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Optical spectroscopy and langmuir probe diagnostics of microwave plasma in synthesis of graphene-based nanomaterials

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PURDUE UNIVERSITY
GRADUATE SCHOOL
Thesis/Dissertation Acceptance
7KLVLVWRFHUWLI\WKDWWKHWKHVLVGLVVHUWDWLRQSUHSDUHG
%\ Alfredo David Tuesta
(QWLWOHG
OPTICAL SPECTROSCOPY AND LANGMUIR PROBE DIAGNOSTICS OF MICROWAVE
PLASMA IN SYNTHESIS OF GRAPHENE-BASED NANOSTRUCTURES
)RUWKHGHJUHHRI
Doctor of Philosophy
,VDSSURYHGE\WKHILQDOH[DPLQLQJFRPPLWWHH
Sameer Naik
Fisher
Timothy
Robert Lucht
Jay Gore
Maxim Lyutikov
To the best of my knowledge and as understood by the student in the Thesis/Dissertation Agreement,
Publication Delay, and Certification/Disclaimer (Graduate School Form 32), this thesis/dissertation
adheres to the provisions of Purdue University’s “Policy on Integrity in Research” and the use of
copyrighted material.
Timothy Fisher
$SSURYHGE\0DMRU3URIHVVRUVBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB
Robert Lucht
BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB
$SSURYHGE\ Ganesh Subbarayan
+HDGRIWKHDepartment *UDGXDWH3URJUDP
10/10/2014
'DWH
OPTICAL SPECTROSCOPY AND LANGMUIR PROBE DIAGNOSTICS
OF MICROWAVE PLASMA IN SYNTHESIS
OF GRAPHENE-BASED NANOMATERIALS
A Dissertation
Submitted to the Faculty
of
Purdue University
by
Alfredo D. Tuesta
In Partial Fulfillment of the
Requirements for the Degree
of
Doctor of Philosophy
December 2014
Purdue University
West Lafayette, Indiana
UMI Number: 3702113
All rights reserved
INFORMATION TO ALL USERS
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and there are missing pages, these will be noted. Also, if material had to be removed,
a note will indicate the deletion.
UMI 3702113
Published by ProQuest LLC (2015). Copyright in the Dissertation held by the Aut
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unauthorized copying under Title 17, United States Code
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ii
To my loving and caring family here and abroad.
iii
ACKNOWLEDGMENTS
My journey at Purdue University has been rather fulfilling but not without its
academic and personal challenges. I have much to thank my advisers, Professor Timothy Fisher and Professor Lucht, for their guidance and often critically constructive
review of my ideas and methods regarding my scientific approach. I do not leave
this campus a competent researcher without their training. My coworkers at the
Birck Nanotechnology Center have also contributed to my success. The staff and my
research colleagues have provided the skills and support I’ve needed to push forward.
Professor Sergey Macheret from the School of Aeronautics and Astronautics deserves a special acknowledgment for his time in helping me derive some of the conceptual theories in this work regarding the chemical kinetics of nonthermal plasmas.
His expertise has facilitated my understanding of some of the results found in this
work. I am thankful for his time and kindness in this matter.
My friends around campus and especially those from St. Thomas Aquinas Catholic
Center have provided me with the moral support a man needs to find his way in life
through thick and thin. They have become my home away from home and I am
grateful to them for their presence in my life. Likewise, my family and friends in New
Jersey and abroad have never doubted my skills and competence and have always
given me love and encouragement. When I lacked resilience, they provided comfort.
When I lacked direction, they provided a light. I thank them for their patience,
kindness and understanding but most of all their love.
Finally and most deservedly, I thank my God without whose grace none of this
would have been possible. I thank Him for the blessings and challenges He has
provided me which have made me who I am today. May my life and work be spent
for the benefit of His greater honor and glory.
iv
TABLE OF CONTENTS
Page
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
vi
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
vii
SYMBOLS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xi
ABBREVIATIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xv
ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xvi
1. INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
2. TEMPERATURE OF HYDROGEN VIA CARS . . . . . . . . . . . . . .
2.1 Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2.1 Carbon Nanostructures and Nanodevices . . . . . . . . . . .
2.2.2 Methods for Synthesis . . . . . . . . . . . . . . . . . . . . .
2.2.3 Coherent Anti-Stokes Raman Scattering Spectroscopy . . . .
2.3 Experimental System . . . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 Microwave Plasma Chemical Vapor Deposition . . . . . . . .
2.3.2 H2 CARS Spectroscopy . . . . . . . . . . . . . . . . . . . . .
2.3.3 Conditions for Graphitic Growth . . . . . . . . . . . . . . .
2.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4.1 Temperature through the Plasma Sheath and Presheath . .
2.4.2 Temperature with the Introduction of CH4 and N2 . . . . .
2.4.3 Analytical Explanation of the Increase in Rotational Temperature with Pressure . . . . . . . . . . . . . . . . . . . . . . .
2.4.4 Analytical Explanation for the Increase in Rotational Temperature with the Introduction of N2 and CH4 . . . . . . . . . .
2.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
3
3
3
4
5
9
9
12
15
18
18
20
3. MOLE FRACTION OF HYDROGEN VIA CARS . . . .
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . .
3.2 Experimental Setup . . . . . . . . . . . . . . . . . .
3.2.1 Optical System . . . . . . . . . . . . . . . .
3.2.2 Determination of H2 Mole Fraction . . . . .
3.2.3 CARS Measurements of the ν 00 = 1 → ν 0 = 2
3.2.4 Radial Temperature Profile in the Plasma .
3.2.5 Intensity Effects . . . . . . . . . . . . . . . .
26
26
27
27
29
35
37
38
. . . . . .
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Hot Band
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20
23
25
v
3.3
3.4
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Discussion and Conclusion . . . . . . . . . . . . . . . . . . . . . . .
4. ELECTRON DENSITY FUNCTION OF MICROWAVE PLASMA
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2 Probe Theory . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3 Druyvesteyn Method . . . . . . . . . . . . . . . . . . . . . . .
4.4 Electron Energy Distributions . . . . . . . . . . . . . . . . . .
4.5 Sources of Error . . . . . . . . . . . . . . . . . . . . . . . . . .
4.6 Numerical Approximation of the Second Derivative . . . . . .
4.7 Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . .
4.8 Results and Discussion . . . . . . . . . . . . . . . . . . . . . .
4.9 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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40
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45
45
45
49
50
52
53
55
58
65
5. SUMMARY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1 Additional Work and Recommendations . . . . . . . . . . . . . . .
67
68
LIST OF REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . .
69
APPENDICES
A. LANGMUIR PROBE MEASUREMENTS . . . . . . . . . . . . . . . . .
77
B. MATLAB CODE FOR SMOOTHING CURRENT-VOLTAGE PROFILE
AND SECOND DERIVATIVE . . . . . . . . . . . . . . . . . . . . . . . .
83
VITA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
93
vi
LIST OF TABLES
Table
4.1
Page
The effect of varying the scan resolution of the Langmuir probe on plasma
parameters (TeV , ne , and Vpl ) at 300 W, 10 Torr. The corresponding
smoothing parameters β and λ are also shown. . . . . . . . . . . . . . .
60
Plasma parameters derived from experimental EEDFs at 300, 400 and 500
W at 10 Torr with a scan resolution of 0.1 V. . . . . . . . . . . . . . .
63
A.1 The effect of varying the scan resolution of the Langmuir probe on plasma
parameters (TeV , ne , Vpl , and Vpk ) at 300 W, 10 Torr. The corresponding
smoothing parameters β and λ are also shown. . . . . . . . . . . . . . .
78
A.2 The effect of varying the scan resolution of the Langmuir probe on plasma
parameters (TeV , ne , Vpl , and Vpk ) at 400 W, 10 Torr. The corresponding
smoothing parameters β and λ are also shown. . . . . . . . . . . . . . .
78
A.3 The effect of varying the scan resolution of the Langmuir probe on plasma
parameters (TeV , ne , Vpl , and Vpk ) at 500 W, 10 Torr. The corresponding
smoothing parameters β and λ are also shown. . . . . . . . . . . . . . .
78
4.2
vii
LIST OF FIGURES
Figure
2.1
Page
Illustration of plasma bulk over molybdenum puck and plasma sheath
boundary region. The plasma bulk couples with the molybdenum puck
creating a sheath region. . . . . . . . . . . . . . . . . . . . . . . . . . .
5
Schematic diagram of CARS beams and energy diagram. The Pump,
Stokes and Probe beams are focused into a probe volume and a coherent,
anti-Stokes signal is generated. . . . . . . . . . . . . . . . . . . . . . .
6
Population distribution for the rotational energy mode of H2 . Intensity
alteration in the rotational fine structure is due to nuclear spin statistics.
8
Schematic diagram of the microwave plasma chemical vapor deposition
reactor at two stage positions: (A) susceptor stage ready to accept substrate and (B) the susceptor stage raised to 53 mm for the ignition of the
plasma. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
Raman spectrum from two positions on the top surface of the copper disc
directly exposed to the plasma. . . . . . . . . . . . . . . . . . . . . . .
11
2.6
Raman spectrum from the bottom surface of the copper disc. . . . . . .
12
2.7
Schematic diagram of the CARS system. BS: beam splitter; TP: thin film
polarizer; λ/2: half-wave plate. . . . . . . . . . . . . . . . . . . . . . .
13
Theoretical fit to room temperature spectrum. CARSFT code converges
to a temperature of 272 K. . . . . . . . . . . . . . . . . . . . . . . . . .
16
Theoretical fit to a H2 spectrum from plasma. CARSFT code converges
to a temperature of 1341 K. . . . . . . . . . . . . . . . . . . . . . . . .
16
2.10 Rotational temperature of H2 at 10 Torr with varying plasma generator
powers, with and without CH4 (10 sccm). Filled triangles correspond to
a precursor mixture of H2 (50 sccm). Empty triangles correspond to a
precursor mixture of H2 (50 sccm) and CH4 (sccm). Downward pointing
triangles correspond to a generator power of 300 W. Upward pointing
triangles correspond to a generator power of 500 W. . . . . . . . . . . .
18
2.2
2.3
2.4
2.5
2.8
2.9
viii
Figure
Page
2.11 Rotational temperature of H2 at 30 Torr with varying plasma generator
powers, with and without CH4 (10 sccm). Filled triangles correspond to
a precursor mixture of H2 (50 sccm). Empty triangles correspond to a
precursor mixture of H2 (50 sccm) and CH4 (sccm). Downward pointing
triangles correspond to a generator power of 300 W. Upward pointing
triangles correspond to a generator power of 500 W. . . . . . . . . . . .
19
2.12 H2 rotational temperature distributions at 10 Torr, 400 W with and without CH4 and N2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
Experimental spectra with a reactor pressure of 10 Torr at (A) room temperature and (B) plasma temperature. The plasma generator power was
500 W and it was entirely composed of H2 . . . . . . . . . . . . . . . . .
32
Theoretical fits to experimental data by use of CARSFT code with a mole
fraction equal to 1 for data at (A) room temperature and (B) plasma
temperature. Fitting of the spectra with the CARSFT code resulted in a
temperature of 277 and 1133 K for room temperature and plasma spectra,
respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
Experimental Rex and theoretical Rth ratios at 10 Torr. Experimental
ratios, shown by the squares, were produced by heating the susceptor
stage of the MPCVD reactor. Theoretical ratios, shown by dots, were
produced by use of the CARSFT code. The dashed line connecting the
dots illustrates the results from the spline interpolation. . . . . . . . . .
34
Experimental and theoretical ratios converted to mole fractions. Error
bars were calculated by use of Equation 3.1. . . . . . . . . . . . . . . .
34
Theoretical spectra generated by the CARSFT code for a thermal system
at 1000 K. Transitions from the cold band are labeled Q01 (J). For clarity,
only the odd J transitions were labeled. . . . . . . . . . . . . . . . . .
36
Theoretical spectra generated by the CARSFT code for a thermal system
at 2000 K. Transitions from the cold band are labeled Q01 (J) and transitions from the hot band are labeled Q12 (J). For clarity, only the odd J
transitions were labeled. . . . . . . . . . . . . . . . . . . . . . . . . . .
36
Radial temperature profile from the center of the plasma. Triangles correspond to measurements with a plasma. Circles correspond to measurements without a plasma at room temperature. . . . . . . . . . . . . . .
38
Square root of the intensity of H2 Q(1) line as a function of beam intensity.
The Stokes beam energy was 4.8 mJ/pulse. . . . . . . . . . . . . . . . .
39
Square root of the intensity of H2 Q(1) line as a function of reactor pressure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
39
ix
Figure
Page
3.10 Experimental mole fraction of H2 (50 sccm) in microwave plasma at various temperatures with CH4 (10 sccm) and N2 (20 sccm) for a reactor
pressure of 10 Torr and a generator power of 300 and 500 W. Dashed
lines indicate the theoretical mole fraction for the corresponding mixture
without a plasma. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
41
3.11 Experimental mole fraction of H2 (50 sccm) in microwave plasma at various temperatures with CH4 (10 sccm) and N2 (20 sccm) for a reactor
pressure of 30 Torr and a generator power of 500 and 700 W. Dashed
lines indicate the theoretical mole fraction for the corresponding mixture
without a plasma. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
42
3.12 Results for the attempt at measuring transitions in the hot band. . . .
43
4.1
Ideal case for ion, electron and probe current produced by Equations 4.1
and 4.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
47
4.2
Ideal probe current plotted in logarithm scale showing linear portion. .
48
4.3
Area-normalized Maxwellian and Druyvesteyn electron energy distributions at TeV = 2 eV. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
51
Raw data shown in red circles and the results of the smoothed filter in blue
curve. The green curve is the second derivative of the smoothed data. The
scan resolution is 0.1 V. . . . . . . . . . . . . . . . . . . . . . . . . . .
56
Hiden ESPion Langmuir probe attached to the right side optical arm of
the MPCVD reactor. The probe is controlled via a computer and is able
to extend and retract into the reactor. . . . . . . . . . . . . . . . . . .
56
Tungsten probe and probe holder in the reactor. The probe has been extended to 21.5 mm corresponding to the center of the reactor and plasma.
57
Horizontal shift of current-voltage profile with pre-cleaning potential. The
direction and amount shifted is proportional to the parity and amplitude
of pre-cleaning potential. . . . . . . . . . . . . . . . . . . . . . . . . . .
59
Change in the shape of the current-voltage profile with scan resolution.
The generator power was 300 W and reactor pressure was 10 Torr. . . .
59
Experimental EEDF at 500 W with a scan resolution of 0.1 V and its corresponding Maxwellian and Druyvesteyn best fit. The experimental temperature obtained is 4 eV while the best fit attempts with the Maxwellian
and Druyvesteyn EEDFs were 4.8 and 3.9 eV, respectively. . . . . . . .
61
4.4
4.5
4.6
4.7
4.8
4.9
x
Figure
Page
4.10 Experimental EEDF at 500 W with a scan resolution of 0.75 V and its corresponding Maxwellian and Druyvesteyn best fit. The experimental temperature obtained is 4.3 eV while the best fit attempts with the Maxwellian
and Druyvesteyn EEDFs were 5.5 and 4.4 eV, respectively. . . . . . . .
62
A.1 Current-voltage profiles, results of smoothing procedure and second derivative at 300 W for a scan resolution of (a) 0.1, (b) 0.2, and (c) 0.5. . . .
79
A.2 Current-voltage profiles, results of smoothing procedure and second derivative at 300 W for a scan resolution of (a) 0.75 and (b) 1.0. . . . . . . .
80
A.3 Current-voltage profiles, results of smoothing procedure and second derivative at 400 W for a scan resolution of (a) 0.1, (b) 0.5 and (c) 0.75. . . .
81
A.4 Current-voltage profiles, results of smoothing procedure and second derivative at 500 W for a scan resolution of (a) 0.1, (b) 0.5 and (c) 0.75. . . .
