ANALYSIS OF A DIAMOND CVD PROCESS USING
COMPUTER SIMULATION
by
Masaki Nagai
B.E. Nuclear Engineering
Osaka University 1988
M.E. Nuclear Engineering
Osaka University 1990
Submitted to the
DEPARTMENT OF MATERIALS SCIENCE AND ENGINEERING
in partial fulfillment of the requirements for the degree of
MASTER OF SCIENCE IN MATERIALS SCIENCE AND ENGINEERING
at the
MASSACHUSETTS INSTITUTE OF TECHNOLOGY
September 1996
© 1996 Massachusetts Institute of Technology
All rights reserved
Signature of Author
Department of Materials Sc4Ence and Engineering
August 9, 1996
Certified by
Profe sor Davia A. movyance
Professor of Materials Engineering
Thesis Supervisor
Accepted by
Professor Linn W. Hobbs
Chairman, Departmental Committee on Graduate Students
996
SEP- 2 17
ANALYSIS OF A DIAMOND CVD PROCESS USING
COMPUTER SIMULATION
by
Masaki Nagai
Submitted to the Department of Materials Science and Engineering on
August 9, 1996 in partial fulfillment of the requirements for the
degree of Master of Science in Materials Science and Engineering
Abstract
A mathematical model was customized to represent fluid flow, heat
transfer, and chemical kinetics in a hot filament diamond CVD
reactor. Computed results from a two dimensional system suggest
that heterogeneous effects of the filament surface should be
included in the model for a more realistic representation of the
system. An assumption was made to consider the effects of the
filament surface as a catalytic factor for the hydrogen (H2 )
dissociation reaction. The computational results for the gas
species concentrations then gave a good agreements with
measurements reported in the literature. By examining the
characteristic diffusion lengths for the different species in the
system, it was found that the concentration of methyl radicals
(CH3), which are the precursors for diamond growth, as well as
methane (CH4) molecules are determined by chemical kinetics in
the gas phase. In contract, the concentration of atomic hydrogen
(H) was not affected very much by chemical kinetics since atomic
hydrogen created at the filament can diffuse very fast to the
substrate. The reaction, CH4 + H = CH 3 + H 2, was found to play a
key role in determining the concentrations of CH 4 and CH3 in the
reactor. General trends involving the effects of several process
parameters were identified in the analysis.
Thesis Supervisor:
Title:
Professor David K. Roylance
Professor of Materials Engineering
Contents
Abstract .. .................................................... 2
List of Symbols...............................................
4
List of Figures...............................................
7
List of Tables ................................................ 9
Acknowledgements .............................................10
1. INTRODUCTION ............................................... 11
1-1. Description of the diamond CVD process and
applications of diamond films ......................... 12
1-2. Literature review .................................... 16
1-2-1. Hot filament diamond CVD ........................ 16
1-2-2. Experimental measurements on the hot filament
CVD process ..................................... 19
1-2-3. Modelling of the hot filament CVD process........28
2. DESCRIPTION OF THE MATHEMATICAL MODEL .................... 37
2-1. General assumptions...................................37
2-2. Balance equations for mass, momentum, and energy.....39
2-3. Gas species transport equations......................40
2-4. Thermodynamic properties of gas mixtures..............42
2-5. Transport properties of gas mixtures.................42
2-6. Gas phase reactions ................................. 48
52
2-7. Surface reactions.................................
........ 52
2-8. Boundary conditions .........................
2-9. Numerical solution method (finite-volume method).. ..55
2-10. Dimensionless numbers ................................ 57
3. SIMULATION OF A TWO DIMENSIONAL AXISYMMETRIC REACTOR ...... 60
3-1. Gas phase reaction mechanisms........................60
3-2. Surface reaction mechanisms ........................... 63
3-3. Defect generation model..............................70
3-4. Simulation of the hot filament diamond CVD process...72
3-5. Results and discussion ...............................76
3-6. Predictions of the general trends in the hot
filament diamond CVD process ........................ 92
4. CONCLUSIONS
....................................
..............
95
Bibliography................................................98
Appendix A.
Qualitative behavior of the hot filament
diamond CVD process............................103
Appendix B. Thermodynamic data and transport properties
of the gas phase species........................114
List of Symbols
Ain
aj
C,
cp
Cp
Dii
D
D
DT
EA
f,
Go
AG ok
g
Hi
Ho
I
j
k
kk,
kk.b
K
Kk
L
id
m,
M
M
N
N,
n
P
P0
P,
Q
Q
cross section area of inflow, m2
thermal diffusion factor for gas pair i-j
molar concentration of gas species i, mole m- 3
specific heat per unit mass, J-kg-1.K-1
molar specific heat, J-mole-1.K-i
binary ordinary diffusion coefficient, m2 "s- I
effective ordinary diffusion coefficient, m 2 .s- 1
Wilke effective ordinary diffusion coefficient, m2 .s-1
thermal diffusion coefficient, kg-m-1.s-'
activation energy, J0mole-1
species mole fraction for gas species i
standard Gibbs energy change of formation for species i,
J mole-1
standard Gibbs energy change for reaction k, J'mole-1
gravity constant, m's- 2
molar enthalpy for species i, J-mole-1
standard heat of formation, J'mole-1
unit tensor
diffusive mass flux, kg-m- 2 "s- I
Boltzmann's constant, 1.38x10-23 J.K-1
forward reaction rate constant for kth gas reaction
reverse reaction rate constant for kth gas reaction
number of gas reactions
equilibrium constant for the kth gas reaction
number of surface reactions
characteristic diffusion length, m
molar mass, kg-mole-i
average molar mass, kg-mole-i
number of surface species
number of gas species
Avogadro's number 6.023x10 23 mole-'
unit vector
pressure, Pa
standard pressure, 1.0135x10 5 Pa
net mass production rate at the surface
volumetric flow rate at standard conditions, sim
total radiative heat source
r
R
Rg,
Rg b
Rs
Rd
Req
So
t
T
T*
To
V
radial distance, m
gas constant, 8.314 Jimole-1.K-1
forward reaction rate for the kth gas reaction, mole m- 3 .s-'
reverse reaction rate for the kth gas reaction, mole m-3 .s-1
reaction rate for the lth surface reaction, molerm-2 .s-'
molar destruction rate, mole-m- 3 .s-1
partial equilibrium ratio
standard entropy, J'mole-1.K-1
time, s
absolute temperature, K
reduced temperature = kT/E
standard temperature, 273.15 K
velocity vector, m's- 1
Greek symbols
X,
6
E/k
K
S
vik
p
0
0
stoichiometric coefficient for the jth surface species in
the ith surface reaction
Kronecker delta function
ratio of maximum energy of attraction and Boltzmann
constant, K
thermal conductivity, W-m- I .K- 1
dynamic viscosity, Pa's
stoichiometric coefficient for ith gas species in kth gas
reaction
density, kg.m -3
collision diameter, A
stoichiometric coefficient for ith gas species in Ith
t
surface reaction
chemical destruction time, s
viscous stress tensor, N.m-2
Q)
mass fraction
Q1
tabulated function of T*
Q
tabulated function of T*
,e
W
D
intermolecular potential energy, J
Subscripts
with respect to the ith species
with respect to gas pair i-j
k
with respect to kth gas reaction
with respect to Ith surface reaction
Superscripts
0
at standard temperature and pressure
due to ordinary diffusion
due to thermal diffusion
LIST OF FIGURES
Figure 1 : A schematic diagram of the diamond CVD process .....
13
Figure 2 : Production process of a diamond film for cutting
tools..............................................15
Figure 3 : Basic setup of a hot filament CVD reactor..........17
Figure 4 : Experimental setup of Martin and Hill [11,12] ...... 24
Figure 5 : Experimental configurations of Debroy [22].........31
Figure 6 : Grid cells and staggered grids for finite volume
method .............................................. 56
Figure 7 : A simple 2D axisymmetric reactor....................61
Figure 8 : A schematic diagram of the surface of a diamond
film ....................................................
65
Figure 9 : Experimental setup of Hsu [9]......................74
Figure 10: Computed gas flow in the region between the
filament and the substrate in Hsu's reactor........77
Figure 11: A schematic diagram of the filament zone...........79
Figure 12: Temperature gradient between the filament and
the substrate in Hsu's reactor.....................82
Figure 13: Gas species mole fraction between the filament
and the substrate in Hsu's reactor.................83
Figure 14: Partial equilibrium ratio of the reaction
CH4 + H = CH3 + H2 . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . .
84
Figure 15: Relative concentrations of H, CH3 , and CH4 between
the filament and the substrate ..................... 86
Figure 16: Characteristic diffusion lengths for H, CH3,
and CH4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . .
88
Figure 17: Relative concentration of H and CH3 above
the substrate .......................................90
Figure 18: Relative deposition rate and defect density above
the substrate......................................91
Figure 19:
Effect of filament zone temperature on H and CH 3
mole fractions ..................................... 107
Figure 20: Effect of temperature in the filament zone
on the growth rate .................................108
Figure 21:
Effect of temperature in the filament zone
on the defect density..............................109
Figure 22:
Effect of total pressure on H and CH3 mole
fractions ......................................... 110
Figure 23: Effect of total pressure on the growth rate.......111
Figure 24: Effect of total pressure on the defect density.... 112
Figure 25: Effect of total pressure on the uniformity of
the diamond film .................................. 113
LIST OF TABLES
Table 1 : Typical experimental conditions for the hot
filament diamond CVD process.........................19
Table 2 : Dimensionless numbers for the hot filament diamond
CVD process .........................................59
Table 3 : Gas phase reaction mechanisms........................62
Table 4 : Surface reaction mechanisms .........................69
Table 5 : Experimental information of quantitative
measurements for gas phase species .............
......
73
Table 6 : Experimental conditions of Hsu [9]...................75
Table 7 : Computational conditions for Hsu's reactor........75
Table 8 : Comparison between measured and calculated gas
species mole fractions...............................78
Table 9 : Comparison between measured and calculated gas
species mole fractions assuming the heterogeneous
effects of the filament surface ..................... 79
Table 10: Predictions on the general trends of the hot
filament diamond CVD process.........................94
Table 11: Computational conditions for qualitative analysis..104
Table 12: Thermodynamic data of the gas species used in the
diamond CVD modeling
Table 13: Transport properties of the gas species used in the
diamond CVD modeling
ACKNOWLEDGEMENTS
I would like to express my grateful appreciations to
Professor Julian Szekely for his warm welcome to his research
group and for his advice on this work.
I would like to thank Professor David K. Roylance for being
my advisor under short notice and for his encouragement and
discussion.
I am indebted to Dr. Gerardo Trapaga for many helpful
discussions and his continuing support.
Special thanks are given to people in the Mathematical
Modelling Group of the Department of Materials Science and
Engineering: Christy Choi, Robert Hyers, Nicole Lazo, Liping Li,
Patricio Mendez, Adam Powell, Kris Schwenke, and Hirokazu Shima.
I owe much to my parents in Kobe, Japan for their great
love and support in my life in U.S.A.
CHAPTER 1.
INTRODUCTION
The ultimate objectives of this research are to develop a
mathematical model for a filament assisted diamond chemical vapor
deposition (CVD) reactor and use such model to investigate the
behavior of the reactor under various operating conditions and to
understand the role of different process parameters, such as the
filament temperature, substrate temperature, and inlet gas
composition.
In this thesis, as a first step towards achieving
the objectives, we developed a two dimensional mathematical model
with detailed chemical kinetics to represent a typical
experimental reactor and to grasp the general behavior of the hot
filament CVD process.
Detailed numerical modeling of diamond CVD process can work
as an effective complement to the actual production operations,
providing information of the fluid flow, heat transfer, and
chemical kinetics.
A comprehensive computational model for the
diamond CVD process requires solving for the temperature,
velocity, and species concentration fields, including both
convective and diffusive species transport, together with complex
gas phase and surface reaction mechanisms.
In addition, the
model should be capable of estimating the quality of the
resulting diamond films.
Once the model is solved and understood, we can predict the
various kinetics inside the reactor together with the gas phase
concentration, diamond film deposition rate, diamond film
quality, and uniformity of the diamond films.
We then are able
to utilize it as the best tool to better understand the behavior
of the system and to optimize the production conditions.
1-1 Description
of the diamond CVD process and
applications
of diamond
films
Since the first publication by Matsumoto et al [1]
describing diamond CVD processes in detail at pressures and
temperatures where diamond is metastable with respect to
graphite, diamond CVD processes have been intensively
investigated and a multitude of diamond film production methods
has been developed.
A diamond CVD process is schematically sketched in figure 1.
The general understanding of diamond CVD processes can be
described as follows:
Gaseous reactants, typically methane (CH4)
and hydrogen (H2 ), flow into a reactor under reduced pressure (1
torr (133 Pa) - 200 torr (26600 Pa)).
They are activated and
decomposed to carbon-containing reactive gas species by thermal
or electromagnetic energy.
