-3, 3.851e-3,
2 .8 3 5 e-3 , 2 .1 5 le -3 , 1.672e-3, 1.325e-3, 1.070e-3, 8.764e-4, 7.270e-4,
6.099e-4, 5 .1 6 7 e-4 , 4 .4 1 6 e-4 },
{1, 1 ,1 ,0 .8 4 2 1 ,0 .1 5 0 6 ,5 .5 8 4 e-2 ,2 .7 6 8 e-2 ,1 .6 0 4 e-2 ,1 .0 2 3 e-2 ,6 .9 8 0 e-3 , 4.996e-3,
3.71 le -3 , 2 .8 3 9 e-3 , 2 .2 2 4 e-3 , 1.776e-3, 1.443e-3, 1.188e-3, 9.916e-4,
8.361e-4, 7 .1 18e-4},
{1, 1 ,1 ,1 ,1 .0 3 8 ,0 .1 7 9 3 ,6.549e-2,3.23e-2, 1.87e-2, I.196e-2, 8.187e-3, 5.886e-3,
4 .3 9 3 e-3 , 3 .3 7 5 e-3 , 2.6 5 6 e-3 , 2 .1 3 1 e-3 , 1.739e-3, 1.439e-3, 1.204e-3,
1.019 e -3 },
{1 , 1, 1, 1, 1, 1.231, 0 .2 0 6 9 , 7.4 4 8 e-2 , 3 .6 4 5 e-2 , 2.104e-2, 1.344e-2, 9.209e-3,
6 .6 3 le -3 , 4 .9 5 9 e-3 , 3 .8 2 1 e-3 , 3.0 1 4 e-3 , 2.425e-3, 1.984e-3, 1.646e-3,
1.382e-3},
{1 , 1, 1, 1, 1, 1, 1 .4 2 4 ,0 .2 3 4 0 , 8 .3 1 5 e -2 ,4 .0 3 8 e -2 ,2 .3 2 e -2 , 1.479e-2, l.0 1 2 e-2 ,
7.289e-3, 5.4 5 5 e-3 , 4.2 0 7 e-3 , 3 .3 2 4 e-3 ,2 .6 7 9 e-3 , 2.1 9 6 e-3 , 1.825e-3},
{1, 1,1,1, 1, 1 , 1, 1 .6 1 6 ,0 .2 6 0 9 ,9 .1 6 3 e-2 ,4 .4 1 6 e-2 , 2.525e-2, 1.605e-2, 1.097e-2,
7.891e-3, 5.905e-3, 4.5 5 6 e-3 , 3 .6 0 2 e -3 ,2 .9 0 5 e -3 ,2 .3 8 3 e -3 } ,
(1, 1, 1,1,1, 1,1, 1, 1 .8 0 7 ,0 .2 8 7 6 ,0 .1 0 0 ,4 .7 8 7 e-2 , 2.724e-2, 1.726e-2, 1.177e-2,
8 .4 5 6 e -3 ,6 .3 2 3 e -3 ,4 .8 7 7 e -3 , 3.8 5 6 e-3 , 3 .1 12e-3},
{1, 1, 1, 1, 1, 1, 1, 1, 1, 1.999, 0 .3 143, 0 .1 0 8 3 , 5.152e-2, 2.9 1 8 e-2 , 1.843e-2,
1.254e-2, 8.9 9 5 e-3 , 6.7 1 9 e-3 , 5 .1 8 0 e -3 ,4 .0 9 4 e -3 },
{1 , 1, 1, 1, 1, 1, 1, 1, 1, 1 ,2 .1 9 0 ,0 .3 4 0 8 ,0 .1 1 6 6 , 5.513e-2, 3.109e-2, 1.958e-2,
1.328e-2, 9.515e-3, 7.099e-3, 5 .4 6 8 e-3 },
{1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ,2 .3 8 1 ,0 .3 6 7 3 ,0 .1 2 4 8 , 5.872e-2, 3.298e-2, 2.070e-2,
1.402e-2, 1.002e-2, 7 .4 6 8 e-3 },
{1, 1,1,1, 1, 1,1, 1, 1 ,1 ,1 ,1 ,2 .5 7 2 ,0 .3 9 3 8 ,0 .1 3 3 0 ,6 .2 2 8 e -2 ,3 .4 8 6 e -2 ,2 .1 8 2 e -2 ,
1.474e-2, 1 .052e-2},
{1 , 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2.763, 0 .4 2 0 2 , 0.1412, 6.5 8 4 e-2 , 3.672e-2,
2.292e-2, 1.545e-2},
{1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2. 954, 0 .4 467, 0.1494, 6.9 3 8 e-2 , 3.838e-2,
2.402e-2},
{1, 1,1,1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ,3 .1 4 5 ,0 .4 7 3 1 ,0 .1 5 7 5 ,7 .2 9 2 e -2 ,4 .0 4 3 e -2 },
{1, 1, 1,1, 1 ,1, 1, 1 ,1 ,1, 1, I, 1 ,1, 1, 1 ,3 .3 3 6 ,0 .4 9 9 5 ,0 .1 6 5 7 ,7 .6 4 4 e-2 },
{1, 1, 1,1, 1, 1, 1, 1, 1 ,1, 1, 1, 1, 1, I, 1, 1 ,3 .5 2 7 ,0 .5 2 5 9 ,0 .1 7 3 8 } ,
{1, 1, 1,1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ,3 .7 1 8 ,0 .5 5 2 3 } ,
{1, 1, 1,1, 1. 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ,3 .9 0 9 }
{ {1, 1, 1, 1, 1, 1, 1, 1, 1, 1,1, 1, 1, 1, 1, 1, 1, 1, 1, 1},
{4.699e8, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, I, 1, 1},
{5.575e7, 4.410e7, 1, 1, 1, 1, 1, 1, 1, I, 1, I. 1, 1 ,1, 1, 1, 1, 1, 1},
{1.278e7, 8.419e6, 8.9 8 6 e6 , 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ,1, 1 ,1, 1, 1, 1, 1},
( 4 . 125e6, 2.530e6, 2 .2 0 le 6 , 2.699e6, 1, 1, I, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1},
{1.644e6, 9 .7 3 2 e 5 ,7 .7 8 3 e 5 ,7.71 le 5 , 1.025e6, 1. 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1},
{7.568e5, 4.389e5, 3.3 5 8 e5 , 3.041e5, 3 .2 5 3 e5 ,4 .5 6 1 e5 , 1, 1, 1, 1, 1, 1, 1, 1, 1, 1,
1, 1. 1, 1}.
{3.869e5,2.215e5,1.651e5, 1.424e5, 1.388e5, 1 .5 6 1 e5 ,2 .2 7 2 e5 , 1, 1, 1, 1, 1, 1, 1,
163
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
1, 1, 1, 1. 1. 1},
{2 .1 4 3 e5 ,1 .2 l6 e5 , 8.9 0 5 e4 , 7 .4 5 9 e 4 ,6 .9 0 8 e 4 ,7 .0 6 5 e 4 . 8 .2 3 7 e 4 ,2 .2 3 3 e 5 , 1, 1, 1.
1, 1, 1, 1, 1, 1, 1, 1. 1}.
{ 1 .2 6 3 e5 ,7 .1 2 2 e 4 ,5 .1 5 6 e 4 ,4 .2 3 5 e 4 ,3 .8 0 0 e 4 ,3 .6 8 8 e 4 ,3 .9 0 5 e 4 ,4 .6 7 6 e 4 ,7 .1 5 1 e 4 ,