82
xi
SYMBOLS
ai
radius of ith atom or molecule
c
speed of light
cj
filter coefficient
e
electron charge
Eeff
effective electric field
Ei (ωα )
ith component of the electric field with frequency ωα
fe
electron collisional frequency
f0
electron density function
fp
electron probability density function
F ()
electron energy density function
FD ()
Druyvesteyn electron energy distribution
FM ()
Maxwellian electron energy distribution
go
degeneracy
h
step width for non-recursive digital filter
iBohm
Bohm ion current
ie
electron probe current
ies
electron saturation current
ii
ion probe current
iis
ion saturation current
ip
total probe current
I(ωα )
intensity of beam with frequency ωα
k
wavevector
jez
electron current density
J
rotational quantum number
kB
Boltzmann constant
xii
L
length of the interaction of the beams
me
electron mass
mi
ion mass
n
degree of differentiation in non-recursive digital filter
na
index of refraction at frequency ωa
ne
electron number density
nH2
H2 number density
ni
ion number density
N
population number density
Na
Avogadro’s number
Pabs
absorbed power by electrons
Pe
electric power
Pia (ωa )
anti-Stokes signal polarization
ptot
pressure
∆pn
momentum transfer by electron-neutral collisions
re
electron hard-sphere radius
rt
probe tip radius
rh
probe holder radius
rH2
H2 hard-sphere radius
R
universal gas constant
Rex
ratio of the peak intensities of the Q(1) transition from experimental spectrum
Rth
ratio of the peak intensities of the Q(1) transition from theoretical
spectrum
Sp
probe surface area
T
temperature
Te
electron temperature in Kelvin
TeV
electron temperature in eV
Ti
ion temperature
xiii
Tpl
plasma temperature
Trt
room temperature
ve
velocity of electron
vez
velocity of electron along the z-direction
vi,th
ion thermal speed
V
probe bias with respect to plasma
Vp
probe bias
Vpl
plasma potential
Vpk
peak location of EEDF
w(j)
window function in filter coefficient
xH 2
mole fraction of H2
xth
theoretical mole fraction value
(n)
yl
non-recursive digital filter of degree of differentiation n
Zr
rotational-translational relaxation rate
Zrot,nuc
product of rotational and nuclear partition functions
β
factor accounting for transfer of energy by inelastic collisions
`t
length of probe tip
0
permittivity of free space
electron energy in plasma
∆eT
kinetic energy transfer by electrons
θrot
characteristic rotational temperature
λr
mean free path
λe−H2
electron mean free path
λDe
Debye length
ν
mean collision frequency
σe−H2
electron-H2 hard-sphere cross section
σstd
standard deviation of the samples with respect to the theoretical
mole fraction xth
(a)
χikjl
dielectric susceptibility tensor
xiv
ωp
pump and probe beam frequency
ωS
Stokes beam frequency
ωa
anti-Stokes beam frequency
ωE
electric field frequency
xv
ABBREVIATIONS
CARS
coherent anti-Stokes Raman scattering
CVD
chemical vapor deposition
EEDF
electron energy density function
MPCVD
microwave plasma chemical vapor deposition
PACVD
plasma-assisted chemical vapor deposition
UHP
ultra-high purity
xvi
ABSTRACT
Tuesta, Alfredo D. Ph.D., Purdue University, December 2014. Optical Spectroscopy
and Langmuir Probe Diagnostics of Microwave Plasma in Synthesis of Graphenebased Nanomaterials. Major Professors: Timothy S. Fisher and Robert P. Lucht,
School of Mechanical Engineering.
Along with the revolutionary discovery and development of carbon nanostructures, such as carbon nanotubes and graphitic sheets, has arrived the potential for
their application in the fields of medicine, bioscience and engineering due to their
exceptional structural, thermal and electrical properties. As roll-to-roll plasma deposition systems begin to provide means for large scale production of these nanodevices,
a detailed understanding of the environment responsible for their synthesis is imperative in order to more accurately design and control the growth of carbon nanodevices.
To date, the understanding of the chemistry and kinetics that govern the synthesis
of carbon nanodevices is only partially understood. In response to this need, the
plasma environment of a microwave plasma chemical vapor deposition reactor has
been studied.
Coherent anti-Stokes Raman scattering spectroscopy was used to probe the H2
molecules in the plasma under various parametric conditions. The rotational temperature of H2 was found to increase with reactor pressure, plasma generator power, and
distance from the deposition surface. At 10 Torr, the temperature range is approximately 700 to 1200 K while at 30 Torr it is 1200 to 2000 K. Also, the introduction of
CH4 and N2 to the plasma increases the rotational temperature consistently. However, the number density of H2 in the plasma does not significantly deviate from
theoretical values corresponding to conditions without the plasma indicating that the
microwave plasma is weakly ionized and that the rotational temperatures obtained
approximate the translation temperature of H2 . The spectral region of the vibrational
xvii
hot band was also inspected but no transitions were found indicating that there is
little vibrational nonequilibrium.
Additionally, a Langmuir probe was used to obtain the electron energy density
function of the plasma. Due to the mismatch between the probe and microwave
plasma system, however, the electron energy density function was distorted resulting
in an overestimation of the electron temperature and an underestimation of its number
density in what is known as the Druyvesteynization effect. In spite of this, an upper
limit for the electron temperature was established at 4 eV and a lower limit for its
number density at 1e10 m−3 . Furthermore, the collisional frequency was estimated
to be on the same order as the microwave electric field frequency indicating that the
plasma is dominated by collisions.
In conclusion, the plasma in the microwave plasma chemical vapor deposition
reactor was found to be weakly ionized and not collisionless. These findings help
to identify the plasma and understand its kinetics. Gas temperature profiles were
established at 10 and 30 Torr based on the rotational temperature of H2 at parametric
conditions conducive to the growth of graphene-based nanomaterials. An upper limit
to for the electron temperature was also obtained.
1
1. INTRODUCTION
The discoveries and developments in nanoscience over the past decade have made
a significant impact in various fields of science and technology. As the time comes
to translate products from laboratory to market, the reliable, efficient and controllable large scale production of nanodevices is of paramount importance. A promising
technique in this effort is the roll-to-roll deposition system utilizing a plasma for the
synthesis of graphene and graphene-based nanostructures. In spite of its well established and successful application in the growth of nanostructures, the environment
in plasma chemical vapor deposition systems remains to be fully understood. It is,
therefore, the effort of this dissertation to elucidate the chemical environment in a
plasma chemical vapor deposition reactor so as to understand the kinetics that govern
the growth of graphene-based nanodevices.
The microwave plasma chemical vapor deposition reactor in the Birck Nanotechnology Center at Purdue University has been used to synthesize single-wall and multiwall carbon nanotubes, graphitic petals and few-layer graphene on various substrates
for a large number of applications. The plasma environment was studied via optical
spectroscopy of the H2 molecule and by use of a Langmuir probe. More specifically,
coherent anti-Stokes Raman scattering spectroscopy was the spatially resolved optical technique used to study the temperature and mole fraction of the most abundant
molecule in the plasma mixture, H2 . Also, a Hiden ESPion Langmuir probe with
a tungsten tip was used to examine the electron energy distribution in the plasma
environment.
In Chapter 2, a technical background for the technique of coherent anti-Stokes
Raman scattering spectroscopy is presented as well as details of the microwave plasma
chemical vapor deposition reactor. Results for the temperature measurements of H2
at parametric conditions conducive to the growth of graphene-based nanostructures
2
are illustrated. A qualitative explanation for the results is also provided based on the
absorption of energy of the electron from the microwaves and energy transfer to the
H2 molecules.
In Chapter 3, a technique for the measurement of the mole fraction of H2 under the same parametric conditions as those in Chapter 2 is presented and verified.
Transitions from the vibrational hot band are also tested. Results are reported and a
discussion as to the implication of the mole fractions found and the hot band tested
are presented.
The theory and background of Langmuir probes are presented in Chapter 4. Typical sources of error in Langmuir probe measurements are also described. The resulting
electron energy density functions for a number of plasma conditions are presented. A
discussion of the results and a deeper understanding of the plasma is also provided.
Finally, a summary of the findings of this dissertation is provided in Chapter 5.
A recommendation for future works is also discussed.
3
2. TEMPERATURE OF HYDROGEN VIA CARS
2.1
Abstract
Rotational temperature profiles of H2 in a microwave plasma chemical vapor depo-
sition reactor were measured via coherent anti-Stokes Raman scattering spectroscopy.
The temperature was found to increase with reactor pressure, plasma generator power,
and distance from the deposition surface. At 10 Torr, the measured temperature range
was approximately 700 - 1200 K while at 30 Torr it was 1200 - 2000 K under the conditions studied. The introduction of CH4 and N2 to the plasma increased the rotational
temperature consistently. These findings will aid in understanding the function of
the chemical composition and reactions in the plasma environment of these reactors
which, to date, remains obscure.
2.2
Introduction
2.2.1
Carbon Nanostructures and Nanodevices
Carbon nanostructures such as carbon nanotubes (CNTs), graphene and graphitic
nanopetals have received much attention in the scientific and engineering communities
over the past few decades due to their exceptional structural [1, 2], thermal [3, 4] and
electrical [5–8] properties. Included in the already vast and ever growing fields of
applications are biosensors [9, 10], hydrogen storage devices [11, 12] and field emitters
[13]. Few-layer graphene has been shown to be an effective ultra-thin oxidation barrier
coating in air [14] and under vigorous flow boiling conditions [15]. CNT arrays coated
by graphitic petals have been found to be efficient nanostructures for maximizing the
0
The contents of this chapter are published in ASME’s Journal of Micro- and Nano-Manufacturing
under the title Laser Diagnostics of Plasma in Synthesis of Graphene-Based Materials. They have
been replicated here with permission from the journal and the authors. DOI: 10.1115/1.4027547
4
electrochemical performace of MnO2 thereby achieving high specific capacitance and
energy density for supercapacitor applications [16].
In a short period of time, numerous research endeavors have demonstrated the remarkable versatility of carbon nanostructures and their derivative nanodevices. The
potential applications and impact in technology and society emphasize the importance of controlling their synthesis and manufacture on a mass scale [17, 18]. A
full understanding of the environment responsible for their synthesis is crucial for
the development of future nanodevices. Yet in comparison to material and device
characterization, studies of process characterization have been much less common
and therefore the present work seeks to elucidate the thermochemical environment of
graphene synthesis via microwave plasma chemical vapor deposition.
2.2.2
Methods for Synthesis
Just as various applications exist for carbon nanostructures, numerous techniques
for their growth have been invented such as exfoliation and cleavage [19], arc-discharge
[20,21], laser ablation [22], thermal chemical vapor deposition (CVD) [23], and plasmaassisted chemical vapor deposition (PACVD) [24]. The latter technique allows for efficient, replicable, controllable and relatively low-temperature synthesis [25]; these are
highly desirable traits in the manufacturing of nanodevices [26]. Consequently, the environments of PACVD reactors have been studied so as to elucidate the growth mechanism and to investigate the scalable manufacturing of carbon nanomaterials [27–37].
In this investigation a SEKI AX5200S microwave plasma chemical vapor deposition (MPCVD) reactor capable of synthesizing carbon nanotubes, graphene and
graphitic nanopetals over a variety of substrates was investigated under parametrically controlled conditions. A MPCVD system differs from a PACVD system primarily in that the source of the plasma is a microwave generator that plays a vital
role in the synthesis mechanism of the carbon nanostructures. Several studies have
reported the temperature and concentration of various chemical species in similar re-
5
actors [35, 38–40]. Furthermore, H2 has been identified as a primary chemical species
in the synthesis process and its temperature and number density have been measured
in various CVD and PACVD systems [28,35,38,39,41,42]. In spite of these endeavors,
the deposition process and growth mechanism remain obscure because of complexity.
Figure 2.1. Illustration of plasma bulk over molybdenum puck and
plasma sheath boundary region. The plasma bulk couples with the
molybdenum puck creating a sheath region.
The plasma-substrate boundary, known as the plasma sheath and illustrated in
Figure 2.1, is of particular interest due to the complex interactions of electrons, ions
and neutral chemical species in this region [43]. In addition, a more complete understanding of the gas phase in the plasma sheath region is essential for the modeling and
ultimately control, development and manufacturing of future nanodevices. Therefore,
the studied conditions in this investigation simulate the environment responsible for
the synthesis of graphene-based materials.
2.2.3
Coherent Anti-Stokes Raman Scattering Spectroscopy
In order to measure the plasma temperature in the presheath and sheath regions,
a non-intrusive technique is required so as not to disturb the plasma chemistry and
physics in the vicinity. Spectroscopic techniques, such as coherent anti-Stokes Raman
scattering (CARS) spectroscopy, have been utilized predominantly for this purpose.
CARS is a non-intrusive, spatially resolved optical technique whereby pump and
6
Stokes beams induce a Raman polarization of the molecule under investigation, and
a probe beam is then scattered from the induced polarization. The scattered beam is
the CARS signal. CARS has been extensively studied and utilized to probe molecules
such as N2 [44], O2 , CO2 [45] and H2 [28, 35, 39] in combustion and plasma systems.
This four-wave mixing process results in a highly coherent signal. Its geometric
projection from the probe volume can be predicted, collected and analyzed. The
theory of CARS has been previously outlined in detail [46] and therefore is only
briefly summarized below.
Figure 2.2. Schematic diagram of CARS beams and energy diagram.
The Pump, Stokes and Probe beams are focused into a probe volume
and a coherent, anti-Stokes signal is generated.
Figure 2.2 illustrates the CARS beams used to generate the anti-Stokes signal
and the energy diagram corresponding to this process. Solid lines indicate rotational
energy levels while dashed lines correspond to virtual levels. The Raman transition
of the molecule under investigation is excited by the pump ωp and Stokes ωS beams.
The probe beam ωp is scattered from the Raman polarization thus generating an
7
anti-Stokes signal ωa with frequency ωa = 2ωp − ωS [41, 47]. In this investigation, the
pump and probe beams are at the same frequency ωp .
The polarization of the anti-Stokes signal Pia (ωa ) is proportional to the product
of the CARS beams’ electric fields [41]
(3)
Pia (ωa ) ∼χijkl Ej (ωp )Ek (ωp )El∗ (ωS ).
(2.1)
(3)
The term χikjl in Equation 2.1 is the nonlinear third-order susceptibility, a fourthrank tensor, with subscripts ijkl representing the Cartesian coordinates (x, y and z)
of the pump, Stokes, probe and anti-Stokes signal beams. The quantity Ei (ωα ) is the
ith component of the electric field with frequency ωα while the asterisk represents the
complex conjugate of the field.
The formation of the anti-Stokes signal depends on the proper alignment and
propagation of the CARS beams into the probe volume and must satisfy the relation
ka = 2kp − kS for maximum phase-matching. Its intensity I(ωa ) for the case of plane
waves and parallel polarizations is given by [41]
I(ωa ) =
4π 2 ωa
c2 n a
2
I(ωp )2 I(ωS )|3χ(3) |2 L2
(2.2)
where L is the length of interaction of the beams.
The Q-branch transitions used in this investigation are characterized by an unchanged quantum rotational number J, such that ∆J = 0 [48]. The population of the
molecules under scrutiny is concentrated near the ground rotational energy states corresponding to v = 0, J = 0, 1, 2, 3, 4 and 5 for the range of temperatures from 300 to
2000 K as illustrated in Figure 2.3. The intensities of the transitions are proportional
to the square of the number density of molecules in the corresponding rotational energy states. As a result, at room-temperature the higher transitions are not as visible
as those near the ground transition. At higher temperatures, the population of higher
rotational energy states increases and the relative amplitudes of higher transitions,
such as Q(4) and Q(5), increase in comparison to lower J transitions.
8
Figure 2.3. Population distribution for the rotational energy mode
of H2 . Intensity alteration in the rotational fine structure is due to
nuclear spin statistics.