Convection and diffusion mechanisms
transport the reactive species to the substrate where they
decompose to diamond, together with other species (hydrogen,
other hydrocarbons, graphite as an impurity), by means of
heterogeneous reactions.
Atomic hydrogen is also generated and
transported to the substrate, where it activates surface sites
for the incorporation of reactive species, and promotes etching
REACTANTS
H2
CH4
ACTIVATION
H
CH4
2H
+
H -
CH3 + H2
FLOW AND REACTION
DIFFUSIO N
DIAMMOND FILM
C"r1-
Z*3UD,3
Figure
1.
A schematic
I
A -r1C
MIt
diagram of diamond
CVD process
of non-diamond species, such as graphite, from the deposition
surface.
This function of the atomic hydrogen is considered as a
key process to produce a high quality diamond film in diamond
CVD.
Diamond is currently grown by many different techniques,
using hot filaments, microwave plasmas, combustion flames, and
high speed direct current arc-jets, as the activation methods.
Among these techniques, the hot filament diamond CVD method is
the most widely used production technique because of its simple
setup, controllability of
production parameters and flexibility
for the scaling up of the reactor size.
Several companies commercialize diamond CVD products. The
diamond film produced by CVD method has, of course, the typical
properties of diamond, such as high hardness, high thermal
conductivity, and high degree of chemical inertness.
It can be
applied for heat sinks, cutting tools and wear resistant tools.
A production process for cutting tools is sketched in figure 2
[2].
First, a diamond film is synthesized in a reactor (a).
After the surface of the diamond on the substrate is polished to
meet surface roughness required for cutting tools, the film is
cut by laser (b).
Next, the substrate is dissolved by acid (c)
to get freestanding CVD diamond pieces (d).
The freestanding CVD
diamond pieces is brazed to cemented carbides to be the final
products (e).
In order to achieve high performance for these application,
Figure
2.
Production process
cutting tools [2]
of a diamond
film for
the diamond film must be of high quality (low defect density).
In terms of cost assessment, fast deposition of the film is
desired.
Since the diamond film surface has to be polished
during the production process, good uniformity of the film
thickness is also desired to reduce the polishing cost.
As a general rule, two properties, low defect density and
high deposition rate, represent conflicting issues in diamond CVD
operations.
We believe that there should be the optimum
production condition and geometry that satisfies an optimal
combination for both properties.
Numerical simulations should
help to find such conditions.
1-2 Literature
review
In this section, important research efforts related to the
hot filament diamond CVD will be reviewed.
First, a typical
experimental setup for the hot filament method will be described.
Next, we will focus on diagnostic techniques that intended to
identify important gas species inside the reactor.
Finally,
several modeling investigations on hot filament reactors will be
discussed.
1-2-1
Hot filament diamond CVD
Figure 3 shows a typical hot filament diamond CVD reactor
that was described by Matsumoto et al. in the first publication
on diamond CVD in 1982 [1,3].
They used Raman spectroscopy and
To vacuum pump
_ _
Figure 3. Basic setup of a hot filament CVD reactor
scanning electron microscopy to identify the deposits as diamond.
All major components necessary to achieve diamond deposition
are included in figure 3.
In a reduced pressured reactor, a
coiled or uncoiled refractory metal filament (usually tungsten or
tantalum), which provides thermal energy to activate a gas
mixture of methane (CH4 ) and hydrogen (H2 ), is located at a
distance of 5 - 20 mm from the substrate.
The filament are
resistively heated up to about 2000 - 2600 K and the substrate
temperature is controlled at 1000 - 1300 K.
Although the feed
gases are typically Methane (CH4 ) diluted to about 1.0 % with
hydrogen, other species such as organic compounds + oxygen [4],
acetylene (C2 H2 ) + hydrogen + oxygen [5] have also been reported
as feed gas mixtures.
Molybdenum, silicon, or silicon carbide
are used as a substrate.
With this simple set up, diamond growth rates in the range
0.1- 1.0 !im/h can be achieved.
listed in Table 1.
Typical experimental data are
Table 1. Typical experimental conditions
filament diamond CVD process
for the hot
Filament temperature
2000 K - 2600 K
Substrate temperature
1000 K - 1300 K
Filament-substrate distance
5 mm - 20 mm
Feed gas mixture
0.2 - 1.0 % CH4 in H2
Pressure
1 - 200 torr
Filament material
W , Ta
Substrate material
Si, SiC, Mo
Typical growth rate
1-2-2 Experimental
process
measurements
0.1 - 1.0
m/hour
on the hot filament CVD
Many gas phase diagnostic studies have been carried out on
the hot filament process intending to analyze the spatial
distribution of gas phase species, such as hydrogen (H2 ), atomic
hydrogen (H), methyl radical (CH3 ), acetylene (C2H2 ), methane
(CH4 ), existing in the region between the filament and the
substrate.
These studies have been also aimed at understanding
the mechanism of diamond formation under conditions of
metastability.
(a) Measurement of carbon containing species
The first study of gas phase species in the region between
the filament and substrate was performed by Celii et al. [8],
employing infrared laser absorption spectroscopy to detect
methane (CH4 ), methyl radical (CH3 ), acetylene (C2H2 ), and
ethylene (C2H4 ).
During film growth, portions of the gas mixture
in the region above the substrate were scanned by the laser, and
the species concentrations were estimated from the frequencies of
the infrared absorption spectrum of the gas.
(C2 H2 ) , methane (CH4 ),
detected.
ethylene (C 2H4 ),
Ethane (C2 H6 ),
Traces of acetylene
methyl radical (CH3 ) were
various C3 hydrocarbon species, and
methylene (CH2 ) were below the sensitivity level.
Methane (CH4) and acetylene (C2 H2 ) mole fractions in the
region immediately above the substrate were measured by Harris et
al. [7] as a function of filament-substrate distance with the aid
of a quartz sampling probe and on-line mass spectrometry.
gases were methane (CH4) and hydrogen (H2 ).
Feed
They reported that
the ratio of the concentration of CH4 and C2H2 reached at 1:1 in
the vicinity of the filament.
A temperature difference of 600 K
was also measured between the filament temperature (2600 K) and
that in the vicinity of the filament (2000 K).
Harris and Weiner [8] estimated methyl radical (CH3 ) and
atomic hydrogen (H) concentrations at the surface of the diamond
film.
Using a quartz sampling probe 3-5 mm from the hot tungsten
filament and gas chromatography of the sampled gas, they measured
ethylene (C2H4) and ethane (C2 6).
They assumed that these
species were formed by the following reactions involving the
methyl radical (CH3) inside the probe:
CH
3
+ CH 3
C2H 4+ H 2 and CH 3 + CH 3 = C 2 H 6
The mole fraction of methyl radicals in the reactor was therefore
calculated from the sum of ethylene (C2H4) and ethane (C2H6 ) mole
fractions.
From the methyl (CH3 ) mole fraction, atomic hydrogen
(H) concentrations in the reactor were also calculated using the
assumption of partial equilibrium for the reaction:
CH 4 + H - CH3 + H2
The methyl radical (CH3) mole fraction increased from 2x10-4
to 7x10-4 , when the methane (CH4 ) concentration was raised from
0.5 to 3.2 % volume.
Simultaneously, the atomic hydrogen
concentration decreased slightly.
Hsu [9,10] used molecular beam mass spectrometry, which is
the most direct measurement technique for the gas phase species,
to determine the concentrations of atomic hydrogen (H), methyl
radical (CH3 ),
acetylene (C2H2 ), and methane (CH4 ) in a hot
filament reactor under diamond growth conditions.
The gas
species were sampled through a 300 pm diameter orifice on the
center of the substrate and analyzed by the molecular beam
quadruple mass spectrometry.
With a small amount of argon added
to the methane/hydrogen mixture as the reference species, mole
fractions of H, H2, CH3, CH4 , and C2H 2 were determined with an
estimated error of
20 %.
With methane (CH4) fractions in the feed gas smaller than
1%, as usually employed in the hot filament diamond CVD,
acetylene (C2H2) was found to be the most dominant hydrocarbon
species at the substrate surface.
With initial CH4 concentration
higher than 2%, CH4 became the dominant species over C2H 2.
The atomic hydrogen mole fraction showed a sudden decrease
from about 2x10-3 at 2% methane concentration to 1x10 - 4 at 7%
methane concentration.
(b) What are the "growth species" in diamond CVD?
From the above diagnostic studies, the question "What is the
precursor that is responsible for diamond growth?" was left
unsolved.
However, several groups have performed experiments to
determine the growth species and now there appear to be
considerable evidences that the methyl radical (CH3 ) is the
precursor for diamond growth.
These studies include the "flow
tube" experiments by Martin et al. [11-13], experiments with free
and forced convective flow done by Schafer et al. [14], and
carbon-13 isotope studies performed at Rice university [15-17].
Martin and Hill [11] used microwave plasma only to generate
a stream of atomic hydrogen in a flow tube system (figure 4).
Then hydrocarbon gas species are injected to the downstream of
the atomic hydrogen.
After injecting a small amount about 2 % of
methane (CH4) to the flowing 90% Ar/10% H2 mixture, within a
furnace at 1120 K downstream from the microwave plasma; the
deposition of diamond was observed on the surface of the
substrates, which covered about 1-2 cm area, corresponding to 1
ms reaction time.
In a subsequent publication [12], additions of
methane (CH4 ) and acetylene (C2H2 ) were compared.
Methane was
found to be about an order of magnitude more effective for
diamond deposition than acetylene.
In their third paper [13] the
flow tube system was modelled including the gas phase reaction
mechanism, and convective and diffusive mass transport.
According to their analysis, when methane (CH4) was added to the
gas mixture, only methane (CH4 ) and methyl radical (CH3) were
present in significant quantities that was able to account for a
reasonable deposition rate.
A reaction probability of 10-3 for
_
~
_
~
--
I
Injector for
C-H species
Tube furnace
ýF-
To pump
Substrat s
Micro
wave
H2/Ar Flow
-
Figure
--
4.
---
---
Experimental
- -
-
-
-
Setup of Martin and Hill
[13,14]
diamond growth from methyl radical (CH3) was deduced from the
model, while 10-5 would apply to the growth from acetylene (C H ).
2 2
This reactivity for acetylene is too small to account for the
growth rate observed in hot filament CVD, implying that the
methyl radical (CH3) is the major growth species under such
conditions.
Schdfer et al.[14] performed hot filament growth experiments
with four parallel uncoiled tantalum wires, 11.5 mm apart from
each other, at a filament substrate distance of 8 mm.
Under the
free convection conditions used in this study (filament
temperature = 2470 K, 0.5 % methane in hydrogen), they observed a
strong decrease of the diamond film growth rate at the center
compared to the edges of the substrate.
In order to explain this
effect, they hypothesized that (1) a complete conversion of
carbon species to acetylene (C2H2) occurs in the filament region,
and (2) acetylene does not act as a "growth species" for diamond.
In the subsequent experiments, feed gas (CH4/H2 mixture or C2H2/H
2
mixture) was applied as a jet striking the substrate vertically
with velocities of several 1000 cm/s.
In the CH4/H2 mixture
case, bell shaped growth rate profiles with maximum growth rates
up to 2 pm/h were found in the methane case.
In the C2H2/H 2
mixture case, a slight growth rate depression in the wafer center
was observed.
These result showed that methyl radical (CH3 )
formed within the gas jet in CH4/H
2
mixture was responsible for
maximum growth rates.
Isotropic competition experiments using 13CH4/
12 C
H2 feed
gases in the hot filament environment were performed by Chu et
al.
[15, 16] and Evelyn et al. [17],
for polyclystalline film
growth, and for homoepitaxial growth on diamond (100), (111) and
(110) planes.
for
12 C
The first order Raman peak frequency at 1332 cm-1
diamond was found to shift to 1282 cm-'
Since this Raman peak shifts as a function of
for 13C diamond.
13 C
fraction, it
was able to be used to determine the isotopic composition of the
diamond films grown.
orientation, the
13C
Irrespective of crystallographic
mole fractions in the films turned out to be
equal to the corresponding mole fractions of methane (CH4 ), which
was sampled directly above the growth surface.
reaction,
CH 4 - CH3 + H,
methane (CH4)
Due to the
can be assumed to
represent the isotopic composition of the methyl radical (CH3 ).
Similar experiments were performed in microwave plasma CVD, also
showing the methyl radical to be about an order of magnitude more
efficient in diamond formation than acetylene (C2 H ).
2
(c) Measurements of atomic hydrogen
The first direct and noninvasive H atom detection in the hot
filament diamond CVD environment was the study of Celii and
Butler [18].
The dependence of atomic hydrogen concentration on
the filament temperature and CH4 /H2 ratio was determined
employing the resonance enhanced multiphoton ionization (REMPI).
They used one coiled tungsten filament and measured the gas
sample 8 mm away from the filament.
At each CH4/H 2 ratio the
REMPI signal increased with temperature, in agreement with
thermodynamics of hydrogen dissociation.