I. 1 , 1 , 1 , 1 , 1 , 1 , 1 . 1 , 1 . ! } .
{7.834e4,4 .3 9 7 e4 ,3 .1 5 6 e4 ,2 .5 5 6 e4 ,2 .2 4 6 e4 ,2 .1 1 0 e 4 ,2 .1 17e4, 2 .3 0 le 4 ,2 .8 1 3 e 4 ,
4 .3 7 7 e4 , 1, 1, 1, 1, 1 ,1, 1, 1, 1, 1},
{5.066e4,2.834 e4 ,2 .0 2 1 e4 ,1 .6 2 0 e4 ,1 .4 0 2 e4 ,1 .2 8 8 e4 , 1.250e4, 1 .2 8 7 e 4 ,1 .4 2 7 e 4 ,
I.7 7 4 e4 , 2.7 9 9 e4 , 1, 1, 1, 1. 1, 1, 1, 1, 1},
{3 .3 9 3 e4 ,1 .8 9 3 e4 ,1 .3 4 3 e4 ,1 .0 6 9 e4 ,9 1 4 8 ,8 2 7 1 ,7 8 4 5 ,7 8 0 4 ,8 1 9 2 , 9 2 3 1 , 1.163e4,
1.857e4, 1, 1, 1, 1, 1, 1, 1, 1},
{2.34 le 4 ,1 .3 0 3 e4 , 92 1 1 , 7 2 8 8 ,6 1 8 5 , 5526, 5156, 5 0 1 0 , 5080, 5 4 1 7 ,6 1 8 6 ,7 8 8 4 ,
1.271e4, 1, 1, 1, 1, 1, 1, 1},
{ 1.657e4, 9 2 1 0 , 6 4 9 0 ,5 1 1 0 ,4 3 0 8 , 3815, 35 1 6 , 3 3 5 9 , 3325, 3 4 2 4 , 3 6 9 9 ,4 2 7 1 ,
5496, 8 9 3 3 , 1, 1, I, 1, 1 ,1 },
{ 1.2e4,6 6 5 8 ,4 6 8 0 ,3 6 7 1 ,3 0 7 9 ,2 7 0 7 , 2471, 2 3 3 1 ,2 2 6 8 , 2 2 8 0 ,2 3 7 7 , 2 5 9 6 ,3 0 2 6 ,
3 9 2 6 ,6 4 2 9 , 1, 1, 1, 1, 1},
{8 8 5 8 ,4 9 1 0 ,3 4 4 4 ,2 6 9 3 ,2 2 4 9 , 1966, 1781, 1664, 1598, 1578, 1606, 1693, 1866,
2192, 2 8 6 4 ,4 7 2 0 , 1, 1, 1, 1},
{6 6 5 4 ,3 6 8 5 ,2 5 8 0 ,2 0 1 3 , 1675, 1457, 1312, 1216, 1156, 1127, 1159, 1232, 1369,
1 6 2 0 ,2 1 3 0 ,3 5 3 0 , 1, 1, 1},
{5 0 7 7 , 2809, 1964, 1529, 1268, 1099, 984.9, 9 06.9, 855.5, 825.2, 8 1 4 .1 , 8 22.3,
8 5 3 .2 ,9 1 4 .4 , 1023, 1217, 1 6 1 0 ,2 6 8 0 , 1, 1},
{3928, 2172, 1517, 1178, 9 7 5 .1 ,8 4 2 .4 , 7 5 1 .7 ,6 8 8 .6 ,6 4 5 .2 , 6 17.3, 6 0 2 .6 ,6 0 0 .5 ,
611.9, 6 3 9 .7 ,6 9 0 .3 ,7 7 6 .7 ,9 2 9 .6 , 1235, 2 0 6 7 , 1}
/♦float escfac[20]
=
/♦float escfac[20]
=
{0, 1.16e-4, 8 .1 8 e-4 , 2.57e-3, 5.90e-3, 1.14e-2, 0.0198, 0 .0 3 2 , 0 .049,
0 .0 7 2 5 ,0 .1 0 3 4 ,0 .1 4 3 9 ,0 .1 9 6 6 ,0 .2 6 4 6 3 ,0 .3 5 3 ,0 .4 6 7 ,0 .6 1 7 ,0 .8 2 0 ,1 .1 0 4 ,
1.531};*/
{0, I, 1, 1. 1, 1, 1, 1, 1, 1, 1, 1 ,1 ,1 ,1, 1, 1, 1, 1, 1};*/
/* * * * * g et the EEDF and input parameters from ELENDIF****/
void hydnewm ain_ (TaO, TeO, NeO, pretO, EEO, feO, M, ebynO)
double *TaO, *TeO, *N e0, *pretO, *ebynO, E E 0[101], fe0[101], M[3];
{
/♦FILE *df;*/
FILE *fptr.
FILE *input;
FILE *ef|)tr,
int i j , k, flagl, flag2;
char item l [100];
double value 1, value2, value3;
double deltaEE, vel, sumea, sumaa, sumsa, su m eal, sum aal, sumoterm, term 1, error, erroref;
double CHP1, CH3P1, CHP2, CH3P2, CONHP, CONH3P, su m l, sum2, sum3, sum4;
164
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
/****R ead the input the data from the file******/
sub_input();
/♦printf ("%g\t%g\t % g\t %g\t °/og\t %g\t °/og\t %g\t %g\t%g\t%g\t°/og\t %g\t %g\t %g\t %g\t % g\t %g\t
%g\t °/og\t %g\t 0/og\n",s\vitchi,switcha,switchr, iter, dist, CO, P 2, normfac, N , Numlev, Pret, N e, Te, Ta,
MaxEE,radius, guessef, radius, L,gamm,Q);
exitO;*/
Ta=*TaO;
Te=*TeO;
N e= *N e0*1.0e6;
Pret=*pretO;
EbyN=*ebynO;
/♦♦♦♦♦Calculate N um ber o f ground state particles for use in atom-atom collisions*********/
NP = Pret* 1013 2 5 /(7 6 0 * 1,38e-23*Ta);
vol = pi*pow (radius,2.0)*L;
/♦printf ("°/og\n”, vol);
exitO;*/
/♦♦♦♦♦♦♦♦♦♦Choose the distribution function 0=M axw ellian, 1=Non-M axwellian, 2 = S h aw ********/
if (dist==0)
{
EE[0] = 0 .0 0 1 ;
deltaEE = M axE E /(N -1);
for (i=0; i<N ; i++)
{
vel = sqrt(2*EE[i] * 1,6 0 9 e-19/9.11 e - 3 1);
chiC [i] = 4*p i*p ow ((9.11 e-3 l/(2 * p i* 1.3 8 e-2 3 *T e)), 1.5)*pow (vel,2.0)
*ex p ((-9 .1 le-3 l/(2 .0*1.38e-23*T e))*p ow (vel,2.0));
E E [i+ l] = EE[i]+deltaEE;
}
else
{
if ( d is t = 2 )
{
alphal = l/(sqrt(CO+0.25)+0.5);
alpha2 = l/(sqrt(C0-K).25)-0.5);
EE[0] = 0.001;
deltaEE = M axE E /(N -1);
/♦printf ("%g\t %g\n", alphal, alpha2);*/
for (i=0; i<N ; i++)
{
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Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
vel = sqrt(2*EE[i]* 1.6 0 9 e -19/9.11e -3 1);
if (E E [i]< I0 .1 9 )
{
chiC[i] = 4*pi*pow((9.11 e-3 I/(2*pi* 1.3 8e-23 *Te)), 1.5)
*pow (veU .0)*(exp((-9.1 le - 3 1/(2* 1.38e-23*Te))
*pow (vel,2.0))-( 1-P 2 )* ( 1-alpha 1)
*exp(-10.19 /(8 .6 15e-5*T e)));
E E [i+l ] = EE[i]+deltaEE;
}
else
{
chiC[i] = 4*pi*pow((9.1 le-3 l/(2*pi*1.38e-23*T e)),1.5)
*pow (vel,2.0)*exp(-10.19 /(8 .6 15e-5*Te))
*(P2*exp(( 10.19-EE[i])/(8.615e-5*Te))+alphal
*( 1-P2)*exp(( 10.19-EE[i])/(alpha 1 *
8.615e-5*T e))+ P 2*(l -P 2 )* ex p (-1 0 .19/
(8.615e-5*Te))*((alpha 1-1 )+f(pow(alphal ,2.0)
/(alpha 1+alpha2))-P 2*(l -a lp h a l))
*exp((EE[i]-2* 10.19)/(alpha2*8.615e-5*Te))));
E E [i+ l] = EE[i]+deltaEE;
};
}
{
for (i=0; i<N; i++)
{
EE[i]=EE0[i]; fe[i]=fe0[i];
}
EE[0] = le -5 ;
/*for (i=0; i<N; i++)
f
printf ("E[%d]=%g\tfe[%d]=%g\n">itEE[i],i, fe[i]);
)*!