The molecular distribution plotted in Figure 2.3 is given by
NJ
go (go ± 1)(2J + 1)
θrot
=
exp −J(J + 1)
N
2Zrot,nuc
T
(2.3)
where NJ is the population number density at level J, N is the total population
number density, g0 is the degeneracy, Zrot,nuc is the product of the rotational and
nuclear partition functions and θrot is the characteristic rotational temperature which
is approximately 87.55 K for H2 [49, 50]. The ± symbol in the numerator varies
depending on the parity of the rotational quantum number J; + for odd J and − for
even J.
9
2.3
Experimental System
2.3.1
Microwave Plasma Chemical Vapor Deposition
The SEKI AX5200S microwave MPCVD reactor includes a 1.5 kW (2.45 GHz)
ASTeX AX2100 microwave generator and 3.5 kW radio-frequency power supply with
a graphite susceptor. The graphite susceptor can be translated along the axis of the
reactor toward and away from the microwave generator and also serves as a heating
stage regulated with temperature feedback control by a shielded K-type thermocouple
located 2.5 mm below the surface. The surface temperature is monitored with a
Williamson dual wavelength pyrometer (model 90) [51]. A schematic diagram of this
MPCVD reactor is shown in Figure 2.4.
A substrate is placed on a 57.2 mm diameter (2 1/4 in.), 6.8 mm thick molybdenum
puck, introduced into the reactor through a front hatch and placed on the graphite
susceptor stage. As illustrated in Figure 2.4, the stage is accessible at a stage height
of 0 mm. However, ignition of the plasma is conducted at 53 mm above this position.
Therefore, the stage is raised from 0 to 53 mm while evacuating the reactor by slowly
decreasing its pressure to 2.5 Torr using an external mechanical pump. H2 (Ultra
High Purity 5.0) is then allowed to flow at 50 standard cubic centimeters (sccm) from
a valve at the top of the reactor, reaching a pressure of 10 Torr. The mechanical pump
combined with a control valve automatically purges the influx of hydrogen from the
bottom side of the reactor to maintain the desired pressure.
The microwave generator is calibrated to couple with the reactor system at a
stage height of 53 mm, a reactor pressure of 10 Torr and a plasma generator power of
300 W. The plasma is ready to be ignited once these conditions have been satisfied.
The graphite susceptor stage, however, obstructs the line-of-sight through the lateral
optical arms at a stage height of 53 mm and therefore must be lowered to allow the
diagnostic laser beams to pass. The plasma generator power and reactor pressure may
be altered to meet the recipe appropriate for the synthesis of carbon nanostructures
10
Figure 2.4. Schematic diagram of the microwave plasma chemical
vapor deposition reactor at two stage positions: (A) susceptor stage
ready to accept substrate and (B) the susceptor stage raised to 53
mm for the ignition of the plasma.
including the introduction of CH4 (Ultra High Purity 4.0) and N2 (Ultra High Purity
5.0).
Once the plasma has been ignited at a pressure of 10 Torr, the plasma generator
power may be varied from 100 to 500 W. In order to raise the power beyond 500
W, the pressure must first be elevated to 30 Torr to avoid arc discharges. At this
pressure, the power may be raised from 500 W to 1000 W. Whenever the stage height,
generator power or reactor pressure are changed from their calibrated positions, the
impedance of the waveguide-reactor system is unbalanced corresponding to an increase in reflected wattage. Therefore, the impedance must be adjusted via three
stub-tuners to reduce the reflected microwave power.
Although the conditions for growth are changed by the lowered position of the
stage within the MPCVD reactor, graphitic growth can still be accomplished over a
copper substrate. To demonstrate this, a 25 mm diameter, 0.5 mm thick copper disc
was introduced into the MPCVD reactor, carefully placed over a ceramic stilt and
exposed to a 400 W H2 plasma at 10 Torr. In order to simulate the experimental
conditions in this investigation and to clear the optical path through the lateral optical
11
windows, the stage height was lowered from 53 to 28.5 mm, and CH4 was introduced
at 10 sccm for approximately 1 minute. The ceramic stilt allows for the plasma to
couple around the copper disc and under it, exposing the bottom surface.
Figure 2.5 and Figure 2.6 show the Raman spectra for the top side and bottom
side of the copper disc, respectively, confirming the growth of a graphitic film. Each
spectrum exhibits D, G and 2D peaks. The D peak is caused by defects, and the G
peak shows the graphitic nature on the surface. On the top side of the copper disc,
two positions were sampled, both of which contain a higher G peak intensity than
D peak. The bottom side, however, contained comparable G and D peak intensities,
illustrating that better graphitic growth was achieved on the top side as compared
to the bottom side of the copper disc. The 2D peak is the second harmonic of the D
peak and always appears in graphene and graphitic layers [52, 53].
Figure 2.5. Raman spectrum from two positions on the top surface
of the copper disc directly exposed to the plasma.
12
Figure 2.6. Raman spectrum from the bottom surface of the copper disc.
2.3.2
H2 CARS Spectroscopy
A schematic diagram of the CARS system is shown in Figure 2.7. As previously
discussed, pump and Stokes beams are used to create a Raman polarization for a
specific molecule and the probe beam is coherently scattered to create the CARS
signal. In this study, the second harmonic, 532 nm, of a pulsed Nd:YAG laser was split
to generate two narrowband beams: the pump and probe beams at approximately
13.5 mJ/pulse. A broadband Stokes beam at 685 nm of approximately 9 mJ/pulse
was generated by the 532 nm beam after pumping a dye solution mixture of LDS 698
(46.2 mg/L of ethanol) and DCM (18.4 mg/L of ethanol). The broadband beam was
amplified with a similar solution.
The anti-Stokes signal was filtered and collected using a spectrometer with a
diffraction grating and detected using a CCD camera (ANDOR DU440-BU). The
camera was configured to record data with an exposure time of 10 seconds, with 15
images collected, for each data set. Consequently, the results in this investigation
are averaged. The data sets were acquired at steady-state after the plasma has been
configured with the desired parametric conditions. Before the acquisition of each data
13
BS
BS
BS
532 nm
Seeded Nd: YAG
685 nm
Amplifier
Broadband
Dye Laser
f = - 100 mm
f = + 200 mm
λ/2
Lens
CCD
Camera System
TP
Spectrometer
435 nm
f = + 250 mm
f = + 200 mm
Figure 2.7. Schematic diagram of the CARS system. BS: beam
splitter; TP: thin film polarizer; λ/2: half-wave plate.
set, the beams were checked for alignment into the MPCVD reactor and also into the
spectrometer. The spectrometer slit spacing was approximately 70 µm.
Rotational Temperature Measurements
When recording each data set, the spectrum of H2 and the non-resonant spectrum
(NRS) were first acquired at room temperature. The spectrum at room temperature
was acquired after purging the reactor to 2.5 Torr and then introducing H2 until the
pressure rose to no more than 10 Torr. The intensity of the signal is proportional to the
number density, and consequently the pressure, of the molecule under investigation.
Care was taken to keep the signal low enough that the CCD camera did not saturate.
If the pressure is too high, the signal saturates the CCD camera, yielding erroneous
results.
14
The plasma was not ignited for this procedure. There are three reasons to measure
the H2 spectrum at room temperature. First, a theoretical fit of the spectrum must
consistently yield a temperature near 293 K in order to validate the subsequent theoretical fits of the data pertaining to the plasma. Second, an appropriate instrument
function must be found for the data set for the CARSFT code. Finally, recording
this spectrum enables accurate concentration measurements in the plasma; this will
be discussed in a future publication.
Similarly, the NRS was acquired by first purging the reactor and then raising its
pressure to approximately 500 Torr with N2 . Acquisition of the NRS measures the
spectral distribution of the Stokes dye laser [41]. The H2 spectrum must be divided
by the NRS before analysis.
The CARSFT code was written in FORTRAN and provided by Sandia National
Laboratories, Version 29-04-03. It was used to obtain the best fit theoretical spectrum
to the experimental spectrum [54]. The parameters varied in the least-square fitting
routine were the horizontal shift, vertical shift, wavenumber expansion, intensity expansion and temperature. The pressure was fixed to the appropriate value depending
on the data set. The CARSFT code also allows for a user-defined instrument function
to be incorporated when fitting the spectrum. The instrument function compensates
for broadening effects due to the spectrometer, CCD camera, and the remaining components of the optical system independent of the Raman line broadening for the H2
CARS transitions. The room-temperature H2 spectrum for each data set was used
to obtain the instrument function. The sum of a Gaussian and Lorentzian profile
was used as the instrument function by varying the half-width at half-maximum and
relative amplitude of each profile.
The procedure for preparing a raw H2 CARS spectrum for the CARSFT code
analysis was:
1. The CARS H2 spectrum and CARS background spectrum were individually
averaged before subtracting the CARS background spectrum from the H2 CARS
15
spectrum pixel-by-pixel. The CARS background was obtained by blocking the
Stokes beam.
2. The NRS spectrum and NRS background spectrum were individually averaged
before subtracting the NRS background spectrum from the NRS spectrum pixelby-pixel.
3. The background-subtracted CARS spectrum was divided by the backgroundsubtracted NRS spectrum pixel-by-pixel.
4. The square-root of the NRS-normalized H2 CARS spectrum was taken on a
pixel-by-pixel basis.
5. An average filter, such as the SMOOTH function in MATLAB, was applied to
the spectrum.
An example of the room-temperature spectrum and its theoretical fit is illustrated
in Figure 2.8 for which the CARSFT code converged to a temperature of 272 K. In
this investigation, the CARSFT code typically underestimated the temperature of
the room-temperature spectrum of H2 by 20 K. An example of the theoretical fit of
a H2 spectrum acquired from the plasma is shown in Figure 2.9. The CARSFT code
in this profile converges to a temperature of 1341 K.
2.3.3
Conditions for Graphitic Growth
This study focuses on the environment responsible for the growth of carbon nanostructures such as carbon nanotubes, graphene and graphitic petals. Carbon nanotubes have been synthesized in the subject chamber at 10 Torr in a mixture of H2
and CH4 with plasma generator powers ranging from 150 to 300 W [7,55,56]. Likewise,
few-layer graphene has been synthesized on a copper foil at 10 Torr, 400 W [57], and
graphitic petals have been grown at 30 Torr, 700 W [58, 59]. In order to investigate
this range of parametric MPCVD conditions, the following sets of experiments were
16
Figure 2.8. Theoretical fit to room temperature spectrum. CARSFT
code converges to a temperature of 272 K.
Figure 2.9. Theoretical fit to a H2 spectrum from plasma. CARSFT
code converges to a temperature of 1341 K.
conducted for the measurement of the rotational temperature of H2 . All data sets
were acquired at steady state, approximately 4 minutes after the plasma parameters
were changed.
17
1. Rotational temperature through the plasma sheath and presheath at various parametric conditions. The rotational temperature of H2 was measured from the
closest position of the probe volume to the puck surface at 2 mm up to 6
mm into the plasma bulk. The pressure was varied from 10 to 30 Torr, and
the plasma generator power varied from 300 to 700 W. Furthermore, CH4 (10
sccm) was introduced to reveal its influence on the rotational temperature. In
this set of experiments, the position of the probe volume, the reactor pressure,
the microwave generator power and the chemical composition of the plasma are
varied.
2. Rotational temperature through the plasma sheath and presheath with the introduction of CH4 and N2 . The influence of heavier species on the rotational
temperature of H2 was explored by taking measurements through the plasma
sheath and presheath regions (2 - 6 mm) at 10 Torr, 400 W. No other parameters
were varied.
Because it was concentrated directly above the molybdenum puck, the plasma
could be translated vertically by moving the susceptor stage up and down along the
axis of the reactor. As shown in Figure 2.7, the CARS beams were focused into the
reactor using a +250 mm focusing lens and collimated with a +200 mm lens after
diverting from the rector. The probe volume was formed by the beams entering the
plasma at an angle to the horizontal, limiting the proximity of the probe volume to
the molybdenum puck. Consequently, the probe volume was located approximately
2 mm above the molybdenum puck at its closest approach. Measurements through
the plasma sheath and presheath regions were performed in increments of 1 mm.
18
2.4
Results
2.4.1
Temperature through the Plasma Sheath and Presheath
The rotational temperature of H2 increased nearly linearly with distance of the
probe volume normal to the surface of the puck at every reactor pressure and plasma
power studied, as illustrated in Figure 2.10 and Figure 2.11 with results at 10 Torr
and 30 Torr, respectively. At 10 Torr, measurements were conducted with a plasma
generator power of 300 and 500 W while at 30 Torr they were conducted with a plasma
generator power of 500 and 700 W. The nearly linear relationship was maintained even
after the introduction of CH4 into the reactor.
Figure 2.10. Rotational temperature of H2 at 10 Torr with varying
plasma generator powers, with and without CH4 (10 sccm). Filled
triangles correspond to a precursor mixture of H2 (50 sccm). Empty
triangles correspond to a precursor mixture of H2 (50 sccm) and CH4
(sccm). Downward pointing triangles correspond to a generator power
of 300 W. Upward pointing triangles correspond to a generator power
of 500 W.
At 10 Torr the rotational temperature of H2 for the positions examined fell in the
range of 700 - 1200 K while at 30 Torr the temperature range was 1200 - 2000 K.
There was clearly an increase of rotational temperature with pressure. For each set
19
Figure 2.11. Rotational temperature of H2 at 30 Torr with varying
plasma generator powers, with and without CH4 (10 sccm). Filled
triangles correspond to a precursor mixture of H2 (50 sccm). Empty
triangles correspond to a precursor mixture of H2 (50 sccm) and CH4
(sccm). Downward pointing triangles correspond to a generator power
of 300 W. Upward pointing triangles correspond to a generator power
of 500 W.
of experiments at each pressure of a mixture containing H2 only, the temperature was
higher by nearly 100 K when increasing the power by 200 W. The introduction of
CH4 increased the temperature particularly at lower plasma generator powers such
as 300 W at 10 Torr and 500 W at 30 Torr.
The increase in temperature as the generator power is increased is consistent with
results of other studies of H2 conducted in similar reactors [31,32,35,60]. This trend is
to be expected because an increase in plasma energy density enhances intramolecular
activity and, therefore, temperature. Similarly, the increase in temperature as the
probe volume moves away from the puck surface corresponds to an approach toward
the center of the plasma bulk where a more energetic environment is to be expected.
As previously mentioned, the temperature of H2 increases substantially when the
pressure increases. At a generator power of 500 W, the temperature increased from
a range of 850 - 1150 K to 1200 - 1650 K with increasing reactor pressure from 10
Torr to 30 Torr. This trend is also in agreement with other similar studies [35, 60].
20
An analytical explanation for the increase of temperature with pressure is provided
in Section 2.4.3.
2.4.2
Temperature with the Introduction of CH4 and N2
As shown in the results of Section 2.4.1, the rotational temperature of H2 increased
with the introduction of CH4 . The synthesis of various carbon nanostructures in
the MPCVD reactor requires the use of CH4 and N2 typically at 10 and 5 sscm,
respectively. In order to evaluate more carefully the effect of the introduction of
heavier chemical species into the plasma on the rotational temperature of H2 , CH4
and N2 were introduced at 10 and 5 sccm at 10 Torr, 400 W.
As shown in Figure 2.12, the rotational temperature increased with the addition
of each substance and reached a maximum with the introduction of a combination of
both CH4 and N2 . The range of the rotational temperature of H2 at 10 Torr, 400 W
was 800 - 1100 K and increased to 900 - 1250 K with the addition of CH4 and N2 .
The introduction of these chemical species alters the collisional frequencies and the
kinematics of the collisions which appear to enhance the transfer of electron kinetic
energy to the H2 molecules. This interesting observation is discussed in section 2.4.4.
2.4.3
Analytical Explanation of the Increase in Rotational Temperature
with Pressure
The mechanism for energy transfer from the microwave field to the temperature of
neutral species such as H2 in a plasma system begins with the absorption of electromagnetic energy by the electrons. As highly mobile and low mass particles, electrons
and their temperature Te are highly affected by the electric field of the microwave
generator. Once they have been energized, electrons are able to transmit energy to
other plasma components for ionization, excitation, dissociation and other plasmachemical reactions by means of elastic and inelastic collisions among electrons, ions
and neutral species [61].