At a fixed temperature,
the signal decreased with increasing CH4/H2 ratio, especially
pronounced at the ratio = 3%, where a sudden drop of the signal
of atomic hydrogen (H) was observed below 2200 K. The authors
attributed this observation to some sort of surface poisoning or
phase change of the filament surface, rather than to gas phase
reactions.
From the result of the spatial distribution of atomic
hydrogen, they concluded that atomic hydrogen was transported to
the deposition region by diffusion.
Schafer and Klages measured the atomic hydrogen
concentration in the vicinity of hot filament by two photon
excited laser-induced fluorescence (LIF) [19,20].
Radial atomic
hydrogen concentration profiles were measured from the filament
surface to a distance 25 mm beyond.
At 1.5 mbar (1.125 torr),
the near surface concentration measured was about 50% of the
equilibrium concentration calculated for the measured filament
temperature (2640 K).
Between 10 mbar (7.5 torr) and 100 mbar
(75 torr) the atomic hydrogen concentrations were found to be
pressure independent.
Frenklach and Wang [21] discussed in detail several roles of
atomic hydrogen for diamond growth, namely: (1) preferential
gasification of graphite, (2) stabilization of sp3 hybridization
of surface carbon atoms against transformation sp2 or spi, and
(3) formation of gaseous diamond precursors, i.e., methyl radical
(CH3).
Based on their own modeling studies, the authors concluded
that the most important role of atomic hydrogen was to control
the concentration of activated sites on the growing diamond
surface.
The high activation energy for the surface site
activation explained the low temperature limit (< 700 K) of
diamond deposition.
At such low temperatures, the concentration
of the activated site was extremely small.
In addition, H atoms serve to transform sp2 carbon sites on
the surface to sp3 bonded carbon, thus preventing the formation
of non-diamond phases.
Preferential etching by H atoms, however,
although occurring at high temperatures, was a relatively slow
process below 1000 K and can be neglected.
1-2-3
Modelling of the hot filament
CVD process
Various chemical kinetics models have been developed to
determine steady state concentrations and reaction paths of
chemical species involved in diamond CVD processes [3,21-26].
In
these previous models, heterogeneous reactions at the filament
were completely neglected.
As for the heterogeneous reactions on
the substrate surface, although in early calculations they were
also neglected, recent studies have successfully included both in
a simple and complex way [27-29, 33-36].
(a) Species transport
Species are transported by convection (i.e., natural and
forced) and diffusion mechanisms.
In natural convection, the
driving force is provided by gravity due to density differences
in the fluid, caused either by temperature or by concentration
gradients within the system.
In forced convection, the driving
force is provided externally or mechanically, for example by a
In a hot filament diamond CVD reactor, natural convection
pump.
occurs to transport gas phase species.
Diffusion also involves
two types of mechanisms, one of which is ordinary diffusion
caused by concentration gradients in the fluid and the other is
thermal diffusion, which is due to temperature gradients in the
fluid.
In fluid dynamics, the dimensionless Peclet number
Pe -
vxl
D
(v: convective velocity, 1: characteristic length, D: diffusion
coefficient of a species considered), which is a measure of the
importance of convective mass transport relative to diffusive
mass transport, is used to characterize CVD processes.
The
peclet number is usually much larger than 1 for the convection
dominated high rate diamond CVD processes, such as dc arcjet
diamond CVD, while for diffusion dominated CVD process, Peclet
number is less than 1. In the hot filament diamond CVD process,
the Peclet number in the region between the filament and the
substrate is well below 1, thus diffusion plays a more important
role than convection in affecting gas species concentrations and
film growth.
Debroy et al. [22] showed this fact experimentally.
Their
set up was designed to have four configurations, which involved
placing the substrate below or above the filament with two
different flow directions, upward and downward gas flow as
sketched in figure 5.
In any configuration, there was little
difference in diamond growth rate, which showed the domination of
diffusive mass transport, and hence, a less important role of
natural convection in the hot filament CVD process.
_
~I_
_
%--
~
_
II
_
_1_1_
4:1....
Tube
f•urnace
7
configuration 1
configuration 2
I
configuration 3
II
I
_
_
_
Figure
5.
_
~
_
I
Experimental
configuration 4
_
__II__
___
configuration
______
_
of Debroy
_
_
[22]
(b) Modeling of gas phase reaction mechanism
Early kinetic calculations of the hot filament environment
were usually zero dimensional approximations, in which the gas
phase composition was calculated, using an assumed temperature
history [23,24].
More refined calculations are based on the assumption of one
dimensional gas flow, including ordinary and thermal diffusion,
with a prescribed temperature profile between the filament and
substrate.
Flow velocities of 1 cm/s were assumed around the
filament region.
The exact values of this velocity are not
critical for the results of the calculation, as long as Peclet
number is much less than 1 (diffusion dominated flow).
Harris
and Weiner [8] used a detailed gas phase chemistry model with 92
reactions, including oxygen containing species, in order to
develop a model to compare with their experiments.
Heterogeneous
chemistry at the diamond surface was completely ignored in their
calculations.
The calculated values of methane (CH4 ), acetylene
(C2 H2 ), methyl radical (CH3 ), and atomic hydrogen (H) showed very
good agreement with the measured value in their experiments.
A detailed surface and gas phase kinetics model of diamond
deposition was solved under an assumed temperature profile by
Frenklach and Wang [21].
Results of the gas phase calculation
including 158 reactions among 50 species were used as an input
for their detailed surface kinetics model.
Most dominant carbon
containing species (mole fractions > 10-4 ) in the gas phase were
methane (CH4 ), methyl radical (CH3 ), and acetylene (C2 H2 ). The
model predicted that both increasing methane concentration and
substrate temperature increased the growth rate and deteriorated
the film quality.
A diamond CVD reactor simulation by Goodwin and Gavillet
[25] used a two dimensional axisymmetric stagnation flow field at
the substrate, produced by a uniform z-directional axial inflow.
The resulting temperature and mole fraction distributions were
radially independent, thus this was essentially a one dimensional
calculation.
The gas phase reaction mechanism took 25 species
and 56 reactions into consideration.
The temperature was
specified only at the gas inlet and the substrate (2000 K and
1000 K, respectively) and varied almost linearly between these
boundaries.
As in the model of Frenklach and Wang [21], CH4,
CH3 , and C2H 2 were the only carbon containing species with mole
fractions over 10-4 . Although several other species were found
to have enough carbon concentration to meet typical growth rates,
their gas phase production rates were not sufficient to
compensate the depletion of the species at the substrate surface.
Reaction probabilities of 4x10 - 3 and 4x10 - 4 was required for
methyl radical (CH3 ) and acetylene (C2 H2 ),
growth rate of 0.5 Vm/hour.
respectively, to give a
Kondoh et al. [26] performed a study on a hot filament
process employing a rapid gas flow toward the substrate.
their model, the Peclet number was between 1 and 10.
In
Due to the
slow reaction of methyl radical (CH3 ) to C2H x species, the
acetylene (C2H2) mole fraction at the substrate could be
suppressed to below 10-5 , more than an order of magnitude below
the methyl radical
concentration.
With increasing filament
substrate distance the calculated methyl radical (CH )
3
concentration decreased parallel to experimental growth rate,
while the acetylene (C2H 2) concentration increased.
This
calculation also supported that the precursor for diamond growth
was the methyl radical (CH3).
(c) Modeling of the surface reaction mechanism
Several elementary mechanisms have been proposed [15-17, 2729] for diamond growth from methyl radical (CH3 ) and acetylene
(C2H 2) which consider how the growth monomer (CH3 or CA
2 2) can be
added to a particular site on a diamond surface through a series
of elementary reactions.
These models have assumed that
chemistry on the diamond surfaces could be understood in terms of
well-known chemistry of alkanes since there is still little
information about the elementary mechanisms of diamond deposition
themselves.
34
Coltrin et.al [30] used the SURFACE CHEMKIN package [31,
32], which was designed to handle the kinetics of a complex set
of elementary reactions at the gas/surface interface for their
modeling of dc plasma gun reactors.
The surface mechanism was
based on the dimer mechanism proposed by Garrison et al [33],
which included the pathways for the incorporation of methyl
radical (CH3 ), acetylene (C2H2 ), and carbon atom (C) from the gas
phase.
Alternatively, several groups have proposed reduced
mechanisms [34, 35], which do not attempt to describe each
elementary step in detail, but seek to capture the correct
qualitative behavior.
In this way, Goodwin [36] proposed an
interesting reduced mechanism in which entire classes of
elementary reactions are grouped into a single, approximate
overall reaction.
His reduced mechanism consists of four steps.
(1) Establishment of a steady state surface radical site coverage
(2) Attachment of reactive hydrocarbon species to the surface at
these sites
(3) Removal back to the gas phase of the surface adsorbates,
either by thermal desorption or attack by atomic hydrogen
(4) Incorporation of the adsorbate into the diamond lattice
By using this mechanism, the author was able to derive
simple analytical expressions for the growth rate and defect
density, which may be compared to experiments.
As a defect generation mechanism, he proposed that defects
were generated when an adsorbed hydrocarbon species reacted with
a nearby adsorbate before the species was fully incorporated into
the lattice.
According to his assumption, the defect density was
simply expressed in the following equation:
Defect density
Growth rate
Growth rate
2
(Atomic hydrogen concentration)
This defect generation mechanism will be adopted to the model
used in this study.
A two dimensional axisymmetric model will be developed in
this work.
In chapter 2, mathematical background used for the
model will be described.
Based on the experimental and
theoretical investigations reviewed in this chapter, the model
that includes ways to calculate the growth rate and the defect
density will be developed in chapter 3.
CHAPTER 2.
DESCRIPTION OF THE MATHEMATICAL MODEL
The objective of CVD reactor modeling is to relate operating
parameters and reactor geometry parameters to measures of film
quality (purity, uniformity) and to use the improved
understanding of the underlying physics and chemistry for
practical advantages, such as process optimization, performance
prediction and reactor design.
In this study, a software package, PHOENICS, is used to
model one of the experimental studies reviewed in chapter 1. The
program is designed to model the behavior of CVD reactors,
including fluid flow, heat transfer relating to a multi-component
gas, and both gas-phase and surface chemical reactions.
The
software can simulate up to 30 gas species undergoing multicomponent diffusion in addition to the conventional convective
and diffusive transport.
Heat transfer is linked with the view
factor surface radiation model.
A list of used symbols in this chapter is provided on page
4-6.
2-1 General assumptions
In order to reduce the complexity of the problem and the
computational expense for the solution of CVD modeling equations,
several general assumptions can be made within certain limits so
that the accuracy and applicability of the model are not
affected.
1. The gas mixture can usually be treated as a continuum.
The
dimensionless Knudsen number
K =
mean free path length (1)
characteristic length of the geometry (L)
is calculated to check the validity of this assumption.
(1)
When K
is small (<0.1), this assumption is good.
2. For the pressures and temperatures used in CVD techniques, the
gases may be treated as ideal gases, behaving in accordance with
the ideal-gas law and newton's law of viscosity.
3. The gas flow is assumed to be laminar.
In general, a fluid
flow becomes turbulent when either the Reynolds number (shear
driven turbulence) or the Grashof number (buoyancy driven
turbulence) of the flow becomes large.
These dimensionless
numbers and their underlying assumptions will be examined in
section 2-11 in connection to hot filament diamond reactors.
4. Gas mixture in CVD reactors may generally be treated as
transparent for heat radiation from heated walls and substrates.
5. The heating of the gas mixture due to the viscous dissipation,
which is the irreversible energy conversion to internal energy
for gases undergoing sudden expansion or compression, may be
neglected since no large gradients of velocity and pressure
appear in CVD reactors.
2-2 Balance equations
for mass, momentum, and energy
With the assumptions outlined in section 2-1, the gas flow
in CVD reactors is described by the conservation equations
ap
at
for mass
for momentum
ct
=
- Vo(p0)
= -V(pVV)
+
(2)
-
V-.
+
VP
pg (3)
(convective) (diffusive) (pressure) (gravity)
2
(i
(4)
t =p(VV + (VV)f) - y2(V0V)
3
* transposed vector
for energy.
cP-t-(pT)
= -
c V(pT) +
(convective)
V(kVT) +
(diffusive)
H
+
V.
N
1i-=
i
(inter diffusion)
aD
V* RT
M- V(Infi)
(Dufour effect)
(5)
N K
RHiVik(Rf
n
1=1
R' )
k=1
(chemical reaction)
The last term of equation (5) represents the heat
generation/consumption due to chemical reactions in the gas
mixture.
2-3.
Gas species
transport
equations
Since reactions in the gas phase cause the destruction and
creation of gaseous species, we have to include a species balance
term in the species balance equation.
The balance equation for
the ith gas species is
a(pwo)
K
dt
=
-V *(p9i)
(convective)
2-3-1
Ordinary
+
V*,
-
(diffusive)
mi k=t vik(Rp f
R-b)
(6)
(chemical reaction)
diffusion
In a binary gas mixture (N = 2), the ordinary diffusion
fluxes are given by Fick's Law:
C
C
S= -I 2 = -pDJoD
2
=
pD 2 V1o,
(7)
A general expression for the ordinary diffusion fluxes in
multicomponent gas mixtures is given by Stefan-Maxwell equation
which relate the diffusive fluxes of all species to the
concentration gradients of all species.