sub_convert (EE, fe);
/* d f= fopen ("b:df.out”, "w");*/
/♦for (i=0; i< N; i++)
printf (“%g\t % g\n”, EE [i], chiC [i]);*/
/*fclose(df);*/
166
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
pointer = fopen("hydnew.out", "a+");
sub_avgEE (EE, cbiC);
/*fprintf(pointer,"Conditions are:\n");
fprintf(pointer, "Electron number density=%g\nElectron temperature=%g\n
A tom temperature=%g\nAverage electron energy=%g\nlntegral chiC*dc=%g\n", N e, Te, Ta, avgEE,
area);
fprintf(pointer,"--------------------------------------------------------\n");*/
if (s w itc h a = 0 && switchr==0)
{
fprintf(pomter, "****Atom-atom and radiative reactions are neglected****\n");
f^rintf(pomter, "\n");
}
if (sw itc h a = 0 )
{
fprintf(pointer, ''****Atom-atom reactions are neglected* * * *\n");
f^rintf(pointer, ”\nn);
}
if (sw itc h r = 0 )
{
fprintf(pointer, "****Radiative reactions are neglected* ***\n");
fprintf(pointer, "\n”);
}
if (s\v itc h i= 0 )
{
fprintf(pointer, "****Ionization from the excited states are neglected****\n");
fj)rintf(pointer, "\n");
}
for (efcount=0; efcount<100; efcount++)
{
vzero = sqrt(2* I.38e-23*T a/I.67e-27);
for ( i = l ; i<Num lev; i++)
{
lambda [i][0] = 1.24e-6/E[i];
/*printf ("%g\n", vzero);*/
kzero[i] = povv (lam bda[i][0],3.0)*guessef*g[i]*A [i][0]/89.0932/vzero;
if (kzero[i]*radius < 0.05)
escfac[i] = 1;
else
{
if (kzero[i]*radius >=4)
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Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
escfac[i] = 1.6/(kzero[i] ♦radius)/
sqrt(pi*log(kzero[i] ♦radius));
else
escfac[i] =-0.0173 ♦pow (kzero[i]
♦radius, 4 .0 )+ 0 .1 14♦pow(kzero[i]♦radius,3.0)
-0 .2 0 4 ♦pow (kzero[i] ♦radius, 2.0)-0.161
♦kzero[i] ♦radius-K).983;
}
/♦printf("%g\n", escfac[i]);+/
}
/♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦♦C all subroutines to calculate various reaction rates
*************************/
sub_KEea (EE, chiC);
sub_KDea (EE, chiC);
sub_KIea (EE, chiC);
sub_KTe 0 ;
sub_KRe (EE, chiC);
sub_KSa 0 ;
sub_Kaa (EE, chiC);
sub_KITaa (EE, chiC);
sub_disso (EE, chiC);
/♦fprintf(pointer,"Iterations start:\n");
fprintf(pointer, "No.\t GuessVt H [0]\t Per. Errorin”);
fprintf(pointer,"--------------------------------------------------------\n " ) //
C hoose the reactions to consider
*******************«************«****************************************/
if (sw itcha==0 && sw itc h r = 0 )
{
/♦fprind'(pointer, "♦♦♦♦Atom-atom and radiative reactions are neglected+#^ \n " );
fprintf(pomter, "\n");*/
for (i=0; i<Numlev; i-H-)
{
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Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
for (j=0; j<Numlev; j++)
{
Kaa[i][j]=0;
KSa[i](j]=0;
}
BCTa[i]=0;
Klaa[i]=0;
KRe[i]=0;
}
}
if ( s w itc h a = 0 )
{
/*fpnntf(pointer, "****Atom-atom reactions are neglected****\n");
fprintf(pointer, "\n");*/
for (i=0; i<Numlev; t++)
{
for (j=0; j<Num lev; j++)
Kaa[i] [j]=0;
KTa[i]=0;
Klaa[i]=0;
}
}
if (sw itchr==0)
;t
/*fprintf(pointer, "****Radiative reactions are neglected****\n");
fprintf(pointer, "\n");*/
for (i=0; i<Numlev; i++)
for (j=0; j<Num lev; j++)
K S a [i][j]= 0 ;
for (i=0; i<Numlev; i++)
KRe[i]=0;
}
169
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
if (sw itchi==0)
{
/♦fprintf(pointer, "♦•"‘♦Ionization from the excited states are neglected^**^\n");
fprintf(pointer, "\n");^/
for ( i=l; i<Num lev; i++)
{
BCIea[i]=0;
Klaa[i]=0;
}
}
/♦fprintf (pointer, "Loop:%d\n", elcount-rl);’ /
dum[0]=0.0;
dum[ 1]=Pret/760.0^ 1013 25.0/2.0;
for (count=I; count<iter, count+-i-)
{
H[0] = dum [count]/1.38e-23/Ta;
NH2 = N P-H [0]-2.0^N e;
/************************************* ***********************************************
Formation o f lhs[Numlev-1 ] [N um lev-1] and rhs[N um lev-1] matrices
*************************************************************************************/
for ( i = l ; i<Numlev; i++)
{
for (j= 1; j<Num lev; j++)
{
if (j<i)
lh s[i-1] [j-1]=Ne"Kea[j] [i]+NP*Kaa[j] [i];
if (j>i)
lh s[i-l]Q -l]= N e*K ea[j][i]+ N P #Kaa(j][i]+KSa[j][i];
sumea=0;
sumaa=0;
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Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
for (k=0; k<Numlev; k++)
{
sumea+=Kea[i][k];
sumaa+=Kaa[i] [k];
}
sumsa=0;
for (k=0; k<i; k++)
{
sumsa+=KSa[i][k];
}
lh s [i-1] [ i- 1]=-Ne*(sumea+KIea[i])-sumsa-NP*(smnaa+KIaa[i])-fudge*sqrt(8* 1.3
8e-23*Ta/pi/1.67e-27)/4.0*(2*pi*radius*L/(pi*pow(radius,2.0)*L));
i
i
rhs[i-I]=-pow(Ne,2.0)*(KTe[i]+KRe[i])-H[0]*(Ne*Kea[0][i]+NP*Kaa[0][i])-NP*Ne*KTa[i];
}
/*for ( i = l ; i<Numlev; i++)
{
{
for (j= 1; j<Numlev; j+ + )
printf (M
% d\t% d\t% g\n’\ i-1, j-1, lh s[i-l][j-l]);
}
pnntf("%g\n'', rh s[i-l]);
}*/
MatsoIver(lhs, rhs, X, N um lev-1);
/*printf ("OK after Matsolverin");*/
for (i= 1; i<Numlev; i-h -)
{
H[i] = X [i-1];
/*printf("%d\t %g\n”, i, H [i]);*/
}
/************************************************************************************
Calculate ionic concentrations (NHp, N H 2p, NH 3p)
* * * * * * * * * * * * * * * * 4
171
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
N H 2p = (K 13*N e*N H 2+ 2e-22*N e*N H 2)/(K 7*N e+ K 9*N H 2+ K 2*N e+K 4*N H 2+ l/(0.814*taul));
/♦printf ("% g\n\ NH2p);
exitO;*/
N H 3p = ((-K 5*pow (NH 2,2.0)+K 9*N H 2)*N H 2p+K 5
*p ow (N H 2,2.0)*N e)/(K 5*p ow (N H 2,2.0)+ K 3*N e+ K 8a*N e+ K 8b*N e+l/(0.745*tauI));
N H p = Ne-NH2p-NH3p;
/♦printf ("%g\t %g\t %g\t %g\n", NHp, N H 2p, NH3p, NH p+N H 2p+N H 3p);^/
su m eal= 0;
sum aa 1=0;
for (k = l; k<Numlev; k-t-+)
{
sumea 1+=Kea[0] [k];
sumaa 1+=Kaa[0] [k];
}
term 1=-N e*(sum ea 1+KIea[0])-NP*(sumaa I +KIaa[0]);
sumoterm=0;
for (k=L; k<Numlev; k+-r)
sum otenn+=H[k]*(Ne*Kea[k][0]+NP^Kaa[k][0]+KSa[k][0]);
dhdti=K2^Ne^NH2p+2^K3^Ne^NH3p+K4+N H 2#N H 2p+
2^K7^Ne^NH2p+K8a^Ne#N H 3p +3+K 8b#N e #N H 3p+K 9^NH2p^NH2;
/♦printf ("%g\t %g\t %g\t %g\n", NHp, NH2p, NH3p, dhdti);V
dhdt=H[0] ♦term 1+(N H p*N e+(KTe[0]+KRe[0])+sumoterm+NP
♦Nhp#KTa[0])+2#Kde, N H 2^ N e+2+KdH2+pow (N H 2,2.0)
+ 2 #KdH%N H 2+H [0]-2+Kre^Ne+pow(H [0],2.0)-2^K rH 2
♦N H 2+pow(H[0],2.0)-2^K rH#p o w (H [0 ],3 .0 )-2 M .4 8 el7
♦Q^(H [0]/N P)/(2-(H [0]/N P))/voI-gam m +K w #H[0]+dhdti;
/♦printf ("%g\t %g\n”, term l, H [0]);V
/♦ exitQ ;♦/
if(d h dt< 0.0)
{
dum [count+1]=dum[count]-fabs((dum[count]-dum[count-1])/2.0);
172
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
}
else
{
dum [count+l]=dum [count]+fabs((dum [count]-dum [count-l])/2.0);
};
if (fabs(dhdt/NP)< 1e - 10)
break;
/♦printf ("%d\t %g\t %g\t %g\t %g\n",count,dum[count],dum[count+l],dhdt, H [0]);*/
}/* iteration loop ends*/