21
1 3 0 0
R o ta tio n a l T e m p e r a tu r e o f H
H
2
(5 0 s c c m ) & C H
H
2
(5 0 s c c m ) & N
H
2
(5 0 s c c m ) & C H
H
2
(5 0 s c c m )
1 2 0 0
2
(K )
1 3 0 0
2
4
(1 0 s c c m ) & N
2
(5 s c c m )
(5 s c c m )
4
1 2 0 0
(1 0 s c c m )
1 1 0 0
1 1 0 0
1 0 0 0
1 0 0 0
9 0 0
9 0 0
8 0 0
8 0 0
2
3
4
5
6
D is ta n c e N o r m a l to P u c k S u r fa c e ( m m )
Figure 2.12. H2 rotational temperature distributions at 10 Torr, 400
W with and without CH4 and N2 .
The increase in H2 temperature with generator power and reactor pressure can be
qualitatively explained by considering the absorption of the microwave power by the
electrons in the plasma. First, it is necessary to consider the oscillation of electrons
in a high-frequency harmonic electric field E(t) = E0 exp(−iωE t) in which E0 is the
electric field and ωE is the electric field frequency. The 1D equation of motion in the
z-direction for the electrons in the electric field is
me
dvez
= −eE0 exp(−iωE t) − me fe vez
dt
(2.4)
where me is the mass of the electron, vez is the velocity of the electron along the
z-direction, e is the electron charge, and fe is the electron collisional frequency as a
result of elastic collisions of electrons with neutral species [43].
The solution of the differential equation in Equation 2.4 yields the velocity term
vez =
−e(fe + iωE )
E0 exp(−iωE t).
me (ωE2 + fe2 )
(2.5)
The electron current density jez may then be written as
jez = −ene vez
ne e2 (fe + iωE )
=
E0 exp(−iωE t)
me (ωE2 + fe2 )
(2.6)
22
where ne is the electron number density and the electric power Pe may be expressed as
Pe = je E. The real part of the electric power is the absorbed power by the electrons
in a high-frequency electric field
Re(Pe ) = Pabs =
ne e2 fe
2
Eeff
2
2
me (ωE + fe )
(2.7)
with an effective electric field Eeff .
Furthermore, the electron temperature Te dependence on the absorbed power Pabs
can be made explicit when considering the transfer of kinetic energy of electrons ∆eT
to neutral species by means of the momentum of elastic electron-neutral collisions
∆pn = me ∆ve . The kinetic energy of the neutral species may be expressed as
∆p2n
(me ∆ve )2
me me ∆ve2
me e
=
=
=
∆
(2.8)
2m
2m
m
2
m T
where m is the mass of the neutral species. The total rate of kinetic electron energy
variation can then be equated to the power absorbed by the electrons Pabs such that
me e
∆ = Pabs
m T
me
ne e2 fe
2
ne fe 3kB (Te − T ) =
Eeff
2
2
m
me (ωE + fe )
me2
Te − T =
E2
3kB m2e (ωE2 + fe2 ) eff
ne f e
(2.9)
and a relation for the difference in electron temperature and neutral species temperature (Te − T ) may be obtained.
The mean collision frequency is directly proportional to the total pressure (fe ∝
ptot ) and, therefore, the expression in Equation 2.9 is inversely proportional to the
square of the total pressure
Te − T =
2
e2 mEeff
3kB m2e (ωE2 + c1 p2tot )
(2.10)
where c1 is a proportionality coefficient. As pressure increases, the neutral species
temperature rises toward the electron temperature as seen in the change in temperature ranges from 10 to 30 Torr in Figure 2.10 and Figure 2.11.
23
2.4.4
Analytical Explanation for the Increase in Rotational Temperature
with the Introduction of N2 and CH4
Equation 2.10 serves as a qualitative approximation for the relationship among
the difference between the electron and neutral species temperatures and plasma
parameters such as the effective electric field and reactor pressure. Its derivation
considers elastic scattering of electrons and neutral species in a weakly ionized and
collision-dominated plasma.
The increase in the rotational temperature of H2 with the introduction of N2 and
CH4 , however, involves various other mechanisms with many variables. Among these
is the inelastic scattering of electrons and the neutral species in the system from which
a plausible explanation for the increase in the rotational temperature of H2 with the
introduction of N2 and CH4 may be derived.
Elastic and inelastic collisions of electrons with neutral and ionized species play
a significant role in the chemical kinetics of plasmas. They are responsible for the
various dissipation processes of translational and internal energies [43]. Particularly,
inelastic collisions are responsible for the vibrational excitation of neutral species. In
fact, approximately 90% of the power input into the plasma is consumed by vibrational excitation in RF discharges at moderate pressures [62].
Vibrational excitation of neutral species occurs by electron impact proceeding
through the formation of intermediate unstable negative ions. The first vibrational
level of the H2 molecule has an energy of 0.53 eV. That of the N2 molecule, on the
other hand, has a lower energy of 0.29 eV. Consequently, the total cross section for
the vibrational excitation of N2 is higher than that of H2 . It is almost an order
of magnitude higher at mean electron energies of 1-3 eV typical for RF discharges
in molecular gases at intermediate pressures. As a result of the relatively higher
energies required to excite the vibrational levels of H2 , only the first vibrational level
is typically excited in nonthermal plasmas whereas the first eight vibrational levels of
N2 are likely to be excited.
24
In the case of a microwave, H2 nonthermal plasma, the addition of a chemical
species like N2 to the mixture means that more energy from the electrons will be
absorbed to excite the vibrational levels of that chemical species. This is not to
say that more power is absorbed by the electrons from the microwaves. Rather, the
energy from the electrons which would have been dissipated in other mechanisms
is used for the vibrational excitation of several levels in the molecules introduced.
The H2 molecules continue to absorb relatively the same amount of energy from the
electrons but, with the introduction of N2 , undergo additional heating due to collisions
with vibrationally excited N2 molecules. The vibrational energy of N2 molecules is
converted into rotational and translational energy in the process N2 (v) + H2 → N2
+ H2 + ∆ where N2 (v) represents the vibrationally excited N2 molecule and ∆ the
resulting kinetic energy of the molecules.
So as to express this mechanism mathematically, a factor β can be introduced into
Equation 2.8
β
∆p2n
me
= β ∆eT
2m
m
(2.11)
such that it represents the additional energy transferred to the molecules by inelastic electron scattering resulting in vibrational excitation of the molecules (β > 1).
Proceeding with the rest of the derivation, a new a relationship for the difference in
electron and neutral species temperature may be obtained
Te − T =
me2
E2 .
3kB m2e β(ωE2 + fe2 ) eff
(2.12)
As shown in Equation 2.12, an increase in vibrational excitation of the molecules in
the mixture results in a decrease in the difference between the electron and neutral
species temperatures. In the case of the H2 plasma, this results in an increase in the
temperature of H2 .
Lastly, the introduction of CH4 into the H2 plasma is a much more complex case
involving the dissociation of the CH4 molecules and their chemical reactions with H2
molecules forming new molecules and ions. Nevertheless, the addition of CH4 will
25
have a similar affect as that of N2 because the CH4 molecule has many vibrational
levels which can absorb more electron energy than the H2 molecule.
2.5
Conclusions
The rotational temperature of H2 has been successfully measured in the plasma
of a MPCVD reactor via CARS. Specific environments conducive to the growth of
carbon nanotubes, graphene and graphitic petals have been targeted, providing a
unique insight into their kinetics. The rotational temperature was found to increase
with distance normal to the surface of the puck, reactor pressure, plasma generator
power and the introduction of CH4 and N2 . This improved understanding of the H2
temperature in the vicinity of the plasma-substrate interaction environment provides
a vital piece of information for those seeking to model the plasma environment within
the MPCVD.
26
3. MOLE FRACTION OF HYDROGEN VIA CARS
3.1
Introduction
The transition from benchtop to personal use of nanodevices has begun. The roll-
to-roll synthesis of graphitic sheets has enabled their large scale production bringing
nanodevices closer to commercial applications. Bae et al. in 2010 successfully synthesized monolayer 30-inch graphene on flexible copper substrates via chemical vapor
deposition [18]. Shortly after, Yamada et al. grew few layer graphene films of nanoand micrometer size, 294 mm wide, under microwave plasma chemical vapor deposition (MPCVD) [63]. As the demand for a reliable, efficient and controllable technique
for their industrial mass production increases, so does the necessity for a fuller understanding of the growth mechanism and the environment responsible for the synthesis
of nanodevices increase.
The gas-phase processes responsible for the synthesis of nanostructures in chemical
vapor deposition and plasma-assisted chemical vapor deposition processes have been
heavily studied both from a modeling [29, 36, 37, 64, 65] and experimental [28, 30, 30,
66,67] perspective. Particularly, H2 and the H-atom are considered to play significant
roles in the deposition of nanostructures, for example, by helping to produce radical
hydrocarbons and terminating dangling bonds on the substrate surface [42]. As a
result, their concentration and temperature in H2 plasmas have also been investigated
through optical spectroscopy such as optical emission spectroscopy [42, 60, 68, 69],
coherent anti-Stokes Raman scattering (CARS) spectroscopy [39, 70–74] and others
[75, 76]. Most recently, a study of a MPCVD reactor has shown that the rotational
temperature of the H2 molecule is approximately 700 - 1200 K and 1200 - 2000 K
at 10 and 30 Torr, respectively [70]. In this report, the study of H2 is furthered
0
The contents of this chapter are part of a manuscript to be submitted to a peer-reviewed journal.
27
by measuring H2 mole fractions via CARS spectroscopy under the same parametric
conditions examined in Reference [70].
3.2
Experimental Setup
3.2.1
Optical System
The MPCVD reactor used for this study has been thoroughly described in previous
publications [51,69,70]. It consists of a 1.5 kW (2.45 GHz) ASTeX AX2100 microwave
generator and a 3.5 kW radio-frequency power supply attached at the top of a metallic,
cylindrical reactor. A graphite susceptor translates along the longitudinal axis of the
reactor and is composed of a heating stage regulated by a K-type thermocouple 2.5
mm below its surface. The temperature at the surface of the stage is monitored by
a Williamson dual wavelength pyrometer (Model 90). The walls of the reactor are
water cooled. The precursor gases are H2 , CH4 and N2 . Lastly, two optical arms
across from each other are attached on the sides of the reactor and allow for optical
access of the plasma.
The H2 plasma was probed via CARS spectroscopy. This four-wave mixing, spectroscopic technique makes use of a pump, Stokes and probe beam to excite the vibrational Raman transition of the H2 molecule and scatter an anti-Stokes beam. The
pump and probe beams were provided by the 532 nm second harmonic of an injection
seeded Nd:YAG laser. The Stokes beam was generated using a broadband dye laser
pumped by the 532 nm output of the Nd:YAG laser. The dye laser was a mixture
of LDS 698 (46.2 mg/per L of ethanol) and DCM (18.4 mg per L of ethanol) centered at 685 nm. As a result, the Raman shift produced by these beams is near 4160
cm−1 . The broadband nature of the dye laser allows for the simultaneous excitation
of multiple transitions in the (ν 00 = 0 → ν 0 = 1) Q-branch of H2 .The approximate
energies were 13.5 mJ/pulse for the pump and probe beams and 9 mJ/pulse for the
Stoke beam. Further details and diagrams of the CARS system may be found in Ref-
28
erence [70]. For an extensive review of the theory of CARS spectroscopy, the reader
is referred to Reference [77].
The CARS beams were carefully focused into the plasma through one of the
optical viewing ports and allowed to pass through to the other side. Using a folded
BOXCARS configuration, the projection of the anti-Stokes beam can be predicted
based on an analysis of conservation of momentum. The anti-Stokes beam was then
focused into the entrance slit of a 0.75 m spectrometer Czerny-Turner (SPEX, Model
1800-II). The spectrum was detected using a CCD camera (ANDOR DU440-BU) and
analyzed with a computer. The spectra analyzed in this investigation were averaged
over 15 images collected each with an exposure time of 10 seconds.
In order to prepare the signal for processing there were three other types of spectra
that were necessary: the spectra of H2 at room temperature, the nonresonant spectra,
and the background spectra for the CARS and nonresonant spectra. The H2 spectra
at room temperature for the pressures studied was obtained without a plasma by
purging the reactor to a pressure of approximately 2.5 Torr and introducing H2 . The
room temperature spectra was used to obtain an instrument function for the CARSFT
code, to be discussed, which accounts for the broadening effects of the instruments.
It was also used in the procedure to derive mole fractions from the measurements
as will be discussed. The nonresonant spectra was acquired by purging the reactor
and filling it with N2 at approximately 500 Torr. The nonresonant spectra was used
to measure the fluctuating contribution from the dye laser intensity. The CARS
spectra is divided by the corresponding nonresonant spectra in a procedure described
subsequently. Lastly, the background spectra was collected immediately following the
acquisition of the CARS and nonresonant spectra at the same conditions by blocking
the Stokes beam. Background spectra for the CARS and nonresonant spectra were
necessary in order to remove the contribution of any stray light into the system.
After the necessary spectra were acquired, the following procedure was conducted:
29
1. The CARS H2 spectra and the CARS background spectra were averaged. The
averaged CARS background spectrum was then subtracted from the averaged
H2 CARS spectrum pixel by pixel.
2. The nonresonant spectra and non-resonant background spectra were averaged.
The averaged nonresonant background spectrum was subtracted from the averaged nonresonant spectra pixel by pixel.
3. The background-substracted CARS spectrum was divided by the backgroundsubtracted nonresonant spectrum pixel by pixel.
4. The square-root of the normalized H2 CARS spectrum was calculated on a pixel
by pixel basis.
After the spectra were treated with this procedure, the CARSFT code written
in FORTRAN and provided by Sandia National Laboratories, Version 29-04-03, was
used to extract the rotational temperature and mole fraction of H2 [54]. Broadening
effects due to the spectrometer, CCD camera and other optical components in the
system were be taken into account using an instrument function. The code is used
to find the best fit of the theoretical spectrum to the experimental spectrum by
varying parameters such as the horizontal shift, vertical shift, wavenumber expansion,
intensity expansion, pressure, mole fraction and temperature.
Although the code allows mole fraction as one of the parameters in the fitting
routine, it was not used because of the error introduced by the instrument function. Instead, a separate procedure involving the spectra at room temperature was
conducted as described below.
3.2.2
Determination of H2 Mole Fraction
The intensity of a transition in the Q-branch of H2 is proportional to various
parameters including the instrument function and number density of the molecule.
In order to determine a mole fraction, all parameters influencing the intensity of the
30
transition except the number density must be eliminated. This was accomplished
by dividing the peak intensity of the Q(1) transition at the plasma temperature Tpl
by the intensity of the Q(1) transition at a fixed and known temperature; in this
case, room temperature Trt was chosen. The Q(1) transition was chosen because it is
present in all of the spectra studied. The peak intensity of the Q(1) transition from
the experimental spectra was compared to the peak intensity of the Q(1) transition
from the theoretical spectra by conducting the following procedure:
1. The peak intensity of the Q(1) transition of the experimental spectrum IQ(1),ex at
the plasma temperature was divided by the peak intensity of the Q(1) transition
of the experimental spectrum at room temperature to form what will be referred
as the experimental ratio, Rex = IQ(1),ex (Tpl )/IQ(1),ex (Trt ).
2. The peak intensity of the Q(1) transition of the theoretical spectrum IQ(1),th at
the plasma temperature was divided by the peak intensity of the Q(1) transition
of the theoretical spectrum at room temperature to form what will be referred
as the theoretical ratio, Rth = IQ(1),th (Tpl )/IQ(1),th (Trt ).
3. The experimental and theoretical ratios are compared and a mole fraction is
derived.