In terms of mass
fractions and mass fluxes, the Stefan-Maxwell equations [37] are:
M
2
V(oM)-
N
-C
-C
mD
(i = 1, N)
(8)
where M is the average mole mass of the mixture.
N
M = Z f,m,
(9)
I=1
For easy implementation of the Stefan-Maxwell equations to the
computer code, equation (7) should be rewritten as follows:
1, = -pD
V-
pm,D,
p
M
M
'J
+ Mo)I
vDC.
J-1 i j
(i
=
1, N)
(10)
where De
is the effective diffusion coefficient
(i = 1, N)
Di
(11)
As an alternative to the Stefan-Maxwell equations, an
approximate expression for the diffusive fluxes in a
multicomponent gas mixture has been derived by Wilke [38].
In
this case, the diffusion for the ith species in a multicomponent
mixture is written in the form of Fick's law of diffusion,
Ji
= -pD , Vw
i
(i
=1, N)
(12)
with a mixture-composition-dependent effective-diffusion
coefficient of the ith species:
D,
= (1-f)
(i
= 1, N)
(13)
2-3-2 Thermal Diffusion
The diffusive mass fluxes due to thermal diffusion are given
by
S- = - D
V(InT)
(14)
in which D T is the multicomponent thermal diffusion coefficient
for the ith species.
2-4 Thermodynamic
properties
of gas mixtures
Following the conventions used in the Chemkin thermodynamic
database [39], in PHOENICS, thermodynamic properties are defined
as a function of the absolute temperature by seven polynomial
fitting constants, al to a7, and expressed as
C,(T)
4
2 + a T3 + a T4
T
a
+
T
a
5
a%+
4
R
3
2
RT
-
2 T+
a+
(T)
4
3
2
So(T)
a ln(T)+aT+
R
4 T+
3 T+
2
a
2
T +
6(16)
T
5
3T2 +
a
3
4T
(15)
+
a5
4
T +a
(17)
The thermodynamic data of ten gas species used in this study
are extracted from the database and listed in Appendix B.
2-5
2-5-1
Transport
properties
Lennard-Jones
of gas mixtures
potential
The transport properties of gas species and of gas mixtures
may be calculated from kinetic theory with some assumptions for
the intermolecular potential energy function of the gas molecules
'P(r).
The most simple form is the so-called rigid elastic
sphere (r.e.s.) potential in which
W(r) =
for
r
< Y, I(r) = 0
for
r
> o
(18)
A more accurate potential function, commonly used for nonpolar molecules, is the Lennard-jones potential
4I(r) = 4e
--
r
(
i
r.
(19)
The Lennard-Jones parameters are taken from the CHEMKIN
transport property database [40] which contains values of a and
e/k for over 180 gas species.
The transport properties of the
gas species used in this study are summarized in Appendix B.
We need the integral functions, Wm(T*), WD(T*), A*(T*), B*(T*)
and C*(T*) for the calculation of the ordinary and thermal
diffusion coefficients.
In order to characterize their
temperature dependence, a reduced temperature
*
T
is defined.
=
kT
(20)
For rigid elastic sphere potential, the value of
these functions is identical to 1.
For the Lennard-Jones
potential, these functions vary slightly with temperature and
their value is of order one.
In PHOENICS, accurate polynomial
fits for their dependence on T* are adopted.
2-5-2 Density,
viscosity
and thermal
conductivity
The density of an ideal gas species is simply given by the
equation of state
Pi
Pm.
RT
RT
(21)
Similarly, the density of the gas mixture follows from
PM
P -T RT
(22)
From the kinetic theory of gases, the dynamic viscosity of a
single gas is given by
5
;amiRT
1•(23)
16 322(TN
NA
i-
Following the recommendations in ref. [41], in PHOENICS, the
mixture viscosity is calculated from
=
N
Xil
(24)
with
0.0.52
-05
-
=i-
++
{I +
j
(25)
The thermal conductivity of a monatomic gas species
predicted by kinetic theory is
15 R
m1
4 mi
-
(26)
(
For polyatomic gases, several semi empirical corrections
predicted by kinetic theory have been proposed in order to
account for the transfer of energy between internal degrees of
freedom and translational motion.
A reasonably accurate
expression is given by the modified Eucken correction [42]
S=15
+ 1.32
4 mR
_
5
R(
2
m
The mixture thermal conductivity is calculated from
(27)
=i
N
i
(28)
j-1
with
2-5-3 Ordinary diffusion
i 0.252
i0.s<
1+i
+ mi-05 1 +
-
(29)
coefficients
The binary diffusion coefficient of a gas pair i and j can
be also calculated from kinetic theory.
The value depends on the
temperature and the pressure, but it is virtually independent of
the mixture composition.
Defining Lennard Jones force parameters
for a gas pair i and j as
oi+
ij =
oj
2
E
'
kT
E ij
Ei1),
=
Tij-
(30)
the binary ordinary diffusion coefficient is obtained as
ia
D,
3
mi +7mm
16
mc mj
2
)1/2
23R-T3
2R3T
pNAn
oirciT
(31)fc
Wile and Lee [43] suggested a correction factor for a fixed
numerical constant in equation (31), the value of which varies
with the molecular weight of the diffusing gases.
1/ 2
3(mi+m
D - 16 1.15 - 0.00837
2-5-4
Thermal
diffusion
mimT
imi
coefficients
m ++m
mimf
j
/2
2n3RT3
2
PN noT2(32)
PNfA
ij D(Te)
PHOENICS provides two ways of calculating the thermal
diffusion coefficients.
One is an exact model to get multi-
component thermal diffusion coefficients from kinetic theory, and
the other is an accurate approximate model called Clark-Jones
approximate model, which adopted in this study.
The latter model
is explained in this section.
Clark-Jones
approximate model
An approximate model for multicomponent thermal diffusion
This model is
coefficients was suggested by Clark-Jones [44].
based on an extension of the exact equation for binary mixtures
to multicomponent mixtures.
In a binary gas mixture, the thermal
diffusion coefficients for each of the species is given by
DT =
P
T
-D 2 -
T
m1 m2 D1 2 a1 2 f 1 f 2 = pDAa2a1 21 2 2
where a12 is the thermal diffusion factor.
(33)
From kinetic theory,
a12 is given by
1 SM)fI - S(2)f 2
12x
12
-
(6C12 - 5)
(34)
with
S~m) m
m,
S,
2m2
2)
m, + m2
2m,
12
X
12
15
;
4A* 2
15
%2 2 4A122,
m2-m
2mi
-
m, - m2
2m22•
1
1
1
(35)
(35)
(36)
I
T
K12 = 0.00263
12
"
12
= 0.00263 -
f2
-
f2
Y= -
2f f 2
+
fY
-
(mm +m2)2
4
U =1- A*
15 12 4mm2
4
S
15
X~,I,k'
f2
U2(Y +
2U (2)
(40)
U(1)
2ff
+
2
U2_
(41)
+
A
12
1
AA*
1
2k
1•2
(39)
K
°.
U
(38)
+- f2
2f, f 2
Uc( +
f1
Y,
(37)
,(T 12)
12, ,(
K12 = 0.00263
oalO (T*)
XX
J2+
--
B
12i
12 5
1 (12
B
12 5 12
1
*+
'2
J
mI
1 (m, - m2)
M
1 (m
-m 2
2
m1 m 2
2
2
(42)
(mm)2
12
5
m1 m
B*,-5
32A*, 5
)
Im,m
1
(43)
Using the binary thermal diffusion factors, an approximate
expression for the multicomponent thermal diffusion coefficients,
suggested by Clark-Jones is given by
_T
r
S
_
-
='
m mjDiaifif j =
j= 1,= 1 M T
(44)
pD ja, oi (1)
I
j= ,j
I
which is a multicomponent extension of equation (33).
In the
above equation, aij
1J is calculated from equations (34)-(43),
replacing f. with fi/(f+fj.) and f. with f./(fi+fj).
2-6 Gas phase reactions
If we assume that K reversible homogeneous chemical
reactions take place between N gaseous species, we can use the
following general notation
N
- Vikj
1=k.b
A t
kk
N
V
-vik
Aij
(k = 1,
K)
(45)
i=
The stoichiometric coefficients, Vik, are taken positive for
reaction products and negative for reactants.
The operator IIII
in equation (59) is defined as
ik - L vik
ik
(46)
Viki2
When the forward reaction rate constant kk,f and the reverse
reaction rate constant kk,b are known, the forward and reverse
reaction rates Rk,f and Rk,b can be obtained from
R:,
= kk.f C"
,
R k.b
I=1
=
(47)
k .b IT
l=1
Thus, the total reaction rate RO may then be obtained from
N
N
R= R
-
.b
kk.f
I=1
IC
-
k k.b
LIC
1=1
,=
NfP i
kk.f I T il=1
I
- k l.bk.
bi.• 1 RTp
V
(48)
Using mass fractions wm,
equation (48) can be written as,
(wjV
R = kkp
I
(IVk
- k kb
N
kI
'
(49)
In general, the value of kk,f, kk,b depend strongly on the
temperature and are independent of the pressure for sufficiently
high pressures.
In this high pressure region, the generalized
Arrhenius expression can be combined with a pressure term, where
a pressure coefficient ck is fitted to experimental data to
describe the pressure dependence.
kk,f
= AkTexp - REAT
P
.P
(50)
At low pressures, however, kk,f and kk,b may enter the so
called "pressure fall off regime", where the reaction rates
depend linearly on pressure.
Two methods of representing the
rate expressions in the pressure fall-off regime have been
included in PHOENICS.
[45].
The simpler one was given by Lindemann
It requires three Arrhenius parameters for each of high
pressure limit (k_)
and low pressure limit (ko) dependent
expression for the rate constant
ko = A oT exp
R
k = AoT
RTk
expl-
RT
(51)
The rate constant at any pressure is then taken to be
k = k
F
rP
(52)
where the reduced pressure, Pr, is given by
P,= [m]kk
(53)
with [m] representing the molar concentration of the mixture.
The Lindemann form corresponds to F = 1. Therefore, six
parameters Ao,
A , B
p0, EA,
and EA,
are needed for the
Lindemann form.
When F is given by
0logP,+
logF = 1+
c
(54)
log F
Ln - d(log P,+ c)
:_n cent
with
Fcen
cent
c = -0.4- 0.67log Fent
(55)
n= 0.75- 1.271og Fcent
(56)
d=0.14
(57)
(1 - a) exp
T*"
the Troe form [46] is obtained.
+ a expT-
TP
+ exp
T
(58)
Here four additional parameters
a, T***, T*, and T** must be specified.
When either the forward reaction rate constant kk,f or the
reverse reaction rate constant kk,b is known, the reaction rate
constant in the opposite direction can be calculated from the
equilibrium reaction.
The equilibrium constant Kk for this
reaction, defined as
(f.eP
KkN
(59)
may be calculated from
Kk(T) = ex-
SAH 0(T) - TAS (T)
RT
IVk
(60)
where
N
AH (T)
=
N
E vikH (T), and
ASO(T) =
l=1
v~kS(T)
O
(61)
,=1
From equation (61) and (73), it may be deduced that kkf(P,T) and
kk,b (P,T) are related through
kk.b(P,T)
=
kkf(P,T)
Kk(T)
RT I'
Po'
(62)
Tabulated thermodynamic properties of specific heats, heats of
formation, and standard entropies of the individual species were
taken from the CHEMKIN database [39], as described in section 2-
4.