/♦fprintf (pointer, "Number o f Iterations Required=%d\n", count);
fprintf (pointer, "Percentage error=%g\n", error);
fprintf (pointer, "Ground state density used for calculating escape factors=%g\n Actual Ground state
density=°/og\n", guessef, H [0]);
printf (pointer,"----------------------------------------------------- \n");*/
erroref = fabs((guessef-H [0])* 100.0/guesset);
if (erroref < 0 . 1 )
break;
else
gu essef = exp((log(guessef)-rlog(H [0]))/2.0);
/*check=0;
for ( i=l ; i<Num lev; i++)
{
{
for (j= 1; j<Numlev; j++)
{
check+=lhs[i-l ] [j-1]*H[j];
fprintf (pointer, "%g\t", lh is[i-l][j-l]);
}
173
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
fprintf(pointer, "%g\n\ rhs[i-l]);
}*/
density balance*************.****/
dndt=0;
for (i=0; i<Num lev; i++)
dndt+=KIea[i]*H[i]*Ne+KIaa[i]*H[i]"‘NP-KTe[i]
*pow (N e,2.0)-K R e[i]*pow (N e,2.0)-K Ta[i]*N e*N P;
dnedt=dndt-N e*(K 7*N H 2p+K 8a*N H 3p+K 8b*N H 3p)+K 13*N e*N H 2+2e-22*N e*N H 2;
/♦printf ("°/og\n\ dndt);*/
/♦for (i=0; i<Num lev; i++)
{
iea += KIea[i]#H [i]+Ne;
r3e + = K T e[i]+pow(N e,2.0);
rei + = K R e[i]^pow(Ne,2.0);
iaa + = KIaa[i]#H [i]+NP;
r3a + = K Ta[i]+N e%NP;
}
printf ("e-a ionization=%g\ne-3b recombination=%g\ne-radiativerecombination=
%g\na-a iomzation=%g\na-3b recombination=%g\n", iea, r3e, rei, iaa, r3a);V
/**********************************«*****************************/
/♦for (i=0; i<Num lev; i++)
printf ("%d\t %g\t %g\n", l + l, H[i], E [i]);V
for (i=0; i<Numlev; i++)
lnnog[i]=log(H [i]/(2^pow (i+1,2.0)));
if(T e< 1 0 0 0 0 .0 )
{
deltah = 8.43e-13^pow (T e, 3.0)-l.41 e-8 ^ p o w (T e,2 .0 )+ 8 .5 8 e-4 #T eH .4 3 ;
}
else
{
deltah = 1.0 le -1 2 +p ow (T e,3.0)-4.09e-8#p ow (T e,2.0)+ 7.77e-4#Te+5.33;
}
printf ("dnedt=%g\tAmb_Diff=%g\n",dnedt,Ne/taue);
174
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
printf ("XH=%g\tXHa=%g\n\n", H[0]/NP,M[ 1]);
fprintf (pointer,"dnedt=%g\tAmb_DiS=%g\n">dnedt,Ne/taue);
fprintf (pointer,"XH=%g\tXHa=%g\n\n”, H [0]/N P,M [1]);
/♦ if (dnedt > Ne/taue && M [ 1] > H [0])
{
if (dnedt* taue/Ne < 2.0 && M [1]/H [0] < 1.2)
{
•/
fprintf(pointer, "r3=%g\tr4=%g\tr5=:% g\n\n",H [2]/g[2]/4.11 e 10
tH [3]/g[3]/l .24e 10,H [4]/g[4]/2.97e9);
fprintf(pointer,"Conditions are:\n");
f^nntf(pomter,"Tg= %g'nXH= °/og\nXHa=°/og\nne= %g\nEbyN=% g\nT e=%g\n", Ta, H [0]/N P ,
M [l], N e. EbyN, Te);
fprintf(pointer, "dnedt= %g\nAmb_Diflf= %g\n\dnedt,N e/taue);
frrintf(pointer,"Pabs=%g\n\( 1-M [ 1])* N e* 1.5*(T e-T a)*2*9.11 e - 3 1/2/1,67e-27
*deltah*Pret/760* 101325/Ta*sqrt(8* 1.38e-23*T e/pi/9.1 l e - 3 1)* 1.7 e-l 9*vol);
fprintf(pointer, "NHp= % g\tNH2p= °/og\tNH3p= % g\n",NHp,NH2p,NH3p);
fprintf (pointer,"--------------------------------------------------------\n");
/♦fprintf (pointer, "Excited state atomic hydrogen populations are:\n");
for (i=0; i<Numlev; i++)
fprintf (pointer, "%d\t %g\t %g\t %g\n*, i+1, H[i], E[i], H [i]/g[i]);
fpnntf(pointer,''--------------------------------------------------------\n\n”);*/
/♦sub_output();*/
i f (dnedt > Ne/taue && M [ l ] < H[0])
{
if (dnedt*taue/Ne < 2.0 && M[1 ]/H[0] > 0 .8 )
{
fprintf(pomter, "r3=%g\tr4=%g\tr5=%g\n\n",H[2]/g[2]/4.1 lelO
,H [3]/g[3]/1.24el0,H [4]/g[4]/2.97e9);
fprintf(pointer,"Conditions are:\n”);
f^nntf(pomter,"Tg= %g\nXH= %g\nXHa=°/og\nne= %g\nEbyN=%g\nTe=%g\n", Ta, H [0]/N P ,
M [l],N e , EbyN, Te);
fprintf(pointer, ”dnedt= % g\nAmb_Diff= %g\n",dnedt,Ne/taue);
f^rintf(pointer,”Pabs=%g\n",( 1-M [ 1])* N e* 1,5*(T e-T a)*2*9.11 e - 3 1/2/1.67e-27
♦deltah*P ret/760*10l325/T a*sqrt(8*l.38e-23*T e/pi/9.11e-31)*1.7e-19*vol);
fprintf(pointer, "NHp= %g\tNH2p= % g\tNH3p= %g\n",NHp,NH2p,NH3p);
fjirintf (pointer,"------------------------------------------------------- \n");
/♦fprintf (pointer, "Excited state atomic hydrogen populations are:\n");
for (i=0; i<Numlev; i++)
fprintf (pointer, "%d\t %g\t %g\t %g\n”, i+1, H[i], E[i], H[i]/g[i]);
fjjnntf(pointer,"--------------------------------------------------------\n\n");^/
/♦sub output/);*/
}
if (dnedt < Ne/taue && M[1 ] > H[0])
{
175
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
if (dnedt*taue/Ne > 0 .5 && M[ 1]/H[0] < 1.2)
{
fprintf(pointer, "r3=%g\tr4=%g\tr5=%g\n\n“J i[2 ]/g [2 ]/4 .1 le l O
•H [3]/g[3]/1.24el0,H [4]/g[4]/2.97e9);
fprintf(pointer,"Conditions are:\n");
fj)rintf(pointer,"Tg= %g\nXH= % g\nXHa=°/o g\nne= %g\n
EbyN=%g\nTe=%g\n", Ta, H[0]/NP. M [l],N e , EbyN , Te);
fprintf(pointer, "dnedt= %g\nAmb_Di£E= %g\n",dnedt,Ne/taue);
fj)rintf(pointer,"Pabs=%g\n",( 1-M [ 1])* N e* 1.5 * (T e-T a )* 2 * 9 .11 e - 3 1/2/1,67e-27
*deltah*Pret/760* 101325/Ta*sqrt(8* 1.38e-23 *T e/p i/9.11 e - 3 1)* 1,7 e -19*vol);
fprintf(pointer, "NHp= %g\tNH2p= % g\tN H 3p= % g\n",NHp,NH2p,NH3p);
fprintf (p oin ter,"-------------------------------------------------------- \n “);
/♦fprintf (pointer, "Excited state atomic hydrogen populations are:\n");
for (i=0; i<N um lev; i++)
fprintf (pointer, "%d\t %g\t %g\t %g\n", i+ 1, H [i], E [i], H [i]/g[i]);
f^nntf(pomter,"-------------------------------------------------------- \n\n");*/
/*sub_outputO;*/
}
}
if (dnedt < Ne/taue && M [ 1] < H[0])
{
if (dnedt*taue/Ne > 0.5 && M[1 ]/H[0] > 0.8)
{
fprintf(pointer, "r3=%g\tr4=%g\tr5=%g\n\n",H[2]/g[2]/4.1 lelO
,H[3 ]/g[3 ]/l .2 4 e 10,H [4]/g[4]/2.97e9);
fprintf(pointer,"Conditions are:\n");
§irintf(pointer,'’Tg= %g\nXH= %g\nXHa=%g\nne= %g\nEbyN=%g\nTe=%g\n", Ta, H [0]/N P,
M [l],N e , EbyN, Te);
fprintf(pointer, "dnedt= %g\nAmb_Diff= °/og\n",dnedt,Ne/taue);
fprintf(pointer,"Pabs=%g\n",( 1-M[ 1])*N e* 1,5 * (T e-T a )* 2 * 9 .11 e - 3 1/2/1.67e-27
*deltah*Pret/76 0 * 101325/Ta*sqrt(8* 1,3 8 e-2 3 * T e/p i/9 .1 l e - 3 1)* 1,7 e -19*vol);
fprintf(pointer, "NHp= %g\tNH2p= % g\tNH3p= % g\n",NHp,NH2p,NH3p);
fprintf (p oin ter,"-------------------------------------------------------- \n ”);
/♦fprintf (pointer, "Excited state atomic hydrogen populations are:\n");
for (i=0; i<Num !ev; i++)
fprintf (pointer, "%d\t %g\t %g\t %g\n", i+1, H [i], E [i], H [i]/g[i]);
fj}rintf'(pomter,"-------------------------------------------------------- \n\n");*/
/*sub_outputO;*/
}
if ( r o u t = l)
sub_outputQ;
fclose(pointer);
}
176
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Read inputs from file
void sub_input 0
{
FILE * input;
int i;
char item [100];
double value;
input = fopen ("hydnew.inp", V ) ;
do
{
fscanf (input, "%s %Lf', item, &value);
if (strcmp (item, "*")== 0)
continue;
if (strcmp (item, "Switch^on") =
switchi=value,
0)
if (strcmp (item, "Switch_Atom”) =
switcha=value;
if (strcmp (item, "Switch_Rad") =
switchr=value;
if (strcmp (item, ”N_Iter") =
iter=value;
0)
0)
0)