In Step 1 of this procedure, the experimental ratio Rex is calculated with the prepared data before fitting with the CARSFT code. This means that the backgroundsubtracted CARS signal is divided by the background-subtracted nonresonant spectrum and the square root function is applied for both, the spectrum from the plasma
and the spectrum at room temperature. It is important to use the same nonresonant
spectrum for each spectrum so as to main consistency.
In Step 2 of this procedure, the ratio Rth is obtained from theoretical spectra
generated by the CARSFT code which allows for the temperature and mole fraction
to be chosen. It is important to use the same instrument function that was used to
fit the spectra for the experimental data being compared.
31
Although a mole fraction may be chosen to exactly match the experimental ratio Rex by varying the mole fraction of the theoretical spectra, it was convenient to
generate a series of theoretical ratios at three different mole fractions for various temperatures with which the experimental ratio may be compared. The three theoretical
ratios chosen in this study were 1, 5/6 and 5/7 corresponding to the mole fractions for
the cases without a plasma studied and the temperatures varied from 700 to 2200 K
in increments of 100 K. First, using spline interpolation, estimated values for Rth were
obtained at the plasma temperatures studied. Second, a second-order polynomial was
fitted to the three Rth values for each mole fraction so that a mole fraction may be
directly extracted by comparing the Rex value at the plasma temperature studied.
For instance, this procedure may be applied to the data shown in Figure 3.1 which
illustrates the experimental room temperature and plasma temperature spectra taken
at 10 Torr. The peak intensity of the Q(1) transition for the plasma temperature
and room temperature are approximately 1.5 and 30 arbitrary units, respectively,
after subtracting their corresponding baselines. Therefore, the experimental ratio is
Rex ≈ 1.5/30 ' 0.05.
In Figure 3.2, the experimental data has been fitted by use of the CARSFT
code with a mole fraction equal to 1. The peak intensity of the Q(1) transition
for the plasma and room temperature are approximately 9.5 and 150 arbitrary units,
respectively, after subtracting their corresponding baselines. The theoretical ratio
Rth ≈ 9.5/150 ' 0.06 is close to the experimental ratio of 0.05 previously calculated.
Instead of varying the mole fraction in the theoretical spectra to match that of the
experimental ratio, a series of three theoretical spectra are calculated as described.
At 30 Torr, the CARS signal was sufficiently strong to saturate the CCD camera.
A neutral density filter was used which reduced the intensity of the signal proportionally. It was necessary, therefore, to compensate for this effect. Before each experiment
at 30 Torr, the reactor was filled with H2 without a plasma at room temperature and
the peak intensity of the Q(1) transition was measured and compared with and without the neutral density filter at a reactor pressure of approximately 10 Torr. The
32
Figure 3.1. Experimental spectra with a reactor pressure of 10 Torr
at (A) room temperature and (B) plasma temperature. The plasma
generator power was 500 W and it was entirely composed of H2 .
factor obtained from this ratio was used to account for the reduction in intensity due
to the neutral density filter at 30 Torr.
The technique to determine the mole fraction of the experimental data has been
verified and the results are illustrated in Figures 3.3 and 3.4. H2 at 50 sccm was
introduced into the MPCVD reactor, and the stage was heated in increments of
100 ◦ C from 700 to 1000 ◦ C and continuously purged to maintain a pressure of 10
Torr. Measurements were taken approximately 2 mm above the molybdenum puck
without the plasma ignited such that these measurements corresponded to H2 without
dissociation or ionization due to the plasma.
Measurements with only H2 in the reactor correspond to a mole fraction of 1
(xH2 = 1). After these measurements were taken, N2 was introduced at 10 sccm to
reduce the mole fraction of H2 from 1 to 5/6. Lastly, the flow rate of N2 was raised
from 10 to 20 sccm, reducing the mole fraction to 5/7. Accordingly, measurements
at mole fractions of 5/6 and 5/7 are taken as well.
33
Figure 3.2. Theoretical fits to experimental data by use of CARSFT
code with a mole fraction equal to 1 for data at (A) room temperature and (B) plasma temperature. Fitting of the spectra with the
CARSFT code resulted in a temperature of 277 and 1133 K for room
temperature and plasma spectra, respectively.
Figure 3.3 illustrates the results of the procedure. The square points on the
plot correspond to the experimental ratios Rex obtained from measurements and the
dashed lines to the spline interpolation of the theoretical ratios Rth obtained from the
CARSFT code. Because three mole fractions were chosen per temperature, the mole
fractions of the experimental ratios may be estimated with a second-order polynomial.
Therefore, the results of the experimental and theoretical ratios in Figure 3.3 may be
converted into mole fractions as shown in Figure 3.4.
The error bars in Figure 3.4 were calculated with the following equation
#1/2
" 4
X
1
σstd =
(xi − xth )2
3 i=1
(3.1)
where σstd is a standard deviation of the samples with respect to the expected, theoretical mole fraction xth . The error bars represent a sample standard deviation based
on the expected value. For a mixture entirely of H2 , σstd = 0.0438. For a mixture
composed of 5/6 and 5/7 of H2 , σstd = 0.0205 and 0.03, respectively. Although the
34
Figure 3.3. Experimental Rex and theoretical Rth ratios at 10 Torr.
Experimental ratios, shown by the squares, were produced by heating the susceptor stage of the MPCVD reactor. Theoretical ratios,
shown by dots, were produced by use of the CARSFT code. The
dashed line connecting the dots illustrates the results from the spline
interpolation.
Figure 3.4. Experimental and theoretical ratios converted to mole
fractions. Error bars were calculated by use of Equation 3.1.
35
error is expected to increase with temperature, these error bars were applied to the
measurements from the plasma to the corresponding mixture of H2 introduced into
the reactor.
The measurements for the verification of this technique shown in Figure 3.4 are
relatively accurate within error. One of the main sources of error in this procedure
is the decreased signal-to-noise ratio at higher temperatures. As illustrated in Figure
3.1, the overall intensity of the signal decreases with temperatures. However, the noise
remains the same. Because of this, determination of the baseline is less accurate.
The temperatures in each set of measurements appear to deviate from the stage
values of 700, 800, 900 and 1000 K. This deviation is because the H2 molecules
probed are 2 mm away from the surface of the stage and will not be at the same
temperature as the stage. As a result, the temperature of H2 will be less than the
temperature of the stage. In addition, error in the temperature measurements is due
to the fluctuation of the profile and intensity in the Stoke’s beam. Because the dye
solution flows continuously in a cell, the spectral profile and intensity that is generated
will vary with time.
3.2.3
CARS Measurements of the ν 00 = 1 → ν 0 = 2 Hot Band
In order to explore the vibrational non-equilibrium of the H2 molecules in the
probe region, the vibrational hot band was investigated. Theoretical spectra were
generated by the CARSFT code at temperatures of 1000 and 2000 K as shown in
Figures 3.5 and 3.6, respectively. The code assumes a thermal system and therefore
is able to illustrate the location and intensities of the ν 00 = 1 → ν 0 = 2 hot band.
Transitions belonging to the cold band are labeled Q01 (J) while transitions belonging
to the hot band are labeled Q12 (J). As shown, transitions from the hot band are
much more apparent at higher temperatures near 2000 K and they appear in the
spectral region near 3900 cm−1 .
36
Figure 3.5. Theoretical spectra generated by the CARSFT code for a
thermal system at 1000 K. Transitions from the cold band are labeled
Q01 (J). For clarity, only the odd J transitions were labeled.
Figure 3.6. Theoretical spectra generated by the CARSFT code for a
thermal system at 2000 K. Transitions from the cold band are labeled
Q01 (J) and transitions from the hot band are labeled Q12 (J). For
clarity, only the odd J transitions were labeled.
So as to excite the Raman vibrational band near 3900 cm−1 , the center wavelength
of the Stokes beam was shifted from 685 nm to approximately 678 nm by adding 20
37
mg of DCM dye to a mixture of the original solution. This had the effect of creating a
second peak near 665 nm in the spectrum of the Stokes beam, thereby broadening the
Stokes beam spectrum to include the region between 685 and 665 nm. The addition
of 20 mg of DCM was conducted in increments of 5 mg such that the change of the
spectrum could be analyzed. The output energy was slightly decreased. The solution
was sonicated for a few minutes to ensure proper mixing.
Transitions from the vibrational hot band of H2 have been previously studied.
Shakhatov et al. [38] studied hydrogen in a RF inductive discharge plasma and found
that the vibrational temperature decreased with pressure while the rotational temperature increased. Therefore, the vibrational hot band was examined where it might
be at its highest temperature and, therefore, intensity. For the MPCVD reactor, this
corresponds to a pressure of 10 Torr which is the lowest pressure the plasma can be
operated in a stable manner.
3.2.4
Radial Temperature Profile in the Plasma
The position of the probe volume within the plasma was investigated. The focusing lenses around the MPCVD reactor were placed such that the probe volume was
visually located at the center of the plasma. In order to test this, they were mounted
on motorized, translational stages (Zaber T-LS Series) so that the temperature could
be probed radially outward. The probe volume was moved horizontally in increments
of 2 mm and the rotational temperature of H2 was measured at each position up to
24 mm away from origin.
Measurements of H2 were taken at room temperature with no plasma at 10 Torr
and with the plasma at 10 Torr, 300 and 500 W. As seen in Figure 3.7, the rotational
temperature of H2 with no plasma remained relatively constant around room temperature values for the entire span of 24 mm. With the plasma ignited, the temperature
remained constant for up to 8 mm away from origin and decreased thereafter for both
38
generator powers examined. The results suggest that the probe volume was indeed
at or near the center of the plasma in agreement with visual inspections.
Figure 3.7. Radial temperature profile from the center of the plasma.
Triangles correspond to measurements with a plasma. Circles correspond to measurements without a plasma at room temperature.
3.2.5
Intensity Effects
At high beam intensities, the H2 transitions may become saturated, and the optical
Stark effect may complicate a proper analysis of the spectra [35, 78]. In order to test
the effect of laser power on the H2 CARS spectrum, the intensity of the Q(1) line of
H2 was recorded as a function of beam energy. The probe and pump beam energies
were varied while the Stokes beam energy was maintained at a constant value of 4.8
mJ/pulse. Figure 3.8 illustrates the response of the Q(1) line intensity. Although
a slight curvature exists in the data points, the profile is nearly linear indicating
that the Raman resonance is not saturated. The Q(1) transition was chosen for this
analysis because it is the most intense at room temperature.
The reactor pressure was varied in order to test the dependence of the anti-Stokes
signal on the species density [35]. As Figure 3.9 shows, a linear relationship exists for
39
Figure 3.8. Square root of the intensity of H2 Q(1) line as a function
of beam intensity. The Stokes beam energy was 4.8 mJ/pulse.
the square root of the intensity of the Q(1) line as the reactor pressure varies from
approximately 5 to 30 Torr with H2 . This is the expected behavior in the absence of
significant saturation or Stark effects.
Figure 3.9. Square root of the intensity of H2 Q(1) line as a function
of reactor pressure.
40
3.3
Results
The experimental mole fractions of H2 in the plasma are shown in Figure 3.10 for
a reactor pressure of 10 Torr and in Figure 3.11 for 30 Torr. Dashed lines indicate
the theoretical mole fraction for the corresponding mixture without a plasma. Error
bars from to the verification procedure illustrated in Figure 3.4 were attached to
the data for the corresponding mixture. However, they do not appear to suffice in
capturing the true error in the measurements. As previously discussed, the error
is expected to increase with temperature because the signal-to-noise ratio increases
making the estimation of the peak intensity and baseline somewhat ambiguous. This
is particularly shown in the trends of mole fractions having a mixture corresponding to
H2 (10 sccm) and N2 (20 sccm) which is nearly at the highest temperatures examined.
At 300 W, all of the experimental mole fractions are above the maximum theoretical
value marked by the dashed line. However, the overall trend for all measurements
appear to indicate that at 10 Torr there is no significant deviation of the experimental
mole fraction in the plasma from theoretical values without the plasma.
Experimental mole fractions at 30 Torr, illustrate a similar trend. At higher temperatures, the experimental mole fractions are sometimes greater than the maximum
theoretical value. This is the case for some measurements corresponding to H2 (50
sccm) and a mixture of H2 (50 sccm) and CH4 (10 sccm). In the case of a mixture
of H2 (50 sccm) and CH4 (10 sccm) there is a possibility of H2 formation due to CH4
dissociation. However, even if this were the case, it does not appear to be significant.
The overall trend, as in a pressure of 10 Torr, is that the experimental mole fractions
do not appear to deviate from their theoretical values without a plasma by much.
The experimental mole fractions should appear to be significantly lower than the
theoretical values without a plasma if any one of the following conditions were met:
1. vibrational nonequilibrium,
2. nonequilibrium of the translational and rotational modes,
3. and dissociation and/or ionization of H2 .
41
Figure 3.10. Experimental mole fraction of H2 (50 sccm) in microwave
plasma at various temperatures with CH4 (10 sccm) and N2 (20 sccm)
for a reactor pressure of 10 Torr and a generator power of 300 and
500 W. Dashed lines indicate the theoretical mole fraction for the
corresponding mixture without a plasma.
Vibrational nonequilibrium would result in the excitation of transitions in the
ν 00 = 1 → ν 0 = 2 hot band as shown in Figure 3.6. In the spectral region near 3900
cm−1 , transitions from the hot band appear. This is because of a population shift
from the cold band to the hot band which ultimately results in a decrease of the peak
intensity of the Q(1) transitions. Therefore, because of the technique described to
measure the mole fraction, the experimental mole fraction would appear much lower.
As shown in Figure 3.12, however, no transitions from the hot band in this region
were detected at 10 Torr, 300 W. This indicates that there is no significant vibrational
42
Figure 3.11. Experimental mole fraction of H2 (50 sccm) in microwave
plasma at various temperatures with CH4 (10 sccm) and N2 (20 sccm)
for a reactor pressure of 30 Torr and a generator power of 500 and
700 W. Dashed lines indicate the theoretical mole fraction for the
corresponding mixture without a plasma.
excitation of the hot band and that, consequently, there is no significant vibrational
nonequilibrium.
Nonequilibrium of the translational and rotational modes would result in a different experimental mole fraction in the plasma than the theoretical mole fraction
without a plasma because the temperatures would be different. By fitting the experimental spectrum to a theoretical spectrum via the CARSFT code, a rotational
temperature is obtained. Therefore, the mole fractions estimated by the technique
presented would result in different values than those of the theoretical mole fractions
without a plasma. This is because the theoretical mole fractions without a plasma
43
Figure 3.12. Results for the attempt at measuring transitions in the hot band.
are calculated for a system in equilibrium where the translational and rotational
temperatures are equal. Other works studying the temperature of H2 have assumed
equilibrium of the translational and rotational modes [39]. However, for microwave
plasmas at low pressures (' 1 Torr), it has been shown that this may not be the
case [76].
Finally, significant dissociation and/or ionization of H2 would result in a lower
experimental mole fraction in the plasma because the number density of H2 would
decrease. The peak intensity of the Q(1) transition is proportional to the number
density of H2 and, therefore, the experimental mole fraction would appear lower.
Because the overall trends of the experimental mole fractions do not deviate significantly from the theoretical mole fractions calculated without a plasma, the results
indicate that the three previously listed conditions are not met. The translational
and rotational modes are in equilibrium and there is no significant dissociation and/or
ionization of H2 . Consequently, the rotational temperatures found in Reference [70]
may be used to approximate the translational temperature of H2 . Furthermore, there
is no detectable vibrational nonequilibrium by the measurements of the mole fraction
which is corroborated by the lack of transitions from the hot band. Lastly, the plasma
44
may be characterized as weakly ionized because the number density of H2 does not
appear to decrease significantly with the ignition of the plasma.
3.4
Discussion and Conclusion
A technique for the measurement of the mole fraction of H2 via CARS spec-
troscopy was introduced and successfully applied to a microwave plasma at 10 and
30 Torr for various generator powers. Results showed that there was no significant
dissociation of H2 in the plasma at any of the parametric conditions studied. A
change in pressure and plasma generator power did not appear to significantly vary
the experimental mole fraction from the theoretical mole fractions without a plasma.