Finally, we can combine the general equations (6) for
species transport and chemical reactions, the expressions for
ordinary diffusion (10,11), the expression for thermal diffusion
(14), and the expressions for the gas phase reactions (47,48) to
solve the following equation for the species concentrations,
51
-• *(p oi)-V *(pDiV mi)+V *(poiDiV(Inm)+V * m iDij.ji
t
vik kkf C-
+ V (D V(In T))+m
k
kk.b
-
I-1
k-l
j
(63)
[C.kI
i=l
2-7 Surface reactions
At the wafer surface, a total number of L surface reactions
will take place, transforming solid and gaseous reactants into
solid and gaseous reaction products
N
M
5- o
M
I-xiJB - V
jAj -
i=1
N
A -
S
Xj Bj
(1
=1, L)
(64)
j=I
1=1
j=1
The growth rate Gs of the bulk species s is given in m/s by,
L
Gs
ms
Ps
E Xs,R
(65)
1=1
2-8 Boundary Conditions
On non-reacting surfaces, the no-slip and impermeability
conditions apply for the velocity
V= 0
(66)
Prescribed temperature boundary conditions and zerotemperature gradients normal to the wall apply for isothermal and
adiabatic walls:
T = Twal, (isothermal walls), fi.VT = 0 (adiabatic walls) (67)
The total mass flux vector normal to a non-reacting surface
must be zero for each of the species:
=0
.(Jc +
(68)
Due to the surface reactions (78), there will be a net mass
production rate Pi of the ith gaseous species of the wafer
surface
Pi = m, Y o,R
(69)
Thus, the velocity component normal to the wafer surface is
expressed by
p
N
L
I=1
1=1
(70)
Assuming a no-slip condition, the tangential component is
zero:
(71)
fixV =O
The reacting surface (substrate) are assumed to be
isothermal, leading to
(72)
T = Tsubstrate
The net total mass flux of the ith species normal to the
substrate surface must be equal to P.:
1
n-(po9 +
+ j ) = m,
cR,,
(73)
If an inflow of Qi slm (liters per minute at standard
temperature TO = 273.15 K and pressure PO = 1 atm) is prescribed
in the inflow of the reactor for each species; the inflow
velocity is given by
V -
10-3 pOO T i n
60
P
60
T
1 N
AiiE Qi
in
(74)
ini-i
Then, the boundary conditions for the velocity vector in the
inflow are given by
fixV =
v = vin ,
(75)
The total mass flow of the ith gas species into the reactor must
correspond to Qi according to
ni (po
iV
+
- ec
i
+
T
)
=
10 3 PO Qimi
60 TO RAin
(76)
At the reactor exit, zero gradients for the total mass flux
vector in the direction normal to the outflow opening may be
assumed, as well as zero heat and species diffusion fluxes.
Furthermore, we assume that the direction of the velocity is
normal to the outflow opening
fi*(VpV) = 0, fixV = 0
(77)
fi.(X VT) = 0
C
(78)
-T
fri(ji + JE) = 0
2-9 Numerical
solution
method
(finite-volume
(79)
method)
PHOENICS applies the finite volume method to solve the
differential equations described in equations (2)-(5).
The
differential equations contain similar terms, which are a
transient term, a convection term, a diffusion term and a
"source" term, which contains additional contributions that
cannot be included in the previous terms.
We can rewrite the
equations to the following general form:
at-(p)
= -V*(pV()
transient
here,
+ V(TFV ) + S.
convection
diffusion
(80)
"source"
is the variable to be solved, F is a diffusion coefficient
and S is the source term.
The solution domain, where the
differential equations apply, is divided into a number of
adjoining rectangular control volumes, or grid cells, each of
which are surrounded one grid point in which all scaler variables
are calculated.
The grid is structured in a sense that each
control volumes has a fixed number of neighbors.
The vector
quantities (velocity V, and species diffusion fluxes) are
calculated in points located at the cell walls, halfway between
the scaler grid points, using a so-called staggered grid.
dimensional grid is illustrated in figure 6.
Two
In figure 6, a
control volume surrounding the grid point P has four neighboring
grid points indicated by N (north), S (south), E (east), and W
Control volume
I
N cell
O
Vector quantities are
calculated at the walls.
Scalar quantities are
calculated at the center
of the cell.
o
I
W cell
i
E cell
o0
w
AY
0
"
all
I-IUI
I
0
O
0
S cell
AX
AX
Figure 6. Grid cells and staggered grid for finite
volume method
(west).
The finite volume equations are obtained by integrating
equation (80) over the control volume P.
fJf [(p,)+V.(p, ) +V.(FV)
dV
=
fSJJSdV
(81)
dV
(82)
By using the Gauss theorem, we have
=
f
S
8t
Several discretization schemes, such as central scheme, up wind
56
scheme, and hybrid scheme, can be used to get the value of the
variable 41.
After integration with one of the scheme,
the
finite volume equation has the following form.
ap(,p = aN,(•N+ asp, + aE4(IE+ aW ~
+aT(
T + source terms
(83)
with the subscripts P,E,W,N,S denoting the locations at which the
variable is computed, and the subscript T denotes the time.
The
a's are coefficients, temporarily treated as if they were
constants during each iteration.
Those terms with subscripts N,
S, E, W express the interactions between neighboring cells by way
of diffusion and convection, while aT denotes the time dependence
effect.
Equation (83) is obtained for each cell and each
variables to be solved.
Finally the variables are calculated
from
•p -
)
terms
aNc)N+ as( ) S + aE(c ) E+ aW 1 + aT( T+source
ap (=aN+as + aE + a + aT)
2-10 Dimensionless
(84)
numbers
Several dimensionless numbers describe the flow, heat and
mass transfer regimes in CVD reactors.
The key dimensionless
numbers are listed in table 2, together with typical values for
the region between the filament and the substrate in hot filament
diamond CVD reactors.
The physical quantities represented are
explained as follows.
The Knudsen number is the ratio of the mean free path length
of the gas molecules and a characteristic dimension length of the
reactor.
The typical distance between the filament and the
substrate (1 cm) is chosen as the characteristic length.
Knudsen
number is used as a measure to see if the gas mixture can be
treated as a continuum.
Since the value of 0.05 is below 0.1,
which is the limit of continuum, the assumption made in section
2-1 is validated for the reactor.
Reynolds number is a measure of the shear driven turbulence,
and Grashof number is also a measure of turbulence due to
buoyancy.
Both of them show very small numbers, which indicates
the flow is laminar.
Therefore, the flow in the region between
the filament and the substrate can be assumed a laminar flow as
described in section 2-1.
Mass Peclet number is a ratio of convective mass transfer
and diffusive mass transfer.
The low value (0.01) means the
species are transported mainly by diffusion.
Thermal Peclet
number for heat transfer is a counterpart of mass peclet number
for mass transfer.
It is a ratio of heat transfer by convection
and conductive heat transfer.
It also have a very low number
(0.001), which leads to the heat is transferred by diffusion
(conduction) not by bulk motion (convection).
Table 2.
Dimensionless numbers
diamond CVD process
Dimensionless number
Knudsen number
Reynolds number
for the hot filament
Formula
1
K
L
Lup
Value
0.05
0.01
U
Grashof number
gp L3 PAT
0.02
Mass Peclet number
Lu
D
0.01
Thermal Peclet number
LupCp
k
0.001
L:characteristic length, l:mean free path, u:velocity, p:density
t:viscosity, g:gravity, P:thermal volume expansion coefficient
D:diffusion coefficient, Cp:specific heat, k:thermal conductivity
CHAPTER 3.
SIMULATION OF A TWO DIMENSIONAL
AXISYMMETRIC REACTOR
As a starting point in grasping some fundamental
understanding on diamond CVD modeling, a simple two dimensional
axisymmetric reactor is simulated.
Figure 7 shows an schematic
sketch of the domain represented in the model.
To define the system, we have to specify the temperature in
the filament zone, substrate temperature, homogeneous reactions
taking place between the filament zone and the substrate, and
heterogeneous reactions on the substrate in order to calculate
velocity and thermal fields, as well as gas species
concentrations.
As for heterogeneous reactions on the filament
surface, they are not well understood and there is little
information available.
In this study, we will propose a way to
model the heterogeneous effects of the filament surface that
provides better agreement with previously measured results.
3-1 Gas phase reaction mechanisms
The gas-phase reaction mechanisms in diamond CVD reactor is
modeled using PHOENICS's chemical reaction code.
Twelve chemical
reactions were chosen based on the kinetic mechanism suggested by
Harris [7] which includes hydrogen, atomic hydrogen, and
hydrocarbon species that contains one or two carbon atoms.
This
mechanism should be accurate enough to model diamond CVD
environment because all major hydrocarbon species that have been
measured in the diagnostic studies are included in this
mechanism.
The reactions used in this study are listed in Table 3. Ten
species, H2 , H, CH2 , CH3 , CH4 , C2 H2 , C2 H 3 , C2H 4 , C2H5 , and C2H 6, are
included.
The forward reaction rates are also listed in Table 3.
Most of them are expressed in the modified Arrhenius expression.
Lindemann reaction mechanism [45] is used only for reaction No. 6
and 7. Troe mechanism [46] is used only for reaction No. 6. All
reactions are reversible and the reverse reaction rates are
obtained from the thermodynamic equilibrium data that is
implemented in PHOENICS.
-I
Filament zone
(Inlet)
..............
K
'''
l
Outlet
Figure
7.
A simple
''
Substrate
2D axisymmetric
reactor
Table
3.
Gas phase reaction mechanisms
Reactions
k = AT exp I-Eaj)
A
(1)
H + H + M
(2)
CH3
(3)
= H2 + M
Ea
Ref.
1. Ox 1018
-1.00
0
30
+ H + M = CH4 + M
6.0x 1016
-1.00
0
30
CH 4
+ H
= CH 3 + H2
2.2X 104
3.00
8,750
30
(4)
C2H6
+ H
= C2 H5
+ H2
5.4x 102
3.50
5,210
30
(5)
C2H 6
+ CH3 = C2H5
+ CH 4
5.5x 10-1
4.00
8,300
30
9.03X 1016
3.18X 1041
0.6041
-1.18
-7.03
6927
654
30
2,762
132
30
30
2.21x 1013
6.37x 1027
0.00
-2.76
2,066
-54
30
30
(6)
CH 3 + CH 3 + M = C2 H6 + M
low pressure limit
Troe parameters, a,
(7)
T***,
T*
C 2 H4
+ H + M = C 2H5 + M
low pressure limit
(8)
C 2H5
+ CH 3 = C 2H4
+ CH 4
7.9x 1011
0.00
0
47
(9)
C2H4
+ M
= C2H2
+ H2 + M 1.5x 1015
0.00
55,800
30
+ H + M
1.4x 1016
0.00
82,360
30
(10)
C2H4
+ M
= C2H3
(11)
C2H4
+ H
= C2H 3 + H2
1. 1x 1014
0.00
8,500
30
(12)
C2H3
+ H
= C2H 2 + H2
4. 0x 1013
0.00
0
30
The units of A depend on the reaction order, and they are given in terms of
moles, cm3 , and seconds. Ea is given in cal/mol.
3-2 Surface reaction mechanisms
An schematic diagram of the surface of the depositing
diamond surface is shown in figure 8.
In this study, a reduced
surface reaction mechanism proposed by Goodwin [36] has been
adopted in the model.
This mechanism classifies all elementary
surface reactions into four steps:
[A] diamond surface
activation, [B] attachment of reactive hydrocarbon species to the
surface at these sites, [C] removal back to the gas phase of the
surface adsorbates, [D] incorporation of the adsorbate into the
diamond lattice.
Each class is explained in detail as follows.
[A] Diamond surface activation
In the diamond structure each carbon atom is surrounded by
four other carbon atoms forming a tetrahedron.
On the surface of
the diamond film, one of the carbon atom is replaced by an
hydrogen atom and the hydrogen can be removed back to the gas
phase. This situation is so-called hydrogen terminated surface
(figure 8).
The hydrogen in figure 7 can be attacked by atomic
hydrogen in the gas phase, which leaves an "activated carbon" at
the surface.
This surface activation is characterized by the
following 2 reactions.
[abstraction of terminating hydrogens by atomic hydrogen]
(1)
CdH + H - C* + H 2
k_, k,
[refilling the activated sites by atomic hydrogen]
CC + H = CdH
(2)
where CdH represents a generic hydrogen-terminated
surface site
on the diamond surface and Cd* represents an activated site due
to the removal of the hydrogen.
k, and k_1 are the forward and
reverse reaction rate constant for the reversible reaction (1).
k 2 is the forward reaction rate constant for reaction (2).
Due
to the large negative change in free energy for reaction (2), the
reverse process is negligible.
At steady state, the rate of creation of radical sites by
reaction (1) balances their rate of destruction by the reactions
(1) and (2).
d[C' I
dtd - 0
=
kl[CdHIIHI + k,[ICd][H 2 1-k 2 [Cd][H]
(3)
Solving for the steady state radical fraction f* leads to
IC*I+IH
[C~j +ICHI
In
n,
k, IH1
(k,+k,2)[H + k_ 1[H 2
1
-
k, X
(k,+k,)XH + k
(4)
where XH is the atomic hydrogen mole fraction on the surface
IH1i
(XH -
[Hi2), and ns is the site density on the diamond film
surface (n
= 5.22x10-9 mol/cm 2 , which is from the bulk diamond
carbon density raised to the two third power).
In case of high
quality CVD diamond production, XH is sufficiently large compared
.
to k1
-1
Thus, equation (4) shows that f* reaches to a limiting
value given by
I
lC
f*
ns
k,
(5)
k,+k,
Since k and k2 are functions of temperature, f* is also a
function of temperature only.
______
_~______
_____
_I
---
-
·I~·I~·--~I~-----· · · ·
gas species
·-----C--
·
_
__
H
_
H
CH3
H
diamond
or
defect
H
I
H
H
CH3
CH3
I
I
I
I
I
C -C -C-C-C
-C --- C -CI 41
S l 1 1 -C
-C -iC
1~
C
/ I
I
_
C-
I
~
--
z/-
--
-3--
i
diamond film
-
--
--
_r_
CdH
H-terminated site
Activated site
-I
·
Figure
--
_---_-
8.