if (strcmp (item, "Dist_Fimc_Type") =
dist=value;
if (strcmp (item, "CNOT") =
C0=value;
0)
if (strcmp (item, "PTWO") =
P2=value;
0)
if (strcmp (item, "Norm_Factor”) =
normfac=value;
0)
if (strcmp (item, "N_Dist_Func”) =
N=value;
0)
if (strcmp (item, "N_Levels") =
0)
0)
177
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Numlev=value;
if (strcmp (item, "Pre_Torr") =
Pret=value;
0)
if (strcmp (item, "N_Electrons") =
Ne=value;
0)
if (strcmp (item, "T_EIectron") == 0)
Te=value;
if (strcmp (item, "T_Gas") =
Ta=value;
0)
if (strcmp (item, ',M ax_EE“) =
MaxEE=vaiue;
if (strcmp (item, "Radius") =
radius=value*0.0254;
0)
0)
if (strcmp (item, "Guess_Escape") =
guessef=value;
if (strcmp (item, "Rates_Output") =
rout=value;
if (strcmp (item, "Height") =
L=value,'0.0254;
0)
0)
0)
if (strcmp (item, WalI_Recomb_H") == 0)
gamm=value;
if (strcmp (item, "Flow_Rate") =
Q=value;
if (strcmp (item, "Diff_Factor") =
fudge=vaiue;
0)
0)
}
while (strcmp(item, "Stop") != 0);
fclose (input);
}
178
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
/****************************************************************************************
Converting evA-3/2 to s/m (f(e) to chiC)
void sub_convert (double EE[size], double fe[size])
{
intk;
double ck, ckm 1, sumarea, fck, fckm 1;
sumarea =0;
for (k= 1; k<N; k++)
{
ck = sqrt((2* 1,609e-19*E E [k ]/9.1 le-3 1 ));
ckm 1 = sqrt((2* 1 ,6 0 9 e-19*EE [k-l ]/9 .11 e - 3 1));
/*printf ("%g\t %g\n", ck, ck m l);* /
fck = 9.5257e-18*pow (ck,2.0)*fe[k];
fckml = 9 .5 2 5 7 e-l8 * p o w (ck m l,2 .0 )* fe[k -l];
sumarea + = (fckt-fckm l)/2*(ck-ckm l);
/*prmtf (”% g\n“, sumarea);*/
norm = 1/sumarea;
/*p nn tf (”%g\t %g\n", sumarea, norm);*/
for (k=0; k<N; k++)
{
ck = sqrt((2* 1 609e-19*E E [k ]/9.11 e - 3 1));
chiC[k] = 9.5257e-l8*pow (ck,2.0)*fe[k]*norm ;
/♦printf ("%g\t %g\n”, EE[k], chiC[k]);*/
Calculation o f average energy (avgEE) and checking integral chiC*dc = 1
I* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * :* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * /
void sub_avgEE (double EE[size], double chiC [size])
179
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
{
intk;
double ck, ck m l, sumavgEE, sumarea;
sumavgEE=0;
sumarea=0;
for (k= I; k<N ; k++)
{
ck = sqrt((2* 1,6 0 9 e -19*EE [k]/9.11 e - 3 1));
ckm 1 = sqrt((2* 1.609e-19*EE [k-1]/9 .11 e - 3 1));
sumarea + = (chiC [k]+chiC [k-l])/2*(ck-ckm l)*norm fac;
sumavgEE +=(chiC[k]*povv(ck,2.0)
+chiC [k-1] *pow(ckm 1,2.0))/2*(ck -ck m 1);
av g E E = 0 .5 * 9 .1 le - 3 1*sum avgEE/1.609e-19;
area=sumarea;
/*printt'("%g\t %g\n", avgEE, area);*/
Calculation o f electron-atom excitation reaction rates
****************************************************************************************/
void sub_KEea (double EE[size], double chiC [sizej)
{
int ij,k ;
double ekl, qk, ck, uk, gu k,ck m l, u k m l, gu km l.q k m l, sum;
for (i=0; i<Numlev; i++)
{
for (j= i+ l; j<Num lev; j+ + )
{
sum =0;
180
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
for ( k = l; k<N; k++)
{
ck = sqrt((2* 1,6 0 9 e-1 9 * E E [k ]/9 .1 1e - 3 1));
ckm 1 = sqrt((2* 1,6 0 9 e- 19*E E [k-1]/9 .11 e - 3 1));
ekl = E[j] - E[i];
if (EE[k]>ekl)
{
uk = EE[k]/ekl;
guk = ((u k -1)/pow(uk,2.0))*log(uk);
qk=4*pi*pow(0.529e-10^.0)*pow((13.595/ekl)^.0)*fTi]D']*guk;
}
else
qk = 0;
if (E E [k-l]>ekl)
{
ukml = E E [k -l]/ek l;
gukm 1 =((ukm 1-1 )/pow(ukm 1,2.0))*log(ukm 1);
qkml = 4 * p i* p o w (0 .5 2 9 e -10,2.0)
* p ow (( 13.595/ekI),2.0)*f[i] [j]*gukm l;
}
else
qkml = 0 ;
sum += (chiC [k] * ck* qk+chiC [k -1] * ckm 1* qkm 1)/2 *(ck-ckm 1);
/♦printf ("%d\t %g\t %g\t %g\n”, k, EE[k], qk, sum );*/
}
Kea[i][j] = sum;
}
}
}
/****************************************************************************************
Calculation o f electron-atom de-excitation reaction rates
void sub_K D ea (double EE[size], double chiC[size])
{
int i jjc;
double ekl, qk, ck, uk, guk,ckm 1, ukm 1, gukm 1,qkm 1, sum;
for (i=0; i<Num lev; i++)
{
for (j=i+1; j<Numlev; j++)
181
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
{
sum =0;
for ( k = l; k<N; k-*-+)
{
ck = sqrt((2* 1,609e-19*E E [k]/9.11 e - 3 1));
ckm I = sqrt((2* 1,6 0 9 e-19*E E [k-1] /9 .11 e - 3 1));
ekl = E[j] - E[i];
uk = (EE[k]+ekl)/ekl;
guk =((uk-1)/pow (uk,2.0))*log(uk);
qk =(g[i] *(EE[k]+ekl)/EE[k]/g[j])*4*pi
*pow (0.529e-10^.0)V >w ((13.595/ekl)^.0)*fIi][j]*guk;
ukm 1 = (EE[k-1 ]+ekl)/ekl;
gukm I =((ukm 1-1 )/pow(ukm 1,2.0))*log(u km 1);
qkm I =(g[i] *(EE[k-1]+ekI)/E E[k-1]/gQ'])*4*pi
*povv(0.529e-10,2.0)*pow (( 13.595/ek l),2.0)*f[i] (j]*gukm l;
sum + = (chiC [k]*ck*qk+chiC [k-l]*ckm l *qkm l )/2*(ck-ckm l);
/♦printf ("%d\t %g\t %g\t %g\n", k, EE[k], qk, sum );*/
Kea[j][i] = sum;
}
}
/****************************************************************************************
Calculation o f electron-atom ionization reaction rates
void sub_KIea (double EE[size], double chiC [size])
int i,k;
double ekl, qk, ck, uk, guk,ckm 1, ukm 1, gukm 1,qkm 1, sum;
for (i=0; i<Numlev; i++)
{
182
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
sum =0;
for (k = l; k<N; k++)
{
ck = sqrt((2* 1.6 0 9 e -19*EE [k]/9.11 e - 3 1));
ckm 1 = sqrt((2* 1.6 0 9 e -19*E E [k-1]/9 .11 e - 3 1));
ekl = 13.595 - E[i];
if (EE[k]>ekl)
{
uk = EE[k]/ekl;
guk = ((u k -1)/pow(uk,2.0))*log(uk);
q k = 2 .66*p i*p ow (0.529e-l0,2.0)*p ow ((13.595/ek l),2.0)*gu k ;
}
else
qk = 0;
if (E E [k -l]> ek l)
{
ukml = EE[k-l]/ekl;
gukm 1 =((ukm I -1 )/pow(ukm 1,2.0))*log(ukm 1);
qkml =2.66*p i*p ow (0.529e-10,2.0)
*pow(( 13.595/ek l),2.0)*gu km l;
}
else
qkm 1 = 0;
sum + = (chiC [k]*ck*qk+chiC [k-1] *ckm 1*qkm 1)/2*(ck-ckm 1);
/♦printf ("%d\t %g\t %g\l % g\n \ k, EE[k], qk, sum );*/
}
KIea[i] = sum;
/****************************************************************************************
Calculation o f three body electron recombination reaction rates
The subroutine calculates N e*K Te[i]
void sub_KTe 0
{
in ti.j;
for (i=0; i<Numlev; i++)
183
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
KTe[i] =N e*(K Iea[i]*g[i]/2)*p ow ((pow (6.626e-34,2.0)
/(2 * p i* 9 .IIe-3 I* 1 .3 8 e-2 3 * T e)),1 .5 )* ex p ((1 3 .5 9 5 -E [i])/(8 .6 15e-5*T e));
/********************************«*******«********v* * * * * * * * * * * * * * « * * * * * * * * * * * * * * * * * * * * * * *
Calculation o f radiative electron recombination reaction rates
void sub_K R e (double EE[size], double chiC[size])
{
int i,k;
double ekl, qk, ck, ckm 1, qkm 1, sum;
for (i=0; i<Numlev; i++)
{
sum =0;
for ( k = l; k<N; k-H-)
{
ck = sqrt((2* 1,6 0 9 e -19 * E E [k ]/9 .1 le -3 1 ));