Likewise, the introduction of CH4 and N2 had no effect. Furthermore, there were
no detectable transitions of the vibrational hot band near 3900 cm−1 which indicate
no significant vibrational nonequilibrium. In conclusion, the results show that the
microwave plasma is weakly ionized and that the rotational and translational modes
are in equilibrium. The rotational temperatures found in Reference [70] may be used
to approximate the translational temperature of H2 .
45
4. ELECTRON DENSITY FUNCTION OF MICROWAVE PLASMA
4.1
Introduction
The study of industrial plasmas for the production of micro- and nanomaterials is
of paramount importance. The etching properties, surface modification and thin-film
deposition capabilities of plasmas due to their non-equilibrium chemistry must be
understood so as to provide reliable processes for mass production [79, 80]. Particularly, the behavior of electrons and ions near the deposition surface must be explored.
Langmuir probes have been successfully used for this application ever since Langmuir
reported that biased probes inserted into plasmas are able to provide information
as to the plasma potential Vpl , electron temperature Te and electron number density
ne [81]. Over the past century, various probes have been designed to explore DC
discharges [80], RF discharges [82, 83], inductively coupled plasmas [84], microwave
plasmas [79] and others [85, 86]. Furthermore, they have been used in conjunction
with optical spectroscopy to validate measurements of electron kinetics and enrich
the understanding of the processes therein [84].
Because of its utility to explore the electron energy distribution in a plasma, a
Langmuir probe was used to examine the electron temperature and number density
in a microwave plasma chemical vapor deposition reactor in conditions favorable to
the growth of graphene-based nanomaterials.
4.2
Probe Theory
Langmuir probes work by applying a negatively biased potential on the probe and
increasing it to a positively biased potential. While the probe is negatively biased
with respect to the plasma potential, positive ions are attracted to its surface and
electrons are repelled producing an ion current ii on the probe. Likewise, when
46
the probe is positively biased, electrons are attracted and ions repelled producing an
electron current ie . The total current (ip = ii +ie ) produced is recorded and the shape
of the current-voltage profile provides information about the plasma parameters.
In many plasmas, the electron velocity distribution may be described by the
Maxwellian distribution. In these cases, the dependence of the ion current on the
probe bias ii (Vp ) is be described as


−iis exp [e(Vpl − Vp )/kB Ti ], Vp ≥ Vpl ,
ii (Vp ) =

−iis ,
Vp < Vpl ,
(4.1)
where e is the electron charge, kB is the Boltzmann constant and Ti is the ion temperature [87].
When Ti ∼ Te , the ion saturation current iis is given by iis = eni vi,th Sp /4 where ni
p
is the ion density, vi,th is the ion thermal speed (vi,th = 8kB Ti /πmi ), mi is the ion
mass and Sp is the surface area of the probe. However when Te Ti , as is the case in
nonthermal plasmas, the ion saturation current is defined by the Bohm ion current,
p
iis = iBohm = 0.6eni Sp kB Te /mi , which is dependent on the electron temperature
rather than the ion temperature. This is because, as a protective sheath forms around
the biased probe shielding the plasma from its potential, the ions and electrons travel
at different speeds due to their differences in mass. Therefore, in order to produce a
sheath of positive charge, ions must approach the surface of the probe with a speed
greater than the Bohm velocity, (kB Te /mi )1/2 , which is achieved across relatively long
distances in the plasma dependent on the electron temperature [87].
The dependence of the electron current on the probe bias ie (Vp ) in Maxwellian
plasmas, on the other hand, is expressed as


ies exp [−e(Vpl − Vp )/kB Te ], Vp ≤ Vpl ,
ie (Vp ) =

ies ,
Vp > Vpl .
(4.2)
The electron saturation current ies is provided by ies = ene ve,th Sp /4 and the electron
p
thermal speed by ve,th = 8kB Te /πme where me is the electron mass. Furthermore,
47
because me mi and Te Ti , the electron saturation current is much greater than
the ion saturation current. As a result, the influence of the ion saturation current is
only detectable in the region of negative probe bias as shown in Figure 4.1.
Figure 4.1. Ideal case for ion, electron and probe current produced
by Equations 4.1 and 4.2.
The ideal case illustrated in Figure 4.1 shows that, if the probe bias Vp is sufficiently negative with respect to the plasma potential Vpl , the ion saturation current
may be obtained. On the other hand, the electron saturation current is obtained once
the probe bias is greater than the plasma potential.
For Maxwellian distributions, the electron temperature may be easily calculated
by plotting the probe current in logarithmic scale and fitting a straight line to the
slope as shown in Figure 4.2. The slope of the line in the transition region is inversely
proportional to the electron temperature as shown in Equation 4.2. Additionally, if
another line is fitted to the electron saturation current, the point where both lines
meet corresponds to the plasma potential. The electron number density may be
derived from the electron saturation current.
There are additional theories describing the transition region of the current-voltage
profile from a Langmuir probe. For instance, the Thin Sheath Collection Theory
48
Figure 4.2. Ideal probe current plotted in logarithm scale showing linear portion.
assumes a Maxwellian velocity distribution of electrons, sheath size much smaller
than the probe radius and a collisionless plasma. In a plot in logarithmic scale, the
curve-voltage profile appears linear due to the exponential dependence of the probe
current on the probe voltage. The electron temperature may be determined from
the slope of the straight line and the electron number density from the equation for
the electron saturation current ies in a similar manner as previously described. On
the other hand, when the sheath thickness is comparable or greater than the probe
radius, the Orbital Motion Limit Collection Theory better describes the currentvoltage profile in the transition region. In this case, a Maxwellian distribution of the
velocity of the electrons is still assumed but the square of the current-voltage profile
is a straight line because of the square-root dependence of the probe current on the
√
probe voltage, ip ∝ V . As a result, the electron number density is obtained from
the slope of the straight line while the electron temperature from its intercept. For
more details on these theories, the reader is referred to Reference [88].
There is a technique commonly used, irrespective of the relative size of the sheath
thickness to the probe radius and the shape of the electron distribution function, to
obtain the electron energy density function (EEDF) from which the electron temper-
49
ature, electron number density and plasma potential may be derived. The technique
is known as the Druyvesteyn method and is described below [89].
4.3
Druyvesteyn Method
The second derivative of the probe current with respect to the probe bias is re-
lated to the electron energy density function of the plasma examined as shown by
Druyvesteyn [89]. This is true for Maxwellian and non-Maxwellian electron distributions but is particularly useful in non-Maxwellian distributions because from the
EEDF the electron temperature and number density may be derived. The following
review of the Druyvesteyn Method was taken from References [90] and [86].
The EEDF, F (), represents the number of electrons in the differential volume
with energies between and d. It is related to the electron density function f0 by
√
4 2π √
F () = 3/2 f0 .
(4.3)
me
However, in some reviews, it is sometimes replaced by the electron energy probability
density function fp expressed as
F ()
fp () = √ .
(4.4)
In this review, F () will be used.
The electron number density may be directly obtained, in an isotropic plasma, by
integrating the EEDF,
∞
Z
ne =
F ()d
(4.5)
0
and the electron temperature is given by
TeV
2
=
3ne
Z
∞
F ()d.
(4.6)
0
where TeV represents the electron temperature in units of eV (1 eV ≈ 11600 K). As
shown in Equations 4.5 and 4.6, all that is needed is a proper expression for the EEDF
in order to obtain the electron number density and temperature.
50
In a weakly ionized, low pressure (< 1 Torr), isotropic plasma examined with a
nonintrusive probe, the electron current of the probe is related to the electron density
function, and in turn to the EEDF, by the following expression
Z ∞
eSp
F ()
ie = √
( − V ) √ d.
2 2me V
(4.7)
Differentiating this expression twice with respect to the probe potential V and rearranging the equation yields the Druyvesteyn formula as shown in Equation 4.8.
√
8me √ d2
V
F () =
ie
(4.8)
Sp e3/2
dV 2
The probe potential refers to the probe bias with respect to the plasma potential
(V = Vp − Vpl ). The ion probe current is insignificant in the probe potential region.
Therefore, the second derivative of the total probe current approximates the second
derivative of the electron probe current (d2 Ip /dV 2 ' d2 ie /dV 2 ) as seen in Equation
4.9. Lastly, the probe potential may be obtained from the zero intercept of the
second derivative of the probe current or the maximum of the first derivative. Thus,
the plasma potential, electron temperature and number density may be derived.
√
F () '
4.4
8me √ d2
ip
V
Sp e3/2
dV 2
(4.9)
Electron Energy Distributions
Once it has been obtained via the Druyvesteyn method, the EEDF from a plasma
may be compared to the Maxwellian distribution. The Maxwellian EEDF, FM (), is
defined by
2ne
FM () =
eTeV
r
exp −
.
πeTeV
eTeV
(4.10)
It is derived from the Maxwellian electron velocity distribution by mapping the function into energy space by use of the kinetic energy term of the electrons, = me ve2 /2
51
Figure 4.3. Area-normalized Maxwellian and Druyvesteyn electron
energy distributions at TeV = 2 eV.
where ve is the electron velocity [91]. As mentioned, not all electron energy distributions are Maxwellian, however. An example of a non-Maxwellian distribution is the
Druyvesteyn distribution given by
"
2 #
√
FD () = AD ne exp −B D
eTeV
4[Γ(1/4)]4
√
π(12 2πeTeV )3/2
[Γ(1/4)]4
BD =
≈ 0.243.
72π 2
(4.11)
AD =
(4.12)
The Druyvesteyn EEDF was derived for the case of non-equilibrium plasma (Te Ti ).
The function accounts for a shift of the electron density to higher energies. This is
shown in the area-normalized EEDFs, F ()/ne , in Figure 4.3 in which the Maxwellian
EEDF and Druyvesteyn EEDF have been calculated at the same temperature of 2
eV. The bulk of the electrons represented by the Maxwellian distribution reside at
lower energies than the Druyvesteyn distribution. Also, the tail of the Druyvesteyn
function drops faster than the Maxwellian distribution at higher energies.
52
4.5
Sources of Error
Although the theoretical technique and analysis of ideal cases is simple, inter-
pretation of the current-voltage curve may be complicated. In actual cases for a
Maxwellian distribution, the probe current will not illustrate lines parallel to the
horizontal at the ion and electron saturation current regions as shown in Figure 4.1.
Instead, these straight lines will be at an angle to the horizontal due to the change
in size of the sheath around the probe caused by the change in probe bias. As the
probe bias is scanned away from the plasma potential in either direction, the probe’s
potential grows and, as a result, the sheath grows thereby also increasing the ion and
electron saturation current. Furthermore, the discontinuity of the plot in logarithmic
scale shown in Figure 4.2 from which the plasma potential may be derived is rounded
and the two-line fitting technique becomes much less reliable.
Some additional inaccuracies arise from inadequate probe design. A true noninstrusive probe consists of a probe and probe holder that are small enough so as not to
affect the ionization and energy balance of the plasma. Godyak et al. have provided a
relation among the probe dimensions, electron Debye length and electron path length
for a nonintrusive probe. The relation is
π`t
rt ln
, rh , λDe λe ,
4rt
(4.13)
where rt is the probe tip radius, rh is the probe holder radius, `t is the length of the
probe tip, λDe is the electron Debye length and λe is the electron mean free path [90].
The electron Debye length is given by
r
λDe =
0 kB Te
e2 ne
(4.14)
where 0 is the permittivity of free space [87]. The Debye length is a characteristic
length scale of plasmas and it describes the screening distance of charged particles in
the plasma. The above expression for the electron Debye length may be rewritten by
calculating the constants yielding
r
λDe ' 7.4e3 ×
TeV
ne
(4.15)
53
where TeV and ne are given in units of eV and m−3 , respectively. This results in units
of a meter for the electron Debye length.
The majority of electrons in a plasma have low energies in the range 0 ≤ ≤ Te
[92]. However, the EEDF is sensitive to the circuitry and electric resistance of the
Langmuir probe, the size of the probe and probe holder in the plasma and the purity
of the probe surface. Therefore, the shape of the EEDF may be distorted as a
result of any design flaw in the Langmuir probe and plasma system thus shifting
the majority of electrons away from this range. This artificial distortion is known
as Druyveteynization of the EEDF and ultimately results in an underestimation of
the electron number density and an overestimation of the electron temperature when
applying Equations 4.5 and 4.6. Accordingly, Godyak et al. provide a requirement
for the location of the peak of the EEDF and the electron temperature so as to ensure
that this effect is not present in the shape of the EEDF [90]. The peak location Vpk
must be less than the electron temperature
Vpk < TeV .
(4.16)
The peak location is the position corresponding to the maximum of the EEDF or,
equivalently, the difference of the maximum of the second derivative of the probe
current and the plasma potential in units of eV. When this condition is not met, the
EEDF may be unreliable and so too the electron number density and temperature.
4.6
Numerical Approximation of the Second Derivative
The second derivative of the curve-voltage profile may be obtained via numerical
differentiation after smoothing by means of a non-recursive digital filter proposed by
(n)
Hannemann [80, 93]. The filter works by replacing each data value yl
by a weighted
average of values in its vicinity as shown in Equation 4.17. Here, h is the step width
of the equispaced data and n is the desired degree of derivation. The same filter is
used to smooth the data and to take its second derivative. However, different filter
54
coefficients cj are used and n = 0 is used for smoothing while n = 2 for the second
derivative.
(n)
yl
=
m
1 X
cj yl+j
hn j=−m
(4.17)
The filter coefficient cj for the smoothing of the raw data is given by Equation 4.18
where w(j) is the a window function specifically designed to smooth current-voltage
profiles from Langmuir probes.
cj =
sin(jπβ)
w(j)
jπ
(4.18)
Like other window functions, such as the Hanning and Blackman filter functions, w(j)
is a generalized cosine window. The window and its coefficients is given in Equation
4.19.
w(j) = Aw + Bw cos(πj/m) + Cw cos(2πj/m)
for − m ≤ j ≤ m
Aw = 0.5 − p
Bw = 0.5
Cw = p
(4.19)
The variables m and p that define the window and its coefficients are given in
Equation 4.20. They depend on smoothing parameters δ, β, and λ which are restricted
as shown in Equation 4.21.
m = IntegerPart
0.07527λ − 0.1934 + 4000/λ3
δ
p = 0.08865(λ − 24.5)0.32 − 0.23
(4.20)
55
The smoothing parameters are chosen so as to provide the minimal amount of smoothing necessary and preserve the details of the EEDF. Care must be taken not to apply
the smoothing filter too aggressively.
β ≤ δ ≤ 2β
β ≤ 0.3
27 ≤ λ ≤ 80
(4.21)
Once the smoothing filter has been applied to the raw data, the second derivative
may be obtained by use of Equation 4.17. In this case, the filter coefficients ci are given
by Equation 4.22 based on interpolation polynomials of second or fourth degree [93].
c−1,0,1 = (1, −2, 1)
c−2,−1,0,−1,2 = (−1, 16, −30, 16, −1)/12
(4.22)
An example of the raw data obtained in this report as well as the performance
of the smoothing filter and the numerical differentiation is illustrated in Figure 4.4.
The data was filtered with the smoothing parameters set as β = 0.03, λ = 80 and
δ = 1.5β.
4.7
Experimental Setup
A Hiden ESPion Langmuir probe was used to study the MPCVD plasma. It was
attached to the right side of the MPCVD reactor’s optical arm as shown in Figure
4.5. The probe was attached via a 2-3/4 CF flange and was able to extend and retract
into the plasma when controlled by a computer. The software that controls the unit
was provided by Hiden.
In its fully retracted state it was located in the reactor’s optical arm away from the
plasma. With an extension of 21.5 mm from its fully retracted state, the probe reached
the center of the reactor and plasma as shown in Figure 4.6. When the susceptor
stage was raised for plasma ignition, the molybdenum puck on the susceptor stage
56
Figure 4.4. Raw data shown in red circles and the results of the
smoothed filter in blue curve. The green curve is the second derivative
of the smoothed data. The scan resolution is 0.1 V.