~
-_-----
A schematic
film
CH3 adsorbed site
-
----
_
_---
Diagram of the surface
of diamond
[B] Attachment of a reactive hydrocarbon species (the precursor
of the diamond) to the surface at these sites
Cd + CnHm = CdA
(6)
ka
with k the reaction constant, CnH
the precursor of the diamond.
[C] removal back to the gas phase of the surface adsorbates, by
thermal desorption
CdA - C
+ CnHm
(7)
kd
or attack by atomic hydrogen (etching)
CdA + H - C* + CnHm+I
(8)
ke
[D] Incorporation of the adsorbate into the diamond lattice with
abstraction of adsorbate hydrogens by H
CdA + (m- 1)H = CdH + (m- 1)H2
ki
(9)
This incorporation step is assumed to be first order in atomic
hydrogen concentration [H] for simplicity.
Similarly, it is
assumed that the etching step in (3) is first order in the
concentration of atomic hydrogen [H].
With these assumptions, A
reduced mechanism provides a simple kinetic expression which can
be written for the rate of change of the surface concentration of
the adsorbed hydrocarbons [CdA].
The rate of change of the
adsorbed hydrocarbons is balanced by the creation by reaction
(6), and destruction by reaction (7),
(8), and (9).
dICdAj
dt
= ka[CnHm][C ]-
kd CdAl-(ke+k,)LCdAI[HJ
(10)
At steady state d[CdA]/dt=0, thus we have
[CdAI =
(11)
k IC Hm]
kd+(ke+ki)IHI
By using this equation and equation (9),
the carbon
incorporation rate RC in mol/cm 2 /s is
1R = k,ICdHIIH]
(12)
Dividing Rc by the molar density of diamond nd (0.2939 mol/cm 3 )
gives the linear growth rate
ki [CdAllHI
G =
s) =3
m/k,ICAIHI
(cm/s) = 3.6x 10'
(inn/h)
(13)
nd
Substituting from equation (11) for [CdA] results in
G = 3.6x 107 kikaICnHml[HICdI
nd [kd + (ke+ ki)[lHj
(utm / h)
(14)
and from equation (5),
[C;] = f* Xnn
(15)
where f* is the radical site fraction and ns is the total surface
site density, we have
G = 3.6x 10f x f x-
n,
n
d
k,kaICnHmln[H
Jkd+
(k e +k,)[Hl]
(um/h)
(16)
As reviewed in chapter 1, there is enough evidence to state that
CnHm corresponds to the methyl radical (CH ). Thus, this reduced
mechanism enable us to calculate the growth rate from the
concentration of two gas species, [H] and [CH 3 ], at the diamond
surface, as:
G = 3.6 x 107
x
n,
kika[CH 3 ][H ]
nd
(kd + (k,+ki)[HIP
f* x -n(k+kHl
(im / h)
(17)
which is the equation adopted in this study to calculate the
diamond growth rate.
Equation (17) implies several significant
points about diamond growth.
First, at constant substrate
temperature the growth rate is a linear function of [CH 3 ].
Secondly, when the concentration of atomic hydrogen [H] is low,
which leads to kd >> (ke+ki)[H], the growth rate increases
linearly with increasing [H]; and when [H] is significantly high,
which leads to kd << (ke+ki)[H], the growth rate does not depend
on [H], only on the concentration of [CH 3 ]. Thirdly, in order to
produce a diamond film that has a good uniformity we must achieve
uniform [H] and [CH3] concentrations simultaneously all over the
substrate.
Goodwin [36] estimated the rate constants
(kl,k2,ka,kd,ke and
ki) at the surface temperature of 1200 K based on the growth
mechanism proposed by Harris, which assumed bycyclononane (C H )
9 4
as a diamond surface [48].
Their values are listed in table 4.
This reaction mechanism has been incorporated into the model
using Phoenix's surface reaction code.
Table 4.
Surface
reaction
Reaction
mechanisms
k
Ref.
(1) C H + H - Cd* + H
k
= 2.9x1012
36
(2) Cd* + H
k 2 = 1.7x1013
36
ka = 3.3x1012
36
(4) CdA - Cd*+CH3
kd = 1i.0x10 4
36
(5) CdA + H - Cd* + CH4
ke =
36
d
(3)
Cd
2
CdH
+ CH 3 - CdA
1
(6) C A + (m-1)H - CH + (m-1)H
0
k. = 2.0x1012
The units of k are given in terms of moles,
The units of k are given in
36
terms of moles, cm3, and seconds.
69
3-3 Defect generation model
There are many types of point and extended defects present
in CVD diamond films including sp2 carbon, interstitials,
vacancies, and dislocations.
At present there is little
quantitative information relating defect densities to growth
conditions.
Goodwin presents a generic model of defect formation which
captures the qualitative behavior of defect generation.
The
basic assumption of his model is that defects are generated when
an adsorbate reacts with a nearby adsorbate before it is fully
incorporated into the lattice.
For example, two neighboring
adsorbed methyl groups could react to form an sp2 ethylene (C2H4)like group, which then could form an sp2 defect.
It is assumed
that the deposition conditions are such that high quality diamond
is being grown, and defect incorporation is a rare event so that
we may ignore the formation of a non-diamond (graphite) species.
The rate of defect generation Rdef is assumed to be
proportional to the concentration of adsorbate pairs on the
surface.
Assuming they are randomly distributed
Rdef = kdefICdAI2
(18)
here kdef is the temperature dependent rate constant for defect
formation.
The defect fraction in the diamond film Xdef is given
by the defect formation rate divided by the rate of diamond
incorporation, i.e.,
Rdcf
Xdef
(19)
Rdf
Rc
From equation (13),
[CdA -G
(20)
k,IH)
and from equation (12)
Rc = G x nd
(21)
Substituting equation (20) and equation (21) into equation (19),
we have
X def
kdefLCd A 2
Gxnd
kdefnd
2
k
G
H2
(22)
At constant substrate temperature,
Xdef
C
2
IH1
(23)
Equation (23) implies the importance of atomic hydrogen
concentration over the substrate to reduce the defect fraction.
This simple expression is adopted and used to calculate the
defect density in the simulation.
3-4 Simulation
of the hot
filament diamond CVD process
Quantitative gas phase species concentrations in hot
filament diamond CVD reactors have been measured by 3 groups,
Celii [6], Harris [7],
experiments.
Hsu [9, 10].
Table 5 summaries their
Unfortunately, none of them provided the complete
information of the geometry of their setups, such as the diameter
of the reactor, distance from the gas inlet to the filament, and
the size of the outlet.
Since they gave only the information of
the filament-substrate distance, we will apply a two dimensional
axisymmetric reactor model between the region from the filament
and the substrate.
Among the diagnostic studies, only Hsu [11]
directly measured four important species, atomic hydrogen (H),
acetylene (C2 H2 ), methyl radical (CH3 ), and methane (CH4 ) using
molecular beam mass spectrometry.
We, therefore, have applied
our model to simulate such experimental conditions.
Hsu's
experimental conditions are summarized in table 6 and his
experimental setup is shown in figure 9.
However, Hsu did not
measure the gas composition in the filament, which would have
been very convenient as an inlet boundary condition in the model.
Harris [7] measured the gas composition in the filament region in
his experiment that had similar experimental conditions as Hsu's.
According to Harris, the ratio of C2H 2 and CH4 at the filament is
1:1.
Thus, we assume this mole ratio at the inlet in Hsu's
experiment for the calculation (table 7).
Table 5.
Experimental
measurements
information of quantitative
for gas phase species
available
Celli et al.
Harris et al.
information
[6]
[7]
reactor
geometry
reactor6
inch diameter
no information
no information
gas flow
rate
sccm
gas
100 flow
.100 sccm
*100 sccm
filament
-tungsten
*2673 K
-tungsten
*2600 K
-temperature drop
of 600 k at the
filament position
-tungsten
-2600 K
substrate
silicon wafer
1173-1273 K
silicon wafer
<1000 K
1073 K
filamentsubstrate
distance
15-25 mm
10-20 mm
13 mm
pressure
25 torr
20 torr
20.75 torr
initial gas
composition
0.5 % CH4 in H2
0.29 % CH 4 , 1000
ppm Ne in H2
0.4-7.2 % CH 4,
measured
gas species
I CH3, C
C2 H4 ,
CH4,
C2H2
Hsu [9]
7.0 % Ar in H 2
H, CH3
C 4 , C2H2
C2H 6 (estimated
from the spectrum)
measurement
method
infrared laser
absorption
I spectroscopy
on-line mass
spectrometry
molecular beam
mass
spectrometry
measuring
point
i somewhere between
the filament and
the substrate
from the filament
zone to the
vicinity of the
vicinity of the
substrate
In a typical hot filament CVD reactor, several publications
[9][51][52] reported that there is a significant temperature drop
(600 K - 900 K) between filament temperature and the vicinity of
the filament.
Since Hsu didn't give any temperature information
except the filament temperature and the substrate temperature, we
estimate a temperature drop of 600 K, which is consistent with
Harris's experiment.
An inlet velocity of 1 cm/s, which is
suggested as a typical value in CVD reactor [28], is also
assumed.
Full set of equation (2)-(6) in chapter 2 are solved with
the computational conditions in table 7 for Hsu's reactor.
calculation was done with a SunSpark 10 workstation and the
convergence was obtained after about 48 hours of CPU time.
Figure
9.
Experimental
0.25mm
m
setup of Hsu
tungsten filament
13 mm
0.3 mm
[9]
orifice
SSubstrate
molecular beam
mass spectrometry
The
Table
6.
Experimental
conditions
of Hsu
Filament temperature
2600 K
Substrate temperature
1073 K
Filament-substrate distance
Feed gas mixture
Pressure
Table
[9]
13 mm
0.4-7.2 % CH 4 and 7.0 % Ar in H2
20.25 torr (2700 Pa)
7.
Computational
Inlet temperature
conditions
for Hsu's reactor
2000 K*
Substrate temperature
1073 K
Filament-substrate distance
13 mm
Feed gas mixture
0.4 % CH4 and 7.0 % Ar in H2
(CH4 : C2H2 = 1:1 at the inlet)*
Total pressure
20.25 torr (2700 Pa)
Flow velocity
1.0 cm/s
* from Harris's experiment [7]
3-5.
Results
and Discussion
Figure 10 shows the computed velocity field on a 2dimensional axisymmetric domain representing half of the reactor.
It is clear from this figure the presence of an stagnation point
on the center of the reactor, with a predominant flow direction
towards the exit of the reactor, representing the convective
contribution in the system.
A comparison between the computed results and the
experimental data reported by Hsu are given in table 8.
The
value for H is one order of magnitude lower than the measured
value, and that for CH 4 is one order higher than the measured
value.
This result suggests that considering only homogeneous
reactions cannot describe the gas phase chemistry within the
region between the filament and the substrate.
We assume that this discrepancy is due to the heterogeneous
effects occurring on the surface of the filament, which are
ignored in this calculation.
The heterogeneous effects may
relate to the dissociation reactions of H2 , and the filament
surface can act as a catalytic effect to accelerate the
dissociation reactions.
In order to model the heterogeneous
effects we assume that within the filament zone (figure 11),
which has the same thickness as that of the filament,
the
reaction constant for the dissociation of hydrogen (No. 1 in
table 3) becomes a bigger value by an acceleration factor (Fa),
7
-'•t
L-
v
0. 05 m/s
Figure
10. Computed
filament
-
-- - -I
gas flow in the region between the
and the substrate of Hsu's reactor
Table 8.
Comparison between measured
gas species mole fractions
Species
Measured mole
fraction
I
and calculated
Calculated mole
fraction
__
H
2.0x10
3
2.29x10-4
CH3
3.5x10 -5
9.71x10-6
CH4
3.0x 10 - 4
1.71x10-3
C2 2
1.5x10-3
1.93x10-3
due to the heterogeneous effects.
Assuming this acceleration
factor, additional calculations were performed for the condition
in Hsu's experiment.
By trial and error, we found that the
acceleration factor of 80 gave a better agreement with Hsu's
experimental results.
The results are compared in table 8.
Every species concentration now has the same order as the
measured values and shows difference only by a factor of 2-4.
filament
zone (heterogeneous effects)fet)- f
zoe(eeoeeu
u-fiamen
1
Substrate
--
Figure
Table 9.
11.
A schematic diagram of the filament zone
Comparison between measured and calculated gas
species mole fractions assuming the
heterogeneous effects of the filament surface
Species
H
Measured mole
fraction
Calculated mole
fraction
2.0x 10 - 3
1.34x10-3
CH3
3.5x 10
CH 4
3.0x 10 - 4
3.32x10-4
C22H2
1.5x 10
-3
1.99x10-3
-5
1.05x10-5
Goodwin and Gavillet [25] took a different approach in order
to get a good agreement with their two dimensional model.
They
first allowed the inlet gas mixture to stay at a constant
temperature of 2000 K for some time, which they call "the
incubation time".
The resultant mixture was then used as the
starting mixture for their calculation.