ckm 1 = sqrt((2* 1.6 0 9 e -19 * E E [k -1] /9 .11 e - 3 1));
ekl = 13.595 - E[i],
qk = 1.0 l e - 13/(2*EE[k]* 1 .6 0 9 e -19 * (i+ l ) /9 .1 le-31)/(1 +(EE[k]/ekl));
qkml = 1.01 e - 13/(2*EE[k-1]* 1.6 0 9 e-19 * (i+ l )/9 .11 e -3 1)/(l+ (E E [k-1]/ekl));
sum
t=
(chiC [k] *ck*qk+chiC [k -1] *ckm 1*qkm 1)/2*(ck-ckm 1);
KRe[i] = sum;
Calculation o f radiative deexcitation reaction rates
*************************************************************
184
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
void sub_KSa 0
{
in tij;
for (i= l; i<Num lev; i++)
K Sa[i][0] = escfac[i]*A [i][0];
for ( i= I ; i<Num!ev; i-t-t-)
{
for (j= i+ 1; j<Num lev; j++)
K S a [j][i]= A [j][i];
}
}
/I* * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * '
Calculation o f atom-atom excitation and deexcitation reaction rates
void sub_Kaa Q
int i j;
double p2, ekl;
for (i=0; i<Num lev; i++)
{
for ( j= i+ l, j<Num lev; j++)
{
ekl = E[j]-E[i],
p2 =( 1+(2*8 6 1 5e-5*Ta/ekl)>'( 1+ pow ((2*9.1 le -3 1*8.615e-5*Ta/l ,67e-27/ekl),2.0));
Kaa[i][j] =(4.06e-28*g[j]/g[i])*A [j][i]*p ow ((8.615e-5
*T a/13.595),0.5)*pow ((13.595/ekl), 4.0)*p 2*exp (-ek l/8.615e-5/T a);
Kaa[j][i] = 4.06e-28*A [j][i]*p ow ((8.615e-5*T a/13.595),0.5)
*pow(( 13,595/ekl),4.0)*p2;
}
}
/*******Q u enChing rates from Burshtein *********«*■»♦/
K aa[l][0]= 3e-15;
K aa[2][0]=2e-15;
K aa[3][0]=1.5e-15;
K aa[4][0]= le-15;
185
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Calculation o f atom-atom ionization and three body atom recombination
reaction rates (For three body atom actually K Ta[i]*Ne is calculated
***********************************************
void sub_KITaa (double EE [size], double chiC [size])
{
int i,k;
double te, ekl, ue, ua, p2, qk, ck, nuk, nuOk, ck m l, nukm l, nuOkml, q k m l, sum;
double oscst[20] = {0.8, 0 .9 ,0 .9 5 , 1, 1.1, 1.3, 1.6, 2.1, 3 ,4 .2 , 5 . 1 ,5 .8 ,6 .5 ,7 ,7 .5 ,8 , 8 .5 ,9 , 9 .5 ,1 0 } ;
for (i=0; i<N um lev; i++)
{
te = 2*avgE E /3/8.615e-5;
ekl = 13.595-E[i];
if (i< 20)
{
p2 =( 1 + (2 * 8 .6 15e-5*Ta/ekl))/( 1+ p o w ((2 * 9 .1 l e - 3 1*8.615e-5*T a/l ,67e-27/ek l),2.0));
ue = (0 -e k l)/8 .6 15e-5/te;
ua = (0 -ek l)/8 .6 15e-5/Ta;
KIaa[i] = 3 .2 2 e -18*oscst[i]*(p ow ((8.615e-5*T a),l .5)/(8.615e-5*te)
/sqrt( 13.595))*p2*exp(ua+ue*pow ((sqrt(te/Ta)-l),2.0));
}
else
sum =0;
for (k= 1; k<N; k++)
If
ck = sqrt((2* 1.6 0 9 e -19*E E [k]/9.11 e -3 1));
ckm I = sqrt((2* 1,6 0 9 e -19*E E [k-1 ]/9 .11 e - 3 1));
if (EE[k]>ekl)
{
nuk = sqrt(2*ekl* 1.6 0 9 e -19/9.11 e - 3 1);
nuOk = 0.7244e5*ekl;
qk = 2.07e-19*pow ((pow((nuk/nu0k), 0 .2 )-1 ), 2);
}
else
qk = 0;
if (E E [k-l]>ekl)
{
nukm l = sqrt(2*ekl* 1.609e-19/9.1 l e - 3 1);
nuOkml = 0.7244e5*ekl;
qkm l =2.07e-19*pow ((pow((nuk/nu0k), 0 .2 ) - l) , 2.0);
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}
else
qkm l = 0;
sum + = (ch iC [k ]*ck*q k+ chiC [k-l]*ck m l*qk m l)/2*(ck-ck m l);
/♦printf ("%d\t %g\t %g\t %g\n", k, EE[k], qk, sum );*/
}
KIaa[i] = sum;
KTa[i] = pow ((p ow (6.626e-34,2.0)/(2*p i*9.1 l e - 3 1 * 1.38e-23*te)),l .5)
♦e.\p(ekl/8.615e-5/Ta)*K Iaa[i]*N e;
}
/* * * * * * * * * * * * * * * * * * * * * * v * * * * * * * * * * v * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * * *
Hydrogen Dissociation rates
void sub_disso (double EE [size], double chiC[size])
{
int i j , k;
double vol. ck, uk, ck m l, u km l, qk, qkm l, sum, chick, ch ick m l, N H 2, dhdt, delta;
/♦double qd[29] = {0, 0, 0 , 0, 0, 0, 0, 0, 0, 0, 0, 0, 0. le -2 2 , 9 e-2 2 , 1.7e-21, 2 e-2 1 , 2 .6 e -2 1 ,2 .9 e -2 1 ,
3 .5 5 e-2 1, 3 .6 e -2 1, 3 .7 5 e - 2 1, 3,9 e -2 1, 4 .0 5 e-2 1 ,4 .1 5 e -2 1 ,4 .1 5 e - 2 1 ,4 .1 O e-21, 4 e - 2 1};*/
/♦double e[29] = {0 .0 0 0 1 ,0 .2 ,0 .4 ,0 .6 ,0 .8 , 1.6, 2 .4 ,3 .2 ,4 ,4 .8 , 5 .6 ,6 .4 , 7 .2 , 8, 8.8, 9 .6 ,1 0 .4 , 11.2, 12,
12.8, 13.6, 14.4, 15.2, 16, 16.8, 17.6, 18.4, 19.2, 20};*/
/* * * * * * * * * * * * * * * * * £ ajcujate e [ectron impact dissociation*********
sum =0;
for (k= 1; k<N; k++)
{
ck = sqrt((2* 1.6 0 9 e-19*E E [k]/9.11 e - 3 1));
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ckm l = sqrt((2* 1,6 0 9 e -19 * E E [k -1] /9 .1 1e -3 1));
if (EE[k]>9)
{
qk = -2 .18e-26*p ow (E E [k ],4.0)+ 2.96e-24*p ow (E E [k ],3.0)
-1 ,4 7 e-2 2 * p o w (E E [k ]^ .0 )+ 2 .9 6 e -2 1 * E E [k ]-1.64e-20;
qkml —2.18e-26*pow (H E [k-l ],4.0)+ 2.96e-24*pow (E E [k-1],3.0)
-1 ,4 7 e-2 2 * p o w (E E [k -1],2 .0 )+ 2 .9 6 e-2 1
♦ E E [k -l]-1 .6 4 e -2 0 ;
}
else
{
qk = 0;
qkm l = 0;
};
sum += (chiC [k] *ck* qk+chiC [k -1] ♦ckm 1♦qkm 1)/2*(ck-ckm 1);
/♦printf ("%d\t %g\t % g\t %g\n", k, EE[k], chiC[k], sum );*/
Kde = sum;
K r e= K d e/(((4 /sq r t(2 .0 )* 1 .3 8 e-2 3 * 8 4 * (l-ex p (-6 5 5 5 .1 2 /T e))* p o w ((2 * p i
1 6e-2 7 * 1,3 8 e-2 3 * T e/p o w (6 .6 3 4 e-3 4 ,2 .0 )),l .5)
♦e.\p (-52083.5/T e)))/1.38e-23/T e);
KdH2
= 1,5 e -15*p ow (T a,0.0)*exp (-48400.0/T a);
dissociation—Hyman et al.****/
KrH2 = 2.7e-43*pow (T a,-0.6)*exp(-0.0/T a);
/* * * H 2
impact
KrH = 2 .6 7 4 e-4 3 * p o w (T a ,-0 .6 )y * * * H impact dissociation—Hyman et
al^***V
KdH = K rH *6.022e23/8.314/T a*(4/l/sqrt(2.0)*1.38e-23
*84*(1 -e x p (-6 5 5 5 .12/T a))*pow ((2*pi* 1,6 e-2 7 * 1,38e-23
♦ T a/p o w (6 .6 3 4 e-3 4 ,2 .0 )), 1.5)*exp(-52083.5/T a));
/♦K w = (-1.54e-8*pow (T a,3,0)+ 2.73e-4*pow (T a,2.0)+ 0.191
*Ta-27.9)* le-4/(radius);*/
Kw=sqrt(8* 1.3 8e-23 *T a/pi/1.67 e-27)/4.0*2.0/radius;
/♦printf ("%g\t °/og\t % g\t %g\t %g\t %g\t %g\n", Kde, Kre, KdH2, KrH2,
KdH, KrH, K \v);V
/I**********************************************************************
188
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Reactions involving hydrogen ions
/* * * ******* e + U2+ = e + H+ + H * * * * * (K 2 )* * * * * * * * * * * * * * /
sum =0;
for (k = 1; k<N; k++)
{
ck = sqrt((2* 1.6 0 9 e -19 * E E [k ]/9 .1 1e - 3 1));
ckm 1 = sqrtt(2* 1.609e- 19*E E [k-1] /9 .11 e - 3 1));
if (EE[k]>2.5)
{
qk = 2e-20;
qkml = 2e-20;
}
else
{
qk = 1e -2 0 * (0 .19 4 * p o w (E E [k ],5 .0 )+ 3 .0 1*pow(EE[k],4.0)-17.7
*pow(EE[k] ,3 .0)+ 49.4 *pow (E E [k],2.0)-67.7
*EE [k]+40.3);
qkm 1 = 1e-20*(0.194*pow (EE [k-1],5 .0 )+ 3 .01 *pow (EE [k-1],4.0)
-17 .7 * p o w (E E [k -1],3.0)+ 49.4*p ow (E E [k -1],2.0)-67.7
*E E [k -l]+ 40.3);
};
sum += (chiC[kj *ck*qk+chiC [k-1] *ckm 1*qkm 1)/2*(ck-ckm 1);
/*printf ("%d\t %g\t %g\t % g \n \ k, EE[k], chiCfk], sum);*/
)
K2 = sum;
/********** e + H 3+ = e + H+ +
* * * * * (X 3 )» * * * * * * * * * * * * * /
K3 = le-17;
I********** U2+ + H2 = H + H+ + H 2 * * * * * (K 4 )* * * * * * * * * * * * * * /
K4 = 0;
/********** H + + 2112 = H 3+ + H2 * * * * * (K 5 )* * * * * * * * * * * * * * /