Figure 4.5. Hiden ESPion Langmuir probe attached to the right side
optical arm of the MPCVD reactor. The probe is controlled via a
computer and is able to extend and retract into the reactor.
57
was nearly 5 mm from the probe. The tungsten probe was 10 mm long and 0.15 mm
in diameter. The probe holder was 8 mm in diameter.
Figure 4.6. Tungsten probe and probe holder in the reactor. The
probe has been extended to 21.5 mm corresponding to the center of
the reactor and plasma.
The software allowed for the control of the acquisition of the current-voltage profile
by varying the probe bias range, bias resolution, pre-cleaning potential, pre-cleaning
duration and the number of scans to be averaged. Pre-cleaning of the probe is sometimes important in deposition plasmas to remove particles from the surface of the
probe. This is conducted by applying a relatively high positive or negative bias on
the probe for a short amount of time before data acquisition. Once pre-cleaning
is conducted, the scan may follow. The effect of the pre-cleaning potential on the
current-voltage profile was examined.
Comparative tests would have been conducted at every parametric condition previously studied with laser diagnostics of the plasma. The Langmuir probe, however,
was extremely sensitive to its location within the plasma as well as the reactor pressure. As a result, the tests were limited to a reactor pressure of 10 Torr and plasma
generator powers of 300, 400 and 500 W. In the scans performed, the probe was extended to 19 mm where it penetrated the plasma without strong coupling with the
58
probe holder. This position also corresponded to the edge of the plasma. A deviation
from this position by even just 1 mm in either direction resulted in erroneous data.
Consequently, the tests were limited to this location. The optimal probe bias range
was found to be nearly -30 V to 30 V. This range was sufficient to detect the ion and
electron saturation currents. Lastly, 10 scans were averaged in each test.
4.8
Results and Discussion
Attempts at pre-cleaning the probe with various potentials for a duration of 100
mS unexpectedly resulted in a horizontal shift of the current-voltage profile. The
pre-cleaning potential was varied from -10 V to 10 V in increments of 5 V and scans
were taken after each pre-cleaning. As illustrated in Figure 4.7, the current-voltage
profile is shifted depending on the pre-cleaning potential. The shift correlates with
the parity and amplitude of the pre-cleaning potential applied. For instance, when the
pre-cleaning potential is -10 V, the current-voltage profile shifts horizontally leftward.
When the potential is 10 V, it shifts rightward. Also, the amount shifted depends
on the amplitude of the pre-cleaning potential applied. At 10 V, the current-voltage
profile is shifter further right than at 5 V. Because of this unexpected and unreliable dependence of the position of the profile on the pre-cleaning potential, the
pre-cleaning potential was left at 0 V for the scans but with a duration of 100 mS.
After various scans, it was found that a duration of 100 mS and 0 V of pre-cleaning
potential provided more repeatable tests. It is unclear as to why this is the case. The
unreliability of the horizontal position of the current-voltage profile may ultimately
result in erroneous plasma potential values.
Upon varying the resolution of the scans, the data was also found to vary unexpectedly. As illustrated in Figure 4.8, although the ion saturation current remains
relatively the same for all scans, the electron saturation current appears to increase
with scan resolution. The scan resolutions tested were 0.1, 0.2, 0.5, 0.75 and 1.0 V.
The plasma potential also appears to increase with scan resolution.
59
Figure 4.7.
Horizontal shift of current-voltage profile with precleaning potential. The direction and amount shifted is proportional
to the parity and amplitude of pre-cleaning potential.
Figure 4.8. Change in the shape of the current-voltage profile with
scan resolution. The generator power was 300 W and reactor pressure
was 10 Torr.
The current-voltage profiles obtained by varying the scan resolution were analyzed
for a generator power of 300 W and a reactor pressure of 10 Torr. The results along
with the smoothing parameters used are provided in Table 4.1. In weakly ionized
60
nonthermal plasmas, the electron temperature and number density are found to be
in the range of 2-10 eV and 10−16 -10−18 m−3 , respectively [43, 84, 86]. For the scan
resolutions studied, the results for the MPCVD plasma provide electron temperatures
fluctuating between 3.9 and 4.8 eV. The number density decreases with scan resolution
from 7.3e10 to 1.4e9 m−3 whereas the plasma potential increases from 4.2 to 7 eV.
Table 4.1. The effect of varying the scan resolution of the Langmuir
probe on plasma parameters (TeV , ne , and Vpl ) at 300 W, 10 Torr.
The corresponding smoothing parameters β and λ are also shown.
Scan Resolution (V)
β
λ TeV (eV) ne (1/m3 ) Vpl (eV)
0.1
0.03 80
4.2
7.3e10
4.2
0.2
0.05 50
4.8
2.5e10
5.2
0.5
0.1 30
4.6
4.9e9
6.6
0.75
0.14 50
3.9
2.3e9
6.7
1
0.17 60
4.3
1.4e9
7
The relatively large changes in electron temperature, number density and plasma
potential with scan resolution are indicative of a serious problem with the Langmuir
probe system. The current-voltage profile should not vary in shape with scan resolution. Furthermore, although the electron temperature is within the expected range,
the number density appears to have been underestimated by the derived EEDF in
comparison to the cited literature.
Similar trends were found at 400 and 500 W of generator power. The electron
number density decreased with scan resolution while the plasma potential increased.
The electron temperature also fluctuated about a mean value. Representatives from
Hiden were contacted about these abnormalities but they were unable to provide an
explanation as to why they occurred and how to remedy them.
In order to address this issue, the EEDFs for each scan resolution were compared.
The experimental EEDFs that appeared to comply with the condition Vpk < TeV
most closely were found to be at a resolution of 0.1 V. For instance, at a generator
61
power of 500 W, the peak location Vpk of the EEDF appears to be under 5 eV, as
illustrated in Figure 4.9, and its estimated electron temperature is 4 eV. On the other
Figure 4.9. Experimental EEDF at 500 W with a scan resolution
of 0.1 V and its corresponding Maxwellian and Druyvesteyn best fit.
The experimental temperature obtained is 4 eV while the best fit
attempts with the Maxwellian and Druyvesteyn EEDFs were 4.8 and
3.9 eV, respectively.
hand, the experimental EEDF with a scan resolution of 0.75 V, shown in Figure 4.10,
appears to have a peak greater than 5 eV with an estimated electron temperature of
4.3 eV. Therefore, the EEDF that appears to most closely comply with the condition
corresponds to a scan resolution of 0.1 V even though the condition is not exactly
met. This detail may be indicative of a lesser distortion of the profile at smaller scan
resolutions. Additionally, experimental EEDFs with smaller scan resolutions appear
more Druyvesteyn-like and less likely to be compromised by the Druyvesteynization
effect. This is illustrated in Figures 4.9 and 4.10. The EEDF with a scan resolution
of 0.1 V looks more like the Druyvesteyn EEDF fitted. This behaviour was true for
scans conducted with a generator power of 300 and 400 W as well.
As a result of the unreliability of the Langmuir probe to produce consistent
current-voltage profiles when certain parameters were varied, the electron temper-
62
Figure 4.10. Experimental EEDF at 500 W with a scan resolution
of 0.75 V and its corresponding Maxwellian and Druyvesteyn best fit.
The experimental temperature obtained is 4.3 eV while the best fit
attempts with the Maxwellian and Druyvesteyn EEDFs were 5.5 and
4.4 eV, respectively.
ature, number density and plasma potential cannot be directly derived from the measurements obtained. However, some general trends may be useful in understanding
the plasma which the probe measurements may provide.
Table 4.2 shows the results of the experimental EEDFs at 300, 400 and 500 W
with a scan resolution of 0.1 V. The peak location of the EEDF was calculated at its
maximum value which may not accurately represent the true peak location because of
its undulating features. However, they do serve to provide a relative approximation.
The electron temperatures calculated were relatively close to each other and within
error for each generator power studied. This is consistent with results of a high
current electron cyclotron resonance proton ion source utilizing a microwave generator
in Reference [86]. In that study, the temperature remained relatively constant across
a large range of generator powers. Likewise, the electron number density increased
which was also consistent with the results from the cited study. The plasma potential
decreased with generator power which was not consistent with the cited study.
63
Table 4.2. Plasma parameters derived from experimental EEDFs at
300, 400 and 500 W at 10 Torr with a scan resolution of 0.1 V.
Generator Power (W)
Vpk (eV) TeV (eV) ne (1/m3 ) Vpl (eV)
300
3.7
4.2
7.2e10
4.2
400
4.0
4.2
9.5e10
3.5
500
4.0
4.0
1.0e11
3.6
The various limitations of the Langmuir probe on the MPCVD plasma have prevented exact values for the electron temperature and number density from being
determined. Typically, these probes are used in RF plasmas in pressures on the order of a milli-Torr. The high frequency field and higher pressure of a microwave
plasma at 10 Torr may have contributed to the limitations of the probe in the system. The number density appears to have been underestimated which is consistent
with the Druyvesteynization effect. Also, varying the scan resolution of the measurements revealed a distortion of the EEDF that manifested itself as a depletion of
the low-energy electrons. This is also consistent with the Druyvesteynization effect.
If the Druyvesteynization effect played a role in the distortion of the plasma, then
the electron temperature was likely overestimated. As such, the results from these
experiments may serve to provide an upper limit for the electron temperature of the
MPCVD microwave plasma at approximately 4 eV and a lower limit for the number
density at approximately 1e10 m−3 for the conditions studied.
With these limits for the electron temperature and number density, some plasma
parameters may be estimated. According to Equation 4.15, the Debye length of the
MPCVD plasma with a temperature of 4 eV and number density of 1e10 m−3 is on
the order of 0.1 m (or 10 cm) which is unrealistic for a plasma that is approximately
5-8 cm wide. Instead, if the number density is approximated at 1e18 m−3 as the
cited literature suggests might approximately be the true number density, the Debye
64
length would be on the order of 1e-5 m (or 10 µ m) which is reasonable for the size
of the microwave plasma.
Furthermore, the applicability of the Langmuir probe as a nonintrusive probe
under Godyak’s condition provided in Equation 4.13 may be estimated. The electron
mean free path must first be calculated. Assuming a Maxwellian velocity distribution
for the electrons, binary collisions among electrons and H2 molecules and gas kinetic
hard-sphere cross sections, the mean free path of electrons colliding with H2 molecules
may be roughly estimated [43]
λe−H2 =
1
,
nH2 σe−H2
σe−H2 = π(re + rH2 )2 ,
nH 2 =
ptot Na
RT
(4.23)
(4.24)
(4.25)
where σe−H2 is the hard-sphere collisional cross section, re ' 5e-17 m is the electron
hard-sphere radius, rH2 ' 1.38e-10 m is the H2 hard-sphere radius, nH2 is the number
density of H2 . Thus, the hard sphere cross section is calculated at approximately
6e-20 m2 . The number density of H2 is approximated by the ideal gas law with
reactor pressure ptot , Avogadro’s number Na , the universal gas constant R and the
temperature T of H2 . With the results from Chapter 2, temperature of H2 is estimated
at 1000 K for a pressure of 10 Torr. As such, the number density of H2 is calculated
and approximated on the order of 1e23 m−3 resulting in an electron mean free path
on the order of 1e-4 m.
Therefore, the Debye length is less than the electron mean free path, λDe < λe−H2 .
However, the dimensions of the Langmuir probe are not. The relation for the probe tip
radius and probe length results in 5.9e-4 m which is on the order as the electron mean
free path, rt ln[π`t /4rt ] ∼ λe−H2 . Also, the probe holder radius at 2e-3 m is greater
than the electron mean free path, rh > λe−H2 . Because of these calculations, the
tungsten probe used may not have been nonintrusive, as defined by Godyak et al., and
may have contributed to the distortion of the EEDF in the plasma. Unfortunately, the
65
dimensional inadequacy of the probe and typical use of the Hiden ESPion Langmuir
probe in the milli-Torr range were unknown prior to conducting these experiments.
In spite of the shortcomings of the Langmuir probe system, the upper limit of
the electron temperature at 4 eV appears to be reasonable because of its consistency
with the cited literature and in estimating other plasma parameters. For instance,
the electron collisional frequency fe may be estimated
fe ∼
with electron velocity ve =
ve
λe−H2
(4.26)
p
2kB Te /me on the order of 1e6 m/s. With a mean free
path approximately 1e-4 m, the electron collisional frequency may be estimated to
be on the order of 1 GHz. This is significant because the microwave electric field
frequency is 2.45 GHz which means that the electron collisions frequency and electric
field frequency are on the same order. In this case, the plasma cannot be considered
collisionless but rather dominated by collisions providing additional information as
to the kinetics of the electrons in the plasma [94].
Lastly, the upper limit on the electron temperature may be used to approximate
the ionization procedure for the H2 molecule. When the electron energy is comparable
to the ionization energy of the neutral chemical species, a single electron impact is
sufficient to ionize the neutral species. When the electron energy is lower, on the
other hand, a stepwise ionization procedure is necessary in which the first electronneutral collision excites the neutral species and a second collision of a much lower
energy ionizes the neutral species [94]. The ionization energy for H2 is approximately
15 eV [95]. The upper limit of the electron temperature is 4 eV. Therefore, the H2
molecules in the microwave plasma are likely to ionized via the stepwise ionization
procedure.
4.9
Conclusion
Although exact values for the electron temperature, number density and plasma
potential at various parametric conditions were not obtained, the trends for the elec-
66
tron temperature and number density were similar to other works. An upper limit
for the electron temperature was estimated at 4 eV and a lower limit for the number
density around 1e10 m−3 . The microwave plasma in the MPCVD reactor is found to
be dominated by collisions because of the high electron collisional frequency on the
order of 1 GHz. Also, H2 molecules are likely to be ionized via a stepwise ionization
procedure described.
67
5. SUMMARY
The results from the studies conducted in this dissertation provide interesting insights
into the chemistry and kinetics of the microwave plasma environment responsible for
the growth of carbon nanostructures. The rotational temperature of H2 increases with
reactor pressure, plasma generator power, and distance from the deposition surface.
At 10 Torr, the temperature range is approximately 700 to 1200 K while at 30 Torr
it is 1200 to 2000 K. Also, the introduction of CH4 and N2 to the plasma increases
the rotational temperature consistently. These results illustrate the difference of the
plasma environment with various growth recipes.
The measured mole fractions are not significantly different from the theoretical
values without a plasma for the corresponding mixtures studied. The results indicate
that there is no significant dissociation and/or ionization of H2 in the plasma and
that the translational and rotational modes are in equilibrium. This indicates that
the rotational temperatures found approximate the translational temperature of H2
and, therefore, the gas temperature of the plasma environment. Furthermore, this
along with the lack of transitions detected in the spectral region of the hot band
suggest that there is little vibrational nonequilibrium of the H2 molecule.
Finally, although measurements of the electron energy density function of the
plasma with the Langmuir probe were distorted, an upper limit to the electron temperature is estimated at 4 eV. This estimate results in a high collisional rate for the
electrons with the H2 molecules indicating that the microwave plasma is dominated
by collisions. Additionally, the ionization procedure is expected to be a stepwise procedure in which multiple electron collisions are needed to ionize H2 . Lastly, a lower
limit for the electron number density is estimated at 1e10 m−3 .
68
5.1
Additional Work and Recommendations
The understanding of the chemistry and kinetics of the plasma must be furthered
with additional experiments. This ongoing effort must address other important chemical species and parameters in the plasma. The CH radical, for instance, is a major
product of the dissociation of hydrocarbons introduced into the plasma. Furthermore,
it is expected to reside within the plasma and not diffuse outside its boundaries. As
such, a laser absorption system is currently being designed to measure its temperature
and concentration in the plasma so as to determine its role in the deposition process.