They adjusted the
incubation time by trial and error in order to get a good
agreement.
This was purely thermodynamic calculation.
Our approach uses an alternative way to define the inlet gas
composition.
Although there is no physical evidences for our
approach as well as Goodwin's, it should be stressed that an
assumption must be taken to track the problem due to the lack of
experimental information.
Furthermore, only with appropriate
experimental measurements, it will be possible to better
understand the controlling kinetic mechanism that determine the
gas composition in the vicinity of the filament, i.e.,
heterogeneous reactions on the surface of the filaments vs. gas
phase kinetics, or a mixture of both.
The computed temperature distribution in the region between
the filaments and the substrate is shown in figure 12.
Within
this region, the thermal peclet number, which is the ratio of the
convective heat transfer and the conductive heat transfer, is
0.001.
Thus the dominant mechanism for heat transfer is
conduction.
Gas species mole fractions above values of 10-8 in the
region between the filament and the substrate are shown in figure
13.
Only three hydrocarbon species (C2H2 , CH4 , CH3) out of ten
have values of mole fraction greater than 10-5 .
Thermodynamics
states that the equilibrium mole fraction of atomic hydrogen, at
temperatures close to that on the substrate surface (1073 K) is
order of 10-6, which is far below the computed value.
This super
equilibrium is achieved by the fact that atomic hydrogen
generated at the filament is transported to the substrate by
diffusion and convection.
The reaction
CH 4 + H = CH3 + H 2
(1)
is found to be in equilibrium at every points in the domain.
The
value of the equilibrium constant Keg is 20.5 at 1000 K and 26.3
at 2000 K.
The partial equilibrium ratio
ICH 3I[H 2 1 1
[CH 4 ][H1 Keq
(2)
between the filament and the substrate is calculated and shown in
figure 14.
The ratio is almost one, thus this equilibrium is
expected to control the local concentration of these 4 species.
ICf%^^
2000
v
14
1500
1000
0.0
5.0
10.0
Distance from the filament (mm)
Figure
12. Temperature gradient between the filament
and the substrate in Hsu's reactor
_
_
__
_P_
_
~
~
~I~
1.OE+00
H2
1.OE-01
1.OE-02
C2H2
1.OE-03
H
CH4
1.OE-04
CH3
1.OE-05
C2H4
1.OE-06
1.OE-07
C2H3
1.OE-08 I
I
Distance from the filament (mm)
-
--
--
Figure
---
-
13.
- -
Gas species mole fractions between the
filament and the substrate in Hsu's
reactor
1
Figur e 14.
n
Partial equilibrium
CH4 + H = CH3 + H
2
ratio
of the reaction
Figure 15 shows the change of relative concentration of
atomic hydrogen (H), methyl radical (CH3 ), and methane (CH4),
between the filament and the substrate above the center of the
substrate.
Atomic hydrogen shows little decrease from the
filament toward the substrate while CH4 and CH3 decreases 2 % and
20 %, respectively.
The results in figure 15 can be explained by examining the
diffusion lengths for the species, which give a measure of the
relative effects of diffusion and the local gas phase kinetics.
A local chemical timescale for each species is given by the
chemical destruction time
c, which is defined as the molar
concentration of the species i, divided by its molar destruction
rate.
Ci
SRd
(Ci: molar concentration, R d : molar destruction rate)
survive
This represents the average time that the molecule can (3)
before being consumed by chemical reactions in the gas phase
chemistry.
During this time, the molecule can diffuse an average
distance
1d
length.
This is given by the equation:
normal to the substrate, which is the diffusion
Id =
ýD,
where Di is the ordinary diffusion coefficient of the ith
(4)
~_ _~ _
__ __
100
I_
_
_ __~
_ _ ~
_ _1_1 _
_ I
_
1
H
CH4
95-
90-
CH3
85-
Distance from the filament (mm)
-
Figure 15.
--
I-----
-- ~111--
---
I--
Relative concentrations of H, CH3 , and CH4
between the filament and the substrate
species, and represents the length that the molecule can reach
before being consumed by chemical reactions.
When 1d is shorter
than the filament-substrate distance, the molecule is consumed
before it reaches to the substrate.
In this case diffusion is
considered "slow", and the concentration profile is determined by
chemical kinetics in the gas phase.
When Id is longer than the
separation distance, the molecule can freely diffuse to the
substrate, and therefore the concentration is not affected very
much by chemical kinetics.
Figure 16 shows diffusion length profiles for atomic
hydrogen (H), methyl radical (CH3), and methane (CH4 ) in the
region between the filament and the substrate.
The length of
atomic hydrogen is about 1 cm, which is comparable to the
filament-substrate distance (13 mm), thus atomic hydrogen
generated at the filament can diffuse to the substrate.
On the
other hand, the length of methyl radical and methane is about
0.1-0.01 cm.
CH3 and CH4 species are affected by the local gas
phase kinetics, mainly by the equilibrium reaction (1).
CH4 + H = CH3 + H,
(1)
_~_
~__~_~~
__
1.OE+00
1.OE-01
1.OE-02
0
10
5
Distance from the filament (mm)
__
Figure
~_
16.
Characteristic
CH 3 , and CH 4
___
diffusion
lengths
for H,
The relative concentrations of atomic hydrogen and methyl
radicals above the substrate are shown in figure 17.
This figure
clearly shows the effect of the convective flow for CH3 . The
concentration of H is not affected by the convective flow since
the diffusion of atomic hydrogen is very fast.
The growth rate and the defect fraction are calculated from
the concentration of atomic hydrogen and the methyl radical,
i.e.,
Growth rate
Defect density
G =3.6 x 107 x f* x--nd [kd+ (ke+ 3ki)[H] (kikaCH
/ h)
] (m/h)
Xdef
G
[H]
2
(17)
(17)
(23)
The growth rate is proportional to the concentration of CH3 and
the defect fraction is proportional to the growth rate.
As shown
in figure 18, their profiles calculated with the above equations
che same trends as the methyl radical concentration profile.
105
100
95
0
1
2
3
4
5
Distance from the center of the substrate (cm)
Figure
17.
Relative concentration
above the substrate
of H and CH3
__
_
~
~~_
_I
_
105
*
0m
4J
0(9
Cu
10
95
1
0
2
3
4
Distance from the center of the substrate (cm)
_
Figure
~
18.
Relative deposition
above the substrate
rate
and defect density
3-6 Predictions on the general
diamond CVD process
trends
of the hot filament
From the results of the calculation for Hsu's reactor and
the surface reaction mechanism assumed in the model, we can make
several predictions on the behavior of the hot filament diamond
CVD reactor, which are presented in this section.
(1) Effect of filament temperature
With increasing filament temperature, we postulate that an
hydrogen dissociation reaction takes place, i.e.,
H2
=12H
and this leads to an increase of the concentration of atomic
hydrogen above the substrate.
From the defect generation model,
the defect fraction is calculated from equation (23), which
implies the fraction is inversely proportional to the square of
the concentration of atomic hydrogen;
X def
G
(HI2
(23)
Therefore, the increase of filament temperature should improve
the diamond film quality.
(2) Effect of the total pressure
Since most of the gas phase reaction constants are
independent of pressure, it is considered that the mole fraction
of the gas phase species are not affected by the change of
pressure very much.
Increasing the total pressure increases the
partial pressure of the gas species, which results in increases
of the concentration of the gas species, one of which is methyl
radical (CH3 ) that is the precursor for diamond growth.
The
increase of the total pressure therefore is expected to increase
the growth rate.
Since the diffusion coefficient is inversely proportional to
the total pressure, the increase of total pressure decreases the
diffusion coefficient.
Therefore, the relative importance of
convective transport becomes more significant as the pressure
increases, which means that the film uniformity may deteriorate.
(3) Effect of the methane concentration
Since the concentrations of CH4 and CH3 are tightly coupled
through partial equilibrium in the equation:
CH
4
+ H = CH 3 + H 2
, increasing the concentration of CH4 requires also an increase
of that of CH3 ; at the same tune, decreasing CH4 also requires a
decrease of CH3 . Thus, the growth rate increases with increasing
methane concentration.
From the defect generation model,
XdefO
G
(23)
, increasing growth rate leads to an increase of the defect
fraction.
Table 10 summarizes the analysis presented above, on the
effect of some critical parameters in the hot filament diamond
CVD reactor.
Table 10.
Predictions on the general trends of the hot
filament diamond CVD process
Increase of
Methane concentration
CHAPTER 4. CONCLUSIONS
A two dimensional axisymmetric model including detailed
kinetic factors in the gas phase and substrate surface was
developed to simulate a diamond CVD reactor.
A comparison
between the model predictions and experimental measurements
reported in the literature indicated that the region between the
filament and the substrate cannot be described only by
homogeneous chemical kinetics and heterogeneous effects on the
filament surface should be included in the model.
An assumption was made to incorporate heterogeneous effects
as a catalytic factor for the hydrogen dissociation reaction.
Increase of the homogeneous reaction constant of 80 times was
required to obtain a good agreement with the measurements.
In the region between the filament and the substrate, the H
abstraction reaction from CH 4,
CH 4 + H = CH 3 + H
2
was found to be in partial equilibrium.
This reaction,
therefore, is expected to control the local concentration of the
four species involved.
Diffusion lengths for H, CH3, and CH. were obtained from the
concentration fields calculated with the model.
The diffusion
length for H was comparable to the filament-substrate distance.
Therefore, atomic hydrogen created at the filament region can
diffuse to the substrate with little effect from the gas phase
chemical kinetics.
As a result, the concentration profile of
atomic hydrogen above the substrate showed a very flat profile.
The diffusion lengths for CH 3 and CH4 were much smaller compared
to the filament-substrate distance.
Thus, their concentration
profiles were affected by the kinetic factors in the gas phase.
As a resulting effect of the concentration profiles of H and
CH3 , the deposition rate and the defect fraction (film quality)
showed the same tendency to increase towards the edge of the
substrate.
General trends describing the behavior of the reactor in a
qualitative way were predicted from the results of the
calculation.
Favorable conditions for high deposition rates and
low defect fractions are expected to be high filament
temperature, low pressure, and high methane concentration.
Modeling of diamond CVD reactors is indeed very complex task
due to interrelating phenomena taking place, such as fluid flow
and heat transfer together with many different chemical reactions
involving a number of species.
Under such situations, detailed
experimental information is absolutely necessary.
However,
experimental information available in the literature on the study
of this system is rather limited and very narrowly focused to
study specific mechanisms, for example, the species
concentrations in the vicinity of the substrate.
For the purpose of future studies, carefully designed
experiments involving measurements of temperature and
concentration profiles of the species in the specific regions
inside the reactor (the gas inlet, substrate, and filament) and
also in the open regions (between the gas inlet and the filament,
the filament and the substrate, and the substrate and the outlet)
are required to examine the validity of the assumption for the
heterogeneous effects made in this study.
It is the results
obtained from such measurements that enable us to know the
chemical kinetic data which can not be predicted from the theory.
Once we know the kinetic mechanism for the heterogeneous
effects, it can be extended to the modeling of a three
dimensional production reactor, which is the final objective of
this research.
Bibliography
[1]
S. Matsumoto, Y. Sato, M. Kamo, and N. Setaka, Jpn. J. Appl.
Phys. 21, L183 (1982)
[2]
T. Yashiki, T. Nakamura, N. Fujimori, and T. Nakai, 1991 In
Proc. Int. Conf. on Metallurgical Coatings and Thin Films.
San Diego, California
[3]
S. Matsumoto, Y. Sato, M. Tsutsumi, and N. Setaka: Jpn. J.
Mater. Sci. 17, 3106 (1982)
[4]
Y. Hirose, and Y. Terasawa, Jpn. J. Appl. Phys. 25, L519
(1986)
[5]
L.R. Martin, J. Mater. Sci. Lett. 12, 246 (1993)
[6]
F.G. Celii, P.E. Pehrsson, H.T. Wang, and J.E. Butler, Appl.
Phys. Lett. 52, 2043 (1988)
(7]
S.J. Harris, and A.M. Weiner, Appl. Phys. Lett. 53, 1605
(1988)
[8] S.J. Harris, and A.M. Weiner, J. Appl. Phys. 67, 6520 (1990)
[9] W.L. Hsu, Appl. Phys. Lett. 59, 1427 (1991)
[10] W.L. Hsu and D. M. Tung, Rev. Sci. Instrum. 63, 4138 (1992)
[11] L.R. Martin: J. Appl. Phys. 70, 5667 (1991)
[12] L.R. Martin, and M.W. Hill, J. Mater. Sci. Lett. 9, 621
(1990)
[13] S.J. Harris, and L.R. Martin, J. Mater. Res. 5, 2313 (1990)
[14] L. Schdfer, M. Sattler, and C. P. Klages: In Applications of
Diamond and Related Materials, ed. By Y. Tzeng, M.
Yoshikawa., M. Murakawa, A. Feldman (Elsevier, Amsterdam
1991) p. 453
[15] C.J. Chu, M.P. D'Evelyn, R.H. Hauge, and J.L. Margrave, J.