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K5 =0;
/********** e + H2+ = H + H
sum =0;
for (k= I; k<N ; k++)
{
ck = sqrt((2* 1.609e-19*E E [k]/9.11 e - 3 1));
ckm l = sqrt((2* 1.6 0 9 e-19*E E [k-1 ]/9 .1 le -3 1 ));
qk = 3e-19*pow (HE[k], -1.14);
qkm l = 3e-19*pow (E E [k -l], -1.14);
sum += (ch iC [k ]*ck*q k+ chiC [k-l]*ck m l*qk m l)/2*(ck-ck m l);
/*p nn tf ("%d\t %g\t %g\t %g\n", k, EE[k], chiCfk], sum );*/
K7 = sum;
/*printf C'%g\n", K7);*/
/*♦*#*****• e + pj3-i- = pjo + H *****(K8a)**************/
sum =0;
for (k= l; k<N; k++)
{
ck = sqrtC(2* 1.6 0 9 e-19*E E [k]/9.11 e - 3 1));
ckm 1 = sqrt((2* 1.6 0 9 e-19*E E [k-1] /9 .11 e - 3 1));
qk = 2.8e-19*pow (EE[k], -0.87);
qkm l = 2.8e-19*pow (E E [k -l],-0.87);
sum += (chiC [k]*ck*qk+ chiC [k-l]*ckm l*qkm l)/2*(ck-ckm I);
/*printl'("%il\t %g\t %g\t %g\n", k, EE[k], chiC[k], sum );*/
K8a = sum;
/*printf ("°/og\n\ K8a);*/
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K8b = 2*K8a;
/* * * * * * * * * * H 2+ + H2 = H3+ + H * * * * * (K 9 )* * * * * * * * * * * * * * /
K9 = 2e-15;
/* * * * * * * * * * e + H2 = e + H 2+ + e ** * * * (K 1 3)**************/
sum =0;
for ( k = l; k<N; k++)
{
ck = sqrt((2* 1,6 0 9 e -19*E E [k ]/9.1 l e - 3 1));
ckml = sq rt((2 * 1 .6 0 9 e -I9 * E E [k -l]/9 .11 e - 3 1));
if(E E [k]> 15)
{
qk = exp(0.0174*p ow (E E [k ],3.0)-l.04*p ow (E E [k],2.0)
+ 2 1 .6*EE[k]-202);
qkm l =exp(0.0174*pow (E E [k-l ],3 .0 )-l .04*pow (E E [k-l ],2.0)
+ 2 1 .6*E E [k -l]-202);
*»
J
else
{
i-
qk = 0;
qkm l = 0 ;
i*
sum += (ch iC [k ]*ck*q k+ chiC [k-l]*ck m l*qk m l)/2*(ck-ck m l);
/♦print!'("%d\t %g\t %g\t % g \n \ k, EE[k], chiC[k], sum );*/
}
K13 = sum;
/* * * * * * * * * * * * * * * * ^ b ip 0lar diffusion o f ion s*************/
dl = l/sqrt!pow((pi/L),2.0)+povv((2.405/radius),2.0));
tau 1= l/(T a * 8 .6 15e-5* l/pow (dl,2.0)*( 11.1 e-4*760/Pret*Ta/273));
ta u e= l/(T e*8.615e-5*l/p ow (d l,2.0)* (l 1. le-4*760/Pret*Ta/273));
/*printf ("%g\t %g\t %g\n", dl, tau 1, taue);
exitO;*/
/*printf (”K2=%g\t K3=%g\t K4=%g\n K5=% g\t K7=%g\t K8a=%g\n K8b=°/og\t K9=% g\t K13=%g\n",
191
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K2, K3, K4, K5, K 7, K 8a, K8b, K 9, K13);
exitQ;*/
M atrix solver routine (Thanks to M axat Touzeibauv for his help)
void Matsolver (double A [2 0 ][2 0 ], double C [20], double X [2 0 ], int n)
{
elim inate(A,Cji);
/* main funcuon*/
substitute(n,A,C JC) ;
}
void eliminate (double A [2 0 ][2 0 ], double C [20],int m)
{
int i 1,i2 j ;
double b;
/* diagonal elimination*/
for ( i 1=0; il< m - l; i 1
)
{
p ivot(A ,C ,il,m );
for ( i2=i 1+ 1, i2<m ; i2+ + )
{
b = A [i2 ][il];
for ( j= i 1 j< m j+ + )
{
if(A [il][il]!= 0 )
A [i2][j]= A [i2 ][j]-b * A [il](j]/A [il][il];
else
messageO;
C [i2]= C [i2]-b *C [il ]/A [il][il];
}
void pivot (double A [2 0 ][2 0 ], double C [20], int i l , int u)
{
double big, dum, tem p, am; /* seeks the biggest current diag.elem ent*/
intjj.j.i;
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big=fabs(A[i 1] [i 1]);
for (i= i 1+1 ;i<u;i++)
{
am =fabs(A [i][il]);
if (am>big)
{
big=am;
jj=i;
}
}
if(jj ! = i D
{
for(j=i 1'J<U'j++)
{
dum=A[jj][j];
AQj]U]=A[il]Lj];
A[i 1] [j]=clum;
}
temp=C[jj];
C [jj]=C [il];
C [il]=tem p;
}
}
void substitute (int s, double A [20][20], double C [20], double X [2 0 ])
{
/* performs back substituuon*/
int i, j, ix jx ;
double sum;
if( A [ s - l] [ s - l ] = 0 . 0 )
{
X [s-1]= 0;
messageO;
}
else
X [ s - 1]= C [s-1]/A [s-1] [s -1];
for (ix = 0 ;ix < s-1;ix++)
{
sum=0;
i=s-ix-2;
j= i+ l;
for (jx=j jx<s-jx++)
sum=sum+A(i] [jx] *X[jx];
X [i]=(C [i]-sum )/A [i][i];
}
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}
void m essage(void) /* Terminates program if matrix is singular*/
{
puts(”Singular matrix");
}
Write the outputs to hydout
* * * * * * * * * * * * * * * *
void sub_output()
{
int i,j\k;
double eax, eadx, aax, aadx, exl, axl, edxu, adxu, eadxu, aadxu, eai, eatbr, aai, aatbr, sdx, sdxu, rr, diff,
net;
eax=0; eadx=0; aax=0; aadx=0; exl=0; axl=0; edxu=0; adxu=0; eai=0; eatbr=0; aai=0; aatbr=0; sdx=0;
sdxu=0; rr=0; di£f=0;
/*for (i=3; i<Num lev; i++)
printf (”%d\t%g\n'', i, K ea[2][i]*H [2]*N e);*/
fprintf (pointer, "N\t\teawt\teadx\t\taax\t\taadx\t\texl\t\taxl\t\ted'<u\t\
tadxu\t\teai\t\teatbr\t\taai\t\taatbr\t\tsd\\t\tsdxu\t\trr\t\tdiS\n”);
fprintf (p oin ter,"----------------------------------------------------------------
------------------------------------------------------- \n\n-);
for (i=0; i<Num lev; i++)
{
for (j= i+ 1; j<Numlev; j-H-)
{
eax += Kea[i][j]*H [i]*Ne;
aax += Kaa[iJ[j]*H[i]*NP;
edxu += K ea[j][i]*H[j]*Ne;
adxu += Kaa[j][i]*H[j]*NP;
sdxu += KSa[j][i]*H[j];
}
for (j=0; j<i; j++)
{
eadx += Kea[i][j]*H [i]*Ne;
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Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
aadx += K aa[i][j]*H [i]*N P;
exl += K ea[j][i]*H (j]*N e;
axl += Kaa[j][i]*H(j]*MP;
sdx += K S a[i][j]*H [i];
x
eai = KIea[i]*H [i]*Ne;
eatbr = K T e[i]*pow (N e, 2.0);
aai = KIaa[i]*H[i]*NP;
aatbr = K Ta[i]*Ne*NP;
rr = K R e[i]*pow (N e, 2.0);
diff =H[i]*fudge*sqrt(8* 1,38e-23*Ta/pi/l ,67e-27)/4*(2*pi*radius*L/(pi*pow(radius,2.0)*L));
net = -eax-aax-eadx-aadx-sdx+exl+axl+edxu+adxu+sdxu-eai-aai+eatbr+aatbr+rr-diff;
fprintf (pointer, "%d\t\t%g\t\t%g\t\t%g\t\t%gVt\t%g\t\t%g\t\t%g\t\t%g\t\t%
g\t\t% g\t\t% g\t\t°/o g\t\t% g\t\t% g\t\t%g\t\t°/o g\t\t% g\n\n",
i+1, eax, eadx, aax, aadx, exl, axl, edxu, adxu, eai,
eatbr, aai, aatbr, sdx, sdxu, rr, diff, net);
eax=0; eadx=0; aax=0; aadx=0; exl=0; a\l=0; edxu=0; adxu=0; eai=0; eatbr=0; aai=0; aatbr=0;
sdx=0; sdxu=0 , rr=0; diff=0;
fprintf (pointer,"------------------------------------------------------------------------------------------------ \n\n");
fprintf (pointer, "Electron-atom Excitation reaction rates are:\n");
for (i=0; i<Numlev; i++)
{
for (j= i+ 1; j<N um lev; j+-r)
fprintf (pointer, “%d\t %d\t %g\t %g\t % g\n",i+l, j+1, K ea[i][j], K ea[i][j]*H [i]*N e,
l/(K ea[ij[j]*N e)),
}
fprintf(pointer, ”-------------------------------------------------------- \n”);
fprintf (pointer, "Electron-atom D e-excitation reaction rates are:\n");
for (i=0; i<Numlev; i++)
for (j= i+ 1; j<Num lev; j+-i-)
tpnntf (pointer, "°/od\t %d\t %g\t %g\t °/og\n" j + 1 , i+ 1, Kea[j][i], K ea[j][i]*H [i]*N e,
l/(K eaO ][i]*N e));
}
fprintf(pointer,"-------------------------------------------------------- \n");
195
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fprintf (pointer, "Electron-atom ionization reaction rates are:\n");
for (i=0; i<Numlev; i++)