The current design makes use of the second harmonic of a 860 nm laser by frequency
doubling to 430 nm with a BBO crystal to probe transitions from the CH radical.
The system should be designed to conduct measurements at various locations within
the plasma and make use of wavelength modulation spectroscopy, if needed, to detect
small concentrations.
Additionally, the application of a carefully designed Langmuir probe for the microwave plasma would be a very useful tool in studying the electron kinetics. The
results from this dissertation should serve to indicate what is needed and what should
be expected from such a study. The electron temperature, number density and plasma
potential could be adequately measured at various parametric conditions. The results
should allow for a quantitative, and not just qualitative, explanation for the increase of
rotational temperature with pressure as described by Equation 2.10. In addition, the
results should provide more accurate collisional rates for electrons and H2 molecules.
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APPENDICES
77
A. LANGMUIR PROBE MEASUREMENTS
In Chapter 4 the effect of scan resolution on the plasma parameters was studied and
summarized. Scan resolutions of 0.1 V consistently provided EEDFs that were more
like the Druyvesteyn distribution function and came closest to satisfying the condition
Vpk < TeV . In order to demonstrate this in more detail, the results from Table 4.1 at
300 W have been shown again in addition to the approximate peak location for the
EEDF in Table A.1. Also, the results at 400 and 500 W are shown in Tables A.2 and
A.3, respectively. The range of scan resolutions at 400 and 500 W was smaller than
that at 300 W. However, the results are consistent.
In addition, each of the current-voltage profiles obtained in these tests are illustrated in Figures A.1 and A.2 for 300 W and Figures A.3 and A.4 for 400 and 500
W, respectively. The current-voltage profiles are smoother with increasing scan voltage. Also, the EEDF that is derived from them is smoother. However, the EEDF
shifts away from a Druyvesteynlike distribution function and also its peak location
is significantly greater than the corresponding electron temperature. As a result,
the measurements with a scan resolution of 0.1 V were chosen for direct comparison
among the different power generator powers.
78
Table A.1. The effect of varying the scan resolution of the Langmuir
probe on plasma parameters (TeV , ne , Vpl , and Vpk ) at 300 W, 10 Torr.
The corresponding smoothing parameters β and λ are also shown.
Scan Resolution (V)
β
λ TeV (eV) ne (1/m3 )
Vpl (eV) Vpk (eV)
0.1
0.03 80
4.2
7.3e10
4.2
3.7
0.2
0.05 50
4.8
2.5e10
5.2
5.2
0.5
0.1 30
4.6
4.9e9
6.6
4.6
0.75
0.14 50
3.9
2.3e9
6.7
5.2
1
0.17 60
4.3
1.4e9
7
5.0
Table A.2. The effect of varying the scan resolution of the Langmuir
probe on plasma parameters (TeV , ne , Vpl , and Vpk ) at 400 W, 10 Torr.
The corresponding smoothing parameters β and λ are also shown.
Scan Resolution (V)
β
λ TeV (eV) ne (1/m3 )
Vpl (eV) Vpk (eV)
0.1
0.03 80
4.2
9.5e10
3.5
4.0
0.5
0.1 70
4.4
7.4e9
6.5
5.5
0.75
0.17 70
4.3
3.7e9
7.0
5.5
Table A.3. The effect of varying the scan resolution of the Langmuir
probe on plasma parameters (TeV , ne , Vpl , and Vpk ) at 500 W, 10 Torr.
The corresponding smoothing parameters β and λ are also shown.
Scan Resolution (V)
β
λ TeV (eV) ne (1/m3 )
Vpl (eV) Vpk (eV)
0.1
0.03 50
4.0
1.0e11
3.6
4.0
0.5
0.1 70
4.7
9.2e9
7.0
5.5
0.75
0.17 70
4.3
4.6e9
7.5
5.3
79
(a)
(b)
(c)
Figure A.1. Current-voltage profiles, results of smoothing procedure
and second derivative at 300 W for a scan resolution of (a) 0.1, (b)
0.2, and (c) 0.5.
80
(a)
(b)
Figure A.2. Current-voltage profiles, results of smoothing procedure
and second derivative at 300 W for a scan resolution of (a) 0.75 and
(b) 1.0.
81
(a)
(b)
(c)
Figure A.3. Current-voltage profiles, results of smoothing procedure
and second derivative at 400 W for a scan resolution of (a) 0.1, (b)
0.5 and (c) 0.75.
82
(a)
(b)
(c)
Figure A.4. Current-voltage profiles, results of smoothing procedure
and second derivative at 500 W for a scan resolution of (a) 0.1, (b)
0.5 and (c) 0.75.
83
B. MATLAB CODE FOR SMOOTHING CURRENT-VOLTAGE PROFILE AND
SECOND DERIVATIVE
The following is the Matlab code used to read the DAT files generated by the ESPion
software and average the data as necessary.
function IV = LP(filename)
% function used to read DAT files created from ESPION software
% this function averages the profiles provides I-V profile
data = LPread(filename);
s = size(data);
I = data(:,2,1);
V = data(:,1,1);
%figure(1)
%plot(data(:,1,1),data(:,2,1),’Color’,[rand rand rand]);
%hold on
for i = 2:s(3)
%plot(data(:,1,i),data(:,2,i),’Color’,[rand rand rand]);
I = interp1(data(:,1,i),data(:,2,i),V,’linear’,’extrap’) + I;
end
I = I/s(3);
%figure(1);
%plot(data(:,1,1),I,’LineWidth’,3)
IV = [V I];
end
84
The following are the Mabtlab codes used to apply the smoothing procedure and
take the second derivative of the current-voltage profiles.
function [E1, EEDF, Vs, Is, V2, I2, fM, fD] = lang(IV,dV,beta,lambda)
%beta = 0.05;
%delta = 1.5*beta;
%lambda = 30;
Ap = pi*0.15*1E-3*10*1E-3; %4.712E-6;
% probe surface area (m^2)
me = 9.109E-31; % (kg)
e0 = 1.602E-19;
% (C)
delta = 1.5*beta;
%lambda = alpha(3);
[V,I,Vs,Is,V1,I1,V2,I2,RMSE] = HB(IV,dV,beta,delta,lambda);
RMSE
figure(1)
%subplot(1,2,1)
n1 = round(max(I)/max(I1));
n2 = round(max(I)/max(I2));
plot(V,I,’.’,Vs,Is,’c’,V1,I1*n1,’-’,V2,I2*n2,’-’,’LineWidth’,2)
legend(’Raw Data’,’Smoothed Data’,[’First Derative ...
(x’,num2str(n1),’)’],[’Second Derivative (x’,num2str(n2),’)’]...
,’Location’,’SouthWest’)
xlabel(’Probe Voltage, V [V]’)
ylabel(’Prove Current, I [A]’)
grid on
85
%%
% clear xi
% [xi, ~] = ginput(2);
% xi = round(xi);
% [~, ix] = find(I2(xi(1):xi(2)) => 0);
% I1pos and I2pos are approximation positions where plasma
% voltage is zero
[~,I1pos] = max(I1);
I2pos = find(V2>=V1(I1pos),1);
% finding actual position where I2 crosses zero
r = round(length(I2)/8);
newpos2 = find(I2(I2pos-r:I2pos+r)<=0,1);
I2pos = I2pos-r+newpos2-2;
disp([’plasma potential: ’,num2str(round(V2(I2pos)*10)/10)])
E1 = V2(I2pos) - V2(1:I2pos);
%E1 = V2(I2pos) - E1;
%EEDF = -I2(I2pos:length(I2));
% not normalized as defined by Heidenreich 1988
EED = sqrt(E1).*I2(1:I2pos);
EEP = I2(1:I2pos); % as defined by Godyak 1993
E1 = fliplr(E1);
EED = fliplr(EED);
EEP = fliplr(EEP);
86
% selection for first zero crossing point
% ve1 = find(E1>5,1);
% vi2 = find(EED(ve1:length(EED))<=0,1);
% E1 = E1(1:vi2+ve1);
% EED = EED(1:vi2+ve1);
% following procedure from Meichsner, pg 193
meanE = trapz(E1,E1.*EED)/trapz(E1,EED);
ne = sqrt(8*me)/(e0^(3/2)*Ap)*trapz(E1,EED)
Te = meanE*2/3;
Te = round(Te*10)/10;
%disp([’Dru approach, mean energy (eV): ’,num2str(meanE)])
disp([’Dru approach, electron temp (eV): ’,num2str(Te)])
%disp([’Dru approach, electron density (1/m^3): ’])
%fprintf(regexprep(sprintf(’%8.2E’,ne),’E-0’,’E-’));
% trapz(E2,2^(3/2)*sqrt(me)*EED2/(e0^(3/2)*ne*Ap)) = ne
EEDF = sqrt(8*me)*EED/(e0^(3/2)*Ap);
[~,ploc] = max(EEDF);
disp([’EEDF peak location: ’,num2str(E1(ploc)),’ eV’])
disp([’difference (+): ’,num2str(Te-E1(ploc))])
E1 = E1’;
EEDF = EEDF’;
Vs = Vs’;
Is = Is’;
V2 = V2’;
I2 = I2’;
87
% fit fMax & fDru with independent temperature accroding to Li (1999)
%
equations these are area normalized functions so the EEDF must also
%
be area normalzied
fMax_Li = inline(’2*sqrt(E)/(beta*sqrt(pi*beta)) .* exp(-E/beta)’,...
’beta’,’E’);
fDru_Li = inline(’4*gamma(1/4)^4/(pi*(12*sqrt(2)*pi*beta)^(3/2)) ...
* sqrt(E).*exp(-0.2432 *(E/beta).^2)’,’beta’,’E’);
% Heidenreich 1988
%f_He = inline(’beta(1)*sqrt(E)*exp(-beta(2)*E.^beta(3))’);
EEDFnorm = EEDF/ne;
beta_Max = 3.3;
beta_Dru = 3.3;
beta_Max = nlinfit(E1,EEDFnorm,fMax_Li,beta_Max);
beta_Dru = nlinfit(E1,EEDFnorm,fDru_Li,beta_Dru);
beta_Max = round(beta_Max*10)/10;
beta_Dru = round(beta_Dru*10)/10;
disp([’Li, Max temp: ’,num2str(beta_Max)])
disp([’Li, Dru temp: ’,num2str(beta_Dru)])
% %% Maxwellian and Druyvesteyn distributions (according to Li)
%
fitted to Te by second derivative
% fM = 2*sqrt(E1)/(Te*sqrt(pi*Te)) .* exp(-E1/Te); % confirmed
% with Fridman pg 13
% Ad = 4*gamma(1/4)^4/(pi*(12*sqrt(2)*pi*Te)^(3/2));
% Bd = 0.2432;
88
% fD = Ad*sqrt(E1).*exp(-Bd *(E1/Te).^2);
%%
figure(3)
plot(E1,EEDFnorm,’k.’,E1,fMax_Li(beta_Max,E1),’b-’,E1,...
fDru_Li(beta_Dru,E1),’r-’)%,E1,fM,’b:’,E1,fD,’r:’,’LineWidth’,2)
%semilogy(E1,EEDF,’k.’,E1,fMax_Li(beta_Max,E1),’b-’,E1,
%fDru_Li(beta_Dru,E1),’r-’,’LineWidth’,2)
legend([’Twofold Differentiation, temp (eV): ’,num2str(Te)]...
,[’Maxwellian Fit, temp (eV): ’,num2str(beta_Max)],...
[’Druyvesteyn Fit, temp (eV): ’,num2str(beta_Dru)],...
[’Maxwellian distribution at ’,num2str(Te),’ (eV)’],...
[’Druyvesteyn distribution at ’,num2str(Te),’ (eV)’],...
’Location’,’SouthWest’)
xlabel(’Energy, E [eV]’)
ylabel(’EEDF’)
grid on
fM = fMax_Li(beta_Max,E1);
fD = fDru_Li(beta_Dru,E1);
end
function [V,I,Vs,Is,V1,I1,V2,I2,RMSE] = HB(IV,dV,beta,delta,lambda)
% Perfoms the smoothing algorithm procedure by
% Hannemann (2013) and applies a two-fold differentiation.
% Input parameters:
% IV -> voltage and current as columns in that order
% dV -> difference between V1, V2, V3, ... measurements
89
% beta -> smoothing parameter,
% greater smoothing with smaller values, < 0.3
% delta -> beta <= delta <= 2*beta
% lambda -> 27 <= lambda <= 80
% Output parameters:
% V,I -> original voltage-current data spaced dV
% Vs,Is -> smoothed data
% V1,I1 -> first differentiation
% V2,I2 -> twofold differentiation
% beta = alpha(1);
% delta = 1.5*alpha(1);
% lambda = alpha(2);
V = IV(:,1);
V = round(min(V)):dV:round(max(V));
I = IV(:,2);
% previously working values
% beta = 0.06;
% delta = 1.5*beta;
% lambda = 50;
%% smoothing filter
m = floor((0.07527*lambda - 0.1934 + 4000/lambda^3)/delta);
p = 0.08865*(lambda-24.5)^0.32 - 0.23;
% if exist(’p’) == 0
%
p = 0.08865*(lambda-24.5)^0.32 - 0.23;
90
% else
%
lambda2 = exp(log((p+0.23)/0.08865)/0.32)+24.5;
% end
c = zeros(size(1:2*m+1));
w = c;
cw = c;
for i = -m:1:m
c = sinc(i*beta)*beta;
ci(i+m+1) = c;
w = (0.5-p) + 0.5*cos(pi*i/m) + p*cos(2*pi*i/m);
cw(i+m+1) = c*w;
end
cw = cw/sum(cw);
Is = zeros(size((m+1):length(I)-m));
Vs = Is;
for i = (m+1):length(I)-m
suma = 0;
for j = -m:1:m
suma = suma + cw(j+m+1)*I(i+j);
end
Is(i-m) = suma;
Vs(i-m) = V(i);
end
%% first differentiation
d = [-1/2 0 1/2];
I1 = zeros(size(2:length(Is)-1));
91
V1 = I1;
for k = 2:length(Is)-1
suma = 0;
for g = -1:1:1
suma = d(g+2)*Is(k+g) + suma;
end
I1(k-1) = suma/dV;
V1(k-1) = Vs(k);
end
%% two-fold differentiation
clear d
d = [-1/12 4/3 -5/2 4/3 -1/12];
Vs_spline = Vs(1):0.001:Vs(length(Vs));
Is_spline = spline(Vs,Is,Vs_spline);
I2 = zeros(size(3:length(Is_spline)-2));
V2 = I2;
for k = 3:length(Is_spline)-2
suma = 0;
for g = -2:1:2
suma = d(g+3)*Is_spline(k+g) + suma;
end
I2(k-2) = suma/dV^2;
V2(k-2) = Vs_spline(k);
end
%% RMS error
RMSE = sqrt(sum( ( Is’-I( find(Vs(1)==V):...
92
find(Vs(length(Vs))==V) ) ).^2 ));
% making vectors same size
% Is2 = I;
% Is2(find(Vs(1)==V):find(Vs(length(Vs))==V)) = Is;
end
VITA
93
VITA
Alfredo D. Tuesta was born in Lima, Peru. At the age of ten, he and his family
moved to Paterson, New Jersey where he finished his primary and secondary schooling
at St. Anthony of Padua School and Don Bosco Technical High School. He enjoyed
and excelled in math and science giving rise to his desire to learn engineering. After
high school, he attended the University of Notre Dame du Lac and earned a Bachelors of Science in Mechanical Engineering in 2007. That fall, he began his graduate
education at Purdue University under the tutelage of Professor Timothy Fisher and
Professor Robert Lucht by exploring the temperature and concentration of acetylene
in a microwave plasma chemical vapor deposition. In 2010, he earned a Masters of
Science in Mechanical Engineering based on his studies via direct absorption spectroscopy. His doctoral work focused on the study of hydrogen via coherent anti-Stokes
Raman scattering spectroscopy and the study of the electron density function by use
of a Lanmguir probe. He plans to continue research in laser diagnostics by joining a
national laboratory.
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