Mater. Res. 5, 2405 (1990)
[16] C.J. Chu, M.P. D'Evelyn, R.H. Hauge, and J.L. Margrave, J.
Appl. Phys. 70, 1695 (1991)
[17] M.P. D'Evelyn, C.J. Chu, R.H. Hauge, and J.L. Margrave, J.
Appl. Phys. 71, 1528 (1992)
[18] F.G. Celii, and J.E. Butler, Appl. Phys. Lett. 54, 1031
(1989)
[19] L. Schafer, C.P. Klages, U. Meier, and W.F. Banholzer, Appl.
Phys. Lett. 58, 571 (1992)
[20] L. Schafer, U. Bringmann, C.P. Klages, U. Meier, and K.
KohseH6inghaus: In Diamond and Diamond Like Films and
Coatings, ed. by R.E. Clausing, L.L. Horton, J.C. Angus, P.
Koidl (Plenum, New York 1991) p. 643
[21] M. Frenklach, and H. Wang, Phys. Rev. B43, 1520 (1990)
[22] T. DebRoy, K. Tankala, W.A. Yarbrough, and R. Messier, J.
Appl. Phys. 68, 2424 (1990)
[23] S.J. Harris and A.M. Weiner, Appl. Phys. Lett. 55, 2179
(1989)
[24] M. Frenklach, J. Appl. Phys. 65, 5142 (1989)
(25] D.G. Goodwin, and G.G. Gavillet, J. Appl. Phys. 68, 6393
(1990)
[26] E. Kondoh, T. Ohta, T. Mitomo, and K. Ohtsuka, J. Appl.
Phys. 72, 705 (1992)
[27] S. J. Harris, and D. N. Belto, Thin Solid Films 212, 193
(1992)
[28] S.J. Harris, and A. M. Weiner, J. Appl. Phys. 70, 1385
(1991)
[29] C.E. Johnson, W.A. Weimer, and F.M. Cerio, J. Mater. Res. 7,
1427 (1992)
[30] M.E. Coltrin, and D.S. Dandy, J. Appl. Phys. 74, 5803 (1993)
[31] M.E. Coltrin, R.J. Kee, and F. M. Rupley, Sandia National
Laboratories Report, SAND 90-8003 (1990)
[32] M.E. Coltrin, R.J. Kee, and F. M. Rupley, Int. J. Chem.
Kinet. 23, 1111 (1991)
[33] B. J. Garrison, E. J. Dawnkaski, D. Srivastava, and D.W.
Brenner, Science 255, 835 (1992)
[34] J.E. Butler, and R.L. Woodin, Philos. Trans. R. Soc. London
342, 209 (1993).
[35] M.H. Loh, and M.A. Cappelli, Diamond Related Mater. 2, 454
(1993).
[36] D.G. Goodwin, J. Appl. Phys. 74, 6888 (1993)
[37] C.R. Kleijn, T.H. Van der Meer, and C.J. Hoogendoorn, J.
Electrochem. Soc., 136 (11):3423-3432 (1989)
[38] C.R. Wilke, Chemical Engineering Progress, 46 (2):95-104
(1950)
[39] R.J. Kee, F.M. Rupley, and J.A. Miller, Technical Report
SAND 89-8009B.UC-706, SANDIA national Laboratories
[40] R.J. Kee, G. Dixon-Lewis, J. Warnatz, M.E Coltrin, and J.A.
Miller, Technical Report SAND 86-8246.UC-401, SANDIA
National Laboratories (1986)
[41] R.C. Reid, J.M. Prausnitz, and B.E. Poling, The properties
of gases and liquids. McGraw-Hill, New York, second edition,
1987
[42] R.A. Svehla, Technical Report R-132, NASA (1962)
[43] C.R. Wilke and C.Y. Lee, Industrial and Engineering
Chemistry, 47 (6):1253-1257, (1955)
[44] R. Clark Jones, Physical Reviews, 59:1019-1033 (1941)
[45] F. Lindemann, Trans. Faraday. Soc.,
17:598 (1922)
[46] R.G. Gilbert, K. Luther, and J. Troe, Ber. Bunsenges. Phys.
Chem., 87:169-177 (1983)
[47] J. Warnats, Combustion Chemistry, edited by W. C. Gardiner
(Springer, New York, 1965), Vol. 1, Chapter 5
[48] S.J. Harris, Appl. Phys. Lett. 56, 2298 (1990)
100
Appendix A.
Qualitative behavior of the hot
filament diamond CVD process
In order to investigate the general behavior of the hot
filament diamond CVD process qualitatively, several calculations
have been done as preliminary calculations.
The 2D axisymmetric
geometry used in these calculations is the same as the one
described in chapter 3. The results presented in this appendix
represent the overall behavior of a "virtual" hot filament
reactor.
The reason to refer to the system as a "virtual"
reactor is because the calculations were not allowed to reach
full convergence, due to the excessive requirements in CPU time
per run (about 48 hours).
The calculations were stopped when the concentration of the
important species, atomic hydrogen (H), methyl radical (CH3 ),
methane (CH4 ), and acetylene (C2H2 ), reached constant values at
critical points of interest (i.e., in the vicinity of the
substrate) even if all the other variables did not reach complete
convergence.
In the results presented here, we ignore the heterogeneous
reactions at the filament region.
Also, we specify a set of base
conditions, in which the initial methane (CH4 ) concentration is
1.0 %, the temperature in the filament zone is 2200 K, and the
102
total pressure is 20 torr.
cm/s at the inlet.
The initial flow velocity is set at 1
The computational conditions used are
summarized in table 11.
The effects of filament temperature,
total pressure, and methane concentration on gas species
concentration, growth rate, and defect fraction are then
evaluated in this preliminary study.
Table
11.
Calculational
analysis
conditions
filament zone temperature
for qualitative
2200, 2400, 2600 K
Substrate temperature
1200 K
Filament-substrate distance
10 mm
Feed gas mixture
0.4, 1.0, 2.0 % CH4 in H 2
Pressure
20, 50, 80 torr
1.0 cm/s
Inlet flow velocity
Results
(1) Effect of filament temperature
Figure 19 shows calculated atomic hydrogen (H), and methyl
radical (CH3 ) mole fraction at the center of the substrate with
different inlet temperature conditions.
With increasing
temperature H mole fraction increases and CH 3 mole fraction
103
decreases.
Linear growth rates with different temperature conditions
are shown in figure 20.
The values do not change very much with
increasing the filament temperature.
Since the defect fraction is calculated using the following
equation,
X def
G
[HIH
it is apparent that the defect density decreases drastically with
the increasing temperature as shown in figure 21.
(2) Effect of total pressure
Figure 22 shows the variation of H and CH 3 mole fractions
for different total pressure conditions.
Both mole fractions
decrease slightly as the pressure increases.
Figure 23 and 24
show the computed linear growth rate and defect fraction for
different pressure conditions.
In figure 23, the growth rate
increases with increasing pressure.
Although the defect fraction
increases with decreasing pressure, the differences between
different pressure conditions are much smaller than those
produced when different filament zone temperatures are employed.
(3) Effect of initial CH 4 concentration
The effect of the initial CH
104
concentration on the growth
rate is shown in figures 20 and 23 as a function of the filament
zone temperature and reactor pressure respectively.
Similarly,
the effect of the initial CH 4 concentration on the defect
fraction is presented in figures 21 and 24.
The growth rate and
the defect fraction increase with increasing the initial CH
4
concentration.
(4) Effect on the uniformity of the diamond film.
Figure 25 shows the effect of the total pressure on the
uniformity of the diamond film.
We can see from the figure that
decreasing the reactor pressure improves the uniformity of the
film.
105
1.OE-01
0
0
1.OE-02 -
0
O
H (2600K)
O
H (2400K)
-o
O
H (2200K)
1.OE-03 -
1.OE-04 -
D
CH3 (2200K)
CH3 (2400K)
1.OE-05CH3 (2600K)
1.OE-061
2
CH4 Concentration (%)
Figure
19.
Effect of filament
CH3 mole fractions
106
zone temperature on H and
_
__I
__^~ 1_ ~I_~__·
0.5
__
-
_
__~_
_____
_~
__
~
0.4-
2600K
0.3-
3
2400K
)
2200K
0.2-
0.1-
0t
A.
-
--
Figure
__
L
0
0
20.
--
-
Effect
growth
II
~
___
__
CH4 Concentration (%)
--- - - - - - -- -- - - -
of filament
rate
107
zone temperature
I
I--
on the
_
___
__
1000
-
~
~
~~
0
2200K
O]
2400K
100 -
1010-
ý
1-
~
Figure 21.
_II_
Effect
defect
0
2600K
CH4 Concentration (%)
I
___I
__
~I
of filament
fraction
108
zone temperature
on the
1.0E-017
1.0E-02 -
O
H (20torr)
O
H (50torr)
O
H (80torr)
*
CH3 (20torr)
ES
CH3 (50torr)
1.0E-03 -
1.0E-04 -
1.01-05 -
1
3 (80torr)
1.0E-06
-
CH4 Concentration (%)
Figure 22.
Effect of total pressure
fractions
109
on H and CH3 mole
Figure
23.
Effect of total
pressure
110
on the growth
rate
250
-
200-
O
20torr
o
o50torr
O
80torr
150 -
100 -
50-
i
3
1
2
CH4 Concentration (%)
Figure 24.
Effect of total pressure
fraction
111
on the defect
140
130
120
110
100
90
0
1
2
3
4
5
Position from the center of the substrate
Figure 25.
Effect of total pressure on the uniformity of
the diamond film
112
APPENDIX B.
Table 12.
al
CH2
Thermodynamic
calculation
a2
a3
data of the gas species for the
a4
a5
a6
a7
Temp.
range
2.500000
0.000000
3.298124
8.249441e-4
-8.143015e-7
-9.475434e-11
4.134872e-13
-1.012521e+3
-3.294094
H2
10005000
2.991423
7.000644e-4
-5.633828e-8
-9.231578e-12
1.582752e-15
-8.350340e+2
-1.355110
3001000
3.636407
1.933056e-3
-1.687016e-7
-1.009899e-10
1.808255e-14
4.534134e+4
2.156560
10004000
3.762237
1.159819e-3
2.489585e-7
8.800836e-10
1 .5381 9 e-3
-7.332435e-13
4.536790e+4
1.712577
S1000
1025000
2.844051
6.137974e-3
-2.230345e-6
3.785161e-10 I-2.452159e-14 1-1.643781e+4
5.452697
2.430442
1.112410e-2
-1.680220e-5
1.621829e-8
-5.864952e-12
1.642378e+4
6.789794
3001000
1.683478
1.023724e-2
-3.875128e-6
6.785585e-10
-4.503423e-14
-1.008079e+4
9.623395
10005000
0.778742
1.747668e-2
-2.783409e-5
3.049708e-8
-1.223931e-11
-9.825229e+3
1.372220e+1
0.000000
0.000000
0.000000
2.547162e+4 1-4.601176e-1
3005000
1000-
5000
CH3
300-
1000
4.436770
5.376039e-3
-1.912816e-6
3.286379e-10
-2.156709e-14
2.566766e+4 1-2.800338
1000-
5000
C2H2
C213
2.013562
1.519045e-2
-1.616319e-5
9.078992e-9
-1.912746e-12
2.612444e+4
8.805378
3001000
5.933468
4.017745e-3
-3.966739e-7
-1.441267e-10
2.378643e-14
3.185434e+4
-8.530313
10005000
2.459276
7.371476e-3
2.109872e-6
-1.321642e-9
-1.184784e-12
3.335225e+4
3.528418
1.148519e-2
-4.418385e-6
-0.861488
7.190480
300-
1000
2.796162e-2 I -3.388677e-5
6.484077e-3
1.155620e+1
-6.428064e-7
7.844600e-10
-5.266848e-14
4.428288e+3 I 2.230389
10005000
2.785152e-8
-9.737879e-12
5.573046e+3
3001000
-2.347879e-10
2.421148
{
3.880877e-14 [ 1.067455e+4 1-1.478089e+1
1000S5000
2.690701
8.719133e-3 I
4.825938
1.384043e-2
4.419838e-6
-4.557258e-6
I 9.338703e-10 I-3.927773e-12
1.287040e+4
-3.598161e-14
-1.271779e+4
6.724967e-10
1.213820e+1
-5.239506
300S1000
1000-
4000
CH
1.462539
1.549467e-2
5.780507e-6
1.2578319e-8
113
4.586267e-12
-1.123918e+4
1.443230e+1
3001000
Table 12.
species
Transport properties of the gas
species for the calculation
Lennard-Jones parameters (E/k)
Collision diameter (A)
H
H2
145.0
38.0
2.050
2.920
CH 2
3.800
CH3
144.0
144.0
CH4
141.4
3.746
C2H2
209.0
4.100
C 2H 3
209.0
4.100
C2H4
280.8
3.971
C2H5
252.3
252.3
4.302
C2H6
2
6
3.800
4.302
114