fprintf (pointer, "%d\t % g\t% g\t% g\n",i+l, KIea[i], K Iea[i]*H [i]*N e, l/(K Iea[i]*N e));
fprintf(pointer,"---------------------------------------------------------\n");
fprintf (pointer, "Three body electron recom bination reaction rates are:\n");
for (i=0; i<Numlev; i++)
fprintf (pointer, "%d\t %g\t % g\t % g\n",i+lJCTe[i],
K T e[i]*pow (N e,2.0), 1/(K T e[i] *Ne));
fprintf(pointer,"-------------------------------------------------------- \n");
fprintf (pointer, "Electron-ion radiative recombination reaction rates are:\n");
for (i=0; i<Numlev; i++)
fpnntf (pointer, "%d\t %g\t %g\t %g\n",
i+l,K R e[i], K R e[i]*po\v(N e,2.0), l/(K R e[i]*N e));
fprintf(pointer,''---------------------------------------------------------\n");
fprintf (pointer, "Atom radiative deexcitation reaction rates are:\n");
for (i=0; i<Numlev; i++)
{
for (j= i+ l; j<Num lev; j+ + )
fprintf (pomter, "%d\t %d\t %g\t %g\t % g\n",j+l, i+1, KSa[j][i], KSa0'][i]*H[j],
l/(KSa(J][i])),
it
fprintf(pointer, ”---------------------------------------------------------\n");
fprintf (pointer, "Atom-atom Excitation reaction rates are:\n");
for (i=0; i<Numlev; i++)
{
for (j=i+1; j<Num lev; j++j
tprintf (pointer, "%d\t %d\t %g\t %g\t % g\n",i+l, j+ 1 , Kaa[i][j], K aa[i][j]*H[i]*NP,
l/(K aa[i](j]*N P)); '
fprintf(pointer,"-------------------------------------------------------- \n");
fprintf (pointer, "Atom-atom Deexcitation reaction rates are:\n");
196
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
for (i=0; i<Num!ev; i++)
{
for (j= i+ lj< N u m le v ; j++)
fprintf (pointer, "%d\t %d\t %g\t %g\t % g\n " j+ l, i+1, Kaa[j][i], Kaa[j][i]*H [j]*NP,
l/(K aa[j][i]*N P));
}
fprintf(pointer,"-------------------------------------------------------- \n “);
fprintf (pointer, "Atom-atom ionization reaction rates are:\n");
for (i=0; i<Num lev; i++-)
fprintf (pointer, ,,0/od\t %g\t %g\t % g\n",i+l, KIaa[i],
K Iaa[i]*H [i]*N P, l/(KIaa[i]*NP));
fprintf(pointer,"-------------------------------------------------------- \n");
fprintf (pointer, "Three body atom recombination reaction rates are:\n");
for (i=0; i<Num lev; i++)
fprintf (pointer, "%d\t %g\t %g\t %g\n",i+1, KTa[i], K Ta[i]*N e*N P, l/(K T a[i]*N P));
fprintf(pointer,"-------------------------------------------------------- \n");
/♦fprintf (pointer, "Dissociation Model:\n");
fprintf (pointer, "Electron Impact Dissociation=% g\t %g\t
%g\nElectron Impact Recombinauon=%git %g\t % g\nH2 Impact Dissociation=% g\t
%g\t %g\nH2 Impact Recombination=%g\t %g\t % g\nH Impact Dissociation=% g\t
%g\t %g\nH Impact Recombination=%g\t %g\t %g\nDiflfiision loss o f H=°/og\t % g\t %g\n"
Kde, K d e*N e*(N P -N H -2*N e),l/(K de*N e), Kre, K re*pow (N H ,2.0)*N e,
l/(K re*N H *N e), K dH 2,K dH 2*po\v((N P-N H -2*N e),2.0),
l/(K dH 2*(N P-N H -2*N e)),K rH 2T C rH 2*pow (N H ,2.0)*(N P-N H -2*N e),
l/(K rH 2*N H *(N P -N H -2*N e)), KdH,KdH *NH *(NP-NH-2*Ne),
l/(K dH *N H ), KrH, K rH*pow(NH,3.0), l/(K rH *pow (N H ,2.0)),
4/D /0.0254*K w ,4/D /0.0254*K w *N H , l/(4/D /0.0254*K w ));
fprintf(pointer,"------------------------------------------------------- \n");
fprintf (pointer, "Diffusion loss o f excited states:\n");
for (i=0; i<Num lev; i++)
fprintf (pointer, "%d\t %g\n",i+l, H [i]*487899.75);
fprintf(pointer, “------------------------------------------------------- \n”);*/
197
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
DIAGNOSTICS AND MODELING IN NONEQUILIBRIUM
MICROWAVE PLASMAS
Abstract of dissertation submitted in partial fulfillment
of the requirements for the degree of
Doctor of Philosophy
by
UMESH M. KELKAR, B.S.M .E., M.S.M.E.
University of Bombay, India, 1991
University of Arkansas, 1994
December 1997
University of Arkansas
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
This abstract is approved by:
DISSERTATION DIRECTOR:
Dr. Matthew Gordon
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
ABSTRACT
Self-consistent zero-dimensional modeling and optical emission spectroscopic
(OES) measurements o f absolute line and continuum emission are perform ed in
hydrogen/methane (1600-3700 W, 40-70 Torr, 300 seem hydrogen and 3 seem methane)
and argon (680 W, 5 Torr, and 250 seem argon) microwave plasmas. The experiments
are conducted in a 2.45 GHz, 6 kW W AVEMAT (Model M PD R -3135) microwave
plasma reactor typically used for diamond deposition. The self-consistent model is
developed by coupling the solutions o f the Boltzmann equation, electron conservation
equation, and power balance equation to a n-level collisional-radiative model (CRM). A
parametric study o f the hydrogen/methane plasma indicates that electron num ber densities
between 6 * 1016 and 3* 1017 m'3, electric field strengths between 40 and 55 Td, and average
electron temperatures between 21,000 and 24,000 K are required to obtain reasonable
agreement between the numerical and experimental data. A similar param etric study in
our argon plasma indicates that electron number densities between 3 and 7*1 0 17 m'3,
electric field strengths betw een 0.05 and 0.35 Td, and average electron temperatures
betw een 9000 and 14,000 K are required to obtain reasonable agreem ent between the
numerical and experimental data.
1
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
In argon plasmas, step-wise ionization from the excited states is the dominant
electron production mechanism, and neglecting it leads to a 30-60% overprediction in the
self-sustaining average electron energy and electric field. In hydrogen/methane plasmas,
the step-wise ionization from the atomic hydrogen excited states is negligible because o f
collisional quenching o f the excited states by molecular hydrogen, and the majority o f
electrons are produced by direct ionization o f molecular and atomic hydrogen ground
states.
In hydrogen/methane plasmas, all the numerical solutions are found to be
inconsistent with the experimentally measured continuum emission unless the electron-H,
cross-section, used for calculating free-free electron-neutral emission, is increased by a
factor between 10 and 20 above the momentum cross-section value.
A comprehensive energy balance study o f our reactor demonstrates that the input
power is only partially deposited in the plasma, and the remaining energy is directly
absorbed by the base-plate and cavity walls.
2
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission.
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Phone: 716/4824)300
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Fax: 716/288-5989
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